Elm Wealth Research

By Victor Haghani, James White and Jerry Bell ¹
Estimated reading time: 10 min.

Introduction: Back to the Future

In the 1989 blockbuster Back to the Future II, time travel enables Michael J. Fox’s nemesis, Biff, to become a gazillionaire by bringing an almanac with sports match outcomes back from the future. We thought it might be instructive, and certainly entertaining, to make a less fanciful version of this dream a reality – for a few lucky people.

In November 2023, we ran an in-person, proctored experiment involving 118 young adults trained in finance. We called the experiment “The Crystal Ball Challenge.” We gave each participant $50 and the opportunity to grow that stake by trading in the S&P 500 index and 30-year US Treasury bonds with the information on the front page of the Wall Street Journal (WSJ) one day in advance, but with stock and bond price data blacked out. The game covered 15 days, one day for each year from 2008 to 2022.

You can play this game for yourself here: Crystal Ball Trading Challenge – though without the pecuniary component. As of the time of writing, over 1,500 people have tested their skill and luck by playing the game on our website.

Summary of results

The players in the proctored experiment did not do very well, despite having the front page of the newspaper 36 hours ahead of time. About half of them lost money, and one in six actually went bust. The average payout was just $51.62 (a gain of just 3.2%), which is statistically indistinguishable from breaking even. The poor results were a product of: 1) not guessing the direction of stocks and bonds very well, and 2) poor trade-sizing. The players guessed the direction of stocks and bonds correctly on just 51.5% of the roughly 2,000 trades they made. They guessed the direction of bonds correctly 56% of the time, but bet less of their capital on bonds than on stocks (if you’re planning a career as a proprietary macro trader, consider putting your focus on bonds).

Perhaps the front page of the WSJ isn’t a particularly clear crystal ball, or our players weren’t very adept at reading it. As former Goldman Sachs CEO Lloyd Blankfein reminded us in a widely-circulated tweet, sometimes the markets don’t react to the news as even seasoned experts expect – an important lesson all successful traders learn, eventually.

It didn’t help that the players also did not seem to know how to size their bets well. On eight of the 30 trading opportunities,² the players in aggregate displayed 2-to-1 odds of being correct in their bets, but they did not bet more heavily on those occasions. Overall, they did not display trade-sizing that bore any relation to their propensity to guess the price moves of stocks or bonds correctly.

Many of the players used excessive leverage relative to their exhibited edge in guessing market direction. On about 30% of the total number of days on which players traded, they used leverage of greater than 20x capital. On 4% of the total occasions, they used leverage of 60x or higher, which carried a very high probability of being wiped out if they guessed wrong. In sum, there was little discernible logic or rationale to their trade-sizing decisions.

See Appendix I for a detailed analysis of player results.

Perhaps this excessive risk-taking by some of the players is partially explained by the finding that most investors tend to overestimate the predictive value of news on market outcomes. For example, a recent survey of 11,000 investors by Andre et al.³ found that about 70% of investors (but not finance academics) believed that stale, four-week-old, good (or bad) news was predictive of high (or low) future stock returns.

However, our sample of 118 staked and proctored players did better than the roughly 1,500 people who have played the game for fun on our website. The median outcome among these players was a loss of about 30% of their capital. Only 40% finished with a profit, and 36% went bust.

We were tickled to see that six players devoted themselves to achieving the maximum possible payout, growing their initial wealth 70,575-fold. They did this by repeatedly playing the game to see what stocks and bonds did on each day, and then using that information to put up the perfect score by correctly betting the maximum size on each trade. We were elated to see our game spark so much passion in some players!

Some of the world’s best traders show how to do better

We invited five seasoned and successful macro traders – four men, one woman – to play the game, with markedly better results. This was a very select group of traders: head of trading at a top-five US bank, founder of a top-ten macro hedge fund, senior trader at a top-ten macro fund, former senior government bond trader at top-three US primary dealer, and former senior Jane Street trader.

These players all finished with gains. On average, they grew their starting wealth by 130%, with a median gain of 60%. All of the players were selective and highly variable in their trade-sizing. They did not bet at all on about 1/3 of the trading opportunities, but bet big on days when they presumably felt confident in the impact of the news on stock or bond prices.⁴

These veteran traders predicted the direction of the markets significantly better than our 118 younger, less experienced participants (63% vs 51.5%), but mostly we ascribe the dramatically different results to the much more rational trade-sizing displayed by the experienced traders. One important conclusion we reach is that there is little value in this crystal ball without sensible trade-sizing.

Motivations

In addition to our curiosity in testing Taleb’s hypothesis, we had four further motivations for conducting this experiment:

1. We are deeply interested in learning how people approach the sizing of attractive investment opportunities, having researched and written extensively on this topic. In 2016, we conducted an experiment (also involving financial rewards) where we invited participants to bet on a digital coin flip that was programmed to have a 60% probability of landing on heads, and published our findings in “Betting on a Biased Coin.” We recently wrote a book, The Missing Billionaires: A Guide to Better Financial Decisions, that is focused on investment sizing.
2. We wanted to quantify the value of macro-economic information. How often would people guess the direction of markets from the information on the front page of the WSJ?
3. We hoped that the tool we developed for the experiment could be productively used to educate and train professional risk-takers.
4. It was going to be a lot of fun!

The Game

Over 90% of the participants were in graduate programs in finance or MBA programs with finance modules at four east coast US universities with low admission rates. The participants were not told in advance that they’d be invited to participate in this experiment. Any who did not want to participate were allowed to leave (though no one did).

Here’s how we explained the rules of the game:

Below are two pictures of the screens that players engaged with. In the first screen, the player can expand the picture of the front page of the WSJ on the left to be able to read it more clearly, then revert to the trading page.

Then, after clicking “Trade” the result of the trade is revealed, showing the market move in stocks and bonds, and the resultant profit or loss. The player’s bankroll is expressed based off of a starting value of $1 million, but the players understood that their payout would be $50 times the ending wealth divided by $1 million, with a minimum of $0 and a maximum of $100.

All the front pages can be seen here, in chronological order: https://elmwealth.com/crystal-ball-gallery/

Conclusion

“He who lives by the crystal ball will eat shattered glass.” — Ray Dalio

Was Taleb correct in his conjecture that “If you give an investor the next day’s news 24 hours in advance, he would go bust in less than a year”? While our experiment didn’t test his statement precisely – we only gave players 15 days of front pages, players were risking just $100 in the game, etc. – by and large we think Taleb is right. His counterintuitive proposition is both insightful and instructive.

The financial industry is replete with individuals and organizations constantly working to develop their own proprietary crystal balls. We hope that the experiment and results described herein convince crystal ball makers that sensible investment-sizing is essential to realizing the value of what they are trying to build.

The poor aggregate showing of our 118 financially-trained participants highlights the importance of educating young, aspiring finance industry professionals in decision-making under uncertainty, and particularly the theory and art of investment-sizing. We hope our Crystal Ball game will be a helpful tool – or a prototype for a better one – that educators and financial firms can use to teach these concepts and skills. Perhaps it may even become part of the hedge fund boot camp training programs at Citadel, Point72, Balyasny, and Jane Street that have been in the news recently.⁵ The uniformly positive results of the five experienced macro traders we invited to play the game suggest that there are teachable skills involved in successful discretionary investing.

Perhaps Matt Levine foresaw the results of our experiment with his article titled: “Knowing the Future Isn’t That Helpful.”⁶ He describes a delightful academic study that analyzed the trading results of a cartel of investors with an excellent, albeit illicit, crystal ball.⁷ The traders bought earnings announcements before they were released from an international hacker group that illegally obtained access to the servers of three commercial newswire companies. These traders were sophisticated and their crystal ball was gem quality, but their batting average was far from perfect – though it was still good enough to make a decent return on their capital…before they were caught by the SEC!

Most stories involving people seeing into the future, like that of the trading cartel above, don’t have “happily ever after” endings. There are usually unintended consequences that come with perfect prescience – a reminder that even prophets can’t escape risk and uncertainty. The best we mortals can do is make our decisions with a framework that explicitly accounts for the presence of risk in just about every big choice we face.

If you haven’t already, you can play the game here: Crystal Ball Trading Challenge

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Appendix I: Detailed Analysis of Player Results

Below is a chart showing the distribution of payouts to the players. The average payout was $51.62 per player, representing a weighted average return across all the players of 3.2%.⁸ About half (45%) the players lost money, and 16% went bust, about the same as the 20% that maxed out at $100. We suspect most readers will agree with us in rating this performance “not very impressive.” It seems that getting the front page of not-any-old-newspaper, but the WSJ, 36 hours ahead of time (albeit with market moves redacted) may not be as valuable as many of us might have imagined.

An experienced market participant is likely able to extract more information from the front page of the WSJ than someone with less experience, such as the participants in our experiment. Even though direct reporting of market moves was blacked out, journalists often report the news biased by how markets reacted after the news. For example, they’ll refer to an employment report as “weak” if the bond market rallies after the report, even if the actual report is more ambiguous – for example, a slightly low number of jobs created, countered by a drop in the unemployment rate and a rise in hourly earnings.

The players forecast the correct direction of stocks and bonds 51.5% of the time. With stocks, they got the correct direction on 48.2% of their trades, and on 56.2% of the bond trades. Notice there were four days where more than 70% of the players forecast the correct direction of the market, and 10 days when more than 60% were correct. However, the players placed 40% more trades in stocks than bonds, which is unfortunate given the players had a better realized edge with bonds than stocks. The table below describes each of the 15 trading opportunities and shows how many trades the players placed on each of the days and what percentage of the trades were placed in the correct direction: long when the market went up and short when it went down.

The next table puts the focus on trade-sizing. It shows the average leverage used for each trading day for stock and bond trades, and also the averages conditional on being higher than 5x leverage. Average leverage used in stock and bond trades was 13x and 10x respectively, and two times as much – 22x and 20x – for trades where leverage was greater than 5x.

We calculated the correlation of leverage (i.e. trade size) for each day versus the win percent for each day, and found a zero correlation in the case of stock trades and a -0.1 correlation for bond trades, along with a +0.2 and -0.1 conditional on leverage used being greater than 5x. It seems that our players on average did not follow a strategy of placing bigger trades on those that they had a higher probability of getting right. Perhaps this is due to them not knowing which ones they had a higher probability of getting right, or perhaps they were not following a disciplined sizing strategy.

The players traded 2,067 times, for an average of about 18 trades per player. The maximum number of trades each player could have made is 30, which would entail doing a stock and bond trade for every front page. Some of the 40% shortfall versus the maximum number of trades is due to 16% of the players going bust, but most of the shortfall is from players abstaining from trading opportunities.

Players were more apt to take long positions in stock and bonds; they traded stocks 62.5% of the time as a long position, and 59.6% of the time for bonds. About 10% and 8% of the players were long stocks and bonds, respectively, for every trade they made.⁹

Unpacking player performance

What accounts for the underwhelming 3.2% return of the players? The fact that our players guessed the direction of stocks and bonds correctly only 51.5% of the time seems to be a pretty big handicap to overcome. It seems the front page of the WSJ wasn’t a particularly clear type of crystal ball for our players to read, and/or they weren’t very good at reading it.

However, even with their weak ability to read the tea leaves, the players could have done quite a bit better if they applied a sensible and constant amount of leverage to all their trades. It would have been reasonable for the players to have estimated the daily standard deviation of stocks and bonds, given our description of how we chose the 15 days, at around 1.5% – 2% for stocks and 1% – 1.5% for bonds. They might have then considered that there could easily be a two or three sigma event in the sample, and so the maximum amount of leverage they could use with a low likelihood of being wiped out might have been 8x for equities and 12x for bonds, if betting on both at the same time.

With that maximum leverage in mind, the next step would be to find the optimal size, subject to the maximum constraint above, given their view of the expected return and risk of the trades.¹⁰ One reasonable choice would have been to apply the Kelly criterion. While the implicit risk-aversion embedded in the Kelly criterion is lower than most people’s risk-aversion with regard to their total wealth, it is reasonable to use it here given the amount of money involved was small relative to the players’ total wealth.

If the players felt they had a 55% chance of being right (an overestimate, as it turned out), that would have suggested something like 6x leverage for stocks and 8x leverage for bonds,¹¹ assuming profits on these trades would be uncorrelated. The optimal size would be a bit lower assuming some positive, but not perfect, correlation in trade outcomes.

If the players had all changed their trade-sizing such that they leveraged all their trades as suggested above (6x for stocks and 8x for bonds), they’d have generated an average return of 10%. Another sign that this is better sizing of trades is that outcomes amongst the players would have had about 45% less dispersion: only 4% of the players would have lost more than 50% of their stake, compared to about 28% in the actual trial. None would have lost more than 75%, and hence, none would have gone bust – recall that 16% of our participants did just that. So, the participants in aggregate would have done better with more reasoned trade-sizing, particularly on the downside…but not tremendously better (and we’d have been 7% more out-of-pocket).

The actual leverage used by participants was, on average, much higher than 6x and 8x for stocks and bonds – generally, people tended to use more leverage in their stock trades than in their bond trades, which is inconsistent with stocks being more volatile than bonds combined with their ability to forecast stock movements being weaker than their skill in guessing bond movements.

As can be seen in the chart below, on about 30% of the days that players traded, they used leverage greater than 20x, and on 4% of the days, they used total leverage of 60x or higher. And on 17 occasions – just over 1% of days traded – players went for 100x leverage, which exposed the player to close to a 50% chance of total loss of capital. It seems clear that there was a fair amount of over-sizing of trades.

However, a much bigger improvement could have been attained from the players doing a better job discerning when they had a more accurate reading of the future. If they’d scaled their trades bigger when they were more likely to be right, they’d have done much better, but it’s hard to know if, on the days when a high percentage of players put on the correct trades, if they really did have a stronger conviction that they were going to be right. It was clear which days were employment and Fed announcement days, and it turns out that the players were more accurate in their readings for bond movements on those days with a 58% hit ratio, while on the other third of the days they only had a 50% hit ratio.

The players also could have done much better if they had based their forecasts on a simple set of rules around the news that was on those front pages. They’d have been correct about 60% of the time if they had shorted bonds whenever the balance of news items was in the direction of a stronger economy, higher inflation, higher energy prices, a stronger Euro or a less accommodating Fed, and vice versa for the opposite news. For stocks, players also would have been correct about 60% of the time if they went long stocks when the balance of news items was in the direction of stronger economy, lower inflation, higher energy prices, a stronger Euro or a more accommodating Fed, and if they went short stocks when the balance of news items was in the opposite direction. There were several days when there was no news of the above variety or there were an equal number of news items on each side of the ledger. In those cases, abstaining from trading would have been a sensible decision.

The table below shows the results from applying the simple trade decision rules described above for direction and sizing. This approach had a success rate of 58% for stocks and 64% for bonds, resulting in an average 6.1% return on each trading day and a 2.4x growth of the bankroll.

As a further test of our hypothesis that this Crystal Ball would bear fruit for players with more experience in connecting news to markets and in sensible trade-sizing, we had five senior bank and hedge fund traders play this game.¹² Their average end wealth was 2.3x their starting wealth, ranging from 1.2x to 5.6x. None went bust, and their average trade success ratio was about 63%. As was the case with the main sample of players, these experienced traders also did substantially better with bonds (71% correct) than with stocks (56%).

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Appendix II: Putting a value on the crystal ball

How much should an investor be willing to pay for a crystal ball that gives them the front page of the WSJ one day in advance, on 15 high-volatility days? In general, when investors are allocating their capital to attractive opportunities with optimal sizing, the risk-adjusted return they are expecting to earn is roughly one half of the expected return.¹³ Let’s say that, when we do a trade, the average expected return to risk ratio of the trades is 0.2. This is twice as high as we were suggesting for our participants, and consistent with a 60% chance of getting the direction of the market correct.

For an investor with a typical degree of risk-aversion,¹⁴ he should risk 10% of his capital on a one standard deviation outcome of such a trade, assuming the trade is uncorrelated with the rest of his investments.¹⁵ His expected return on the trade is then 2% (0.2 * 10% = 2%), and his risk-adjusted return is about 1% per trade. In practice, there needs to be a further reduction for managing the leverage that will be employed, but let’s leave that to the side for the purposes of this example.

The final step is to estimate how many times we expect the crystal ball to give us useful news. Let’s say it’s 75% of the time. Then, our risk-adjusted wealth grows by 1% for each of the 24 trades we expect to do, resulting in certainty-equivalent wealth 1.27x our initial wealth (1.01²⁴ = 1.27). This tells us that the most we can pay for this crystal ball is 21% of our wealth (1 – 1/1.27 = 21%).

Another useful perspective on the value of the crystal ball is to compare the risk-adjusted value of getting the newspaper in advance once per year versus being able to invest in the stock market for the whole year. The Sharpe ratio of one day’s trades driven by the crystal ball reading is around 0.2 – 0.3 (perhaps much less depending on who is doing the reading) which is less than what most people believe is the typical Sharpe ratio of one year’s worth of investing in the stock market.

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Appendix III: Caveats and shortcomings of this study

As with most studies involving paying relatively nominal sums to university students, it’s natural that players’ behavior with a $50 starting bankroll would be very different from how they would use this crystal ball if they could trade on their total wealth.

Players may have done much better if they’d been given company-specific information ahead of time and been allowed to trade individual stocks with that information. Several players told us that they felt the crystal ball would have been much more useful if they knew more about the context of the front page news, in particular what the market was primarily concerned about at the time.

The maximum leverage allowed in our game is higher than most investors can access through futures. However, out-of-the-money options do provide a viable alternative, though with considerably higher costs. Also, we assumed that the players could hold on to their trades until the following day’s close. It’s possible that in some cases, the intraday move in the markets may have wiped out the player’s capital before the next close.

In practice, it is unlikely investors would leverage their total wealth without a limit on the worst-case outcome they could experience. Two effective ways to limit losses from leveraged investments are: 1) putting the trades in a limited liability vehicle, where losses are limited to the capital therein, or 2) buy short-term out-of-the-options to limit the maximum loss on the leveraged trades. Option 1 has the drawback that it may not be possible to get the desired amount of leverage, and Option 2 involves the costs associated with the options contracts.

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Appendix IV: The redacted front pages used in the experiment

All the front pages can be seen here, in chronological order: https://elmwealth.com/crystal-ball-gallery/

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Further Reading & References

• Andre, P., Schirmer, P. and Wohlfart, J. (2023). “Mental models of the stock market.” SAFE Working Paper No. 406. SSRN.
• Haghani, V. and Dewey, R. (2017). “Rational decision making under uncertainty: Observed betting patterns on a biased coin.” Journal of Portfolio Management.
• Haghani, V. and White, J. (2023). The Missing Billionaires: A Guide to Better Financial Decisions. New York: Wiley.
• Hwang, J. and Lee, D. (2024). “Economic valuation of becoming a superhero.” Journal of Cultural Economics.
• Kelly, J. L. (1956). “A New interpretation of information rate.” Bell System Technical Journal 35 (4). 917-926.
• Koudijsy , P. (2014), “Those who know most: Insider trading in 18th Century Amsterdam.” NBER.
• Kumar, N. and Tetley, L. (June 19, 2024). “Hedge Fund Talent Schools Are Looking for the Perfect Trader.” Bloomberg.
• Levine, M. (November 29, 2024). “Knowing the Future Isn’t That Helpful.” Money Stuff. Bloomberg.
• Merton, R. (1969). “Lifetime portfolio selection under uncertainty: The continuous-time case.” The Review of Economics and Statistics 51 (3). 247-257.
• Nowell, A. (2022). The most desired superpowers around the U.S. TransImpact.
• Xie, C. (2020). “The Signal Quality of Earnings Announcements: Evidence from an Informed Trading Cartel.”

