Inference for the Sharpe Ratio Using a Likelihood-Based Approach

Abstract

The Sharpe ratio is the prominent risk-adjusted performance measure used by practitioners. Statistical testing of this ratio using its asymptotic distribution has lagged behind its use. In this paper, highly accurate likelihood analysis is applied for inference on the Sharpe ratio. Both the one- and two-sample problems are considered. The methodology has 𝑂(𝑛^−3/2) distributional accuracy and can be implemented using any parametric return distribution structure. Simulations are provided to demonstrate the method's superior accuracy over existing methods used for testing in the literature.

1. Introduction

The measurement of fund performance is an integral part of investment analysis. Investments are often ranked and evaluated on the basis of their risk-adjusted returns. Several risk-adjusted performance measures are available to money managers of which the Sharpe ratio is the most popular. Introduced by William Sharpe in 1966 [1], this ratio provides a measure of a fund’s excess returns relative to its volatility. Expressed in its usual form, the Sharpe ratio for an asset with an expected return given by 𝜇 and standard deviation given by 𝜎 is given by the following: SR=𝜇−𝑅_𝑓𝜎, (1.1) where 𝑅_𝑓 is the risk-free rate of return. From this expression, it is clear to see how this ratio provides a measure of a fund’s excess return per unit of risk.

The Sharpe ratio has been extensively studied in the literature. The main criticism leveled against this measure concerns its reliance on only the first two moments of the returns distribution. If investment returns are normally distributed then the Sharpe ratio can be justified. On the other hand, if returns are asymmetric then it can be argued that the measure may not accurately describe the fund’s performance as moments reflecting skewness and kurtosis are not captured by the ratio. To address this issue, several measures exist in the literature which integrate higher moments into the performance measure. The Omega measure is one such measure that uses all the available information in the returns distribution. Keating and Shadwick [2] provide an introduction to this measure. While various methods are available, they are also more complex and often very difficult to implement in practice. To gauge the trade-off between the attractiveness of such measures and their cost, Eling and Schuhmacher [3] compared the Sharpe ratio with 12 other approaches to performance measurement. Eling and Schuhmacher [3] focussed on the returns of 2,763 hedge funds. Hedge funds are known to have return distributions which differ significantly from the normal distribution and, as such, provide a rich and relevant environment for such a comparison. Their study concluded that the Sharpe ratio produced rankings that were largely identical to those obtained from the 12 other performance measures. In other words, the choice of performance measure did not matter much in terms of ranking funds. From a practical perspective, this imparts credibility to the use of the Sharpe ratio for ranking funds with highly nonnormal returns.

Our focus in this paper is on the statistical properties of the Sharpe ratio. While this ratio may be the most widely known and used measure of risk-adjusted performance for an investment fund, fund managers rarely (if ever) indicate any measure of associated statistical significance of their rankings. This point was made in Opdyke [4]. We use a likelihood-based statistical method to obtain highly accurate 𝑝 values and confidence intervals for evaluating the significance of the Sharpe ratio. While the methodology we use is applicable under any parametric distributional assumption, for expositional clarity we demonstrate the use of the method under the assumption of normal returns. We investigate inference for both the one-sample problem as well as the two-sample problem. We compare our method to currently existing methods in the literature. Simulations indicate the superior performance of the method compared to existing methods in the literature.

The paper is organized as follows. Section 2 provides an overview of the asymptotic distribution of the Sharpe ratio under various assumptions. Section 3 lays out the framework for the likelihood-based approach we will use for testing. Section 4 applies this methodology for inference on the Sharpe ratio. Examples and simulations for this one-sample case are then presented in Section 5. The two-sample problem is given in Section 6, with example and simulations presented in Section 7. Section 8 concludes.

2. Distribution of the Sharpe Ratio

Jobson and Korkie [5] and Lo [6] derived the asymptotic distribution of the Sharpe ratio assuming identically and independently normally distributed returns. Consider a fund with a return at time 𝑡 given by 𝑅_𝑡, 𝑡=1,…,𝑇. Further assume that the returns are identically and independently distributed (IID) as 𝑁(𝜇,𝜎²). If we let the vector 𝜃=(𝜇,𝜎²)^′, then as given in Lo [6] we have the following asymptotically normal distribution for 𝜃: √𝑇𝜃−𝜃→𝑁(0,𝑉), (2.1) where 𝜎𝑉≡²002𝜎⁴, (2.2) and 𝜃=(𝜇,𝜎²)^′ is the maximum likelihood estimator (MLE) with 𝜇 and 𝜎² given by the following formulas 1𝜇=𝑇^𝑇_𝑡=1𝑅_𝑡,𝜎²=1𝑇^𝑇_𝑡=1𝑅_𝑡−𝜇². (2.3) Lo’s result is straightforward to prove. For large samples, by the central limit theorem, we have 𝜃−𝜃^′var𝜃⁻¹𝜃−𝜃→𝜒²_𝑘, (2.4) where 𝑘=dim(𝜃) and 𝜃=𝜇𝜎varvarcov𝜇,²cov𝜇,𝜎²var𝜎². (2.5) See for instance, Cox and Hinkley [7] for this result. Given 𝜇=(1,0)𝜃 and 𝜎²=(0,1)𝜃, we have 𝜇−𝜇𝜇(),𝜎var→𝑁0,1²−𝜎²var𝜎²→𝑁(0,1). (2.6)

