Elm Wealth Research

Wall Street Journal writer Sam Goldfarb writes about Elm Partners and Victor for the cover of the B Section on November 11th.

“Since 2011, Mr. Haghani has run, from a small office near his home in London, Elm Partners Management LLC, an investment firm that now manages around $550 million of assets. Using a simple algorithm, the firm takes into account valuations and momentum to invest in index and exchange-traded funds across different asset classes.”

Read the full article here.

By Victor Haghani ¹

A few nights ago at the dinner table, I was ritually lamenting my unsuccessful efforts to lose a few pounds. My son Mark, now a high school senior, posed a good question – the kind I’ve learned to think about very carefully. If I lost 5 pounds of fat, where would it go? After letting me flail around a bit, he gave me the answer (hint: neither heat release nor bathroom visits are correct answers.²). Turns out we lose most of the weight by exhaling CO₂. Simple. So, if you want to lose weight, don’t hold your breath!

This reminded me of a similarly “simple” question my friend Larry Hilibrand asked me about 10 years ago that also involved a “conservation of mass” principle. In a world with two assets – cash and equities – how can investors in aggregate increase their allocation to equities, given that for every buyer there has to be a seller (assuming companies aren’t issuing or buying back shares), and so cash can’t come into or leave the market?

I recall some floundering back then too, before the obvious came into view: the price of equities must go up. I thought I was off the hook, but there was one more question: Just how much do equities need to go up if investors in aggregate have 50% in equities and decide they want to increase that to 60%? I encourage you to give it a quick guess before doing the math.

There are a few ways to figure it out, one being to consider that an investor holding $50 in equities and $50 in cash will need the original holding of cash to become 40% of the new total portfolio value after equities go up in price. So, her new total portfolio value would need to be $50 / 0.4 = $125 . The only way for that to happen is for equities to rise in value to be worth $75. That’s a 50% gain, as you can see in the chart below. I don’t know about you, but my guess was a lot lower than 50%.

And, notice from the chart below that a 50% starting allocation is where this impact is the smallest (see the footnote below if you want a formula).³

Of course, the same math holds if investors all want to reduce their allocation to equities. From the same starting allocation of 50%, equities would need to drop by 33.3% for everyone to wind up with a 40/60 equity/cash allocation.⁴

In practice, the impact would be dampened by companies issuing or buying back shares, although it’s worth noting that IPOs in the US have averaged only 0.25% of total US market capitalization annually over the past ten years.⁵

However, in the short term, return-chasing investors, who estimate future expected returns based on historical performance, may have the opposite impact and actually put more fuel on the fire.⁶

This paradigm is a gross over-simplification of reality, but I’ve found it useful, and unexpected.

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1. Victor is the Founder and CIO of Elm Partners. Past returns are not indicative of future performance. This not is not an offer or solicitation to invest.
   A big thank you to Larry Hilibrand, who provided the idea for this note, as well as valuable comments in drafting it, and to James White for helping me make it shorter.
   
2. Heat release would imply a bodily nuclear reaction, and the stuff that leaves us was largely never part of us to lose. And here’s a YouTube video titled Mathematics of Weight Loss if you want a more complete answer. Also, thank you to Lasse Pedersen who pointed me to the even more beautiful chemistry and physics of the question, “Where does a tree get its mass from?” In this clip from BBC series Fun to Imagine (1983), the late Richard Feynman, with his infectious exuberance, explains
   “People look at a tree and think it comes out of the ground, that plants grow out of the ground…if you ask, where does the substance [of the tree] come from? You find out…trees come out of the air!”
3. The equity price change needed to change aggregate asset allocation by ∆% from a starting allocation of E₀ is: ((1 – E₀)/(1 – (E₀ + ∆)) – (1 – E₀)) / E₀ – 1 ≈ for small ∆ , and min occurs at E₀ = 50% .
   
4. The answer is symmetric in log-space, as ln(1.5) = -ln(0.67) , suggesting a 33.3% drop in is the same amount as a 50% gain in log space.
   
5. “Value of initial public offerings (IPOs) in the United States from 2000 to 2021”.
   
6. At Elm, we strongly believe in estimating equity returns looking forward, rather than backwards, as discussed in this video, The Most Important Number You Won’t Find in the WSJ. Even investors who use a long historical window for their estimate will find that a doubling of the market will increase the historical 20-year return by over 3.5% per annum.
   

By Victor Haghani ¹

“…I don’t know.”

That was the only answer I had to each of my dad’s questions. It was very dark as we wandered the streets near my Upper East Side apartment in the early hours of October 20^th, 1987, discussing the events of the prior day – Black Monday.

I was 25 years old and had been on John Meriwether’s Arb Desk for just over a year, following two years working in Salomon’s Bond Portfolio Analysis Group. My memories from the day the US stock market dropped 22% didn’t seem very noteworthy when my friend Rich Dewey asked to interview me for an article, Black Monday Revisited, he was writing for Bloomberg to mark its 30^th anniversary. After all, compared to the other people Rich was interviewing—Paul Tudor Jones, Howard Marks, Stanley Druckenmiller, Ed Thorp and my former Salomon colleagues Eric Rosenfeld and Michael Lewis – what could I add?

I’ve been thinking that my recollection of some things that didn’t happen might be interesting. First of all, Salomon’s Arb Group didn’t do a single trade on October 19^th, or pretty much for that whole week. It wasn’t that we didn’t see great opportunities, but rather that it was clear to everyone that this was a crisis and a time to preserve capital. Salomon was first and foremost a financial intermediary. Our capital was limited, and it would be sorely needed for providing liquidity to clients and projecting financial strength to all our counterparties. Salomon had about $3.5 billion of capital supporting a balance sheet of $100 billion.

