Elm Wealth Research

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.
   

Victor was recently a guest on Bloomberg’s Trillions podcast with Joel Weber and Eric Balchunas, discussing his long road from the trading desks at Salomon Brothers to LTCM to Elm’s inception and subsequent expansion.

You can also listen to the episode on iTunes.

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For more on the topics discussed in the episode:

• ETFs: Better Than Mutual Funds for Long Term Investors too?
• Do Index Buyers Make Over-Valued Stocks More Over-Valued?
• A Penny Saved is Two Pennies Earned
• Home Biased: A Case for More Indexing
• What’s all the Hoopla? Truly Passive Indexers are Still a Rare Breed
• Mind the Gap: Inequality and Diversification
• How well do global market-cap weighted indexes represent the true ‘market portfolio’?

By Victor Haghani and James White ¹

If you’ve been asking yourself the question: “To BTC, or not to BTC?” the market, in its infinite wisdom, may have just decided for you: we’re all HODLRs² now, whether you like it or not.³

In a recent Bloomberg Odd Lots podcast interview with Joe Weisenthal and Tracy Alloway, Sam Bankman-Fried – 28-year old MIT grad, $10bb Crypto Whale and founder of FTX crypto exchange – explained a relatively new development in the relationship between Bitcoin and the stock market:

“If you rewind to 2018, you wouldn’t be sure whether stocks and crypto were positively or negatively correlated…[Some] people thought Bitcoin is the flight-to-safety asset – when stocks crash people would turn to Bitcoin – but others said it was a risk-on asset…Summary…wasn’t totally clear. Then after March 2020, [when Bitcoin and the stock market were both down big in the first two weeks of the month] now we know the answer. It’s clear that – at least right now – people see crypto as something with huge upside potential and the sort of the thing that does well in risk-on markets, not risk-off markets. Because when you’ve got a lot of money to play around with and you’re looking for something to do with it, something with huge upside sounds really appealing and you’re willing to take the risk that it goes to zero – but when everything is crashing and you’ve lost all your money, a lot of people are going to put it into what they think is going to be safe. That’s why the correlation [between equities and crytpo-currencies] has become really robustly positive.”

We took a look at the data, and they do support Bankman-Fried’s observations. From the start of 2018 until now, Bitcoin’s Equity Beta – the standard measure of the systematic, market risk of an individual stock – has been 1.0.⁴ By contrast, in the early days of Bitcoin, from 2014 to the end of 2017, it was only 0.3, with extended periods when it behaved like a safe-haven asset, going up when the stock market was falling.⁵ This was consistent with the popular “digital gold” view of Bitcoin – a protection against a meltdown of the Western financial system. Meanwhile, real, shiny gold exhibited an Equity Beta of -0.2 and 0.2 over the same two periods, and as you’d expect, generated a much lower return as well.

From a purely statistical perspective, the last three years of data is telling us that if we wake up a year from now – or a month from now, the way things have been going – and Bitcoin has quadrupled to $200,000, the odds would be heavily in your favor to guess that the stock market will have gone up in value too – although far less than fourfold. Similarly, if Bitcoin collapses to $2,000, it’s pretty likely – as a matter of statistical inference – that the stock market will be lower. This argument doesn’t depend on knowing the direction of causality (if any) between changes in the stock market and Bitcoin. In fact, our best guess of what’s going on is that both the stock market and Bitcoin are being driven by some common forces, rather than Bitcoin driving price action in the stock market or the other way around.

As Bitcoin has gotten bigger, more widely-held and more integrated into the global financial system, it has been behaving more and more like a stock – a big, techie, super-volatile stock – and likewise, some stocks have been acting a little more like crypto-currencies. Here’s Bankman-Fried in the same interview, this time on how stocks are starting to behave more like coins:

“In the last year, equities have started to look more like crypto. You look at GameStop…There’s a word for it in crypto, it’s called a ‘shit-coin.’ The beautiful moment of this [was] when RobinHood banned the buying of GameStop…GameStop crashed…They stopped buying GameStop and they bought what is in retrospect the only possible answer to this question. They bought Dogecoin! As soon as GameStop started crashing, Dogecoin ten-x’d [went up tenfold]. Absolutely beautiful!”

Depending on your outlook, there are varying implications of Bitcoin’s current incarnation as a risk-on asset.

• If you think the expected return on Bitcoin is close to that of the stock market, you may want a small holding – Bitcoin’s current size relative to the global equity market is about 1%, and the total digital coin market amounts to a bit under 2% – just as you would want to own all the stocks in the market portfolio. However, many investors, ourselves included, feel owning the global stock market through low cost, liquid index funds and ETFs provides sufficient diversified exposure to risk-on assets, and don’t feel the need to own all the other assets that don’t come with a global stock market index fund.
• If you think the expected return is significantly higher than that of the stock market, then a more substantial holding would be consistent with that view.
• However, if you think it has an expected return lower than that of the stock market, as would be typical of a safe-haven asset, then you probably don’t want to own any of it: with a Beta of close to 1, it’s not offering the risk-mitigation to warrant the lower return.⁶ Even if you’re more worried about inflation than financial crisis risk, we suspect that owning a combination of inflation-protected Treasury bonds (TIPS), real estate and equities,⁷ might be a better way to get the job done.
• Finally, for those who are outright bearish on Bitcoin and its brethren, the higher correlation between the stock market and Bitcoin might be the canary in the coal mine – a warning that we should not expect the stock market to be unconnected to swings in the crypto-currency markets. Thinking back to the buildup to the dot.com crisis and the subprime mortgage debacle, many market observers (your authors included) wrongly believed that the broad stock market would be minimally affected by a crash in these relatively small pockets of speculative fever, as each amounted to roughly 4% of global stock market capitalization.⁸ Currently, Bitcoin, other digital coins and related businesses are about 3% of the size of the stock market.

