Elm Wealth Research

A Volatile Mix of MicroStrategy, 2x Leveraged ETFs and Bitcoin

Elm Admin — Thu, 05 Dec 2024 05:00:00 GMT

December 5, 2024

Risk and Return

Moonshot or Shooting Star? A Volatile Mix of MicroStrategy, 2x Leveraged ETFs and Bitcoin

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

Chase Furey (25) has turned $700,000 of his parents’ retirement assets into $1.8 million by investing in MicroStrategy and a related leveraged ETF.

…He moved all of his investments, worth about $112,000, into the Defiance ETF instead and has grown his portfolio to about $400,000.

The Harvard graduate, who studied economics in college, convinced his parents to let him manage $700,000 of their retirement assets. He said he came up with a “less dangerous and smarter” plan for them, investing 27% of their portfolio in the Defiance [2x leveraged MSTR] ETF and the rest in MicroStrategy shares. The money has more than doubled to $1.8 million, he said.

“I think bitcoin could hit $400,000 and I think MicroStrategy could possibly 10x from where it is now by the end of next year, so that’s kind of my game plan with that,” he said.

[Authors’ Note: we hope he won’t be too Furey-ous if things don’t go according to his game plan.]

George Bodine, a 69-year-old retired airline captain in Covington KY, said he bought a modest stake in MicroStrategy in January that has since grown into a seven-figure position. Bodine, a die-hard bitcoin fan, said he is willing to pay up for the stock because of something called the BTC yield.

The term, which MicroStrategy introduced to investors in August, measures the percentage change in how many bitcoins per share MicroStrategy owns. As of Sunday, the company held 1.45 bitcoins for every 1,000 of its shares outstanding, using a share count that assumed all its convertible debt was turned into stock. That ratio was up 59.3% since Dec. 31 — and the increase is what MicroStrategy calls its year-to-date BTC yield. Based on that stat, Bodine said he now owns more bitcoin per share than he did at the start of the year.

“So in my mind, I’m getting more bitcoin than I could even in the market just buying spot [bitcoin],” he said.

Peter Duan, a 35-year-old wealth adviser in Los Angeles, went all in on MicroStrategy in September after selling his bitcoin and Tesla holdings. Yet he says he wouldn’t recommend MicroStrategy stock to his clients.

“Unless you do the requisite 100-plus hours of studying bitcoin on top of 100-plus hours of MicroStrategy, you should not enter this trade,” he said. “Because it is a very sophisticated trade that 99.99% of Wall Street doesn’t even understand.”

“This is not a YOLO thing, this is not a GameStop thing,” he said. “This is a rigorous exercise that takes a lot of deep first principles thinking.”

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

When Dan Hillery, a graduate student at Brown University, began investing in MicroStrategy in April, it accounted for about 40% of his portfolio. Hundreds of bullish options trades later, his returns have ballooned so much that MicroStrategy now makes up about 90%.

Hillery became familiar with options trading as an undergraduate, when he studied for quantitative finance tests in the hopes of landing a job at Citadel or Jane Street. “I never got hired at any of those places,” said Hillery, now 23. “So I ended up taking matters into my own hands.”

Hillery now holds a seven-figure position in MicroStrategy after scoring a 1,300% return in the past three months. He said he believes in the company’s long-term prospects but may consider selling some of his shares down the line.

Rajat Soni wasn’t able to purchase crypto directly with the pension fund he received after leaving his job as a fixed-income analyst at TD Bank. The 32-year-old from Toronto fully invested the money in bitcoin exchange-traded funds instead. Then he went down the MicroStrategy rabbit hole.

“Once you see it, you can’t unsee it,” Soni said of MicroStrategy founder Michael Saylor’s vision to turn his company into a bitcoin buying machine.

In August, Soni moved his entire pension fund into MicroStrategy. He has also dabbled in…an ETF that aims to provide double the daily return of MicroStrategy shares…

Soni said he has scored a 200% return on the stock in about seven months. MicroStrategy now makes up about 40% of his entire portfolio, and bitcoin about 60%. He said TD Bank is his only other investment — he was issued some shares as an employee. “If I could, I would dump it for MicroStrategy,” he said.

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.
With adjustments for convertible bonds outstanding, but unadjusted for potential corporate capital gains tax liability.
This is shown in the tool’s full output, though not in the summarized output we show above.
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.
Specifically, we chose the union of the top 1000 stocks as of 1995, 2005, and 2015, and filtered slightly to ensure good data quality.
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.
This was discussed in greater detail in this WSJ article from December 2^nd, 2024.
Not including the volatility drag from rebalancing, but just adding up the daily “miss” vs 2x MSTX.
From these two WSJ articles, here and here.
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.
Taken from the same WSJ articles previously referenced, and also from X.

Leverage It or Leave It? Making Sense of Turbo-charged ETFs Elm Partners

Elm Admin — Tue, 26 Nov 2024 05:00:00 GMT

November 26, 2024

Risk and Return

Leverage It or Leave It? Making Sense of Turbo-charged ETFs

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:

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%.
While the expected return of the ETF is 5.9%, the median return is a loss of 75.4%.
To a 1-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 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)

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.

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.
Returns taken from the Proshares factsheet.
T-bills averaged about 2.25% over the period
filed with the SEC by Tidal Investments and awaiting response
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.”
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.

Elm Wealth Webinar, Nov. 19th, 2024 - Elm Partners

Admin — Wed, 20 Nov 2024 05:00:00 GMT

Elm Wealth Webinar, Nov. 19th 2024

Victor Meets the Bogleheads Elm Partners

Elm Admin — Wed, 23 Oct 2024 04:00:00 GMT

October 23, 2024

Investing 101

Victor Meets the Bogleheads

By Victor Haghani, James White and Jerry Bell ¹
Estimated reading time: 6 min.

“Bogleheads” are DIY investors who are passionate about index investing. They gather each year to share ideas about sensible investing, and to celebrate the life and contributions of John Bogle, the founder of Vanguard and arguably the person who has done more than anyone to improve investor welfare. Victor was very pleased to attend their recent annual conference in Minneapolis, and do a Q&A session with Morningstar’s Christine Benz. There were also about a dozen authors of excellent personal finance books and blogs who gave presentations, including Christine Benz, Rick Ferri, William Bernstein, Allan Roth, Mike Piper, Jackie Cummings Koski, Karsten Jeske and Sarah-Catherine Gutierrez. Victor thoroughly enjoyed the experience, and hopes he’ll be invited back to next year’s conference in Austin!

We agree 100% with almost everything discussed over the course of the three-day conference. However, one area where we noticed our opinions diverge from the Boglehead consensus view was on asset allocation. At Elm, one of our core beliefs is that optimal asset allocation should depend on the expected return and the riskiness of the assets being invested in, and on the individual’s degree of risk aversion. Expected returns and risk change over time, and therefore, so too should one’s asset allocation.² We call our particular approach – which uses low-cost, broad coverage index ETFs to build client portfolios – Dynamic Index Investing®.

The consensus among Bogleheads – and among the vast majority of respected personal finance authors such as Charlie Ellis, Burton Malkiel, David Swensen and John Bogle – is that static asset allocation is the better approach.³ They believe that an investor should choose the percentage of their savings that they want to have in equities and then stick to that percentage through time.⁴ As John Bogle wrote in The Little Book of Common Sense Investing, “In general, investors should not engage in tactical allocation [varying the stock/bond ratio as market conditions change].” We respect these views, and know they have merit in many circumstances.

In this note, we want to explore the conditions under which the Bogleheads and like-minded investors are justified in following the static asset allocation approach – but before we dive into the details of this analysis, we want to say up front that on the broad spectrum of investment options ranging from utter folly to reasoned prudence (illustrated in the diagram below), both static and dynamic index investing are nearly on top of each other way over on the far right, sensible end of the continuum. We’re nearly as fond of Bogleheads-style static asset allocation as we are of Elm’s Dynamic Index Investing®, and in fact we do offer our clients a static index investing option if desired. So in this note, we’re really focusing a powerful magnifying glass at a very small strip of the investing spectrum.

The spectrum of investing styles for individual investors

The case for dynamic asset allocation

The theoretical case for dynamic asset allocation dates back to 1969 research by MIT economists Paul Samuelson and his student Robert Merton. One thing that came out of their research is a rule of thumb, known as the Merton share, which gives the optimal fraction of wealth (κ) that an investor should allocate to the stock market.⁵ The formula below has three inputs: the expected return of stocks over safe assets (μ), the riskiness of stocks (σ), and the investor’s individual degree of risk aversion (γ).

