This post is available as a PDF download here.
Summary
- Traditional portfolio management focuses explicitly on the trade-off between risk and return.
- Anecdotally, investors often care more about the growth of their wealth. Due to compounding effects, wealth is a convex function of realized returns.
- Within, we explore geometric mean maximization, an alternative to the traditional Sharpe ratio maximization that seeks to maximize the long-term growth rate of a portfolio.
- Due to compounding effects, volatility plays a critical role in the growth of wealth. Seemingly lower return portfolios may actually lead to higher expected terminal wealth if volatility is low enough.
- Maximizing for long-term growth rates may be incompatible with short-term investor needs. More explicit accounting for horizon risk may be prudent.
In 1956, J.L. Kelly published “A New Interpretation of Information Rate,” a seminal paper in betting theory that built off the work of Claude Shannon. Within, Kelly derived an optimal betting strategy (called the Kelly criterion) for maximizing the long-term growth rate of a gambler’s wealth over a sequence of bets. Key in this breakthrough was the acknowledgement of cumulative effects: the gambler would be reinvesting gains and losses, such that too large a bet would lead to ruin before any probabilistic advantage was ever realized.
Around the same time, Markowitz was laying the foundations of Modern Portfolio Theory, which relied upon mean and variance for the selection of portfolios. Later work by Sharpe and others would identify the notion of the tangency portfolio: the portfolio that maximizes excess return per unit of risk.
Without leverage, however, investors cannot “eat” risk-adjusted returns. Nor do they, anecdotally, really seem to care about it. We, for example, have never heard of anyone opening their statement to look at their Sharpe ratio.
More academically, part of the problem with Markowitz’s work, as identified by Henry Latane in 1959, was that it did not provide an objective measure for selecting a portfolio along the efficient frontier. Latane argued that for an investor looking to maximize terminal wealth (assuming a sequence of uncertain and compounding choices), one optimal strategy was to select the portfolio that maximized geometric mean return.
The Math Behind Growth-Optimal Portfolios
We start with the idea that the geometric mean return, g, of a portfolio – the value we want to maximize – will be equal to the annualized compound return:
With some slight manipulation, we find:
For[1],
We can use a Taylor expansion to approximate the log returns around their mean:
Dropping higher order terms and taking the expected value of both sides, we get:
Which can be expressed using the geometric mean return as:
Where sigma is the volatility of the linear returns.
Multi-Period Investing: Volatility is a Drag
At the end of the last section, we found that the geometric mean return is a function of the arithmetic mean return and variance, with variance reducing the growth rate. This relationship may already be familiar to some under the notion of volatility drag.[2]
Volatility drag is the idea that the arithmetic mean return is greater than the geometric mean return – with the difference being due to volatility. Consider this simple, albeit extreme, example: on the first day, you make 100%; on the second day you lose 50%.
The arithmetic mean of these two returns is 25%, yet after both periods, your true compound return is 0%.
For less extreme examples, a larger number of periods is required. Nevertheless, the effect remains: “volatility” causes a divergence between the arithmetic and geometric mean.
From a pure definition perspective, this is true for returns. It is, perhaps, somewhat misleading when it comes to thinking about wealth.
Note that in finance, we often assume that wealth is log-normally distributed (implying that the log returns are normally distributed). This is important, as wealth should only vary between [0, ∞) while returns can technically vary between (-∞, ∞).
If we hold this assumption, we can say that the compounded return over T periods (assuming constant expected returns and volatilities) – is[3]:
Where is the random return shock at time t.
Using this framework, for large T, the median compounded return is:
What about the mean compounded return? We can re-write our above framework as:
Note that the random variable is log-normal, the two terms are independent of one another, and that
Thus,
The important takeaway here is that volatility does not affect our expected level of wealth. It does, however, drive the mean and median further apart.
The intuition here is that while returns are generally assumed to be symmetric, wealth is highly skewed: we can only lose 100% of our money but can theoretically make an infinite amount. Therefore, the mean is pushed upwards by the return shocks.