Given the Sharpe ratio is a function of 𝜃, the Delta method can be used to derive its asymptotic distribution. Let SR=𝑔(𝜃) be a continuous function with Jacobian denoted by 𝜕𝑔(𝜃)/𝜕𝜃 and let the Sharpe ratio estimate be given by SR=𝑔(𝜃). Applying a first-order Taylor expansion on 𝑔(𝜃) around the true parameter value 𝜃, we have 𝑔𝜃(𝜃)+(𝜃)≐𝑔𝜕𝑔𝜕𝜃^′𝜃−𝜃. (2.7) Rearranging terms and multiplying by √𝑇 yields √𝑇𝑔𝜃(𝜃≐(𝜃)−𝑔)𝜕𝑔𝜕𝜃^′√𝑇𝜃−𝜃. (2.8) Using the above result and applying the Delta method, we have the asymptotic distribution of 𝑔(𝜃): √𝑇𝑔𝜃(𝜃≐(𝜃)−𝑔)𝜕𝑔𝜕𝜃^′√𝑇(𝜃)𝜃−𝜃→𝑁0,𝜕𝑔𝜕𝜃^′𝑉𝜕𝑔(𝜃).𝜕𝜃 (2.9) Note that the Jacobian term is readily computed as follows: 𝜕𝑔(𝜃)=(𝜃)𝜕𝜃𝜕𝑔(𝜃)𝜕𝜇𝜕𝑔𝜕𝜎²=1𝜎−𝜇−𝑅_𝑓2𝜎³. (2.10) Combining terms yields the variance of the asymptotic distribution: 𝜕𝑔(𝜃)𝜕𝜃^′𝑉𝜕𝑔(𝜃)𝜕𝜃=1+𝜇−𝑅_𝑓²2𝜎²1=1+2SR². (2.11) By the central limit theorem, the distribution of the Sharpe ratio assuming identically and independently distributed normal returns is therefore √𝑇SR−SR1+(1/2)SR²→𝑁(0,1). (2.12) From this distribution, 100(1−𝛼)% confidence intervals for the Sharpe ratio can be constructed in the usual fashion: SR±𝑧_𝛼/21+(1/2)SR²𝑇, (2.13) where 𝑧_𝛼/2 is the 100×(1−𝛼/2)th percentile of the standard normal distribution.

The above derivation only holds under the IID normal assumption. Under the IID assumption but without assuming normally distributed returns, Mertens [8] derived the following asymptotic distribution of the Sharpe ratio: √𝑇SR−SR1+(1/2)SR²−SR⋅𝛾₃+SR²⋅𝛾₄(),−3/4→𝑁0,1 (2.14) where 𝛾₃ and 𝛾₄ are defined as follows: 𝛾₃=𝐸𝑅_𝑡−𝜇³𝜎³,𝛾₄=𝐸𝑅_𝑡−𝜇⁴𝜎⁴with𝜎²1=1+2SR². (2.15)

Christie [9] derived the asymptotic distribution of the Sharpe ratio under the more relaxed assumption of stationarity and ergodicity. Opdyke [4] interestingly showed that the derivation provided by Christie [9] under the non-IID returns condition was in fact identical to the one provided by Mertens [8]. For a complete discussion and proof see Opdyke [4]. For our purposes, we will use asymptotic distributions (2.12) and (2.14) to compare to our proposed likelihood-based approach. In the next section below we review the likelihood-based approach to inference and in the following section we apply the methodology to the Sharpe ratio.

3. Likelihood-Based Approach to Inference

In this section we review standard first-order likelihood methods and then turn our attention to higher-order methods. For this section, the underlying model under consideration is as follows: 𝑌=(𝑌₁,…,𝑌_𝑛) is a vector of IID random variables with density given by 𝑓(⋅;𝜃), where 𝜃 is a 𝑘-dimensional vector of parameters. Statistical inference concerns drawing conclusions about 𝜃 or a function of 𝜃, namely, 𝜓=𝜓(𝜃), given an observed sample 𝑦=(𝑦₁,…,𝑦_𝑛) and in this paper we assume that the interest parameter is 𝜓(𝜃). We further assume that this interest parameter is scalar. Let 𝜆=𝜆(𝜃) be a (𝑘−1)-dimensional nuisance parameter vector. The log-likelihood function of 𝜃 is 𝑙(𝜃)=𝑙𝜃;𝑦=log^𝑛_𝑖=1𝑓𝑦_𝑖;𝜃. (3.1)

Likelihood analysis typically involves the maximum likelihood estimator (MLE) 𝜃=(𝜓,𝜆^′)^′=argmax_𝜃𝑙(𝜃;𝑦) and the constrained maximum likelihood estimator 𝜃=(𝜓,𝜆^′)^′ = argmax_𝜆𝑙(𝜃;𝑦), where the maximum is taken for fixed values of 𝜓. The constrained MLE can be solved by maximizing 𝑙(𝜃) subject to 𝜓(𝜃)=𝜓. This is often done using the Lagrange multiplier method where the Lagrangean is given by the following: 𝐻(𝜃,𝛼)=𝑙𝜃;𝑦+𝛼𝜓(𝜃)−𝜓. (3.2) The Lagrange multiplier is given by 𝛼. The tilted log-likelihood is defined as follows: 𝑙̃(𝜃)=𝑙(𝜃)+𝛼𝜓(𝜃)−𝜓. (3.3)