The Merton share
κ = μ γ σ 2

The fraction of wealth to invest in equities would be constant if changes in the expected risk premium were always balanced by changes of the same proportion in risk (measured as variance). Historically, the stock market’s expected risk premium and risk have tended to move in the same direction, but not in such a precise way as to keep the optimal allocation to equities constant. In fact, they sometimes move in opposite directions, and those are the times the Merton share calls for large changes in asset allocation.

Of course, any asset allocation – dynamic or static – needs estimates for the expected return and risk of major asset classes. We’ve written frequently (here, here, and here) about why we think reasonable estimates do exist and how we come up with them – and, if you don’t want to do it yourself, there are many sources online you can use. A web search for “capital market assumptions” of Vanguard, Blackrock, JP Morgan, or most other investment management firms will provide long-term expected return and risk estimates for major stock and bond markets. We’ve found that estimates from these different sources are usually clustered together, so it shouldn’t matter too much which one you choose, although you may want to average several together – or you can just use the ones we provide on our Elm Wealth website.

We have written about the Merton share and its applications in our book, The Missing Billionaires: A Guide to Better Financial Decisions. In Chapters 2 and 3, we discuss its theoretical underpinnings; in Chapter 5, we present long-term historical simulations suggesting that dynamic asset allocation, using US equities and US treasury bonds, generated a compound return 2.5% above a static asset allocation with similar risk. We believe that future excess returns will be lower. Over a long horizon, we’d suggest 1.5% as the expected pretax extra return from a dynamic approach, and around 1% on an after-tax basis, both with less risk than a static approach.

The case for static asset allocation

Now let’s make the case for static asset allocation. We’ll lay out the circumstances and beliefs about markets under which keeping your asset allocation constant is the better choice.

1) You’re a DIY investor whose cost of time is high relative to capital deployed
Changing your asset allocation in response to varying market conditions is time-consuming, and doing it tax-efficiently is doubly so. It’s certainly more challenging than keeping your asset allocation constant over time, and some DIY investors may simply not have the wherewithal to implement a dynamic approach.

If you’re a committed DIY investor, you may find the static approach more attractive if you value your time higher than your expected excess risk-adjusted return from a dynamic approach. You’ll have to put in extra hours to follow a dynamic approach, collecting and processing expected return and risk data and deciding on and executing trades. Depending on your skill level and how you value your marginal leisure time, we could see reasonably deciding in favor of static asset allocation for portfolios up to around $250,000, even if you completely agree with our views around dynamic allocation.⁶

2) If you expect the stock market risk premium and risk to stay in a narrow range
Even if the cost to you of dynamically managing your asset allocation is low, either because you have a lot of wealth or you’re willing to outsource to a low-cost manager, a static weight portfolio might be the better choice if you believe that the equity risk premium and risk are likely to fluctuate in a narrow range. While that has not been the case in the past and isn’t what we expect, it’s not a completely unsupportable view. If expected risk premium and risk are relatively stable, and if they tend to move together when they do change, then you won’t be giving up very much by opting for a static asset allocation. Furthermore, if you expect to be adding to your investment portfolio over time, you may reasonably expect that you’ll sometimes be buying equities when they’re more attractive and sometimes when they’re less attractive, and so this can have a stabilizing effect as well. Of course, it is still important to set your equity allocation in line with the average risk premium and riskiness level that you expect, and your personal level of risk-aversion.⁷

3) Your “optimal” allocation to equities is more than 100%, but you decide against leverage
A static asset allocation can make sense when your human capital is stable and large relative to your financial capital. You are likely to conclude that your optimal asset allocation is to hold a leveraged position in stocks over a broad range for the equity risk premium and risk. However, if your cost of leverage is significantly above the risk-free rate, and/or you cannot continuously rebalance your portfolio to keep this leverage constant, then you may reasonably choose to avoid leverage. In this case, you’d be right to follow a static asset allocation holding close to 100% in equities.

A few arguments in favor of static asset allocation that we don’t agree with

1) Dynamic asset allocation is market timing
We sometimes hear that dynamic asset allocation is bad because it is a form of “market timing” and market timing is bad. But what is market timing? We asked ChatGPT “what is market timing?” and here’s what we got:

Market timing is an investment strategy that involves making buying or selling decisions of financial assets, typically stocks, based on attempts to predict future market price movements… it frequently involves making short-term trading decisions based on expected near-term price fluctuations.

Defined like that, we’re not fans of market timing either – but we maintain that Dynamic Index Investing® is not market timing as defined above. It is important to recognize that this style of dynamic asset allocation – driven by changes in expected risk premia and riskiness – does not rely on any market inefficiency. It just relies on changes in the supply and demand of capital over time.

In contrast, stock picking or factor investing – such as overweighting exposure to small cap stocks, value stocks, etc. – are approaches which do primarily rely on market inefficiency for investors to earn extra returns above and beyond what’s needed to offset the higher costs and risks involved in such strategies. All that extra return can only come from other investors who are taking the opposite active exposures. For such investing to make sense, you need to believe that you are profiting from market inefficiencies arising from the mistakes or preferences of the investors on the other side of your concentrated bets.

2) There aren’t good risk and return estimates
We often hear people say that our Dynamic Index Investing® approach doesn’t make sense because it is not possible to estimate the expected return and riskiness of stock markets. As we explained earlier in this note, we strongly believe that it is possible to reasonably make those estimates, and that they’re readily available online from many large investment management firms and our Elm Wealth website.

Furthermore, the argument that it’s not possible to estimate return and risk does not specifically favor static asset allocation because you still need those estimates in order to arrive at the weights of your static asset allocation. Indeed, in the true absence of any estimates for expected returns and risk, how is it possible to make investment decisions at all?

3) Or, it’s best to use historical returns, and they don’t change much
We suspect that, in many cases where an investor’s estimated returns and risk are quite constant over time, what’s implicitly happening is that those estimates are primarily being anchored to very long-term historical returns. Long-term historical returns are pretty constant over time, and so an asset allocation using them as inputs will be pretty static too. However, we caution against estimating the prospective return of the stock market based on historical returns.

To see why, consider using the historical return of a 30-year treasury bond to measure its expected return. Imagine that five years ago, it was trading at a 10% yield to maturity. Over the next five years, its yield declined by half, to 5%. The historical return on this bond will look fantastic at 20% per annum – but it won’t provide any clue that the forward-looking expected return from this bond, if held to maturity, is actually only 5%, not 20%.

Broad equity markets are obviously not completely bond-like, but they’re more similar to bonds than one might think. Earnings yield provides a decent predictor of future long-term returns because corporate earnings look somewhat bond-like when viewed across an entire large economy. Accordingly, you can view the earnings you’re getting divided by the price you’re paying as a good (though imperfect) estimate for the real return you should expect. In contrast, short-term or even long-term history just doesn’t provide the forward-looking information we need.

4) Dynamic asset allocation isn’t tax efficient
Dynamic asset allocation is indeed less tax-efficient than static asset allocation, which in turn is less tax-efficient than buy-and-hold.⁸ This is not an issue for investors whose wealth is primarily in non-taxable accounts like 401ks and IRAs. For taxable accounts, the tax inefficiency of the higher volume of trading in a dynamic approach can be mitigated through tax-loss harvesting and tax-aware rebalancing.

Conclusion

Despite the intuitively appealing nature of dynamic asset allocation, for many DIY investors, static asset allocation can be a better choice. Simplicity, comfort and ease of implementation are really important features of an investment strategy, and a static asset allocation scores high in those dimensions. While we believe that dynamic asset allocation is theoretically optimal, it is important to follow an investment approach that you are sufficiently comfortable with to stick with over long periods of time and different market conditions.

While static asset allocation has fewer moving parts and might seem easier than the dynamic approach, we have observed that a static asset allocation is more difficult to stick with over time and through changing market environments. We know very few investors who have maintained their chosen static asset allocation for more than several years before they read some news that makes them uncomfortable with their level of exposure, and they move to a new “static” allocation.

For us personally, dynamic asset allocation is more stress-free because it is both intuitively appealing and theoretically sound. Importantly, it allows us to own more equities over time and worry less. Whichever approach you choose, as long as you’re like the Bogleheads Victor met in Minneapolis who build their portfolios with low-cost, broad index funds while keeping an eye on taxes and other fees, you’ll be at the very best end of the spectrum of investment choices.

An example of when dynamic asset allocation worked well
Our choice of this example is for illustrative purposes only, and not to suggest that dynamic asset allocation will always or usually outperform a static asset allocation approach. As we’ve discussed in the body of this note, we recognize that dynamic asset allocation may not be appropriate for many investors. There are many 10-year periods over which dynamic asset allocation would have resulted in a return below and/or a risk above that of a static asset allocation. However, we believe that changing your asset allocation over time as the expected excess return and risk of stocks change, is a more logical approach than keeping your allocation constant through time.