Over the long run, however, the annualized compound return does not approach the mean: rather, it approaches the median. Consider that the annualized compounded return can be written:
Taking the limit as T goes to infinity, the second term approaches 1, leaving only:
Which is the annualized median compounded return. Hence, over the long run, over one single realized return path, the investor’s growth rate should approach the median, not the mean, meaning that volatility plays a crucial role in long-term wealth levels.
The Many Benefits of Growth-Optimal Portfolios
The works of Markowitz et al. and Latane have subtle differences.
- Sharpe Ratio Maximization (“SRM”) is a single-period framework; Geometric Mean Maximization (“GMM”) is a multi-period framework.
- SRM maximizes the expected utility of terminal wealth; GMM maximizes the expected level of terminal wealth.
Over time, a number of attributes regarding GMM have been proved.
- Breiman (1961) – GMM minimizes the expected time to reach a pre-assigned monetary target V asymptotically as V tends to infinity.
- Hakansson (1971) – GMM is myopic; the current composition depends only on the distribution of returns over the next rebalancing period.
- Hakansson and Miller (1975) – GMM investors never risk ruin.
- Algoet and Cover (1988) – Assumptions requiring the independence of returns between periods can be relaxed.
- Ethier (2004) – GMM maximizes the median of an investor’s fortune.
- Dempster et al. (2008) – GMM can create value even in the case where every tradeable asset becomes almost surely worthless.
With all these provable benefits, it would seem that for any investor with a sufficiently long investment horizon, the GMM strategy is superior. Even Markowitz was an early supporter, dedicating an entire chapter of his book Portfolio Selection: Efficient Diversification of Investments, to it.
Why, then, has GMM largely been ignored in favor of SRM?
A Theoretical Debate
The most significant early challenger to GMM was Paul Samuelson who argued that maximizing geometric mean return was not necessarily consistent with maximizing an investor’s expected utility. This is an important distinction, as financial theory generally requires decision making be based on expected utility maximization. If care is not taken, the maximization of other objective functions can lead to irrational decision making: a violation of basic finance principles.
Practical Issues with GMM
Just because the GMM provably dominates the value of any other portfolio over a long-horizon does not mean that it is “better” for investors over all horizons.
We use quotation marks around better because the definition is largely subjective – though economists would have us believe we can be packaged nicely into utility functions. Regardless,
- Estrada (2010) shows that GMM portfolios are empirically less diversified and more volatile than SRM portfolios.
- Rubinstein (1991) shows that it may take 208 years to be 95% confident that a Kelly strategy beats an all-cash strategy, and 4700 years to be 95% sure that it beats an all-stock strategy.
A horizon of 208 years, and especially 4700 years, has little applicability to nearly all investors. For finite horizons, however, maximizing the long-term geometric growth rate may not be equivalent to maximizing the expected geometric return.
Consider a simple case with an asset that returns either 100% or -50% for a given year. Below we plot the expected geometric growth rate of our portfolio, depending on how many years we hold the asset.
We can see that for finite periods, the expected geometric return is not zero, but rather asymptotically approaches zero as the number of years increases.
Finite Period Growth-Optimal Portfolios
Since most investors do not have 4700 hundred years to wait, a more explicit acknowledgement of holding period may be useful. There are a variety of approximations available to describe the distribution of geometric returns with a finite period (with complexity trading off with accuracy); one such approximation is:
Rujeerapaiboon, Kuhn, Wiesemann (2014)[4] propose a “robust” solution for fixed-mix portfolios (i.e. those that rebalance back to a fixed set of weights at the end of each period) and finite horizons. Specifically, they seek to maximize the worst-case geometric growth rate (where “worst case” is defined by some probability threshold), under all probability distributions (consistent with an investor’s prior information).
If we simplify a bit and assume a single distribution for asset returns, then for a variety of worst-case probability thresholds, we can solve for the maximum growth rate.
As we would expect, the more certain we need to be of our returns, the lower our growth rate will be. Thus, our uncertainty parameter, , can serve, in a way, as a risk-aversion parameter.