Using the above information, standard first-order departure measures can be obtained. These measures include the Wald departure (the standardized maximum likelihood estimator) and the signed square root log-likelihood ratio statistic: 𝑞≡𝑞𝜓=𝜓−𝜓var𝜓^−1/2,𝑅≡𝑅𝜓=sgn𝜓−𝜓2𝑙𝜃;𝑦−𝑙𝜃;𝑦^1/2. (3.4)(3.5) Note that var(𝜓) can be estimated by using the Delta method var𝜓≈𝜓^′_𝜃𝜃var𝜃𝜓_𝜃𝜃, (3.6) where the variance of 𝜃, var(𝜃), can be estimated by using either the Fisher expected information matrix 𝐸−𝜕²𝑙(𝜃)𝜕𝜃𝜕𝜃^′, (3.7) or the observed information matrix evaluated at the MLE 𝑗_𝜃𝜃′𝜃𝜕=−²𝑙(𝜃)𝜕𝜃𝜕𝜃^′_𝜃=𝜃. (3.8) The latter is preferred however because of the simplicity in calculation. The 𝑝 value function for these methods are 𝑝(𝜓)=Φ(𝑞) and 𝑝(𝜓)=Φ(𝑅), where Φ represents the standard normal cumulative distribution function. These 𝑝 value functions have order of convergence 𝑂(𝑛^−1/2) and are correspondingly referred to as first-order methods (note that (3.5) is invariant to reparameterization whereas (3.4) is not. However, in practice, (3.4) is preferred because of its simplicity of use. Doganaksoy and Schmee [10] illustrate that (3.5) has better coverage than (3.4) in the cases they examined). A 100(1−𝛼)% confidence interval is then given by the following: min𝑝⁻¹𝛼2,𝑝⁻¹𝛼1−2,max𝑝⁻¹𝛼2,𝑝⁻¹𝛼1−2. (3.9)

Approximations aimed at improving the accuracy of the first-order methods have been worked on during the past three decades. Higher-order asymptotic approximations for the distribution of an estimate can be constructed using expansion methods of which Edgeworth expansions and saddlepoint approximations are most common. Edgeworth approximations tend to produce good approximations near the center of a distribution, with relatively poorer tail approximations, while saddlepoint approximations produce remarkably accurate tail approximations. Given we are interested in tail probabilities, our focus will thus be on approximation methods that use the saddlepoint method. We will not review the saddlepoint approximation literature here but the interested reader is directed to Daniels [11, 12] and Barndorff-Nielsen and Cox [13].

For a canonical exponential family model with canonical parameter vector 𝜃=(𝜓,𝜆^′)^′ and density 𝑓𝑦;𝜃=exp𝜓𝑢𝑦+𝜆^′𝑣𝑦−𝑐(𝜃)+𝑑𝑦, (3.10) where (𝑈(𝑌),𝑉(𝑌)) is a minimal sufficient statistic for 𝜃, the saddlepoint-based approximation for the density of 𝜃 is given as follows: 𝑓𝜃;𝜃≐𝑐𝑗_𝜃𝜃′𝜃^1/2exp𝑙𝜃;𝑦−𝑙𝜃;𝑦, (3.11) where 𝑐 is a normalizing constant. This extremely important approximation to the density of the maximum likelihood estimator is referred to as Barndorff-Nielsen’s 𝑝^∗ formula and can be found in Barndorff-Nielsen [14] (this approximation is valid for more general models but requires the existence of an ancillary statistic). Thus the marginal density for 𝜓 can be obtained. In general however, the closed form of the marginal density is not available.

Fraser et al. [15] showed that the result by Lugannani and Rice [16] can be applied to approximate the 𝑝 value function and it takes the form: 𝑝𝜓(𝑅)(𝑅)1≐Φ+𝜙𝑅−1𝑄. (3.12) The statistic 𝑄 is given by the following: 𝑄≡𝑄𝜓=𝜓−𝜓𝑗_𝜃𝜃′𝜃^1/2𝑗_𝜆𝜆′𝜃^−1/2, (3.13) where 𝑗_𝜃𝜃′(𝜃) is the observed information matrix and 𝑗_𝜆𝜆′(𝜃) is the observed nuisance information matrix defined as follows: 𝑗_𝜃𝜃′𝜃𝜕=−²𝑙𝜃;𝑦𝜕𝜃𝜕𝜃^′_𝜃=𝜃,𝑗_𝜆𝜆′𝜃𝜕=−²𝑙𝜃;𝑦𝜕𝜆𝜕𝜆′_𝜃=𝜃. (3.14)(3.15) The term |𝑗_𝜃𝜃′(𝜃)||𝑗_𝜆𝜆′(𝜃)|⁻¹ can be viewed as an estimate of the variance of 𝜓, where the elimination of the nuisance parameter 𝜆 has been taken into consideration. Note that 𝑄 is a standardized maximum likelihood departure in the canonical parameter scale. The statistic 𝑅 is the log-likelihood ratio given by (3.5). An alternate approximation to (3.12) is given by Barndorff-Nielsen [17]: 𝑝𝜓1≐Φ𝑅−𝑅𝑅log𝑄≐Φ𝑅^∗. (3.16) These approximations are asymptotically equivalent (see [18]) and have order of convergence 𝑂(𝑛^−3/2). They are correspondingly referred to as third-order methods. Hence 100(1−𝛼)% confidence interval for 𝜓 can be obtained from (3.9).

