Elm Wealth Research

By Victor Haghani, Vladimir Ragulin and Richard Dewey¹
This article is also available for download in PDF form on SSRN.

When we first heard about the overnight effect – the propensity for stocks to deliver all their returns when the market is closed and no returns during the trading day – our first reaction was: that can’t be right! After some preliminary reading, our follow-on reaction was: who cares!? In talking with market participants and academics, we found that most people shared our reaction, if they knew about the effect at all. “Bad data,” “quack analysis,” “a statistical fluke” were the common refrains.

But after some more digging, we found plenty of evidence for the overnight effect and more vexing features to the puzzle. We found that not only did the effect exist at the index level as previously reported (see chart of S&P 500 returns below), but it also shows up in a suggestively clustered pattern in individual stocks returns, and is particularly strong in “Meme” stocks. Moreover, there are plenty of reasons to care about this market anomaly.

We think this research is important for three reasons. First and foremost is that retail traders are potentially missing out on billions of dollars of returns due to mistimed trades. Second, there is speculation that the overnight effect might have implications for the long-term valuation of the entire equity market. And finally, assuming our findings and those of others who have studied the effect are correct, this is one of the most consistent, significant and overlooked anomalies in finance, which can contribute to our understanding of the limits of market efficiency.

Hiding in Plain Sight

The “overnight effect” is observed across most major equity markets and is consistent back to the 1990s at least. And yet, just a smattering of papers over the years have explored this phenomenon – a tiny fraction of the attention devoted to other potential market anomalies such as momentum, value, or a host of other return factors – leaving the overnight effect an overlooked and largely unexplained anomaly in financial markets research.

Much of the search for an explanation for the overnight-versus-intraday effect in the broad stock market has focused on aggregate characteristics of the market, such as whether the level and nature of overnight risk compared to intraday risk warrants a higher overnight return. In search of clues to the underlying causes of the superior performance of the stock market overnight, we examined the behavior of individual stocks in the intraday versus overnight sessions.

Our primary line of research was to explore the return pattern of a long-short portfolio constructed to test for persistence in overnight-versus-intraday return patterns at the individual stock level.² The portfolio generated a return of about 38% per annum (exclusive of estimated transaction costs and without leverage³), with a Sharpe Ratio high enough to make an efficient-markets economist blush. The reason we report the return excluding transactions costs is to emphasize that the purpose of this note is to understand why the stock market has delivered almost all its return when the lights are out, rather than to propose a potentially profitable trading strategy. Yes, a 38% return is very special and later in the paper we show how an investor could have made a modest amount of money from this anomaly. However, to have made a lot of money, an investor would have needed to think hard about employing leverage, improving the signal to noise ratio further, reducing risk through diversification and achieving low transactions costs (including market impact).

Background

The divergence between overnight and intraday returns of the aggregate stock market was first reported in 1986 by Ken French and Richard Roll. We found valuable insights in about a dozen academic papers thereafter, namely Cliff et al. (2008), Barardehi et al. (2021 and 2022), Berkman et al. (2012), Bogousslavsky (2021), Bondarenko (2020), Branch and Ma (2012), Hendershott et al. (2020), Kelly and Clark (2011), Lachance (2015, 2021), Lou et al. (2019), and Boyarchenko et al.(2020) of the NY Federal Reserve.

Perhaps the most widely discussed research on this puzzle is the half-dozen notes published on SSRN by Bruce Knuteson, a particle physicist and former employee of the pioneering quantitative investment firm, DE Shaw. Knuteson brings attention to Lachance’s (2015) finding that this return pattern was present in nearly every other stock market around the world.⁴ Notably, Knuteson has called on regulators and journalists to investigate the cause, which he argues will be found to be large quant funds pushing stock prices in favor of their existing leveraged long-short portfolios.

Hard to Explain

Potential explanations for the relatively high overnight return of the stock market include:

1. Markets are riskier closed than open. For example, many economic indicators, earnings reports and merger announcements are released when the stock market is closed.
2. Net of trading costs, including balance sheet, risk charges and funding costs for holding positions overnight, the anomaly goes away. If we think about the stock index as a risk-free bond, then we’d expect all the return to come overnight, when interest would be credited, with no return earned during the day.
3. There are so many possible weird price patterns that we might observe, including many calendar-related ones that disappeared shortly after being “discovered,” that when we find one, we shouldn’t be surprised.⁵ It’s like bumping into a high school friend on a visit to Paris, and mistakenly concluding such a coincidence was one in a billion, without acknowledging that there are about a billion things that could happen every day that we’d find equally weird.

These sound plausible, but there is evidence in our research and others that #1 is empirically wide of the mark,⁶ and #2 explains some (note that the average T-bill rate was 2.4% over the period) but far from all of the overnight return, and does not explain why there was no return during the risky intraday session. Finally, the persistence of the effect, although weakened over the fifteen years since it was first reported, weighs against #3.

The Grandmother of All Anomalies?

Inspired by the research of Lachance (2015), we constructed a portfolio that was long, but only during the period the market was closed, the 20% of S&P 500 stocks⁷ that had the highest overnight-minus-intraday return over the prior two years,⁸ and short, again, only during the time the market was closed, the bottom 20% of stocks by that measure. The stock positions were equally weighted.

The chart below shows the growth of $1 invested in the long-short portfolio since 1995. The gross return is 38% per annum, exclusive of transactions costs. To get to a net-of-transactions cost figure, note that round-trip transactions costs of 1 basis point on each of the long and short side of the portfolio reduce returns by about 5% per annum, and a 1% borrow fee on shorts reduces the portfolio return by 1%.

We find these results striking, and thought-provoking. How ever much one sees the overnight-versus-intraday returns of the S&P 500 to be anomalous and an affront to efficient markets theory, the effect observed at the individual stock level is significantly less likely to have been generated from the standard random walks associated with the Efficient Markets Hypothesis.⁹ Efficient markets champion Eugene Fama famously referred to the Momentum effect in individual stocks as “The Mother of All Anomalies.” Given the Sharpe Ratio of this long-short overnight strategy is about ten times higher than that generated historically by long-short momentum portfolios,¹⁰ perhaps this overnight long-short phenomenon should be known as the “Grandmother of All Anomalies”?

A Clue from Meme Stocks?¹¹

If we can understand what’s behind the return of this long-short portfolio, perhaps it will lead to an explanation for the overnight-versus-intraday long only performance of the stock market at the index level. We believe the return patterns of the following Meme stocks provide a valuable clue:

The first chart is saying that a day-trader who bought AMC Entertainment at the open and sold it at the close every day from the start of 2019 to late May 2022 would have suffered a 99.6% loss of capital – but, during the night, would have made a return of 30,000% over the same period (both ignoring transaction costs).¹²

Could it be that the more an investment appeals to retail investors, the more it shows a better overnight return relative to its intraday return? A look at two investments – which are not stocks, but have strong appeal to retail investors – gives further weight to this idea. They are the ARK Innovation ETF (ARKK) run by Cathie Wood, and the Grayscale Bitcoin Trust (GBTC) which gives retail investors exposure to Bitcoin. The divergence between their overnight returns and their intraday returns is remarkable.

Attention Stocks

A striking feature of the stocks that tended to perform best at night was their popularity with retail investors, in that they were well-known brand names, frequently mentioned in the news (as quantified by Google Trends data), heavily traded in the options markets, and/or had high recent returns. Such stocks are aptly referred to as “Attention Stocks”, e.g. in Barber and Odean (2008) and Berkman et al. (2012). The table below shows the top sixty out of one hundred stocks by market capitalization that the strategy was on average net long or net short most frequently.¹³ We have highlighted in yellow the stocks that we think fit the Attention Stock definition best, and also show the Google Trends score. Notice there are twice as many Attention Stocks among the long overnight names as among the short overnight names, their Google Trend average score is about ten times higher, and the long overnight names are on average two and a half times larger in market capitalization than the short overnight names.¹⁴ This is consistent with results by Lou et al. (2014), Bogousslavsky (2021), and Barardehi (2022) who correlated overnight performance with popular style factors and found ¹⁵ that good overnight performers tend to have higher Price/Book ratios, higher Beta and volatility, lower quality of earnings, and higher momentum. Most of these characteristics fit an a priori notion of what retail investors would be attracted to, and are also just the characteristics that would raise some concerns with traditional real-money institutions who would otherwise have the capacity to hold these stocks overnight.

Connecting the Dots

In our view, an explanation based on retail investors pushing up opening prices seems the most plausible, as in Berkman et al. (2012). The specific mechanism that might be causing these return patterns is suggested by the following two generally accepted characteristics of market micro-structure:

1. Stock market liquidity is deeper at the close of the trading day, and shallower at the open. A given size trade executed at the open has a bigger price impact than at the close.
2. Retail investors place their orders more at the open, and institutional investors more at the close. This is seen from studies of brokerage trading records and analysis of the timing of small and large trades over the course of the day. Small trade sizes occur more towards the beginning of the day, and large trades later in the day.¹⁶ It seems reasonable that retail investors tend to make their single stock investment selections at leisure in the evenings or over the weekends, and then place their orders before going to work, which will often be executed at the open. When selling one stock to buy another, they may do that at the open too, but they may be more price sensitive and use limit orders to sell stocks out of their portfolios. In the 1990s, before the rise of online brokers, trading on the open was standard advice for retail traders as a way to avoid the bid/ask spread and ensure a reasonable execution at a price that can be checked against the Wall Street Journal. Conversely, institutions and/or professional money managers tend to like to execute at market closing prices, where there is more liquidity and also as they tend to have their performance judged against market closing prices.

The liquidity differential over the course of the day means this retail buying at the open hurts intraday returns, and helps overnight returns, while selling at the open has the opposite impact.¹⁷ The two critical links between retail impact at the open and the positive overnight drift in the broad market index are:

1. The Attention stocks that comprise the long side of the portfolio returned 29% versus only 6% for the stocks on the short side of the portfolio. This asymmetry seems plausible, by which the force of attraction to Attention stocks is stronger than the more diffuse force of retail being repulsed by the “anti-Attention” or “Neglected” stocks, which make up the short side of the portfolio.
2. The Attention stocks that make up the long side of the portfolio have larger market capitalizations than the Neglected stocks on the short side, and so the impact of a given percentage increase in price in those long stocks would outweigh an opposite percentage price impact on the short stocks for a market capitalization weighted index, like the S&P 500. So, even if the force of retail’s attraction to Attention stocks were equal to the force of repulsion from Neglected stocks, we should still see a net upward drift in the overall broad stock market index.

The proposition that retail investors think about what trades they want to do when the market is closed and then put in orders on the open is supported by noting that average returns to the long-short portfolio over weekends were about one and a half times higher than returns between weekdays.¹⁸ It seems that retail investors have spent some of the extra time afforded by the weekends to read Barron’s and think about what trades they want to do when the markets open again – although given the ratio of the size of the effect is not quite proportional with the extra free time, it’s comforting to know that retail investors also spend some of their weekends playing golf and watching Netflix!

Weekends also present a natural experiment testing the alternative hypothesis that the overnight effect was caused by companies on average announcing better-than-expected earnings after trading hours over the past thirty years. Since US companies release earnings much less frequently just before weekends and holidays, the fact that weekends exhibited more overnight drift discounts this earnings-driven explanation.

