Elm Wealth Research

By Victor Haghani and James White¹
ESTIMATED READING TIME: 7 min.

A shooting star’s beauty is its curse — it burns itself out to be seen.
  – Anonymous

Introduction

We’ve enjoyed lots of feedback since distributing our Leveraged ETF calculator and accompanying research note last week. Most of the incoming questions have asked our views on the 2x leveraged long ETFs MSTX and MSTU, based on the shares of MicroStrategy, Inc. (MSTR). With the caveat that we do not have domain expertise in MicroStrategy, Inc. nor in Bitcoin, we’re excited to share our thoughts on this fascinating situation.

In this note, we’ll:

• Explain why we think 2x leveraged MSTR ETFs are not an attractive investment vehicle using return distributions from our Leveraged ETF tool.
• Estimate the probability of the 2x leveraged MicroStrategy ETFs going bust in the next year at between 20% and 50%.
• Question whether market liquidity is sufficient to support the current size of these leveraged MSTR ETFs.
• Give our perspective on why MSTR trades at such a big premium to its underlying holdings of Bitcoin, and why we think that’s not likely to persist.
• Dive into the concept of “Bitcoin yield” in the context of MSTR, and also explain the “power law” as pertains to estimating BTC’s future return.
• Discuss why we wouldn’t invest in a long BTC vs short MSTR ETF if such an ETF were to be brought to market.

Background

MSTR has received tremendous media and investor attention over the past month. The company is primarily a leveraged-long holder of Bitcoin. Over the past month, MSTR stock is up 95%, while Bitcoin has increased by about 50%, breaking through the newsworthy $100,000 price level on the very day that we’re publishing this article. MSTR stock trades at a substantial premium to the value of the Bitcoin it owns, with the common equity having a market value of about $100 billion, which is about 2.4x the roughly $40 billion of Bitcoin it owns.²

The outspoken founder and executive chairman of MSTR, Michael Saylor, describes his company as a “Bitcoin development company,” while others, such as the Wall Street Journal, refer to it as a “Bitcoin buying machine.”

Saylor has laid out ambitious plans to significantly increase the company’s Bitcoin holdings through a combination of equity and debt financing. In October, he announced the “21/21 Plan,” aiming to raise $42 billion over the next three years — $21 billion through equity issuance and $21 billion via (mostly convertible) bonds. Such issuance would allow MSTR to buy 420,000 additional Bitcoins at the current price, representing about 2% of the total amount of Bitcoins ever created, and a much higher percentage of Bitcoins that are freely traded.

There are several ETFs that provide 2x (and even 3x) leveraged long exposure to MSTR. The two largest right now are MSTU ($3 billion) and MSTX ($1.8 billion). Between them, they own about $10 billion of MSTR common stock exposure through margined longs, swaps or options positions.

What follows are answers to some of the questions our readers have sent our way.

How can I use your tool to analyze the 2x leveraged MSTR ETFs?

Our tool can be found here: Leveraged ETF Tool. For the Long Side, simply input Ticker = MSTR and Leverage = 2, then delete the default Ticker for the Short Side.

Info-tips in the tool give details on the inputs and how it works. The output uses volatility calculated using the past two years of daily data – which, for MSTR was 90% per annum or roughly 5.6% per day. The default input for the risk-free rate is 5%. As an admittedly arbitrary starting point, we set the underlying expected return (of MSTR, in this case) to 8%, which we expect many users will want to override with their own expectations.

A few things to note in the output for this case, using the default assumptions above:

• While the expected return of the ETF is 8.9%, the median return is a loss of 79%.
• To a one-year horizon, there is a 67% probability of loss and a 56% chance of losing more than 50%.
• There’s a roughly 8% probability of the ETF going up more than four-fold, making the return distribution of this ETF much like a lottery ticket or an out-of-the-money option.
• If the one-year return of MSTR turned out to be 8%, the expected return on the ETF would be a loss of 55%.³

While the past two years’ MSTR realized volatility was about 90%, the options market – which provides a more forward looking estimate of a stock’s volatility – is currently suggesting a much higher variability of about 160% per annum or roughly 10% per day. We provide output from our tool with the same assumptions as above, but with 160% used as the volatility estimate.

If MSTR bounces around at this extreme level of volatility over the next year, the most likely outcome is that investors will lose 99% of their investment.

What is the probability that a 2x leveraged MSTR ETF goes bust in the next year?

We just saw that, at 160% MSTR volatility – the level implied by the options market – the one-year median return for the 2x leveraged ETF is -99%. This means that there’s a 50% probability of return outcomes being either better or worse – so we can say that, from this perspective, the probability of going bust in this case is about 50% per year. At the 90% two-year historical volatility, the probability of losing 99% or more is about 5%, and there’s nearly a 25% chance of losing 95% or more.

Now let’s use some data to look at the probability of going bust just from a single really bad day. The price of a 2x leveraged ETF should go to zero if the price of the stock underlying the ETF goes down by 50% or more in a single day.⁴ The probability of such an event is a function of the variability of the MSTR stock price. If we assume the volatility of MSTR will be about 90% (or 5.6% per day), then we could think of a 50% decline in the stock price in one day as being a roughly 9x daily volatility move. A natural question is how often do stocks with very elevated variability, like MSTR, experience days when they decline by 9x their daily variability in returns?