Consider now a general full rank exponential family model with parameter vector 𝜃 and density given by the following: 𝑓𝑦;𝜃=exp𝜑^′(𝜃)𝑡𝑦−𝑐(𝜃)+ℎ𝑡𝑦, (3.17) where 𝜑(𝜃) is the canonical parameter and 𝑡(𝑦) is the canonical variable. In order to use either the Lugannani and Rice approximation given in (3.12) or the Barndorff-Nielsen approximation given in (3.16) for inference on our interest parameter 𝜓(𝜃), we need to calculate both 𝑅 and 𝑄 in 𝜑(𝜃) scale. Note that 𝑅 calculated in the original 𝜃 scale is equivalent to 𝑅 calculated in 𝜑(𝜃) scale. Moreover, Fraser and Reid [19] derive a 𝑄 in the 𝜑(𝜃) scale. Their methodology involves replacing the parameter of interest 𝜓 by a linear function of the 𝜑(𝜃) coordinates (for a detailed discussion and derivation see Fraser and Reid [19]). This newly calibrated parameter is given by the following: 𝜒(𝜃)=𝜓_𝜃′𝜃𝜑_𝜃⁻¹_′𝜃𝜑(𝜃). (3.18) Then |𝜒(𝜃)−𝜒(𝜃)| measures the departure of |𝜓−𝜓| in the 𝜑(𝜃) scale. The function 𝑄 is then constructed as follows: 𝑄=sgn𝜓𝜃−𝜓𝜒𝜃−𝜒𝜃var𝜒(𝜃)−𝜒(𝜃)^−1/2. (3.19) Fraser and Reid [19] show that 𝜒𝜃𝜃≈𝜓var−𝜒^′𝜃′𝜃𝚥̃⁻¹_𝜃𝜃′𝜃𝜓_𝜃′𝜃𝚥̃_𝜃𝜃′𝜃𝜑_𝜃′𝜃⁻²𝑗_𝜃𝜃′𝜃𝜑_𝜃′𝜃⁻², (3.20) where 𝑗_𝜃𝜃′(𝜃) is defined in (3.14) and 𝚥̃_𝜃𝜃′(𝜃) is defined as follows: 𝚥̃_𝜃𝜃′𝜃𝜕=−²𝑙̃(𝜃)𝜕𝜃𝜕𝜃^′_𝜃=𝜃, (3.21) where 𝑙(̃𝜃) is the tilted log-likelihood function defined in (3.3). The function 𝑄 in (3.19) along with 𝑅 is used in the Lugannani and Rice approximation in (3.12) or the Barndorff-Nielsen expression in (3.16) which implicitly yields a new 𝑅^∗.

In the next section we apply the above methodology and use Barndorff-Nielsen’s approximation to obtain highly accurate 𝑝 values to test particular hypothesized values of the Sharpe ratio. We further note that for all our examples and simulations in this paper, the Lugannani and Rice expression given in (3.12) was also used but we do not report results from this method as they are virtually identical to those produced by the Barndorff-Nielsen expression.

4. Methodology for the Sharpe Ratio

We emphasize that while the third-order methodology discussed in the previous section is applicable under any parametric distributional assumptions, for expositional clarity we demonstrate the use of the method under the assumption of IID normal returns (see Fraser et al. [20] for the general model setup). Consider a fund with return at time 𝑡 given by 𝑅_𝑡, (𝑡=1,…,𝑇). And let 𝑅_𝑓 be the mean return for the risk-free asset. We assume that 𝑅₁,…𝑅_𝑇 IID 𝑁(𝜇,𝜎²). The interest parameter is the Sharpe Ratio: SR=𝜇−𝑅_𝑓𝜎=𝜓, (4.1) and 𝜃=(𝜇,𝜎²)^′. The log-likelihood and related first and second derivatives, denoted by 𝑙_⋅(𝜃) and 𝑙_⋅⋅(𝜃), respectively, are given as follows: 𝑙(𝜃)=𝑙𝜇,𝜎²𝑇=−2log𝜎²−12𝜎²Σ𝑅_𝑡−𝜇²,𝑙_𝜇(𝜃)=1𝜎²Σ𝑅_𝑡,𝑙−𝜇_𝜎2(𝜃)𝑇=−2𝜎²+12𝜎⁴Σ𝑅_𝑡−𝜇²,𝑙_𝜇𝜇(𝜃)𝑇=−𝜎²,𝑙_𝜇𝜎2(𝜃)1=−𝜎⁴Σ𝑅_𝑡,𝑙−𝜇_𝜎2𝜎2(𝜃)=𝑇2𝜎⁴−1𝜎⁶Σ𝑅_𝑡−𝜇². (4.2)

To obtain the maximum likelihood estimators, the first order conditions 𝑙_𝜇(𝜃)𝑙=0,_𝜎2(𝜃)=0 (4.3) are solved, and we have 𝜇=𝜎𝑅,₂=Σ𝑅_𝑡−𝑅²𝑇. (4.4) The estimated Sharpe ratio is then SR=𝜇−𝑅_𝑓𝜎. (4.5) The observed information matrix is given as 𝑗_𝜃𝜃′(𝜃)=−𝑙_𝜃𝜃′(𝜃) which can be expressed as follows: 𝑗_𝜃𝜃′(𝜃)=−𝑙_𝜇𝜇(𝜃)−𝑙_𝜇𝜎2(𝜃)−𝑙_𝜎2𝜇(𝜃)−𝑙_𝜎2𝜎2(𝜃)=𝑇𝜎²1𝜎⁴Σ𝑅_𝑡1−𝜇𝜎⁴Σ𝑅_𝑡−𝑇−𝜇2𝜎⁴+1𝜎⁶Σ𝑅_𝑡−𝜇². (4.6) The observed information matrix is then evaluated at the MLE: 𝑗_𝜃𝜃′𝜃=𝑇𝜎²00𝑇2𝜎² (4.7) with determinant equal to 𝑗_𝜃𝜃′𝜃=𝑇𝜎²𝑇2𝜎⁴=𝑇²2𝜎⁶. (4.8)