While we agree with Sam Bankman-Fried that, currently, Bitcoin is mostly behaving like a risk-on, speculative asset, we also recognize that it has gone through many rebirths and transformations. In early November 2020, Odd Lots co-host Tracy Alloway made the most cogent case we’ve heard for why we should expect the Bitcoin narrative to continue evolving, which will force investors to reassess its place in the matrix of investment choices:

“I have a confession to make: I am now bullish on Bitcoin. I’m bullish on Bitcoin because I’m bullish on cognitive dissonance in a complex society, and on people’s ability to produce endless narratives for cryptocurrency – even ones that are, at times, contradictory. Since its creation back in 2009, Bitcoin has been lauded and promoted as so many things. It’s a method of payment (you can buy pizza!) but it’s also a speculative financial asset whose value is destined to go up (so you should save it!). It’s a hedge against inflation (because central banks are printing money!) but it’s also a financial asset that benefits when interest rates are close to zero and there’s less opportunity cost to hold it (it’s digital gold!). Bitcoin is a way of disintermediating the existing financial system (because you can’t trust the bank!), but it’s also something that would benefit from a flood of institutional money (asset managers are diving in!).”

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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 Simon Bowden, Richard Dewey, Costas Kaplanis, John Karubian and Ricky Moezinia for their suggestions.

2. According to Investopedia.com:
   “HODL is a term derived from a misspelling of “hold” that refers to buy-and-hold strategies in the context of bitcoin and other crypto-currencies.”

   More recently, it’s been used as an acronym for “Hold On for Dear Life,” in recognition of the wild ride that most crypto-currency investors experience.
   

3. Not that you’ll find Bitcoin has magically appeared in your brokerage account, although for every $1,000,000 invested in the broad US stock market, you own over $100 of Bitcoin through your ownership of Tesla (TSLA), Microstrategy (MSTR) and Square (SQ), and another $2,000 or so in crypto-currency focused companies such as Coinbase (Coin).
   
4. Note that a Beta of 1.0 does not imply a correlation of 1.0. Highly volatile assets can have a high Beta with a low correlation. β_i = ρ_i,m σ_i / σ_m , where i refers to the asset whose Beta we are measuring and m refers to the broad stock market.
   
5. Beta measured using weekly overlapping weekly returns for the S&P500 and Bitcoin from 9/30/2013 to 4/19/2021.
   
6. Another serious challenge to the notion of Bitcoin as a safe-haven asset is the fact that under most plausible assumptions, the combination of a low expected return, high volatility, and a conviction that its price cannot go below zero, results in a strong downward drift in the expected median value of Bitcoin over time.
    
   For example, with Bitcoin’s 85% annual volatility and a 0% expected return, the median price of Bitcoin in four year’s time would be about 80% lower than today’s price. This effect is known as “volatility drag.”
   
7. For example, German equities actually increased in value in US dollars through the hyperinflation experienced from 1921 – 1923, while all nominal claims were essentially wiped out. See “The Economics Of Inflation – A Study Of Currency Depreciation In Post War Germany”, by Costantino Bresciani-Turroni, 2008.
   
8. Total loss of value in dot.com companies was estimated at $1.7 trillion, with stock market capitalization around $38 trillion (Wikipedia: Dot-com bubble). Subprime mortgage market at its peak was $1.3 trillion (Wikipedia: Subprime crisis background information), and the global stock market was $40 trillion.
   

By Victor Haghani and James White ¹

Introduction

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

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

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

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

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

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

Three Spending Policy Options

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

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

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

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

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

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

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

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

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

Chart 2: Spending 5.1% a year

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

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

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

Chart 3: Spending 4.2% a year

How to Compare Different Spending Policies

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

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

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

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

Utility

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

Time Preference: Weighing a Better Present Against a Better Future

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

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

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

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

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

Put Your Faith in DEUS: Discounted Expected Utility of Spending

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

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

In Search of the Optimal Spending Policy

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

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

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

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

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

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

k^* = μ – r γσ²

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

C^* = r_ce – r_ce – r_tp γ

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

Where Merton Meets the Road

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

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

Chart 4: Comparing Merton Optimal vs. Sustainable Spending Rules

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

Conclusion

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

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

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

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

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

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

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

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

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Appendix: Extensions

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

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

Family Wealth

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

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

Other Popular Spending Policies

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

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

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

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

Endowment Growth Through Ongoing Contributions

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

For Investors Expecting Higher Returns Than Implied by Their Risk Taking

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

r_ce = r + k(μ – r) – γ k² σ² / 2

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

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

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Further Reading and References

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

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

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

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

Last year, the Journal of Portfolio Management published our paper on factor investing, ‘Smart Beta: The Good, the Bad and the Muddy.’