We looked at about 1500 US stocks over the past 50 years, chosen so that at some point they were within the top 1000 stocks by market-cap.⁵ We found that the annual probability of such stocks experiencing a one-day price decline of 9x daily volatility was about 6%. This isn’t quite the final answer though, as we need the probability of a stock dropping by that much some time during the day, rather than just close-to-close. The usual estimate for the probability of touching a level over some time interval is to simply double the probability of being below that level at the end.⁶ So, assuming MSTR volatility of 90% per annum, the probability of a down 50% intra-day move occurring at least once over the next year is about 12%.

If we use the MSTR volatility implied by the options market of 160%, then down 50% is only 5x daily volatility. The same data as above yields a close-to-close annual probability of about 30%, which we estimate as about a 60% probability of an intra-day drop that would send the ETF to 0.

There are a number of alternative perspectives one could take in trying to estimate this probability: for instance, trying to estimate the probability of a large one-day drop in Bitcoin and how that might impact the MSTR premium to BTC. For example, a 25% one day drop in BTC and a 33% collapse of the MSTR premium would imply a 50% drop in the MSTR share price.

A more complex analysis might try to estimate whether it is possible for these leveraged ETFs to become large enough that their daily rebalancing trades could themselves drive the price down 50% in one day. For example, imagine that MSTR rapidly triples in price due to some combination of BTC rally and an increase in MSTR’s premium to the BTC it owns, and the assets in the MSTR leveraged ETFs go from $5 billion to $30 billion. The market capitalization of MSTR could be about $270 billion and the leveraged ETFs would be owning $60 billion, or 22%, of MSTR stock outstanding.

Now imagine for some reason, MSTR stock drops 15% during the day – which, given MSTR volatility, would not be unusual. The leveraged ETFs would need to sell $9 billion of MSTR stock at the closing price. Recently, MSTR daily average trading volume at the close of the day has been about $2 billion, so this would be quite an impactful amount of MSTR to sell at the end of the day. For every 1% the price declines further than the 15%, the ETFs will need to sell another $500 million of MSTR, and if that pushes the price down by another 1%…well, you can see this doesn’t have a happy ending for owners of the leveraged ETF or MSTR.

Bottom line, we think there’s a pretty decent probability – somewhere in the range of 15% to 50% – that these 2x leveraged MSTR ETFs are effectively wiped out in any given year if they are not voluntarily deleveraged or otherwise de-risked sooner.

Can the market support the positions and trading activity of the roughly $5 billion of MSTR leveraged long ETFs?

There are signs that these leveraged ETFs are already starting to hit practical market capacity constraints. For example: recently, the daily returns of the ETFs have started to diverge in troubling ways from 2x the daily return of MSTR as displayed in the chart below. We understand that the ETF sponsors are having difficulty managing the advertised exposures in the conventional fashion and have started to rely on using options on MSTR to deliver the desired leveraged exposure.⁷

Below we can see that the cumulative daily gap between MSTX and 2x MSTR has widened considerably in recent weeks:⁸

Why is MSTR trading at such a large premium to the value of Bitcoin that it holds?

Some observers suggest that there are many investors, primarily based outside the US, who are unable to own Bitcoin ETFs and are unwilling to own BTC directly or through an exchange such as Coinbase. These investors choose to get their BTC exposure by buying MicroStrategy, hence driving MSTR to a premium versus its BTC holdings. We don’t really think this is what has driven MSTR to its significant premium to its holdings of Bitcoin.

Other investors believe that there’s a decent probability that MSTR becomes a member of the NASDAQ-100 index and/or the S&P 500 index, which would give a price boost to the stock due to demand from index investors.

The below quotes convey a few other perspectives and motivations of buyers of MSTR and the 2x leveraged MSTR ETFs.⁹

What is the meaning of “Bitcoin yield” in the context of shares of MSTR?

The concept of “Bitcoin yield” in the context of investing in MicroStrategy (MSTR) stock refers to the increase in Bitcoin ownership per share over time. This is not a traditional yield like dividends or interest, but rather a measure of value creation unique to MicroStrategy’s Bitcoin-focused strategy. The company issues both equity and convertible debt to acquire more Bitcoin, and if it can issue this capital at a higher and higher premium to the underlying Bitcoin it owns,¹⁰ then investors will see a growth in the amount of Bitcoin they indirectly own per share of MSTR. For example, MicroStrategy reported a year-to-date Bitcoin yield of 41.8% as of November 17, 2024.

This effect can work in reverse too, generating a negative Bitcoin yield if the company issues more MSTR shares at a lower premium to its BTC holdings or if outstanding convertible bonds are redeemed rather than converted into more shares of MSTR.

We don’t think buying MSTR for its “Bitcoin yield” is a sensible investing approach.

What do Bitcoin investors mean when they talk about using the ‘Power Law’ to estimate the return of Bitcoin?

Many Bitcoin investors believe that the price path of BTC is well described by a regression line fitted to the logarithm of time since BTC inception versus the logarithm of the price of BTC. Such a regression calls for BTC to appreciate about 40-50% or so over the coming year. We don’t believe that extrapolating future prices based on past prices (whether directly or after taking their logarithms) makes sense for most financial assets, and we are skeptical that it is a sound approach for estimating the expected return of BTC.