We borrowed money to finance our long positions in bonds primarily through the repo market, but we also had to borrow securities to support our short positions. When we shorted a bond, such as the 9.25% of 2/15/2016 that we were running a big position in at the time, we needed to borrow that specific bond from someone who held it in order to make good on the sale. We could borrow money from anyone– a client, a bank or as a last resort even the Fed. But the only party who could lend you a security was someone who owned it free and clear, and hadn’t lent it already. And the lending market was mostly overnight, so we had to roll every day.²

Our biggest fear was that clients who were lending us the securities we were short might ask for them back, forcing us to cover our shorts and unwind our trades. It’s almost axiomatic that when you’re forced to unwind a trade, you lose money on it. With these concerns in mind, John holed us up in a room off the desk, and we put our efforts into methodically triaging our portfolio. We kept the trades we’d be most able to hold to convergence, and cut the ones that were least defensible. We had been having a solidly profitable year up until October, but gave back most of our gains in the few days around Black Monday. The firm’s decision to let us keep our best positions was rewarded with a very profitable 1988.

The defensive orientation on our desk was echoed across the whole firm, and it was probably the same at Goldman, Bankers Trust and all the other trading houses on the Street. As a highly levered financial firm, dependent on short-term funding of our balance sheet, our crisis mentality was about surviving the storm, not trying to profit from it. While the most popular 1987 crash stories celebrate the trading acumen of the likes of Tudor Jones, Druckenmiller or Taleb, the less publicized story of how so much capital was constrained or frozen—an essential ingredient in all market panics– might be the more important takeaway.

Another thing that I don’t remember happening was an economic depression following the stock market crash. Well, I guess that’s because it didn’t! It was supposed to though, just as the Great Depression followed the Black Tuesday of October 1929. In fact, in December 1987, 33 prominent economists (5 with Nobel prizes) issued a statement predicting that “the next few years could be the most troubled since the 1930s.”³ The fact that we moved forward with barely a blip to the real economy makes us feel that October’s stock market crash was bogus, a market move that had nothing to do with fundamentals. But it didn’t have to turn out that way. It’s important to remember what didn’t happen: an alternative future in which the Fed didn’t act as it did (would Volcker have reacted as Greenspan did?), the stock markets fell even further, financial firms started failing, and we got a long and deep recession.⁴

Could it happen again? Of course it could. And it has. Extreme market moves of the magnitude of Black Monday’s 20-times normal daily move have occurred periodically since then, just not in the US equity market or on the one-day time scale. For example, two years ago, the Swiss Franc put in a 40 times daily upward move against the Euro when the Swiss National Bank suddenly abandoned the policy of capping its value.⁵ If we look at more arcane, but still important, markets, we find further examples, such as changes in swap spreads or long-dated equity volatility in October 1998 (what is it with October anyway?), or diversified equity momentum trading strategies that lost close to 90% in 2009, or the melt-down of equity quant strategies the week of August 6^th, 2007.

Plus ça change. Humans, with all our behavioral foibles, are still important players in the markets. While the presumed cause of Black Monday, Portfolio Insurance,⁶ is now defunct, it has been superseded by vast amounts of capital dedicated to algorithmic trading or trend-following, both strategies expressly designed to make money, not to stabilize markets. Risk management systems based on VaR (recent volatility of positions) or a tight stop-loss discipline are inherently destabilizing too. And then we have the Volcker Rule and other post-financial crisis regulatory changes, which have the unintended consequence of dramatically reducing the ability and incentive of the banks to provide liquidity in normal market conditions, let alone in a crisis. What do we have against all this? More circuit-breakers and a tradition of Central Bank intervention to stabilize markets in every crisis since Black Monday.⁷ Hopefully, they will continue to do so, but we should be prepared for when they don’t.

You may wonder, how did this prepare me for another tumultuous October, eleven years later, when I was a partner at LTCM? Stay tuned, its 20^th anniversary is just twelve months away.

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1. Victor is the founder and CIO of Elm Partners. Past returns are not indicative of future performance. This not is not an offer or solicitation to invest.
   
2. We always appreciated the important role played by the repo desk, but during the crisis these guys, who sat at the edge of the trading floor and who at other firms on the Street were treated as back-office operational staff, earned hero status and a mainstream position on the trading floor thereafter.
   
3. “Group of 7, Meet the Group of 33”
   
4. One more thing that didn’t happen in those tumultuous days was our normal afternoon Liar’s Poker session, but it did come back before long.
   
5. January 15, 2015. Within hours it had recovered to being down ‘just’ 20%. Other examples: over a longer horizon, we have the equity market sell-off from late 2007 to early 2009, which was a 60% drop (88% in lognormal terms), a very unlikely move given the typical annual variability of equity markets. In 2013 the US bond market experienced its “Taper Tantrum” when 10-year US Treasury rates almost doubled, to 3%, over a 6-month period. There was also the ‘flash rally’ of October 15, 2014, when the 10-year Treasury rate declined by 0.37% in about 30 minutes, before quickly going most of the way back to where it had been.
   
6. For my younger readers, Portfolio Insurance. The Brady Commission Report concluded that Portfolio Insurance was the proximate cause of the stock market decline. It was estimated (here) that $60-90 BB of assets were following a portfolio insurance strategy, and they needed to sell $10-15 BB of equities on Black Monday. To put this in perspective, the $ value of US equity trading volume is 15x higher and the market capitalization of US publicly listed companies is 10x larger today than it was 30 years ago (World Bank).
   