It’s since become one of our most popular pieces, and we’re happy to announce that the paper has been awarded the Bernstein Fabozzi/Jacobs Levy Award for Outstanding Article.

For anyone who may have missed it, you can view the full paper here.

By Victor Haghani and James White ¹

The US Treasury effectively “owns” about 24% of the stocks held by high income US taxable investors. Through the capital gains tax, Uncle Sam has an effective exposure of more than $1 trillion of equities.² And this huge-but-silent investor might be about to get a lot bigger if capital gains taxation is increased. But it’s not all bad news; luckily there is something you can do to lessen the blow. In this note, we’ll explain how adjusting your asset allocation to account for capital gains taxation can improve your after-tax expected welfare.

Last month, we wrote about a framework for deciding how much to accelerate the realization of capital gains if tax rates are likely to increase. The idea is to maximize your Expected Risk-Adjusted Wealth, which can often lead to a different decision than one based solely on the ‘static’ analysis of the single most likely scenario.³ Linked to the decision of how much to accelerate is the question of how capital gains taxation impacts how much risk to take, which is the question we address in this note. Throughout this note, we’ll more generally use equities as a proxy for risky assets.

The primary take-aways for long-term investors are:

• increases in the capital gains tax rate often (but not always) lead to higher optimal equity allocations,
• in general, investors should respond to changes that increase the bite of capital gains tax, such as higher inflation or larger embedded unrealized gains, by increasing their exposure to equities,
• for a given dollar amount of expected tax, capital gains taxation is the kindest form it can take, as it reduces Expected Risk-Adjusted Wealth less than alternatives such as income or wealth taxes.

The optimal risk decision for a taxable investor is a function of too many variables to be reducible into a rule-of-thumb, which is why we have provided a calculator you can use along with the note.

You can find our Decision Calculator here.
You can find the math and code behind the framework here.

The Impact of Capital Gains Tax on the Optimal Allocation to Equities

Let’s start with looking at whether a higher future capital gains rate should change how much to allocate to equities, which we expect to generate capital gains in the future but may also generate capital losses. Happily, there’s a long history of economists studying the effect of capital taxation on risk-taking. Economists have noted that if the government applied capital gains taxation symmetrically to gains and losses, then investors shouldn’t care about the tax rate at all; they would simply respond to increases in tax rates by increasing their exposure to the risky asset so as to keep the post-tax expected return constant.⁴

Unfortunately, we don’t live in a world of symmetric taxation, and it’s unlikely we’ll wake up in that world any time soon. Instead, we’re taxed on realized gains, with realized losses creating a tax-loss carry-forward credit which can only be used against future realized gains.⁵ But understanding how things would work in the symmetric case is a good starting point for investors with very long horizons and/or who hold assets that are very appreciated versus their cost basis.

To fully answer the question of how the capital gains tax rate should impact an investor’s risk-taking, we’ll again reach for our Risk-Adjusted Wealth Maximization tool, which we used to crack the problem posed in our To Realize or Not To Realize note. We’ll assume market expected returns and risk as set out in Table 1, such that in a zero-tax world our hypothetical investor (with a standard level of risk aversion) would maximize their Expected Risk-Adjusted Wealth by choosing a 70% equity/ 30% bond allocation.⁶ Then, for any given set of tax rates, investment horizon and initial unrealized gains, we can find the optimal allocation to equities that results in the highest expected Risk-Adjusted Wealth. In Chart 1, we compare the optimal allocation to equities as a function of horizon, under four taxation assumptions:

• no taxation at all (dotted blue)
• taxation of interest at 50%, dividends at 30% and capital gains at 0% – for investors expecting to avoid paying capital gains tax completely (orange)⁷
• taxation of interest at 50% and dividends and capital gains at 30%, with no value on loss carry-forwards (green)
• taxation of interest at 50% and dividends and capital gains at 30%, with 50% value on loss carry-forwards (red)

If only interest and dividends are taxed, the optimal allocation to equities would be just a bit higher than in the no-taxation case. This is because, with our Base Case assumptions, taxing dividends reduces equity returns by slightly less than taxing interest reduces the return on the risk-free asset, making equities look slightly more attractive after-tax. When we introduce taxation of capital gains (the red and green lines) for investment horizons greater than 10 years, we see the predicted result of a higher optimal allocation to equities – although the effect is not as large as it would be in the pure symmetric case.

At horizons under 10 years, the presence of capital gains tax causes our investor to want a smaller equity allocation than in a tax-free world, if they place no value on loss carry-forwards. To see why, we can think of capital gains tax as being a call option sold for free by the investor to the government. For short horizons, the value of the call option takes a big chunk out of the expected return offered by equities, so owning less is optimal.⁸ As the horizon lengthens, the optionality of the capital gains tax liability decreases as a per annum cost and the benefit of deferring the payment of the tax increases, resulting in an increasing optimal equity allocation. Figuring out exactly what is optimal for a given investor will depend significantly on how they value the potential tax-loss carryforward that may arise from a realized loss.⁹

Sizing Up the Impacts of Taxation

In the idealized world from econ 101, with only a risk-free and a risky asset to choose from, no taxation, no transactions costs and asset prices that follow random walks, an investor’s optimal allocation to the risky asset is a function of just three variables: the risky asset’s excess expected return, its risk, and the risk-preferences of the individual. When we bring taxes into the problem, the optimal asset allocation becomes a function of many variables we can ignore in the zero-tax world. In Table 2 below, we use our tax calculator to see how the optimal allocation to equities changes based on changes in a range of variables, and also how these changes affect investor welfare. We use the assumptions in Table 1 as the Base Case, with a 20-year horizon.