As an example, we can employ J.P. Morgan’s current capital market assumptions, our simulation-based optimizer, the above estimates for E[g] and V[g], and vary the probability threshold to find “robust” growth-optimal portfolios. We will assume a 5-year holding period.
Source: Capital market assumptions from J.P. Morgan. Optimization performed by Newfound Research using a simulation-based process to account for parameter uncertainty. Certain asset classes listed in J.P. Morgan’s capital market assumptions were not considered because they were either (i) redundant due to other asset classes that were included or (ii) difficult to access outside of private or non-liquid investment vehicles.
To make interpretation easier, we have color coded the categories, with equities in blue, fixed income in green, credit in orange, and alternatives in yellow.
We can see that even with our uncertainty constraints relaxed to 20% (i.e. our growth rate will only beat the worst-case growth rate 80% of the time), the portfolio remains fairly diversified, with large exposures to credit, alternatives, and even long-dated Treasuries largely used to offset equity risk from emerging markets.
While this is partly due to the generally bearish view most firms have on traditional equities, this also highlights the important role that volatility plays in dampening geometric return expectations.
Low Volatility: A Geometric Mean Anomaly?
By now, most investors are aware of the low volatility anomaly, whereby strategies that focus on low volatility or low beta securities persistently outperform expectations given by models like CAPM.
To date, there have been three behavioral arguments:
- Asset managers prefer to buy higher risk stocks in effort to beat the benchmark on an absolute basis;
- Investors are constrained (either legally or preferentially) from using leverage, and therefore buy higher risk stocks;
- Investors have a deep-seeded preference for lottery-type payoffs, and so buy riskier stocks.
In all three cases, investors overbid higher risk stocks and leave low-risk stocks underbid.
In Low Volatility Equity Investing: Anomaly or Algebraic Artifact, Dan diBartolomeo offers another possibility.[5] He notes that while the CAPM says there is a linear relationship between systematic risk (beta) and reward, the CAPM is a single-period model. In a multi-period model, there would be convex relationship between geometric return and systematic risk.
Assuming the CAPM holds, diBartolomeo seeks to solve for the optimal beta that maximizes the geometric growth rate of a portfolio. In doing so, he addresses several differences between theory and reality:
- The traditional market portfolio consists of all risky assets, not just stocks. Therefore, an all stock portfolio likely has a very high relative beta.
- The true market portfolio would contain a number of illiquid assets. In adjusting volatility for this illiquidity – which in some cases can triple risk values – the optimal beta would likely go down.
- In adjusting for skew and kurtosis exhibited by financial time series, the optimal beta would likely go down.
- In general, investors tend to be more risk averse than they are growth optimal, which may further cause a lower optimal beta level.
- Beta and market volatility are estimated, not known. This causes an increase in measured asset class volatility and further reduces the optimal beta value.
With these adjustments, the compound growth rate of low volatility securities may not be an anomaly at all: rather, perception of outperformance may be simply due to a poor interpretation of the CAPM.
This is both good and bad news. The bad news is that if the performance of low volatility is entirely rational, it’s hard for a manager to demand compensation for it. The good news is that if this is the case, and there is no anomaly, then the performance cannot be arbitraged away.
Conclusion: Volatility Matters for Wealth Accumulation
While traditional portfolio theory leads to an explicit trade-off of risk and return, the realized multi-period wealth of an investor will have a non-linear response – i.e. compounding – to the single-period realizations.
For investors who care about the maximization of terminal wealth, a reduction of volatility, even at the expense of a lower expected return, can lead to a higher level of wealth accumulation.
This can be non-intuitive. After all, how can a lower expected return lead to a higher level of wealth? To invoke Nassim Taleb, in non-linear systems, volatility matters more than expected return. Since wealth is a convex function of return, a single bad, outlier return can be disastrous. A 100% gain is great, but a 100% loss puts you out of business.
With compounding, slow and steady may truly win the race.
It is worth noting, however, that the portfolio that maximizes long-run return may not necessarily best meet an investor’s needs (e.g. liabilities). In many cases, short-run stability may be preferred at the expense of both long-run average returns and long-term wealth.