At the end of the year 2000, the cyclically-adjusted earnings yield of US equities was 2.9% and US inflation protected bonds (TIPS) had a real yield of 3.75%. At the end of 2010, the earnings yield of US equities was 6% and TIPS offered a 1% real yield. If you believe, as we do, that the earnings yield of the equity market is a decent estimate of its long-term real return, then you would not have wanted the same asset allocation at the end of 2010 as you had at the end of 2000. And you would have been justified in owning less equities and more TIPS in 2000, and more equities and less TIPS in 2010. Over the first decade of this century, US equities under-performed 10-year maturity TIPS by over 4% pa, while in the second decade, it was the other way around, with equities outperforming TIPS by 10% pa.

An investor who kept 60% in US stocks and 40% in bonds over the two decades enjoyed a compound return of 7.1%, while an investor who was 30%/70% in stocks/bonds for the first ten years, and then 90%/10% in stocks/bonds for the next ten years – for an average exposure of 60/40 – would have earned a compound return of 9.2%, 2.1% higher, with roughly the same risk. The dynamic asset allocator’s realized Sharpe ratio would have been 27% higher than the Sharpe ratio of the 60/40 static weight investor.

Appendix: How some respected personal finance books line up on the static versus dynamic asset allocation debate

Books advocating static asset allocation	Books advocating dynamic asset allocation
The Little Book of Common Sense Investing by John Bogle	Dynamic Asset Allocation by James Picerno
Unconventional Success by David Swensen	Strategic Asset Allocation by John Campbell and Luis Viceira
The Four Pillars of Investing by William Bernstein	Continuous-Time Finance by Robert Merton
If You Can by William Bernstein	Expected Return by Antti Ilmanen
The Little Book of Safe Money by Jason Zweig	The Missing Billionaires by Victor Haghani and James White
Morningstar’s 30-Minute Money Solutions by Christine Benz
All About Asset Allocation by Rick Ferri
A Random Walk Down Wall Street by Burton Malkiel
The Elements of Investing by Charles Ellis and Burton Malkiel
Winning the Loser’s Game by Charles Ellis
Stocks for the Long Run by Jeremy Seigel
How to Think About Money by Jonathan Clements
Enrich Your Future by Larry Swedroe
Lifecycle Investing by Barry Nalebuff and Ian Ayres
Risk Less and Prosper by Rachelle Taqqu and Zvi Bodie
Personal Finance for Dummies by Eric Tyson
The Only Investment Guide You’ll Ever Need by Andrew Tobias
The Index Card by Helaine Olen and Harold Pollack
The Intelligent Investor by Benjamin Graham
Global Asset Allocation by Mr Meb Faber

When a Crystal Ball Isn't Enough to Make You Rich - Elm Partners

Elm Admin — Thu, 26 Sep 2024 04:00:00 GMT

September 26, 2024

Featured Insights

When a Crystal Ball Isn’t Enough to Make You Rich

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

Introduction: Back to the Future

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

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

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

Summary of results

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

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

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

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

See Appendix I for a detailed analysis of player results.

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

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

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

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

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

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

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

Motivations

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

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

The Game

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

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

We are giving you $50 to play our “Crystal Ball” game. The object is to see how well you can do trading stocks and bonds if you know the news from the front page of the WSJ one day in advance. In other words, you’ll be in that dreamed-of position of being a trader who “knows the future.” For example, you will be shown the front page of the WSJ for a Wednesday, and be able to take a long or short position in the stock market and in the bond market at prices prevailing at Monday’s close (that is, two days earlier). Your trades will be liquidated at Tuesday’s closing prices. Note, on each front page we’ve blacked out anything that tells you explicitly what market prices actually did that day – leaving that information in would make this game too easy and no fun at all!

You will be trading the S&P 500 stock market index and a 30-year US Treasury bond futures contract, and you can use as much leverage as you’d like to, up to 50x. Use the sliders to choose the positions you want and then click the “Trade” button. Remember that for the 30-year Treasury bonds, prices go down when yields go up, and you are trading on price.

You are starting off with $50 of bankroll, and we will pay you however much this has grown to, or shrunk to, with a maximum payout of $100. You will have 45 minutes to play the game.

We have not chosen these days to try to trick you – they are randomly chosen. You will be able to trade on 15 different days, once per year over the past 15 years. The days will be presented to you in a randomized order. We’ve chosen these days randomly from a set of days where one third of them are days of employment reports, one third from days of Fed announcements, and the other third purely randomly, all taken from days that are in the top half of days ranked by overall market volatility. You can use the “Skip” button to skip any day you don’t feel like trading, and you can trade stocks, bonds, or both, each day. You can use the “Finish” button if you want to stop before being presented with all 15 days. Leverage of 1x means your position size is equal to your capital size. There are no transaction costs or overnight financing costs or rebates on your trades.

Good luck, and have fun – you may never have this opportunity again!

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

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

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

Conclusion

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

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

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

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

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

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

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

Appendix I: Detailed Analysis of Player Results

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

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

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

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

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

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

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

Unpacking player performance

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

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

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

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

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

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

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

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

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

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

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

Appendix II: Putting a value on the crystal ball

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

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

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

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

Appendix III: Caveats and shortcomings of this study

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

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

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

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

Appendix IV: The redacted front pages used in the experiment

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

The Most Important Number Not in The Wall Street Journal Elm Partners

Elm Admin — Mon, 09 Sep 2024 04:00:00 GMT

September 9, 2024

Investing 101

The Most Important Number Not in The Wall Street Journal

By Victor Haghani, James White and Jerry Bell
Estimated reading time: 3 min.

Announcing a new quarterly market snapshot going out to our research subscribers with long-term expected return estimates for US and global equity markets, plus some other useful bits of information which are also hard to find.

If you’d like to receive it, you can sign up here – and yes, it’s free.

When we think broadly about how to invest, a good starting point is to figure out how much of our wealth we want to invest in major stock markets and how much to keep in safe investments. To make that decision thoughtfully, two major pieces of information we need are the expected return of equities and the return offered on safe assets.¹ Even those of us who plan to pick just a handful of stocks, or to invest in private or alternative investments, are well-served by starting off with a clear picture of the returns offered by broad equity markets and safe assets.

From conversations with friends and clients, we’ve become aware of how hard it is to find the information we need to make this most fundamental investing decision. Unfortunately, the essential information investors need is not printed amongst the thousands of prices, returns and other numbers published daily in the Wall Street Journal, or in any other major, broadly-available financial publication. We thought it would be interesting to see whether finance professionals had figured out how to find this information. If we found that even these investors were mostly flying blind, we could reasonably infer that most other investors would be too.

So, a few weeks ago we took the opportunity of a dinner party to poll 19 of our friends, all finance mavens, including three university professors of finance. Our fellow diners had an average of four decades of financial market experience. Spoiler alert: based on the results, we’ve decided to begin publishing this essential information at the end of each quarter.

The Survey

Our first question was to ask for the diners’ estimates of the Cyclically-Adjusted Earnings Yield (CAEY) for the broad US equity market, and the broad non-US equity market.

This question is motivated by our view that the best and most popular estimate of the long-term real return of the broad equity market is its CAEY.² This metric might be more recognizable expressed as 1/CAPE, where CAPE is the Cyclically-Adjusted Price-earnings Ratio popularized by Professors Robert Shiller and John Campbell. While there’s no universal agreement that CAEY is the best predictor of long-term real equity returns, we’re pretty sure that it is the most popular one used in practice.

At Elm Wealth, our metric of choice is P-CAEY, a variant of CAEY we’ve developed that accounts for most companies not paying out all their earnings as dividends. We’ve written a note about P-CAEY with the full background and details here.

Our dining companions were not especially well-informed about current CAEY levels, as illustrated in the two charts below. While about 40% of the respondents gave a pretty accurate reading, 60% were disturbingly wide of the mark. The standard deviation of the estimates was about 2.8%. This is roughly equal to the standard deviation you’d get from people throwing darts at a board with equal likelihood of all numbers across a range of 0% to 10%.³ In other words, it was as if people had no idea where the CAEY was other than that it was positive and less than about 10%.

Our second question was asking our diners for their estimate of the real yield on 10-year US Treasury inflation protected bonds (TIPS).

We chose the 10-year TIPS yield because it is directly comparable to the stock market return estimate provided by CAEY, which is also a real (inflation-adjusted) return. Also, TIPS are particularly relevant for long-term investors who care more about the lifetime inflation-adjusted spending their wealth can support than about the present value of their wealth. This makes long-term TIPS the safest asset they can buy. We’ve written more about this in our note here.