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1. Many people were instrumental in bringing this experiment and research article to life. Foremost are the contributions of our research associate James Cross, who helped design and single-handedly programmed the Crystal Ball game during the summer of 2023 while still an undergraduate at Princeton. We thank our long-time research collaborator Richard Dewey for his guidance in designing the study and interpreting the results. Jason Zweig of the Wall Street Journal got us off the ground, and introduced us to ASU Professor Rawley Heimer, whose experience in designing and running studies similar to ours was invaluable. We owe a debt of gratitude to our many friends and colleagues, who as always, did their best to clarify and vet our analysis, and in many cases to be guinea pigs for the study: Jerry Bell, Larry Bernstein, Mimi Duff, Fash Golchin, Jessica Haghani, Joshua Haghani, Mark Haghani, Larry Hilibrand, Alex Imas, Spencer Jakab, Agustin Lebron, Saman Majd, Bill Montgomery, Andy Morton, Vladimir Ragulin, Chris Rokos, Jeffrey Rosenbluth and Steven Schneider. If this research has merit, much of the credit goes to them, although all errors are our own. We thank Nassim Nicholas Taleb for his insightful observation that gave birth to this line of inquiry. Finally, our heartfelt thanks go to the roughly 1,500 people who took time from their busy lives to pit their wits and luck against our Crystal Ball challenge.
   
2. 15 days with one stock and bond trading opportunity each.
   
3. Andre, P. et al. (2023). “Mental Models of the Stock Market.” SSRN.
   
4. Despite their relatively strong performance, several of these traders told us they found the game much more challenging than they thought it would be.
   
5. Kumar, N. and Tetley, L. (June 19, 2024). “Hedge Fund Talent Schools Are Looking for the Perfect Trader.” Bloomberg.
   
6. Levine, M. (November 29, 2024). “Knowing the Future Isn’t That Helpful.” Money Stuff. Bloomberg.
   
7. Xie, C. (2020). “The Signal Quality of Earnings Announcements: Evidence from an Informed Trading Cartel.”
   
8. The players who went bust actually finished with a negative balance. The average player return is 0% if we account for the busted players finishing with a debit balance, but of no more than 25% of their starting capital. The average return would go from 0% to 3.8% if we also capped player outcomes at +125% rather than +100%.
   
9. These players seemed to be expressing a view that the WSJ front page from the future held no valuable information. Or perhaps they were heeding another warning from Taleb: “To bankrupt a fool, give him information.” from The Bed of Procrustes: Philosophical and Practical Aphorisms.
   
10. For a fuller discussion of trade-sizing, see chapters 2 – 7 of our book, The Missing Billionaires: A Guide to Better Financial Decisions. Wiley. (2023).
    
11. For Kelly, we use SR / σ, using SR = 0.1, daily σ_stocks = 1.75% and σ_bonds = 1.25%, giving us stocks at 0.1 / .0175 = 6x, and bonds at 0.1 / .0125 = 8x.
    
12. Unproctored, but we are confident we can rely on their integrity, and their natural curiosity, to have played the game straight.
    
13. Both expressed in excess of the safe asset return, and with a few other assumptions about random walks, ability to rebalance positions continuously and frictionlessly, and ignoring taxes.
    
14. Twice as risk-averse as implied by the Kelly criterion.
    
15. And that the outcomes are normally distributed, and that he can rebalance his exposure to keep his leverage constant.
    

By Victor Haghani and James White ¹

The Cyclically-Adjusted Price Earnings ratio, known as CAPE, is the most commonly used metric for estimating the long-term expected real return of the stock market. Although the merits of using cyclically-adjusted earnings were first suggested by Graham and Dodd in their magisterial book Security Analysis, Professors John Campbell and Robert Shiller share the primary credit for introducing the use of CAPE to forecast long-term stock market returns in their seminal 1988 research article “Stock Prices, Earnings, and Expected Dividends.”²

They suggested using a moving average of earnings “because yearly earnings are quite noisy as measures of fundamental value; they could even be negative while fundamental value cannot be negative.” And they concluded: “…we think that it can be argued that a long moving average of earnings is a very natural variable to use to represent fundamental value, and that there are not many competitors for this role.” While Campbell and Shiller used a 30-year moving average of earnings in their 1988 paper, over time, CAPE became defined as the price of an index of stocks divided by the index’s inflation-adjusted average earnings over the past ten years.

The reciprocal of CAPE (1/CAPE), known as the Cyclically-Adjusted Earnings Yield (CAEY), is the metric many investors use to estimate the long-term expected real return of the stock market. The rationale for this estimate is the supposition that, if companies paid out all their earnings to investors (whether as dividends or share buybacks), they would be able to maintain a constant level of real earnings in perpetuity – therefore, that earnings yield is also the expected real return. As we’ll see, this model of corporate earnings is not only appealing in terms of simplicity and intuition, but is also supported by the past 140 years of US public company experience, and is readily available for non-US equity markets over the shorter histories.

The chart below shows a common illustration of the relationship between CAEY and the next 10 years’ real return of the US stock market.

A shortcoming of Shiller and Campbell’s definition of cyclically-adjusted earnings is that it doesn’t take account of the fact that, in general, companies don’t pay out all their earnings as dividends each year. The fraction of earnings not paid out in dividends is either reinvested in the business or paid out via stock buybacks. Reinvesting earnings in the business is done in the expectation of growing future earnings, and this earnings growth should ideally be accounted for when smoothing earnings over the previous ten years for the purpose of predicting long-term future earnings. An empirical hint of why this might be an issue can be seen in the chart above, where using the traditional CAEY results in the average realized real return being about 1.5% higher than the average CAEY for the four central buckets.

To estimate how much retained earnings will grow future earnings, we prefer the use of cyclically-adjusted earnings yield rather than the default textbook metric of return on book equity. Return on book equity is a measure of the average return on equity in place, rather than the more relevant return on incremental equity – and, as an accounting metric, is subject to a variety of distortions. For example, book equity often ignores intangible assets (which have been growing rapidly as a fraction of total corporate assets in recent decades) and generally ignores upward mark-to-market on assets. Earnings yield (while imperfect) is, by contrast is a market-based signal – and empirically, it’s more consistent with historical earnings growth.

Buying back stock doesn’t grow top line earnings, but it does reduce shares outstanding and hence increases earnings per share. Again, the application of the market’s cyclically-adjusted earnings yield is the best simple estimate we have for how earnings used for stock buybacks will increase earnings per share over time.

To the extent that Campbell and Shiller’s definition of cyclically-adjusted earnings is meant to provide an estimate of future average earnings – smoothing out the peaks and troughs in the business cycle – then the measure really should take account of the dividend payout ratio, particularly when it is low. From 1880 to 1988, when Campbell and Shiller published their CAPE article, the average dividend payout ratio in the US was 65%, which perhaps wasn’t low enough to warrant an adjustment – but from 1988 to the end of 2024, the average dividend payout ratio fell to just 45%, and it’s dropped to 35% over the past two years.³

We believe a better measure of cyclically-adjusted earnings should directly account for the logic that retained earnings should increase earnings per share over time (whether through investment in the business or through share buybacks) in addition to the inflation adjustment already part of Campbell and Shiller’s measure. We believe such an adjustment is simple to implement and, when used to compute earnings yield, should provide a better measure of the long-term expected real return of the stock market. We call this modified measure “payout and cyclically-adjusted earnings,” or P-CAE. The earnings yield it’s used to compute we call “P-CAEY,” and the price-earnings multiple “P-CAPE.”

The way we compute payout and cyclically-adjusted earnings is to take each year’s inflation-adjusted earnings for the past ten years and bring forward the earnings not paid out as dividends at a growth rate equal to the CAEY at the time of those earnings.⁴ For example: if, ten years ago, the dividend payout ratio was 60%, real earnings on the stock market index were $10, and the CAEY was 6%, the payout-adjusted earnings we’d use would be:

60% * $10 + 40% * $10 * (1 + 6%)¹⁰ = $13.2

We then take the average of each of those ten years of payout and inflation-adjusted earnings.

Note that, for dividend payout ratios of less than 100% and for positive earnings yields, P-CAE will be higher than Shiller and Campbell’s cyclically-adjusted earnings, which are only adjusted for inflation. From 1890 to 2024, this new payout and cyclically-adjusted earnings metric was, on average, 19% higher than the standard cyclically-adjusted earnings metric.

How should we decide whether this is an improvement, and big enough to warrant its adoption? First and foremost, does it make sense? Doing something that has a stronger logical foundation is usually worth it, and we think this adjustment passes that first test. This is particularly important to think about before looking at the empirical results, as we just don’t have enough historical data to draw strong statistically-based conclusions.⁵

Second, with the caveat that 140 years of data isn’t that much when looking at 10-year stock market returns, we’ll want to compare how each metric has done in forecasting future earnings and returns.⁶ The table below shows a few summary statistics, which are supportive of the hypothesis that our suggested P-CAE metric is more useful than the same metric without the payout adjustment.

	Cyclically-Adjusted Earnings (CAE)	Payout AND Cyclically-Adjusted Earnings (P-CAE)
Window	10 years	10 years
Inflation-adjusted	Yes	Yes
Dividend Payout-Adjusted	No	Yes
	1890 – 2024 (full Shiller dataset)
Avg Cyclically-Adjusted Earnings vs Next Year’s Actual Earnings	-13%	2%
Average of Earnings Yield minus 10yr Prospective Market Real Return (Arithmetic per annum)	-1.4%	0.1%
% of Variance of 10yr Prospective Real Return Explained	24%	35%
	1950 – 2024 (post WWII)
Avg Cyclically-Adjusted Earnings vs Next Year’s Actual Earnings	-15%	0%
Average of Earnings Yield minus 10yr Prospective Market Real Return (Arithmetic per annum)	-1.9%	-0.5%
% of Variance of 10yr Prospective Real Return Explained	15%	33%

Notice that the standard Shiller and Campbell metric underestimates future earnings by 13% and 15% in the two periods, which we’d expect since that metric is not taking account of companies retaining earnings or repurchasing shares. Also, the shortfall is bigger in the more recent period, which is consistent with dividend payout ratios being lower over the second half of the 1890 – 2024 sample period. It’s also supportive that P-CAEY explains more of the next ten years of real returns, and by a decent margin in both samples.⁷ The chart below shows P-CAEY and the next ten-year US real stock market return.

Not adjusting for dividend payout rates leads to an increasingly poor raw earnings estimate as the window used for the estimate lengthens. For example: using the full US stock market earnings history from 1880 to 2024 as the window rather than ten years, Campbell and Shiller’s CAE would be $45 per S&P500 unit today, while the P-CAE estimate would be $210. Actual S&P500 earnings at the end of 2023 were $197.⁸ We are not suggesting that such a long window is preferable to a ten-year window, but rather to illustrate how the Shiller and Campbell metric diverges from the P-CAE suggested here, and it’s reassuring that P-CAE actually stands up pretty well to such a long window. It’s also comforting to note that the 2.1% annual rate of growth of US real corporate earnings experienced from 1890 to 2024 is what we’d expect from our simple model using an average dividend payout ratio of 65% and average growth rate of 6% on earnings not paid as dividends, assumptions which are not too far from actual experience.

Again, we must stress that the historical data available does not provide enough statistical power to make a decision solely on the basis of the data – but, if you’re already attracted to P-CAPE over CAPE based on the structural rationale, the empirical evidence does provide some further comfort.

In 2014, Bunn and Shiller suggested an adjustment to CAPE, which they called “Total Return CAPE (TR CAPE).” It adjusts for the changing dividend payout ratio over time. However, Bunn and Shiller’s TR CAPE makes its adjustment in such a way that in effect retained earnings are assumed to make future earnings lower, which is counter to economic logic. We believe that TR CAPE was constructed to be used as a statistical signal in a regression analysis context. In contrast, we are proposing P-CAEY as a direct forecast of the future real return of the stock market.

In the above analysis, we’ve used data graciously provided by Professor Shiller on his website. We have also studied the dividend payout adjustment using an alternative dataset for US stock market returns and earnings, and we’ve assessed non-US stock market datasets. We found results similar to those shown above, as can be seen in the table below. Another benefit of P-CAEY is that it gives a more consistent measure across international equity markets with different dividend payout ratios.

The one exception from a historical perspective has been the Canadian stock market. Neither CAEY nor P-CAEY has proven a useful return estimate over the past 35 years. Perhaps this is because Canada has had greater industry concentration than the other larger regional equity markets, or perhaps it’s due to 35 years of overlapping 10-year periods being such a sparse dataset.

We recognize that there are many who are critical of the use of CAEY as an estimator of future stock market real returns. We find most of these criticisms take the form of: “Twenty years ago, the CAEY of the US equity market was about 4.5% – but over the next 10 years, the actual was so much higher, coming in at 9.3% pa.”⁹ We don’t think the modification we’re suggesting in P-CAEY will go very far in changing the minds of such critics. We also think this isn’t a particularly valid criticism, as CAEY being a useful estimator doesn’t require that it explains all (or even most) long-term return variation. But, if you were attracted to the logic of Campbell and Shiller’s CAPE to begin with, we think you’ll find their measure adjusted for dividend payouts a worthwhile improvement.

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Further Reading & References:

• “Elm Wealth Capital Market Assumptions.” (2024). Elm Wealth.
• Ang, A. and Bekaert, G. (2007), “Stock Return Predictability: Is it There?” Review of Financial Studies.
• Asness, C. (2012), “An Old Friend: The Stock Market’s Shiller P/E.” AQR.
• “Historic CAPE Ratio by country.” (2024). https://indices.cib.barclays/IM/21/en/indices/static/historic-cape.app Barclays.
• Boudoukh, J., Israel, R. and Richardson, M. (2019). “Long Horizon Predictability: A Cautionary Tale.” Financial Analysts Journal.
• Bunn, O., and Shiller, R. (2014). “Changing Times, Changing Values: A Historical Analysis of Sectors within the US Stock Market 1872-2013.” SSRN.
• Campbell, J. and Shiller, R. (1988). “Stock Prices, Earnings and Expected Dividends.” Journal of Finance.
• Campbell, J. and Thompson, S. (2007). “Predicting Excess Stock Returns Out of Sample: Can Anything Beat the Historical Average?” Review of Financial Studies.
• Cochrane, J. (1992). “Explaining the Variance of Price-Dividend Ratios.” Review of Financial Studies.
• Cochrane, J. (2011). “Presidential Address: Discount Rates.” Journal of Finance.
• Gavin, M. (2014). “Introducing the SCAPE: Why US equities are less expensive than they seem.” Barclays.
• Graham, B. and Dodd, D. (1934). Security Analysis. McGraw-Hill.
• Keimling, N. (2016). “Predicting Stock Market Returns Using the Shiller CAPE — An Improvement Towards Traditional Value Indicators?” SSRN.
• Shiller, R. and Jivraj, F. (2017). “The Many Colors of CAPE.” Barclays.
• Siegel, J. (2016). “The Shiller CAPE Ratio: A New Look.” Financial Analyst Journal.

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1. This not is not an offer or solicitation to invest. Past returns are not indicative of future performance. Thank you to Larry Hilibrand, Vlad Ragulin, Samir Bouaoudia, Antti Ilmanen, John Campbell, Steve Mobbs, Andy Morton, Rich Dewey, Jeffrey Rosenbluth, Joshua Haghani and our partner Jerry Bell for reading drafts of this note and sharing their comments with us.
   
2. Graham and Dodd recommend an approach that “shifts the original point of departure, or basis of computation, from the current earnings to the average earnings, which should cover a period of not less than five years, and preferably seven to ten years.” (Security Analysis, page 452).
   
3. Some investment commentators like to estimate the long-term expected return of the stock market as the sum of current dividend yield plus expected dividend growth. The variability in corporate dividend policy over time poses a big challenge to that approach.
   
4. Ideally, we would like to use P-CAEY, reduced by .5 * σ² (σ = standard deviation of stock market returns) to convert from an arithmetic to a compound return. However, for simplicity and to avoid getting slightly different values as a function of the starting point, we use CAEY without the arithmetic-to-compound return adjustment.
   
5. Even if we assumed the data we have came from a stable distribution, which is almost certainly not the case.
   
6. Looking at monthly overlapping ten-year periods over 140 years has about the same statistical power as looking at just 14 to 17 back-to-back ten-year periods (far, far from the nearly 1,700 overlapping observations) depending on one’s assumptions about the underlying processes. See Boudoukh et al. (2019), “Long Horizon Predictability: A Cautionary Tale.”
   
7. We calculate “percent of variance explained” as 1 – f² / s² where f² is the average of the squared differences between CAEY (or P-CAEY) and realized real return, and s² is the variance of realized returns.
   
8. We view the small 7% difference as more of a coincidence than an argument for using a super-long window. Earnings from a century ago just don’t seem that relevant to estimating future earnings today.
   
9. There are many other criticisms to be sure, such as the arbitrary nature of the ten-year equal-weight window, and the failure to account for changing industry weights, but we think such criticisms are not the main ones. For a few examples of CAPE critiques, see: Armstrong, Robert. “The valuation mystery,” Financial Times, May 27, 2024, or Shah, Devesh. “Asset Allocation & International Equities, Part I: What is the right percentage allocation?” Mutual Fund Observer, June 2024.
   

By Victor Haghani, Vladimir Ragulin and James White ¹

In Lake Wobegon, all the women are strong, all the men are good-looking, and all the children are above average.
  – Garrison Keillor

In 1991, William Sharpe made perhaps the strongest argument to date for market capitalization-weighted index investing in a three page article titled, “The Arithmetic of Active Investing”:

If “active” and “passive” management styles are defined in sensible ways, it must be the case that

(1) before costs, the return on the average actively-managed dollar will equal the return on the average passively-managed dollar and

(2) after costs, the return on the average actively-managed dollar will be less than the return on the average passively-managed dollar

These assertions will hold for any time period. Moreover, they depend only on the laws of addition, subtraction, multiplication and division. Nothing else is required.

The key insight of this idea is that, if we add all non-market capitalization-weighted portfolios together into one big portfolio, it must be identical to the market capitalization-weighted portfolio, i.e. the “market portfolio.” While the practical implications of Sharpe’s Arithmetic have been debated, its logic has been broadly accepted, and many see it as being one of the main drivers of the massive growth of index investing over the past three decades.²

Sharpe’s seminal paper landed a body blow on the stock-picking industry. Perhaps he felt his argument packed more than enough punch to make investors rethink their stance on stock-picking – but whatever his reasons, he stopped short of laying out a corollary to his “Arithmetic of Active Management”, which is just as powerful an indictment.

The corollary requires a bit more explanation than his main argument, but it is almost as simple and rests on the same basic insight laid out in his 1991 paper, that all active portfolios aggregate to the market portfolio:

(1) the average risk across all actively-managed portfolios of stocks will be greater than the risk of the market portfolio, and

(2) the average risk-adjusted excess return across all active portfolios will be less than the risk-adjusted excess return of the market portfolio, before taking account of fees and trading costs

We can see why the average risk across all active portfolios is greater than the risk of the market portfolio by seeing that every active portfolio can be expressed as holding the market portfolio plus an “active exposures” portfolio of longs and shorts in all the constituents, such that the market portfolio plus the active exposures portfolio equals the given active portfolio. Further, each active portfolio requires that there be someone(s) holding an active portfolio with the opposite active exposures, which we’ll call the mirror portfolio.

The average of the risk of any active portfolio and its mirror will be greater than the risk of the market portfolio. The key to seeing why is to notice that the two portfolios of active exposures have the same risk, but their correlations to the market portfolio will have opposite signs. As a result, when the portfolios are averaged together the correlation terms will cancel each other out, leaving just the extra tracking risk from the active exposures as an addition to the risk of the market portfolio. Since this holds for any active portfolio, it follows that averaging across any number of active portfolios gives the result that the average risk across all active portfolios must be greater than the risk of the market portfolio.

We cannot easily say how much higher the average risk of active portfolios will be versus the risk of the market portfolio, as it depends on the concentration of each active portfolio. However, we can get a sense for the magnitude by considering randomly constructed portfolios holding different numbers of stocks, such that in aggregate all the portfolios equal the market portfolio.

In the table below, we compare the risk of the market portfolio with the average risk of portfolios randomly constructed of 5, 25, and 100 stocks, selected so that they aggregate as closely as possible to the market portfolio.⁴ These concentrated portfolios have between 4% and 30% more risk than the market portfolio (see furthest right column). These active portfolios of N stocks are riskier than one might naively estimate by assuming that portfolio idiosyncratic risk decreases with √_N¹ . This is because 1) many of the idiosyncratic risks of individual stocks are correlated with each other (e.g. through being in the same industry sector or sharing factor exposures), and, 2) the uneven market capitalization weights result in greater concentration in portfolios than would arise from portfolios in which each stock had the same weight.