[1] Note that we are using here to represent the mean of the linear returns. In Geometric Brownian Motion, is the mean of the log returns.
[2] For those well-versed in pure mathematics, this is an example of the AM-GM inequality.
[3] For a more general derivation with time-varying expected returns and volatilities, please see http://investmentmath.com/finance/2014/03/04/volatility-drag.html.
[4] https://doi.org/10.1287/mnsc.2015.2228
[5] http://www.northinfo.com/documents/559.pdf
Tax-Managed Models & Asset Location
By Corey Hoffstein
On September 11, 2017
In Portfolio Construction, Weekly Commentary
This post is available for download as a PDF here.
Summary
Before we begin, please note that we are not Certified Public Accountants, Tax Attorneys, nor do we specialize in tax management. Tax law is complicated and this commentary will employ sweeping generalizations and assumptions that will certainly not apply to every individual’s specific situation. This commentary is not meant as advice, simply research. Before making any tax-related changes to your investment process, please consult an expert.
Tax-Managed Thinking
We’ve been writing a lot, recently, about the difficulties investors face going forward.[1][2][3] It is our perspective that the combination of higher-than-average valuations in U.S. stocks and low interest rates in core U.S. bonds indicates a muted return environment for traditionally allocated investors going forward.
There is no silver bullet to this problem. Our perspective is that investors will likely have to work hard to make many marginal, but compounding, improvements. Improvements may include reducing fees, thinking outside of traditional asset classes, saving more, and, for investors in retirement, enacting a dynamic withdrawal plan.
Another potential opportunity is in tax management.
I once heard Dan Egan, Director of Behavioral Finance at Betterment, explain tax management as an orthogonal improvement: i.e. one which could seek to add value regardless of how the underlying portfolio performed. I like this description for two reasons.
First, it fits nicely into our framework of compounding marginal improvements that do not necessarily require just “investing better.” Second, Dan is the only person, besides me, to use the word “orthogonal” outside of a math class.
Two popular tax management techniques are tax-loss harvesting and asset location. While we expect that tax-loss harvesting is well known to most (selling investments at a loss to offset gains taken), asset location may be less familiar. Simply put, asset location is how investments are divided among different accounts (taxable, tax-deferred, and tax-exempt) in an effort to maximize post-tax returns.
Asset Location in a Perfect World
Taxes are a highly personal subject. In a perfect world, asset location optimization would be applied to each investor individually, taking into account:
Such information would allow us to run a very simple portfolio optimization that could take into account asset location.
Simply, for each asset, we would have three sets of expected returns: an after-tax expected return, a tax-deferred expected return, and a tax-exempt expected return. For all intents and purposes, the optimizer would treat these three sets of returns as completely different asset classes.
So, as a simple example, let’s assume we only want to build a portfolio of U.S. stocks and bonds. For each, we would create three “versions”: Taxable, Tax-Deferred, and Tax-Exempt. We would calculate expected returns for U.S. Stocks – Taxable, U.S. Stocks – Tax-Deferred, and U.S. Stocks – Tax-Exempt. We would do the same for bonds.
We would then run a portfolio optimization. To the optimizer, it would look like six asset classes instead of two (since there are three versions of stocks and bonds). We would add the constraint that the sum of the weights to Taxable, Tax-Deferred, and Tax-Exempt groups could not exceed the percentage of our wealth in each respective account type. For example, if we only have 10% of our wealth in Tax-Exempt accounts, then U.S. Stocks – Tax Exempt + U.S. Bonds – Tax Exempt must be equal to 10%.
Such an approach allows for the explicit consideration of an individual’s tax rates (which are taken into account in the adjustment of expected returns) as well as the distribution of their wealth among different account types.
Case closed.[4]
Asset Location in a Less Than Perfect World
Unfortunately, the technology – and expertise – required to enable such an optimization is not readily available for many investors.
As an industry, the division of labor can significantly limit the availability of important information. While financial advisors may have access to an investor’s goals, risk tolerances, specific tax situation, and asset location break-down, asset managers do not. Therefore, asset managers are often left to make sweeping assumptions, like infinite investment horizons, defined and constant risk tolerances, and tax indifference.
