Coherent portfolio performance ratios

Abstract

In Quantitative Finance 2016, Chen, Hu and Lin (CHL) claimed the following: ‘ … there is yet no coherent risk measure related to investment performance.’ (p. 682). Our paper suggests and analyzes four coherence axioms that portfolio performance ratios should satisfy.

Our Portfolio Riskless Translation Invariance axiom must be satisfied to assure separation of the objective decision to optimize a portfolio’s risky composition from the subjective decision to optimize the weight of the portfolio's level of risk-free asset. Performance ratios with fixed thresholds other than the risk-free rate do not satisfy this axiom, allowing portfolio managers to affect an ex-ante performance ratio merely by changing the proportion of the risk-free asset in the portfolio rather than by improving the composition of the portfolio’s risky components. The magnitude of this potential drawback is examined using S&P-500 stock index data.

Replacing the fixed threshold, T, with a threshold $T(\gamma ,\alpha )$ that equals γ times the portfolio’s risk premium plus (1-γ) times the risk-free rate, eliminates the above shortcoming for any selected γ. In addition, using performance ratios with threshold $T(\gamma ,\alpha )$ rather than fixed T, assures consistency of performance ratios of effective stochastic dominance and risk-free asset rules.

Keywords:

• Coherent performance ratios
• Coherent risk measures
• Downside risk measures
• Lower partial moments
• Sortino ratio
• Omega ratio
• Kappa ratio
• Stochastic dominance
• FSDR and SSDR rules

JEL Classification:

• G11
• D81

Acknowledgment

We wish to acknowledge the very helpful comments and suggestions of two anonymous referees.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 The stochastic dominance rules for all rational investors (First degree Stochastic Dominance rule - FSD) and for all rational risk averse investors (Second degree Stochastic Dominance rule - SSD), respectively, provide the necessary and sufficient (optimal) efficiency rules for portfolio preference. However, the practical application of these rules for constructing optimal portfolios and obtaining market equilibrium conditions is quite limited.

2 For a review of the history of downside risk measures, see Nawrocki (1999).

3 We interchangeably refer to the ‘risky component’ as the ‘equity component’ or the ‘equity level’.

4 FSD applies to investors whose utility functions are not decreasing with wealth (positive first derivative). SSD applies to the utility functions of risk avert investors (positive first derivative and negative second derivative). In general, SD rules of the n-th degree apply to utilities with positive odd derivatives and negative even derivatives. (For more details, see Levy (2006), Theorem 3.5, page 131.) In Section 7 of this paper we show that when borrowing and lending at the riskless rate is allowed, the monotonicity requirement is less restrictive than would seem at first glance, since the inefficient set tends to increase with the use of Stochastic Dominance with Riskless Asset Rules. 

5 The case where $\alpha =0$ is redundant since in this case the entire portfolio’s capital is invested in the riskless asset, there is no risk and a ratio of reward to variability is undefined.

6 This axiom is related only to the expected strategic level of the riskless asset in the portfolio. It does not preclude the potential gains and a higher performance ratio due to successful timing in entering or exiting the risky market, as well as selecting a high beta portfolio before a bullish market, even though several empirical studies indicate that investment professionals lack a return timing ability (see: Friesen and Sapp (2007), Cuthbertson et al. (2010), Bodson et al. (2013), Sherman et al. (2017)).

7 The monetary separation theorem is the basis of the Sharpe-Lintner-Mossin Capital Asset Pricing Model (CAPM, Sharpe (1964), Lintner (1965), Mossin (1966)).

8 For example, assume two uniform distributions as follows: ${\tilde{R}}_y\sim U(0.05,0.07)\text{and}{\tilde{R}}_x\sim U(0.10,0.20)$and R_f = 0. Clearly, x dominates y according to FSD since even the lowest return of x is higher than the highest return of y, leading to the preference of x over y by every rational investor. However, calculating the Sharpe ratios, we obtain the following: $SR\left({\tilde{R}}_y\right)=\frac{0.06}{0.01}=6>3=\frac{0.15}{0.05}=SR\left({\tilde{R}}_x\right)$. Namely, according to the Sharpe ratio, ${\tilde{R}}_y$ is a better performing investment than ${\tilde{R}}_x$.

9 Note that as α decreases, the portfolio’s risk-free asset weight increases and the portfolio’s return gets closer to the risk-free rate. If $T>R_f$, the ratios are zero at some low level of α and negative at even lower α levels. 

10 We use this notation as a homage to the well-known “Coefficient of Variation” risk measure which is defined as a variable’s standard deviation divided by its expected return.

11 The quantile function is the inverse of the cumulative distribution function (CDF) and the q order quantile$X\left(q\right)$ of a random variate $\tilde{X}$ satisfies the following probability condition: $Pr\left(\tilde{X}\le X\left(q\right)\right)=q$.

12 With respect to SOR(T), Balder and Schweizer (2017) showed that if $X\underset{SSD}{D}Y$and $E\left(X\right)\ge T\ge E\left(Y\right)$, then $X\underset{\text{SOR}\left(T\right)}{D}Y$.

13 Note that propositions 4 through 7 hold for the Kappa ratio of all $n\ge 1$ degrees. Recall that the Omega ratio of the first degree is identical to 1 plus the Kappa ratio of the first degree and the Sortino ratio is identical to the Kappa ratio of the second degree.

14 Additionally, the diversification of the risky assets with other risky assets can reduce the number of conflicts, but the analysis of these diversification possibilities is related more to the determination of the parametric optimal portfolio.