If, as was the case in the 1960s, the median number of stocks in an individual’s brokerage account was just two, the average riskiness of these highly concentrated portfolios would be 1.5x that of the market portfolio. A more recent 2005 study showed that stock investors with liquid assets over $1mm directly hold on average 15 stocks.⁵

Another viewpoint we can take is to compare the average risk of a set of non-market capitalization-weighted ETFs and mutual funds to the risk of the S&P 500 over the past 10 years. Even though these actively-managed funds hold about 200 stocks each, their average risk was 7% higher than the risk of the relevant market portfolio. In particular, it is very interesting to note that Vanguard’s growth and value funds – which each own over 200 stocks, and represent mirror active portfolios of each other – have an average risk that is 7% higher than the S&P 500.

You might say, what’s the big deal if the risk on a typical actively-managed portfolio is 10% higher than the risk of the market portfolio? Well, we think it is a big deal! Assuming it comprises most of the risky part of a portfolio, to be indifferent between the active portfolio and the market capitalization-indexed portfolio, you’d want them to have the same Sharpe ratio. This means the active portfolio would need to have 10% more expected return net of fees in excess of the safe asset than the market portfolio. If, for example, you think the market portfolio offers a 4% return in excess of the safe asset, then the active portfolio would need to offer 0.4% more, or a 4.4% excess return, just to be equally as attractive on a risk-adjusted basis.⁶

For years, investors and commentators have bemoaned the roughly 0.6% per annum difference between the average expense ratios on US actively-managed equity mutual funds and US equity index funds.⁷ We think they should be just as concerned, if not more so, by the extra cost of risk involved in holding concentrated portfolios in aggregate. This cost of risk of active management can easily be as large as or, in extreme cases of concentration, dwarf the extra fees that have garnered investor attention for so long.

Conclusion

Vanguard founder John Bogle was profoundly impacted by Sharpe’s Arithmetic, which he developed into his “Cost Matters Hypothesis” (CMH) presented in the same journal that published Sharpe’s Arithmetic 14 years earlier:⁸

Gross returns in the financial markets minus the costs of financial intermediation equal the net returns actually delivered to investors…To explain the dire odds that investors face in their quest to beat the market, however, we don’t need the EMH (Efficient Markets Hypothesis); we need only the CMH [Cost Matters Hypothesis]. No matter how efficient or inefficient markets may be, the returns earned by investors as a group must fall short of the market returns by precisely the amount of the aggregate costs they incur. It is the central fact of investing.

In the spirit of the late and great John Bogle, we would like to offer the “Risk Matters Hypothesis” (RMH), as an addition to the EMH and CMH in warning investors of the challenge they face in adding value through stock-picking:

The average risk-adjusted excess return across all active portfolios will be less than the risk-adjusted excess return of the market portfolio, before taking account of fees and trading costs.

As we discuss in more detail in our book, The Missing Billionaires: A Guide to Better Financial Decisions, it is natural that investors should and do require compensation for bearing risk. However, all too often we don’t adequately account for it in our investment decisions.

If there were no extra fees, taxes or other monetary costs associated with active management, Sharpe’s 1991 argument may not have been as influential as it has proved to be. In the past 30 years since Sharpe laid out his arithmetic, there has been a dramatic decrease in the fees charged by active stock managers, commissions for retail stock trades have gone to zero, and the inside bid-ask spread on equities has decreased. Taken together, these have reduced – but not eliminated – the importance of Sharpe’s original argument.

However, in the risk corollary to Sharpe’s Arithmetic described in this note, active investors are engaged in a negative sum activity even if there are no extra fees involved.⁹ Logic dictates that investors cannot in aggregate be rewarded for the extra risk they incur in owning concentrated stock portfolios.

Using Sharpe’s insightful observation that the portfolios of all active investors must equal the market portfolio and applying it in the dimension of risk, his original warning still rings true that active stock investors in aggregate need to overcome a substantial threshold of extra return in order to improve their welfare.

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Further Reading and References

• AQR. (2012). “Why Do Most Investors Choose Concentration Over Leverage?” Alternative Thinking 2Q 2012.
• Black, F. 1986. “Noise”. Journal of Finance 41, 529–543.
• Bogle, J. 2005. “The Relentless Rules of Humble Arithmetic.” Financial Analysts Journal 61 (6), 22-35.
• Bogle, J. 2014. “The Arithmetic of the ‘All-In’ Investment Expenses.” Financial Analyst Journal 70 (1), 1-9.
• Campbell, J. 2018. Financial Decisions and Markets: A Course in Asset Pricing. Princeton University Press.
• Chen, H., Noronha, G. and Singal, V. 2006. “Index Changes and Losses to Index Fund Investors.” Financial Analysts Journal 62 (4), 31–47.
• Dick-Nielsen, J. 2012. “Index Driven Price Pressure in Corporate Bonds.” Working paper, Copenhagen Business School.
• Haghani, V. and White, J. 2023. The Missing Billionaires; A Guide to Better Financial Decisions. Wiley.
• Malkiel, B. 2023. A Random Walk Down Wall Street. W. W. Norton & Company.
• Merton, R. 1969. “Lifetime Portfolio Selection under Uncertainty: The Continuous-Time Case.” The Review of Economics and Statistics 51 (3), 247–57.
• Pedersen, L. 2018. “Sharpening the Arithmetic of Active Management.” Financial Analysts Journal 74 (1), 21-36.
• Sharpe, William F. 1991. “The Arithmetic of Active Management.” Financial Analysts Journal 47 (1), 7–9.
• Stambaugh, RF. 2014. “Presidential Address: Investment Noise and Trends.” The Journal of Finance 69, 1415-1453.

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1. This not is not an offer or solicitation to invest. Past returns are not indicative of future performance.
    

   We thank John Campbell, Jeffrey Rosenbluth and Mark Grinblatt for their help and encouragement. All errors are our own.
   

2. For discussions of where Sharpe’s Arithmetic may not be a good model of reality, see Pedersen (2018), Chen et al. (2006), or Dick-Nielsen (2012) for the analysis of frictions in bond index funds.
   
3. We recognize this is laid out informally. We hope to come back to this at a later date with a more rigorous treatment, including all assumptions needed. In the same spirit, this result also holds for the average standard deviation of returns, for -1 < ρ < 1, but the math is not as neat and tidy as it is for the average variances.
   
4. We use the past 10 years of weekly return data, and current weights of the Bloomberg 500 US stock index. Our simulated investor portfolios hold stocks with market-cap weights, which is different from the standard calculation of diversification benefits which assumes equal weights, e.g. Malkiel (2023). Once the stock has been selected, we include it with the weight proportional to its market cap. This is because with the equal-weighted approach, it is not possible for the aggregated holdings to match the market weight of the mega-caps like AAPL by aggregating equal-weighted portfolios of more than 15 stocks, since even if each concentrated portfolio holds AAPL (which it wouldn’t), adding them together only gives a 6.7% AAPL weight ( = 1/15) for the aggregated portfolio – below the actual 7% weight. With our approach, an investor holds larger positions in the mega-caps, and therefore needs more stocks to achieve the same risk reduction vs. the standard equal-weighted approach.
   
5. Stambaugh (2014).
   
6. You may also ask, what’s the big deal about a 10% difference in Sharpe ratio, if you don’t expect a higher return on your actively-managed portfolio? The answer is that a 10% lower Sharpe ratio causes a 20% reduction in your risk-adjusted return, as we describe in The Missing Billionaires, Chapter 5, page 59.
   
7. Average Equity and Bond Mutual Fund Expense Ratios Continue to Decline (2022).
   
8. Bogle, J. 2015. “The Relentless Rules of Humble Arithmetic.” Financial Analysts Journal, 61 (6), 22-35.
   
9. Ignoring some possible, though hard to observe or heavily weigh, risk transfer arguments.
   

By Victor Haghani, Vladimir Ragulin and James White ¹

“You know, some clichés are clichés because they are true.”
  – George Carlin

The first book that many new arrivals on the trading floors of banks and hedge funds are encouraged to read is Reminiscences of a Stock Operator by Edwin Lefevre (1923). The story highlights the importance of controlling one’s emotions in trading while being attuned to the herd mentality that often drives markets. The following two quotations reflect these twin themes of individual discipline and the pack-like movement of crowds:

“Cutting losses quickly is the foremost rule of speculating.”
“The big money is not in the buying or the selling, but in the waiting.”

For many successful speculators, “Cut your losses early; let your profits run” tops the list of advice they offer to the next generation of traders, as was the case with 12 out of the 14 renowned money men interviewed by Jack Schwager in Market Wizards: Interviews With Top Traders (1989).² Here’s how one of the interviewees, Paul Tudor Jones, expressed it: “If I have positions going against me, I get right out; if they are going for me, I keep them.” This dictum is not lost on today’s most successful money managers, such as the Millennium, Balyasny and Exodus Point hedge fund groups, who put the tenet of cutting losses quickly at the very core of their investment and risk management policies.

Of course, this advice is far from universally accepted, and there are many situations where there is agreement that it’s not the right thing to do. Many people believe the qualities of perseverance, loyalty and determination are essential ingredients of success. The well-known author and psychologist Angela Duckworth wrote a popular book dedicated to this idea, aptly titled Grit. In it, she writes, “People who accomplished great things…often combined a passion for a single mission with an unswerving dedication to achieve that mission, whatever the obstacles and however long it might take.”

In the narrower sphere of investing, a number of financial economists – such as 2022 Nobel prize winner Phil Dybvig – have criticized the idea of a preset exit from an investment as being suboptimal at best and indefensibly irrational at worst. His 1988 paper “Inefficient Dynamic Portfolio Strategies or How to Throw Away a Million Dollars in the Stock Market” argues that strategies that cut risk to zero following losses are strictly suboptimal if asset prices follow random walks and investors exhibit smoothly decreasing marginal utility of wealth.³

However, most practitioners believe that there are times when markets don’t follow the well-behaved random walks which underlie many elegant formulas of modern finance. The rest of this note will explore two cases in the context of investing where the policy of cutting losses quickly and letting profits run (which we’ll abbreviate to CLE-LPR) can make sense. If you’re interested in the question of whether CLE-LPR makes sense in the broader context of other life decisions, read “Heads or Tails: The Impact of a Coin Toss on Major Life Decisions and Subsequent Happiness.” (Levitt, 2021), or watch Victor’s short TEDx talk: Quitting is for Losers Winners. (Haghani, 2017).

In the chart below, we see how a CLE-LPR strategy would have fared if applied to US stock market investing over roughly the past 100 years. It shows historical returns of investing in the S&P500 index with a CLE-LPR strategy versus a static weight (70% stocks/30% T-bills) portfolio of equal risk. For the dynamic strategy, we sell our S&P 500 long position and invest in T-Bills as soon as the trailing 12-month return drops below -5%, stay out of the market for at least three months, and reinstate the long as soon as the 12-month return is better than -5%.

This approach means losses are cut early, while a profitable position is held as long as it keeps going up. The CLE-LPR strategy generated a 1.4% higher annual return over the period with the same risk measured as standard deviation of daily returns, and roughly the same average exposure to the stock market of 70%.⁴ This translates into a 25% higher Sharpe Ratio for the dynamic strategy than for the static stock/T-bill portfolio (0.53 vs. 0.43).

Also, the Cut-Losses-Early-and-Let-Profits-Run strategy outperformed the static portfolio during the two worst market sell-offs in 1929-32 and 2007-09, when the static portfolio’s max drawdowns were 70% and 41% respectively, while the CLE-LPR’s largest drops during the same crises were 42% and 21%.

Why Does Cut-Losses-Early-and-Let-Profits-Run Work?

One way to answer this question might be to notice that this investment approach is very similar to momentum-based investing,⁵ which has been very successful over a wide range of asset markets and time periods. For example, in “Time Series Momentum” (2012), Moskowitz et al. found “… significant time series momentum in equity index, currency, commodity, and bond futures for each of the 58 liquid instruments we consider.”⁶ Over the period of our simulation, a momentum signal based on the past year of stock market performance would have lined up with the positioning from the CLE-LPR strategy 94% of the time.

Of course, pointing out that CLE-LPR is very similar to momentum investing doesn’t really answer the question, it just reformulates it into: “Why do momentum strategies work?” It’s hard to know for sure, but there are many theories to choose from. They range from underreaction to new information, to overreliance on recent returns in forming expectations of future returns (a.k.a “return chasing”), to slow recognition of regime shifts in markets, just to name a few. In most markets, positive momentum is usually associated with periods of lower risk, which in turn may encourage investors to want to own more of that asset, thus increasing the price.⁷ Whatever the reason, there is little doubt that momentum in asset prices has been present for a long time, across a wide range of traded assets.

Why Some Hedge Fund Managers Love CLE-LPR

Imagine you run a hedge fund and your job is to allocate capital to a set of traders. To keep things simple, let’s assume you’re managing just one trader, and he tells you that he can generate $2 of profit for every $1 of risk he takes on an annual basis.⁸ That is, he is telling you that the distribution of his trading PnL has a Sharpe Ratio of 2.

You feel that the biggest loss your investors will tolerate is about 10% in a given year, so you want to make sure there’s a very low probability that your Fund loses more than 10%. You’re considering two ways of managing your trader.

Under the first, you tell him that his risk, measured in annual standard deviation of returns, can be no higher than 10% of the Fund’s capital. If his trading has a Sharpe Ratio of 2, you reason that the probability of losing 10% at some point during the year is about 1.3%, which you feel is tolerable.

Under the second regime, you tell him that he should take risk in proportion to how far away he is from losing 10% of the Fund’s capital. To begin with, when he’s 10% away, he can take 10% risk. Later on, if he’s made 5% profits, and so he’s 15% away – he can take 15% risk. If he’s close to losing 10%, then he has to have cut his positions down close to zero. This pattern of position sizing is the embodiment of “cut your losses early and let your profits run.”⁹

Let’s compare the implications of these two ways of managing the trader’s risk-taking. Under both regimes, assuming the trader believes his trading has a Sharpe Ratio of 2, he will start off running risk equal to 10% of the Fund’s capital.¹⁰ The trader under the first system will keep his risk constant at 10% of the Fund’s capital as his PnL evolves, while the trader under the CLE-LPR regime will be dramatically increasing and decreasing his risk as his trading profit and loss fluctuates over time, in proportion to how far away he is from losing 10% of the Fund’s capital.

The chart below shows the probability distribution of outcomes for the Fund’s return under the two regimes, in both cases assuming the trader’s PnL has a Sharpe Ratio of 2. We believe everyone involved – you as the hedge fund manager, the trader who works for you, and your investors – will prefer the return pattern from the CLE-LPR regime to the return pattern arising from taking constant risk.

The CLE-LPR regime produces a higher expected net return (50% vs 18%),¹¹ with only a slightly higher probability of loss (6% vs 2%) and a much higher expected fee for the hedge fund manager and trader to share (12.5% vs 4.5% of capital, assuming a 20% incentive fee). Perhaps the biggest advantage is that, if you are wrong about the skill of the trader and it turns out that his true trading Sharpe Ratio is 0, then under the constant risk regime, there’s a 1-in-3 chance of hitting -10% during the year, while the risk of a -10% return in the CLE-LPR regime remains zero, by construction.¹² In addition to this being a very valuable protection for the hedge fund manager and the hedge fund investors, a trader who suspects he has little skill has a strong incentive to avoid working for a hedge fund that employs a tight CLE-LPR risk-management regime.¹³

Conclusion

Most people are naturally inclined to patiently, and painfully, stick with losing decisions for too long while cashing out of winning decisions too quickly. Indeed, this was an early finding in the field of behavioral economics, and was important enough to be given a name: the “Disposition Effect.”¹⁴ It is often the case that controlling our natural instincts can be rewarding. We suspect that much of the power of cutting losses quickly and letting profits run, at least when it comes to investing, lies in the difficulty of overcoming our propensity to do the exact opposite.

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Further Reading and References

• Baur, DG & Dimpfl, T. (2023). “Cut Your Losses and Let Your Profits Run.” Journal of Portfolio Management (forthcoming), SSRN.
• Duckworth, A. (2017). Grit: Why Passion and Resilience are the Secrets to Success. Vermilion.
• Dybvig, PH. (1988). “Inefficient Dynamic Portfolio Strategies or How to Throw Away a Million Dollars in the Stock Market.” The Review of Financial Studies.
• Forsyth, PA, & Vetzal, KR. (2023). “Multi-period Mean Expected-Shortfall Strategies: ‘Cut Your Losses and Ride Your Gains’.” Applied Mathematical Finance, 29(5).
• Geczy, C., & Samonov, M. (2017). “Two Centuries of Multi-Asset Momentum (Equities, Bonds, Currencies, Sectors and Stocks).” SSRN.com.
• Haghani, V. (2017). Quitting is for Winners. TedX.
• Haghani, V., & McBride, S. (2016). “Return Chasing Can be Hazardous to Your Wealth.” Elm Wealth.
• Lefevre, E. (1923). Reminiscences of a Stock Operator: The Story of Jesse Livermore, Wall Street’s Legendary Investor. Cosimo Classics.
• Levitt, SD. (2021). “Heads or Tails: The Impact of a Coin Toss on Major Life Decisions and Subsequent Happiness.” The Review of Economic Studies, 88(1), 378-405.
• Moskowitz, T., Yao Hua Ooi, YH, & Pedersen, LH, (2012). Time series momentum.” Journal of Financial Economics, 104(2), 228-250.
• Shefrin, H., & Statman, M. (1985). “The Disposition to Sell Winners Too Early and Ride Losers Too Long: Theory and Evidence.” Journal of Finance, 40(3), 777-790.
• Stone, M., Michalow, D., & Beck, T. (2011 May). “Blind ambitions.” Institutional Investor Magazine.
• Kaminski, K. & Lo, A. (2014). “When do stop-loss rules stop losses?” Journal of Financial Markets 18(C), 234-254.

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1. This not is not an offer or solicitation to invest. Past returns are not indicative of future performance.
    

   We thank Andy Morton and Rich Dewey for their valuable input.
   

2. ChatGPT4 agrees: “One of the most popular sayings related to managing trading profit and loss is: ‘Cut your losses short and let your profits run.’ This phrase emphasizes the importance of having a well-defined exit strategy to limit losses when a trade goes against you and to allow winning trades to continue capturing profits.”
   
3. A simple intuition for this result is that, in the absence of regime changes or private market forecasts, the optimal position is given by the well-known Merton Rule – and any deviation from it, via a stop-loss or another overlay, only results in the investor leaving money, or more accurately, ‘expected risk-adjusted return’ on the table.
   
4. Ignoring transaction costs. The t-stat of the excess return is 1.3, which represents a confidence level of 90%.
   
5. Time series momentum is a phenomenon where assets that have exhibited strong (poor) past performance over a certain time period, usually six months to one year, tend to continue performing well (poorly) in the near future.
   
6. Also see: Geczy and Samonov,. (2017). “Two Centuries of Multi-Asset Momentum: (Equities, Bonds, Currencies, Commodities, Sectors and Stocks).”
   
7. This explanation doesn’t fit so well in the case of momentum in bond prices. Bonds tend to be more volatile in a financial crisis, when bond price momentum tends to be positive.
   
8. We’re making further idealistic assumptions that the return distribution of the trader is normally distributed and that trading takes place continuously with no transaction costs. We are also assuming the risk-free interest rate is 0, and the fund charges no management fee.
   
9. This risk-taking approach would naturally arise if the trader were trying to maximize his Expected Utility, with a Constant Relative Risk Aversion Utility function taking his buffer capital as his wealth. With the numbers in this example, the implied coefficient of risk aversion would be 2, a typical value in the personal investing context.
   
10. We’re also assuming his personal risk aversion is low enough that he’ll be happy using his full risk budget.
    
11. This may seem like a strange and surprising result – in fact, we haven’t seen it written about before. Under the two regimes, the starting risk and hence position is the same, but over time the trading is different. There will be a larger expected position size under the CLE-LPR regime, and hence a higher expected gain.
    