Have you ever seen anything like this before?

While no two situations are ever exactly the same, and this MSTR narrative is most definitely highly unusual, we have seen some similar situations over the years. The example that comes to mind most directly is that of the Grayscale Bitcoin Trust (ticker GBTC): a Canadian closed-end fund that held Bitcoin, and at its peak, had close to $30 billion of assets. At some point in 2017, the trust traded at 2.3x the value of the Bitcoins it owned. More recently, the trust traded at a significant discount, of as much as 50% during the first half of 2023. Many vehicles that lock up investor capital often trade at a premium to underlying assets to begin with, but in the longer-term trade at a discount. This is a typical pattern for closed-end funds, SPACs and many corporate holding companies.

The end-of-day trades that leveraged ETFs must execute, buying when the market goes up and selling when it goes down, is reminiscent of the algorithmic trading associated with “Portfolio Insurance” in late 1987. Portfolio Insurance flows are generally accepted as the proximate cause of the October 19^th, 1987 “Black Monday” stock market crash. According to the Brady Commission, sales of $10 – 15 billion of equities created a downward spiraling, self-reinforcing feedback loop which ultimately resulted in the US stock market dropping 22% that day. While markets are much larger today than they were in 1987, the flows associated with leveraged ETFs (which tend to be concentrated in the final minutes of the trading day) could have a significant, destabilizing market impact.

What is your view of the MSTR-BTC premium in the future?

We will be surprised if MSTR is not at a discount in five years, particularly if the company follows through on its “21/21 plan” of issuing over $40 billion of stock and convertible bonds. Owning BTC in a corporate entity strikes us as less efficient than the ETF structure, as there will be capital gains tax liability on Bitcoin sold at a profit, and also the very real possibility of MSTR having to pay 15% tax on unrealized gains as part of the recent introduction of the minimum corporate tax rules.

Given your view of the premium going down over time, are you shorting MSTR versus buying BTC yourselves?

No, we’re not. For starters:

• Frictions are very high.
• We hate being short with unlimited downside, and in this case, the likelihood of getting squeezed out seems especially high.
• Given the return and risk characteristics of the trade, the optimal sizing would be too small to move the needle.
• We like the simplicity of being long-only investors in low-cost, highly-diversified index funds.
• It would be exciting if an ETF sponsor created a new long-short ETF that was long 1x Bitcoin and short 1x MSTR. Despite the convenience – and limited downside – of putting the long BTC vs short MSTR trade on via such an ETF, we would not invest in it. You can see why from the output of our Leveraged ETF tool below. The return pattern of such a long-short ETF is not very enticing at all, even with the assumption that MSTR underperforms BTC by 15% per annum.

Appendix: A few more examples of why people are buying MSTR, 2x leveraged long MSTR ETFs, and BTC¹¹

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1. Thank you to our friends Andy Constan, Dave Blob and Samir Bouaoudia for doing their best to help us think clearly about this fascinating topic. As always, we thank our colleagues Jerry Bell and Steven Schneider for making the whole process of publishing research fun and fast.
   
2. With adjustments for convertible bonds outstanding, but unadjusted for potential corporate capital gains tax liability.
   
3. This is shown in the tool’s full output, though not in the summarized output we show above.
   
4. It is possible that the ETF manager would intervene before the value of the ETF hits zero, but we would expect the value of the ETF at the end of such a day to be close to zero, and likely on the path to full liquidation and return of any remaining capital to investors.
   
5. Specifically, we chose the union of the top 1000 stocks as of 1995, 2005, and 2015, and filtered slightly to ensure good data quality.
   
6. To see why this is true in a simple random walk without drift, note that for every path that finishes below the level at the end of the period, there is another path where it hit the level and then followed a path that was a mirror of the path that finished below the level. So, for every path that finished below the relevant level (here a 50% drop), there’s another path that touched the level but then reflected and wound up above the level at the end.
   
7. This was discussed in greater detail in this WSJ article from December 2^nd, 2024.
   
8. Not including the volatility drag from rebalancing, but just adding up the daily “miss” vs 2x MSTX.
   
9. From these two WSJ articles, here and here.
   
10. The calculation usually assumes that the convertible bonds will be converted into equity when they mature, which assumes MSTR stock will appreciate over the life of the convertible bond.
    
11. Taken from the same WSJ articles previously referenced, and also from X.
    

By Victor Haghani and James White¹
Estimated reading time: 6 min.

Investors can now choose from about $100 billion in ETFs that provide leveraged long or short exposure to a broad range of popular stock indexes and individual companies. These ETFs are designed to deliver a daily return that is a multiple of the daily return of the underlying index or stock on which the ETF is based, less fees, frictions and the cost of leverage. For these leveraged ETFs, 2x and 3x are the most common multiples. It is well known – and stated in the prospectus and fact sheets – that beyond one day their return, even adjusted for fees and costs, will not be equal to the leverage multiple times the return of the underlying asset. The cause of this difference in longer-term returns is the daily rebalancing trades that the ETF needs to execute to keep its leverage constant through time. In general, the more volatile the underlying asset, the higher the leverage ratio, and the more time that goes by, the bigger the difference will be between the ETF’s return and the “multiplied” return of the underlying asset.