To solve for the constrained MLE, the Lagrange multiplier method from (3.2) is used 𝐻(𝜃,𝛼)=𝑙(𝜃)+𝛼𝜇−𝑅_𝑓𝜎−𝜓, (4.9) with first derivatives equal to 𝐻_𝜇(𝜃,𝛼)=𝑙_𝜇(𝜃)+𝛼𝜎,𝐻_𝜎2(𝜃,𝛼)=𝑙_𝜎2(𝜃)−𝛼𝜇−𝑅_𝑓2𝜎³,𝐻_𝛼(𝜃,𝛼)=𝜇−𝑅_𝑓𝜎−𝜓. (4.10) Setting these first derivatives equal to zero, we have: 𝛼=−𝜎𝑙_𝜇𝜃,𝜇=𝑅_𝑓𝑙+𝜓𝜎,_𝜎2𝜃+𝜎𝑙_𝜇𝜃𝜇−𝑅_𝑓2𝜎³=0. (4.11)(4.12)(4.13) Working on (4.13) we have −𝑇2𝜎²+12𝜎⁴Σ𝑅_𝑡−𝜇²+𝜎1𝜎²Σ𝑅_𝑡−𝜇𝜇−𝑅_𝑓2𝜎³=0 (4.14) which further simplifies to −𝑇𝜎²−𝜓Σ𝑅_𝑡−𝑅_𝑓𝜎+Σ𝑅_𝑡−𝑅_𝑓²=0. (4.15) If we let 𝑟_𝑡=𝑅_𝑡−𝑅_𝑓 and 𝑟=Σ𝑟_𝑡/𝑇, then we have 𝜎²+𝜓𝑟𝜎−Σ𝑟²_𝑡𝑇=0 (4.16) with 𝜎=−𝜓𝑟+𝜓𝑟²+4Σ𝑟²_𝑡/𝑇2,since𝜎>0. (4.17)

The constrained MLE are therefore given by: 𝜎=−𝜓𝑟+𝜓𝑟²+4Σ𝑟²_𝑡/𝑇2,𝜇=𝑅_𝑓+𝜓𝜎,𝛼=𝑇𝜓−𝑟𝜎, (4.18) so that 𝜃=(𝜇,𝜎²)^′. Using these constrained maximum likelihood estimators we are able to obtain the titled log-likelihood as defined in (3.3), and the corresponding first and second derivatives are: 𝑙̃(𝜃)=𝑙(𝜃)+𝛼𝜇−𝑅_𝑓𝜎,𝑙̃−𝜓_𝜇(𝜃)=𝑙_𝜇(𝜃)+𝛼𝜎,𝑙̃_𝜎2(𝜃)=𝑙_𝜎2(𝜃)−𝛼𝜇−𝑅_𝑓2𝜎³,𝑙̃_𝜇𝜇(𝜃)=𝑙_𝜇𝜇(𝜃),𝑙̃_𝜇𝜎2(𝜃)=𝑙_𝜇𝜎2(𝜃)−𝛼2𝜎³,𝑙̃_𝜎2𝜎2(𝜃)=𝑙_𝜎2𝜎2(𝜃)+34𝛼𝜇−𝑅_𝑓𝜎⁵. (4.19) We can use these second derivatives to calculate 𝚥̃_𝜃𝜃′(𝜃) as 𝚥̃_𝜃𝜃′𝜃=−𝑙̃_𝜇𝜇𝜃−𝑙̃_𝜇𝜎2𝜃−𝑙̃_𝜎2𝜇𝜃−𝑙̃_𝜎2𝜎2𝜃. (4.20) For the normal model, the canonical parameter is given by: 𝜑(𝜃)=𝜇𝜎²,1𝜎^2′. (4.21) Two related matrices derived from the canonical parameter are given below 𝜑_𝜃(𝜃)=1𝜎²−𝜇𝜎⁴10−𝜎⁴𝜑_𝜃−1,(𝜃)=𝜎²−𝜇𝜎²0−𝜎⁴. (4.22)(4.23) With our parameter of interest 𝜓(𝜃)=𝜇−𝑅_𝑓𝜎 (4.24) we have 𝜓_𝜃(𝜃)=𝜓_𝜇(𝜃),𝜓_𝜎2(𝜃)=1𝜎,−𝜇−𝑅_𝑓2𝜎³. (4.25) Quantities (4.21), (4.23), and (4.25) can be used to calculate (3.18). The quantity var(𝜒(𝜃)−𝜒(𝜃)) can further be calculated. We now have all the ingredients to calculate 𝑅^∗(𝜓) given in (3.16).

5. Examples and Simulations

In this section, we provide examples and simulations for inference on the Sharpe ratio. We compute confidence intervals and 𝑝 values using our proposed third-order method given in (3.16). We label this method “proposed” in the tables below. We compare our results to those obtained by using the existing methods, in particular, using both the Jobson and Korkie [5] asymptotic distribution in (2.12) and the Mertens [8] distribution given in (2.14). These methods are labeled as “Jobson and Korkie” and “Mertens,” respectively. Results from the signed square root log-likelihood ratio statistic in (3.5) are additionally provided and labeled “likelihood ratio.”