7. Except for the Swiss, who didn’t get the memo.
   

By Victor Haghani and James White ¹
This post was first published on Bloomberg Prophets.

Google “risk parity” and you’ll see a grab bag of conflicting results: articles and posts trying to explain what it means, why it reduces stock-market volatility, why it increases stock-market volatility, why it’s less risky than a traditional portfolio, or why it’s more risky, among other things.² We’ll try to cut through this confusion to show that risk parity and traditional portfolios are closely related in philosophy.

Risk parity is all about how an investor allocates risk, not capital, typically with the use of leverage and with the idea that an equal risk allocation to various asset classes increases the benefits of diversification. Risk parity and traditional portfolios are usually presented as being philosophically miles apart, and hard to compare or analyze side by side except by looking at the historical record, such as in the chart below. The problem, though, is that financial market history is limited in that it reflects only a very specific set of historical conditions and, as we know, past performance isn’t indicative of future results.

What’s an investor to think? Both types of portfolios come out of the same theory of portfolio construction, but with different sets of basic starting assumptions. The theoretical toolkit we’re talking about here is the “Optimal Expected Utility” framework applied to financial markets by Paul Samuelson and his protégé Robert Merton starting in the 1960s.³ Their work helped them both garner a Nobel Prize, and produced a set of practical tools for determining how much of one’s wealth should be allocated to different investments with the understanding that the future is uncertain. The tools are primarily based on an investment’s expected excess return, volatility, and the investor’s personal level of risk-aversion.⁴

The basic idea is that an investor’s utility doesn’t keep going up as investment size – and thus risk – increases, but instead there’s an optimal investment size that maximizes expected utility given one’s personal level of risk-aversion. This simple idea leads to some powerful results. With the five assumptions below, the utility toolkit tells us that the portfolio that maximizes expected utility is the one with the highest Sharpe ratio – a common measure of risk-adjusted returns⁵ – levered or de-levered to an optimal level of risk.

• Assumptions for Optimality of Basic Risk Parity Portfolio:
• All assets follow a random walk and are continually tradeable
• All assets have equal pair-wise correlation with each other
• All assets have equal Sharpe ratio over a long horizon⁶
• Unlimited leverage is available at the risk-free rate
• No fees, transactions costs, or other drags on return

To build that portfolio with uncorrelated, equal-Sharpe ratio assets, we’d hold an amount of each asset that is inversely proportional to its volatility, resulting in each asset contributing an equal amount of risk to the portfolio. This is why it’s called “risk parity” investing.⁷ Using some stylized risk/return assumptions, the table below shows how this works with two risky assets for an investor with a “typical” amount of risk aversion.⁸ The first three rows show arbitrary allocations, and the final three rows show utility-optimal allocations corresponding to the given portfolio assumptions.

Portfolio	Stocks	Bonds	Expected Excess Return	Risk	Sharpe Ratio	Risk-Adjusted Return
Stocks	100%	0%	4.0%	16.0%	0.25	0.2%
Bonds	0%	100%	1.0%	4.0%	0.25	0.8%
Traditional	60%	40%	2.8%	9.7%	0.29	1.4%
Risk Parity Unlevered	20%	80%	1.6%	4.5%	0.35	1.3%
Risk Parity Levered	50%	200%	4.1%	11.6%	0.35	2.1%
Risk Parity + 0.6% Extra Borrow Cost	45%	85%	2.5%	8.0%	0.31	1.5%

The Merton toolkit suggests our investor would optimally want to own, via leverage, $250 of the equal-risk portfolio for every $100 of savings, resulting in the “Risk Parity Levered” portfolio in the table. The performance of this portfolio is quite a bit better on a risk-adjusted basis, 0.7 percent a year to be exact, than the traditional 60/40 stock/bond portfolio.⁹

We made some strong assumptions to get this result. Let’s see what happens when we loosen just one of them and assume that leverage isn’t available at the risk-free rate, but at a rate 0.6 percent higher? That cuts the risk-adjusted return of the optimal portfolio to 1.5 percent per year. This portfolio is very close, both in risk-adjusted return and Sharpe ratio, to the traditional 60/40 portfolio, making the traditional portfolio functionally optimal given the assumptions. We get the same result if we assume the investor doesn’t want to use leverage, regardless of the rate. Changing this assumption isn’t some abstract technicality: Leverage in real markets is not freely available at all times or at consistent rates, and there are many reasons an investor may choose to eschew leverage.¹⁰

The portfolios we see here represent two ends of a spectrum – but the range is surprisingly narrow. In our admittedly stylized two-asset example, only 0.7 percent per year of risk-adjusted return separates the fully-levered risk parity portfolio from the unlevered traditional one, which gives a sense for the level of fees, trading costs and extra borrowing expense a risk parity strategy could plausibly support.¹¹ If the five assumptions above seem reasonable, risk parity portfolios may make sense for you, but if not, a more traditional portfolio may be a better fit, and is just as consistent with good finance theory.¹²

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1. Victor is the founder and CEO of Elm Partners, a HNW Robo-investment manager. James works with Elm Partners in addition to pursuing his own research and investment interests.
   
2. There are a number of overviews of risk parity online. Here’s one: “Understanding Risk Parity”.
   
3. 3 One of the most seminal papers is Robert C. Merton’s “Lifetime Portfolio Selection under Uncertainty: the Continuous-Time Case.” The Review of Economics and Statistics (51), 1969.
   
4. A core result is that, for one risky asset following a random walk, the optimal investment size is where (μ – r)/(n * σ²) is the asset’s excess return, σ its volatility, and n the investor’s coefficient of risk-aversion. We often refer to this result as the “Merton Rule.”
   