We have written about leveraged ETFs and this phenomenon twice before, featuring the hapless character of George Costanza, here and here. As a reminder of what’s going on here, let’s take a look at today’s largest leveraged ETF, the $25 billion Proshares Ultrapro QQQ (ticker TQQQ). It aims to deliver 3x the daily return of the Nasdaq 100 index (QQQ). Over the five years to September 30, 2024, the compound return on the underlying Nasdaq 100 index was 21.9%. If an investor expected to get a return close to 3x that 21.9%, he’d have been pretty disappointed. TQQQ generated a return of only 37.2%, not even two times the return of the underlying asset.² Some of this shortfall is due to the cost of leverage³ and the 0.84% annual fees. But most of the shortfall is due to the daily rebalancing trades that must be executed to keep its leverage at the 3x target – buying the underlying QQQ every day it goes up and selling it every day it goes down.

Battle Stations!

“… these ETFs are likely designed for the type of investor that is probably a lot more active than they should be.”
  – Dan Sotiroff, Morningstar analyst

Now that you know how these “simple” leveraged long and short ETFs work, it’s time to wrap our minds around the latest version of this structure: the Battleshares leveraged long and short ETFs.⁴ These ETFs are designed to bet on the continued success of bold, disruptive companies while wagering against the old-school giants they’re set to replace. Each ETF will provide a leveraged long exposure of about 2x on the trailblazing company and a 1x short position on the legacy competitor. For instance, the “COIN vs WFC” ETF will take a turbo-charged 2x long position on crypto-asset bank Coinbase (COIN) while shorting 1x of traditional bank Wells Fargo (WFC). A table in the Appendix shows the potential lineup of Battleshare ETFs.

A Tool for a Fuller Picture of Returns

Given the increasing variety, complexity and growing interest in leveraged ETFs, we decided to build a tool that would generate a return distribution for any leveraged long, short, or long-short ETF structure, which you can find here. An example of the inputs and outputs of the tool are shown below, for the as-yet-to-be-launched Battleshares 2x long COIN vs 1x short WFC ETF.

Info-tips in the tool give details on the inputs and how it works. The output above uses volatility and correlation between the long and short assets calculated using the past two years of daily data. The default inputs for the risk-free rate is 5%. As an admittedly arbitrary starting point, we set the expected return of the assets to 8%, which we expect many users will want to override with their own expectations.

A few things to note in the output for this case, which uses the default assumptions above:

1. If the 1-year return of COIN and WFC turned out to be 8%, the expected return on the ETF would be a loss of 49%.
2. While the expected return of the ETF is 5.9%, the median return is a loss of 75.4%.
3. To a 1-year horizon there is a 67% probability of loss, and a 56% chance of losing more than 50%.
4. There’s a roughly 8% probability of the ETF going up more than 4-fold, making the return distribution of this ETF much like a lottery ticket or an out-of-the-money option.

It’s also noteworthy that an “opposite” ETF that would be structured to be 2x long WFC and 1x short COIN would also lose 50% conditional on the two stocks returning 8% for the year. We discussed how you can lose (a lot of) money on a trade and on its opposite in our 2019 note, “If George Costanza Were a Hedge Fund Manager.”

The tool can also be used for leveraged long or leveraged short ETFs, by leaving the ticker for the short or long asset blank or setting its leverage to zero. For example, the MSTX ETF gives a 2x leveraged long exposure to MicroStrategy Inc (MSTR), a large holder of Bitcoin with a side business in software. See output below.

In the Appendix, we show a table of the 60 largest leveraged ETFs linked to stocks or digital assets, along with their expected one-year median return. It’s interesting to note that there are no leveraged short ETFs among these 60 largest ETFs at the current time. This is mostly due to the market rally over the past few years comprehensively vaporizing the assets in those ETFs.

When do leveraged ETFs make sense for individual investors?

There’s a line of reasoning taken by some who say as long as there’s full disclosure, any voluntary trading between consenting adults is fine and good, and so leveraged ETFs always make sense. We don’t find this reductionist argument terribly persuasive, but even if we did, we don’t feel that investors benefit from full and clear disclosure with these ETFs. For example, in the few prospectuses we’ve inspected, we couldn’t figure out the cost of leverage involved, nor the potential cost associated with the daily rebalancing trades required.

But even with the fullest disclosure possible, we struggle to find cases where these ETFs, particularly those based on single stocks, would make sense as investment vehicles for investors with typical risk preferences.⁵

Conclusion

Leveraged ETFs have fascinating longer term return distributions, which at least some investors are likely to find surprising. These longer term returns are highly relevant. Investors in aggregate cannot escape them, even if every single individual investor had a one day holding period.

We hope the leveraged ETF tool we have made available on our website will help investors, commentators and researchers more easily visualize the highly asymmetric return distributions that arise from many of these ETF structures.

p.s.