5.1. Examples

The dataset for our examples consists of monthly return prices for three time series. The first series represents return prices for a large-cap mutual fund (Fund), the second for a market index (Market), and the third for 90-day Treasury bills (Cash). This data spans a period of one year. The data is listed in Table 1 and originates from the Matlab Financial Toolbox User’s Guide (data in the User's Guide is provided for a 5-year period. We focus on the most recent year of data. See Section 4: Investment Performance Metrics of the manual for further information).

Table 1 

Monthly return prices.

Table 2 reports 95% confidence intervals for the Sharpe ratio separately for the large-cap mutual fund and the market index for the four methods discussed previously. Monthly returns are calculated from the return prices listed in Table 1 and are used to calculate the Sharpe ratio. The mean of the 90-day Treasury returns is used as the risk-free rate. Table 2 shows that the confidence intervals obtained from the four methods produce rather different results. Theoretically, the proposed method has third-order accuracy whereas the remaining three methods do not. Although the confidence intervals produced by the Jobson and Korkie [5] approximation have the best concordance with those produced by the proposed method, the difference is still noticeable. This result will be borne out in the simulations as well.

Table 2 

95% Confidence Intervals for the sharpe Ratio.

The 𝑝 value functions calculated from the methods discussed in this paper for the Sharpe ratio for the large-cap mutual fund and the Sharpe ratio for the market index are plotted in Figures 1 and 2, respectively. These significance functions can be used to obtain 𝑝 values for specific hypothesized values of the Sharpe ratio. As we are typically interested in tail probabilities which tend to be small, it is important to estimate such probabilities with precision. A few values with their corresponding 𝑝 values are provided in Tables 3 and 4 for the market fund and market index, respectively. From these tables we can see that the 𝑝 values vary across the methods. If for instance, interest is on testing whether SR=1 for either fund, then the corresponding 𝑝 values for such a test are given in the last column of Tables 3 and 4. Focussing on the market index and using a 5% level of significance, the Sharpe ratio may or may not be statistically significant depending on the method chosen for the hypothesis test.

Table 3 

Fund: 𝑝 values for SR.

Table 4 

Market: 𝑝 values for SR.

Figure 1 

𝑝value function for Fund.

Figure 2 

𝑝 value function for Market.

5.2. Simulations

Two simulation studies of size 10,000 were performed to compare the three existing methods to the proposed third-order method. The first simulation was constructed to mimic the mutual fund return data from the above example and the second to mimic the market index returns: 𝑋∼𝑁(0.007378393,0.01137605), 𝑌∼𝑁(0.01211267,0.01789324) and 𝑅_𝑓=0.003181247. 90%, 95%, and 99% confidence intervals for the Sharpe ratio were obtained for each sample and the lower probability error, the upper probability error, and the central coverage were recorded. Lower (upper) probability error refers to the proportion of times the true parameter value falls below (above) the confidence interval. Central coverage is defined as the proportion of times the true parameter values falls within the confidence intervals. Tables 5 and 6 report the results from these simulations for the fund and market returns, respectively. For reference, the nominal values and the corresponding standard errors have been included for each the lower probability error, the upper probability error, and the central coverage.

Table 5 

Results for simulation study for fund for 𝑛=12.

Table 6 

Results for simulation study for market for 𝑛=12.

From these results tables it is clear that the third-order method outperforms the other three methods based on the criteria we examined. It must be noted however, that the Jobson and Korkie [5] method provides surprisingly good results. We note that other values for the parameters of the simulation were chosen (but not reported) and the results were consistent with the reported results. Standard errors for the simulations can easily be computed. This information has been included in the tables and labeled “standard error.” From these standard errors, it can be seen that the proposed method produces results that are uniformly within three standard deviations of the nominal value. The other three methods produce less satisfactory results.

6. The Two-Sample Case

Suppose one is interested in testing hypotheses concerning the Sharpe ratios of two funds 𝑋 and 𝑌. For instance, one may be interested in testing the null hypothesis SR_𝑋≥SR_𝑌 or SR_𝑋=SR_𝑌 against the alternative hypothesis SR_𝑋<SR_𝑌 or SR_𝑋≠SR_𝑌, respectively. In this section we apply the third-order methodology described in Section 3 to test the difference between the Sharpe ratios of two funds. Consider two funds with returns given by (𝑥₁,…,𝑥_𝑛) and (𝑦₁,…,𝑦_𝑚). Further assume for expositional clarity that these returns are identically and independently distributed as 𝑁(𝜇_𝑋,𝜎²_𝑋) and 𝑁(𝜇_𝑌,𝜎²_𝑌), respectively. The mean return for the risk-free asset is given by 𝑅_𝑓. Our interest is in testing the difference in Sharpe ratios of the two funds. The interest parameter 𝜓 is then defined as follows: 𝜇𝜓=^𝑋−𝑅_𝑓𝜎_𝑋−𝜇_𝑌−𝑅_𝑓𝜎_𝑌. (6.1) The log-likelihood function for this problem is written as follows: 𝑙(𝜃)=𝑙𝜇_𝑋,𝜎²_𝑋,𝜇_𝑌,𝜎²_𝑌𝑛=−2log𝜎²_𝑋−12𝜎²_𝑋Σ𝑥_𝑖−𝜇_𝑋²−𝑚2log𝜎²_𝑌−12𝜎²_𝑌Σ𝑦_𝑖−𝜇_𝑌². (6.2) The corresponding first derivatives are 𝑙_𝜇𝑋=1𝜎²_𝑋Σ𝑥_𝑖−𝜇_𝑋,𝑙_𝜎²_𝑋𝑛=−2𝜎²_𝑋+12𝜎⁴_𝑋Σ𝑥_𝑖−𝜇_𝑋²,𝑙_𝜇𝑌=1𝜎²_𝑌Σ𝑦_𝑗−𝜇_𝑌,𝑙_𝜎²_𝑌𝑚=−2𝜎²_𝑌+12𝜎⁴_𝑌Σ𝑦_𝑗−𝜇_𝑌². (6.3) These first derivatives are set equal to zero and simultaneously solved for the overall MLE: 𝜇_𝑋=𝜎𝑥,_2𝑋=Σ𝑥_𝑖−𝑥²𝑛,𝜇_𝑌=𝜎𝑦,_2𝑌=Σ𝑦_𝑗−𝑦²𝑚. (6.4) The second derivatives are required to obtain the observed information matrix: 𝑗_𝜃𝜃(𝜃)=𝑛𝜎²_𝑋1𝜎⁴_𝑋Σ𝑥_𝑖−𝜇_𝑋100𝜎⁴_𝑋Σ𝑥_𝑖−𝜇_𝑋−𝑛2𝜎⁴_𝑋+1𝜎⁶_𝑋Σ𝑥_𝑖−𝜇_𝑋²𝑚0000𝜎²_𝑌1𝜎⁴_𝑌Σ𝑦_𝑗−𝜇_𝑌100𝜎⁴_𝑌Σ𝑦_𝑗−𝜇_𝑌−𝑚2𝜎⁴_𝑌+1𝜎⁶_𝑌Σ𝑦_𝑗−𝜇_𝑌². (6.5) Evaluating this observed information matrix at the MLE produces 𝑗_𝜃𝜃(𝜃) with corresponding determinant denoted by |𝑗_𝜃𝜃(𝜃)|.