Any explanation should also account for similar divergences between overnight and intraday returns that have been witnessed in other assets, particularly when packaged in ETFs. For example, Vanguard’s popular broad bond market ETF, BND, and Invesco’s Commodity Index tracker, DBC, have had a higher return overnight than over the full day over the past 15 years. Perhaps this is explained by observing that in some sense ETFs are like popular stocks, since they are mostly retail-oriented with market-makers providing liquidity. A more detailed explanation is provided by Lachance (2021) “ETFs’ High Overnight Returns: The Early Liquidity Provider Gets the Worm” in which she argues:

“At the root of this distortion [of unusually high ETF overnight returns] is the phenomenal growth of the ETF market, which the paper shows is associated with highly positive order imbalances [at the market open] that exceed 10% on average. As the saying goes, ‘The early bird gets the worm’ and this paper shows that it can be particularly rewarding to provide liquidity around the open as the spreads are three to four times higher than during the day.”

Kelly and Clark (2011) suggest that the overnight effect arises from day-traders who are averse to taking overnight risk and therefore liquidate their portfolios at the close and reestablish them in the morning. Similarly, high frequency market makers may choose to avoid holding overnight risk even if it means sacrificing some expected return to flatten their books at the close of trading.

This Attention Stock story does not mean other explanations (e.g. Lou et al.’s, Kelly and Clark’s, Knuteson’s, Mamayski’s, and others) cannot also provide part of the answer, perhaps even a bigger part, but we do believe that our explanation likely provides one important piece of the puzzle.

Of course, there is a more prosaic explanation for the observed overnight-versus-intraday behavior, which would be that the opening and closing prices which we, and other researchers, have used in this area of study are simply not reflective of actual prices available to market participants.

The $100 Bill That Shouldn’t Be There

It’s not enough to describe the potential cause of the overnight phenomena; we also must explain why sophisticated capital hasn’t traded against it in sufficient size to make it go away.

The most common and reasonable explanation is transaction costs, with particular emphasis on market impact. Trading at the opening and closing auctions does not involve crossing a bid-ask spread. However, price impact – the amount a trade moves the market – will be a meaningful consideration for an investor contemplating dedicating significant capital to taking the other side of this anomaly. Brokmann et al. (2014) and Frazzini et al. (2018) estimate transaction costs including price impact for large traders and generally agree with the “square root of fraction of volume” function put forward by Kyle (1985) and Barra (1997). This function suggests that trading just 1% of daily volume of a stock that exhibits 2% daily price volatility would result in 40 basis points of round-trip price impact, which would reduce annual returns of a long-short overnight portfolio by 200% per annum.

Commissions and exchange fees might also have played a role. At the beginning of our sample, which starts in 1995, commissions were significantly higher. A 10 basis point per trade commission level, not uncommon in early 1990s, would have translated into a 100% per annum performance drag. And even a trader executing small sizes and paying only 1bps per trade would not have made any money in the last 8 years, even though the strategy’s gross returns remained very attractive.

Very sophisticated trading firms might be able to mitigate the above frictions. However there are several reasons why these firms might be reticent to trade the overnight effect. Some of the reasons include:

• Risk covariance: The overnight-versus-intraday drift may be one of a number of anomalies caused by retail flows, making it a less attractive opportunity as part of a portfolio that already has a lot of exposure to strong retail flows. For example, we note that during the period in early 2020 when there was a reversal in the direction of the overnight-versus-intraday effect at the index level, some prominent quant funds (including DE Shaw) suffered significant losses.
• Size constraint: There may be a limit on the size of trade informed capital can do in the opening auction, which may be a constraint if the number of players able to take the other side of retail is limited.
• Risk tolerance: HFT and other market makers exhibit a strong preference to end the day with flat positions. They like to be able to manage their exposure minute to minute, and are averse to being locked in for hours or days (i.e. weekends and holidays). Similarly, mid-frequency statistical arbitrage firms like to end the day without significant factor exposures and are willing to pay to close positions.
• Infrequent, but severe drawdowns: This is often cited as a possible explanation for cross-sectional Momentum returns in equities as described by Daniel et al. (2016). However, the long-short overnight individual stock strategy’s most severe drawdown was just 4%, so this explanation does not apply to historical returns of the strategy examined herein, although perhaps large drawdowns are latent.

Going, Going, Gone?

These “limits to arbitrage” arguments explain why a tsunami of capital has not washed the overnight-versus-intraday anomaly away. However, these limits are not enough to stop a decay of the inefficiency, and indeed we observe that the effect has been waning following the circulation of a half dozen relevant papers between 2008 and 2015. The chart below shows five-year rolling returns to the long-short portfolio, which suggests, but doesn’t quite prove, that something has changed in the past five to ten years. We also show the rolling five-year overnight minus intraday returns of the S&P 500, which shows the same general pattern of very high realized returns until about ten years ago. It is interesting that several ETFs are in the works which are structured to only invest in the stock market when it is closed.¹⁹

Macro Worries

Is it possible that the upward price impact at the open on Attention Stocks has had a cumulative impact of pushing the overall stock market to a higher level than would otherwise be the case? Even if our description of some of the market forces at play is valid, the answer depends on the speed of decay of price impact and the effectiveness of active investors to lean against non-fundamentally driven price changes. It is a topic which has already received attention, such as Shiller (1984), but perhaps in light of the overnight-versus-intraday behavior that we’ve discussed herein, it is a question worthy of further research. These overnight-versus-intraday patterns just don’t look like what we’d expect from a healthy, well-functioning market where all investors are getting a fair shake.

Putting the Overnight Effect to Bed… for Now

Our research on the overnight effect in single name stocks suggests that the hypothesis of retail trading likely goes a long way toward explaining the phenomenon at both the level of the overall stock market, and that of individual stocks. There is likely some additional interplay with other market participants, along with balance sheet, risk and funding charges that also account for some of the observed effect.

It’s plausible that well-known mid-frequency statistical arbitrage funds are indeed taking advantage of this effect as has been previously suggested, but due to transaction costs and portfolio construction constraints, they are unable to completely arbitrage the effect away. Without arbitrage players taking the other side of retail, the overnight effect might have been much larger than what we observed.

It has been suggested that ordinary investors should take heed of this phenomenon and execute their buys at the end of the day, and sales at the open. That advice is not wrong, but it’s worth bearing in mind that even for a relatively active investor who turns over 100% of their portfolio per year, the improvement in return they should expect is only a few basis points. And, for non-professional investors who are so actively trading that this would make a big difference, our suggestion would be to trade less rather than trade at the close!²⁰

Sadly, it is possible that retail investors who actively trade individual stocks have left many billions of dollars of stock market return on the table. Of course, this is not a novel insight. For example, Barber and Odean’s seminal paper, “Trading is Hazardous to Your Wealth” (2000), found that households that traded the most at a large discount broker from 1991 to 1996 underperformed the market by 6.5% per year. A more recent survey of the retail stock trading scene in Spencer Jakab’s “The Revolution That Wasn’t: GameStop, Reddit, and the Fleecing of Small Investors,” (2022) vividly documents the herd behavior that likely generated the massive divergence between overnight and intraday returns in Meme stocks. Alas, it is a tale that does not have a happy ending.

We hope that our research on individual stocks goes a little way toward explaining this curious and puzzling phenomenon. While we can’t be sure of the exact drivers, we believe that the overnight effect merits further study given its potentially large implications for retail traders, policy makers, academics and ultimately all curious market denizens.

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PS…

Perhaps the Bloomberg financial journalist and humorist Matt Levine is on to something when he suggests,

“…the stock market should have much, much shorter hours; 15 or 30 minutes a day should suffice for everyone who wants liquidity to find it, and traders could spend the rest of their days researching companies… or reading poetry or hanging out with their families…Go home! Read a book! Walk the dog!”
  – Money Stuff, Feb 6^th 2020 and Oct 6^th 2021

And for retail investors, we might add: “And get better returns!”

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Further Reading and References

• Barardehi, Y., D. Bernhardt, T. Ruchti, and M. Weidenmier. (2021). “The Night and Day of Amihud’s (2002) Liquidity Measure.” Review of Asset Pricing Studies.
• Barardehi, Y, V. Bogousslavsky, and D. Muravyev. (2022). “What Drives Momentum and Reversal? Evidence from Day and Night Signals.” Working Paper.
• Barber, B. and T. Odean. (2000). “Trading is Hazardous to Your Wealth: The Common Stock Investment Performance of Individual Investors.” The Journal of Finance.
• Barber, B. and T. Odean. (2008). “All That Glitters: The Effect of Attention and News on the Buying Behavior of Individual and Institutional Investors.” The Review of Financial Studies.
• Barra. (1997). “Market Impact Model Handbook.” MSCI.
• Berkman, H., PD Koch, L. Tuttle, and YJ Zhang. (2012). “Paying Attention: Overnight Returns and the Hidden Cost of Buying at the Open.” Journal of Financial and Quantitative Analysis.
• Bogousslavsky, V. (2021). “The Cross-Section of Intraday and Overnight Returns.” Journal of Financial Economics.
• Bondarenko, O. and D. Muravyev. (2020). “Market Return Around the Clock: A Puzzle.” SSRN.
• Boyarchenko, N, L. Larsen, and P. Whelan. (2020). “The Overnight Drift.” Federal Reserve Bank of New York.
• Branch, B. and A. Ma. (2012). “Overnight Return, The Invisible Hand Behind Intraday Returns?” Journal of Applied Finance.
• Brokmann, X., E. Serie, J. Kockelkoren and JP Bouchaud. (2014). “Slow Decay of Impact in Equity Markets,” ArXiv.
• Cliff, M., MJ Cooper and H. Gulen. (2008). “Return Differences between Trading and Non-Trading Hours: Like Night and Day.” Unpublished Working Paper, University of Utah.
• Daniel, K. and T. Moskowitz. (2016). “Momentum Crashes.” Journal of Financial Economics.
• Frazzini, A., R. Israel and T. Moskowitz. (2018). “Trading Costs.” SSRN.
• French, K. and R. Roll. (1986). “Stock Return Variances: The Arrival of Information and the Reaction of Traders.” Journal of Financial Economics.
• Hajric, Vildana and Lu Wang. (2022). “New ETFs Aim to Capture Those Overnight Returns in Stock Market.” Bloomberg.
• Hendershott, T., D. Livdan and D. Rosch. (2020). “Asset Pricing: A Tale of Night and Day.” Journal of Financial Economics.
• Jakab, Spencer. (2022). The Revolution That Wasn’t: GameStop, Reddit, and the Fleecing of Small Investors, Portfolio.
• Kelly, MA and Clark, SP. (2011). “Returns in Trading versus Non-Trading Hours: The Difference is Day and Night.” Journal of Asset Management.
• Knuteson, B. (2016). “Information, Impact, Ignorance, Illegality, Investing, and Inequality.” SSRN.
• Knuteson, B. (2018). “How to Increase Global Wealth Inequality for Fun and Profit.”SSRN.
• Knuteson, B. (2019). “Celebrating Three Decades of Worldwide Stock Market Manipulation.”SSRN.
• Knuteson, B. (2020). “Strikingly Suspicious Overnight and Intraday Returns.”SSRN.
• Knuteson, B. (2021). “They Chose to Not Tell You.”SSRN.
• Knuteson, B. (2022). “They Still Haven’t Told You.”SSRN.
• Kyle, A. S. (1985). “Continuous auctions and insider trading.” Econometrica.
• Lachance, Marie-Eve. (2021). “ETFs’ high overnight returns: The Early Liquidity Provider Gets the Worm.” Journal of Financial Markets.
• Lachance, Marie-Eve. (2015). “Night Trading: Lower Risk But Higher Returns?” SSRN.
• Levine, Matt. (2018). “Is Everything Manipulated.” Bloomberg.
• Lou, D., C. Polk and S. Skouras. (2019). “A Tug of War: Overnight Versus Intraday Expected Returns.” Journal of Financial Economics.
• Mamaysky, forthcoming.
• Qiao, K. and L. Dam. (2019). “The Overnight Return Puzzle and the ‘T+1’ Trading Rule in Chinese Stock Markets”. SSRN.
• Shiller, R. (1984). “Stock Prices and Social Dynamics.” Brookings Papers on Economic Activity.