12. And with the assumption that the positions of the trader are perfectly liquid.
    
13. The annual expected P&L of a strategy assuming optimal risk allocation is proportional to the square of the Sharpe Ratio since, from Merton’s analysis, a better trader is both comfortable with taking more risk and expects to earn a higher return on each unit of risk he takes. The formula is Expected P&L at Optimal Risk = Capital * SR² / risk-aversion. With this rule of thumb, the hedge fund manager can select risk allocation parameters such that potential earnings for a high SR trader considering joining the fund just exceed compensation from alternative jobs.
    
14. See Shefrin and Statman’s 1985 paper, “The Disposition to Sell Winners Too Early and Ride Losers Too Long: Theory and Evidence.”
    

By Victor Haghani and James White ¹

The Big Question

The most popular indicator of the attractiveness of the stock market – Shiller’s Cyclically-Adjusted Price Earnings ratio (CAPE) — is currently at 39x in the US, higher than it’s been 98% of the time for the past 120 years. What’s a thinking investor to make of this? Should he stay clear of the US stock market, or stick to some pre-set strategic allocation to equities, or is there something else going on? In this note, we’ll argue that CAPE is far from irrelevant, but on its own, it doesn’t tell an investor how much stock exposure to have.

Listen to Victor discuss this research on Bloomberg’s “What Goes Up?” podcast:

CAPE Basics

When the CAPE ratio is high, the prospective return of the stock market is low. This finding makes logical and intuitive sense, and is borne out in historical data. We can say something more specific and powerful: 1/CAPE is a pretty good, though imperfect, predictor of the inflation-adjusted return of the stock market.² The measure of 1/CAPE is known as the Cyclically-Adjusted Earnings Yield (or just “Earnings Yield”), because it’s calculated as Earnings divided by Price. If you invest in the stock market when the Earnings Yield is 6%, your best expectation is that you’ll earn a long-term return (after inflation) of 6%. This is telling us that, contrary to popular belief, when Earnings Yield is low we shouldn’t expect to lose money from the Earnings Yield reverting to some average higher level, and vice versa. In other words, the predictive power of Earnings Yield over a long horizon is not improved by assuming that it is mean-reverting (For a deeper dive, see our 2017 article “Market Multiple Mean-Reversion: Red Light or Red Herring?”). The chart below illustrates this, using a horizon of ten years:

Chart 1: Next 10-Year Real Return vs. Earnings Yield at Start
US Equities, 1900 – 2021

The first reaction of everyone who has seen this chart — your authors included — is: “Ooooeeee! Investor Hall of Fame, here I come!”

Alas, a backtest – as shown in the chart below – pours ice-cold water on these dreams: a simple dynamic approach based on Earnings Yield failed to deliver a higher Sharpe Ratio³ than a static allocation over the entire 120-year period for which we have data, and has actually under-performed since 1943.⁴ This result is one reason you will find so few investment products that offer a dynamic asset allocation strategy based on Earnings Yield or similar metrics. See Asness et al. “Market Timing: Sin a Little” (2017) for a lively and detailed description of the wellspring of this cold water.

Chart 2: Static vs. Conventional Dynamic Asset Allocation:
US Stocks and T-Bills, 1900 – 2021

Worth Another Look?

This result is puzzling though, because it just seems like basic common sense that it should be better to have more exposure when the market is offering higher expected returns, and we’ve seen that Earnings Yield has some power as an indicator of when to expect those higher returns.

The rest of this note is devoted to exploring an alternative, more internally consistent approach to dynamic asset allocation using Earnings Yield as the driver, which has historically delivered the improved performance we’d expect.

When Occam’s Razor Shaves Too Close

The historical analysis of the use of Earnings Yield to dynamically allocate between US stocks and US T-bills presented in Chart 2 is done in Occam’s spirit of maximum simplicity. The rule sets the equity allocation to:

1. Be proportional to the Earnings Yield at each point in time,
2. Average 65% over the whole sample, the same as for the benchmark Static Strategy, and
3. Never be negative or in excess of 100% (i.e. no shorting and no leverage).⁵

We’ll call this the “Conventional Dynamic Strategy.”

There are a number of problems with this approach, including:

1. The asset allocation decision in this strategy is comparing the attractiveness of equities to the attractiveness of T-bills. However, Earnings Yield is not a predictor of the relative attractiveness of stocks versus T-bills; it is only a predictor of the future real return of equities.⁶
2. Changes in riskiness of the stock market are ignored. It is intuitive that all else equal, an investor would want to have less allocated to equities when they are expected to be more volatile.
3. For the equity allocation to average 65% over the period requires knowing what the Earnings Yield of the stock market was over the whole sample, which a non-clairvoyant investor who wanted to follow this strategy could not have known.

A More Consistent Asset Allocation Rule Based on Excess Earnings Yield

If we want to use Earnings Yield to decide how much to invest in the stock market, to be consistent we also need to evaluate alternatives to stocks in terms of their expected real return. It seems natural to turn to US Inflation-indexed bonds (TIPS) as the relevant low-risk alternative to stocks, since the yield on TIPS is a measure of their expected real return, and so provides a directly comparable measurement to the Earnings Yield of equities. Indeed, there are strong arguments that “the (inflation) indexed perpetuity is the riskless asset for a long-term investor, since it finances a constant consumption stream over time,” as suggested by Harvard professors Campbell and Viceira in “Who Should Buy Long-Term Bonds” (2001).⁷ For practical purposes, long-term TIPS are pretty close to the inflation-indexed perpetuity they suggest.

It seems more natural to think about the attractiveness of stocks relative to inflation-protected bonds, rather than just by considering the level of Earnings Yield in isolation. For example, if the Earnings Yield of the stock market were 4% and the real yield on TIPS were also 4%, why would we want to own any equities?⁸ Or for a historical illustration, consider that the Earnings Yield of the US stock market was about 2.7% at the end of 2000 and also at the end of 2021 – but the ten-year TIPS yield was 3.6% back in 2000 and -0.7% at the end of 2021. Would a rational investor choosing between equities and TIPS want to have the same exposure to equities at both points in time, just because the Earnings Yield was the same? We think most investors would agree that they should want to own more equities at the end of 2021 than 21 years earlier. And yet, the conventional analysis that uses the market’s Earnings Yield without reference to the real return offered by safe assets suggests owning the same amount of equities in both cases.

We propose three changes to the (disappointing) Conventional Dynamic Strategy presented in Chart 2, setting the allocation to equities at each point in time to be:

1. Proportional to the excess of the stock market’s Earnings Yield above the real yield of inflation-protected bonds (US TIPS). We’ll refer to this measure as “Excess Earnings Yield.”
2. Inversely proportional to the risk (measured as variance) of the stock market as might reasonably have been estimated by an investor at the point in time of the asset allocation decision.
3. At the level a Utility-Maximizing investor, with a typical and stable degree of Constant Relative Risk-Aversion (CRRA), would choose based on the estimates of Excess Expected Return and Risk set out in 1 and 2 above.⁹ By constructing the allocation decision from first principles in this way, we address the problem in the Conventional Strategy of needing to know the average level of Earnings Yield over the whole period.

We will call this the “Excess Earnings Yield Dynamic Strategy.”

Determining exposure to equities as a function of expected excess return, risk, and investor risk-aversion to maximize Expected Utility under standard assumptions is known as the Merton Rule (see footnote 9). For this analysis, we’ll assume an investor with a degree of risk-aversion we found typical in a survey we conducted in 2018. This level of risk-aversion is such that an investor would choose to allocate 62.5% to equities if faced with an excess expected equity market return of 5% per annum, and equity riskiness of 20% per annum.¹⁰ The chart below shows the allocation to equities resulting from the Merton Rule, which we use in our historical analysis from the end of 1997 (the start of the TIPS market) to the end of 2021.¹¹ In the first 3 1/2 years of the study, the desired allocation to equities was zero, because the Earnings Yield of the stock market was relatively low, TIPS yields were high, and therefore the Excess Earnings Yield was negative.

Chart 3: Allocation to US Equities Based on Merton Share vs. 10-Year TIPS
1997 – 2021

The chart below shows the performance that these allocations would have generated, compared to a static allocation of 65% in stocks and 35% in bonds (TIPS being the type of bond). The Excess Earnings Yield Dynamic Strategy performed much better, delivering a return about 2% pa higher than the Static Strategy, with lower risk and a nearly 50% higher Sharpe Ratio.¹² By contrast, over the same period starting in 1997, the Conventional Dynamic Strategy illustrated in Chart 2 generated a return about 1.5% pa lower than the comparable Static Strategy but with roughly the same Sharpe Ratio. Of course, this is a very short window, and we certainly are not suggesting that you should follow this approach or avoid the conventional approach solely based on this back-test.

Chart 4: Excess Earnings Yield, Dynamic vs. Static Allocation
US Equities and 10-Year TIPS
1997-2021, Logarithmic Scale

It would be nice to see this analysis taken back further – unfortunately, the US Treasury has only been issuing TIPS since 1997. However, we do think it’s possible to construct a decent hypothetical history of long-term US real interest rates going all the way back to 1900, which we present in the chart below.¹³

Chart 5: Ten-Year Real Yield Series
Actual & Hypothetical

Using this hypothetical history of US real rates, we get the chart below, which shows the performance results back to 1900. Bottom line: the Excess Earnings Yield Dynamic Strategy did a lot better than a Static Strategy. Not only did the Excess Earnings Yield Dynamic Strategy do much better in terms of absolute return and quality of return than the 65/35 Static Strategy, but perhaps even more remarkable, it outperformed being 100% in US equities over the entire period, which generated a lower total return of 10.0% with 40% more risk.

Chart 6: Excess Earnings Yield, Dynamic vs. Static Allocation
US Equities and 10-Year TIPS
1900-2021, Logarithmic Scale

One more thing to consider is that it’s hard to say how an investor would have decided on a 65/35 stock/bond asset allocation to begin with, without use of some sort of framework, such as the Merton Rule, that put a price on risk. If an investor in 1900 were thinking about how much to invest in equities based on some other objective – such as maximizing Expected Wealth – he would have tried to invest the most that he could in equities with maximum leverage. Following such an approach, the investor would have likely gone bust in the 1929-1933 stock market meltdown of over 85%, and possibly in the several other greater-than-50% market declines experienced over this period.

In the Appendix, we provide details of all our assumptions and sources of data, and find that the Base-Case historical result just outlined is robust to changes in many of the assumptions. We also show the significant improvement delivered historically from including Time-Series Momentum as an additional indicator of the expected risk and/or the expected return of equities.

It’s Risk-Adjusted Returns That Matter

As pointed out earlier, the rule underlying the Conventional Dynamic Strategy presented in Chart 2 is focused solely on expected return, as it does not use the changing risk of the stock market as an input. If looking only at expected returns, then 100% in equities (or more if leverage is available) will always be the best allocation for any period where equities beat bonds. But intuitively, that can’t be right, as we need to make an adjustment for risk. Consider a situation where an investor expects equities to outperform TIPS by 2% pa, and given the meager expected excess return of equities, he decides to allocate just 25% of his portfolio to equities. Then, over the next ten years, equities do outperform TIPS by exactly the 2% per year he expected. An analysis not including risk would conclude that he’d have been better off with 100% in equities, because he’d have made more money after ten years with the higher allocation – but that is a flawed conclusion. He chose the 25% in equities because that allocation maximized his Expected Risk-Adjusted Return, and since the realized return was equal to the Expected Return, his decision should also be optimal ex post – which is to say that he would have experienced a lower Risk-Adjusted Return by holding a higher equity allocation.¹⁴

Improvement in Sharpe Ratio is a Twofer (Squared!)

Since 1900, the Excess Earnings Yield Dynamic Strategy has generated a Sharpe Ratio about 25% higher than that of the Static Strategy. Just how big a deal is a one-quarter increase in the Sharpe Ratio on one’s investment portfolio?¹⁵ A very big deal indeed! A one-quarter increase, whether it comes from a higher expected excess return or lower risk, delivers a compound benefit to an investor in that it:

1. Provides a one-quarter higher return per unit of risk, and,
2. It also increases the optimal allocation to equities by one-quarter, generating an additional one-quarter improvement.

So, a one-quarter increase in Sharpe Ratio generates roughly double that improvement (a 56% improvement, to be exact!) in the Risk-Adjusted Return of the investor’s portfolio.¹⁶ An improvement of this magnitude in expected Risk-Adjusted Return, compounded over the long-term horizons over which individuals typically save and invest for retirement, can make truly life-changing enhancements to investor outcomes.

Can Everyone Be a Dynamic Asset Allocator?

An Excess Earnings Yield Dynamic Strategy is not an approach that all investors can pursue at the same time. Economists would say that such a strategy is not macro-consistent. This is a pretty stringent test of an investment approach. Even a static asset allocation strategy that aims to keep a fixed fraction of wealth in equities would fall foul of this test. In fact, the only strategy that all investors can pursue at the same time is buy-and-hold at global market-cap weights. Whenever an investor is considering pursuing a strategy that not everyone can follow, he needs to have a good look in the mirror and ask why he is different from the average investor.¹⁷ An Excess Earnings Yield Dynamic Strategy is probably a good fit for long-term investors who expect their risk-aversion to remain steady through time, and who are willing and able to estimate expected real returns and risk offered by their investments.

Conclusion

We believe it’s never a good idea to adopt an investment strategy based primarily on historical simulations. However, when you believe a strategy makes sense a priori, it is worthwhile to challenge and update the strength of that belief with a look at the empirical evidence. Before looking at the historical record, we firmly believed it made intuitive sense to dynamically change one’s allocation to equities based on their expected return relative to the appropriate safe asset, and the empirical record reinforced that belief. It is time to correct the record regarding the efficacy of Dynamic Asset Allocation using the market’s Earnings Yield as a key input. And it is also time to differentiate this disciplined approach grounded in theory from the many seat-of-the-pants dynamic approaches that go under the pejorative heading of “Market Timing.” The magnitude of improvement in welfare that is available to investors who are willing and well-suited to vary their exposure to equities as their expected excess real return and risk change over time is too big to be left on the table.

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Appendix

Data and Sources

S&P 500 Stock Index Prices (1870 – 2021 monthly)	Standard and Poors, Online Data: Robert Shiller
S&P 500 Earnings and Dividends	Online Data: Robert Shiller
US T-Bill Rates	Online Data: Robert Shiller, St. Louis Federal Reserve
US Ten-Year Treasury Yield	St. Louis Federal Reserve, US Department of the Treasury
UK Ten-Year Inflation-Linked Bond Yield (1985-1997)	King and Low (2014)
US Ten-Year TIPS Yield (1997 – 2021)	St. Louis Federal Reserve, US Department of the Treasury
US CPI Inflation (1880 – 2021)	Online Data: Robert Shiller
Implied Inflation Forecasts (1955 – 1970)	Kozicki-Tinsley (2006), Ilmanen (2011)
Survey-based Inflation Forecasts (May 1970 – November 1984)	Philadelphia Fed, Cleveland Fed, Blue Chip Economic Indicators, Livingston Survey

Construction of US Ten-Year TIPS Yields and Total Return Series from 1900 to 2021
From 1997 to 2021, we use Ten-Year TIPS yields directly. From 1985 to 1997, we use Ten-Year UK Inflation-Linked Bond yields. From 1900 to 1984, we use the US nominal Ten-Year bond yield minus an estimate for the ten years of US inflation, and we subtract a further 0.5% from the resultant real yield as a representation of a risk-premium that investors are likely to have demanded to bear inflation risk. The prospective inflation forecast we use from 1900 to 1984 is the average of survey data from 1970 to 1984, implied inflation forecasts from Kozicki-Tinsley (2006) from 1955 to 1970, a weighted average of realized 20-year, 10-year, 5-year and 1-year inflation with weights of 40%, 30%, 20% and 10% respectively from 1933 to 1955, and the weighted average of realized inflation itself averaged with 0 from 1900 to 1933 to represent some bounding of expectations at 0 inflation during the period the US was on the gold standard. Our approach to constructing this series owes a debt to Antti Ilmanen (2011).

Construction of Equity Market Volatility Forecast 1900 – 2021
We calculate a series of rolling 10-year equity volatility and rolling 2-year equity volatility from monthly closing prices of the S&P500. We then take a weighted average of the life-to-date average of the rolling 10-year volatility and the most recent 2-year volatility. We put 75% and 25% weight on the 10-year and 2-year volatility measures, both expressed as variances, and then take the square root of that weighted average to arrive at the spot estimate of equity volatility that an investor might reasonably have used in deciding how much equity exposure to take using the Merton Rule. The chart below shows the volatility estimate we used in the historical simulation.

Chart 7: Equity Market Volatility Estimate Used in Historical Simulation

Shiller Cyclically-Adjusted Earnings Yield and Excess Earnings Yield Histories

Chart 8: Shiller Cyclically-Adjusted Earnings Yield
US, Dec. 1899 – Dec. 2021

Just a Good Draw?
We cannot say whether or not the past 120 years were just a favorable period of time for dynamic asset allocation. However, we can answer the question of how much better we would have expected dynamic asset allocation to perform given the range of expected excess returns equities offered at different times. To do this, we ran a simulation in which half the time, the excess expected return of equities was 1% and the other half of the time it was 9%, which roughly matched the spread of expected excess returns experienced in the past 120 years.¹⁸ We found that dynamically scaling the exposure to equities over many simulated histories delivered a roughly 30% average improvement in the Sharpe Ratio versus a static strategy. Against this backdrop, the historical experience of the past 120 years appears to be just a little bit worse than we’d have expected. The simulation also suggests that over a shorter horizon of 40 years, the dynamic asset allocation has an 85% probability of generating a higher return and a 65% chance of resulting in a higher Sharpe Ratio than a static weight strategy.

Robustness of Simulation Results to Different Assumptions
There are many other popular metrics used in dynamic asset allocation strategies, such as Tobin’s Q, Equity Market Value to GDP and Aggregate Investor Allocation to Equities (AIAE), to name a few. We prefer Earnings Yield because it directly gives an estimate for the long-term real return of the equity market, whereas all the other metrics need to be regressed against their historical averages in order to provide a return estimate. A survey-based forecast of future earnings may be better than using the past ten years of inflation-adjusted earnings as done in the Cyclically-Adjusted Earnings Yield, but we do not have that survey data going back very far and so could not run the historical simulation on that basis. Another metric, Cyclically-Adjusted Dividend yield plus dividend growth, closely relates to Earnings Yield and might be effectively used in conjunction with it, but this metric suffers from requiring an estimate of growth and being more sensitive to changes over time in corporate earnings payout policies.

We consider Time Series Momentum an indicator of prospective risk (it can also be thought of as a return indicator with much the same practical effect), which can be effectively used in combination with Earnings Yield to significantly improve risk-adjusted returns. We give results for the joint application of Excess Earnings Yield and Momentum in the table below, and also in the chart below.

We explored a range of different assumptions applied to the historical simulation. Below we describe each change in assumptions and the resultant Sharpe Ratio for the dynamic and static strategies over the entire period and the period since the introduction of inflation-protected bonds in 1985 in the UK.

Chart 9: Excess Earnings Yield, Dynamic vs. Static Allocation
Using Momentum as Risk Proxy
US Equities and 10-Year TIPS
1900 – 2021, Logarithmic Scale

Decade by Decade Results

The Excess Earnings Yield Dynamic Strategy experienced a lower Sharpe ratio in 3 of the 12 decades examined.

Higher Turnover
A dynamic strategy is likely to experience higher turnover than a static strategy, and hence will incur higher transactions costs and possibly a higher tax cost as well. In our simulation with monthly rebalancing, the average turnover of the dynamic strategy was 29% per annum, versus 10% for the static weight strategy. Both of these turnover figures could be reduced by rebalancing less frequently and less fully to targets. Implementing a dynamic strategy is more complex and takes more of an investor’s attention, although on the other hand, a rules-based dynamic approach may be easier for an investor to stick with as it can scratch the investor’s itch to feel responsive in the face of a changing world.

