Inference for the Difference of Two Independent KS Sharpe Ratios under Lognormal Returns

<jats:p>A higher-order likelihood-based asymptotic method to obtain inference for the difference between two KS Sharpe ratios when gross returns of an investment are assumed to be lognormally distributed is proposed. Theoretically, our proposed method has <jats:inline-formula>
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                  </jats:inline-formula> distributional accuracy, whereas conventional methods for inference have <jats:inline-formula>
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                  </jats:inline-formula> distributional accuracy. Using an example, we show how discordant confidence interval results can be depending on the methodology used. We are able to demonstrate the accuracy of our proposed method through simulation studies.</jats:p>


Introduction
Let P t be the price of an investment at time t, and assume this investment does not pay out dividends. e net return R t , of this investment between time t − 1 and time t is given as where g t is the gross return or relative price of that investment. Moreover, let r t � log g t (2) denote the log return at time t which represents the continuously compounded return. One of the most common measures used to assess an investment's performance is the Sharpe ratio. is ratio, proposed by Sharpe [1], takes the form where r f is a risk-free return. SR then measures the excess expected return, or risk premium, relative to its volatility. When data are given in terms of relative prices rather than returns, Knight and Satchell [2] propose the following extension to the Sharpe ratio: is extension is known as the KS Sharpe ratio. In statistical literature, it is common to assume that log returns (i.e., the r t ) are identically and independently distributed as normal with mean μ and variance σ 2 . Equivalently, the g t are identically and independently distributed as lognormal with mean and variance given by respectively. Under this assumption, the Sharpe ratio and the KS Sharpe ratio can be written as follows: Liu et al. [3] applied the standard likelihood method to obtain inference for the Sharpe ratio with independent data, and Ji et al. [4] extended the methodology to obtain inference for this ratio with autocorrelated return data. Fu et al. [5] relaxed the distributional assumption and applied the adjusted empirical likelihood ratio method to obtain inference for the Sharpe ratio. Despite work on the Sharpe ratio, there does not appear to be any statistical studies on the KS Sharpe ratio except for the original paper by Knight and Satchell [2], in which they obtained the uniformly minimum variance unbiased estimator for the ratio.
As the KS Sharpe ratio depends only on σ, one sample exact inference is straightforward. However, inference for either the difference or the ratio of two independent KS Sharpe ratios is complicated and no exact analysis is available. In this paper, we propose to use a higher-order likelihood-based asymptotic method to obtain inference for the difference between two independent KS Sharpe ratios. e proposed method provides an important indicator to practitioners who are interested in comparing the performance of two assets. We compare our method to two standard asymptotic methods through an example and simulations. While the example shows the differences between the three methods, the simulation studies demonstrate the extreme accuracy of our proposed method. Accuracy is important as sample sizes can be limited in practice. e proposed method is straightforward and easy to use in practice. e methodology can also be applied to obtain inference for the ratio of two independent KS Sharpe ratios.

Likelihood-Based Inference
In this section, we review first-order likelihood-based inferential procedures. We then propose a modification to the first-order procedures such that the resulting inference procedure has thirdorder accuracy. Let (y 1 , . . . , y n ) be a sample from a population with probability density function f(y i ; θ), where θ is a k-dimensional parameter. Let ψ � ψ(θ) be the scalar parameter of interest and λ � λ(θ) be the (k − 1)-dimensional nuisance parameter. As defined by Kalbfleisch [6], the log-likelihood function for θ based on the observed sample (y 1 , . . . , y n ) is where a is an arbitrary additive constant. Reid [7] notes that the value of the likelihood function is only meaningful in relative terms. e likelihood function contains almost everything that a model has to say about the observed data. Among the possible values of θ, the overall maximum likelihood estimator (mle) of θ, maximizes the likelihood function regardless of the value of the additive constant a. Hence, without loss of generality, we set a to be zero. Moreover, let be the observed information evaluated at mle.