Our dinner companions did pretty well with this question. The average answer was 1.9%, which was pretty close to the going yield of 2.1% at dinner-time, with 70% of the respondents giving an answer in the range of 2% to 2.5%. The other 30% of the guesses were all lower than 2%, averaging 1.25%. Altogether, it seems like these finance professionals had a pretty good handle on this important piece of information.⁴

Our plans for making this information available to you

Starting in early October, we’ve been sending a quarterly email to our research subscribers with Cyclically-Adjusted Earnings Yields for broad equity markets and TIPS yields (if you’re receiving this, you’re already signed up. If you think others might enjoy, they can sign up here – and yes, it’s free). Over time, we may add other useful bits which are also hard to find. We also plan to make this information, updated daily, available on our website.

You can piece this information together from other sources, which we describe in the shaded box below. It is our hope that, before long, this information will become widely and freely accessible via the Wall Street Journal, Bloomberg and other popular media sources. It’s hard to see how investors can make sensible, informed decisions without having forward-looking estimates of stock market and safe asset returns.

Other sources for TIPS yield and CAEY

Bloomberg: United States Rates & Bonds, or a Google search for “10 year TIPS yield” should take you to the St. Louis Federal Reserve page or to Treasury Direct, which both give a pretty up-to-date reading.

You can find Shiller’s version of CAPE in a spreadsheet he makes available on his website.

For non-US equity markets, Barclays publishes current and historical CAPE values here. The CAEY for these markets is equal to the reciprocal of CAPE figures provided.

Metrics of risk and risk-aversion are important as well, as we discuss in our book The Missing Billionaires. In this note, we focus on the return of equities and safe assets because we believe those are inputs for which generally-accepted metrics are available using public information, but are not regularly published in major media outlets.
An alternative approach taken by some investors is to extrapolate from past stock market returns. We aren’t fans of this approach, as we don’t believe the past is generally a good predictor of the future in the domain of investments, especially when a forward-looking estimate of the expected return of the stock market is available in the form of CAEY. Another approach, embraced by most wall street stock research departments, is to make an estimate by blending a range of subjective factors, including estimates from the first two approaches above. This might be the best approach if only the experts could agree (or if an average of these estimates were published on a consistent basis). Another short-coming is that these experts tend to focus on estimating short-term stock market returns, which is extra challenging, and if followed would lead to wild swings in investor asset allocations. It would also help if the experts making these estimates were free of the conflicts that arise from them working at firms that are more profitable when their clients invest more in stocks and when the stock market goes up.
The standard deviation of a uniform distribution is (U – L)/√12, where U is the upper end of the range and L the lower end. (10% – 0%)/√12 = 2.9%.
Our dinner took place in London, and so we also asked for estimates for the yield on 10-year UK inflation-linked bonds. Our friends’ guesses were far less accurate, which reflects the fact that these yields are surprisingly difficult to find in the public domain.

Direct Indexed Tax Loss Harvesting: Is the Juice Worth the Squeeze?

Elm Admin — Tue, 27 Aug 2024 04:00:00 GMT

August 27, 2024

Tax Matters

Direct Indexed Tax Loss Harvesting: Is the Juice Worth the Squeeze?

By Victor Haghani and James White ¹
Estimated reading time: 6 min.

The content on this page is being provided as general market commentary and for educational purposes only. It does not constitute any form of investment advice, or any form of recommendation to buy or sell any securities or adopt any investment strategy mentioned herein, or any form of advertisement for Elm Wealth services or strategies. Any investment strategies and investment results discussed herein are for illustration purposes only in the context of the commentary, and do not reflect actual or hypothetical Elm Wealth strategies or results, or an offer to provide such strategies or results. This content is intended only to provide observations and views of the author(s) at the time of writing, both of which are subject to change at any time without prior notice. The information contained in the commentaries is derived from sources deemed by Elm Wealth to be reliable but its accuracy and completeness cannot be guaranteed. This material does not have regard to specific investment objectives, financial situation and the particular needs of any specific reader. Any views regarding future prospects may or may not be realized. Past performance is no guarantee of future results.

Introduction

Most people have come to agree that investing in low cost, diversified index funds is a good idea. Most people also like paying less in taxes and deferring their payment into the future.² Direct Indexed Tax Loss Harvesting (DI) programs combine these benefits into one package. And wealth managers like them too, as they can charge extra fees by putting their clients into DI programs rather than steering them into index funds offered by Vanguard or Blackrock. Investors have been so convinced of the benefits of DI that Goldman Sachs, Morgan Stanley and JPMorgan manage upward of $300 billion in these programs.³

In this note, we’re going to throw some cold water on the DI love-fest by explaining why most tax-sensitive investors would be better off with a simpler approach to tax loss harvesting.⁴ We call this approach Segmented ETF investing, and it involves building your desired portfolio using low-cost sector and (optionally) international ETFs. Segmented ETF investing is expected to generate a similar amount of tax losses as Direct Indexing, but without DI’s costs, risks and limits on diversification. The main problems of Direct Indexing stem from the difficulties of attempting to harvest single-name losses, while tightly tracking a benchmark index and complying with the Wash Sale rule.

How does Direct Indexed Tax Loss Harvesting work?

Most DI programs involve a separately managed brokerage account in which the investment manager buys a portfolio of hundreds of individual US stocks, with the goal of matching an index such as the S&P 500.⁵ Then, the manager routinely sells any stocks below their cost basis in order to realize those capital losses.⁶ Once a loss is realized, 30 days must elapse before the same stock can be repurchased – the “Wash Sale” rule – so the manager must buy different stocks to replace those which were sold. In doing so, they attempt to keep the portfolio statistically matching its benchmark index as closely as possible – but it’s impossible to perfectly match the benchmark once harvesting has commenced.

How much capital losses should you expect to realize?

To evaluate DI against alternatives, we need a sense of the magnitude of losses we can expect to realize, and what it depends on.

Let’s say there was no Wash Sale rule, so you can realize losses anytime they’re available without changing your portfolio composition. For each stock, the amount of losses we should expect is related to the value of a put option on the stock.⁷ Just as with a put option, the expected harvest will depend on the stock’s volatility, its dividend yield, the risk-free rate and the horizon. In the chart below, we show an estimate of expected harvesting over different horizons and volatility levels.⁸

The average volatility of individual US stocks over the past five years has been 35%, as shown on the chart above.

Notice that the amount of losses harvested is roughly proportional to the stock’s volatility. This makes it pretty clear why DI was developed: individual stocks are much more volatile than broad stock indexes. Notice also that as horizon lengthens, the amount of expected harvesting increases (as shown by how much each line rises above the line below it), but by less and less. We’ll come back to discuss the implications of these characteristics shortly.

Direct Indexing headwinds

There are three major consequences from complying with the Wash Sale rule, which together reduce the amount of harvesting expected in real-world DI programs:

Over time, the portfolio will not match the index exactly and may earn higher or lower returns than the index. This is known as “tracking error.”
Managers and clients are worried about too much tracking error – with reason, because it can be hard or impossible to distinguish “random” tracking error from systematic underperformance. As a result, managers generally operate under a relatively tight tracking-error constraint, which can significantly limit the amount of harvesting available.
In the long term, some of the tracking error will be hard-baked into the portfolio, since bringing the portfolio back to the index would realize capital gains which is at odds with the tax-loss harvesting objective.

One particularly disturbing aspect of DI tracking error is that there is reason to fear it has a negative expected return. Whenever an investment strategy veers away from the market portfolio, it is engaging in a zero-sum activity requiring some other market participant to be on the other side of the trades it is doing. The trades that DI portfolios need to make are largely predictable by sophisticated market participants. While it’s not possible to know exactly who is on the other side of DI trades, we do know that there are a number of hedge funds and trading firms – Citadel, RenTec, Virtu and DE Shaw, to name just a few – that make a pretty steady and sumptuous living by being at the top of the stock-trading food chain.

With the passage of time and the general expected upward drift of the stock market, there will be fewer and fewer tax loss harvesting opportunities for the portfolio. Eventually the portfolio will have investments almost entirely with unrealized gains. This is mitigated to a small degree by the potential of investing incoming dividends, although with the 1.25% US stock market dividend yield, this doesn’t help much over typical horizons. Unfortunately, even though a “seasoned” DI portfolio may not be generating significant harvesting opportunities, it’s still a complex portfolio of hundreds of stocks which must be monitored and re-balanced, and will usually continue to be subject to DI management fees.

A better way: Segmented ETF portfolios

We believe that for most investors, Segmented ETF investing is a better way to combine the benefits of index investing with tax-loss harvesting.

Start by choosing the baseline portfolio of asset classes that is right for you.

Next, build your portfolio using the most segmented, broadest range of low-cost index ETFs to represent that baseline. For example, the broad US stock market can be segmented into eleven low-cost sector ETFs, and regional international ETFs can be used for the broad non-US stock market if desired.