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1. Victor is the founder and CIO of Elm Wealth, Vlad is a consultant with Elm and Rich is a co-founder of Raposa Research and an advisor to Royal Bridge Capital.
    

   Thank you to Alexey Bachurin, Larry Bernstein, Simon Bowden, Samir Bouaoudia, Aneet Chachra, Adrian Eterovic, Ian Hall, Larry Hilibrand, Antti Ilmanen, Costas Kaplanis, Bruce Knuteson, Marie-Eve Lachance, Harry Mamaysky, Steve Mobbs, David Modest, Terrance Odean, Vladimir Pakhomov, James White, and Anna Wroblewska for taking a look and sharing their reactions, comments, and experiences with us.

    

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

2. All data in this note is from Yahoo Finance. For S&P 500 chart, includes reinvested dividends, and net of fees charged on the SPY ETF, but without trading costs. As an additional check, we re-ran the analysis using price data from another data vendor, eodhistoricaldata.com and using our own Elm index of US Liquid Equities instead of S&P 500 (to control for systematic patterns around S&P rebalances). The results were very similar. We are happy to make the Python code we used to collect the data and perform the calculations available.
   
3. For $1mm of capital our simulated portfolio is long and short $1mm each.
   
4. With the notable exception of China, which Qiao and Dam (2019) discuss in a way that Knuteson argues perfectly fits his thesis.
   
5. Ignoring the selection bias aspect of the problem, just how unlikely an outcome is the observed overnight versus intraday return pattern? An intuitive perspective on the problem is to imagine putting all the intraday returns in an unbroken sequence. If intraday volatility was 70% of close-to-close volatility on average (in fact, it was a bit higher), then 30 years of intraday experience would be equivalent to 15 years of normal stock market volatility. Assume the same for the overnight as well.
    
   Now we can think about the question like this: how weird is it to have had a 15-year period when the stock market had a roughly 0% return, followed by a 15-year period when it returned about 20% a year? Based on Professor Robert Shiller’s 150 years of US stock market data, that never occurred, as there was no 15-year period of a 20% annual total return. So, yes, the overnight versus intraday return of the S&P 500 is quite unexpected.
   
6. While the standard deviation of overnight returns from 1993-2022 was lower than that of intraday returns, overnight returns exhibited greater negative skew and kurtosis (fat-tailedness). However, for an investor with typical risk-preferences, the lower standard deviation far outweighed the impact of the difference in the higher moments.
   
7. That is, stocks that were in the index in May 2022. We also ran the analysis based on the largest stocks that were trading at each point in time, as detailed in footnote 2, and the results were essentially the same.
   
8. The indicator we use is a simplified version of the Overnight Bias Parameter (OBP) used by Lachance, which had a two-year lookback window. This is only one of several factors that showed statistically significant ability to predict overnight excess return in her study.
    

   We also examined performance using a window of six months, one year and three years, and qualitatively similar results, with the six month and one year windows generating somewhat more positive historical simulated performance.
   

9. For example, the t-statistic for the long-short strategy is about 17 versus 2 to 3 for the overnight-versus-intraday effect in the S&P 500 depending on choice of the null hypothesis being tested.
   
10. See Kenneth French data library here.
    
11. From Wikipedia:
    “A Meme stock is a stock that gains popularity among retail investors through social media. The popularity of meme stocks is generally based on internet memes shared among traders, on platforms such as Reddit’s r/wallstreetbets. Investors in such stocks are often young and inexperienced investors. As a result of their popularity, meme stocks often trade at prices that are above their estimated value based on fundamental analysis, and are known for being extremely speculative and volatile.”

12. Note that three of the four Meme stock examples above were not part of our long-short strategy, as they were not members of the S&P 500, although they did make appearances in our alternative dataset, described in footnote 2.
    
13. For the Google Trends Index, we took the average of the top 60 names by time spent in long/short portfolios, which is more representative of our equal-weighted portfolio simulation. Taking top longs/shorts by market cap produces similar results.
    
14. That the stocks in the overnight portfolio were bigger than those in the short overnight portfolio was the case for almost the entire period from 1995 – but, from around March 2020, the average size of the short portfolio became larger.
    
15. See Table II Lou et al. (2019).
    
16. See Figure 3 in Lou et al. (2014).
    
17. It is also believed that retail traders overreact to news that arrives when the market is closed, but it is not clear how this would cause an upward drift in the long-short portfolio or the broad market.
    
18. The standard deviation of returns of the long-short portfolio between weekdays and over weekends was roughly the same at about 0.5%
    
19. See “New ETFs Aim to Capture Those Overnight Returns in Stock Market”, Bloomberg, March 4, 2022.
    
20. This excludes professional quant funds who don’t need our advice. Their ability to make money from such weak and decayed signals is much greater due to super-low trading costs, cutting-edge techniques to separate alpha from noise, and ability to diversify on an industrial scale.
    

By Victor Haghani and James White ¹

finsplain – /finˈsplān/: Explaining finance, crypto and NFTs to someone who does not give a sh*t and thinks you should get a life already.
  – Urban Dictionary

What are we to make of the recent near-total wipeout of the stablecoin Terra and its companion coin Luna, in which around $50 billion of value vaporized in less than a week? ² One perspective that might be useful comes from a parable beloved by Vic’s dad:³

Hassan and Ali are antique dealers in the bazaar. Hassan notices a beautiful antique bowl in Ali’s shop, which Ali was given by an itinerant dervish in return for a cup of rice. Hassan asks Ali for a price to buy it, but Ali doesn’t want to part with it completely, so offers to sell Hassan a 3/4 share for 75 Rials. The next day, Ali drops by for tea and expresses regret at having sold a majority interest in the bowl. Ali proposes buying back half his previous sale, but Hassan is reluctant. Now Ali has to offer 75 Rials for a 3/8 share to get the deal done. The next day, Hassan is having some FOMO,⁴ so he wants to buy back a 3/16 interest in the bowl – and again, it takes 75 Rials to get Ali to the table.

After thirty days of this very sociable pattern of trading, Ali and Hassan congratulate each other on discovering such a valuable business. Based on their last trade, in which Ali bought back a roughly 1.5 billionth interest in the bowl for 75 Rials, they valued the bowl at about 50 Billion Rials, with ownership split almost exactly 50/50, and their bank accounts exactly as they were before the very first trade.⁵

They spend the evening drinking to their success, hatching plans to sell a small interest in the bowl to passing tourists, or posting the bowl as collateral for a large loan that will give them a chance to enjoy their new-found riches.⁶ But that very night, the bowl is stolen. Down on his luck, the thief exchanges it for a bowl of rice in the bazaar of a distant city.

What Was the Bowl Worth?

Were Hassan and Ali the victims of a colossal theft which robbed them of 99.99% of their wealth? Looking at the price at which they last traded a miniscule fraction of the bowl, the answer would seem to be “yes”, in line with the modern market convention of computing value by taking the last marginal trade price and multiplying by the number of shares. But we suggest a more sensible way to think about the wealth generated by an asset is as the present value of the future consumption it can support. Hassan and Ali may have been able to sell a few fractional shares to unsuspecting tourists, or hoodwink a hapless loan officer into giving them a small loan against the bowl, but it’s clear there’s no way this bowl could support 50 billion Rials of future consumption or anywhere close to it. The true wealth stolen from them was a lot closer to one cup of rice.

Let’s contrast this to an asset like the common stock of General Motors, currently worth about $50 billion. If every holder of GM decided to sell simultaneously, they might receive less than $50 billion, or they might receive more (controlling stakes often trade at a premium) – but the ownership of an interest in GM is sufficiently deep and broad-based that $50 billion is a reasonable estimate for the wealth that would be lost by stockholders in aggregate if the GM stock price suddenly went to zero.

There’s a continuum between assets like GM’s common stock and Hassan and Ali’s bowl. Your evaluation of how much wealth was created and destroyed by the rise and fall of Terra and Luna will depend on where along this continuum these assets lie.

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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.
    

   Thanks to John Karubian, Vildana Hajric and Anna Wroblewska for their helpful comments.
   

2. Terra and Luna were “worth” about $20 billion and $40 billion when they were changing hands at their highest prices, in early April 2022. See Matt Levine’s excellent post from May 11^th, 2022 for more on specifics: Terra Flops.
   
3. We talked about this parable before, in a slightly different form in 2017: A Brainteaser Double-Feature for the Holidays
   
4. FOMO = Fear of Missing Out.
   
5. The final share of the bowl traded was 3 / 2³¹ , and the cup’s exact “value” using the last trade price was 75 /(3 / 2³¹) , or 53,687,091,200 Rials.
   
6. Or selling some of the bowl to yield-seekers by offering to pay a 20% rate of interest on their bowl investment – in kind, of course.
   

Victor was recently featured on MoneyTree’s podcast, which focuses on providing education and information for investors. The following is an excerpt from his conversation with host Kirk Chisholm. You can also listen to the audio on Spotify or via MoneyTree (Victor is on for the first 35 minutes or so).

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Kirk Chisholm: Victor, tell us a little bit about your background, you know, how you got to where you are today.

Victor Haghani: I started off straight out of university going to work for Salomon Brothers in New York in 1984, working in fixed income research, the bond portfolio analysis group, and eventually wound up on the government arbitrage desk and working for John Meriwether. And when John went to set up LTCM, I followed him and was a founding partner there, and moved with my wife to London where we lived and raised our family until a few years ago. In 2001, I decided to take a long sabbatical from finance for about 10 years.

And I realized that I was just not doing a very good job of managing my family’s wealth. I tried out a couple of different approaches. First I tried to do what I saw a lot of friends who I respected doing, which was to try to beat the market, investing in hedge funds and private equity and angel deals in VC and various one-off trades, and so on.

And I just felt that it wasn’t for me. It was too much work. It was very tax inefficient. I think that the risk/return wasn’t even that great. So, I was attracted back to the basics of wanting to be invested in a very diversified, long-only portfolio. I started to move my family’s savings into a portfolio of low cost, broad market index funds. As I was doing that, I realized that there were a few questions that anybody moving in that direction needs to answer for themselves.