For an investor starting out with large unrealized gains in their portfolio (row 6), we find that a larger equity allocation of 81% is optimal.¹⁰ The increase in the optimal allocation to equities that results from higher unrealized gains suggests that as the market goes up, assuming expected returns remain the same, investors should want to increase their exposure to the market. The framework also suggests, as shown in row 5, that an increase in the capital gains tax rate should lead taxable, long-term investors to want to increase their allocation to equities, a conclusion that may run counter to the analysis of many market observers who view an increase in the capital gains rate as making equities less attractive. Of course, both of these effects are predicated on investors thinking about the impact of capital gains taxes in a framework similar to the one we are proposing, and the impact on markets will be driven by the characteristics of marginal investors who may look quite different from our long-term, taxable, utility-maximizing exemplar.

In the rightmost column of the table, we can see that an increase in the capital gains tax rate from 30% to 50% (row 5) is roughly equivalent to the harm caused by each of the following:

• An increase in unrealized gains in the portfolio of $35 per $100
• An increase in inflation of 1.25%¹¹
• A decrease in the risk-free real rate of 0.25%
• A wealth tax of 0.2% per annum

Perhaps most surprising of these observations is the last one, that a 0.2% wealth tax decreases a long-term investor’s welfare as much as an increase in the capital gains tax from 30% to 50%, even though a 20% increase in capital gains tax would result in about three times as much expected taxes paid over the 20-year horizon as the wealth tax.¹² There are two reasons for this surprising equivalence: 1) the capital gains tax is paid at the end of 20 years while the wealth tax payments are spread out over the whole period, and 2) more significantly, the capital gains tax is paid only when the investments have done well, whereas the wealth tax is owed regardless. The same reasoning explains why just a 0.25% decrease in the risk-free real rate does the same harm to our investor’s welfare as a 20% hike in the capital gains tax rate. It is sobering to realize that the 4% drop in long-term US TIPS yields over the past 20 years has done far more damage to the welfare of current investors than any possible change to the long-term capital gains tax rate could inflict.

Conclusion

The taxation of investment income and capital gains not only reduces the welfare of individual investors relative to a world of zero taxes on our savings (ignoring possible societal benefits of the taxes collected), but it also can materially affect our asset allocation decisions. Capital gains tax, due to its character as a one-sided tax, requires a framework that takes account of risk in outcomes and an investor’s personal risk-aversion. It is also notable that the level of real interest rates and inflation have a material impact on both our risk-adjusted wealth and our tax-adjusted optimal asset allocation.

From Table 2, we see that almost every change that hurts investor welfare is also associated with an increase in the optimal allocation to equities. In general, the greater allocation to equities softens the blow from the taxman taking a bigger slice of the investor’s pie.

We encourage you to spend some time with our calculator, and please do get in touch with us if you’d like to discuss how this framework might be applied to your individual circumstances and decisions.

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Disclaimer:

In this note, we’ve provided a framework for thinking about tax-related investment decisions. Nothing in this note should be construed as tax advice pertaining to any individual’s specific circumstances, and your authors are not tax experts. We have simplified the problems we have discussed considerably, ignoring many important tax rules, including but not limited to: discussions of the impact of different rates for long-term versus short-term capital gains tax, different rates at different income levels, the value of capital gains tax loss carryforwards, estate, trust and charitable considerations, and many, many more.

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Further Reading and References:

• Domar, Evsey D. and Richard A. Musgrave. “Proportional Income Taxation and Risk-Taking.” The Quarterly Journal of Economics. Vol. 58, issue 3, 388-422. 1944.
• Feldstein, Martin. Capital Taxation. Harvard University Press. 1983.
• Feldstein, Martin. “The Effects of Taxation on Risk Taking.” Journal of Political Economy. Vol. 77, No. 5, 755-764. 1969.
• Haghani, Victor, Larry Hilibrand and James White. “When it Pays to Pay Capital Gains.” 2019.
• Haghani, Victor and James White. “How Much Should the Tax Tail Wag the Asset Allocation Dog?” 2017.
• Haghani, Victor and James White. “Measuring the Fabric of Felicity.” 2018.
• Haghani, Victor and James White. “To Realize, or Not to Realize: Capital Gains Tax and Portfolio Choice.” 2020.
• Stiglitz, J. E. “The Effects of Income, Wealth, and Capital Gains Taxation on Risk-Taking.” The Quarterly Journal of Economics. Volume 83, Issue 2, 263–283. 1969.

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1. This not is not an offer or solicitation to invest. We are not tax experts and nothing in this note should be construed as tax advice. Past returns are not indicative of future performance.
    