Finally, realize losses in that portfolio of ETFs whenever available. The key here is that this can typically be done with considerably less tracking risk – and effort – than in Direct Indexing. This is because multiple ETFs tracking similar, but not identical, indexes are available for most asset buckets. For example, there are multiple “flavors” available in ETF form for the US Real-estate sector – they’re different enough that moving between them doesn’t trigger the Wash-Sale rule, but similar enough that switches introduce much less tracking error compared to Direct Indexing.

Recall that the expected amount of harvested losses is roughly proportional to the volatility of the assets in the portfolio. Industry sector ETFs and regional international ETFs are, on average, more volatile than the broad stock market, and have exhibited about 70% of the volatility as the average individual US stock. The table below compares realized volatility over the past five years.⁹

	Realized Annual Volatility
US Single Stock Average	35.1%
US Sector Average	25.2%
S&P 500	21.2%
Intl. Region Average	21.1%
Intl. Total Market	19.6%

To a first approximation, a TLH program using sector ETFs would deliver about 70% of the capital losses that a DI program could potentially generate – but remember that most DI programs do not deliver on their full potential, because most programs put a limit on tracking risk. From our own experience and from discussions with people who have been in popular DI programs, we believe these programs typically harvest less than 70% of their potential harvesting, were tracking risk not a concern.

In summary: you should expect roughly the same amount of loss harvesting from a portfolio of sector and regional international ETFs as can be expected from Direct Indexing with typical limits on tracking risk.

Segmented ETF investing: have your cake, and eat it too

Segmented ETF investing is superior to Direct Indexing in almost every dimension beyond the roughly equivalent expected harvesting generated by the two approaches.

Tax-loss harvesting an ETF portfolio is usually offered for zero additional fee from most ETF portfolio managers. The weighted average expense ratio of the underlying ETFs in a segmented ETF portfolio is 0.1% or lower. This compares to fees for DI of around 0.35% for very large accounts, and in many cases more than 1%. For most investors, the effective cost of these fees is considerably higher due to IRS limits on deducting managed account fees against income. Such fees can completely eliminate the expected risk-adjusted benefit of Tax-Loss Harvesting for many investors.
Segmented ETF investing involves much lower tracking risk and, more critically, no reason to believe that the tracking error will be negative.
The Segmented ETF portfolio will continue to represent the desired baseline, rather than drift into an unbalanced single stock portfolio with baked-in tracking risk versus the desired index and the need for ongoing management even when the portfolio has limited opportunities for generating capital losses.
Another advantage of the Segmented ETF approach is that it gives you the ability to choose a more diversified baseline, including exposure to international equities, rather than being limited to a portfolio of several hundred of the largest US stocks. For example, Vanguard’s eleven US industry sector ETFs give exposure to over 2,500 individual US stocks, and their international ETFs give exposure to another 10,000 stocks! Recent research has highlighted the importance of investing with the broadest diversification possible by showing that the best-performing 4% of listed companies account for the net gain for the entire US stock market since 1926.¹⁰ A DI program which invests in just 10% or less of available stocks runs the risk of missing out on some of those 4%.

There is one advantage of DI worth noting, which is that it is expected to generate a steadier stream of harvested losses than a Segmented ETF portfolio. This is because the idiosyncratic risk of individual stocks is much higher than that of industry sectors or international regions. Even if the market goes up, there will usually be quite a few individual stocks that will offer the opportunity to realize losses.

If desired, the lumpiness of the capital losses generated by a Segmented ETF approach can be offset by owning a modest extra amount of each ETF. This hedge will compensate you for having fewer losses to realize if the market goes up.

Both approaches can be customized to reflect your desire to underweight a particular sector, such as a finance industry professional wanting to avoid financial stocks. However, there are some customizations – such as wanting to only own companies with headquarters in particular US states – which Segmented ETF investing will not be able to accommodate.

Conclusion: Don’t Let the Tail Wag the Dog

We believe the all-in net benefit of Direct Indexing programs can be improved upon for many investors by instead taking advantage of Segmented ETF investing, and harvesting losses if and when they present themselves. In short, why wouldn’t you prefer an approach which delivers about the same expected amount of capital losses via a more diversified portfolio, but without the high fees, risks and complexity of investing in hundreds of individual stocks?

Investors with an abundance of short-term capital gains

For investors who tend to have a steady stream of short-term capital gains, the conventional tax-loss harvesting approach can be significantly improved upon – and in a seemingly counter-intuitive way.

Instead of trying to defer realizing a long-term gain as long as possible, it can make sense to realize the gain immediately after one year has passed since purchase and the gain becomes subject to the preferential long-term tax rate. If the asset is sold and repurchased, the basis in the asset and the “basis clock” are both reset, thereby creating the option to realize a valuable short-term loss if the asset falls in the year ahead. We call this a “clock reset” strategy.

The benefits of the Segmented ETF approach over the DI approach hold in mostly the same ways discussed above for this flavor of Tax-Loss Harvesting, too. For more details, see our research note: When it Pays to Pay Capital Gains (2019).

Improving Our Favorite Returns Estimator - Elm Partners

Elm Admin — Wed, 19 Jun 2024 04:00:00 GMT

June 19, 2024

Featured Insights

Introducing P-CAPE: Incorporating the Dividend Payout Ratio Improves Our Favorite Estimator of Stock Market Returns

By Victor Haghani and James White ¹

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Deflating Concerns About TIPS - Elm Partners

James White — Tue, 07 May 2024 04:00:00 GMT

May 7, 2024

Investing 101

TIPS Do Offer Valuable Inflation Protection – But You Need to Decide What You’re Protecting

By Victor Haghani and James White ¹

We’ve been hearing a lot of this:

In mid-2020, I got worried about inflation. I decided to protect myself against a jump in my living expenses by buying U.S. Treasury Inflation Protected Securities (TIPS). I was right to worry about inflation – prices are 21% higher today than they were four years ago, rising by over 5% per annum – but I was wrong to buy TIPS. The value of my long-term TIPS ETF is down about 0.5% in nominal terms, and down 18% adjusted for inflation.² If that’s “inflation protection,” give it to somebody else! ³

Do TIPS offer effective protection against inflation?

TIPS are designed to protect the real spending power of your wealth over a given horizon. If the frustrated investor above had bought TIPS maturing in four years, her wealth would have much more closely tracked inflation over that period.⁴ But while that’s tempting, as we’ll see below, it’s not necessarily her best course.

It’s natural that some investors think of inflation protection strictly as ensuring their portfolio value will rise during periods of unexpectedly high inflation. If that’s truly your only goal, you should buy short-dated TIPS. However, your long-term spending power is impacted not just by inflation, but also by long-term real interest rates. $1 million of wealth goes a lot further when real interest rates are at 4% than when they’re at 0%.

If your goal is to protect long-term spending power rather than the narrower goal of protecting inflation-adjusted wealth, then longer-dated TIPS (owned directly or through ETFs) make sense and are effective. However, just as “no man can serve two masters,” TIPS cannot protect inflation-adjusted wealth in the near-term while simultaneously protecting long-term spending power.

In “Back to the Future: Reviving a 19th Century Perspective on Financial Well-Being,” we discuss why we think protecting long-term spending power is the more desirable objective – and, for readers looking for a deeper dive, we discuss a number of related issues in these two notes as well:
How I Learned to Stop Worrying and Love the Bomb
A Sheep in Wolf’s Clothing

How do TIPS work?
TIPS are bonds issued by the US Government, with a fixed maturity (e.g. 10 years) and fixed percentage real coupon (e.g. 2%). Every day, the bond’s redemption value and coupon payments are adjusted based on the headline Consumer Price Inflation (CPI) Index.⁵ If you buy the bond at par and hold it to maturity, you’ll earn a real (i.e. adjusted for inflation) return equal to the percentage coupon, and a nominal return equal to the real return plus CPI. In the meantime, as with any bond, its market value will fluctuate based on the going market real yield for a given maturity.

See TreasuryDirect for more information about the mechanics of TIPS.

Protecting Long-term Spending Power

Let’s look at an example. You have $1 million of savings you want to convert into a 25-year string of constant inflation-adjusted annual cash flows, to “lock in” your real spending power over that time. You buy a portfolio of TIPS with amounts selected so that the interest and principal payments will generate that desired series of real cash flows. To make the math super simple, let’s assume all the bonds are available at a real yield of 0%: then the $1mm you spend on the portfolio of TIPS will provide $40,000 per year of constant inflation-adjusted income.⁶

A year later, your fears of higher inflation are realized with one-year CPI running at 5%! The Fed has hiked short-term interest rates by 4%, and all the bonds you bought are now trading at a real yield of 2%. This isn’t completely fictitious, being close to what actually did happen between April 2022 and August 2023. So now you look at your brokerage statement and see that your $1mm of starting capital plus intermediate payments has turned into just $836k, for a loss in value of 16%. You might be feeling like your inflation protection let you down – but did it?