Indeed, we currently make these very assumptions within our QuBe model portfolios. Yet, we think we can do better.
For example, consider investors at either end of the spectrum of asset location. On the one end, we have investors with the vast majority of their assets in tax-deferred accounts. On the other, investors with the vast majority of their wealth in taxable accounts. Even if two investors at opposite ends of the spectrum have an identical risk tolerance, their optimal portfolios are likely different. Painting with broad strokes, the tax-deferred investor can afford to have a larger percentage of their assets in tax-inefficient asset classes, like fixed income and futures-based alternative strategies. The taxable investor will likely have to rely more heavily on tax-efficient investments, like indexed equities (or active equities, if they are in an ETF wrapper).
Things get much messier in the middle of the spectrum. We believe investors have two primary options:
With all this in mind, we have begun to develop tax-managed versions of our QuBe model portfolios, and expect them to be available at the beginning of Q4.
Adjusting Expected Returns for Taxes
To keep this commentary to a reasonable length (as if that has ever stopped us before…), we’re going to use a fairly simple model of tax impact.
At the highest level, we need to break down our annual expected return into three categories: unrealized, externally realized, and internally realized.
Using this information, we can fill out a table, breaking down for each asset class where we expect returns to come from as well as within that category, what type of tax-rate we can expect. For example:
For example, in the table above we are saying we expect 70% of our annual U.S. equity returns to be unrealized while 30% of them will be realized at a long-term capital gains rate. Note that we also explicitly estimate what we will be receiving in qualified dividends.
On the other hand, we only expect that 35% of our hedge fund returns to be unrealized, while 15% will be realized from turnover (all at a long-term capital gains rate) and the remaining 50% will be internally realized by trading within the fund, split 40% short-term capital gains and 60% long-term capital gains.For example, in the table above we are saying we expect 70% of our annual U.S. equity returns to be unrealized while 30% of them will be realized at a long-term capital gains rate. Note that we also explicitly estimate what we will be receiving in qualified dividends.
Obviously, there is a bit of art in these assumptions. How much the portfolio turns over within a year must be estimated. What types of investments you are making will also have an impact. For example, if you are investing in ETFs, even very active equity strategies can be highly tax efficient. Mutual funds on the other hand, potentially less so. Whether a holding like Gold gets taxed at a Collectible rate or a split between short- and long-term capital gains will depend on the fund structure.
Using this table, we can then adjust the expected return for each asset class using the following equations:
Where,
In English,
As a simple example, let’s assume U.S. equities have a 6% expected return. We’ll assume a 15% qualified dividend rate and a 15% long-term capital gains rate. We’ll ignore state taxes for simplicity.
Our post-tax expected return is, therefore 6% – (6%-2%)*(30%*15%) – 2%*15% = 5.52%.
We can follow the same broad steps for all asset classes, making some assumptions about tax rates and expected sources of realized returns.
(For those looking to take a deeper dive, we recommend Betterment’s Tax-Coordinated Portfolio whitepaper[5], Ashraf Al Zaman’s Tax Adjusted Portfolio Optimization and Asset Location presentation[6], and Geddes, Goldberg, and Bianchi’s What Would Yale Do If It Were Taxable? paper[7].)
How Big of a Difference Does Tax Management Make?
So how much of a difference does taking taxes into account really make in the final recommended portfolio?
We explore this question by – as we have so many times in the past – relying on J.P. Morgan’s capital market assumptions. The first portfolio is constructed using the same method we have used in the past: a simulation-based mean-variance optimization that targets the same risk level as a 60% stock / 40% bond portfolio mix.
For the second portfolio, we run the same optimization, but adjust the expected return[8] for each asset class.
We make the following assumptions about the source of realized returns and tax rates for each asset class (note that we have compressed the above table by combining rates together after multiplying for the amount realized by that category; e.g. realized short below represents externally and internally realized short-term capital gains).
Again, the construction of the below table is as much art as it is science, with many assumptions embedded about the type of turnover the portfolio will have and the strategies that will be used to implement it.