   We are very grateful for the help of Larry Bernstein, Richard Dewey, Larry Hilibrand, Peter Hirsch, Mark Perwien, Jeffrey Rosenbluth and Roberta Sydney. All errors are our own.
   

2. Based on studies by the Federal Reserve on the distribution of stock market ownership by household net worth, and also supported by average US Treasury capital gains tax receipts of roughly $100 billion per year. See here and here. This does not include further exposure of the US Treasury to US equities through corporate tax receipts, which tend to run twice the size of annual capital gains tax receipts.
   
3. For those familiar with Utility theory, Risk-Adjusted Wealth is equivalent to Certainty-Equivalent Wealth.
   
4. For example, see Domar and Musgrave (1944), Stiglitz (1969) and Feldstein (1969). Also assuming zero risk-free interest rates, that the capital asset has a zero dividend rate (which can be achieved for equity risk by using equity index futures), and that the investor is willing and able to borrow to leverage her position in the capital asset, if required, at the risk-free rate.
   
5. There are proposals to start taxing unrealized gains too. We’ll leave discussion of that for a future note, if it becomes more likely. Also, under current rules, taxpayers can allocate $3,000 of capital losses per year to offset earned income.
   
6. This could result, for example, from a view that equities have a 4.5% expected excess return and 18% risk, both per annum, and the investor’s preferences are represented by a CRRA utility function with coefficient of risk-aversion equal to 2. See our note Measuring the Fabric of Felicity for the risk-aversion survey supporting this Base Case choice.
   
7. For example, by donating the assets to a charity or having children inherit the assets and utilize the step-up basis allowance.
   
8. It should be noted this is an odd sort of option contract, whereby the seller determines the expiration date of the option and under current rules the option goes away at death.
   
9. A parameter that we have built into the accompanying calculator.
   
10. The 81% is measured on the basis of $100 of portfolio value, ignoring the capital gains tax liability. If instead we value the starting portfolio at $89.50, net of the current value of the tax liability, then the desired allocation to equities is 90.5%, much closer to the 98.5% allocation that we’d get if capital gains taxation were applied symmetrically.
     

    For a case of the risky asset having a zero basis, we find an optimal equity allocation of 85%, which scales up to 115% as a percentage of starting wealth minus current value of the tax liability, a result higher than the 98.5% optimal under symmetric taxation.
    

11. We assume that stock and bond prices do not fall when inflation rises. The impact of higher inflation on Risk-Adjusted Wealth in this scenario flows through the fact that investors are taxed on nominal income, not income after inflation.
    
12. In the Base Case, equities are expected to grow at 4% a year (6% total return less 2% dividend yield) so after 20 years $76 invested in equities would be expected to grow to $169 for a capital gain of $93 and at 20% tax, an extra $18.60 of tax, compared to about $5.70 of wealth tax payable over the period.
    

By Victor Haghani and James White ¹

This is a note about taxes, however: we are not tax experts and
nothing in this note should be construed as tax advice.

With the US presidential election just a few weeks away, it’s a good time to think about capital gains taxes. A number of our clients have been concerned about the possibility of significantly higher tax rates in the future and have asked us how we think about the investing implications of such a change. In this note, we suggest a framework based on maximizing expected risk-adjusted wealth for answering some of the important questions arising from the interaction of taxes and investing.² We believe this framework is superior to a conventional static analysis, and we’ll explain why as we dive into the central question arising from tax rates potentially increasing in the near future: to realize gains now, or to defer realization to the future.

Investment theory holds that in a world of efficient markets, random walks and no taxes, horizon shouldn’t materially affect many types of investment decisions. However what we have found is that as soon as tax enters the picture, horizon in many situations becomes the single most important input an investor needs to think about. We find that for investors with long horizons and moderate-size unrealized capital gains, it will usually make sense to defer realization even if the investor expects the tax rate to jump by 20 percentage points. We were surprised to find that at intermediate time horizons, the optimal decision can be to realize some, but not all, of the unrealized gains. Investment horizon isn’t the only critical input, and the interaction of all the relevant variables puts the development of a generic rule-of-thumb out of reach. Optimal decision-making in light of taxes is a function of many idiosyncratic and personal inputs, which is why we decided to provide a calculator you can use along with the note.

You can find our Decision Calculator here.
You can find the math and code behind the framework here.

The central question we will address in this note is whether it is better to sell (and re-purchase) appreciated assets now and pay today’s long-term capital gains tax rate, or wait to realize gains in the future and pay a likely higher capital gains tax rate.

At first, this may seem like a fairly straightforward problem of weighing a bigger payment in the future against a smaller payment today: simply a question of the time value of money using an appropriate discount rate.³ In many circumstances, the decision would ride on a choice between different plausible discount rates. For example, in the base case we’ll discuss below, we’d want to realize early if we chose the risk-free interest rate, while we’d defer realization to the horizon if we used the expected return of a plausible risky asset. But as we’ll explain in this note, there is no robust way to choose an appropriate discount rate in a static framework. There are three main problems with the conventional approach, which all arise from the fact that risk is an integral part of the problem:

1. The conventional analysis captures only one of many possible scenarios. For example, the asset may drop below its cost basis resulting in no tax liability in the future, as the government doesn’t pay us “negative taxes” when we have a capital loss.⁴ A sound analysis needs to take account of all possible outcomes. Chart 1 shows how realizing gains now vs. later can result in dramatically different changes in value depending on the final level of the risky asset at the investment horizon.⁵
2. Considering many possible scenarios and making decisions which maximize your expected wealth still may not lead us to the correct decision. The problem is that maximizing expected wealth leads to absurd decisions around risk-taking: for any investment with a positive expected return above the risk-free rate, investing more in that investment will always lead to higher expected wealth, so it can’t help to answer any questions involving trade-offs between risk and return.⁶
3. The conventional static comparison, by construction, cannot find that a partial realization of gain is optimal, even though (as we’ll see below) there are cases when that’s exactly what an investor should do.