We can’t close this note without a few words on the potential impact of the trades that these leveraged ETFs have to execute each day at the market close. There are about $100 billion of ETFs that are on average either 2x long or 1x short stocks or stock indexes. For every 1% that the underlying assets go up (down) in price, the ETFs will need to buy (sell) $2 billion of stocks at that day’s market close. On a very volatile day when the stocks underlying these ETFs move by 3%, there will be $6 billion of buying or selling at the market close, or approximately 1% of daily US stock trading volume. It’s not clear exactly how much price impact that amount of buying or selling would have, but everyone we talked to in the hedge fund equity trading business thought it would be noticeable.⁶

p.p.s.

There’s a new filing from Defiance for a 2x leveraged long ETF with ticker “HOT” that will give 2x exposure to 5 to 20 of the most volatile stocks. We suspect it won’t be long before HOT is overtaken by an even spicier structure.

Appendix: List of Leveraged ETFs

Battleshares Proposed ETFs: Long 180-220% vs Short 80-120%
NVDA vs INTC ETF	Nvidia versus Intel
TSLA vs F ETF	Tesla versus Ford
AMZN vs M ETF	Amazon versus Macy’s
COIN vs WFC ETF	Coinbase Global versus Wells Fargo
MSTR vs JPM ETF	MicroStrategy versus JP Morgan
NFLX vs CMCSA ETF	Netflix versus Comcast
LLY vs YUM ETF	Eli Lilly versus Yum! Brands
GOOGL vs NYT ETF	Google versus New York Times

Source: Gil, D. (2024)

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References

• Gil, D. (2024). “Battleshares ETFs aim to pit innovators against legacy peers.” Financial Times.
• Grinold, R. and Kahn, R. (1994). Active Portfolio Management. McGraw-Hill.
• Haghani, V. and White, J. (2023). The Missing Billionaires: A Guide to Better Financial Decisions. Chapter 23, “The Costanza Trade.” Wiley.
• Haghani, V. and White, J. (2019). If George Costanza Were a Hedge Fund Manager. Elm Wealth.
• Haghani, V. and White, J. (2020). George Costanza At It Again: The Leveraged ETF Episode. Elm Wealth.
• Hajric, V. and Tsekova, D. (2024). “Gamblers Are Sinking Billions Into a Leveraged Market Fringe.” Bloomberg.
• Pessina, CJ. and Whaley, RE. (2020). “Levered and Inverse Exchange-Traded Products: Blessing or Curse?” Financial Analysts Journal.

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1. This not is not an offer or solicitation to invest. Past returns are not indicative of future performance. We thank Aneet Chachra, Richard Dewey, Larry Hilibrand, Vladimir Ragulin and our Elm colleagues Jerry Bell and Steven Schneider for their comments and contributions to this note and the accompanying leveraged ETF tool.
   
2. Returns taken from the Proshares factsheet.
   
3. T-bills averaged about 2.25% over the period
   
4. filed with the SEC by Tidal Investments and awaiting response
   
5. We have a friend who says he’s run short positions in these ETFs, which he says has been quite profitable. Perhaps this qualifies as a sensible use case? We agree with the view of Pessina and Whaley (2020): “Levered and inverse ETPs are neither suitable buy-and-hold investments nor effective hedging tools. They are unstable and exist only as mechanisms for placing short-term directional bets. Levered and inverse products are not, and cannot be, effective investment management tools.”
   
6. A common rule of thumb, as per Kahn and Grinold (1994), for market impact is k 𝝈 daily √(fraction of daily volume), with k usually around 1. So trades of 1% of daily volume with daily price volatility of 2% would be expected to move the market by about 0.2%. To the extent these flows are widely anticipated and occur in the closing auctions, the impact may be less.
   

By Victor Haghani and James White ¹

There’s been a lot of excitement and reporting about a new ETF: the Alpha Architect 1-3 Month Box ETF (ticker BOXX), designed to give investors the return of short-term US Treasury Bills with the tax character of long-term capital gains.

Long-term capital gains are taxed at lower rates than interest income – 20% versus 37% for the top US federal tax rates. With short-term interest rates at about 5%, this 17% difference in tax rates provides 0.85% more annual after-tax return, and you can add another 0.07% if you use BOXX to defer your tax bill for 5 years. That’s pretty good, and worth paying attention to if you can get it.

Two excellent Bloomberg articles were published on Feb 22nd that do a terrific job explaining how BOXX achieves its tax “magic.” The first is by Zachary Mider, “T-Bills Without Tax Bills? and hours later came “Put the Money in the BOXX” by Matt Levine (and a few days later also by Matt there’s “Maybe Don’t Put the Money in the BOXX?”).

Since BOXX went live on December 22, 2022, its annualized return has been 0.09% higher than that of State Street’s SPDR Bloomberg 1-3 Month T-Bill ETF (ticker BIL). BOXX has a stated expense ratio of 0.395%, but currently charges 0.195% and has about $1 billion of assets, while BIL sports an expense ratio of 0.136%, has $31 billion of assets and has been around for 17 years.

However, past returns are not necessarily indicative of future performance, and we think it’s reasonable to expect that BOXX’s prospective post-tax, risk-adjusted return relative to BIL (or direct ownership of Treasury Bills) does not justify owning BOXX. Before diving in, we should qualify what follows by noting that we are not tax experts.