5. Sharpe Ratio is the ratio of an asset’s excess return to volatility: SR = (μ – r)/ σ)
   
6. This is consistent with both the historical record and what many equilibrium models would predict.
   
7. Assuming risk and volatility are interchangeable for the purposes of this discussion. For uncorrelated assets with different Sharpe ratios, the solution is to scale each proportional to Sharpe ratio and inversely proportional to risk.
   
8. We’ll define typical here as that degree of risk aversion that would maximize expected utility by investing 100 percent of savings in a stock/bond portfolio with a 60/40 mix. With the numbers in our illustration, this implies a coefficient of risk aversion in the Merton model of 3. For readers familiar with the Kelly Criterion, this means our investor is 3 times as risk averse as a Kelly bettor. Also, typical risk parity implementations include four or more assets, including commodities and credit.
   
9. Risk-adjusted return = Expected Return – ½ * σ² * n , where n is the coefficient of risk aversion.
   
10. A few reasons that come to mind: 1) real markets may not follow pure random walks but can also gap, 2) leverage may not be easily adjusted once set, 3) terms other than rate may not be attractive, or 4) whenever the investor hears the word “leverage” he or she suffers painful flashbacks.
    
11. And they are separated by only 0.06 in terms of Sharpe ratio, although some back-tests suggest a difference of close to 0.2. As discussed in “What’s Past is Not Prologue,” we cannot rely on historical data on its own to support or reject the existence of this amount of difference in Sharpe ratio.
    
12. The authors would like to thank Larry Hilibrand and Vlad Ragulin for their help.
    

By Victor Haghani and James White ¹
This post was first published on Bloomberg Prophets.

There may be many reasons to worry about the current record price levels of U.S. equities, but agonizing over the fallout from valuations reverting back to their historical averages should not in itself be high on the list. In fact, deviations from the mean for one popular valuation measure – the cyclically adjusted price-to-earnings ratio – don’t actually tell us much about expected market returns.

What makes this particularly important now is that the CAPE stands at 30. The CAPE was equal to higher than it is today in only 59 of the 1,640 months going back to 1881, or less than 3.6 percent of the time. ² On average it takes about five years for the CAPE to move halfway back to its 16.8 historical average, based on a simple regression analysis.

Investors are understandably concerned that if it’s correct to expect the market to revert to a CAPE of about 24 over the next five years, then equities should drop about 22 percent, assuming earnings stay constant. No wonder so many are waiting for a stock market correction before putting more of their savings to work.

Data courtesy of Yale professor Robert Shiller. ³

Although there is statistical evidence that the CAPE is indeed mean-reverting, this process isn’t as strong as it can appear from the chart above. There’s a well-known result referred to as “reversion-to-the-mean bias,” which is that even random events, which by definition have no mean-reversion, often appear to be doing so for a while, sometimes quite a while, if the observer wrongly assumes that the recent level is the true mean. That’s essentially what we’ve done by looking at the CAPE history relative to the mean calculated from today’s vantage point. Elroy Dimson, a London Business School professor and renowned stock market historian, sees this as a big effect: “Much of the evidence for mean reversion is based on optical illusions based on hindsight.”

Also, a significant amount of the CAPE’s mean reversion is realized by earnings catching up with stock prices, rather than the other way around. For example, assuming the S&P 500 and earnings stay roughly where they are, the CAPE will drop from around 30 to about 25 in three years as the depressed earnings of the 2007-2009 period drop out of the 10-year lookback window for the CAPE.⁴

Most importantly, the outright level of CAPE, or more precisely, the cyclically adjusted earnings yield, known as 1/CAPE, is itself a pretty good predictor of future real equity returns.⁵ This makes it hard for another metric based on CAPE, such as its deviation from its mean, to add much information. The cyclically adjusted earnings yield when used by itself in a linear model explains about 25 percent of the variation in realized 10-year equity returns. Adding CAPE’s deviation from its mean as a second factor fails to explain any meaningful additional variation, indicating it has little extra predictive power.⁶

Today’s CAPE is telling us that we should expect equities to give us a real return in the long-term of 1/30, or about 3.33 percentage points, above inflation. This is a lot less attractive than the real return of 6.4 percent delivered by the U.S. stock market since 1881, though it’s somewhat more attractive compared with today’s long-term real rate offered by U.S. Treasuries of just under 1 percent.

As CAPE changes over time, the expected return of equities will also vary. Investors should be dynamic in their asset allocation, sizing exposure roughly in proportion to expected return.⁷ All else equal, investors should want to own less equities given today’s expected return than if CAPE was at 20 and the expected return was 5 percent (1/20). However, most investors would want to own no equities – or even run a short position – if they believed they’d be taking a 22 percent hit caused by CAPE reverting halfway back to its average over the next five years.⁸

History and common sense tell us that paying a higher earnings multiple for equities lowers the return one should expect to earn. History, however, is not telling us that a mean reversion of the CAPE is likely to deliver either a hit, or a windfall, to equity returns. Of course, history is not necessarily indicative of the future, and investors may have other reasons to expect equities to perform worse or better than predicted by 1/CAPE, especially in the short term. A historically driven fear of mean reversion should not be one of them.

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1. Victor is the founder and CEO of Elm Partners, a HNW Robo-investment manager. James works with Elm Partners in addition to pursuing his own research and investment interests.
   
2. Of course, much has changed since 1881, when the U.S. flag had only 38 stars.
   
3. Data can be found here. CAPE was popularized in Campbell and Shiller’s “Stock Prices, Earnings and Expected Dividends,” found here. The idea goes even further back, to Graham and Dodd’s famous work “Security Analysis” (1934).
   