If Expected Equity Returns Are Inversely Related to Changes in Market Level
By construction, when the market drops over a short period of time, the Cyclically-Adjusted Earnings Yield will go up, because Cyclically-Adjusted Earnings is based on the past ten years of earnings, which hardly changes from day to day. If an investor believes that the Expected Return of the stock market goes up when the market falls, then he should want a higher allocation to equities than suggested by the basic Merton Rule. This extra amount of equities was called “hedging demand” by Merton (1971), because it represents a hedge against the investment opportunity set faced by the investor. When the market goes down, the investor’s portfolio value goes down, but the increase in attractiveness of his investment opportunities offsets some of that loss in value, and so he can afford to own more equities. Pushing in the opposite direction of this hedging demand is the tendency for the market to be more volatile when it falls, which the market for options exhibits through the volatility “skew.” We view these phenomena as important, but not changing the basic conclusion that dynamic asset allocation driven by estimated expected return and risk is a sensible approach to investing.

As per the assumptions in the Merton Rule described above, risk-adjusted return is calculated by subtracting from the expected or realized excess return of a portfolio the cost of risk defined as:

γ (f σ)² 2

Where f is the fraction of the portfolio allocated to the risky asset, and γ and σ are as described in the Merton Rule above.

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Further Reading and References:

• “Elm Wealth Capital Market Assumptions.” (2024). Elm Wealth.
• Asness, C., Moskowitz, T., and Pedersen, L. (2013). “Value and Momentum Everywhere.” Journal of Finance 68 (3): 929–985.
• Asness, C., Ilmanen, A., and Maloney, T. (2017). “Market Timing: Sin a Little.” Journal of Investment Management 15 (3): 23-40.
• Campbell, J. and Shiller, R. (1988). “Stock Prices, Earnings and Expected Dividends.” Journal of Finance 43 (3): 661-676.
• Campbell, J. and Viceira, L. (2001). “Who Should Buy Long-Term Bonds?” American Economic Review 91 (1): 99-127.
• Cochrane, J. (2022). “Portfolios for Long-Term Investors.” Review of Finance 26 (1): 1-42.
• Cochrane, J. (2011). “Presidential address: Discount rates.” Journal of Finance 66 (4): 1047–1108.
• Haghani, V. and White, J. (2018). “Measuring the Fabric of Felicity.” Elm Wealth.
• Haghani, V. and White, J. (2017). “What if High Stock Values Revert to Normal Levels?” Bloomberg.
• Haghani, V. and White, J. (2017). “What Our Market Return Forecasts Really Mean: Equity Convexity and Investment Sizing.” Elm Wealth.
• Haghani, V. and White, J. (2017). “Market Multiple Mean-Reversion: Red Light or Red Herring?” Elm Wealth.
• Haghani, V. and White, J. (2020). “Taking Stock.” Elm Wealth.
• Ilmanen, A. (2011) Expected Returns. London: Wiley.
• Keimling, N. (2016). “Predicting Stock Market Returns Using the Shiller CAPE — An Improvement Towards Traditional Value Indicators?” SSRN Electronic Journal.
• King, M. and Low, D. (2014) “Measuring the `World’ Real Interest Rate.” NBER Working Paper Series 19887. https://www.nber.org/papers/w19887
• Kozicki, S. and Tinsley, P. (2006). “Survey-Based Estimates of the Term Structure of Expected U.S. Inflation.” Bank of Canada Working Paper No. 2006-46.
• Merton, R. (1969). “Lifetime Portfolio Selection under Uncertainty: The Continuous-Time Case.” Review of Economics and Statistics 51 (3): 247-257.
• Merton, R. (1971). “Optimum Consumption and Portfolio Rules in a Continuous-Time Model.” Journal of Economic Theory 3 (4): 373-413.
• Merton, R. (1973). “An Intertemporal Capital Asset Pricing Model.” Econometrica 41 (5): 867-887.
• Micaletti, R. (2021) “Market Timing Using Aggregate Equity Allocation Signals.” Alpha Architect.
• Rintamaki, P. (2021) “Total Wealth Portfolio Compositioin and Stock Market Returns.” SSRN Electronic Journal.
• Samuelson, P. (1994). “The long-term case for equities.” Journal of Portfolio Management 21 (1): 15-24.
• Jivraj, F. and Shiller, R. (2018). “The Many Colours of CAPE.” Yale ICF Working Paper No. 2018-22.

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1. This not is not an offer or solicitation to invest, nor should this be construed in any way as tax advice. Past returns are not indicative of future performance.
   
2. A variety of corporate-growth models can produce the result that real equity returns will be centered around the earnings yield. One basic condition under which real returns will equal the earnings yield would be if company earnings can grow with inflation with all earnings paid out currently to shareholders.
    

   While these models are all caricatures of the real world in a variety of ways, they nonetheless provide a solid starting point for thinking about expected stock market returns and making sense of long-term historical data. For a more up-to-date evaluation of CAPE as a predictor of real equity returns, particularly assessed in non-US equity markets, see Keimling (2016). They conclude:

   “Existing research indicates that the cyclically adjusted Shiller CAPE has predicted long-term returns in the S&P500 since 1881 fairly reliably for periods of more than 10 years. Furthermore, the results of this paper indicate that this was also the case for 16 other international equity markets in the period from 1979 to 2015.”

3. A measure of risk-adjusted return.
   
4. Unless otherwise stated, all historical analyses presented in this note are exclusive of trading costs and taxes.
   
5. The simplest asset allocation rule that meets these three requirements is: k^* = min(9.7 * EY, 100%) , where k^* is the allocation to equities, EY is the Earnings Yield of the stock market at the time of the asset allocation decision, and (1 – k^*) will be the allocation to T-bills, and assuming EY > 0 at all times.
   
6. Comparing Earnings Yield to the yield on T-bills also would not be consistent, as Earnings Yield is a real return estimate while the yield on T-bills is a nominal return estimate. Using Earnings Yield minus the T-bill rate as the asset allocation driver results in the same conclusion conveyed by Chart 2. A Dynamic Asset allocation rule based solely on the Earnings Yield of the stock market would make sense if the expected real return of T-bills was constant through time, but we know this is not the case.
   
7. As Stanford economist John Cochrane further elaborates in “Portfolios for Long-Term Investors” (2021), “Their (Campbell and Viceira’s) proposition is obvious if you look at the payoffs. An (inflation) indexed perpetuity gives a perfectly steady stream of real income, which can finance a steady risk-free stream of consumption. It is the risk-free payoff stream.”
   
8. This statement ignores taxes, which generally favors holding equities for taxable US investors. There are other reasons an investor may want to own some equities under these circumstances, such as to avoid putting 100% faith in the Earnings Yield metric, as a form of hedging demand as described in Merton (1971) or as a partial hedge of an affluent investor’s consumption basket.
   
9. The formula we use is known as the Merton Rule and is:   f^* μ γ σ ² Where f^* is the optimal fraction of the portfolio to allocate to equities, μ is the Excess Earnings Yield, γ represents the level of risk-aversion of the investor in CRRA Utility and σ is the expected volatility of equities. γ was set to 2 in the historical simulation, a round number representing an investor slightly more risk tolerant than the average of those surveyed by Haghani and White (2018).
   
10. Using the Merton Rule with γ = 2 we get: k^* = 62.5% = 5% / (2 * 20%²) .
    
11. We assume that the Earnings Yield is an indicator of the real Arithmetic return of equities, although there is a good argument that Earnings Yield is predicting the real Geometric return. See Haghani and White, “What Our Market Return Forecasts Really Mean: Equity Convexity and Investment Sizing,” (2017). We constrain the allocation to equities to be between 0% and 100%, i.e. no shorting, no leverage. Relaxing the no-shorting and no-leverage constraints does not change the results materially.
    
12. In calculating the Sharpe Ratio, we are adding .5 * StDev² to the Geometric realized return to convert to an Arithmetic return to use in the numerator of the ratio.
    
13. Of particular note is the decade following WWII, during which we estimate ten-year TIPS would have traded at an average yield of -1.5%. During this period, the ten-year nominal Treasury bond yield averaged 2.5% and inflation ran at about 5%, touching 20% in the years directly following the end of the war.
    
14. Also assuming his risk assumptions were realized.
    
15. A long-term investor may choose to measure risk in terms of the long-term real annuity value of his wealth – for example, using a perpetual inflation-protected bond as his numeraire. See Appendix for Sharpe Ratio of the Excess Earnings Yield Dynamic Strategy with returns measured relative to 10-year TIPS, which also shows a roughly one-quarter improvement versus a static strategy.
    
16. The improvement is (5/4)² – 1 = 9/16 = 56% .
    
17. See John Cochrane’s “Portfolios for Long-Term Investors” (2021), pp 19-20 for a deeper discussion of the “Average Investor” theorem and the “Look-in-the-Mirror” test.
    
18. Assumptions: γ = 2, σ = 20%, annual rebalancing.
    

By Victor Haghani and James White ¹

Introduction

Many of our readers are involved with various forms of endowments – their own Donor Advised Fund or Foundation, or charitable advisory boards they sit on. Families with wealth in excess of what they expect to spend in their lifetimes may also think of that surplus as an endowment to benefit future generations. Even for those with no connection to endowments, there are valuable lessons to be learned from the question of how one should invest and spend their resources when freed from the complications of taxes and human longevity.

Harvard Professor John Campbell defines an endowment as “a promise of vigorous immortality”. We think he means the endowment should be able to fulfill its mission indefinitely into the future: spending shouldn’t be so profligate that the capital will be exhausted in one generation, nor so miserly that nothing is accomplished and capital accumulation becomes an end in itself.² We’ll expand on this description of an endowment’s mission and explain the important and fascinating result that, under most reasonable sets of assumptions, it is optimal to spend substantially less than the expected real return of the endowment’s investment portfolio.

The challenges of choosing the best investment and spending policies are clearly connected. The conventional approach many endowments, especially large ones, have adopted is to invest like Yale and spend about 4% of the value of the endowment each year, a spending rate chosen so that there’s an arbitrarily low probability of spending falling below an arbitrarily chosen floor.³ There are two problems with this orthodoxy, besides the double dose of arbitrariness: 1) the investing and spending choices are not part of a unified framework, even though in reality they are inexorably connected, and 2) neither policy is explicitly responsive to changes in the investing landscape.

It’s been over twenty years since the Yale investment model was introduced in David Swensen’s book, Pioneering Portfolio Management: An Unconventional Approach to Institutional Investment, which instantly became the de facto endowment operating manual. It was undoubtedly both pioneering and unconventional when Swensen implemented it at Yale in the late 1980s. However, over the past twenty years, there has been nothing short of a sea-change decline in interest rates and expected returns on risky assets. Since 1999, low-risk real interest rates have plummeted from +4% to current levels around -1%. Alternative asset classes can’t make up for the decline in the expected returns offered by public markets, as they are no longer the high-return niche they used to be. Unfortunately, there is little in Swensen’s book that addresses how an endowment’s investing and spending policies should react to such dramatic changes in investment opportunities as we’ve experienced.

Ironically, nestled right inside the universities with some of the largest endowments, finance professors such as Robert Merton (MIT/Harvard) and John Campbell (Yale/Harvard) have developed valuable insights and tools which explicitly take account of changing environments and opportunities. Sadly, these ideas seem not to have made it into the mainstream of endowment practice, a state of affairs we hope this note will help to redress.

We’ve developed some web-based tools you can use to further explore many of the concepts found throughout this note. One is naturally focused on non-taxable endowments, the other on taxable individual investors:
  Endowment Investing and Spending Calculator
  Individual Investing and Spending Calculator

Three Spending Policy Options

It’s easiest to get an appreciation for the problem of choosing a long-horizon spending policy by taking the investment policy as already being chosen. Let’s evaluate three possible annual spending policies, given an investment environment and endowment asset allocation as described in Table 1. We’ll put to the side for now the role future contributions play on both spending and investment policy (see Appendix). Throughout, we’ll work in inflation-adjusted terms.

Table 1: Investment Environment and Policy Assumptions
Long-term risk-free real rate	0%
Expected real return on a well-chosen mix of public and private market risky assets4	6.0%
Risky Assets Annual Volatility of Returns	16%
Endowment asset allocation	85% in risk-assets 15% in risk-free assets
Endowment Expected Return	5.1%5

Policy 1: Spend a Fixed Annual Sum Equal to the Expected Simple Return of the Portfolio

The expected real return on the endowment’s portfolio, as per Table 1, is 5.1% per annum. Let’s first consider a policy of spending a fixed, but inflation-adjusted, $5.10 each year, assuming a starting value of the portfolio of $100. The endowment can get into trouble if its value drops but it keeps on making $5.10 payments each year. And if it experiences excellent returns, then the endowment will get very big and the $5.10 it will be spending each year will seem too meager. Chart 1 shows the median⁶ spending and average spending over time. In the early years, it’s very likely the endowment will have enough assets to meet the $5.10 spending policy, but in about 35 years, there’s a roughly 50% chance that the endowment will have run out of money, and so median spending drops to zero. The average or expected spend also drops over time, although not as dramatically as the median spending amount.

It’s unlikely any endowment is intentionally following this kind of fixed dollar spending policy – however, in personal financial planning, the most prominent spending rule does take exactly this form. It is known as the ‘4% rule,’ and it advises retirees to calculate 4% of their savings at retirement, and spend that inflation-adjusted dollar sum every year (and hope they won’t go broke). We include this as our first rule because it so clearly illustrates the close connection between spending risk and investment risk in the long term.

Chart 1: Spending $5.10 a Year (starting endowment value = $100)

Policy 2: Spend a Fixed Annual Percentage of the Endowment Value Equal to the Expected Simple Return of the Portfolio

The second policy we’ll consider is to spend the expected real simple return of the portfolio each year. From Table 1 that means spending 5.1% per year, and in fact this is close to Yale’s actual spending target policy of 5.25% over the past decade.⁷

Chart 2 uses a heatmap to show the probability of falling below a given level of spending, with the lightest color signifying 100% probability. Under this policy, it may come as an unpleasant surprise that median real spending falls by about 40% over 50 years, and by 2/3rds over 100 years. The median endowment value also falls by these amounts, since the spending rule is a fixed percentage of endowment value. We suspect most endowments would find this profile unattractive. The cause of this problem is often referred to as “volatility drag”, and relates to how volatility in returns makes the median return always lower than the average return.⁸ Following a spending policy equal to the expected portfolio return will keep the average spending amount and the average portfolio value constant, but this average is heavily influenced by a very small probability of extremely good outcomes. The median outcome, which is the most likely outcome, will always be lower, and if the average outcome is constant over time the median must be falling, as we see here.

Chart 2: Spending 5.1% a year

Policy 3: Spend a Fixed Annual Percentage of the Endowment Value Equal to the Expected Compound Return of the Portfolio

This brings us to the third spending policy, which is to spend the expected compound real return of the portfolio. With the assumptions from Table 1 it is 4.2% per annum.⁹ This happens to match the average spending rate across all US college and university endowments.

We can see in the chart below that now median spending stays constant over time, while average spending drifts higher. Early thinking about endowment spending, such as that of Nobel laureate James Tobin (1974), viewed the previous rule, spending the expected simple return of the portfolio, as the ‘Sustainable Spending Rate’ that endowments should adopt.¹⁰ More recently, however, the consensus has shifted to viewing this third rule as a better definition of Sustainable Spending because it keeps median spending and median endowment value constant over time, and medians accord better with what we are “likely” to experience.¹¹

Chart 3: Spending 4.2% a year

How to Compare Different Spending Policies

For many endowment trustees, they may find the third policy more attractive than the first two…but is it the optimal choice? We can see from the charts that Rule 3 produces the highest total median and average dollar spending over the horizon – because spending less in early years allows more growth to fund higher spending later on – but does that make it the best policy? It would be a pretty tall order to identify the best spending policy just by eyeballing the differences between colorful heatmaps. Focusing on medians rather than averages seems reasonable, but it’s a value judgement to which we haven’t yet given a particularly rigorous foundation.

We need a more powerful summary statistic for comparing different spending rules. Such a metric should take account of:

• The level of spending – more is better than less.
• The smoothness of spending over time – consistent spending is better than volatile spending.
• The immediacy of spending – sooner is better than later.

The standard metric which economists use which neatly incorporates all of these criteria in evaluating an uncertain stream of spending over time is Discounted Expected Utility of Spending.

Utility

In order to use this metric, we need to uncover the endowment’s Utility function. This is not as difficult an undertaking as it may seem. First, it has been observed that the amount of risk endowments take does not seem to vary much over a wide range of endowment sizes, which allows us to reasonably use a form of utility called Constant Relative Risk-Aversion (CRRA) Utility. This type of Utility function has only one parameter, which is the degree of risk-aversion.¹² For a particular endowment, its risk-aversion can be deduced from the investment policy it has chosen, if we know the estimates of expected return and risk on which it based its portfolio choice.¹³ Given the investment environment described in Table 1, the endowment’s level of risk-aversion implied by its chosen portfolio is a fairly normal level exhibited by wealthy, financially sophisticated individuals.¹⁴

Time Preference: Weighing a Better Present Against a Better Future

With the endowment’s utility function and the distribution of portfolio returns in hand, we can calculate the Expected Utility of Spending for any spending policy – but to calculate the Discounted Expected Utility, we need to know how the endowment discounts current versus future benefits of spending, that is, the endowment’s “Time Preference.” Economists and philosophers have long noted the general human preference for good things to happen to us sooner rather than later, but does that apply to an endowment too? James Tobin thought that endowments should have zero time preference, stating: ¹⁵

“The trustees of an endowed institution are the guardians of the future against the claims of the present. Their task is to preserve equity among generations…In formal terms, the trustees are supposed to have a zero subjective rate of time preference.”

With respect to Professor Tobin’s view, we wonder if it is plausible or advisable for any social entity – endowment, foundation, family or individual – to exhibit zero time preference. Is it reasonable that an endowment would put an equal value on the social welfare arising from $1 today as it would on the same amount of welfare generated in 1,000 years? Indeed, there are good reasons why it would be rational for endowments to express some degree of time preference, such as a belief that making the world better today will pay dividends in making the world even better in the future, and acknowledging the truth that while endowments expect to exist for a very long time, that’s not the same as forever.¹⁶

How can we help an endowment calibrate its time preference? One suggestion is for the endowment trustees to contemplate how much they would spend if their only investment option were a risk-free asset paying a 0% real return each year. Any spending in this case would run down the endowment value, and the choice of how fast determines the endowment’s time preference.

The U.S. federal government suggests cost-benefit analysis of social programs use a real “social rate of time preference” of 3%.¹⁷ Another data point that garnered much attention was the U.K.’s Stern report on the economics of climate change (2006) which more controversially used a rate of time preference of 0.1% for weighing costs and benefits occurring over many years. We will use a rate of time preference of 2% for the Base Case analysis that follows.¹⁸

Put Your Faith in DEUS: Discounted Expected Utility of Spending

An endowment should prefer one spending policy over another if it generates higher Discounted Expected Utility of Spending (DEUS). In the table below, we compare the three spending rules we’ve already discussed against each other using this metric. What we show for each rule is how many dollars the endowment would need to start with so that it would generate the same amount of DEUS under each spending rule over a one hundred year horizon. We’ve assumed a time preference of 2% per annum, and risk-aversion consistent with risk and return numbers in Table 1. Notice that the endowment would need considerably more assets to start with under rules 1 and 2 to generate the same expected welfare as under rule 3.¹⁹ The endowment’s choice of spending policy matters a lot.

Comparing Spending Rules: Size of Endowment Needed to Generate Equal Welfare Over 100 Years Under Different Spending Policies
Rule 1 Spend $5.10 pa	Rule 2 Spend 5.1% of Endowment pa	Rule 3 Spend 4.2% of Endowment pa
$184	$151	$100

In Search of the Optimal Spending Policy

If we can compare the DEUS for different spending policies we propose, it begs the question: can we find an optimal spending rule? Remarkably, the answer is yes.²⁰ Robert Merton found it, and shared it in his first published economics article in 1969, marking the start of one of the most prolific and creative careers in financial economics.²¹ In fact, Merton did more than solve for the optimal spending rule: he solved for the joint optimal spending rule and optimal investment policy.