Let’s start with an analysis of the expected return of BOXX, and then we’ll move to assessing the relative risk of BOXX versus Treasury Bills. First, BOXX has a higher expense ratio than BIL. The stated long-term expense ratio of BOXX is 0.395% per annum,² compared to 0.136% for BIL. At the long-term expense ratio, that takes 0.21%³ out of the 0.85% of tax savings, leaving 0.64% of net tax savings versus BIL.

Next, there’s the question of what rate of interest will be set by market participants who provide the other side of the BOXX options trades. Historically, as discussed in this article from the NY Fed, the implied interest rate in options boxes has averaged about 0.35% above T-Bills (from January 1996 through April 2023). While this historical positive spread may continue into the future, it is worthwhile to ask yourself how you would price those options if you were being asked to be the counterparty to BOXX’s trades?

Your selling of the options box-spreads generates cash – the cash that the BOXX ETF wants to invest – but at the same time, you’ll need to post collateral to the clearing exchange. You need to do this to give comfort to your counterparty – the Options Clearing Corporation (OCC) – that if you disappear with the cash, they won’t have a loss.

You’ll probably go out and buy a T-Bill of equal maturity to the expiration of the options you traded and post that as collateral to the OCC. For this to be worthwhile, you’ll need to price the options using an interest rate lower than the T-Bills you had to buy to post as collateral, so you can earn a spread.

How much of a spread you’ll want to earn is a difficult question to answer beyond saying “as much as I can get.” Perhaps one place we can look for an indication of what market participants charge for nearly risk-free trades is the much-discussed Treasury bond basis trade. A number of recent articles (here and here) indicate that traders demand an expected return of at least 0.5% per annum on assets.

This required profit margin by counterparties supplying the options trades to BOXX plus the higher expense ratio of BOXX leaves 0.24% of tax benefit for BOXX versus holding BIL. We should also make an allowance for higher transactions costs in all the trades BOXX needs to do, not only in all the various options trades, but also the trades with the Authorized Participants (APs) needed to clean out all the capital gains from the primary options trades used to invest the capital of the ETF.

For investors living in a state with an income and capital gains tax,⁴ state taxes must be considered as well. US T-Bills are exempt from state income tax but the capital gains from BOXX would not be. An investor in a state with a roughly 10% income tax would lose an additional 0.5% of the potential tax benefit of BOXX. Combined with the reduced benefits described above, this would render BOXX significantly less attractive than owning T-Bills through an ETF such as BIL, and even less so versus owning T-Bills directly.

Turning to the risk side, we see one primary risk, and a number of smaller secondary risks. The most notable risk is that the ETF is not investing in T-Bills, but rather it holds a position in options contracts in an omnibus account at its options broker, who in turn is exposed to the OCC, an entity with a AA credit rating. The OCC is an exchange, with a wide range of counterparties of varying credit quality. And we know that exchanges can get into trouble, despite intense regulatory oversight and the requirement that counterparties post collateral to mitigate credit exposure.⁵ For example, the near-insolvency of the London Metal Exchange in the wild nickel price runup in March 2022 illustrates that investing in an exchange is riskier than investing in US T-Bills. How much spread an investor should require for these credit risks is hard to quantify precisely, but we’d suggest something around 0.2% – 0.5% per annum as a reasonable estimate. Note that 0.35% per annum of spread represents a 50% probability of at least one default with a 50% loss every hundred years, which doesn’t feel to us like an overestimate of this risk.

Adding the cost of credit risk to all the other costs already noted gives us total costs greater than the 0.85% potential tax benefit. We might stop here, but a number of other risks are also worth mentioning. These include a negative tax ruling from the IRS on the BOXX mechanics (see Matt Levine’s aforementioned “Maybe Don’t Put the Money in the BOXX?” or Steven Rosenthal’s “Tax Gimmick in a BOXX”), higher transactions costs in executing its strategy, and wider spreads required by the ETF market makers (APs) due to the complexity of the structure and the lower liquidity of BOXX relative to larger and higher volume Treasury Bill ETFs. The latter two risks become especially salient as BOXX increases in size. A further risk for a BOXX investor with a long-term horizon is that if interest rates are lower in the future, the potential tax benefits of BOXX will be proportionately smaller.

Investors are generally attracted to owning US T-Bills as a safe and highly liquid place to keep some of their savings, to be instantly available for unexpected emergencies or investment opportunities. In order to get the tax benefit of investing in BOXX, a holding period of more than one year is needed, since the short-term capital gains rate is equal to the tax rate on interest income, at the highest marginal rates. This seems at odds with the primary rationale of holding T-Bills in the first place. If you have a 30% chance of needing to sell BOXX to raise cash within one year, that reduces the gross tax benefit by 0.25%.⁶

So, we’re not at present planning on using BOXX for our Elm Wealth clients, especially those who are subject to high rates of state taxation. However, what concerns us most about BOXX is the potential harm it may do to the whole ETF marketplace by creating a feeling that it is taking advantage of an ETF “loophole.”