Your objective in buying this portfolio of TIPS was to create an inflation-hedged $40,000 per year of income to spend. While the present value of your portfolio is indeed lower by 16%, the cash-flow stream you created is still intact, and you’ll continue to get $40,000 per year for the next 24 years, adjusted for inflation.

This scenario is not particularly unusual, in that when inflation runs unexpectedly hot, the Fed is likely to hike short-term interest rates at a fast enough pace to eventually slow the economy and reign in inflation, which normally will lead to higher real interest rates on TIPS.

Does this make TIPS a risky investment? While the present value of the portfolio of TIPS you bought fluctuates (wildly, in this example), its long-term spending power remains constant.⁷ If you have a short horizon, buying long-term TIPS is definitely risky – but, if you think about risk with respect to your long-term spending power, long-term TIPS are relatively safe.⁸

TIPS and Taxes
In our discussion above, we assumed an investor owning TIPS in a non-taxable account. For taxable investors, the inflation-protection of TIPS is diluted by taxation of the inflation component of TIPS returns. For example, if you have a 40% marginal tax rate on interest income, and buy long-term TIPS at a real yield of 2%, and if inflation runs at 2.5%, the after-tax return is 2.7%, for a 0.2% real, after-tax yield. However, if inflation instead runs at 5%, the investor will earn a real yield of -0.8%. While the inflation protection of TIPS is weakened by US taxation, TIPS will still usually provide greater inflation protection than T-Bills or nominal bonds.⁹

Is it better to own TIPS via owning bonds directly or through a TIPS ETF?

The question, “Should you buy bonds or bond funds?” gets a lot of discussion in the financial press.¹⁰ We often read that it’s better to buy individual bonds rather than bond funds, as you will never suffer a loss on individual bonds as long as you hold them to maturity. We think this argument is, at best, confused.

If you want to protect against inflation or lock in a real rate of return to a specific date, then you should buy and hold individual bonds, whose maturity will naturally run down as you approach your target date. However, we think this is a relatively rare use-case. Few people, even those getting on in years, have a specific date with their name on it. Instead, many investors either have a medium-to-long and rolling horizon, or are allocating between asset classes.

In either of the latter cases where you’re trying to maintain the duration of a bond portfolio, the mechanics of holding individual bonds versus a bond ETF will be very similar. In both forms, bonds will naturally be running off, and you’ll be replacing them by buying new issues.¹¹ If the ETF is trading close to its Net Asset Value, as TIPS ETFs normally do, the returns will also be very similar between holding the ETF and a similar portfolio of individual bonds. The main difference will be that the ETF charges a management fee (0.03% in the case of SCHP), but is more convenient and likely has lower transaction costs than managing your own portfolio of individual bonds.

This not is not an offer or solicitation to invest. Past returns are not indicative of future performance.
Based on the largest TIPS ETF, SCHP, and including reinvested dividends.
Or, for an expression of these sentiments in the financial press, see this FT article by Toby Nangle: “TIPSplaining a lousy inflation hedge.”
In this case, she’d still have underperformed inflation by about 4% in total, since TIPS maturing in four years were trading at a real yield of -1% in mid-2020. She also could have bought and rolled shorter-dated TIPS.
TIPS at issue come with a nice but small freebee: deflation protection. The ultimate redemption value will not be less than par, even if there has been deflation over the life of the bond.
Assume all the TIPS have 0% coupons, so they’re all trading at par with a 0% real yield. Then you’re just buying $40,000 notional of 25 bonds, with maturities from one year to 25 years from present. The portfolio costs you 25 x $40,000 = $1mm, and that will provide you with an inflation-adjusted $40,000 per year.
Some readers have asked what should they have done if they strongly believed that the yield on long-term TIPS was going to increase from -1% to +2% before long, which actually did come to pass? Are we suggesting that such an investor still buy TIPS since it is the safest way for them to protect the long-term real spending power of their wealth? No, we are not. What we are suggesting is that the investor should specify the return and risk of the various other investments he can make relative to the lowest risk asset for him, which we suggest is TIPS for long-term investors. When the investor assessed each potential investment relative to the -1% yield of long-term TIPS, he may well have decided, based on his views, that rolling T-Bills or owning equities had a sufficiently high return relative to their risk versus TIPS to warrant holding those and owning no TIPS (or even shorting TIPS). Just because TIPS are the safest asset doesn’t mean the investor needed to own them.
Some readers have asked us whether it is better to roll short-dated TIPS, as this will offer the same inflation protection as owning long-term TIPS but without the interest rate risk. We think the contrary is the case for people who are concerned with protecting the long-term spending power of their wealth; rolling short-term TIPS is the strategy with interest rate risk. If you roll one-year TIPS for 10 years and real rates drop over the period, your real spending power has gone down, and vice versa if real rates rise over the period. But if you own 10-year TIPS, you are locking in a known real quantity of spending over the period, regardless of what happens with interest rates in the meantime. In this sense, not relative to your nominal wealth but relative to your spending power, it’s the one-year TIPS which give you rates exposure, while the 10-year TIPS have none. From a “balance sheet” perspective, it’s the opposite – one-year TIPS have nearly no apparent interest rate exposure, while 10-year TIPS have plenty – but for most people, we believe the spending-power perspective is more helpful than the balance-sheet perspective, since it’s maximizing the utility of lifetime spending (and bequesting) that’s the most sensible overall financial objective function.
We wish US taxation on inflation-protected bonds was the same as it is in the UK, where only the coupon income is taxed, but not the inflation-adjustment of the principal repayment at maturity.
For example, this recent article by the WSJ’s Jason Zwieg: “What to Do With Bonds When Inflation Won’t Die.”
For ETFs which sell bonds that fall outside the index, selling and replacing with longer bonds can have a tax impact, realizing capital gains or losses, but is unlikely to have a significant impact on returns as long as the ETF portfolio remains close to the target duration.

Merton Share Derivations: What's in your denominator? Elm Partners

James White — Thu, 02 May 2024 04:00:00 GMT

May 2, 2024

Uncategorized

Merton Share Derivations: What’s in your denominator?

By Jeffrey M. Rosenbluth and James White ¹

1. Introduction

The Review of Economics and Statistics published a pair of companion papers in 1969. “Lifetime Portfolio Selection by Dynamic Stochastic Programming”, by Paul Samuelson and “Lifetime Portfolio Selection under Uncertainty: The Continuous-Time Case”, by Robert Merton. Both deal with the question of how to allocate one’s portfolio between a risk-less and risky asset in a multi-period setting. The Samuelson paper considers the discrete time case and Merton’s the continuous time one. Merton solves this problem and provides a closed form solution of the highly stylized case where the risky asset rate of return follows a Brownian Motion (so that its price follows a Geometric Brownian Motion), the risk-less rate is constant, the utility function is CRRA (Constant Relative Risk Aversion) and the investor re-balances the portfolio continuously. Under these assumptions, he also shows that portfolio selection is myopic, that is, independent of the investment horizon and in fact the investor keeps a constant fraction of her wealth in the risky asset. We call this fraction the Merton Share. It is important to note that the Merton Share formula would be different under alternative sets of assumptions and that some of Merton’s assumptions are unrealistic: in particular, continuous re-balancing and Geometric Brownian Motion for asset prices. Nevertheless, we believe that using the Merton Share as a rule of thumb makes good sense and will often be close to the correct solution.

Merton used the theory of optimal control, and the Bellman principle of optimality in particular, to derive a partial differential equation (the Hamilton-Jacobi-Bellman equation) to solve the problem. In general, finding a closed-form solution to the HJB equation is rare and solutions are typically found numerically. In this note, we motivate and derive the Merton Share several different ways that are hopefully easier mathematically and provide more intuition as to why the formula makes sense. In doing so, we often deal with a single-period model and sometimes need to use approximations to derive the formula. The framework we will be using throughout to derive this formula is Expected Utility maximization; we will also often be assuming CRRA utility.

U ( W ) = { W 1 − γ 1 – γ if γ ≠ 1 log ⁡ ( W ) if γ = 1

The defining feature of CRRA utility functions and hence its name, Constant Relative Risk Aversion, is that relative risk aversion is constant:

R ( W ) = − W U ” ( W ) U ′ ( W ) = γ

We denote by k̂ the optimal fraction of wealth to invest in the risky asset. The Merton Share formula is:

k ^ = μ γ σ 2

where μ is the expected excess return,² that is the return on the risky asset minus the risk-less rate and σ is its standard deviation. This formula certainly passes the smell test, the Merton Share is higher when excess return is higher, and lower when standard deviation and risk aversion are higher. The variance term in the denominator σ² as opposed to perhaps σ may seem less intuitive though.