4. We find that on average, for every 10 percent the CAPE has been above or below its historical average, 10-year real average earnings increase (or decrease) by about 2 percent relative to their long-term average growth rate, which accounts for about 20 percent of the superficial mean-reversion we see in the CAPE data.
   
5. That 1/CAPE is a good predictor of the real return of the equity market is consistent with a story that corporate earnings would grow with inflation if they were paid out in full each year. You won’t be surprised to learn that professors Shiller and Campbell were among the first to write about this too in their 1988 paper. There are other equally powerful predictors, similar in spirit to 1/CAPE, such as ones that use dividend yields and real dividend growth parameters to predict equity returns.
   
6. We’re describing very simple linear models here, but there’s no evidence that more complex models produce different results, and Ockham’s Razor would counsel us to err on the side of simplicity. We’ve also made a choice to focus on a relatively long realized return horizon, namely 10 years. When looking at shorter horizons, 1/CAPE explains much less of the variation in return, and CAPE’s deviation from the mean similarly adds very little.
   
7. Most frameworks for asset allocation, such as the Merton Rule, suggest investment should be proportional to excess expected return.
   
8. Some people feel the expected return should be thought of as a bit higher than that, taking into account of the positive convexity of equities – that is, that equities can go up a lot but cannot go down more than 100 percent. More in this note here.
   

The Journal of Portfolio Management recently published Elm’s research paper “Do Index Buyers Make Overvalued Stocks More Overvalued?”

We must have said something that sounded insightful, because the note was awarded the William F. Sharpe award for Institutional Investor Journals Paper of the Year on indexing and ETFs.

You can view the paper here.

From The Economist:

“Who wants mediocrity? That is what a lot of people say when the subject of index-tracking, or passive fund management, comes up. They would rather choose a fund manager (an active manager in the jargon) who tries to beat the market by picking the best stocks. It does sound like a good idea.”

Read the full article on Economist.com.

By Victor Haghani and James White ¹

For investors who feel the expected return of equities isn’t as attractive today as it was 5 to 10 years ago when PEs were lower, it’s natural to be thinking about reducing exposure to the market. For taxable investors, reducing your allocation will also mean paying capital gains tax.² As we’ve done in past notes on coin flipping and optimal trade-sizing, we’ll use a few basic assumptions about individual risk aversion to suggest a simple, intuitive rule-of-thumb for weighing up the trade-off between saving on tax versus achieving your target equity allocation.

Our general finding is that if you’ve already held your equities long enough to qualify for long-term capital gains treatment, then the virtues of rebalancing to your target allocation far outweigh the efficiency from deferring a tax payment. In this case, it’s optimal to rebalance your portfolio most of the way to your target – and simply rebalancing all the way to target is almost as good. However, if waiting to rebalance will deliver the relatively large benefit of converting short-term gains to long-term gains, or you expect a significant cut in capital gains tax rates in the next few years, then you will probably be willing to keep an allocation to equities well above the optimal level you’d desire if you were investing from scratch.

Let’s illustrate the question with an example. Say 6 years ago you had a view that the long-term pre-tax excess return of US equities over cash was 5%,³ and based on that you invested 50% of your taxable savings in US stocks, and the other 50% you kept in cash. Since that time, your equities have doubled in value, excluding dividends, and they’d have grown to represent 67% of your portfolio.⁴ But you now find equities less attractive to own than you did back then, as higher current PEs suggest lower future returns. Let’s say you think equities’ expected excess return is now down from 5% to 3%, and if you were investing your portfolio from scratch you’d want to have an allocation to equities of just 30%.

If you reduced your exposure down to 30% from 67%, you’d have to make a hefty capital gains tax payment of roughly $12.50 per $100 of equities you own, or 4.6% of the value of your portfolio, assuming a long-term capital gains rate of 25%.⁵

…So what to do?

You’d think this is a pretty basic question that would have some well-distilled answers. Well, we asked Google “how to decide whether to realize a capital gain” and we found a lot of answers – more than 15 million in fact. But we didn’t see any in the top 100 that went beyond “it depends” and “consult your tax advisor.”⁶ We are most definitely not qualified tax advisors, and this is not tax advice, but let’s see if we can come up with a more useful analysis than “it depends.”

First, we need to quantify how much tax you’d save by not selling your appreciated equities today. We’ll start off by assuming the long-term tax rate is constant at 25% for your remaining investment horizon – later we’ll discuss results with modified assumptions. In this case, your tax savings arises entirely from deferring the realization of the capital gain – in doing so, you are in effect getting the use of the funds you’d have paid in taxes to invest in your portfolio. If the expected return on your portfolio of equities and T-bills is 1.9% – the blend of 30% in equities with a nominal expected return of 4% and 70% in T-Bills earning 1% – then for every $100 of equities you don’t sell, you get to earn 1.9% on the $12.5 of tax you’re not paying right now. This equates to a benefit of 0.24% per annum.⁷

Great. We’re halfway there. Now we just have to figure out what it costs us to hold more equities than our original target of 30%. Here’s where we have to draw on a bit of our past discussions about optimal trade-sizing (you can find a brief refresher, illustrated with ketchup and fries, here).

Recall that, if you were investing the portfolio from scratch, we assumed that you want to have 30% of your savings in equities based on your view that equities had an expected excess return of 3%. A basic model of personal risk-taking, known as the Merton Rule,⁸ says that your optimal holding of a risky asset should increase in direct proportion to its expected return. That’s as simple a financial formula as you’ll find, and it seems pretty reasonable too. It’s simply saying that if you want to have 30% of your savings in equities when you expect equities to deliver a 3% excess return, then, all else equal, if you feel that the expected return of equities is 4%, you should want to have 40% in equities; more generally, for every extra 1% of return, you’d want to have an extra 10% allocated to equities.