In addition, although of lesser impact, the static analysis doesn’t take into account the effect a higher future tax rate may have on how much of the risky appreciated asset you want to hold going forward, nor can it determine how tax rate uncertainty impacts the optimal allocation decision.

The goal in sound financial decision-making is to make choices which offer the best risk-adjusted result. This requires accounting both for uncertainty in outcomes and for the decision-maker’s personal level of risk-aversion. The standard approach to quantifying an individual’s aversion to risk is through their utility function: a mapping between wealth and the benefits of wealth, its utility. Classical utility functions show utility increasing with wealth – but as wealth increases, it produces diminishing marginal gains in utility, as illustrated in Chart 2. This has a deep connection to risk-aversion and risk-taking: given a risky situation, the more your loss of utility is disproportionately large relative to a potential gain, the more compensation you require to bear that risk. Thus, the more concave the utility curve, the more risk-averse the individual.

To make good risk-adjusted decisions, we need to find choices which maximize an individual’s expected utility, not expected wealth.⁷ There’s a direct correspondence between utility and risk-adjusted wealth (they’re interchangeable) – so going forward, we’ll perform our calculations using utility, and then express our results in terms of risk-adjusted wealth. That will allow us to compare different decisions in dollars rather than units of utility, which some readers may find too abstract. Armed with this tool, and assuming a typical degree of investor risk-aversion, we can determine to what extent we should realize a capital gain at the lower tax rate today.⁸

To do this, we calculate expected risk-adjusted wealth for each possible amount of immediate realization and each possible allocation to the risky asset, as follows:

1. For each possible price of the risky asset at the investment horizon, we calculate how much after-tax wealth the investor would have.⁹
2. We use the investor’s utility function to translate that amount of after-tax wealth into personal utility.
3. We multiply each amount of utility by the probability of the risky asset realizing the corresponding price at the horizon, using the risky asset’s assumed expected return and risk. We add up the probability-weighted utility values, and this gives us expected utility.
4. Using the investor’s utility function, we translate from expected utility into risk-adjusted wealth.

The result is an Expected Risk-Adjusted Wealth curve as a function of how much gain to realize immediately, given an optimal allocation to the risky asset.

To illustrate this analysis, we’ll assume the investor has a 20-year horizon, that the risky asset (a stock ETF, for example) has an expected return of 6% with 18% volatility, and that the long-term capital gains tax rate will go from 30% today to 50% in the future. We’ll also assume that the investor must pay tax at the horizon, without the benefit of tax mitigations such as step-up basis for estate and charitable purposes, 1031 exchanges or moving state tax jurisdictions (although all of these could be modeled within this framework).¹⁰ If the investor has a capital loss at the horizon, we assume a zero value for the loss carryforward. The full set of assumptions are in Table 1.

Chart 3 displays the risk-adjusted wealth curves for different initial levels of current unrealized gain, assuming an optimal risk allocation and a starting portfolio worth $100 (ignoring the tax liability). These curves allow us to find the decision – the percentage of gain to realize today – that maximizes expected risk-adjusted wealth. For an asset with 0 initial unrealized gain (blue), whether we sell the asset now or later doesn’t matter as there is no gain to realize; as expected, we see expected risk-adjusted wealth is constant. For our base-case unrealized gain of 35% (orange), realizing nothing now produces the highest gain in expected risk-adjusted wealth of approximately $20.5, the leftmost point on the curve. If the investor instead chose to realize all gains today, the rightmost point on the orange curve, their expected risk-adjusted wealth would fall to roughly $18.5.

Things get more interesting in the case of a starting unrealized gain of 70% of the portfolio value (green) – an admittedly extreme case of the asset having a zero basis. In this case, we get the intriguing result that it is optimal to realize about half the gain immediately, which delivers about $4 more risk-adjusted wealth than no immediate realization, and $2 more than full immediate realization. In the base case, a discount rate of 3.7% would produce the same conclusion to defer realizing the gain and an increase in value of $2¹¹ – but notice that in the case of a higher unrealized gain of 70%, there exists no discount rate to use in a static analysis which would suggest optimally realizing half the gain, or optimally realizing any fractional amount.

Notice also that the green curve of this 70% gain case gets very flat around the optimal decision to immediately realize 50% of the gain. If the investor decided to realize 40% or 60% of the gain, their expected risk-adjusted wealth would be almost the same. This is a general feature of decisions that come out of this framework: getting close to optimal delivers almost all the expected benefits of the precise optimal decision. But, the further away one moves from optimality, the expected welfare of the investor declines ever more swiftly.