First-Order Methods.
With regularity conditions stated in Cox and Hinkley [8], the asymptotic mean and variance of the mle are given as θ and j −1 (θ). Hence, for a large sample size n, by the multivariate central limit theorem, the limiting distribtion of (θ − θ) is the k th dimensional multivariate normal distribution with mean 0 and variance j −1 (θ). us, is asymptotically distributed as χ 2 k , with first-order accuracy or equivalently, with a rate of convergence O(n −1/2 ). By applying the delta method, we have where and ψ θ (θ) is the derivative of ψ(θ) with respect to θ. e statistic in (10) is asymptotically distributed as a standard normal distribution with first-order accuracy. A (1 − α)100% confidence interval for ψ is thus given by where z α/2 is the (1 − α/2)100 th quantile of the standard normal distribution. e statistic q(ψ) is referred to as the Wald statistic or the standardized mle statistic. Jobson and Korkie [9] applied this method to obtain approximate inference for the Sharpe ratio. Another commonly applied asymptotic method is based on the likelihood ratio method. Since ψ is a scalar parameter of interest, we have where denotes the constrained mle (i.e., the mle of θ for a given ψ). e statistic R(ψ) is likewise asymptotically distributed as standard normal with first-order accuracy. A (1 − α) 100% confidence interval for ψ is e statistic R(ψ) 2 is the Wilks statistic or the log-likelihood ratio statistic and R(ψ) is referred to as the signed log-likelihood ratio statistic. Reid [7] provided a detailed review of the asymptotic distribution of q(ψ) and R(ψ). In practice, the Wald statistic is used more frequently than the signed log-likelihood ratio statistic due to its simplicity in calculation. e determination of the constrained mle required for the signed loglikelihood ratio statistic can be a much more difficult task. However, theoretically, the advantage of the signed log-likelihood ratio statistic lies in its invariance to reparameterization. e Wald statistic does not possess this property and results will vary depending on the parameterization used.

ird-Order Method.
Many methods have emerged in recent years that improve upon the accuracy of the signed loglikelihood ratio statistic, see Skovgaard [10] for a detailed overview. In particular, there are three major improvement methods which are proposed by the following papers: Barlett [11]; Barndorff-Nielsen [12,13]; and Fraser and Reid [14]; and they are reviewed in Skovgaard [10]. Bartlett [11] proposed the Bartlett correction method, a method that has fourthorder accuracy and is applicable to any vector parameter of interest. Except in special cases, the Bartlett correction factor is extremely difficult to calculate. Barndorff-Nielsen [12,13] proposed the modified signed log-likelihood ratio method.
is method has third-order accuracy and is applicable only to a scalar parameter of interest. It further requires the existence of an ancillary statistic which may not exist, and even if it does exist, it may not be unique. Fraser and Reid [14] generalized the modified signed log-likelihood ratio method such that it is applicable to any model and does not require an ancillary statistic. e method achieves third-order accuracy. In this paper, we apply Fraser and Reid's method to obtain inference for the difference of two independent KS Sharpe ratios under lognormal returns.
Barndorff-Nielsen [12,13] defined the modified signed log-likelihood ratio statistic for a scalar parameter ψ to be where R(ψ) is the signed log-likelihood ratio statistic defined in (13), and Q(ψ) is a special statistic that depends on the existence of an ancillary statistic. Fraser and Reid [14] showed that for the natural exponential family model with θ as the canonical parameter and ψ a component of θ, Q(ψ) � q(ψ), which is the standardized mle statistic defined in (10). Fraser and Reid [14] further extended their methodology to the exponential family model where ψ is not a component of the canonical parameter. eir approach takes the following steps: (1) Obtain the canonical parameter φ(θ) (2) e statistic R(ψ) as defined in (13) remains unchanged as the signed log-likelihood ratio statistic is invariant to reparametrization (3) Obtain the standardized mle statistic Q(ψ) in the canonical parameter scale as where is the derivative of φ(θ) with respect to λ evaluated at the constrained mle, and φ θ (θ) is the derivative of φ(θ) with respect to θ evaluated at the overall mle Fraser and Reid [14] showed that R * (ψ) is asymptotically distributed as standard normal with third-order distributional accuracy. Brazzale et al. [15] have a collection of examples where they apply this third-order method and demonstrate the extreme accuracy of the method even when the sample size is small. In what follows, we apply the above methods to obtain confidence intervals for the difference of two independent KS Sharpe ratios.

