Ranking Non-Normal Investment Risks

**Kent Smetters ^{*} Xingtan Zhang â€
This Version: February 3, 2014**

**Abstract**

While the Sharpe ratio is still the dominant measure for ranking risky investments, much effort has been made over the past three decades to find more robust measures that accommodate non- Normal risks (e.g., “fat tails”). But these measures have failed to map to the actual investor problem except under strong restrictions; numerous ad-hoc measures have arisen to fill the void. We derive a generalized ranking measure that correctly ranks risks relative to the original investor problem for a broad utility-and-probability space. Like the Sharpe ratio, the generalized measure maintains wealth separation for the broad HARA utility class. The generalized measure can also correctly rank risks following different probability distributions, making it a foundation for multi-asset class optization. This paper also explores the theoretical foundations of risk ranking, including proving a key impossibility theorem: any ranking measure that is valid for non-Normal distributions cannot generically be free from investor preferences. Finally, we show that approximation measures, which have sometimes been used in the past, fail to closely approximate the generalized ratio, even if those approximations are extended to an infinite number of higher moments.

**Keywords:** Sharpe Ratio, portfolio ranking, infinitely divisible distributions, generalized ranking measure, Maclaurin expansions

**JEL Code: G11**

^{*}Kent Smetters: Professor, The Wharton School at The University of Pennsylvania, Faculty Research Associate at the NBER, and affiliated faculty member of the Penn Graduate Group in Applied Mathematics and Computational Science. By Email: smetters@wharton.upenn.edu.

â€ Xingtan Zhang, PhD (Penn Mathematics), and first-year PhD student in Applied Economics at The Wharton School at The University of Pennsylvania. By email: xingtan@wharton.upenn.edu.

Bill Sharpe’s seminal 1966 paper demonstrated that picking a portfolio with the largest expected risk premium relative to its standard deviation is equivalent to picking the portfolio that maximizes the original investor’s expected utility problem, assuming that portfolio returns are Normally distributed.^{1} This simple mean-variance investment ranking measure – the “Sharpe ratio” – is, therefore, a sufficient statistic for the investor’s problem that does not rely on the investor’s preferences or wealth level.

The immense power of the Sharpe ratio ranking measure stems from the fact that it allows the investment management process to be decoupled from the specific attributes of the heterogeneous investor base. The multi-trillion dollar money management industry relies heavily on this separation. Investors in a mutual fund or hedge fund might differ in their levels of risk aversion and wealth (including assets held outside the fund). Nonetheless, an investment manager only needs to correctly estimate the first two moments of the fund’s return in order to pick the single risky portfolio that is best for each underlying investor.^{2} It is not surprising, therefore, that the Sharpe ratio is tightly integrated into the modern investment management practice and embedded into virtually all institutional investment analytic and trading platforms. Even consumer-facing investment websites like Google Finance reports the Sharpe ratio for most mutual funds along with just a few other basic statistics, including the fund’s alpha, beta, expected return, R^{2} tracking (if an indexed fund), and standard deviation.

Of course, it is well known that investment returns often exhibit “higher order” moments that might differ from Normality (Fama 1965; Brooks and Kat 2002; Agarwal and Naik 2004, and Malkiel and Saha 2005).3 In practice, investment professionals, therefore, often look for investment opportunities that would have historically-that is, in a “back test”-produced unusually large Sharpe ratios under the belief that large values provide some “buffer room” in case the underlying distribution is not Normal. This convention, though, is misguided. Outside of the admissible utility-probability space (“admissible space” for short) where the Sharpe ratio is valid, it is easy to create portfolios with large Sharpe ratios that are actually first-order stochastically dominated by portfolios with smaller Sharpe ratios (Leland 1999; Spurgin 2001; and Ingersoll et al. 2007).^{4} Besides the presence of the usual “fat tails,” it is now well known that non-Normally distributed risks easily emerge at the investment fund level with modern trading strategies and the use of financial derivatives (Section 6), which have exploded in use over time.

The historical debate over the ability of the Sharpe ratio to correctly rank risky investments following non-Normal probability distributions (e.g., Tobin 1958, 1969; Borch 1969; Feldstein 1969) has led to a long-standing interest in producing more robust ranking measures. The first line of work dates back to at least the work by Paul Samuelson (1970), who, at the time, also expressed some skepticism that such extensions were actually needed in practice. (The subsequent explosion of modern trading strategies, novel asset classes and financial derivatives might have changed his mind.) A short list of other contributors include the well-cited paper by Kraus and Litzenberger (1976), Scott and Horvath (1980), Owen and Rabinovitch (1983), Ingersoll’s (1987) classic textbook, Brandt et al (2005), Jurczendo and

^{1}Of course, the Sharpe ratio builds on the pioneering mean-variance work by Markowitz (1952, 1959) and Tobin (1958).

