Comparing Sharpe ratios: So where are the p-values?

Abstract

Until recently, since Jobson and Korkie (1981), derivations of the asymptotic distribution of the Sharpe ratio that are practically useable for generating confidence intervals or for conducting one- and two-sample hypothesis tests have relied on the restrictive, and now widely refuted, assumption of normally distributed returns. This paper presents an easily implemented formula for the asymptotic distribution that is valid under very general conditions — stationary and ergodic returns — thus permitting time-varying conditional volatilities, serial correlation, and other non-iid returns behaviour. It is consistent with that of Christie (2005), but it is more mathematically tractable and intuitive, and simple enough to be used in a spreadsheet. Also generalised beyond the normality assumption is the small sample bias adjustment presented in Christie (2005). A thorough simulation study examines the finite sample behaviour of the derived one- and two-sample estimators under the realistic returns conditions of concurrent leptokurtosis, asymmetry, and importantly (for the two-sample estimator), strong positive correlation between funds, the effects of which have been overlooked in previous studies. The two-sample statistic exhibits reasonable level control and good power under these real-world conditions. This makes its application to the ubiquitous Sharpe ratio rankings of mutual funds and hedge funds very useful, since the implicit pairwise comparisons in these orderings have little inferential value on their own. Using actual returns data from 20 mutual funds, the statistic yields statistically significant results for many such pairwise comparisons of the ranked funds. It should be useful for other purposes as well, wherever Sharpe ratios are used in performance assessment.

Appendices

Appendix A

Alternate derivation of the distribution of under IID via the delta method

According to Stuart and Ord (1994), pp. 350–352, the ‘delta method’ can be used to derive the variance of a function, g(x₁, x₂, x₃, …), of a number of random variables, and if the function is the ratio of two random variables, then:

If x₁=μ and x₁=σ then

since x₁=μ−R _ft yields the same results above, we can treat μ/σ=SR, and so

which is Merten's (2002) result. The more technical conditions required for the valid use of the delta method are discussed in Appendix C.

Appendix B

Equivalence of Merten's (2002) iid, and Christie's (2005) GMM, derivations of the distribution of

Under only the requirements of stationarity and ergodicity, Christie (2005) derives (C21),

which can be simplified as below:²⁸

since E[R _t ²]=σ²+μ²,

since E[R _t ³]=μ₃+3σ²μ+μ³,

So , which is Merten's (2002) result, and that derived in Appendix A.

Appendix C

Variance of the difference between two Sharpe ratios

If SR _a and SR _b are the respective Sharpe ratios for the returns (R _at and R _bt ) of funds ‘a’ and ‘b,’ then use the ‘delta method’²⁹ (see Greene, 1993; Stuart and Ord, 1994) to obtain the asymptotic variance of : Assuming σ _f ²=0, which is always essentially, if not literally true, SR=μ−R _f /σ=f(μ, σ²), and so let u=(μ _a , μ _b , σ _a ², σ _b ²) and û=(μ̂ _a , μ̂ _b , ς̂ _a ², ς̂ _b ²), then, where Ω is the variance–covariance matrix of u:

where σ_a, b=Cov(a, b), μ_3a=E[(a−μ _a )³]=Cov(μ _a , σ _a ²), μ_3b=E[(b−μ _b )³]=Cov(μ _b , σ _b ²) (see Mertens, 2002), μ_1a, 2b=E[(a−μ _a )(a−μ _b )²]=Cov(μ _a , σ _b ²), and μ_1b, 2a=E[(b−μ _b )(a−μ _a )²]=Cov(μ _b , σ _a ²) (see Espejo and Singh, 1999). Now, ,

Since =Var(σ _a ²)=Cov (σ _a ², σ _a ²) =μ_4a−σ _a ⁴=E [(a−E[a])⁴]−σ _a ⁴ =E[(a−E[a])²(a−E[a])²]−σ _a ²σ _a ², then Cov(σ _a ²,  σ _b ²)=E[(a−E[a])²(b−E [b])²]−σ _a ²σ _b ²=μ_2a,_2b−σ _a ²σ _b ², whereμ_2a,_2b is the second central moment of the joint distribution of a and b. The same result can be obtained using Stuart and Ord's (1994) (pp.457–458) result of Cov(ς̂ _a ², ς̂ _b ²)=κ_2a,_2b/n+2κ_1a,_1b²/(n−1), where κ_2a,_2b is the second joint cumulant of the joint distribution of a and b, and κ_1a,_1b is the first joint cumulant, equal to the first joint central moment, μ_1a,_1b, which is the covariance. Dropping the n coefficients due to the use of the estimators ς̂ _a ², ς̂ _b ² for σ _a ², σ _b ² yields Cov(σ _a ², σ _b ²)=κ_2a,_2b/n+2κ_1a,_1b² =κ_2a,_2b+2μ_1a,_1b²=κ_2a,_2b+2σ_a, b². Recognising that the joint cumulant can also be expressed in terms of central moments, κ_2a,_2b=μ_2a,_2b−μ_2a,₀ × μ_0, 2b −2μ_1a,_1b²=μ_2a,_2b−σ _a ², σ _b ²−2σ_a, b² (see Stuart and Ord, 1994, p. 107, and Smith, 1995), we have:

Thus,

Hence, analogous to the variance of the distribution of a single , (6), the variance of the difference between two is

Note that when ρ_a, b=0, μ_2a,_2b=σ _a ²σ _b ², μ_1a,_2b=0, and μ_1b,_2a=0, and so the entire covariance term of Var _diff disappears, as it should.

Minimum variance unbiased estimators of μ_1a,_2b, μ_1b,_2a, & μ_2a,_2b are the respective h-statistics h_1a,_2b, h_1b,_2a, & h_2a,_2b, where h_1, 2=[2s_0, 1²s_1, 0−ns_0, 2s_1, 0−2s_0, 1²s_1, 1 −n²s_1, 2]/[n(n−1)(n−2)], and h_2, 2=[−3s_0, 1²s_1, 0² +ns_0, 2s_1, 0²+4ns_0, 1s_1, 0s_{1,  1}−2(2n−3)s_1, 1²−2(n²−2n+3)s_1, 0s_{1,  2}+s_0, 1²s_2, 0(2n−3)s_0, 2s_2, 0−2(n²−2n+3) s_0, 1s_2, 1+n(n²−2n+3)s_2, 2]/[(n−3)(n−2)(n−1)n], where s_x, y are the simple power sums of s_x, y=∑_i=1ⁿa _i ^xb _i ^y (see Rose and Smith, 2002, pp. 259–260).

This derivation is valid under iid returns, but because the one-sample estimator (6), derived using the same (delta) method (a la Mertens, 2002), was shown in Appendix B to be valid under the more general conditions afforded by its (identical) GMM derivation (a la Christie, 2005), we suspect that those more general conditions of stationarity and ergodicity are the only requirements for the two-sample estimator of (13) as well. Proving this is the topic of continuing research.

Appendix D

Equivalence of Var _diff with Memmel (2003) and Jobson and Korkie (1981)

Under iid normality, Memmel's (2003) correction of Jobson and Korkie's (1981) variance of the two-sample statistic for the difference between two Sharpe ratios is:

Under iid normality, TV is identical to Var _diff , as shown below:

Under iid normality, μ₃/σ₃=0, μ_1, 2=0, μ₄/σ⁴=3, & μ_2a,_2b=(1+2ρ_a, b²)σ _a ²σ _b ² (see Stuart and Ord, 1994, p. 105), so

which is Memmel's (2003) result.

Appendix E

Simulation distributions: APD of Komunjer (2006)

Komunjer (2006) gives the density of the APD below:

where 0<α<1, λ>0, and δ_α, λ=2α^λ(1−α)^λ/α^λ+(1−α)^λ

The α parameter (0<α<1) controls skewness, with symmetry at α=0.5, and λ>0 controls kurtosis, such that when α=0.5, λ= → the uniform distribution, λ=1.0 → the Laplace distribution (with variance=2.0), λ=2.0 → the normal distribution (with variance=0.5) and any λ → the Generalised Power Distribution. When α≠0.5, λ=1.0 → the Asymmetric Laplace distribution of Kozubowski and Podgorski (1999), and λ=2.0 → the two-piece normal distribution (see Johnson et al., 1994, vol. 1, p. 173 and vol. 2, p. 190). This does APD allow simultaneous control over skewness and kurtosis, nesting the normal and Laplace densities, and asymmetric versions of each, as well as any ‘in between’ combination of asymmetry and kurtosis (Figure E1).

Location and scale are accommodated via the simple transformation: X≡θ+φU

APD moments are given by:

(see Table E1). So for example,

and

To standardise the APD for the simulations presented in this study, U is modified by u′=u/sqrt[Var(u)] (because, for example, when α=0.5 and λ=1.0, Var(U)=2.0, and when α=0.5 and λ=2.0, Var(U)=0.5).