Questions such as, what should my allocation to equities be? What should my allocation to non-US equities be? Which different buckets make sense to have? Should I choose a static allocation and stick with that forever? Or should I change the allocation in some way as circumstances change?

As I thought about those questions for myself and my family through discussing it with my friends, I started to realize that other people wanted to move in this direction too. But they didn’t really want to spend a lot of time implementing the sort of framework of managing a portfolio of low-cost ETFs and index funds and doing tax loss harvesting and the like, so a business was born. It started off with my family as the first client, then a dozen or so friends invested. This was back in early 2012.

KC: You mentioned LTCM, and your time with John Meriwether. Maybe you can talk a little bit about your experience there and what you learned from that experience.

VH: I think that for me personally, what came out of it was around the question of what decisions did I personally make, and how did I think about them? The biggest decision for me and my family was how much of my family’s savings to invest in the Fund we were managing. I owned a partnership interest in the management company, but how much did I want to invest in the Fund?

As you know, we had a bad outcome, but it seemed like a really good investment–it was a reasonable position to believe that the expected return of the fund was maybe 10% to 15% and the annual standard deviation was 15%. Something like a sharp ratio of 1.0 or 0.75 seemed pretty reasonable as an expectation for what we were doing. Investing in equities, maybe that has a Sharpe Ratio of just 0.3 or 0.4. On top of that, we didn’t have to pay an incentive fee.

So here’s this great thing. How much should I have invested in it? How should I have thought about that? One possible answer is, well, just invest as much as you can because it’s so good. The more that you invest, the higher is your expected wealth. If you could get leverage, that would lead you to a higher amount of expected wealth in the future.

But what came out of the experience for me was that expected wealth really wasn’t the thing that I should have been trying to optimize. I should have been trying to optimize for some sort of risk-adjusted or certainty-equivalent wealth, in economic speak. I should have been trying to maximize expected utility, which is taking account of the fact that the marginal benefit of more and more money to me was decreasing. And if I lost money, it was going to be increasingly painful.

If I had really thought about it more like that at the time, I would’ve invested less.

I think that when you find a good investment, that’s an important part of successful investing, but it’s possible to own too much of it — even if it ultimately turns out to be a great investment. With LTCM, within about 12 to 18 months after October of ‘98, most of the positions worked out well. Most of them, not all of them, ultimately performed as we expected when we put them on.

So, if you find a really good investment but you own too much of it, you can actually go bankrupt. It’s the same with equity markets: If you’re sitting there in the year 1900, and you know that equities are going to give you a 10% return for the next 120 years, well, that’s great.

What should you do? What if you could get five times leverage on that? And you think, well, that sounds great – I’m going to have all the money in the world after 120 years, or my great-grandchildren will. But no, with five times leverage you would’ve been wiped out several times. So that’s not the right strategy.

KC: So how do you look at position sizing?

VH: I think that the approach to take is to start by calibrating your personal level of risk aversion. And there are a lot of different ways of doing it—let’s say you’ve done that.

A basic principle is don’t take risks that you’re not getting compensated for. Given that it’s nearly costless to own low cost, broad index funds and ETFs from Vanguard or iShares, why not avail yourself of that? Why take idiosyncratic risk from being in a concentrated portfolio if there’s a good chance that you’re probably not getting compensated for it?

Risk is a tangible cost to me, just as much as paying fees. If I have to take more risk, that’s like actual money out the door. It’s not quite the equivalent, I’m not actually paying anybody for the risk I’m taking, but I should almost have a little bank account when I take risk. I should have to put money into it for taking the risk and evaluate the things that I’m doing with the assumption I have to pay for taking risk.

KC: You mentioned asymmetric risk. It’s a topic that the experts don’t talk about enough, but yet it’s kind of the baseline for a lot of investing, certainly on an institutional level or at your level.

VH: It’s reasonable to have a preference for positively asymmetric payoffs versus negatively asymmetric ones. A positively asymmetric payoff pattern is one where there’s a high probability of a small loss and a low probability of a large gain, like buying a lottery ticket or out of the money call option on a risky asset. Given a Sharpe Ratio for that, if you have another trade with the same Sharpe Ratio but it has a small probability of a big loss and a large probability of a small gain, we prefer the former. I think one of the things that you want to try to do is to stay away from catastrophic downside risks.

So that’s the starting point, that it is reasonable and sensible to have a preference for positively asymmetric investments compared to negatively asymmetric ones.

Seems kind of obvious, but many people are tempted to buy a bond if it has a promised extra return of 4% or 5% because people perceive the credit as kind of shaky, but it’s very likely they’re going to pay it back. You’re getting this extra 4% return because they might not, and if they don’t, you’re going to lose a lot of money. But the attraction of the promised yield, the yield that you can almost tangibly put your hands on, leads people to take some of these negatively asymmetric tail bets that are really a problem for wealth accretion.

That brings us into talking about options. They can be helpful sometimes in trying to cut off negative tails or trying to create positive tails, but for individual investors, I think they’re really hard to analyze.

At the end of the day, the options market is a zero-sum activity. You know that everybody who buys a call option is buying it from somebody who’s selling them a call option, same for puts and so on. I think that the idea that you’re getting in on something good with options is something to be a little bit skeptical about. There are times and places where protecting your positions with out of the money put options, or trying to get some positive asymmetry into your portfolio with call options, can make sense. And often it’s a lot better than other alternatives. I’d much rather see somebody buy out of the money calls on the stock market than buy two week call options on Tesla, or even to just buy Tesla for that matter. I’d rather see somebody at least get the diversification of the stock market through the options mechanism, then have a concentrated position in a couple of highly volatile stocks.

In general, I think it makes sense to just have an amount of equities that you’re comfortable with rather than having more than that and then trying to buy and roll put options on the portfolio. It feels more comfortable to me to do that.

KC: When you’re looking for improved asset allocation, how are you doing that? Because everyone looks at asset allocation as a good risk management tool, but how are you doing that differently so that you can say this is a better version of what other people are doing?

VH: As far as we know, outside of Elm there’s no purely algorithmic, dynamic, low-cost asset allocation product available for individual investors out there. If you would use a Betterment or WealthFront, they’ll put you into a static portfolio that they feel meets your risk tolerance and your goals and so on, but they don’t change that as market conditions are changing.

Our view, and it’s really a pretty standard in the research and academic communities, is that the expected return of the stock market changes over time. The earnings yield is a reasonable predictor of the long term, real return of a broad stock market. And the decision that we need to make is how much to put in the stock market and how much to not have in the stock market.

That’s the main decision that I’m making with my family’s savings, and our clients’ savings. I’m not worried about a lot of other asset classes. The main ones are stocks and a safe asset. Now for me, the safe asset really is TIPS. I want to take my wealth and spend it for the rest of my life and give some to my kids and have them spend it for their lives and so on. I want a real, inflation-protected stream of consumption over my life. And so actually I find that long-term TIPS are a safer asset for me than owning Treasury Bills. You’ve lost 25% percent of your purchasing power by being in Treasury Bills over the last 10 years or so. And in the 1940s into the early ‘50s Treasury Bills lost 35% to 40% of their purchasing power.

That’s the same as just losing your money. There’s no difference between that and just losing the money and then keeping up with inflation. At Elm, we say, okay, let’s look at earnings yield minus the yield on TIPS and we’re going to increase or decrease our allocation to equities based on that.

We also care about risk. So, the riskier the stock market is, the less we want to own. The less risky it is, the more we want to own. We need some sort of metric for the riskiness of the market. What we decided to use is a momentum metric: the one year moving average of the market versus today’s level.

That’s a decent, simple proxy for risk. When momentum is negative, generally the market’s riskier and we want to reduce positions, and vice versa. So, for example, since October of 2021 we’ve been reducing our allocation to equities as momentum in more and more of the global equity markets has gone negative. Almost all major risk assets are in this negative momentum territory. So we want to be underweight for that, but we also want to take account of the long term earnings yield minus TIPS rate, which is pretty attractive right now.

That means that altogether, we’re underweight versus our baselines by about 15%. We’re 60% in equities right now, which is this balance between the markets being sort of risky because momentum is negative on the one hand, and the earnings yield minus the TIPS rate on the global equity market being 6% or 5% right now, which is a lot of extra expected return relative to the safe asset.

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For more of the conversation, please listen on Spotify or via MoneyTree (Victor’s segment runs for about the first 35 minutes).

By Victor Haghani and James White ¹

Investors can be forgiven for feeling like 2022 has been a pretty bad year so far. Interest rates and consumer prices have spiked up, and stock prices are sharply down.

But, in terms of what really matters, many investors are better off than they were at the end of 2021. We’re not going to argue that having less wealth is actually good for you,² and it’s not a case of “seeing the cup as half full rather than half empty” either. Rather, it’s about measuring what’s in the cup correctly.³

We can blame our financial institutions and media for leading us astray when it comes to measuring our financial well-being. At some point over the past 150 years, we decided to measure our financial condition in terms of the current, present value of our wealth, rather than as the annual flow of income it can provide. Our forebears knew that what matters is not wealth per se, but rather how much we are able to spend on our needs, wants and gifts to others over the years of our lives.

Any time we want to, we can log on to our financial accounts and get the up-to-date value of all our investments, and how much they’ve gone up or down to the nearest 0.01%⁴ – but there is simply no easy way to get a periodic mark-to-market of our wealth expressed as an annual real income stream for the rest of our lives. There’s a little more math involved, and a few personal details and market assumptions are needed too, but any financial institution that serves us could easily do the necessary calculation.

Measuring wealth in real income stream terms would look like what you see in the table below, assuming a sixty-year old single woman with five million dollars invested in a diversified stock and bond portfolio. In practice, the discount rate we should use for calculating the real income stream equivalent of her wealth would be the risk-adjusted expected return on her portfolio, but for simplicity we will use the relevant long-term TIPS real yield (see footnote for more details):⁵

Real Income Stream Equivalent of Wealth (End of 2021 to End of May 2022)

Because of the 1.18% increase in the long-term real interest rate, the real income stream equivalent of her wealth actually increased by 4.8% over the past four months, even though the real present value of her wealth declined by 12.8%. She is better off, and even though it’s just by a tiny amount in absolute terms, she’s hugely better off relative to the brokerage account headline figure of -9.8%. There is no financial legerdemain or alchemy going on here – just household finance done as it should be.

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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. Although we’ve seen a few investors that seem to be acting as if it is.
   
3. It’s a topic we’ve written about several times before, most recently two years ago in “Back to the Future: Reviving a 19th Century Perspective on Financial Well-Being.”
   
4. For liquid investments, at least.
   
5. Assumptions: 60% allocated to a broadly diversified global stock market ETF (Vanguard’s VT), 25% to a broad bond index fund (Vanguard’s BND) and 15% to short-term bank deposits (State Street’s BIL).
    

   Other assumptions: no taxes, CPI represents the investor’s consumption basket, no bequest desire, no longevity risk, the investor wants a level real income stream. The 30-Year Real Annuity Rate is approximated as the average of the 10-year and 30-year US TIPS yields. A fuller analysis would use the real expected risk-adjusted return of the investor’s portfolio, which would depend on the investor’s estimates of the expected return and risk of each asset class at the start and end of the period, and the investor’s personal level of risk-aversion.