Three variables feed into the optimal amount of risk to take, known as the “Merton Share”:

1. The expected return of the risky portfolio in excess of the risk-free rate. Higher excess returns imply a higher optimal risk setting.
2. The variability of the risky portfolio measured by its variance.²² Higher variance implies lower optimal risk-taking, all else being equal.
3. The risk aversion coefficient – higher risk aversion implies lower optimal risk.

For the optimal spending rule, Merton shows that it must be a proportional rule, spending a fixed fraction of the portfolio each period. This optimal spending fraction is also a function of three inputs:

1. The “risk-adjusted” return, also known as the Certainty-Equivalent return of the total portfolio when invested at the optimal risk level. This is the certain return one would accept in lieu of the risky portfolio’s return. Higher risk-adjusted returns allow for higher spending rates, but generally not on a one-for-one basis.
2. The time preference rate. Higher time preference increases the optimal spending rate.
3. The level of risk aversion. If time preference is lower than the portfolio’s risk-adjusted return (as in our Base Case), then higher risk aversion increases the optimal spending rate, and vice versa.

Merton Optimal Investment and Spending Formulas for an Endowment with Infinite Life

k^* = μ – r γσ²

where k^* is optimal exposure to the risky asset
μ is the expected return on the risky asset
r is the return on the safe asset
σ is the annual standard deviation of returns of the risky asset
γ is the coefficient of CRRA risk aversion

C^* = r_ce – r_ce – r_tp γ

where C^* is the optimal spending rate²³
r_ce is the certainty equivalent return of the optimal portfolio
r_ce = r + k^* 2 (μ – r) r_tp is the investor’s time preference of spending

Where Merton Meets the Road

Knowing the optimal rule to follow is great, but does it deliver much real improvement over Rule 3, the “Sustainable Spending” policy? Staying with the same set of assumptions, Merton’s optimal spending policy would be to spend 2.4% of the value of the endowment each year. An endowment following the Sustainable Spending policy for 100 years, spending 4.2% per year, would need about 33% more in starting assets in order to deliver the same discounted expected utility from following the Merton-optimal rule, and the gap gets bigger as we look at longer horizons. The simplicity of the Sustainable Spending Rule is attractive, but it does not directly take account of the endowment’s risk aversion or time preference, and so in general it will lead to suboptimal spending decisions.

The chart below shows the average and median spending under the two spending policies (Sustainable and Optimal). It is difficult to visually decide which spending policy is more attractive without having a comprehensive metric that takes account of the main contours of the endowment’s preferences over uncertainty and time.

Chart 4: Comparing Merton Optimal vs. Sustainable Spending Rules

There are some preference sets for which the Sustainable Spending rule is quite close to Merton’s optimal spending rule, and others for which it’s even further away than the base case we examine. In our example from Table 1, if the endowment exhibited time preference equal to 7% per year, then Merton’s optimal spending policy would be to spend 4.2% per year, the same as the Sustainable Spending policy. On the other hand, if the endowment had zero time preference the Merton-optimal spending rate would be much lower, at just 1.6%.

Conclusion

The Merton model and its extended family of descendants do not appear to play a central role in shaping the investment and spending policies of major endowments, foundations or other long-lived pools of capital. For example, in David Swensen’s already-mentioned endowment bible, there is no mention of Merton or the cohort of researchers, notably John Campbell, who have extended his work. Swensen’s only mention of Merton is to dismiss his formulation of the problem:

“Economists might suggest that a utility function be employed to identify the appropriate asset allocation. Since few market participants would have any idea how to specify such a function, this technique proves remarkably unhelpful.”

We found several other influential books on endowment and foundation investing equally silent on Merton’s formulation and solution of the problem.²⁴

We don’t agree with Swensen’s criticism that the expected utility framework is too abstract to be useful in guiding, and linking, an endowment’s investment and spending policies. For example, in a survey we conducted two years ago and reported in Measuring the Fabric of Felicity, we found that a sample of financial professionals were comfortable calibrating personal utility functions. In addition, we found that their preferences were consistent with Constant Relative Risk Aversion, the form of utility function underlying the Merton formulation described in this note. Since Merton’s 1969 paper, researchers have extended the model to make it more realistic in many dimensions, and we discuss a partial list of these extensions in the Appendix.

Unlike endowments, individuals are afflicted with tedious burdens like taxes, finite and variable longevity and an uncertain posterity. These factors make finding optimal rules somewhat more complex, and change the details in various ways, but the core principles stay the same:

• risk aversion and time-preference matter
• risk-taking should be proportional to excess expected returns and inversely proportional to variance
• spending should follow a proportional rule and be linked to risk-adjusted return
• investment risk and spending risk are inseparable

We intend to discuss the more complex case of the individual investor in a follow-up note.

Much has changed in the 30-plus years since Swensen arrived at the Yale endowment, and in the 50 years since Merton’s original solution to the endowment investing and spending policy problem. Markets can go through a number of radically different investing environments over the life of any single individual, and even moreso in the case an endowment. For stewards of long-term capital, we think the Merton framework and its extensions provide valuable guidance on navigating these changing waters.

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Appendix: Extensions

While Merton’s 1969 analysis gave us two simple formulas for the optimal spending and investing policy, it was under a stylized and restrictive set of assumptions. However, his formulation of the problem with the objective of maximizing Discounted Expected Utility of Spending is versatile and leads to solutions under a wide array of more realistic assumptions, many of which he provided in subsequent papers.

• The original Merton 1969 formulation was for a two-asset case, but later versions explicitly handle multiple assets. The two-asset case can be used where the risky asset can be considered the optimal portfolio of risky assets. The model handles any choice of risk-free asset, from Treasury Bills to inflation-indexed bonds.
• Although the original 1969 model assumed a constant risk-free rate and a risky asset that followed a random walk with constant expected return and variability, these assumptions can be relaxed. If the risk-free rate and expected excess return of the risky asset themselves follow independent random walks, the results remain substantially the same.
• In another important extension, Merton coined the term ‘hedging demand’ to describe the result that investors should want to own extra amounts of the risky asset if its expected return tends to go up when the price of the asset goes down, and vice versa.
• While we have focused on preferences characterized by Constant Relative Risk Aversion, solutions can be found for any concave and smooth utility functions, including ones that separate relative risk aversion from the elasticity of intertemporal substitution of consumption.²⁵
• Parameter uncertainty leads investors to take less risk than would be optimal based on the point estimate of investment attractiveness. Similarly, learning also leads to conservatism, having the opposite effect as mean-reversion, and generates negative hedging demand.
• Spending policies, such as ‘sustainable spending’ or smoothed spending, can be exogenously specified and then an optimal investment policy given that spending policy can be solved for. John Y. Campbell and Roman Sigalov of Harvard have solved such a model, which suggests that as expected investment returns fall, the optimal endowment investment policy is to take more risk, a phenomenon known as ‘reaching for yield.’

Family Wealth

Taxes on income, capital gains and inheritance result in significantly lower expected returns for private taxable wealth than that experienced by non-taxable endowments or foundations. As a result, optimal spending policies for large pools of private capital will be substantially lower than optimal spending policies for tax-exempt entities, assuming similar degrees of risk-aversion and time preference that we used for our typical endowment. With the investment environment assumptions from Table 1, but with a flat 30% tax on returns, 0% inheritance tax and assuming inflation of 2%, the Merton optimal spending policy for an endowment-like taxable pool of capital would be lower than that for a non-taxable endowment at about 1.5% per annum.

For families who ascribe similar utility to the consumption of future generations, the optimal spending rate would be lower than that of an endowment to take account of the expected growth in the size of the pool of future beneficiaries. For individuals whose savings will be primarily used during their retirement, the Merton optimal spending rule for the infinite horizon can be used to annuitize wealth to a finite horizon. We will discuss this in more detail in an upcoming note that focuses on individuals.

Other Popular Spending Policies

Probably the first long-term spending policy that occurs to most investors is to spend the interest and dividend income they receive on their portfolio, which they hope will leave the earning power of their portfolio constant over time. Currently, for a portfolio 85% invested in global equities and 15% in US inflation-linked bonds, the resulting policy would be to spend 1.4% per year. While the simplicity of this rule is admirable, it is unlikely to be optimal except by coincidence as it does not explicitly take account of the risk or time preference of the investor, and its connection with the expected return of the portfolio is weak due to changes in company dividend policy over time. And of course, a dividend-based rule is not much help for investors who allocate heavily to alternative investments which have distribution policies that are arbitrary and rarely adjusted for inflation. A variant of this rule uses the Cyclically Adjusted Earnings Yield in place of the dividend yield of equities. The problem with this policy is that earnings yield is an estimate of the expected real return of the equity market, and as we’ve seen already, spending the expected return of the portfolio (see spending Rule 2 in the body of the note) is unlikely to be optimal.²⁶

Many endowments apply a percentage spending rule on a smoothed basis. For example, the Yale endowment states:

Spending in a given year sums to 80% of the previous year’s spending and 20% of the targeted long-term spending rate applied to the market value at the start of the prior year. The spending amount determined by the formula is adjusted for inflation and an allowance for taxes, subject to the constraint that the calculated rate is at least 4.0% and not more than 6.5% of the Endowment’s inflation-adjusted market value at the start of the prior year.

Smoothed spending policies pick up the problem of fixed dollar policies, which can result in the endowment running out of money surprisingly quickly.

Endowment Growth Through Ongoing Contributions

Endowments usually expect to receive further donations over time, and this can be integrated into the model for the optimal investment and spending policy. For example, Yale’s endowment has received annual donations of 2% to 2.5% of the value of the endowment over the past decade. However, many donors want their contributions to have long-term impact, and don’t expect them to form part of the annual operating budget. These, and related considerations such as the risk characteristics of the flow of donations over time, have been addressed and modeled by Robert Merton (1991) and others. Foundations and wealthy families can usually think about spending policies without this complication.

For Investors Expecting Higher Returns Than Implied by Their Risk Taking

Some investors may expect much higher returns than are reflected in their portfolio choice, possibly as a response to an aversion to leverage. For example, if an endowment expected a 10% return on a well-chosen portfolio of risky assets with 15% risk, the optimal allocation to those risky investments would be 160% with the level of risk aversion we’ve been using. But let’s say the endowment decides to allocate just 85% to this attractive mix of investments. What is the optimal spending policy the endowment should pursue in this case? We can still use the Merton spending rule as expressed, but we need to define the risk-adjusted return on the portfolio more generally as:

r_ce = r + k(μ – r) – γ k² σ² / 2

where k is the actual allocation to the risky part of the portfolio, and not necessarily the optimal Merton Share allocation

Using this higher risk-adjusted return based on expected risky asset returns of 10% as an input to the Merton rule for optimal spending, but assuming an allocation of just 85% to the risky assets, produces a spending policy of 4.7% as compared to an optimal spending policy of 2.4% if the expected return on risky assets were 6%. Remarkably, the optimal spending percent would only increase from 4.7% to 5.9% if the endowment allocated 160% of its capital to the risky assets, making use of leverage. The relatively small uplift to spending rate doesn’t seem worth the 75% increase in exposure to risky assets, making it understandable why an endowment might choose less than the optimal amount of risk. As is usually the case with decisions based on optimizations, the marginal benefits decline as the optimal point is approached, so investors should focus on getting in the general vicinity of optimality and not be too fixated on getting to the exact optimal point.

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Further Reading and References

• Acharya, Shanta and Elroy Dimson. Endowment Asset Management: Investment Strategies in Oxford and Cambridge. Oxford University Press. 2007.
• Annable, Vince. The Household Endowment Model : Wealth Planning for Affluent Families. Wealth Strategies Advisory Group. 2019.
• Baumeister, Roy, George Loewenstein and Daniel Read. Time and Decision: Economic and Psychological Perspectives of Intertemporal Choice.” Russell Sage Foundation. 2003.
• Benzel, Rick and James E. Demmert. The Sustainable Endowment. New Insights Press. 2019.
• Black, Fischer. “The Investment Policy Spectrum: Individuals, Endowment Funds and Pension Funds.” Financial Analysts Journal. 1976.
• Campbell, John, Y. and Luis M. Viceira. Strategic Asset Allocation. Oxford University Press, (2002).
• Campbell, John, Y. and Roman Sigalov. “Portfolio Choice with Sustainable Spending: A Model of Reaching for Yield.” NBER. 2020.
• Campbell, John, Y. “Investing and Spending: The Twin Challenges of University Endowment Management.” Forum Futures. 2012.
• Dybvig, Philip H. “Dusenberry’s Ratcheting of Consumption: Optimal Dynamic Consumption and Investment Given Intolerance for Any Decline in Standard of Living.” Review of Economic Studies. 1995.
• Dybvig, Philip H. and Zhenjiang Qin. “How to Squander Your Endowment: Pitfalls and Remedies.” Washington University in St. Louis and University of Macao. Unpublished paper, 2019.
• Ennis, Richard and J. Peter Williamson. “Spending Policy for Educational Endowments.” The Common Fund Publications. 1976.
• Ford Foundation Advisory Committee on Endowment Management. “Managing Educational Endowments: Report to the Ford Foundation.” Ford Foundation. 1969.
• Goetzmann and Swensen. Yale Endowment Management Course description.
• Grinold, Richard, David Hopkins, and William Massy. “A Model for Long-Range University Budget Planning Under Uncertainty.” Bell Journal of Economics. 1978.
• Hindy, Ayman and Chi-fu Huang. “On Intertemporal Preferences With a Continuous Time Dimension II: The Case of Uncertainty.” MIT. 1989.
• Kochard, Lawrence E. and Cathleen M. Rittereiser. Foundation and Endowment Investing: Philosophies and Strategies of Top Investors and Institutions. Wiley. 2008.
• Merton, Robert, C. “Lifetime Portfolio Selection under Uncertainty: The Continuous-Time Case.” The Review of Economics and Statistics. Aug 1969.
•      , “Optimum consumption and portfolio rules in a continuous-time model.” Journal of Economic Theory. 1971.
•      , Continuous-Time Finance. Oxford. 1990.
•      , “Optimal Investment Strategies for University Endowment Funds.” NBER. 1991.
• Orr, Leanna. “David Swensen Is Great for Yale. Is He Horrible for Investing? How the Yale Model ate endowments — and everything else.” Institutional Investor. July 2019.
• Swensen, David, F. Pioneering Portfolio Management: An Unconventional Approach to Institutional Investment. Free Press. 2000.
• Tobin, James. “What is Permanent Endowment Income?” American Economic Review. Vol. 2, No. 64, 427-432. 1974.
• Yale Investment Office. Spending Policy, 2019 Update. p24. 2019.

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1. This not is not an offer or solicitation to invest, nor should this be construed in any way as tax advice. Past returns are not indicative of future performance.
    

   We are very grateful for the help of Jamil Baz, Larry Hilibrand, Ayman Hindy, Chi-fu Huang, Antti Ilmanen, Andy Morton, Vlad Ragulin, Jeffrey Rosenbluth, Eric Rosenfeld and Scott Wilson. All errors are our own.
   

2. Campbell (2012).
   
3. 2019 NACUBO-TIAA Study of Endowments, 2010-2019 spending. Here. US College and University Endowments and related foundations. Yale spends about 5.25%.
   
4. This expected return is expressed as an arithmetic pa expected return. We assume the returns of the portfolio of risky assets follow a random walk.
   
5. 5.1% = 85% x 6% + 15% x 0%
   
6. The median value of spending is the outcome that defines the middle of the distribution, and is often useful to think about separate from the average outcome because the median is less heavily influenced by extreme outliers. For a time series of returns with volatility, the median return will always be lower than the average return.
   
7. As we’ll discuss later, Yale’s spending policy makes use of smoothing and also collars of 6.5% and 4%, which in effect makes its spending rule something of a hybrid between a percentage rule and fixed dollar rule.
   
8. An example illustrates this effect. Let’s say that each year, there’s a 50/50 chance that the endowment’s portfolio either increases in value by 18% or decreases in value by 8%. The average of +18% and -8% is the 5% average return of the endowment’s portfolio, with a bit of rounding. But, if the endowment goes up by 18% the first year, and then declines by 8% the second, or the other way around, the value of the portfolio will have returned only 4.2% per annum, 0.8% lower than the 5% expected annual return. This 4.2% return is the compound (or median or geometric average) return of the portfolio, while 5% is the expected arithmetic return of the portfolio.
   
9. Also known as the geometric average return.
   
10. “…The trustees of an endowed university like my own (Yale) assume the institution to be immortal. They want to know, therefore, the rate of consumption from endowment which can be sustained indefinitely. Sustainable consumption is their conception of permanent endowment income…Consuming endowment income so defined means in principle that the existing endowment can continue to support the same set of activities that it is now supporting.”
    
11. Dybvig and Qin, “How to Squander Your Endowment,” 2019, and Campbell and Sigalov, “Portfolio Choice with Sustainable Spending: A Model of Reaching for Yield,” 2020.
    
12. CRRA Utility takes the form: U(C) = (1 – C^{(1 – γ)}) / (γ – 1) , where C is spending and γ is the coefficient of risk aversion.
    
13. Alternatively, for an endowment wishing to decide on an investment policy, there is a very reasonable range of CRRA risk-aversion levels which can serve as a useful starting point without requiring a complicated, idiosyncratic calibration exercise.
    
14. Equal to 2.75 times the risk-aversion of a typical Las Vegas professional, card-counting gambler trying to maximize the growth rate of his bankroll. It’s a level of risk-aversion that would make an investor ambivalent about accepting a gamble with a 50% chance of making 25% versus a 50% chance of losing 15%. See our note, The Fabric of Felicity.
    
15. In the 1974 article already cited, and referenced by David Swensen in describing the Yale endowment’s spending policy choice.
    
16. If the endowment truly had zero time-preference, then in a world in which the only investment available to an endowment were a risk-free asset that paid a 0% real return above inflation, it would be optimal for the endowment to spend zero, thereby preserving the real value of the endowment forever. Another reductio ad absurdum argument observes that an endowment that had no time preference and expected to live forever would be willing to pay an infinite price for a perpetual, risk-free bond that offered a positive real yield. While we present this scenario as a thought-experiment, it is not as far-fetched as it used to be.
    
17. See OMB Circular A-4 and OMB Circular A-94.
    
18. Time preference is equal to the desired spending rate in the zero return scenario multiplied by the coefficient of risk-aversion of the endowment, which we’ve assumed is equal to 2.75 in the Base Case. So, if the trustees felt that they would spend 0.75% of the endowment each year in a zero return environment, time preference would equal 2% pa.
    
19. These results are sensitive to choice of time preference, but robust within a reasonable range of choices. For example, with time preference of 0% , rules 1 and 2 would need to start with capital of $184 and $168, and with time preference of 4%, rules 1 and 2 would need to start with $183 and $131 to generate the same discounted expected utility of spending over 100 years as rule 3.
    
20. Given a set of assumptions that asset prices are well-behaved and investors exhibit Constant Relative Risk Aversion.
    
21. Of course, others also deserve credit for contributing to these insights, including Paul Samuelson, Merton’s mentor and collaborator, and Franco Modigliani, who was awarded a Nobel prize for his pioneering contributions to the field of life-cycle financial decision-making.
    
22. 22 Variance is equal to Standard Deviation squared.
    
23. Merton’s formula for optimal consumption can be arrived at by observing that a consumption policy is optimal only if all along its path the marginal utility of $1 not consumed at a point in time is equal to the marginal benefit of that $1 plus its growth at a later time, adjusted for time preference. This can be expressed as: U'(C_t) = U'(C_t + dt) * e^{(r_ce – r_tp)dt} . This simplifies to: ln((C_t + dt) / C_t) = (r_ce – r_tp) / γ . As ln((C_t + dt) / C_t) is the growth in optimal consumption, and r_ce is the growth in the portfolio, then optimal consumption, C^* = r_ce – (r_ce – r_tp) / γ . This is Merton’s formula for consumption, C^* .
    
24. Such as “Foundation and Endowment Investing: Philosophies and Strategies of Top Investors and Institutions”, “Endowment Asset Management: Investment Strategies in Oxford and Cambridge”, “The Sustainable Endowment” or “The Household Endowment Model: Wealth Planning for Affluent Families.”
    
25. Known as Epstein-Zin utility.
    
26. It is an open question whether CAEY is a predictor of the expected arithmetic or geometric return of the equity market, but in either case, this criticism holds.
    