To solve for the constrained MLE, the Lagrange multiplier method is used 𝐻(𝜃)(𝜃)𝜇=𝑙+𝛼^𝑋−𝑅_𝑓𝜎²_𝑋−𝜇_𝑌−𝑅_𝑓𝜎²_𝑌−𝜓, (6.6) with the following first derivatives: 𝐻_𝜇𝑋=1𝜎²_𝑋Σ𝑥_𝑖−𝜇_𝑋+𝛼𝜎²_𝑋,𝐻_𝜎²_𝑋𝑛=−2𝜎²_𝑋+12𝜎⁴_𝑋Σ𝑥_𝑖−𝜇_𝑋²−𝛼𝜇_𝑋−𝑅_𝑓2𝜎²_𝑋^3/2,𝐻_𝜇𝑌=1𝜎²_𝑌Σ𝑦_𝑗−𝜇_𝑌−𝛼𝜎²_𝑌,𝐻_𝜎²_𝑋𝑚=−2𝜎²_𝑌+12𝜎⁴_𝑌Σ𝑦_𝑗−𝜇_𝑌²+𝛼𝜇_𝑌−𝑅_𝑓2𝜎²_𝑌^3/2,𝐻_𝛼=𝜇_𝑋−𝑅_𝑓𝜎²_𝑋−𝜇_𝑌−𝑅_𝑓𝜎²_𝑌−𝜓. (6.7)

Setting these derivatives equal to zero and solving the resulting system produces the constrained MLE. The constrained MLE can be obtained by solving the following iteratively: 𝜎²_𝑋=Σ𝑥_𝑖−𝜇_𝑋²+𝜇_𝑋−𝑅_𝑓Σ𝑥_𝑖−𝜇_𝑋𝑛,𝜎²_𝑌=Σ𝑦_𝑗−𝜇_𝑌²+𝜇_𝑌−𝑅_𝑓Σ𝑦_𝑗−𝜇_𝑌𝑚,𝜇_𝑋=𝑚𝜎_𝑌𝑅_𝑓+𝑛𝑥𝜎_𝑌+𝑚𝑦𝜎_𝑋−𝑚𝜎_𝑋𝑅_𝑓+𝑚𝜓𝜎_𝑋𝜎_𝑌(𝑛+𝑚)𝜎_𝑋,𝜇_𝑌=𝑛𝑥𝜎_𝑌−𝑛𝜎_𝑌𝑅_𝑓+𝑛𝜎_𝑋𝑅_𝑓−𝑛𝜓𝜎_𝑋𝜎_𝑌+𝑚𝑦𝜎_𝑋(𝑛+𝑚)𝜎_𝑋, (6.8) and Lagrangian Σ𝑦𝛼=^𝑗−𝜇_𝑌𝜎_𝑌. (6.9) So we have 𝜃=(𝜇_𝑋,𝜎²_𝑋,𝜇_𝑌,𝜎²_𝑌)^′. We are now able to define the tilted log-likelihood: 𝑙̃(𝜃)(𝜃)+𝛼𝜇=𝑙^𝑋−𝑅_𝑓𝜎_𝑋−𝜇_𝑌−𝑅_𝑓𝜎_𝑌−𝜓, (6.10) with first derivatives: 𝑙̃_𝜇𝑋=𝑙_𝜇𝑋+𝛼𝜎_𝑋,𝑙̃_𝜎2𝑋=𝑙_𝜎²_𝑋−𝛼𝜇_𝑋−𝑅_𝑓2𝜎³_𝑋,𝑙̃_𝜇𝑌=𝑙_𝜇𝑌−𝛼𝜎_𝑌,𝑙̃_𝜎2𝑌=𝑙_𝜎²_𝑌+𝛼𝜇_𝑌−𝑅_𝑓2𝜎³_𝑌,𝑙̃_{𝜇𝑋𝜇𝑋}=𝑙_{𝜇𝑋𝜇𝑋},𝑙̃_{𝜇𝑋𝜎2𝑋}=𝑙_𝜇𝑋𝜎²_𝑋−𝛼2𝜎³_𝑌,𝑙̃_{𝜇𝑋𝜇𝑌}=0. (6.11) The second derivatives are needed to calculate 𝚥̃_𝜃𝜃(𝜃): 𝚥̃_𝜃𝜃(𝜃)=𝑗_𝜃𝜃(𝜃)+0𝛼2𝜎³_𝑋𝛼002𝜎³_𝑋−3𝛼𝜇_𝑋−𝑅_𝑓4𝜎⁵_𝑋𝛼00000−2𝜎³_𝑌𝛼00−2𝜎³_𝑌3𝛼𝜇_𝑌−𝑅_𝑓4𝜎⁵_𝑌. (6.12)