    

   Our estimate of the result from such an analysis is that the real income stream value of the investor’s wealth would have increased by about 2%, rather than the 4.8% increase we see based on using only the risk-free real rate, but still much less than the 12.8% decline in the real present value of wealth.
   

By Victor Haghani and James White ¹

Judging by retail investor stock option trading volumes, the answer would seem to be an emphatic and resounding “YES!” For the first time since the introduction of listed equity options in 1973, the average daily notional trading volume of options on individual stocks has matched that of the underlying stocks themselves – at about $450 billion per day – with retail investors reckoned to account for about a quarter of this activity. On top of this is another $1 trillion per day in trading in options on stock indexes, plus further volume in non-exchange traded activity in structured products and exotic options.²

In our article just published in the Journal of Derivatives, written with our friend and Elm investor Vlad Ragulin, we ask whether options make sense for a broad class of investors. The Base-Case investor that we use throughout our analysis is risk-averse, with preferences modeled on a typical Elm investor – indeed, pretty close to those held by Vic, James and Vlad.³

Of course, one perspective on whether all this options trading makes sense is to argue that anything people do voluntarily must be in accord with their preferences and therefore is just fine. This has become a popular line of reasoning, but would have resulted in one of the shortest research articles ever.⁴ Instead, we take a less reductive approach, and start by defining what it means for options to “make sense” for our Base-Case investor. The use of options can have a complex impact on an investor’s portfolio – in practice, they can transform the expected return, risk and shape of the distributions of outcomes. As we’ve discussed in many previous articles, Expected Utility is the natural metric for making choices between different combinations of risk and return, and it can handle ranking arbitrary shapes of return distribution as well. So we’ll say that a particular options strategy warrants a place in an individual’s portfolio if it increases his Expected Utility.

Here are our main findings:

• Under classic textbook assumptions, fairly priced options don’t add Expected Utility, and therefore don’t make sense for our Base-Case investor.⁵
• If the investor cannot rebalance his portfolio frequently and/or the risky asset is prone to discontinuous price jumps or high transaction costs, then options can add Expected Utility by effectively replacing the continuous rebalancing that the investor is unable to do directly. However, the benefit from this use of options is very, very small in typical cases. This situation would most commonly call for selling a small amount of options, as that’s the correct options position for rebalancing an unlevered risky asset to a constant portfolio weight.
• When we relax the other textbook assumptions one by one, we find that fair options still don’t add much value for an investor who is willing to periodically rebalance his portfolio to account for changes in the investing environment, such as the return and risk profile of risky assets, and his current degree of risk-aversion.
• Turning to the use of far out-of-the-money put options as portfolio insurance, we compare buying insurance on your portfolio to buying insurance on your home. We conclude that if the probability of the risky part of your portfolio “burning to the ground” is at least as likely as your home going up in smoke, then buying options for protection might make sense, depending on how they are priced. However, it’s harder to know the true odds of a massive stock market decline than of estimating the risk to your house, and also in practice it’s difficult to buy portfolio protection that only covers a huge drop in value without also paying – and perhaps overpaying – for protection against smaller declines as well.
• At first it may seem strange to find a bad investment, like an options strategy which doesn’t add Expected Utility, and also find that the opposite of it is also bad. If we only consider the expected return, we can’t find such a case.⁶ But when we bring risk into the picture, and also require the investment to come in some quantum of size, then we can find many such cases. In the context of options strategies, we find that two popular and opposing strategies are both “bad” from a historical point of view, and are unlikely to make sense prospectively too, due to their both being very inefficient in the dimension of time-diversification. The two strategies are:
  • selling and rolling one-month put options on the stock market, keeping the rest of the portfolio in T-bills, and
  • buying and rolling one month put options on the stock market and keeping the balance of the portfolio invested in the stock market.
  The two strategies can be thought of as opposites in terms of their options exposure, and both have delivered a lower return and lower quality of return–in terms of Sharpe Ratio and maximum drawdowns – compared to a simple portfolio of 50% in stocks and 50% in T-bills over the past 30 years.
• Volatility should not be considered an asset class, as options are a zero-sum game. If it were an asset class, which side of the market would it be?
• The volatility skew⁷ is probably not something investors can materially benefit from. It’s devilishly difficult to determine the fair value for options, particularly for short-term and/or far out-of-the-money options.
• There’s an argument made that many young investors should use equity call options to have leveraged exposure that takes account of being in a state of low financial capital and high human capital.⁸ We’re not convinced there’s a big enough Expected Utility gain to make this strategy worthwhile in practice, but we accept that it’s plausible and in the right direction.
• When an investor trades an option on an individual stock, he engages in two zero-sum games at once: stock picking and options trading. Thus, any benefit would require either winning these zero-sum games or achieving a more efficient risk profile.
• Many investors view far out-of-the-money options as lottery tickets. For example, on many days in late 2021, Tesla options alone represented about one third of options volume, and about 80% of the Tesla options volume was on short-dated, far out-of-the-money call options. It may be that these options are a better deal, dollar for dollar, than a lottery ticket, but our Base-Case investor would be better off spending $10 a week on lottery tickets than spending $50,000 a year on far out-of-the-money options. Americans spend about $70 billion each year on lottery tickets. We think retail investors are spending multiples of that figure on options that are acting as a surrogate for lottery tickets or trips to Vegas.
• Investors should stay away from structured volatility notes, volatility ETFs, and exotic options.

While literally hundreds of books proffer instruction on how to make money trading options,⁹ there is not a single mention of options in many books on personal finance, including those written by such well-regarded experts as Jack Bogle, Charles Ellis, and Burton Malkiel. Our article is an attempt to bridge this gap, and in sum we find that options are unlikely to be welfare-enhancing, let alone a panacea, for individual investors with risk preferences similar to those of our Base-Case investor. This should not come as a surprise, given the zero-sum nature of the options market, in contrast to the generally recognized positive sum activity of investing in the broad stock market.

If you’re interested in diving more deeply into our analysis and conclusions on whether options warrant a place in individual investors’ portfolios, or more specifically whether they make sense in your personal circumstances, we’d love to hear from you.

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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. And all of this is just in the US, not counting non-US equity options volumes.
   
3. We assume the investor has CRRA utility with a standard risk-aversion coefficient. If such an investor is a utility-maximizer, then for bets with the same expected return and variance they will have a preference for positively-skewed bets (high chance of a small losee, low chance of a large gain) over negatively-skewed bets.
   
4. No article can beat Dennis Upper’s “The Unsuccessful Self-Treatment of a Case of ‘Writer’s Block'” in the Journal of Applied Behavioral Analysis (1974) which contained zero words!
   
5. The main assumptions are of a risky asset (or portfolio of assets) following geometric brownian motion, and an investor with CRRA Utility who is able to continuously rebalance without transactions costs.
   
6. Ignoring transactions costs and taxes. By Expected Return we mean the Expected Arithmetic (not compound) Return.
   
7. Low strike options trading at a much higher implied volatility than high strike options.
   
8. Most prominently by Ayres and Nalebuff in Lifecycle Investing: A New, Safe, and Audacious Way to Improve the Performance of Your Retirement Portfolio (2010).
   
9. Just type “options trading” into the Amazon.com search window.
   

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

In general, the more optimistic we are on the prospects of an investment, the more of it we’ll want to own. However, at extreme levels of bullishness, the normal relationship can be turned on its head and it can make sense to own less of an asset the more we like it. It’s hard to think of everyday examples that work like this, so this poses something of a puzzle. It’s a problem which gets little attention in mainstream finance, because we rarely witness high enough forecasts of investment quality to observe this effect in the wild.² We can thank the digital-asset revolution and its band of optimistic crypto-asset enthusiasts for bringing this problem into focus, which also gives us some valuable insights that apply in other domains of financial decision-making.

Risk-Aversion

We’ll assume a representative Investor with a relatively moderate level of risk-aversion.³ For those familiar with the Kelly Criterion, we’re assuming an investor with twice the risk-aversion of a Kelly Bettor.⁴ Many individuals find that while “full” Kelly may be appropriate for cash management in blackjack or poker, it calls for too much risk when applied to total wealth. In gambling circles, our Base-Case investor would be called a “half-Kelly bettor”, and would find himself in the company of quite a few famous investors who also size their risks in line with this level of risk-aversion. More concretely, such an investor would be indifferent to a 50/50 coin flip which could increase his wealth by 50% or decrease it by 25%.⁵

Crypto Risk and Return

It is hard to know how to accurately model the price behavior of digital assets, but most would agree that they do not follow the continuous Geometric Brownian motion of finance textbooks, and their returns are far from being well-described by the standard Normal distribution. Financial assets in general (and digital assets in particular) experience price jumps and fat tails, and their variability and expected returns are difficult to estimate and can change dramatically over time. Additionally in the case of digital assets, many investors recognize some risk of losing their investment through hacking, hard forks, loss of private keys, etc – more like seeing one’s house burn down rather than experiencing a bad run in the market. To cut through all the uncertainty of crypto return distributions, we’re going to radically simplify and assume that to some chosen horizon there’s a 50% chance the asset goes to 0, and a 50% chance it goes up by a factor of P, the “payoff ratio”.

A Puzzling Result

The chart below shows the optimal amount our Investor should allocate given a range of payoff ratios.⁶ As we increase the payoff ratio from 1:1 to 2048:1, the expected return and risk of the asset goes up – and crucially, the ratio of return-to-risk, the Sharpe Ratio⁷, is also going up. That’s what we mean when we say that the quality of the investment is getting better and better.

When the expected payoff ratio is between 4:1 and 8:1, the optimal allocation reaches its peak at about 17%. Then – and this is the whole point of this note – at higher payoff ratios, the optimal bet size declines. For very optimistic investors (and believe us, there are some really optimistic ones out there⁸), an allocation potentially well under 10% may be optimal.

Explaining the Hump

An intuitive explanation for this hump is this:

• It starts at zero: At the 1:1 ratio at the far left of the chart, since the upside and downside are equally likely and equal in size, the optimal allocation starts at zero since the investor shouldn’t take risk without some expected reward.
• Then it goes up: As we increase the payoff ratio by moving to the right, the investment opportunity warrants some allocation of capital.
• Then it eventually goes back down to zero again: At some point (rarely seen in practice), when an asset is sufficiently great, holding even a small amount of it will make us fabulously rich in the good outcome – so there’s no reason to hold a ton of it and risk the bad outcome on a large fraction of wealth. Put another way, since we can get as much cake as we can ever eat for a pittance, why spend a penny more?! This effect takes the optimal allocation back down again towards zero, giving us the hump shape.

While the circumstances that give us this result are something of an oddity, a closer look at what’s going on gives us insights about wealth and risk that are more broadly applicable.

A Different Perspective on Wealth and Risk

Normally, we think of our financial wealth as the sum of the current market values of the assets we own.⁹ This works well enough in most circumstances, but when an investor feels he has come across an outstanding gem of an investment, using the market price for that asset rather than a value that reflects the higher intrinsic risk-adjusted value can lead to suboptimal decision-making. Encountering an asset – a digital asset, in this case – that the investor believes has a 50/50 chance of delivering a 100:1 payoff ratio makes him effectively wealthier than a standard mark-to-market accounting treatment would indicate. If he really believes in the attractiveness of the investment, he would need a substantial compensating payment to forego investing in it. We use the term Certainty-Equivalent Wealth to refer to the level of wealth that he would be equally happy to possess with absolute certainty instead of his current wealth plus the opportunity to invest in the highly-attractive asset.