By Victor Haghani and James White ¹

March 2020 packed 2 ½ years of normal U.S. stock market volatility into one month, making it the most volatile month on record. Daily variability clocked in at 6%, six times higher than the average over the past 90 years. How should an investor respond to such volatility? There are at least three schools of thought:

1. Steadfast: Stay the course and don’t be shaken by short-term swings in volatility. When setting your asset allocation to begin with, assume that the stock market will be much more volatile than normal from time to time, and when that happens take it in stride. As Vanguard founder John Bogle said, “Don’t pay a lot of attention to the volatility in the marketplace. All these noises and jumping up and down along the way are really just emotions that confuse you.” ²
2. Greedy: Increase your exposure to the market, because, as Warren Buffett has counseled, “Be fearful when others are greedy and greedy when others are fearful.”
3. Fearful: Reduce your exposure to the market because it has become riskier right now. This approach, known as “Volatility Targeting,” has been researched and supported by a host of respected academicians, with Ray Dalio, founder of the Bridgewater hedge fund group, its highest profile practitioner.³

Let’s work through these arguments in a world of two investments: a risk-free asset and the stock market. Let’s say you’re an investor who has just retired and has wealth well in excess of what’s needed to provide the basics for your family.⁴ How much you would optimally allocate to the stock market depends on your estimate of its return and risk relative to the risk-free asset, your personal degree of risk aversion, and how much savings you need to set aside to meet basic needs.⁵ As we’ll discuss below, which of the three schools of thought will appeal to you most will depend primarily on where you look for your estimates of expected risk and return.

Steadfast

Many investors and researchers use long-term market history to extrapolate future expected returns. This is the foundation of the Bogle recommendation: assume long-term return and risk are constant and ignore changes in short-term volatility and valuation metrics. We’re generally skeptical of backward-looking, historically-based forecasts of the future, though Bogle’s counsel is easy to follow and has the potential psychological advantage of helping an investor look through times of stress and turmoil. Most proponents of the steadfast approach recommend periodic portfolio rebalancing to maintain static target asset allocation weights.⁶

Greedy

A more adaptive, and forward-looking approach acknowledges that the long-term expected return of the market varies over time. Among the most commonly used, and effective, estimators of the future return of the stock market is the Cyclically-Adjusted Earnings Yield popularized by Robert Shiller. Importantly, it’s a predictor of the very long-term market real return, ignoring the hard-to-predict changes in sentiment that dominate shorter-term returns. If we rely on this long-term estimator, then it follows we should want to use a similarly long-term measure of risk, which will be relatively unaffected by swings in short-term volatility.⁷ This is consistent with Buffett’s advice to be greedy when others are fearful: when the stock market goes haywire and drops precipitously, long-term expected returns have likely gone up, while long-term risk is little changed.

Fearful

“The long run is a misleading guide to current affairs. In the long run, we are all dead. Economists set themselves too easy, too useless a task if in the tempestuous seasons they can only tell us that when the storm is long past the ocean will be flat.”
  – J.M. Keynes, A Tract on Monetary Reform (1923)

While the “Earnings Yield” estimate of the market’s long-term return is based on expected cash flows in the form of earnings and dividends, in the short run, stocks are driven primarily by changes in how market participants discount those future cash flows. As Benjamin Graham put it: “In the short run, the market is a voting machine but in the long run it is a weighing machine.” A Volatility Targeting strategy scales exposure inversely to short-term market volatility, increasing exposure when volatility is low and decreasing it when volatility is high. There are several lines of thought which support Volatility Targeting.⁸ Many practitioners focus on the desirability of keeping portfolio volatility constant over time, which may give investors greater peace of mind and potentially the confidence to take higher levels of risk over time. Indeed, Volatility Targeting is the optimal strategy under the assumption that the Sharpe Ratio (i.e. return-to-risk ratio) of equities stays constant as short-term volatility varies.⁹ Another popular theory suggests the market systematically under-reacts to changes in short-term volatility.¹⁰ The idea is that when an asset’s short-term volatility goes up, investors are slow to mark down its price enough to make the expected short-term return high enough to warrant holding as much of that asset. By reducing exposure when volatility goes up, the volatility-targeter seeks to get ahead of the slow but necessary mark-down process.

In practice, there are many impactful details to implementing a Volatility Targeting strategy, which is why historical studies reach varying conclusions on its effectiveness. We found that studies which assume the least practical implementations for ordinary investors produced the most attractive historical results. The chart below is based on how we imagine an individual investor with a baseline asset allocation of 75% US equities and 25% T-bills might actually apply Volatility Targeting, with no leverage and rebalancing each month-end. Please see the Appendix for full details of all historical backtests. The Volatility Targeting strategy did well from 1985 to the present, but less so over the entire period. We explored a wide range of different implementations and none that we could find was significantly better than what’s displayed below.

A Different Shade of Fearful: Momentum

“…most of the time the trend prevails…. Most of the time we are punished if we go against the trend.”
  – George Soros, Soros on Soros: Staying Ahead of the Curve

There is another indicator of short-term returns that has a loyal group of followers, and has some strong similarities with Volatility Targeting: Momentum. Time Series Momentum is measured by comparing today’s market level to a reference point, usually 6 – 12 months in the past for asset allocation purposes. If Momentum is positive, meaning today’s level is higher than its recent average, it indicates that near-term returns will be higher than if Momentum is negative. Researchers have found that Momentum is predictive of near-term asset price performance across virtually all assets that have been investigated.¹¹ There are many theories for why Momentum has worked; nearly all are based on behavioral foibles, grounded in the tendency of investors to extrapolate recent performance. The result is “return-chasing” behavior, creating trends in asset prices that Momentum indicators identify as they start to unfold. The chart below shows that a Momentum-driven portfolio performed quite a bit better than a static portfolio historically. In all 9 decades from 1930 to 2020, the 10-year return on the Momentum portfolio was higher than the static portfolio, with roughly the same risk (please see Appendix for more detail).

Volatility Targeting versus Momentum

Researchers have long noticed that volatility tends to rise when the stock market falls, and vice versa, which suggests that for equities Volatility Targeting and Momentum signals tend to line up most of the time.¹² We found that from 1928 – 2020 they in fact did point in the same direction 67% of the time. The chart below shows that over this period Momentum pretty consistently out-performed the Volatility Targeted dynamic portfolio. In 8 of the 9 decades from 1930 to 2020, the 10-year return on the Momentum portfolio was higher than the return on the Volatility Targeted dynamic portfolio, with the same risk over the past 50 years, and slightly higher risk over the full period. The Momentum portfolio also had a modestly higher Sharpe ratio.

We emphasize that such backtests do not by themselves provide enough evidence to warrant applying either approach, or preferring one to the other: past returns do not indicate future performance. We always need to ask whether the historical patterns we have found are the result of randomness, structural features, or systematic investor behavior. Even if we believe it’s not the result of chance, we need to also believe it’s likely these behaviors will persist in the future and outweigh the actions of other investors who are trying to take advantage of them.

Looking past the historical data, if you believe investors are focused on short-term return and risk, and are slow to adjust prices to changes in volatility, then you may be more attracted to Volatility Targeting, even though Momentum has historically done better as an indicator of short-run returns. However, if you are more attracted to the paradigm that return-chasers drive market dynamics, then you’ll find Momentum an attractive indicator to use in scaling your exposure to the market. Of course they are not mutually exclusive, and you may want a blend of both strategies. One good question to ask is whether you believe that high volatility in a rising stock market would have the same predictive power as high volatility in a falling market. Is it high volatility or the market’s recent direction that is primarily driving near-term returns? ¹³

Finally, even if you view Volatility Targeting and Momentum as equally likely to improve the risk-adjusted return of your portfolio, you may also want to consider the relative complexity of implementation, which favors Momentum as the strategy with fewer choices to make.

Conclusion

What does all this mean for how you should respond to extreme market volatility? Depending on how you think about long-term and short-term returns, and your desire for simplicity in managing your savings, we see merits in all the schools of thought, individually or in any combination. At Elm, we favor a fusion of Buffett’s approach for determining our long-term allocation to equities with Soros’ advocacy of Momentum to adjust for investors’ penchant to chase returns in the short term.

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Appendix: Details of Historical Performance of Volatility Targeting, Momentum and Static Portfolios

Data
We used daily S&P 500 index price data from finance.yahoo.com using the series ^GSPC. We used dividend and US CPI data from Professor Robert Shiller’s website to create a real total return index for the US stock market from December 31, 1928 to March 27, 2020. For T-bills, we also used Robert Shiller’s online data, using the one-year T-bill rate as a proxy for a daily T-bill rate. For the real return of the static portfolio, we assumed the portfolio was held at a constant asset allocation of 75% equities and 25% T-bills, rebalanced back to those weights at the end of each month. We assumed no transactions costs or frictions of any kind in these historical analyses. The data table below has an estimate of turnover for each of the portfolios.

Volatility Targeting
For the Volatility Targeting portfolio, we used the historical average 60-day rolling US equity volatility over the whole 1928 – 2020 period of 16.5% as the target level of volatility at which the portfolio will be 75% allocated to equities.¹⁴ For our base case, we set the allocation to equities, W* = 75% * 16.5% / Vol , where Vol is equal to annualized volatility of equity returns over the past 60 trading days. One way of thinking about this form of Volatility Targeting is that it assumes that in the short term the Sharpe Ratio of the equity market stays constant by the expected return of equities changing in proportion to changes in short-term volatility. As an illustration of the application of this rule, at the time of writing 60 day realized volatility was 63% pa, calling for an allocation to equities of 20%. We imposed a no-leverage constraint by capping the desired allocation to equities at 100%. The average allocation to equities over the whole period was 82.2%, higher than the 75% static baseline. The quality of historical returns measured by Sharpe Ratio is not materially changed by different choices target level of volatility. We also ran historical simulations using 20-day and 40-day lookback windows, squeezed volatility estimates in the spirit of GARCH analysis, and also we used VIX implied market volatility from 1990 onwards, the period over which VIX data was available. None of these choices for estimating market volatility as an input to the asset allocation rule made the historical returns of Volatility Targeting materially different. It is not surprising that using implied volatility didn’t materially improve the results of Volatility Targeting. Since 1990, one-month implied volatility has explained about 50% of next month’s realized volatility, about the same predictive power we get from predicting next month’s realized volatility using the past month’s realized volatility. We also explored using the Merton Rule at the end of each month to set the target allocation to equities, W* = μ / γ / σ² , which is consistent with the assumption that the short-term expected return of equities remains constant despite changes in short-term market volatility. This is referred to as “Variance Targeting.” We held the expected excess return, μ , constant at 5%, the coefficient of risk aversion, γ constant at 2.5,¹⁵ and volatility, σ , equal to the past 60 business days’ realized stock market volatility. We chose these parameters for their reasonableness and as they result in an average asset allocation not too far from the static baseline target of 75% equities. This approach also did not materially improve the quality of returns versus our baseline Volatility Targeting parameterization. We explored daily rebalancing, which roughly quadrupled turnover, without a material increase in quality of returns. We relaxed the leverage constraint to allow the investor to hold up to a 4x leveraged exposure to equities (an implementation we strongly advise against), but again this did not materially change the Sharpe Ratio of the strategy, although it did materially increase absolute historical returns. The only assumption which substantially improved the performance of Volatility Targeting was the unrealistic presumption that the investor had perfect foresight with regard to future realized volatility.¹⁶

Momentum
Momentum was measured as the current total real return index less the average of the total return index over the past year less 2.5%. The purpose of subtracting 2.5% is to make the incidence of positive and negative Momentum roughly equal. The target allocation for the Momentum portfolio was set at the end of each month as 100% equities/0% T-bills if Momentum was positive, and 50% equities/50% T-bills if Momentum was negative. We also used a Momentum signal based on a six month look-back window, and the results were similar to those from the one-year lookback. The average allocation to equities over the whole period was 79.2%, higher than the 75% static baseline, and lower than the average 82.2% equity exposure in the Volatility Targeting strategy.

The table below provides further details of the historical simulations explored.

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Further Reading and References:

• Black, Fischer. “Studies of Stock Price Volatility Changes.”  Proceedings of the 1976 Meeting of the Business and Economic Statistics Section, American Statistical Association, 177-181. 1976.
• Engle, R. “Autoregressive Conditional Heteroskedasticity with Estimates of the Variance of U.K. Inflation.”  Econometrica 50: 987–1008, (1982).
• Fleming, J., C. Kirby and B. Ostdiek. “The Economic Value of Volatility Timing.”  The Journal of Finance 56 (1): 329–352. 2001.
• Fleming, Jeff, Chris Kirby and Barbara Ostdiek, “The economic value of volatility timing using realized volatility.”  Journal of Financial Economics 67, 473–509. 2003.
• French, Kenneth R., G. William Schwert and Robert F. Stambaugh. “Expected Stock Returns and Volatility.”  Journal of Financial Economics 19, 3–29. 1987.
• Geczy, Christopher and Mikhail Samonov. “Two Centuries of Price Return Momentum.”  Financial Analysts Journal, Vol. 72, No. 5. Sep 2016.
• Harvey, Campbell R., Edward Hoyle, Russell Korgaonkar, Sandy Rattray, Matthew Sargaison and Otto Van Hemert. “The Impact of Volatility Targeting.”  Journal of Portfolio Management. Fall 2018.
• Lochstoer, Lars and Tyler Muir. “Volatility Expectations and Returns.”  INSEAD. 2019.
• Merton, Robert. “Lifetime Portfolio Selection under Uncertainty: The Continuous-Time Case.”  The Review of Economics and Statistics, Vol. 51, No. 3. Aug. 1969.
• Moskowitz, Tobias, Yao Hua Ooi and Lasse Heje Pedersen. “Time Series Momentum.”  Journal of Financial Economics. 2012.
• Tang, Yi, and Robert F. Whitelaw. “Time-varying Sharpe ratios and market timing.”  Quarterly Journal of Finance 1, 465–493. 2011.

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1. This not is not an offer or solicitation to invest, nor should this be construed in any way as tax advice. Past returns are not indicative of future performance.
    

   We thank Antti Ilmanen, Campbell Harvey and Myron Scholes for their helpful comments. Of course, the views, analysis and any errors are solely our own.
   

2. MarketWatch: Why Bogle and Buffett tell investors to ignore market noise
   
3. In a 2010 white paper titled “Engineering Targeted Returns and Risks,” Dalio explained “how to structure a portfolio to target a 10% return with 10-12% risk.” While Bridgewater’s Volatility Targeting implementation is proprietary, it is believed that they use longer-term measures of volatility, which would dampen their reaction to changes in volatility over the short run.
   
4. Or, if you’re still working, you have a secure job that makes your human capital very low risk and stable, like a government bond.
   
5. Robert Merton provided an early solution to this problem:
   W* = (μ – r) / (γσ²)

   where W* is the fraction of wealth in excess of subsistence needs to allocate to the risky asset, μ is the expected return of the risky asset, r is the risk-free rate, σ is the expected variability of the risky asset and γ is the investor’s level of risk aversion. See our note Measuring the Fabric of Felicity for a deeper discussion, particularly of γ .
   

6. Some would argue that maintaining static weights is an active strategy, in that it calls for buying equities when they fall and selling when they rise. One’s choice of benchmark against which to measure other investment approaches is important and can have a significant influence on the selection of the optimal strategy. For example, see our recent note: Back to the Future: Reviving a 19th Century Perspective on Financial Well-Being, in which we argue that an investor’s choice of minimum risk asset will have a profound influence on portfolio choice.
   
7. Let’s take an investor with a 25-year planning horizon. He believes stock market volatility will be 18% a year in the long run, and it’s been running at 18% in the short run. But then all of a sudden, there’s a panic and one-month volatility (i.e. VIX) goes to 50%. But he expects volatility to drop half-way back to 18% in a couple of months. Under these assumptions, the short-term spike in volatility would only raise 25-year expected volatility from 18% to 18.5%.
   
8. See French et al. (1987), Fleming et al. (2003), Tang and Whitelaw (2011), Harvey et al. (2018), Lochstoer and Muir (2019).
   
9. Optimal under the Merton Rule. For a slightly different perspective, invoking the concept of time-diversification, see this interview with Myron Scholes.
   
10. Lochstoer and Muir (2019) state: “Slow moving expectations about volatility lead agents to initially underreact to volatility news followed by a delayed overreaction.”
    
11. See Moskowitz et al. (2012) and Geczy and Samonov (2016).
    
12. First noted in Black (1976). Harvey et al., pp14, 31 (2018):
    “Risk assets exhibit a so-called leverage effect (i.e., a negative relation between returns and volatility), and so volatility scaling effectively introduces some Momentum into strategies. That is, volatility often increases in periods of negative returns, causing positions to be reduced, which is in the same direction as what one would expect from a time-series Momentum strategy. Historically such a Momentum strategy has performed well…we show that it is indeed the Momentumness of volatility scaling that explains a large part of the cross-sectional variation in the Sharpe ratio improvement when using volatility scaling for the various assets considered.”

13. An example of one such period to consider is the five year period starting in late February 1932, during which the S&P 500 experienced a real total return of 23.5% per annum, with realized volatility of 35%.
    
14. Realized volatility over the whole 1928 – 2020 period, measured using daily data was 19.3%, higher than the overlapping 60 day realized volatility was 16.5%. This difference is mostly due to the distribution of daily returns being significantly fat-tailed versus a standard normal distribution.
    
15. See our note Measuring the Fabric of Felicity for a discussion of the coefficient of risk aversion.
    
16. Perfect foresight is highly beneficial in nearly all realms of investing, and we commend its use whenever possible.
    

By Victor Haghani and James White ¹

“Mr. Darcy soon drew the attention of the room by his fine, tall person, handsome features, noble mien, and the report which was in general circulation within five minutes after his entrance, of his having ten thousand a year.“
  – Jane Austen, Pride and Prejudice (1813)

These days, we tend to assess our net worth by tallying up the market value of our financial assets, even though it might be more natural to think of our wealth as a stream of dollars over time given the nature of our income and spending. Perhaps this fixation on lump sum wealth is induced by the media – you won’t find Elon Musk on any rich list appraised at $1 billion per year, even though that’s about what his $40 billion of stock holdings would be worth in annuity form. But if you see your savings as a means to the end of future spending and bequests, the per-year measure seems more appropriate.

Let’s entertain the idea that what we really care about is the long-term, inflation-adjusted purchasing power which $1 can lock in today – we’ll call this the ‘Real Annuity Value’ of $1.² This shift in perspective has some pretty big implications for how we save and invest. For one, we have to rethink the notion that T-bills and other cash proxies, such as money market funds and bank deposits, are the lowest-risk assets we can own. While it’s true that the nominal value of T-bills doesn’t go up or down much day to day, we’ll see them as dramatically more risky once we focus on their Real Annuity Value.

The chart below shows the ‘real’ (i.e. inflation-adjusted) total return history of T-Bills, in terms of both dollars and Real Annuity Value:

As you can see, in dollars T-Bills had very low risk as well as very low real total return. Not so when looking at their performance expressed in Real Annuity Value: if you were invested in T-Bills over the whole period, by the end you could only buy half the Real Annuity you could have bought at the beginning. That’s a really significant loss of long-term purchasing power for a supposedly low-risk investment. The much lower Real Annuity Value delivered by T-bills is not a result of inflation, which we’re accounting for. Instead, it’s because long-term real interest rates, which set the payout of real annuities, dropped from 3.75% in 1997 to about 0% today. Looking further back in time, investments in US T-bills lost about 33% and 40% respectively of their Real Annuity Value over 1916 – 1920 and 1940 – 1948.³

Now let’s do the same comparison for the S&P 500:

We see that, while the total returns differ, equities are about as equally volatile measured in dollars or in Real Annuity Value. T-Bills lost about 50% of their Real Annuity Value, while equities gained about 50% – respectable, though far less than their dollar gains of over 200%.