We also make the following tax rate assumptions:
The results of both optimizations can be seen in the table below.
Broadly speaking, we see a shift away from credit-based asset classes (though, they still command a significant 27% of the portfolio) and towards equity and alternatives.
We would expect that if the outlook for equities improved, or we reduced the expected turnover within the portfolio, this shift would be even more material.
It is important to note that at least some of this difference can be attributed to the simulation-based optimization engine. Percentages can be misleading in their precision: the basis point differences between assets within the bond category, for example, are not statistically significant changes.
And how much difference does all this work make? Using our tax-adjusted expected returns, we estimate a 0.20% increase in expected return between tax-managed and tax-deferred versions right now. As we said: no silver bullets, just marginal improvements.
What About Municipal Bonds?
You may have noticed municipal bonds are missing from the above example. What gives?
Part of the answer is theoretical. Consider the following situation. You have two portfolios that are identical in every which way (e.g. duration, credit risk, liquidity risk, et cetera), except one is comprised of municipal bonds and one of corporate bonds. Which one do you choose?
The one with the higher post-tax yield, right?
This hypothetical highlights two important considerations. First, the idea that municipal bonds are for taxable accounts and corporate bonds are for tax-deferred accounts overlooks the fact that investors should be looking to maximize post-tax return regardless of asset location. If municipal bonds offer a better return, then put them in both accounts! Similarly, if corporate bonds offer a more attractive return after taxes, then they should be held in taxable accounts.
For example, right now the iShares iBoxx $ Investment Grade Corporate Bond ETF (LQD) has a 30-day SEC yield of 3.16%. The VanEck Vectors ATM-Free Intermediate Municipal Index ETF (ITM) has a 30-day SEC yield of just 1.9%. However, this is the taxable equivalent to an investor earning a 3.15% yield at a 39.6% tax rate.
In other words, LQD and ITM offer a nearly identical return within in a taxable account for an investor in the highest tax bracket. Lower tax brackets imply lower taxable equivalent return, meaning that LQD may be a superior investment for these investors. (Of course, we should note that municipal bonds are not corporate bonds. They often are often less liquid, but of higher credit quality.)
Which brings up our second point: taxes are highly personal. For a wealthy investor, an ordinary income tax of 35% could make municipal bonds far more attractive than they are for an investor only paying a 15% ordinary income tax rate.
Simply put: solving the when and where of municipal bonds is not always straight forward. We believe the best approach is account for them as a standalone asset class within the optimization, letting the optimizer figure out how to maximize post-tax returns.
Conclusion
We believe that a low-return world means that many investors will have a tough road ahead when it comes to achieving their financial goals. We see no silver bullet to this problem. We do see, however, many small steps that can be taken that can compound upon each other to have a significant impact. We believe that asset location provides one such opportunity and is therefore a topic that deserves far more attention in a low-return environment.
[1] See The Impact of High Equity Valuations on Safe Withdrawal Rates – https://blog.thinknewfound.com/2017/08/impact-high-equity-valuations-safe-retirement-withdrawal-rates/
[2] See Portfolios in Wonderland & The Weird Portfolio – https://blog.thinknewfound.com/2017/08/portfolios-wonderland-weird-portfolio/
[3] See The Butterfly Effect in Retirement Planning – https://blog.thinknewfound.com/2017/09/butterfly-effect-retirement-planning/
[4] Clearly this glosses over some very important details. For example, an investor that has significant withdrawal needs in the near future, but has the majority of their assets tied up in tax-deferred accounts, would significantly complicate this optimization. The optimizer will likely put tax-efficient assets (e.g. equity ETFs) in taxable accounts, while less tax-efficient assets (e.g. corporate bonds) would end up in tax-deferred accounts. Unfortunately, this would put the investor’s liquidity needs at significant risk. This could be potentially addressed by adding expected drawdown constraints on the taxable account.
[5] https://www.betterment.com/resources/research/tax-coordinated-portfolio-white-paper/
[6] http://www.northinfo.com/documents/337.pdf
[7] https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2447403
[8] We adjust volatility as well.