To illustrate, say our investor decides to invest 17% of his wealth in a crypto-asset that has a 50/50 chance of going up eight-fold or down to zero. The expected return of the asset is 300%, and the investor’s expected wealth is 150% higher than his starting wealth – but his Certainty-Equivalent Wealth is that level of wealth that he’d be just as happy having for sure but without the ability to buy the attractive crypto-asset. Using our Base-Case investor’s preferences, we can calculate that his Certainty-Equivalent Wealth should be about 125% of starting wealth.

We now turn to the question of risk: how much risk is the investor taking in the above illustration? Normally, one might say that his downside is losing 17% of starting wealth, and that is his risk – but that ignores that his “true” wealth, his Certainty-Equivalent Wealth, is 125% of his nominal starting wealth because he bought this great investment that sports a 300% expected return. If that’s his relevant starting wealth, now we see that his downside is much higher, at 34% (the loss from his wealth going from 1.25 to 0.83) if the asset’s downside case is realized. Just like we’d expect, as the investment gets more attractive the investor is indeed taking more downside risk, measured against his Certainty-Equivalent Wealth.

Conclusion

“All models are wrong, but some are useful.”
  – George E.P. Box

The assumptions we’ve made in this analysis have been chosen for simplicity and illustration. In particular, we recognize that modeling outcomes of digital assets in a binary manner is not realistic, though it does capture a certain kind of view that digital assets will either go to the moon or fade away. But the general effect we describe, that the optimal holding of an asset at some point goes down as its attractiveness goes up, holds over a pretty broad range of assumptions about distributions of outcomes and investor risk preferences.¹⁰

Moreover, the insight that we should view our base wealth as inclusive of the value provided by attractive investment opportunities is important and applies in other situations. For example, a hedge fund or private equity manager might be prone to over-investing in the fund he manages if he underestimates the value and risk associated with his ownership interest in his fund management business.

If you think you have an investment opportunity that might benefit from this kind of analysis, we’d love to discuss it with you.

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Further Reading and References

• Black, Fischer. “Noise”. Journal of Finance (1986).
• Haghani, Victor and James White. “Measuring the Fabric of Felicity”. SSRN (2018).
• Kelly, John, L. “A New Interpretation of Information Rate”. Bell System Technical Journal (1956).
• Merton, Robert, C. “Lifetime Portfolio Selection under Uncertainty: the Continuous-Time Case”. The Review of Economics and Statistics (1969).
• Thorp, Edward, O. “Fortune’s Formula: The Game of Blackjack”. American Mathematical Society (1961).

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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. Our thanks to Steve Mobbs for his input on this note. Past returns are not indicative of future performance.
   
2. For example, in his 1986 Presidential address to the AFA titled “Noise”, Fischer Black stated:
   “…We might define an efficient market as one in which price is within a factor of 2 of value, i.e., the price is more than half of value and less than twice value…By this definition, I think almost all markets are efficient almost all of the time…”

3. As per a survey we conducted in 2018 of thirty financially sophisticated investors and described in “Measuring the Fabric of Felicity”, we assume our Base-Case investor is in the bottom quartile of risk-aversion.
   
4. We assume the investor exhibits Constant Relative Risk Aversion (CRRA), with a coefficient of risk aversion, γ , of 2. CRRA Utility can be written as:
    

     U(W) = (1 – W^{(1 – γ)}) / (γ – 1) for γ <> 1 , and
     U(W) = ln(W) for γ = 1 . γ = 1 gives us the Kelly Criterion,

    

   and γ = 2 is referred to as “half-Kelly”.
   

5. Feel free to get in touch with us if you’d like some guidance on calibrating your personal level of risk-aversion.
   
6. The optimal allocation k^* is that which maximizes the investor’s Expected Utility. Given a payoff ratio P upside probability π and Constant Relative Risk-Aversion coefficient g ,
     k^* = (P – π P)^1/g – π^{1 / g} (P – π P)^{1 / g} + π^{1 / g} P

   We are assuming this is the only investment opportunity available.
   

7. Technically, the ratio of expected return to standard deviation, assuming a 0 risk-free rate.
   
8. One example of a highly optimistic investor is Michael Saylor, CEO and founder of MicroStrategy (MSTR), an owner of several billion dollars of digital-assets. On August 18th, 2021, he stated here that he expected Bitcoin to become “a $100 Trillion asset.” Given the price at which Bitcoin was trading at the time of his statement, it implied a hundred-and-twenty-fold increase in the price of Bitcoin.
   
9. Investors in private, illiquid assets often think of them at their cost basis, or somewhere between cost and an estimate of fair market value.
   
10. But not all whales are of the humpback variety. Two notable cases where it may not hold are: 1) if the asset follows a continuous random walk and the investor can rebalance his portfolio continuously without frictions, or 2) for an investor with risk-tolerance equal to or greater than a full Kelly bettor, i.e. whose utility function is at least as risk-tolerant as U(W) = ln(W) .
    

By James White, Mark Haghani and Victor Haghani ¹

Introduction

Scott Fawcett’s “Decade” system of golf decision-making is revolutionizing golf strategy, and we think Scott’s approach to golf has a lot of lessons for good investing too. You may have read about him in this Golf Digest piece, or heard him on Larry Bernstein’s What Happens Next show, or the Wharton Moneyball sports analytics podcast. He’s the golf guru that US Open winner Bryson DeChambeau credits with much of his success, as does 2021 rookie of the year favorite Will Zalatoris, who Scott caddied and coached to multiple amateur titles. Scott’s approach to golf is viewed as so valuable that the NCAA forbids university golf teams to invite Scott to give team seminars, because they feel it bestows an unfair advantage! Scott acknowledges he didn’t invent the approach he advocates – winning golfers have always intuitively played this way and the seminal work of Mark Broadie introduced the metrics and optimization approach on which Scott’s Decade system rests – but he was the first to systematize it and teach it in a form that golfers could use in practice. He calls his system “Decade” because it’s meant to save its users the ten years that an attentive and intelligent golfer would take to arrive organically at the same understanding.

Shot Selection Under Uncertainty

At the center of Scott’s approach are two core ideas: first, our decision-making needs to totally embrace uncertainty. We cannot make good golf decisions by focusing on the shot we want to hit, even if it’s the shot with the single most likely outcome, nor can we make a sound decision by focusing on avoiding the single worst outcome. The reality is that, even for a top pro golfer, there’s a wide range of possible outcomes for any given shot. We need to take account of all the places the ball can wind up after our shot, and their respective probabilities. In the chart below, we reproduce one of Scott’s most powerful exhibits. It’s the shot pattern of a scratch golfer (one of your authors, in fact), hitting 70 balls with a 7-iron at the target shown with a red star at a distance of 180 yards.

Shot Dispersion Pattern: Mark Hitting 70 Shots with a 7-iron, Targeting Red Star

Notice that not one of the shots finished on the target. When even a very good golfer stands over the ball, he needs to know it isn’t going to wind up where he wants it to go, not by a long shot! The standard deviation of this shot pattern is 6.2 yards left-to-right, and 3.2 yards short-to-long. Notice also the oblong pattern, indicating that a right-handed golfer tends to hit long balls to the left and short balls to the right, because the ball goes further when struck with a more closed club face. A golfer can get a shot dispersion pattern for each club, and each situation – fairway, rough or sand, wind or calm – which will provide a probability map of all the places where the ball can go.

The second pillar of the approach is using a more sophisticated way of evaluating where the ball winds up than just measuring how close it is to the hole. Scott’s system relies on research done by Columbia Professor (and very good golfer) Mark Broadie. Mark is the inventor of the “Strokes Gained” method of measuring the value of each shot, which he shared with the golf world in his book Every Shot Counts (2014). He calculated, based on millions of actual shots hit by professional golfers, the average number of shots required to get the ball in the hole from just about every conceivable position on a golf course. For example, he calculated that a ball on the green, eight feet from the hole, takes 1.50 strokes on average to get in the hole – roughly even odds of a one-putt and a two-putt. Or, from the fairway 180 yards from the hole, it takes professionals 3.08 shots to get the ball in the hole.² You’ll find a table showing these, and many more examples, in Appendix II at the end of this note.

Strokes Gained accounting provides a simple and powerful way to put a value on each possible outcome, which is much superior than using a system that is based solely on minimizing your distance from the hole after your shot. For each possible shot outcome from our current position, we just need to take the difference between the number of shots expected to finish the hole from the starting and ending position of our ball. So, if we’re 180 yards from the hole in the fairway, and we hit the ball 8 feet from the hole, we know the impact of that excellent shot was positive 0.58 strokes gained. We improved our position by 1.58 strokes, from 3.08 shots to finish the hole from 180 yards on the fairway to 1.50 shots to finish the hole when we’re on the green eight feet from the hole. It took us just one shot to improve our position by those 1.58 strokes, so this shot was worth 0.58 strokes gained.

The chart below shows the expected strokes to finish a hole for shots landing in various positions on and around the iconic green of the 18th hole at Pebble Beach.³ The green slopes down back to front, and left to right – so, if the ball winds up in the sand trap on the pin-side of the green, the expected number of shots to finish is 2.85, as it would be so difficult to get the ball to stop close enough to the hole to putt it in on the next shot. If the ball lands on the beach or in the ocean to the left, we count that as a 4.0, because a penalty of one stroke is assessed and the golfer will still be in a position from which he’ll need about 3 more shots to get the ball in the hole.⁴

18th Green, Pebble Beach.
Numbers in Green Represent Expected Number of Shots to Finish the Hole From Each Position

Now that we have a way of assigning a value to each possible shot outcome, we can calculate the value of shooting for any target, taking full account of the fact that there’s going to be quite a bit of dispersion in our shot outcome. We should choose the target that will leave us with the lowest expected number of strokes to finish the hole, which is equivalent to saying we should choose the target that gives us the highest expected strokes gained. The three illustrations below compare the expected outcomes for three possible targets that a golfer could select from 180 yards in the fairway.

Target = Pin.
Expected Shots to Finish Hole = 2.01.
Assumes Mark’s Shot Dispersion from Roughly 180 Yards Away in the Fairway

Target = Center of Green.
Expected Shots = 2.03.
Assumes Mark’s Shot Dispersion from Roughly 180 Yards Away in the Fairway

Target = Scott Fawcett Suggested Target.
Expected Shots to Finish Hole = 1.92.
Assumes Mark’s Shot Dispersion from Roughly 180 Yards Away in the Fairway

Choosing the pin as the target would result in the shot pattern represented in the first panel above. Notice that there are lots of balls that finish close to the hole, but that two balls would be on the beach or the ocean, and eleven balls would be in the sand trap left of the green – a tough spot to be in. Averaging over the 70 shot outcomes of Mark’s shots gives us 2.01 expected shots to finish the hole, using the pin as a target. While this is the target which would minimize the expected final distance of the ball from the hole, most seasoned golfers would recognize that it isn’t the optimal target. In the second panel above, we show the shot pattern if the golfer aimed at the dead center of the green, a target which has conventionally been considered the ‘smart’ and conservative shot. Here, the expected shots to finish the hole is 2.03, which is a little bit worse than aiming for the flag. Finally, we show the expected outcome choosing the target that Scott Fawcett’s system would recommend, which is a few yards to the right of and below the pin. Here we get 1.92 expected shots to finish the hole, a pickup of 0.09 expected strokes versus aiming at the flag.

