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 meanreturn, g, of a portfolio – the value we want to maximize – will be equal to the annualized compound return:
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.
Growth Optimal Portfolios
By Corey Hoffstein
On July 5, 2017
In Portfolio Construction, Weekly Commentary
This post is available as a PDF download here.
Summary
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.
Over time, a number of attributes regarding GMM have been proved.
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,
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:
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:
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