We now put the two halves of the analysis together to get our answer. Every extra dollar of appreciated equities you don’t sell has an extra return of 0.24% a year. If you want to hold an extra 10% of equities for every 1% of extra expected return, then it follows that for an extra 0.24% of return, you’d optimally like to own an extra 2.4% of equities. So, your tax-adjusted optimal allocation to equities is 32.4%, very close to your optimal allocation ignoring taxes of 30%.⁹ This suggests you should move about 95% of the way to your 30% target allocation from your current allocation of 67%.¹⁰

Working through this example suggests this very simple rule-of-thumb:

Extra equities to own above your optimal allocation ignoring taxes = B * Q r , where:

  B = annual tax benefit on appreciated equities not sold
  Q = how much equities you’d want to own ignoring taxes
  r = your expected excess return on equities

The bigger the unrealized gain, the higher the tax rate, the higher the return on equities or T-bills or the less risk-averse you are, the further you should be willing to diverge from your optimal allocation ignoring taxes. In our example, the benefit from holding a 32% allocation versus 30% is relatively modest, but the benefit in rebalancing down from 67% to 30-32% is high. At 67% allocation, the portfolio earns a risk-adjusted rate of -0.1%, suggesting you’d be better off owning nothing than maintaining your allocation at that level.

Let’s put this rule-of-thumb to work on a few other salient examples representing commonly encountered tax situations. All cases keep to our Base Case assumptions regarding expected excess return for equities, level of personal risk aversion and amount of capital gain:¹¹

Description	Current Tax Rate	Long-term Tax Rate	Horizon (Years)	Extra Allocation vs. Optimal from “Scratch” Allocation
Base Case: deferring long-term cap gains	25%	25%	1	2%
Hold to death for basis step-up, or donate	25%	0%	30	8%
Move from high-tax to low-tax state (e.g. NY to WY)	31.5%	25%	5	12%
Big Long-Term CG tax cut	25%	15%	5	16%
Hold on for Short-term CG to turn into Long-term CG	44.6%	25%	0.5	37%
Tax rate increase that exactly offsets value of deferring	25%	25.35%	1	0%

What if I Expect Higher Tax Rates in the Future?

The last entry in the above table shows how much tax rates would need to rise over the next year to exactly balance the value of deferral: just 0.35% with the assumptions we’ve used. In practice, the problem is more complex than this, as, for one thing, you would want to consider the analysis to multiple horizons. If the expected increase in tax rates outweighs the value of deferral to the relevant horizon, then this simplified analysis suggests you should realize capital gains entirely, pay the resulting capital gains tax and then reinvest in equities to your desired allocation from scratch.

Conclusions

Our simple rule-of-thumb suggests that in most scenarios involving a realization of long-term capital gains, when you do not expect a reduction in capital gains rates in the next few years and you’re pretty far from where you’d like to be, your optimal move is fairly close, but not all the way, to your desired allocation ignoring taxes completely. However, when it comes to bearing extra risk in order for a short-term capital gain to convert into a long-term capital gain, you should be willing to take quite a lot of extra risk to enjoy the benefit of that lower tax rate.

At Elm Partners, in rebalancing portfolios for both our Fund and SMA clients we are willing to tolerate quite high deviations from target allocations to avoid short-term capital gains, but we are much less tolerant for avoiding long-term gains. Thus, our approach to tax-efficient investing is broadly consistent with the rule-of-thumb we have presented here.

Caveat Emptor!

The purpose of this note was to provide a simple rule-of-thumb to help illustrate some, but not all, of the factors investors should consider in trying to take account of tax effects in their investment decisions. The tax code is much more complex than the simple representation we have used in our examples. For example, we did not give any treatment to the fact that there is an asymmetry in capital gains taxation, in that we pay tax on gains, but we may get very limited refunds from the government on capital losses. Or more succinctly, as some unfortunate soul once said, “Man cannot live on capital loss carry-forwards alone.” We likewise did not treat whether investors form their target allocation views based on pre-tax or post-tax returns, or whether investors view portfolio risk inclusive or exclusive of tax liabilities. Also, state taxation varies from state to state and adds another layer of complexity. And adding insult to injury, tax policy changes over time in hard-to-predict ways. In the words of Albert Einstein, “The hardest thing in the world to understand is the income tax.”

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Further Reading and References:

• Balcer, Y., and Judd, K., 1987, “Effects of Capital Gains Taxation on Life-Cycle Investment and Portfolio Management,” Journal of Finance, 42, 743-761.
• Dammon, R., C. Spatt, and H. Zhang, 2001a, “Optimal Consumption and Investment with Capital Gains Taxes,” Review of Financial Studies, 14, 583-616
• Dybvig, P., and Koo, H. K., 1996, “Investment with Taxes,” Unpublished working paper, Washington University in St. Louis.
• Merton, R., 1971, “Optimum Consumption and Portfolio Rules in a Continuous-Time Model,” Journal of Economic Theory, 3, 373-413.
• Odean, T., 1998, “Are Investors Reluctant to Realize Their Losses?,” Journal of Finance, 53, 1775-1798.

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1. Victor is the Founder and CIO of Elm Partners, and James is Elm’s CEO. Past returns are not indicative of future performance. This not is not an offer or solicitation to invest.
    