An improvement of 0.09 shots may seem like a small amount of improvement from good decision-making – but picking up 0.09 strokes per shot on, say, 20 interesting shot situations per round can add 7 shots in a four-round tournament (0.09 x 20 x 4 = 7.2 shots), and can make the difference between a Top Ten finish versus middle of the pack.⁵

Of course, most golfers don’t carry strokes-gained tables, shot dispersion patterns, and computers around with them for making detailed expected-strokes-gained calculations for each possible shot target. This is where Fawcett’s Decade system comes in, providing decision-making heuristics players can follow in real-time on the course.

Scott the Financial Advisor

Scott’s insistence on taking uncertainty into account in making good decisions applies equally to golf and investing. We cannot reach good decisions by focusing on the base-case return of an investment. Instead, we need to weigh up all possible investment outcomes, the cost or benefit of each one to us, and the probability of each. And, just as we need the “Strokes Gained” accounting system to evaluate each possible shot outcome, we need to evaluate different monetary investment outcomes by the change they bring to our welfare – or Utility, as it’s called in economic theory.

Just as Scott tells us not to make golf decisions based on minimizing the expected distance from the hole after each shot, or alternatively, maximizing the number of birdies per round, so too in investing it’s important to choose the appropriate objective to maximize. In investing, making decisions that maximize the expected amount of money we’ll have can lead to poor, often nonsensical decisions. For example, maximizing expected wealth tells us to always take as much risk as we possibly can – using as much leverage as we can get – in any investment with a positive expected return. This is the policy which results in the highest expected wealth, but also results in a near-certain chance of going bankrupt sooner or later! The better objective is to make decisions that maximize our Expected Utility, which takes into account the fact that increasing amounts of wealth lead to smaller and smaller increases in our welfare.

In both golf and investing, we need to take account of the inherent, uncontrollable dispersion of outcomes, figure out what each outcome is worth to us, either to our golf score or to our personal welfare, and then find the decision which gives us the best expected overall outcome. Just as saving a mere 0.09 strokes on about one in four shots can add up to a very significant improvement over the course of a full golf tournament, in investing it’s also the case that small gains can add up to a big difference in outcomes. For example, a reduction of 0.75% in annual investment management fees over 40 years of saving and investing can result in 20% more savings to spend in retirement.

You might ask, “what if all golfers adopt Scott’s Decade system?” Naturally, it won’t provide the relative advantage it offers when few golfers are using it, but it will still represent the optimal approach to posting low scores. Investing, in contrast to a golf tournament, is not a zero-sum game, and so all investors can “win” from the application of good decision-making in investing. However, many of the cognitive biases that afflict golfers – the illusion of control, extrapolation bias, recency bias, overconfidence bias, and fallacy of the hot hand – also stand in the way of better investment decision-making. Scott has succeeded in giving us a system that makes golf a lot simpler, but he rightly warns us that embracing uncertainty and sticking with the program isn’t easy!

Relax and Enjoy the Ride

There’s a lot more to Scott’s system than what we’ve described here, and the deeper we dig the more of his wisdom and insights we find applicable to investing. Here are a few of Scott’s pearls of wisdom. We’ll let you decide whether they’re equally useful as applied to your investing:

• Eliminate the bogeys and birdies will take care of themselves.
• Don’t abandon your strategy just because you’ve had a run of bad outcomes.
• Keep things simple. Find the shot shape that’s natural for you, and stick with it. Hitting a good golf shot is already difficult enough without trying to put a different spin, trajectory and shape onto each shot.
• Almost never worry about how your opponents are doing. Make decisions that will lead to your best expected strokes gained outcome on each shot. It’s too confusing and complex and you don’t have enough information to strategically modify your play against your opponents.
• “The single most important thing I teach my students is expectation management.”

Players using Scott’s Decade system for reaching golf decisions report a feeling of calm and liberation from knowing they’re following a sound process and recognizing that they cannot control the individual shot outcomes. Accepting that their golf shots are more like the pattern of pellets from a shotgun, rather than a bullet from a rifle, gives them a totally different outlook and experience from a round of golf. It becomes all about making good decisions, and accepting that there’s just going to be a lot of luck – good and bad – in each round. But they know that, if they keep making good decisions, they’ll make the most of whatever technical skill they have in the long-run and be able to accept the ups and downs along the way with greater equanimity.

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Appendix I: Does It Make Sense to Give Up Expected Shots to Add Variability?

As Scott says, “The bottom-line of game theory in golf is that it is exhausting to constantly try to run iterations of finishing positions AND find situations where you can increase your scoring variance WITHOUT destroying your overall expected score.” It may seem like a good idea to accept a worse expected score in exchange for greater score variability, since money payoffs in golf tournaments are a convex function of finishing place. However, we agree with Scott that it’s unlikely to be the right thing to do in most cases. For example, a casual look at the payouts for the 2021 US Open at Torrey Pines suggests that if it cost 0.5 expected shots to generate two shots of extra variability per round, it generally would not pay to go for higher variability.

Bear in mind that the normal variability in a round of golf for a tour player is 2 to 3 strokes per round – holding course, condition and golfer form constant – so intentionally adding two shots of variability would be a pretty big deviation from optimal play. A fuller analysis would also bring risk-aversion – via the player’s Utility function – into the analysis, which will further push against the idea of accepting a worse expected score for higher variability. We agree with Scott’s advice to keep it simple in golf, and we think it applies to investing too: “…saving your energy and just sticking to the system is the optimal play for almost everyone.”

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Appendix II: Average Number of Shots a PGA Tour Player Takes to Hole Out from Various Lies and Distances from the Hole

For example, starting from the tee 400 yards from the hole, PGA Tour pros average 3.99 strokes to hole out. Starting on the green eight feet from the hole, the PGA Tour average strokes to hole out is 1.50. A recovery is an obstructed shot to the hole, e.g., a shot from behind a tree that forces a pitch back to the fairway. Reproduced with permission from Mark Broadie from his book Every Shot Counts (2014).

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Further Reading and References

• Broadie, Mark. Every Shot Counts: Using the Revolutionary Strokes Gained Approach to Improve Your Golf Performance and Strategy. Avery. 2014.
• Broadie, Mark, “Assessing Golfer Performance on the PGA TOUR,” Interfaces, Vol. 42, No. 2, 146-65. 2012.
• Rapaport, Daniel. “Has This Data Nerd Created a Golf Cheat Code?” Golf Digest. 2021.
• Scott Fawcett’s Decade Elite online program.

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1. Victor is the founder and CIO of Elm Partners and James is Elm’s CEO. Mark is pursuing a Masters degree in Data Science and is a captain of the golf team at the University of Pennsylvania.
    

   The authors thank golf strategy experts Mark Broadie and Scott Fawcett for their foundational work that made this note possible, and for their helpeful comments on this note. Thanks also to Larry Bernstein, Jonathan Garrick, Larry Hilibrand, Peter Hirsch and Cade Massey for reading a draft of this note and sharing their comments, and thanks to our Elm colleague Steven Schneider for doing such an artistic job with the exhibits. Of course, any errors rest with the authors. This note is not an offer or solicitation to invest.
   

2. Strokes Gained data for college golfers is also available, but most people use the numbers for the PGA tour players.
   
3. The pin position is from the first round of the 2019 US Open.
   
4. We’ve made a conservative approximation here in that the golfer might be able to hit a shot from the beach if he finds his ball in a good location, or depending on where his ball goes out of play, he may be in a position where his expected strokes to finish the hole could be somewhat below 3.
   
5. Also, seven shots made the difference between winning the 2021 US Open vs tying for 15^th place.
   

By Victor Haghani and James White ¹

Now, more than perhaps at any time in the past 70 years, investors are concerned that there’s no asset they can invest in which is truly risk-free. Worries of unsustainable public policies leading to debasement or default – through inflation, taxation or repudiation – have sapped confidence in the traditional safe assets of bills and bonds issued by the US and other rich countries. Investors are increasingly considering whether other assets – equities, real estate, commodities, crypto-currencies – should be used to construct an ersatz risk-free asset. In this note, we’ll address this problem with a practical definition of the ideal personalized risk-free asset, and then we’ll discuss how to construct an efficient portfolio when that ideal asset doesn’t exist in investable form.

The usual jumping-off point for thinking about portfolio construction is the Two-Asset Case, in which the investor has to decide how to allocate his wealth between a risky and a risk-free asset. The framework can be extended to handle many risky assets based on their expected return and co-variability, with the risk-free asset providing the base unit of account. In addition to the role the risk-free asset plays in portfolio choice, it is also central to optimal lifetime spending policy.²

Ultimately, the benefits of wealth come from spending it on stuff over time, either for ourselves or for others. An asset is “risk-free,” in this sense, if it allows us to lock in today a precise amount of real spending to a given horizon with no uncertainty. For wealth which is likely to be deployed philanthropically or spent by far-future generations, a reasonable choice might be an asset which makes risk-free payments indexed to a broad inflation measure, such as CPI or aggregate per capita income – but for the stuff we want to consume for ourselves and our immediate family, we need an asset indexed to our own personalized inflation rate. There are services that exist to help you calculate a personalized, backward-looking inflation rate.³ This can be a valuable exercise in giving a sense for the relationship between the inflation you expect to experience relative to broader measures such as CPI. Less personalized, but easier to come by is a standard-of-living index of families of your income bracket (see sidebar below).⁴ The expected mix and timing of your personal, inter-generational and philanthropic spending will suggest a blend of a broad inflation measure with a more personalized inflation index.⁵

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Is Per Capita Income Growth More Relevant Than CPI?

Consider a family with income (or wealth) in the top percentile of the population. Would an index of per capita income growth of the top 1% of the population be a more relevant index than national consumer price inflation or aggregate per capita income growth? For example, from 1959-2020, US CPI grew 3.5% per year, while aggregate per capita income growth was 5.9% (2.4% real growth) – and, for the top 1% of the income distribution, growth was 6.4% (2.8% real growth).⁶ Although these growth numbers are relatively close over this 60-year historical period, looking to the future, the behavior of the top end of per capita growth can be much higher or lower than the average for the whole population.

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There generally won’t be any single asset or basket of assets that will make absolutely risk-free payments based on your personalized, blended inflation index. Happily though, even when there is no perfectly risk-free asset, we can still compare different asset allocation choices in terms of the Expected Utility they generate, with the optimal portfolio weights being those that produce the highest Expected Utility. The intuition behind using Expected Utility to make decisions under uncertainty is that it gives us a general way of weighing positive and negative outcomes in terms of how they affect our welfare, or Utility, recognizing that the marginal benefit we derive from each additional dollar of wealth declines as our level of wealth increases.