We believe that the tax treatment of ETFs is more correct and equitable for investors than the tax treatment of traditional mutual funds, which can unfairly accelerate capital gains on long-term investors and create more capital gains than are actually realized by the mutual fund. We explained this in some detail in a note we wrote in 2015, “ETFs: Better Than Mutual Funds for Long Term Investors too?” More than 16 million US households benefit from the diversification, liquidity and fair tax treatment offered by ETFs, according to Investment Company Institute estimates. If the BOXX ETF grows so large and attracts so much attention that it precipitates a change in the rules governing ETF taxation, the result would be a tremendous and lamentable decrease in investor welfare. We truly hope our worries are misplaced.

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1. Thank you to Larry Hilibrand, Charles Wright, Dave Blob and Jon Seed for their helpful comments, and to Wes Gray and Larry Lempert for discussing specifics of their ETF with us. The opinions expressed in the article are not necessarily shared by those who gave us help. Nothing in this note should be taken as tax advice, or investment advice, or an offer or solicitation to invest. Past returns are not indicative of future performance.
   
2. Although currently, the sponsor has a fee waiver in place until January 31, 2025 so that the expense ratio of the fund is 0.195%.
   
3. Expressed in after-tax terms, using a 20% tax rate. We make this adjustment throughout this note where appropriate.
   
4. Every state except Alaska, Florida, New Hampshire, Nevada, South Dakota, Tennessee, Texas, and Wyoming.
   
5. At the end of 2022, the OCC had about $14 billion of assets supported by about $700 million of equity. BOXX’s $1.4 billion of assets represents over 10% of the OCC’s Clearing Fund Deposits as of end of 2022. See here
   
6. An exception to this analysis is an investor for whom capital gains are effectively tax-free because he has capital loss carryforwards so large that they will never be fully used.
   

By Victor Haghani and James White ¹

There are a number of ways an investment can go south, but getting the size of a trade wrong can convert even a good trade into a bad bet. That’s the subject of our new article, “The Big Investment Mistake,” published in this month’s Bloomberg magazine.

I hope you enjoy the article, but at the very least, you can see a gorgeous picture of Elm’s own Chief Happiness Officer – Milo, the soft-coated Wheaten terrier!

If you want to take a deeper dive into these topics, grab yourself a copy of our book, The Missing Billionaires: A Guide to Better Financial Decisions (named to The Economist’s Best Books of 2023 list).

The Missing Billionaires was included on the list for “The Best Books of 2023, as chosen by The Economist”, listed on 12/1/2023 for the time frame of calendar year 2023. This list is not based on any specific, publicly available criteria, it was compiled using The Economist’s own internal criteria and was based solely on the entity’s own thoughts and opinions. Neither Elm Wealth nor the book’s authors provided any form of compensation to be included on the list.

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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.
   

By Victor Haghani and James White ¹

The S&P 500 is down 15% over the past year,² so you’d think this would have been a great time to own some protection on your portfolio. Unfortunately, that’s not how things have turned out in this bear market (at least not yet) and not for what is probably the most popular way of protecting a stock portfolio with options.

Your Money or Your Time? It Hurts to Lose Both

If you bought and rolled one-month put options on the S&P 500 to hedge an investment in that index, over the past year you’d have lost an extra 2% on top of the 15% loss from just holding the S&P 500 unhedged, not to mention the loss of your time spent managing the strategy. The Chicago Board Options Exchange (CBOE) makes this really easy to see, by publishing an index of daily returns on the S&P 500 hedged by buying and rolling 5% out-of-the-money put options. The index ticker is PPUT.³

A more apples-to-apples comparison would be the put-protected S&P 500 versus just owning less stocks. As the chart below shows, if you’d kept 65% of your portfolio in stocks and 35% in T-bills, roughly consistent with the average risk level of 100% stocks fully protected with put options, you’d have outperformed the hedged strategy by about 8% over the past year.

We’re not suggesting that when the stock market goes down, an options-protected portfolio will always do worse than both a 100% long portfolio or one with a reduced equity exposure. But, we are focusing on the experience of the past year to make the point that options are not a silver bullet and do not provide “for-sure protection”. Even in a bear market, a portfolio protected with options will not always do better than an unhedged portfolio, and can do a lot worse than one that protects itself by simply owning less equities.

Just Unlucky?

Yes and no.

Yes, the main reason for the poor relative performance of the put-protected portfolio over the past year is some unlucky timing of when the options were purchased and expired and the exact path the market took over the year.

And no, there’s also a sense in which this outcome wasn’t just a uniquely bad throw of the dice. There are several fundamentally unattractive features of protecting your portfolio with short term put options. One is the fact that it’s nearly impossible to know whether the options you’re buying are fairly-priced. Another is that the options strategy reduces desirable time diversification, which in turn reduces Sharpe Ratio⁴ relative to keeping a constant fraction in equities.⁵

Options and Time Diversification

Time diversification is reduced because the exposure to stocks from a put-hedged portfolio will vary quite dramatically over time – some periods of higher exposure and others of lower exposure mean there are fewer “important” days determining total periods returns, thus lower diversification. When the options are purchased, the effective market exposure will be around 65%. But, if the market drops and the options are deep in-the-money, the exposure will drop to close to nothing – or, if the market is well above the put option strike price the portfolio might have close to 100% exposure. We can see the reduced diversification through risk metrics too: a portfolio that has 100% in the stock market half the time and 30% in the market the other half the time will have volatility about 15% higher than that of a portfolio with a constant 65% exposure.⁶