   A big thank you to Larry Hilibrand, Ayman Hindy and WK, as well as our colleagues at Elm – Jeff, Arjun, Bruce, Gregg and Tara – for your help with this note.
   

2. Assuming the investments are not held in tax-advantaged retirement accounts. Also, we’re assuming the investor is not able to achieve total portfolio level asset allocation changes either through making exaggerated changes in non-taxable retirement accounts or through adding to investment accounts with further savings from earnings each year. Both of these can be tax efficient ways of moving one’s asset allocation without incurring capital gains taxes.
   
3. To be precise, that 5% is the expected arithmetic return in excess of T-bills, a standard proxy for cash. Also, for the purposes of this example, we’re assuming that the T-bill rate = 1%. We are using US equities for this example for simplicity, but as you know, at Elm Partners we believe in global diversification.
   
4. Assuming you spent your investment income. Also, more significantly, we’re ignoring the fact that you have a tax liability (think of it as negative cash) of $12.5 (25% of 50% of $100 of appreciated equities), which, if you think of it as negative cash, you may wish to subtract from your $50 of cash, leaving you with $41.7 of cash. Your allocation to equities taking this tax liability into account is actually 71%, higher than the more conventionally calculated allocation of 67% we’re using here.
   
5. Federal + Obamacare tax + a gross-up for the phase-out of itemized deductions, but assuming you live in a state with no capital gains tax like Wyoming, Florida or Texas. See “How High are Capital Gains Tax Rates in Your State?”. As per the previous footnote, we are not taking the embedded capital gains tax liability into account in doing this calculation of how much you have invested in equities. You’d need to sell even more equities to get to 30% in equities taking the capital gains tax liability into account.
   
6. We did find several thorough treatments of this question in the academic literature, which we listed in the reference section below. For an example of the more typical treatment of this question, see this recent NY Times article here. Perhaps an explanation for why this problem hasn’t gotten the attention it deserves is because we so often delegate to active managers who don’t have an incentive to defer the realization of gains. However, long-term investing in broad index funds, as typified by the type of investing we do at Elm Partners, will force us to confront the question posed in this note more frequently.
   
7. While we have considered plausible arguments that would lead us to value deferral at either the equity expected return or the risk-free rate, it should be noted that your choice of rate will not alter the general conclusion that the value of deferral is not significant enough to move you far from your optimal allocation ignoring taxes, unless your basis is very low. This is the case wherein you defer the capital gains indefinitely and your post-tax portfolio return converges to the pre-tax return, as shown by this formula:
     After tax rate of return = ((1 – τ) * (1 + r)^T + τ)^{(1 / T)}– 1
   where r is the pre-tax return on your portfolio, τ is the tax rate and T is your horizon. You can also see this by realizing that in the very, very long-term, if you never sold your equities, the initial basis would converge to a tiny fraction relative to the value of your holding. So, at that point, you’d be earning the pre-tax return on your entire tax liability which would result in an after-tax return equal to the pre-tax return.
   
8. Merton rule: for a portfolio with cash earning rate r and a risky investment with expected return µ and volatility σ , the optimal fraction of wealth to invest in the risky asset is (µ – r) / (γ σ²) , where γ reflects the investor’s idiosyncratic level of risk-aversion. For an investor who has an optimal holding of equities of 30% for a 3% expected excess return, and assuming equity market annual volatility of 18%, we get a coefficient of risk aversion of 3, signifying the investor is three times as risk averse as a Kelly bettor.
   
9. This is a rule-of-thumb, so we’ve left out the effect of your blended rate at the adjusted allocation being different than the blended rate at the optimal allocation ignoring taxes. This is a relatively small effect, and seems worth putting to the side for a rule-of-thumb.
   
10. If, even after the 100% equity rally, the investor still felt that the expected excess return of the stock market was unchanged at 5%, then 50% would remain her optimal allocation ignoring taxes, and the optimal rebalancing factoring in taxes would be to reduce the allocation to equities from 67% to 54%.
    
11. Calculations for tax benefit in each case are as follows, where g is the gain, r is the blended return on portfolio at desired allocation ignoring taxes, T is the horizon in years, τ₀ is the tax rate at start, and τ_T is the tax rate at horizon:
      • Base Case: g_* τ_{0 *} r
      • Hold-until-you-die: Base Case + g * τ₀ / (1 – τ₀) / T
      • Tax cut or move to lower tax jurisdiction: g_* τ_o* r + g * (τ₀ – τ_T) / (1 – τ₀) / T
      • Converting ST to LT: g _* τ_T* r + g * (τ₀ – τ_T) / (1 – τ_T) / T
    In case of Converting ST to LT, the maximum deviation is the starting allocation minus desired allocation.
    

By Victor Haghani and James White ¹

A few weeks ago, one of our investors told us she had some cash that she wanted to add to her account with us, but that she felt the market was at such a high level that she should wait for a market correction before putting it to work. She was confident that a market drop would happen before long, and she saw little downside in waiting for it. We’ve been hearing this general sentiment from many of our investors, and potential investors, and thought it timely to share some thoughts and analysis on this topic.

While we share our investor’s concern about the current valuation of the US market, we told her that in fact the downside to waiting is more than meets the eye. While the probability of a correction at some point is indeed quite high, the correction may happen from a much higher market level, or may take so long to happen that she stops waiting and gets “pushed in” at a higher level. In fact, under the assumption that the market follows a random walk, as long as you believe the expected return of the market is higher than what you’d earn on your cash, the expected opportunity cost of waiting for a correction exceeds the expected benefit of investing after the correction.