^{2}The Sharpe ratio, however, only ranks risky portfolios in order to determine the best one. The ratio itself does not determine the optimal division of an investor’s wealth between this best risky portfolio and the risk-free instrument. That division must be determined in a second stage using consumer-specific information. The Sharpe ratio, therefore, supports the standard division between the “investment manager,” who determines the best risky portfolio, and the “financial planner” who, knowing each client in more detail, helps decide the share of the client’s wealth that should be invested into this best risky asset based on the client’s specific circumstances.

^{3}A related literature has examined how disaster risk can explain equilibrium pricing within the neoclassical growth model (Barro 2009; Gabaix 2012; Gourio 2012; Wachter 2012)

^{4}In other words, the portfolio with the smaller Sharpe ratio would be preferred by all expected utility maximizers with positive marginal utility in wealth.

Maillet (2006), Zakamouline and Koekebakker (2008), DÃ¡vila (2010) and Pierro and Mosevich (2011). This line of work, however, imposed fairly strong restrictions on investor utility preferences and/or the risk distribution. The current paper contributes to this line of research by deriving a ranking measure that is valid over a broad admissible space.^{5}

A second line of research bypasses the investor’s expected utility problem altogether and produces risk measures that satisfy certain mathematical properties such as “coherence.”^{6} Examples of coherent risk measures include “average VaR,” “entropic VaR,” and the “superhedging price.” While these measures satisfy certain axioms, a portfolio that maximizes one or more of these measures does not necessarily maximize the standard investor expected utility problem, as considered by Sharpe and many others. The application of these measures for the actual investor is, therefore, unclear, which might help explain the continued popularity of the Sharpe ratio.

A third line of work, which is actually the largest line in scope, has evolved more from practitioners. It has produced heuristic measures that have a more “intuitive” interpretation in nature than the axiomatic-based measures. Common heuristic measures include “value at risk (VaR),”^{7} Omega, the Sortino ratio, the Treynor ratio, Jensen’s alpha, Calmar ratio, Kappa, Roy’s safety-first criterion, numerous tail risk measures, various upside-downside capture metrics, and many more.^{8} These metrics, however, tend to be especially problematic. Not only do they fail to satisfy any sort of reasonably mathematical properties, there is no apparent relationship between a reasonable description of the investor problem and these measures. In practice, therefore, investment managers often combine the Sharpe ratio with one or more of these measures when attempting to account for non-Normal risk (e.g., maximize the Sharpe ratio subject to the investment’s “value at risk” being less than some threshold). Despite its limitations, the Sharpe ratio, therefore, remains the gold standard of the investment industry.

This paper makes three contributions. First, as summarized in our Lemma 2, we demonstrate how to solve an infinite-order Maclaurin expansion for its correct asymptotic root when no closed form solution exists. We can then derive a generalized ranking measure (the “generalized ratio”) that correctly ranks risky returns under a much broader admissible utility-probability space consistent with the Sharpe ratio or previous extensions. By “correctly ranks,” we mean it in the tradition of Sharpe: the generalized ratio picks the portfolio preferred by the original investor expected utility problem.

It is easy to motivate the importance of allowing for a broad admissible utility-probability space. A broad utility space captures realistic investor attitudes toward risk. For example, while the common assumption of Constant Absolute Risk Aversion is useful for obtaining various theoretical insights, it is also fairly implausible for modeling risky investment decisions. Similarly, allowing for a broad set of risk distributions is, of course, important for accommodating more extreme risks with “fat tails.”

But our generalized ratio can also rank between risks that follow different probability distributions. The generalized ratio, therefore, can be used as the foundation for multi-asset class optimization. For example, it can pairwise rank a risky portfolio without financial derivatives that follows one distribution

^{5}Throughout this paper, we will write expressions like “ranking measure ABC is valid over admissible space XYZ” even though such terminology is a bit redundant since admissibility implies validity. However, we believe that such terminology is generally understood and more readable than various alternatives.

^{6}A “coherent” risk measure satisfies monotonicity, sub-additivity, homogeneity, and translational invariance (Artzner et al 1999). More recent work has emphasized risk measures that avoid “worst case” scenarios and are monotonic in first-order stochastic dominance. See, for example, Aumann and Serrano (2008); Foster and Hart (2009); and Hart (2011).