2. Motivation

Let’s start to reveal why the denominator in the Merton Share is variance as opposed to standard deviation. We actually have another relation for k that relates it to the standard deviation of the portfolio. If we invest a fraction k of wealth in the risky asset with standard deviation of return σ, then the standard deviation σ_p of the portfolio is kσ.

σ p = k σ

Rearranging, we have:

k = σ p σ

If we choose k = k̂ (the Merton Share), and let σ̂_p denote the standard deviation of the portfolio at k̂, the we obtain:

σ ^ p σ = μ γ σ 2

so that:

σ ^ p = μ γ σ

This, hopefully, provides some intuition for why variance in the denominator of our Merton Share formula makes sense. It says that the risk (standard deviation) of the optimal portfolio is the ratio of excess return to standard deviation of the risky asset divided by the coefficient of relative risk aversion γ. The ratio of excess return to standard deviation is called the Sharpe Ratio, and is a commonly-used metric of the quality of a risky asset or trade.

Our intuition didn’t lead us far astray. It’s the risk of the optimal portfolio – not the optimal fraction – that is proportional to the Sharpe Ratio.

2.1 Myopic Portfolio Choice

Let’s approach the question of “Why variance in the denominator?” from another angle. First, in addition to CRRA with relative risk aversion γ, we also make the more restrictive assumption that asset returns are independent over time.³ Consider an investor with a two-period horizon. At the end of the first period the investor is faced with a single period optimization problem and since her risk aversion does not depend on wealth and returns are independent, the solution to this problem does not depend on how much was invested in the risky asset in Period 1. Now, the time 0 portfolio choice problem does not depend on time 1 wealth. Hence, the investor makes a single period portfolio choice at time 0 as well. When an investor makes the same portfolio decisions regardless of horizon, we say portfolio choice is myopic. By backward induction the above argument can be applied to any number of periods. This shows that with CRRA utility and time independent returns that portfolio choice is myopic. In our case we have in fact an even stronger result, constant portfolio choice over time, in which the investor holds the same fraction of wealth in the risky asset in each period. Let’s state this a a theorem and prove it more formally.

Theorum 1. If returns follow a stochastic process with independent increments, then for investors with CRRA utility of wealth, portfolio choice is constant over time.⁴

Proof. By the scale invariance property of CRRA utility, we know that k̂ does not depend on W_t that is wealth at any time t. From the independent increments assumption, we know that future risky asset prices do not depend on past wealth or past choices of k̂. Therefore, portfolio choice is myopic and k̂ is constant.

How does this help us to motivate the use variance in the denominator of the Merton Share? If portfolio choice is myopic, that means we would invest the same fraction of wealth in the risky asset for any horizon t. Suppose we have the function:

k ( X t ) = μ t ρ ( X t )

and we are choosing between standard deviation and variance for the operator ρ. We know that if our choice is myopic, then k(X_t) will not depend on t. For this to be true, its denominator must be a factor of t so that the t‘s will cancel. If X_t has independent increments, as do the majority of the stochastic processes employed to model excess returns, then StDev(X_t) = σ √t, so ρ can’t be standard deviation. On the other hand, variance Var(X_t) = σ² t works just fine.

3. Derivations

We provide core derivations (and two more in appendix) that are designed to motivate different aspects of the portfolio choice problem as it relates to the Merton Share.

3.1 Static Approximation

In this section, we assume the risky asset excess return is identically distributed over periods of the same length and that they are uncorrelated. In this case, both mean and variance are proportional to the horizon. The utility function U(W) is required to be twice differentiable and concave. We approximate this utility function with a Taylor series, resulting in a formula that is only valid for short horizons ⁵. We then specialize this result to the CRRA utility case.

Let W be the value of the initial portfolio. For a portfolio return Y, let U(W(1 + Y)) be the utility after one period with horizon t. Since U is twice differentiable we can approximate it with a second order Taylor series about Y = 0:

U ( W ( 1 + Y ) ) ≈ U ( W ) + U ′ ( W ) Y W + 1 2 U ” ( W ) ( Y W ) 2 E [ U ( W ( 1 + Y ) ] ≈ U ( W ) + U ′ ( W ) E [ Y ] W + 1 2 U ” ( W ) E [ Y 2 ] W 2 = U ( W ) + U ′ ( W ) E [ Y ] W + 1 2 U ” ( W ) ( Var [ Y ] + E [ Y ] 2 ) W 2

Notice what is happening here, the combination of approximating utility by a Taylor series and taking its expected value introduces the moments of the probability distribution into the equation! If we take more terms of the Taylor series for a better approximation, then we need more moments. This should gives us additional comfort in choosing variance, not standard deviations, in the Merton Share formula.

In our case, the portfolio with a fraction k of wealth invested in the risky asset and the remainder in the risk free asset Y = (r + kX)t, where t is the horizon of the investment. The excess return X has mean μ t and variance σ² t as per our assumption, and r is the risk free rate of return. So E[Y] = (r + kμ)t and Var[Y] = k²σ² t. Since E[Y]² = (r + kμ)² t², it can be ignored for small t.

We want to maximize:

U ( W ) + ( r + k μ ) t U ′ ( W ) W + 1 2 k 2 σ 2 t U ” ( W ) W 2

We differentiate with respect to k to obtain the first order condition:

0 = μ t U ′ ( W ) W + k σ 2 t U ” ( W ) W 2 = μ U ′ ( W ) + k σ 2 U ” ( W ) W

Hence:

k ^ = − μ U ′ ( W ) σ 2 W U ” ( W )

Recall from Section 1 the coefficient of relative risk aversion:

R ( W ) = − W U ” ( W ) U ′ ( W ) 1 R ( W ) = − U ′ ( W ) W U ” ( W )

Substituting this in to the above formula for k, we arrive at:

k ^ = μ R ( W ) σ 2

This is a fairly general result, we have made very few assumptions about the utility function and asset return distribution.

Specializing to the CRRA utility case R(W) = γ so that:

k ^ = μ γ σ 2

the Merton Share.

3.2 Asset Prices follow a Geometric Brownian Motion

In this section, we derive the Merton Share using assumptions similar to the ones Merton himself used. We assume CRRA utility, and have a risky asset S_t that follows a Geometric Brownian Motion (GBM) and a risk-less asset B_t with continuously compounded return r. That is:

d S t S t = ( r + μ ) d t + σ d Z t d B t B t = r d t

where Z_t is a Standard Brownian Motion (i.e μ = 0, σ = 1).

This setup is very common in finance. It is also very different from the derivation above, in that we are now have a dynamic optimization problem. Hence, k̂ is now a stochastic process that depends on the price path of the asset and time t, – call it k̂(S_t, t). Solving for k̂(S_t, t) is a problem in Stochastic Control⁶ which is beyond the scope of this note and requires quite a bit more mathematical machinery ⁷. But by employing Theorem 1, we know k̂ is constant and hence we can side step the stochastic control problem. Note that we still require the portfolio to be re-balanced to contain a fraction of wealth k̂ in the risky asset at every moment in time.

Given the above differential equations we can write down the stochastic differential equation (SDE) for wealth. We can think of this as saying that instantaneous returns on the wealth portfolio are k times the instantaneous return on the risky asset plus 1 – k times the return on the riskless asset.

d W t W t = ( 1 – k ) d B t B t + k d S t S t = ( 1 – k ) r d t + k ( r + μ ) d t + k σ d Z t = ( r + k μ ) d t + k σ d Z t

Just as the risky asset is following Geometric Brownian Motion, we can see that the portfolio also is following GBM, i.e. the portfolio is also expressed as an SDE for GBM. The difference now is that the drift is r + kμ and the diffusion term is kσ, hence:

W t = W exp ⁡ ( ( r + k μ – 1 2 k 2 σ 2 ) t + k σ Z t )

Without loss of generality, we can let W = 1, letting R_t = (r + kμ -½ k² σ²)t + k σ Z_t. We have:

E [ W t 1 – γ 1 – γ ] = E [ exp ⁡ ( ( 1 – γ ) R t ) 1 – γ ] = exp ⁡ ( ( r + k μ – 1 2 k 2 σ 2 ) t + 1 2 ( 1 – γ ) k 2 σ 2 t / 2 )

where we have used the fact that the mean of a log-normal random variable with drift m and diffusion term s is:

exp ⁡ ( m + 1 2 s 2 )

For γ > 1, maximizing this expression is the same as minimizing:

( r + k μ – 1 2 k 2 σ 2 ) + 1 2 ( 1 – γ ) k 2 σ 2

The first order condition is:

μ – k σ 2 + ( 1 – γ ) k σ 2 = μ − γ k σ 2 = 0

Solving for k gives:

k ^ = μ γ σ 2

Appendix

Normal Returns and Constant Absolute Risk Aversion (CARA) Utility

CARA utility and normally-distributed returns provide the only case where the Merton Share is an exact formula in the single-period world. Normal returns are undesirable since they allow negative asset prices and can’t be used for both sub-period and total period returns. The CARA (exponential) utility function exhibits constant absolute risk aversion A, which is also unrealistic. Despite these shortcomings, this case provides an instructive example. The CARA (exponential) utility function is:

U ( W ) = − exp ⁡ ( − A W ) A

To maximize expected utility, we can minimize the negative of U. Letting W be the starting wealth, we have:

min k E [ exp ⁡ ( − A ( 1 + r + k X ) W ) ] = E [ exp ⁡ ( − A ( 1 + r ) W ) exp ⁡ ( − k A X W ) ] = exp ⁡ ( − A ( 1 + r ) W ) E [ exp ⁡ ( − k A X W ) ]

The expectation of the log-normal random variable:

E [ exp ⁡ ( − k A X W ) ] = exp ⁡ ( − k A μ W + k 2 A 2 σ 2 W 2 2 )

So our minimization problem becomes:

min k exp ⁡ ( − A ( 1 + r ) W ) exp ⁡ ( − k A μ W + k 2 A 2 σ 2 W 2 2 )

which is the same as:

max k A μ W – k 2 A 2 σ 2 W 2 2

The first order condition is:

0 = A μ W – k A 2 σ 2 W 2 = μ – k A σ 2 W

Solving for k gives:

k ^ = μ A W σ 2 = μ R ( W ) σ 2

This is effectively the Merton Share formula, and is the same result we obtained in Section 3.1.

Let’s explore this result a bit further. Since we assume that a utility function U is strictly concave, we know from Jensen’s inequality that:

E [ U ( W 1 ) ] < U ( E [ W 1 ] )

We can think of this as an equality:

E [ U ( W 1 ) ] = c U ( E [ W 1 ] )

for some c > 1. For most combinations of utility function and wealth distribution, we do not know what c is explicitly, but for the combination of exponential utility and normal returns we do. It’s exp(k² A² σ² W² /2). This shows that our maximization problem is a trade-off between mean μ and variance σ².

Quadratic Utility

The quadratic utility function

U ( W ) = − 1 2 ( a – W ) 2

is not very realistic in that it has increasing absolute risk aversion and a “satisfaction” point beyond which more wealth lowers utility.

Its Arrow-Pratt Measure of Absolute Risk Aversion is:

A ( W ) = 1 a – W

It’s often used to demonstrate a utility function whose portfolio selection fraction depends only on mean and variance regardless of the distribution of returns.

As usual, we start with the expected utility maximization problem:

max k E [ − 1 2 ( a – ( 1 + r + k X ) W ) 2 ]

Differentiating with respect to k and setting to 0:

0 = E [ W X ( a − ( 1 + r – k X ) W ) ] = a μ W − ( 1 + r ) μ W 2 – k ( σ 2 + μ 2 ) W 2 = μ ( a − ( 1 + r ) W ) – k ( σ 2 + μ 2 ) W k ( σ 2 + μ 2 ) W = μ ( a − ( 1 + r ) W ) k = μ ( a − ( 1 + r ) W ) ( σ 2 + μ 2 ) W = μ σ 2 + μ 2 ( 1 – r W A ( W ) W A ( W ) )

Static Approximation Revisited

When we derived the Merton Share back in Section 3.1, we made the assumptions that excess returns are identically distributed over periods of the same length and that they are uncorrelated. We needed to do this to ensure that both portfolio return and variance scale with horizon t. This is what allowed us to approximate the solution for small t. It turns out we can drop this restriction if instead we assume that the mean excess return is small. We can always write the excess return X as the sum of its expected return and a random variable with zero mean and the same standard deviation as X, say Z:

X = μ + Z

This lets us take k̂ to be a function of μ, k̂(μ) then we can use a first order Taylor expansion about 0 to estimate it.

k ^ ( μ ) ≈ k ^ ( 0 ) + μ k ^ ′ ( 0 )

And since the optimal investment in a risky asset with zero return is 0.

k ^ ( μ ) ≈ μ k ^ ′ ( 0 )

Let W₁ = (1 + r)W and w̃ = W₁ + k̂(μ)(μ + Z)W. At the optimum, k̂, the first order condition must be 0.

E [ ( μ + Z ) W U ′ ( W ( 1 + r + k ^ ( μ ) ( μ + Z ) ) ) ] = E [ ( μ + Z ) W U ′ ( W ~ ) ] = 0

We use this to calculate k̂'(0) by implicit differentiation. Differentiating the first order condition with respect to μ, then setting μ = 0:

0 = E [ ( μ + Z ) W ( k ^ ( μ ) W + k ^ ′ ( μ ) ( μ + Z ) W ) U ” ( W ~ ) + W U ′ ( W ~ ) ] = E [ Z 2 W 2 k ^ ′ ( 0 ) U ” ( W 1 ) + W U ′ ( W 1 ) ] = E [ Z 2 W k ^ ′ ( 0 ) U ” ( W 1 ) + U ′ ( W 1 ) ] = σ 2 W k ^ ′ ( 0 ) U ” ( W 1 ) + U ′ ( W 1 ) k ^ ′ ( 0 ) = − U ′ ( W 1 ) σ 2 W U ” ( W 1 ) μ k ^ ′ ( 0 ) = μ R ( W ) σ 2

which is the same result we found in Section 3.1. In this case, we see that small means a first order Taylor expansion of k̂ is sufficient, i.e. μ is close to 0.

Elm Wealth Research

A Volatile Mix of MicroStrategy, 2x Leveraged ETFs and Bitcoin

Moonshot or Shooting Star? A Volatile Mix of MicroStrategy, 2x Leveraged ETFs and Bitcoin

Introduction

In this note, we’ll:

Background

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

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

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

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

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

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

Have you ever seen anything like this before?

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

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

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

Leverage It or Leave It? Making Sense of Turbo-charged ETFs Elm Partners

Leverage It or Leave It? Making Sense of Turbo-charged ETFs

Battle Stations!

A Tool for a Fuller Picture of Returns

When do leveraged ETFs make sense for individual investors?

Conclusion

p.s.

p.p.s.

Appendix: List of Leveraged ETFs

References

Elm Wealth Webinar, Nov. 19th, 2024 - Elm Partners

Elm Wealth Webinar, Nov. 19th 2024

Victor Meets the Bogleheads Elm Partners

Victor Meets the Bogleheads

The case for dynamic asset allocation

The case for static asset allocation

A few arguments in favor of static asset allocation that we don’t agree with

Conclusion

Appendix: How some respected personal finance books line up on the static versus dynamic asset allocation debate

Further Reading & References

When a Crystal Ball Isn't Enough to Make You Rich - Elm Partners

When a Crystal Ball Isn’t Enough to Make You Rich

Introduction: Back to the Future

Summary of results

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

Motivations

The Game

Conclusion

Appendix I: Detailed Analysis of Player Results

Unpacking player performance

Appendix II: Putting a value on the crystal ball

Appendix III: Caveats and shortcomings of this study

Appendix IV: The redacted front pages used in the experiment

Further Reading & References

The Most Important Number Not in The Wall Street Journal Elm Partners

The Most Important Number Not in The Wall Street Journal

The Survey

Our plans for making this information available to you

Other sources for TIPS yield and CAEY

Direct Indexed Tax Loss Harvesting: Is the Juice Worth the Squeeze?

Direct Indexed Tax Loss Harvesting: Is the Juice Worth the Squeeze?

Introduction

How does Direct Indexed Tax Loss Harvesting work?

How much capital losses should you expect to realize?

Direct Indexing headwinds

A better way: Segmented ETF portfolios

Segmented ETF investing: have your cake, and eat it too

Conclusion: Don’t Let the Tail Wag the Dog

Investors with an abundance of short-term capital gains

Further Reading and References

Improving Our Favorite Returns Estimator - Elm Partners

Introducing P-CAPE: Incorporating the Dividend Payout Ratio Improves Our Favorite Estimator of Stock Market Returns

Further Reading & References:

Deflating Concerns About TIPS - Elm Partners

TIPS Do Offer Valuable Inflation Protection – But You Need to Decide What You’re Protecting

Do TIPS offer effective protection against inflation?

Protecting Long-term Spending Power

Is it better to own TIPS via owning bonds directly or through a TIPS ETF?

Merton Share Derivations: What's in your denominator? Elm Partners

Merton Share Derivations: What’s in your denominator?

1. Introduction

2. Motivation

3. Derivations

Appendix

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