It’s easy to read too much into these charts over any given period, but one interpretation is that equities are intrinsically a bit like a real annuity themselves: they provide an indefinite stream of earnings, which naturally adjust somewhat to inflation. They’re risky and have a volatile risk premium relative to the Real Annuity Value, but nonetheless they’re more like a real annuity than T-Bills are.⁴

Conclusion:

“Our approach to saving is all wrong: We need to think about monthly income, not net worth.”
  – Robert C. Merton, HBR (2014)

We’ve seen that T-Bills and similar cash-like assets are significantly more risky in terms of Real Annuity Value than they appear when viewed in plain dollars. Given their generally low expected real return, this makes cash-like assets look pretty unappealing to hold in excess of amounts needed to cover near-term expenses and contingencies. In contrast, equities are not significantly more risky in this new light, and may be even more attractive if we believe the long-term expected earnings streams they generate makes them a form of a long-term, real annuity. In our recent note Taking Stock, we found that the stock market currently looks more attractive viewed from the Real Annuity Value perspective, both prospectively and relative to historical valuations.

Back in Jane Austen’s day, wealth was harder to value and less liquid than it is today – one reason why it was more common to think about it as an annual flow, rather than an upfront value. While it’s more straightforward to measure your investment portfolio as a current lump sum value, a significant fraction of most peoples’ financial resources – their human capital and future social security and pension benefits – are much more readily thought of as long-term real annuities. If you see your wealth as a reservoir for long-term future consumption, we think it’s well worth the extra mental effort to think about all your financial resources and decisions with the Real Annuity Value perspective.

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Appendix: Real Annuity Value Mechanics

Once we start measuring financial well-being in terms of long-term annual purchasing power, we’ll need to identify a new risk-free asset to take the place of T-bills.⁵ What we need is an asset that pays a real $1 per year for many years, and with the highest assurance of payment possible. US Government Treasury Inflation-Protected bonds (TIPS) are a pretty good candidate, even though their cash flows aren’t quite in the form of a long-term real annuity due to their principal repayment at maturity.

Improving upon TIPS as our risk-free asset to generate the historical return charts in this note, we constructed a new currency: the RA-$ (for Real Annuity Dollar) which represents $0.02 per year for the next 50 years, adjusted for inflation. The value of the RA-$ will fluctuate versus the regular Dollar, driven by long-term real interest rates, which we can get from the market pricing of TIPS. Since long-term real interest rates are about zero right now, the value of RA-$1 currently would be equal to about $1, as 2 cents per year, inflation-adjusted for 50 years equals $1 discounted at a zero real interest rate. In early 1997, at the start of the period in our chart, long-term real interest rates were about 3.75%, making the value of RA-$1 about $0.50 regular Dollars.

Prior to 1997 – when the US Treasury started issuing TIPS – the notion of Real Annuity Value and RA-$’s would have been purely hypothetical. Over the past twenty years, there have been many books and articles from experts in personal finance such as Robert Merton, Zvi Bodie and John Campbell that argue the Real Annuity Value framework is the most consistent and logical one for reaching sound personal financial decisions, and that investors should adopt TIPS as their minimum risk asset in lieu of T-bills and other cash-like investments. If investors were to embrace this shift in perspective more broadly, the long-term TIPS market would have to grow well beyond its current size of $215 billion, which is less than 1% of the total US Treasury and investment grade bond market.

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Further Reading and References:

• Austen, Jane. Pride and Prejudice.  1813. Reprint: Penguin Books, 2003.
• Bodie, Zvi and Rachelle Taqqu. Risk Less and Prosper: Your Guide to Safer Investing.  Wiley. 2011.
• Breeden, Doug. “An Intertemporal Asset Pricing Model with Stochastic Consumption and Investment Opportunities.”  Journal of Financial Economics, pp. 265-96. 1979.
• Campbell, John and Luis Viceira. “Who Should Buy Long-Term Bonds?”  American Economic Review 91.1, pp. 99-127. March 2001.
• Heldman, James. “How Wealthy is Mr. Darcy – Really?”  Jane Austen Society of North America. 1990.
• Merton, Robert, C. “The Crisis in Retirement Planning.”  Harvard Business Review. July–August 2014.
• Merton, Robert C. and Arun Muralidhar. “Time for Retirement ‘SeLFIES’?”  SSRN. 2017.
• White, James and Victor Haghani. “The Equity Risk Premium: A Novel Perspective on the Past Fifty Years.” SSRN. March 2020.

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1. This not is not an offer or solicitation to invest, nor should this be construed in any way as tax advice. Past returns are not indicative of future performance.
   Thank you to Bob Merton for his suggestions and for sharing his thoughts on this topic, to which his decades of research and writing have contributed so much. Thanks also to Vlad Ragulin (one of Bob’s many students) and Rich Dewey for their valuable comments.
   
2. An annuity is a fixed sum of money paid each year, typically for a long period of time. A real annuity pays a fixed inflation-adjusted sum of money each year. US Social Security payments can be thought of as a real annuity, with a start date at entitlement age.
   
3. As per data made available by Professor Robert Shiller here. We assume that the level of real rates did not change over these two periods. Nominal rates changed little, and unfortunately long-term real rate data does not exist for those time periods.
   
4. A laddered portfolio of US Treasury Inflation-Protected Securities (TIPS) are the most like a real annuity that an investor could buy, and is a good candidate for the minimum risk asset an investor can buy.
   
5. In this note, we focus on a risk-free Real Annuity as a benchmark and numeraire for assessing one’s financial resources and making investment decisions. However, an even more accurate metric would take account of the expected return and risk of the full investment opportunity set as well as one’s personal aversion to risk. Such a measure would use the long-term risk-adjusted return of an investor’s desired portfolio instead of the long-term risk-free real rate to compute the Real Annuity Value to be used as one’s personal numeraire.
   

By Victor Haghani and James White ¹

If you’re a US investor, international equity exposure has never been so readily available at such a low cost. Nonetheless, surveys indicate US investors typically allocate 80 – 85% of their equity holdings to US equities, much higher than their proportion of global market value. We recently wrote about how this kind of “Home Bias” can impact expected returns. Here we turn to evaluating 10 arguments often heard in support of high levels of Home Bias.²

1. US equities have outperformed non-US equities by 170% over the past 10 years

Ironically, we really couldn’t make the point for international diversification any better than this. Ten years ago, few would have or did put forward this magnitude of US outperformance as a likely scenario, but it happened.³ Over long periods of time, individual equity markets can significantly outperform or underperform in ways which are very difficult to predict, though easy to explain ex-post with the benefit of hindsight. Ten years sure feels like a long time, and it’s tempting to conclude that it’s long enough to draw some conclusions about what the next ten years will hold, but if ever there was a place to say it, it’s here: past performance is not indicative of future returns.

Sources: Bloomberg, MSCI, FTSE. Emerging Market equities included from December 1989.

2. Over the past 10 years, an internationally diversified portfolio wasn’t anywhere near optimal

The theory of diversification is not that a diversified portfolio is likely to ever look optimal in hindsight, but rather that it has superior risk/return characteristics looking forward given an uncertain future. A concrete example may be helpful:

Over the last 10 years, the portfolio that had the highest realized return-to-risk ratio (i.e. Sharpe Ratio), was a portfolio that had about 20 stocks in it, chosen from all members of the S&P 500.⁴ The reason everyone doesn’t now own just those 20 stocks is that many investors have an accurate sense that it will be a different small group of stocks doing the best over the next 10 years. And indeed, the best risk-adjusted “hindsight” portfolio from 1999-2009 is completely different than one from 2009-2019. The idea behind diversifying to own 500 stocks is not that 500 stocks will beat every combination of 20 stocks over any given period. Instead, the problem is we don’t know which will be the best 20 stocks looking forward, and without this knowledge, the diversified portfolio looks better than choosing a more concentrated portfolio. But, if you know the big winners with hindsight, they’ll always look a lot better than owning the diversified portfolio.

So too with international diversification. Over any given period, there’s very likely to be one or two markets which significantly outperform the globally diversified portfolio, and in recent years amongst major markets, that outperformer has mostly been the US. That doesn’t reflect a flawed theory of diversification, just a recognition that – in both theory and practice – the benefits of diversification are to be seen through the windshield looking at the road ahead rather than in the rear-view mirror.

3. Investors favor the familiar

This is completely understandable, but it’s a cognitive bias that can potentially come at a high cost. For most people still in their earning years, their human capital is often their largest asset, and domestic markets are much more highly-correlated with that human capital than are foreign markets. All other things equal, we’d be better off owning things less correlated with our primary asset, the very opposite of this bias.

US investors may be especially unaware of this subtle cost of concentration because the history of US equity markets has been so benign in our investing lifetimes. However, US investors in the 1930s or 1970s, Japanese investors in the 1980s, or Russian investors in the early 1900s and late 1990s all received a first-hand lesson in the value of international diversification.

4. International equities don’t offer much diversification, because whenever US equities experience a large correction, international equities usually go down as much or more

Over short horizons of days, weeks, and months, large moves are indeed typically shared by nearly all public equity markets. This is also true of equities within the same market, yet we intuitively understand that despite this, owning a portfolio of 500 stocks spread out over all sectors of the economy provides meaningfully more diversification than owning just a handful of stocks. While individual equities may often be highly correlated during large moves over short time periods, such as during the 2008 financial crisis, most investors have longer horizons over which these correlations tend to dissipate. This is true of the correlation between US and international equity markets as well, as we see in the chart below:

Sources: Bloomberg, MSCI, FTSE. Emerging Market equities included from December 1989.

A number of studies support the view that international diversification works over the long term as differences in underlying economic, demographic, political, social and regulatory fundamentals between countries and regions make themselves felt in equity market returns. For example, see International Diversification Works (Eventually)  by Asness, Israelov and Liew (2010): “Over longer horizons, underlying economic growth matters more than short-lived panics with respect to returns, and international diversification does an excellent job of protecting investors.”  Increased globalization has likely led to higher correlations among regional equity markets, but it’s far from clear whether this globalizing trend will continue or start to reverse.

There’s also an important point to make about the limits of the historical record: even a long past fails to plumb the depths of possible futures. Many events are possible which have never happened before and don’t show up in any data series. Diversification offers an effective first line of defense against low-likelihood but large-impact scenarios.

5. US companies earn a meaningful fraction of revenue internationally, thus investors get international diversification just from US equities

US equities do have substantial international earnings. However, all US companies share exposure to a long list of significant US-specific risks — economic, political, financial, regulatory, tax, labor, etc. These are important risks, and non-US stocks can provide diversification from them in a way that US stocks with significant international earnings streams cannot.

According to Morningstar, 35% of the revenue of US public market companies came from outside the US in 2018. While this may seem to provide broad exposure to international economic activity, it offers less diversification than it seems. US companies’ international revenues come from big companies in a concentrated set of industries. For example, in 2018, 60% of the revenues of the information technology sector came from non-US markets while the utilities, real estate and financials sectors received just over 10% of their revenues from overseas.⁵ And there are many segments of the non-US economy which US companies hardly touch at all, particularly in less open economies in the developing world.

6. Non-US equities are riskier than US equities

International equity market returns measured in dollars have been more volatile than US market returns. For example, since 1990, the volatility of US equity returns was 14.5% compared to 17% for non-US equities.⁶ However, this by itself is not an argument against diversification. The three main inputs into the “portfolio optimization” problem are volatility, return, and correlation – and all three matter. There is still a significant benefit from a large non-US allocation if non-US equities are more volatile than US equities, but also offer a higher expected return and diversification benefits through imperfect correlation. The “extra” volatility of non-US equities is known and should already be incorporated into market prices and thus into expected returns.

7. US investors spend their savings in dollars, and so they should only invest in dollars to avoid the currency risk associated with non-US equities

It’s a useful simplification to split investment assets into two buckets: minimum-risk assets (risk-free assets, in theory) and risky assets. An investor’s base level of expected future spending in retirement should ideally be supported by minimum-risk assets. To the extent that spending is going to be in one’s home currency, then the minimum-risk assets must also be in that currency.⁷ For a risky asset though, the main things that matter for how it fits in a portfolio are its expected return, volatility, and correlation with other investments. For a given level of volatility, return and correlation, it doesn’t matter that some of that volatility comes from currency risk rather than some other source. Hence, US investors should not shun foreign equities just because they are not denominated in dollars, as long as their expected return is sufficient given their contribution to overall portfolio risk.⁸ Additionally, although it may be difficult for US investors to imagine, there are circumstances when being diversified away from one’s domestic currency can be beneficial.

8. The US is the greatest place on Earth to invest

We agree that the US has been a terrific environment for business and this is likely to continue, but sadly this is no secret and so should already be reflected in the pricing of US equities. If anything, there’s little room for this common view to be strengthened over time, and significant room for it to be weakened.

9. Investing in non-US equities is difficult and expensive

This certainly used to be the case, but not so much anymore. The expense ratio of Vanguard’s non-US equity index fund (VXUS) stands at 0.09% down from a 0.45% initial average expense ratio for their European, Asian and Emerging Market equity index funds, launched in 1990, 1990 and 1994 respectively. While Vanguard’s US equity index funds with an expense ratio of 0.03% are cheaper than their non-US equity index funds, the gap is quite narrow at just 0.06%. There is still a tax wedge between US and non-US dividends as a smaller fraction of non-US dividends have the preferred “qualified” status, which, for high marginal rate US taxpayers, gives them a roughly 20% lower tax rate than non-qualified dividends. We estimate that the total of expense and tax differences adds up to about a 0.15% extra cost for holding non-US equities. In a simple mean-variance framework, this changes their optimal portfolio weight by about 5%, providing justification for a bit of Home Bias.⁹

10. US equities are about 55% of the MSCI global equity index, so isn’t owning 80-85% of US equities a pretty minor deviation?

The major index providers, MSCI and FTSE, include significant “investability” and free-float adjustment factors in their market weights. These adjustments make sense in the context of creating an index which can accommodate the benchmarking of trillions of dollars of investment, but they do have the effect of exaggerating US market weights. The raw, unadjusted global market value weight of US equities is closer to 35%, and it is expected to decline in the future as the developing world catches up with the US, so an 80-85% US allocation represents a dramatic departure from global market-value weights today, and in the foreseeable future.

The chart below illustrates the expected gain possible from different levels of international diversification from a US investor’s perspective. It assumes 40% in US equities is the optimal weight, based on unadjusted global market value weighting¹⁰ while also taking account of the extra expense and tax costs of owning non-US equities. Moving from 100% in US equities to 85% captures only about 40% of the benefit of optimal diversification. Moving further to 55% in US equities captures more than 90% of the total diversification gain available. Notice that the closer we get to the optimal point, the gain curve becomes flatter and there’s less available gain from each 1% change in allocation.

Conclusion

You’ve probably gathered that we don’t find much merit in most of the arguments supporting a high degree of Home Bias in global equity investing. However, as seen in the chart above, there’s a relatively broad range of choices around the optimal allocation which are reasonable and which involve little sacrifice in portfolio quality. Indeed, Elm’s offerings use a Baseline US equity exposure of about 50%, significantly higher than the 35% raw market-value weight while still delivering the vast majority of expected diversification benefits. Some of this adjustment from 35% to 50% comes from the small extra cost associated with holding non-US equities, as described in [9] above. Most of it, though, comes from taking into account investor preferences, happily in a way which isn’t significantly sub-optimal.

This note reflects how we think about determining our “Baseline” allocation to US and non-US equities. This Baseline serves as the starting point for our dynamic asset allocation approach. Depending on the level of current expected returns for each asset bucket, we vary allocations away from the Baseline.¹¹

Some of the above arguments for US Home Bias have been famously made by Warren Buffett, who has received a lot of attention for taking a strong “no-place-like-home” position. Indeed, he’s instructed his heirs to avoid non-US equities completely by taking all their equity exposure through a low-cost S&P 500 index fund. Perhaps he’ll reconsider his advice after reading this note.

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Further Reading and References

• Asness, Clifford, Roni Israelov, and John Liew. “International Diversification Works (Eventually).”  Financial Analyst Journal. 2011.
• Brinson, Gary and Roger Ibbotson. “Global Investing: The Professional’s Guide to the World Capital Markets.”  McGraw-Hill. 1993.
• Bryan, Alex. “Investing Close to Home is Overrated.”  Morningstar. 2019.
• Dimson, Elroy, Paul Marsh, and Mike Staunton. “Triumph of the Optimists: 101 Years of Global Investment Returns.”  Princeton University Press. 2002.
• French, Kenneth, and James M. Poterba. “Investor Diversification and International Equity Markets.”  NBER. 1991.
• Haghani, Victor, and James White. “Do US Industry-Sector Weights Explain the Higher Valuation of US vs non-US Equities?”  Elm Partners. 2019.
• Scott, Brian, Kimberly Stockton, and Scott Donaldson. “Global equity investing: The benefits of diversification and sizing your allocation.”  Vanguard Research. 2019.

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1. This not is not an offer or solicitation to invest, nor should this be construed in any way as tax advice. Past returns are not indicative of future performance.
   Thank you to Gary Brinson, Jeffrey Rosenbluth, Larry Hilibrand, Antti Ilmanen, Vladimir Ragulin, Rich Dewey, Aneet Chachra and Joshua Haghani for their insightful and helpful comments, and to Paul White of Vanguard for providing us with information about the history of Vanguard’s international equity index offerings.
   
2. We specifically focus on Home Bias as it relates to equities, as there are very different issues related to international fixed-income markets. We also assume the perspective of a US-based equity investor, though much of what’s said is relevant for non-US investors, especially given that no other market has nearly as large a weight in the global portfolio as the US.
   
3. Indeed, many financial commentators were even expecting the US would experience a ‘lost decade’ of low or negative equity returns.
   
4. From 2009-2019, the mean/variance optimal portfolio had 22 stocks in it and a realized, backward-looking Sharpe ratio of 2.4, vs 0.92 for the portfolio of all stocks in the S&P 500 present in the index over the entire period.
   
5. From Factset report here.
   
6. Annualized volatility calculated from monthly dollar returns.
   
7. By this criterion, non-US bonds denominated in foreign currency would not be a suitable holding for this bucket.
   
8. How much expected return one should demand or be willing to give up relating to currency risk depends primarily on the degree to which currency risk impacts the risk of international equities measured in dollars. There are good reasons to expect foreign equity markets to rise when their domestic currency falls, dampening the volatility in dollars, and we tend to see this in normal times. A very mild correlation in the range of 0.15 to 0.25 in this direction is enough to make the impact of currency fluctuations on non-US stock returns in dollars close to zero.
   However, over the past twenty years, that normal relationship has been overwhelmed by global investors treating the US dollar as a safe haven and flocking to it in times of crisis. If global investors continue to behave in that way, then US investors in non-US equities should expect to earn some compensation for bearing the risk of non-safe-haven currencies.
   
9. Using a correlation of 0.7 between US and non-US equities.
   
10. We selected parameters of volatility and correlation of US and non-US equities that would make the unadjusted market value weights be the optimal mean-variance weights.
    
11. For each asset bucket, the expected return forecast has a long-term component based on valuation, and a medium-term component based on momentum. You can read more about our asset allocation methodology here.
    

By James White and Victor Haghani ¹

Abstract:

“Factor investing, or Smart Beta as it’s known in long-only form, has become one of the most popular forms of investing, straddling the active-passive divide. The authors evaluate theoretical and empirically-based arguments for factor investing, concluding that while many of the arguments, particularly the theoretical ones, are sound, there are still reasons for considerable skepticism. They describe the trajectory of the factor investing paradigm, from the cradle of the efficient markets school to its championing by some of the world’s most successful investment management firms. While factor investing is typically discussed using the language and machinery of efficient-markets models, investors are primarily expecting anomalous excess returns more consistent with behavioral explanations and other market inefficiencies. For factors with plausible risk-based explanations, the authors conclude that even in the presence of significant factor premia, the market portfolio is still likely to be optimal for most investors. The authors also provide simple logical arguments to assess claims such as that factor investing delivers gross investment returns similar to traditional active managers, but with lower fees.”

Read the full article in The Journal of Portfolio Management here.

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1. We are grateful for the many helpful comments of Richard Dewey, Chi-fu Huang, Antti Ilmanen, Vladimir Ragulin, Aneet Chachra, John Glazer, Joshua Haghani, Larry Hilibrand, Peter Hirsch, Arjun Krishnamachar, David Modest, Hedi Kallal and Jeffrey Rosenbluth.