The canonical parameter for this model is given by the following: 𝜑_𝜃(𝜃)=𝜇_𝑋𝜎²_𝑋,1𝜎²_𝑋,𝜇_𝑌𝜎²_𝑌,1𝜎²_𝑌^′. (6.13) The matrices associated with this vector can now be calculated: 𝜑_𝜃(𝜃)=1𝜎²_𝑋−𝜇_𝑋𝜎⁴_𝑋1000−𝜎⁴_𝑋10000𝜎²_𝑌−𝜇_𝑌𝜎⁴_𝑌1000−𝜎⁴_𝑌,𝜑_𝜃⁻¹(𝜃)=𝜎²_𝑋−𝜇_𝑋𝜎²_𝑋000−𝜎⁴_𝑋0000𝜎²_𝑌−𝜇_𝑌𝜎²_𝑌000−𝜎⁴_𝑌. (6.14) Given our parameter of interest 𝜓(𝜃)=𝜇_𝑋−𝑅_𝑓𝜎²_𝑋−𝜇_𝑌−𝑅_𝑓𝜎²_𝑌, (6.15) we have 𝜓_𝜃(𝜃)=1𝜎²_𝑋,−𝜇_𝑋−𝑅_𝑓2𝜎²_𝑋^3/2,−1𝜎²_𝑌,𝜇_𝑌−𝑅_𝑓2𝜎²_𝑌^3/2. (6.16) Quantities 𝜑_𝜃(𝜃), 𝜑_𝜃⁻¹(𝜃) and 𝜓_𝜃(𝜃) can be used to calculate the new parameter 𝜒(𝜃) given in (3.18). The additional ingredients necessary for the construction of the standardized maximum likelihood departure 𝑄 given in (3.19) are readily available from the above information.

7. Example and Simulations

In this section we provide an example and a simulation study for the two-sample Sharpe ratio case. We compare the third-order likelihood-based inference method to the classical methods used for testing, namely, the maximum likelihood statistic and the likelihood ratio statistic. These latter two statistics are analogous to expressions (3.4) and (3.5).

7.1. Example

For our example, we use the data presented in Table 1 to compare Sharpe ratios. We may for instance, be interested in whether the mutual fund’s risk-adjusted return as captured by the Sharpe ratio is significantly better than the market’s return. In Table 7 we present the 95% confidence interval for the difference between the Sharpe ratios for the mutual fund and market index. We can see that while the intervals produced using the MLE and likelihood ratio methods are similar, the interval obtained from the third-order method differs from these methods. As this is an example, we cannot comment on which interval is more accurate, but it is relevant to note the differences between the intervals which may be important in real world settings. The 𝑝 values for testing a null hypothesis of a zero difference between the Sharpe ratios of the mutual fund and market index are given by: 0.3675, 0.3675, and 0.3777 for the MLE, likelihood ratio and proposed, respectively. As tail probabilities tend to be small probabilities, it is important to approximate these as accurately as we can.

Table 7 

95% Confidence intervals for Sharpe ratio difference.

7.2. Simulations

In this section we provide a simulation study to assess the performance of the third-order method relative the MLE and likelihood ratio. The size of each simulation is 10,000 and again the parameter values were chosen to mimic the example data: 𝑋∼𝑁(0.007378393,0.01137605), 𝑌∼𝑁(0.01211267,0.01789324), and 𝑅_𝑓=0.003181247. Confidence intervals for the difference between the Sharpe ratios for funds 𝑋 and 𝑌 were obtained for various sample sizes of each fund. Table 8 records the results from this simulation. Lower and upper probability errors are recorded as is central coverage. The nominal values and standard errors of the simulation are also reported for reference. As in the one sample case, these simulation results generally indicate that the propsed method outperforms the other methods based on the criteria we examined.

Table 8 

Results for Simulation Study.

8. Conclusion

A higher order likelihood method was applied for inference on the Sharpe ratio. The two-sample problem for comparing Sharpe ratios was also considered. This statistical method is known to be extremely accurate and has 𝑂(𝑛^−3/2) distributional accuracy. The methodology was demonstrated and worked out explicitly for independently and identically normally distributed returns. Simulations were provided to show the exceptional accuracy of the method even for very small sample sizes. While our assumption of independently and identically normally distributed returns may seem restrictive, we stress that the methodology can be applied more generally for any parametric return distribution structure and our choice of normality was made to clearly illustrate the methodology’s merits to the applied practitioner. A natural next step would be to consider inference when returns are not independently and identically distributed. For the two-sample problem, it would be fruitful to compare the approach taken by Ledoit and Wolf [21] which involves the studentized bootstrap with the proposed methodology.

Acknowledgment

It is noted that part of this paper originates from a chapter of Liu's doctoral dissertation.