For the purposes of illustration, we’ll assume an individual with an ideal, risk-free asset that is a 50/50 blend of US CPI and per capita income growth of the top 1% of the income distribution. We’ll assume that the per capita income index is expected to run 2% per annum higher than CPI with an annual tracking risk of 1.5%. We will ignore the risks of changes in taxation, taxes on higher inflation and sovereign default, although in practice all of these can have a significant impact on results.

We need to transform the expected return and risk of the assets in our opportunity set so they are all expressed in relation to our Ideal Risk-Free Asset, as illustrated in the table below. Adjusting expected returns is straightforward given our assumption that the index of our Ideal Risk-Free Asset grows at 1% above CPI. Estimating the risk of each asset as we change the benchmark against which we measure its returns is more challenging. There is simply not enough historical data to make a sharp estimate with confidence, particularly with regard to tail events. However, analyzing 125 years of US data provided by Yale Professor Robert Shiller suggests that US equities have been about 1/20^th less volatile when measured against a CPI index, and a further 1/20^th less volatile measured against a blended index of inflation and real per capita consumption.⁷ The historical data also suggest T-Bill volatility of 7% and 8% measured against CPI and our assumed ideal index, respectively. For TIPS, we assume they are not risky when measured against a CPI index, but have annual volatility of 3% measured against our assumed Ideal Risk-Free Asset.

A few things to note on expected return and risk relative to the Ideal Risk-Free Asset:

• Expected Returns are all 1% lower.
• In the rightmost pair of columns, there is no riskless asset available to invest in.
• T-Bills trade places in riskiness with TIPS when we switch the Risk-Free asset from T-Bills to a CPI-linked asset.⁸
• Equities are less risky, but not significantly so.

The table below shows the portfolio weights which maximize Expected Utility under each assumption of the risk-free asset. Notice that the optimal allocation to equities goes up quite substantially, from 62% to 79%, which is primarily a result of equities being less risky when measured against the Ideal Risk-Free Asset,⁹ and secondarily a result of the recognition that neither Bills nor TIPS are risk-free. The utility surface is quite flat in the vicinity of optimality, so in practice if actual portfolio weights are off by a bit relative to optimal weights, it isn’t a big concern in terms of overall portfolio quality.¹⁰

We also see a significant change in the real Certainty-Equivalent Return (CER) of the portfolio. This is primarily driven by the use of our personalized inflation index which we assume will run 1% above CPI, rather than by changes in the optimal weights. Switching from CPI to the assumed Personal Inflation Index decreases the real CER by by 0.8%, the main impact of which would be to lower the investor’s optimal long-term spending policy.¹¹

Conclusion

We’re unlikely to find any real-world assets which meet a reasonable definition of “risk-free” for any given investor, but it turns out that’s OK. Rather than the common practice of trying to construct a pseudo-risk-free asset from a set of available assets which don’t really fit the bill, instead we should simply treat the opportunity set as consisting entirely of risky assets and optimize the real risk-adjusted return of the risky portfolio, using our personalized inflation index as the deflator. In practice, this approach to portfolio construction in the absence of a truly risk-free asset is flexible enough to handle an arbitrarily large number of different assets and a broad range of assumptions about possible outcomes in asset prices, including discontinuous jumps as well as changing risk and correlation patterns over time.

The impact of this change in perspective will depend largely on the starting point of the investor’s asset allocation. For investors with low to moderate risk-aversion – who start off with a high allocation to equities and other patently risky assets – treating the safest assets as being risky (but still the least risky of the available options) is likely to have a modest impact on optimal portfolio weights. But for investors who exhibit a high level of risk-aversion, who are mostly allocated to the safest assets to begin with, explicitly accounting for the risk in government bills and bonds can have a significant impact on their asset allocation. And for almost all investors, regardless of their degree of risk-aversion, moving to an Ideal Risk-Free Asset more closely aligned with per capita income growth will reduce the investor’s risk-adjusted real return and thereby call for a lower long-term spending policy.

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Appendix

Here’s what a highly stylized two-asset case looks like when neither asset is risk-free.

• Asset A: the ‘primary’ risk asset with expected return 5% and volatility 18%
• Asset B: the quasi-‘safe’ asset, which is not completely safe

We assume a typical level of risk aversion for which the investor would optimally want the majority of wealth in the primary risk asset.

The blue line shows how the optimal allocation to the primary risk asset rises as the quasi-safe asset’s volatility goes up, and the grey line shows how the optimal portfolio’s certainty-equivalent return (CER) drops accordingly. The chart shows that even if the quasi-safe asset has volatility of up to 5%, it doesn’t make much difference to the asset allocation. Also, the acknowledgement that the “safe” asset isn’t so safe hardly impacts CER, from 1.94% down to 1.85%.

Here are the equations for the optimal allocations k^* :

k_A^* = μ_A – μ_B + γ(σ_B² – σ_A σ_B ρ) γ (σ_A² + σ_B² – 2σ_A σ_B ρ)

k_B^* = 1 – k_A^*

A is the primary risk asset
B is the quasi safe-asset
k^* is the optimal allocation to each asset
μ is expected return
σ is volatility
ρ is correlation of returns between A and B
γ is the Constant Relative Risk Aversion coefficient

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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 Ian Hall and Alan Howard for their comments and suggestions.
   

2. Note that throughout we will use the term “asset” very generally to include any actual or hypothetical collection of returns and risks.
   
3. Here and here are two such services, although neither are particularly targeted to spending patterns of wealthy families.
   
4. Robert Merton and Arun Muralidhar suggest that governments address population retirement needs by issuing securities indexed to national per capita income growth in their idea of Standard-of-Living indexed, Forward-starting, Income only Securities. See SeLFIES: A New Pension Bond and Currency for Retirement (2020).
   
5. And should reflect the currency mix of all forms of your spending.
   
6. While higher income cohorts of the population have seen their income growth dramatically outpace the overall population, this has not always been the case historically, and it might be appropriate for future projections to embrace a degree of reversion to mean. See The U.S. Income Distribution: Trends and Issues and Per capita personal income in the United States.
   
7. Shiller doesn’t provide per capita income data. Calculations based on 10- to 30-year rolling returns, 125 years from 1891 to 2016. Shiller’s data is available here. Note that, in transforming the risk and return of assets to be relative to the ideal risk-free asset, we do not need to alter the correlations between those assets.
   
8. For a deeper dive into the risk of T-Bills for long-term investors, see our note Back to the Future: Reviving a 19th Century Perspective on Financial Well-Being.
   
9. To a first order approximation, the optimal allocation to the risky asset in the standard Two-Asset model is inversely proportional to the variance of the risky asset. The decrease in equity volatility from 18% to 16% would result in a 25% increase   18%² 16%² – 1 in the optimal allocation, which is close to the increase we see from 62% to 79%.
   
10. Currently, this is why in our Elm Global Balanced strategies, we effectively treat T-Bills as if they were the lowest-risk asset. This is an issue we regularly revisit.
    
11. See our note Spending Like You’ll Live Forever.
    

By Victor Haghani and James White ¹

Apple and Microsoft each represent over 5% of the S&P500, with Amazon and Google not far behind. The ten largest US companies add up to 27% of the S&P500. Historically, the top ten companies have represented a smaller fraction of the S&P500; for example, from 2005 to 2019 the average was about 20%. We’re often asked if this unusually high concentration at the top is something to worry about. For investors like us who believe the markets are pretty efficient, we think the answer is “no” on three counts:

1. First, the broad stock market isn’t as concentrated as the S&P500 suggests. The S&P500 covers about 80% of the total US stock market, and less than 50% of the global stock market by value. Indeed, the S&P500 owns less than 6% of all the public companies that a broad, global equity fund such as Vanguard’s VT owns. So a globally diversified portfolio of stocks is roughly half as concentrated as the S&P500, with about 13% in the top ten companies, and Elm’s global equity Baseline is even less concentrated with about 10% in the top ten names,² as shown in the table below. Vanguard’s global market capitalization weighted index fund is invested in over 9,000 individual companies, and the 1,000 largest holdings comprise only 75% of the total index. We think few investors would view this as a portfolio low on diversification, but we’ll expand on this perspective further in the two points below.
2. Let’s consider an alternative to a market capitalization weighted index that invests the same dollar amount in every stock in the index – referred to as an equal-weight index.³ Every investor who allocates to the equal-weight index has to find someone else who is willing to hold even more of the largest stocks and go short the smaller stocks, to in effect take the other side of the active bet that the equal-weight investor is making. Getting someone to take the other side of your trade may well require you to accept a lower return. Taken to the extreme, if all the capital invested in market capitalization index funds wanted to switch to equal-weight index funds, they’d either have to own substantially more than 100% of many of the smaller companies – assuming their prices didn’t change – or they’d wind up forcing all companies to have market capitalizations closer to each other, regardless of the underlying characteristics of their businesses. No wonder the largest equal weight S&P500 index fund (RSP) is only 2% of the size of the largest market capitalization weighted US equity index fund (VTI).
3. The market capitalization weighted portfolio is the only portfolio that all investors can own at the same time. For investors to be happy holding more of the biggest names, they need to deliver just enough extra expected return to compensate for their being such a big part of the market.⁴ How big an assumption this is depends on how concentrated the market is in the biggest companies. The table below gives an estimate for the amount of extra expected return that the largest stocks would need depending on how dominant they are in the index. Notice that, with a size distribution similar to that of today’s global stock market, the largest stock would only need to deliver a tiny bit more expected return (0.08%) than the 100^th largest stock for the market capitalization weighted portfolio to be the most efficient portfolio. In other words, this result is telling us that the current market portfolio, with the top ten stocks representing about 13% of the market, isn’t very concentrated at all. By contrast, if we assumed that the largest stock represented 25% of the market portfolio (a truly extreme degree of concentration), then that stock would need to have a significantly higher expected return to compensate for its outsized impact on the overall market.
   
   In this stylized analysis, we assume that every stock’s idiosynchratic (non-market) risk is independent of that risk in every other stock. In reality, we know that groups of stocks can share common traits, such as the industry they’re in, or other characteristics, often referred to as “factors,” such as size, growth prospects, etc. An equal weight index will always have a bias in favor of small companies versus large ones, and in general will also have a bias to value stocks over growth stocks. These risks can lead to substantial divergences in performance between market capitalization and equal weight indexes, with differences in one-year returns of more than 5% occurring 25% of the time.

Conclusion

While market capitalization weighted indexes are currently more concentrated than usual, they are still very well diversified based on the small amount of extra expected return the biggest stocks would need to offer to compensate for their weight in the index. In sum, market capitalization weighting is the best index design available, offering the most efficient and diversified exposure to the broad stock market.

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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. The concentration figures for Elm’s global equity Baseline are calculated for our Global Balanced strategy for US SMA investors. Figures will vary slightly for our other programs.
   
3. Another alternative to market capitalization weight indexing we could have considered is referred to as “Fundamental Indexing.” We believe the analysis presented here also applies to this form of indexing. See our article published in the Journal of Portfolio Management titled “Do Index Buyers Make Over-Valued Stocks More Over-Valued?” which argues against the proposition of Fundamental Indexers that market capitalization weighted indexing is flawed.
   
4. Indeed, this is a central tenet of Portfolio Theory.