So, even if the portfolio protected with put options has the same expected return as a portfolio that just owns less stocks, the risk of the protected portfolio will be higher and therefore the expected Sharpe Ratio will be lower.⁷ This is borne out by the long-term historical data for the PPUT index. From 1986 to present, the simpler portfolio of 65% in the S&P 500 and 35% in T-Bills had a 0.6% per annum higher return, lower risk, and a Sharpe Ratio which was 22% higher compared to the PPUT strategy.⁸ That 22% higher Sharpe Ratio actually represents nearly 50% more expected welfare, because not only would you have earned a 22% higher excess return for the same risk, but in addition, you should have wanted to have 22% more of the simpler stock/T-Bill portfolio than the put-protected portfolio strategy, since the quality of the former was higher. The result: twice as much improvement in your expected welfare.

When Do Options Help?

We recently co-authored an article with our friend Vladimir Ragulin, titled “Do Options Belong in the Portfolios of Individual Investors?” (Journal of Derivatives Spring 2022)

Our conclusion was that options are unlikely to be welfare-enhancing, let alone a panacea, for affluent individual investors with typical risk preferences in most circumstances. We know this is a pretty strong conclusion, given that options trading now rivals stock trading in daily volume. You’d think that must be a sign that a lot of people are benefiting themselves through all that voluntary trading. But on the other hand, a lot of money gets put down on tables from Vegas to Macau, and so perhaps we shouldn’t find it so shocking that there’s so much options trading even though it is a zero-sum game, or worse after transactions costs.

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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 Vladimir Ragulin for his help with this article.
   

2. To the end of October 2022.
   
3. You can find the data on the CBOE website here.
   
4. A simple measure of the quality of an investment, the Sharpe Ratio is the ratio of an asset’s return in excess of the risk-free rate divided by its standard deviation of returns.
   
5. Assuming stock market returns are normally distributed and the options are fairly priced. This is a pretty strong result which will hold under a fair bit of deviation from these assumptions.
   
6. The portfolio that spends half its time at 30% and half at 100% exposure to the market has volatility, σ, equal to √(.5 * (0.3 * σ)² + .5 * (1.0 * σ)²), which is 14% higher than the 0.65 * σ for the portfolio with a constant 65% in the stock market.
   
7. And, in case you’re wondering whether doing the opposite would be better, this loss of time diversification argument also weighs against the sometimes popular strategy of selling put options.
   
8. The CBOE also publishes an index of returns from selling one-month 2% out-of-the-money put options (PUTY), which can be used to figure out how a portfolio protected with those put options would have done. The result since 1986 is pretty much the same as that for 5% out-of-the-money puts– lower return and lower Sharpe Ratio than a portfolio holding less stocks with about the same average exposure to the market.
   

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

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

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

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

Hiding in Plain Sight

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

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

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

Background

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

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

Hard to Explain

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

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

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

The Grandmother of All Anomalies?

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

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

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

A Clue from Meme Stocks?¹¹

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

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

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

Attention Stocks

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

Connecting the Dots

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

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

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

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

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

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

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

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

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

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

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

The $100 Bill That Shouldn’t Be There

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

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

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

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

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

Going, Going, Gone?

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

Macro Worries

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

Putting the Overnight Effect to Bed… for Now

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

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

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

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

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

---

PS…

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

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

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

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Further Reading and References

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

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

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

    

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

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

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

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

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

By Victor Haghani and James White ¹

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

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

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

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

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

What Was the Bowl Worth?

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

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

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

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

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

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

By Victor Haghani and James White ¹

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

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

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

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

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

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

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

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1. This not is not an offer or solicitation to invest, nor should this be construed in any way as tax advice. Past returns are not indicative of future performance.
   
2. Although we’ve seen a few investors that seem to be acting as if it is.
   
3. It’s a topic we’ve written about several times before, most recently two years ago in “Back to the Future: Reviving a 19th Century Perspective on Financial Well-Being.”
   
4. For liquid investments, at least.
   
5. Assumptions: 60% allocated to a broadly diversified global stock market ETF (Vanguard’s VT), 25% to a broad bond index fund (Vanguard’s BND) and 15% to short-term bank deposits (State Street’s BIL).
    

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

    

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

By Victor Haghani and James White ¹

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

Risk-Aversion

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

Crypto Risk and Return

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

A Puzzling Result

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

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

Explaining the Hump

An intuitive explanation for this hump is this:

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

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

A Different Perspective on Wealth and Risk

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

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

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

Conclusion

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

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

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

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

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Further Reading and References

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

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1. This not is not an offer or solicitation to invest, nor should this be construed in any way as tax advice. Our thanks to Steve Mobbs for his input on this note. Past returns are not indicative of future performance.
   
2. For example, in his 1986 Presidential address to the AFA titled “Noise”, Fischer Black stated:
   “…We might define an efficient market as one in which price is within a factor of 2 of value, i.e., the price is more than half of value and less than twice value…By this definition, I think almost all markets are efficient almost all of the time…”

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

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

    

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

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

   We are assuming this is the only investment opportunity available.
   

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

By 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.