In practice, few investors believe markets efficiently follow a random walk, even though it’s a pillar of finance theory. Investor beliefs do tend to be strongly influenced by history though, so let’s look at Yale Professor Robert Shiller’s US stock market data going back to the late 1800s.² Before describing what we found in this data, we should lay down a few caveats: 1) historical market behavior is not necessarily indicative of the future, 2) even 115 years of market data isn’t enough to draw statistically significant conclusions on many interesting questions, especially those involving relatively rare events, and 3) the US stock market is not the whole stock market, and the rest of the global market, which at Elm forms a significant fraction of client portfolios, looks a lot less frothy than the US market.

With these caveats in mind, the first question we ask the data is: during times when the market has been “expensive” what has been the average cost or benefit of waiting for a correction of 10% from the starting price level, rather than investing right away? We define “expensive” as times when the stock market had a CAPE (Cyclically-adjusted Price-Earnings Ratio) that was more than one standard deviation above its historical average level. While the CAPE of the US Stock Market is currently hovering around two standard deviations above average, as per our second caveat above, there aren’t enough equivalent periods in the historical record to make a statistically significant data analysis conditioned on the current state of the market. We focus on a comparison over a 3-year horizon, which we choose because the investor is unlikely to wait indefinitely for the hoped-for correction. The most salient findings from this analysis are:

• From a given “expensive” starting point, there was a 56% probability that the market had a 10% correction within 3 years, waiting for which would result in about a 10% return benefit vs. having invested right away.³
• In the 44% of cases where the correction doesn’t happen, there’s an average opportunity cost of about 30% – much higher than the average benefit.
• Putting these together, the mean expected cost of waiting for a correction was about 8% versus investing right away.

We note that while the probability of a correction inside the horizon is somewhat likely, it’s far from certain – and that when it doesn’t happen, the expected opportunity cost is much higher than the expected benefit. We suspect that the perception that waiting for a correction is a good strategy arises primarily for three reasons. First, while a correction occurring is indeed more likely than not, investors may confuse the chance of a correction from peak-to-trough with the lower chance of a correction from a fixed price level. For example, the historical probability of a 10% correction happening any time during a 3-year window is 88%, significantly higher than the 56% occurrence of that correction from the market level at the start of the period. Second, the cost of waiting and not achieving the correction is a “hidden” opportunity cost, and we humans have a well-documented bias to underweight opportunity costs relative to realized costs. Finally, investors may believe they can wait indefinitely for the correction to happen, but in practice few investors have that sort of staying power. In fact, as we’ll see below, the longer you’re able to (stubbornly) wait for the correction, the greater the average opportunity cost you would have suffered.

We’ve made some very specific assumptions here: that the investor is waiting for a 10% correction, has a 3-year horizon, etc. So, we repeated this historical analysis with correction ranges from 1% to 10%, horizons of 1-year and 5-years, and with different criteria for what makes the market look “expensive.” ⁴ Each point in the chart below represents a combination of these four assumptions. You can see that across all scenarios there has been a material cost for waiting. The longer the horizon that you’d have been willing to wait for the correction to occur (the red points represent a 5-year horizon), the higher the average cost.

Now shifting focus from the historical record to looking forward, it’s true that the lower one’s expectation of the stock market return, the lower the expected cost of waiting for a correction. If you believe the stock market has a negative expected return to a particular horizon, then waiting for a correction to invest makes sense. However, at least as far as the historical record for the US stock market goes, higher market valuations are consistent with lower prospective long-term returns, but not negative expected returns.

We’re not suggesting that looking at the historical record closes the book on the question of whether or not to wait for a correction. As you know, we firmly believe in the axiom that past returns are not indicative of future returns – and there are many other personal and circumstantial factors an investor should consider before deciding whether entering the market now is the right thing to do. However, we have the impression that some investors, like our client who got us thinking about this question in the first place, are waiting for a correction specifically because they think that history shows that waiting pays. We hope it’s been useful to show that history, for what it’s worth, doesn’t support that view.

For further related reading, you may enjoy our recent note: What’s the Best Way to Get Invested in the Market?

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1. Victor is the Founder and CIO of Elm Partners, and James is Elm’s CEO. This not is not an offer or solicitation to invest, nor should this be construed in any way as tax advice. Past returns are not indicative of future performance.
   Thanks to Larry Hilibrand, Vladimir Ragulin, Chi-fu Huang, Andy Morton and John Glazer for their input.
   
2. Professor Shiller’s S&P data is presented as monthly averages, which we felt might be more relevant in exploring the topic of this note.
   
3. The cost or benefit is calculated as the return difference over the full 3 years between being fully invested for the entire period versus waiting for the 10% correction and re-entering the market when and if it occurs. Carry from dividends and cash yield are both included in the differential return calculation.
   
4. In addition to our CAPE-based definition of “expensive” we also looked at waiting for a correction from times when the market was at an all-time high at the start of the period.
   

The research we conducted into how people bet on a computer-generated coin-flip they are told has a 60% chance of landing on heads was published in the Spring 2017 edition of the Journal of Portfolio Management.

In 1986, Gary Brinson and his co-researchers, in their paper “Determinants of Portfolio Performance,” found that investors’ decisions about asset allocation dominated any choices they made about individual securities or managers.

Over the long term, your decision about how much to allocate to risk assets (e.g. 75%, 50% or 25%) is by far the highest impact investment decision you’ll make.

Our coin-flipping experiment brings attention to the fact that many of us, even those who study finance and investing at university, struggle with how to think about these important decisions.

We first circulated our research on coin flipping back in October 2016, but if you haven’t read the original note, we hope you’ll enjoy this more up-to-date version in the Journal of Portfolio Management here.