To get a better feel for what’s going on, let’s look at how the optimal amount to realize changes with time horizon, as illustrated in Chart 4.

We can see that if you have a short horizon, it’s optimal to realize a high fraction of current gains – but as horizons lengthen out, you want to realize less and less. Why might this be? For a very short horizon, say just one minute, it makes sense to realize 100% of the gain right away, re-establish the position and then liquidate it after the 1 minute has elapsed.¹² Over that very, very short horizon, it’s fair to expect very little price change in the risky asset, so taking the gain now is pretty much a no-risk, no-brainer. However, as the horizon gets longer, risk enters the calculus as the relative benefit of the realization decision is tied to the risky asset price at the horizon, as illustrated in Chart 1. The risk of early realization has the general profile of being short an option: if the asset price doesn’t move much, the investor is better off taking advantage of the lower tax rate by realizing early, but if the asset price goes down or up dramatically, deferring realization will have been the better decision. A big drop in the asset price will leave the investor with a non-refundable tax loss,¹³ while a big rise in the asset price favors deferral as it allows the investor who hasn’t paid tax early to have a larger exposure to the risky asset. At the same time, the time value component of deferring paying taxes goes up with an increasing horizon. These two effects shift the balance towards deferring realization as horizon lengthens.

So far we have treated the future capital gains tax rate as known with certainty, but it’s easy enough to see how tax rate risk impacts the realization decision. In Chart 5, using the assumptions in our base case, we plot the optimal realization decision comparing the case where the capital gains rate increases to 50% versus the case where there’s an equal chance that it’s unchanged at 30%, goes up to 50%, or goes up to 70%. Notice that all we’ve done is introduce tax rate risk, as we’ve left the expectation of a 50% future tax rate the same in both cases. What we find is that tax rate uncertainty pushes the investor to realize more gains at every horizon up to a horizon a bit beyond 20 years, when full deferral is suggested under both sets of future tax rate assumptions. The reason we get this result is that our investor is risk-averse, and uncertainty in the future capital gains tax rate introduces risk into deferral.¹⁴

Conclusion

Making sound financial decisions under uncertainty requires finding the decision that maximizes your expected risk-adjusted wealth. Doing so often involves more than a back-of-the-envelope calculation, but the math is not terribly complex – and we’ve provided a calculator to do the heavy lifting for you. While the cases we’ve dealt with in this note have been simplified, the framework we’ve proposed is flexible enough to handle more realistic setups, including a more general, multi-period lifetime saving, spending and investment framework, and also an integration of correlations between tax rates, asset prices, inflation and interest rates. If you’re interested in building your own calculator to handle more complex or personalized cases, we’ve shared our Jupyter Notebook with the core framework, equations, and Python code.

The core points we’d like to leave you with are:

1. The expected-utility framework is superior to a static analysis for answering tax questions involving uncertain outcomes and produces sensible results which maximize risk-adjusted wealth. This approach can suggest partial realization as optimal, a solution that is not possible within the conventional solution.
2. Horizon is very important: At shorter horizons, it tends to make sense to realize some amount of gains early when capital gains tax rates are expected to rise, while deferring gains is more attractive at longer investment horizons.
3. Under many circumstances, the difference in risk-adjusted wealth between the optimal decision and a bad decision is great enough to warrant an investor spending some time to make a thoughtful decision. However, in the neighborhood of the optimal decision, risk-adjusted wealth differences are small, so it’s reasonable to get in the ballpark of the best decision without worrying about a high degree of precision.

In practice, most investors see themselves as having multiple horizons over which they expect to liquidate assets to fund spending, and portfolios generally have a mix of assets with different cost bases and different characteristics. This adds complexity to the analysis, but doesn’t change its basic character – making decisions that maximize expected risk-adjusted wealth is the best way to improve your expected welfare.

We encourage you to spend some time with our calculator, and please do get in touch with us if you’d like to discuss using this framework or how it might be applied to your individual circumstances and decisions.

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Disclaimer:

In this note, we’ve provided a framework for thinking about tax-related investment decisions. Nothing in this note should be construed as tax advice pertaining to any individual’s specific circumstances, and your authors are not tax experts. We have simplified the problems we have discussed considerably, ignoring many important tax rules, including (but not limited to) discussions of the impact of different rates for long-term versus short-term capital gains tax, different rates at different income levels, the value of capital gains tax loss carryforwards, estate, trust and charitable considerations, and many, many more.

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Technical Appendix

The capital gains tax rate τ₀ will be changing tomorrow to τ . You have some amount of homogenous unrealized gains g₀ , and you want to know whether you should realize your gains now (or more generally, what fraction should be realized), given that you also want to hold the optimal of the risky asset to horizion T , at which point you’ll realize any additional gains and pay any taxes.

We assume the risky asset S follows a GBM with mean return μ and volatility σ , and that you have CRRA utility with elasticity γ . S also pays a dividend at rate δ which is taxed at τ_d .