^{7}Standard VaR is not coherent, whereas the variants on VaR noted in the previous paragraph are coherent.

^{8}Modigliani (1997) proposed a transformation of the Sharpe ratio, which became known as the “risk-adjusted performance measure.” This measure attempts to characterize how well a risk rewards the investor for the amount of risk taken relative to a benchmark portfolio and the risk-free rate. This measure is not included in the list in the text because it mainly provides a way of interpreting the unit-free Sharpe ratio rather than offering an alternative measure in the presence of non- Normally distributed risk.

against another risky portfolio with option overlays following a different distribution. This flexibility is much more powerful than simply assuming that all potential portfolio combinations follow the same probability distribution form, even if that distribution is more extensible than Normality.

Like the original Sharpe ratio, our generalized measure preserves wealth separation under the broad functional form of HARA utility, which includes many standard utility functions as special cases.9 Unlike the original Sharpe ratio, however, our generalized ratio does not preserve separation from investor preferences. But we show that this limitation is not a function of the generalized risk measure. Rather, we prove a key impossibility theorem: preference separation is generically impossible in the presence of non-Normal risk. Fortunately, as a practical matter, the generalized ratio still supports the decoupled investment management process noted above: instead of reporting a single ratio, each fund can report a small tuple of ratios corresponding to different standardized levels of risk aversion (see Section 6). By law, financial advisors must already actively test for the level of risk aversion of each client.

Second, using some of the machinery that we developed, we then “backtrack” to explore the theoretical foundations of the classic Sharpe ratio in more detail. Despite its extensive usage in academics and industry, very little is actually known about the Sharpe ratio beyond the few cases where it is well known to correctly rank risks (e.g., Normally distributed risk or quadratic utility). We show that the Sharpe ratio is actually valid under a larger admissible space than currently understood. We also explore why it is challenging to actually write down a necessary condition for the Sharpe ratio to be a valid ranking measure. In the process, we are also able to generalize the Kraus-Litzenberger (1976) “preference for skewness” result to an unlimited number of higher moments. This generalization is useful because plausible utility functions produce an infinite number of non-zero higher-order derivatives, and there does not exist any probability distribution that can be fully described by any finite number of cumulants greater than two.

Third, we derive a linear approximation of the investor problem in the presence of non-Normal higher-order moments. This formulation accommodates a simple closed-form solution, and it nests some previous attempts to generalize the Sharpe ratio. Our computations, however, show that approximations can be very inaccurate. Accurate ranking, therefore, requires using the generalized ratio.

The paper is organized as follows. Section 2 provides an overview of the standard investor problem. Section 3 derives the generalized ratio described earlier. Section 4 explores the theoretical foundations of the Sharpe ratio in more detail. Section 5 derives the linear approximation. Section 6 provides numerical examples comparing the Sharpe ratio, the generalized ratio, and the linear approximation for a range of potential investment applications. Section 7 concludes. Proofs of lemmas and theorems are provided in the Appendices.

The Sharpe ratio correctly ranks risky investments, consistent with the original investor problem, if risks are Normally distributed. Considerable past effort has been made to develop new measures that are robust to non-Normally distributed risks, which emerge with “fat tails,” modern trading strategies and the modern extensive use of financial derivatives in hedging portfolio risk. Some of this effort has started with the original investor problem and added some higher moments under fairly strong restrictions on the risk distribution and/or utility function. Even more ranking measures have been developed that satisfy certain mathematical properties or are purely ad hoc. Those measures, however, do not map back to an original investor problem, making their interpretation unclear. Not surprisingly, the Sharpe ratio, therefore, remains the gold standard in the industry, despite its lack of robustness to more general risk distributional assumptions.

This paper derives a generalized ranking measure that is valid under a broad admissible utilityprobability space and yet preserves wealth separation for the broad HARA utility class. Our ranking measure can be used with non-Normal distributions. Because it can also pairwise compare composite risks following different distributions, it can also serve as the foundation for multi-asset class portfolio optimization, thereby replacing the mixture of other measures that are currently being used in industry. We demonstrate that the generalized ratio can produce very different optimal allocations than the Sharpe ratio, especially in the context of financial derivatives and other securities that produce non-Normal distributions. Along the way, we prove a key impossibility theorem: any ranking measure that is valid at non-Normal “higher moments” cannot generically be free from investor preferences. But, as a matter of practice, we demonstrate how the generalized ratio can be easily presented at the fund level for different risk tolerances, which already must be legally assessed by financial advisors for each investor.

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