As usual, we want to optimize expected utility:

U(θ,κ) = 𝔼 [u P̂_T]

where:

u(w) { w^{1 – γ} – 1 1 – γ , γ ≠ 1

u(w) = ln(w), γ = 1

P̂_T = P_T  – (φ⁺ + α(1 – T T_c )⁺ φ^–) τ

φ = (1 – θ – ε)g + P_Te^–κ T δ(1 – τ_d) – P₀

ε = 1 κ₀ ((κ₀ – κ)⁺ – κ₀ θ)⁺

P_T = P₀e^{(κ(μ – δ_Td) + (1 – κ)(1 – τ₁)r – ½ κ2 σ2)T + κ σ Z_T}

P₀ = 1 – (θ + ε)g_τ 0

Note: ε is the amount of extra immediate gain realization (if any) additional to θ called for by rebalancing the risk asset from κ₀ to κ . α is the multiple applied to the asset value of a loss carryforward if received today, and T_c is the time after which loss carryforwards have no asset value.

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Further Reading and References:

• Domar, Evsey D. and Richard A. Musgrave. “Proportional Income Taxation and Risk-Taking.” The Quarterly Journal of Economics. Vol. 58, issue 3, 388-422. 1944.
• Feldstein, Martin. Capital Taxation. Harvard University Press. 1983.
• Feldstein, Martin. “The Effects of Taxation on Risk Taking.” Journal of Political Economy. Vol. 77, No. 5, 755-764. 1969.
• Haghani, Victor and James White. “How Much Should the Tax Tail Wag the Asset Allocation Dog?” 2017.
• Haghani, Victor, Larry Hilibrand and James White. “When it Pays to Pay Capital Gains.” 2019.
• Stiglitz, J. E. “The Effects of Income, Wealth, and Capital Gains Taxation on Risk-Taking.” The Quarterly Journal of Economics. Volume 83, Issue 2, 263–283. 1969.

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1. This not is not an offer or solicitation to invest, nor should this be construed in any way as tax advice. Past returns are not indicative of future performance.
   We are very grateful for the help of Larry Bernstein, Larry Hilibrand, Peter Hirsch, Mark Perwien, Marlin Risinger, Jeffrey Rosenbluth and Roberta Sydney. All errors are our own.
   
2. For those familiar with Utility theory, Risk-Adjusted Wealth is equivalent to Certainty-Equivalent Wealth.
   
3. Assuming the price of the asset at the horizon is at or above its current price.
   
4. The taxpayer does get a capital loss carryforward, which can be used against future capital gains. In our analysis in this note, we’ll be assuming that capital loss carryforwards have no value beyond the investor’s chosen horizon. However, the proposed framework can handle multiple horizons and assigning value to capital loss carryforwards, a feature we have built into the calculator.
   
5. The chart assumes the investor makes the same percentage allocation to the risky asset regardless of realizing the gain early.
   
6. Assuming the ability to borrow at the risk-free rate to invest arbitrarily more.
   
7. This decision framework is a superset of the well-known Kelly Criterion, not an opposing framework as it’s sometimes portrayed. Optimizing expected utility given log-utility and binary bets reproduces the Kelly Criterion exactly.
   
8. We’ll assume the investor exhibits Constant Relative Risk-Aversion (CRRA) utility with coefficient of 2, which means the investor would have an optimal allocation to equities of about 70% based on an expected excess return of 4.5% and equity volatility of 18%, ignoring tax effects. See our survey on CRRA risk aversion here.
   
9. In calculating the value of the portfolio at the horizon, we make the unrealistic (but non-impactful) assumption that the investor maintains the chosen percentage asset allocation over the entire period without incurring taxes in the process of rebalancing the portfolio to keep the allocation constant.
   
10. Nor in this note will we consider other tax mitigation approaches such as hedging of appreciated assets, splitting assets between taxable accounts and non-taxable accounts such as IRAs or tax-loss harvesting strategies run on portfolios of many individual equity holdings.
    
11. Explanation of 3.7% discount rate: realizing a $35 gain today at a 30% tax rate creates a tax payable of $10.50, while realizing $35 at a 50% tax rate in 20 years produces a tax payable of $17.50. Discounting $17.50 at 3.7% per annum for 20 years gives a present value of $8.50 which is $2 less than the $10.50 generated by realizing today.
    
12. We assume there are no transaction costs, and there are no wash sale restrictions on realizing gains and re-establishing positions, as far as we know, although as stated already, this is not tax advice.
    
13. For an investor with multiple horizons, a tax-loss at an early horizon will have some value as a carryforward to a longer horizon. This is straightforward to incorporate into the framework described herein, and it is built in to the calculator that accompanies this note.
    
14. Tax rate uncertainty can also support the decision to convert a traditional IRA into a Roth IRA even for an investor who doesn’t expect higher tax rates in the future, as the conversion and resultant tax payment upfront reduces the risk associated with uncertain future tax rates. A number of other variables also significantly drive the conversion decision.
    

By Victor Haghani and James White ¹

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

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

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

Steadfast

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

Greedy

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

Fearful

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

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

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

A Different Shade of Fearful: Momentum

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

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

Volatility Targeting versus Momentum

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

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

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

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

Conclusion

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

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

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

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

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

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

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Further Reading and References:

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

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

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

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

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

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

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

By Victor Haghani and James White ¹

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

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

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

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

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

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

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

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

Conclusion:

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

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

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

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Appendix: Real Annuity Value Mechanics

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

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

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

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Further Reading and References:

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

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