Inference for the Difference of Two Independent KS Sharpe Ratios
As discussed in Section 1, with (r 1 , . . . , r n ) IID normally distributed with mean μ and variance σ 2 , or alternatively,(g 1 , . . . , g n ) � (e r 1 , . . . , e r n ) IID lognormally distributed, the KS Sharpe ratio is a function of σ 2 alone. It is well-known that n i�1 (r i − r) 2 /σ 2 is distributed as χ 2 n−1 . Exact (1 − α)100% confidence intervals for σ 2 can therefore be obtained. Since the KS Sharpe ratio is a one-to-one function of σ 2 , exact (1 − α)100% confidence intervals for KS can also be obtained. However, inference for the difference between two independent KS Sharpe ratios is not as straightforward. Exact results are not available. In this section, we apply the likelihood-based methods discussed to this particular inferential problem.
Consider two independent investments with log returns (r X 1 , . . . , r X n ) from a normal distribution with mean μ X and variance σ 2 X . e (r Y 1 , . . . , r Y m ) are drawn from a normal distribution with mean μ Y and variance σ 2 Y . e parameter of interest is the difference between the two KS Sharpe ratios: e overall mle is Journal of Probability and Statistics e inverse of the observed information evaluated at the mle is used as an approximation to the variance of θ. By applying the delta method, we are able to estimate the variance of ψ as e statistic ψ is asymptotically distributed as normal with mean ψ and approximate variance var(ψ). q(ψ) can be obtained from (10). is can be viewed as an extension of the method discussed by Jobson and Korkie [9]. Moreover, it can be shown that the constrained mle is where (24) e variances given by σ 2 X and σ 2 Y do not have closedform solutions and must be obtained numerically. With this information, R(ψ) can be obtained from (13). Confidence intervals for ψ based on the first-order methods can likewise be obtained. We note that the model is an exponential family model with the canonical parameter given by Hence, Q(ψ) can be obtained from (17) and the modified signed log-likelihood ratio statistic R * (ψ) can be obtained from (16). Confidence intervals for ψ based on third-order methods can thus be obtained.

Example.
To illustrate our proposed method, consider the data reported in Table 1. is table records the monthly relative returns of Tesla, Inc. (TSLA) and Netflix, Inc. (NFLX) downloaded from Yahoo Finance for the period January 2019 to January 2020. Our interest is in comparing the performance of the two stocks using the difference between the two KS Sharpe ratios. In terms of the underlying assumptions, we note that a Shapiro-Wilks test of normality for TSLA and NFLX gives p values of 0.7107 and 0.0664, respectively. Correlation test (p value � 0.1177) between the two series suggests the two populations are independently distributed as normal. e p value from a Durbin Watson test of serial correlation is 0.8834 for TSLA and 0.3769 for NFLX, which suggests no serial correlation. Table 2 records the 95% confidence intervals for the difference between the two KS Sharpe ratios calculated using the three different likelihood methods discussed in this paper. While the intervals all suggest NFLX is the preferred stock, they do not produce quantitatively similar results.

Simulation Results.
e results from the abovementioned example show how the three methodologies can result in quite different confidence intervals. In this section, simulation studies are carried out to assess the performance of these methods. Extensive simulation studies were performed but only a selection of the results is reported in this paper.
e presented results are representative of the findings of all simulations conducted. For each combinations of n, μ, and σ, 10,000 Monte Carlo replications are performed. For each generated sample, the 95% confidence interval for the difference of the KS Sharpe ratios is calculated. e performance of a method is judged using the following three criteria: (1) e central coverage probability (CP): the proportion of the true difference of the KS Sharpe ratios falls within the 95% confidence interval (2) e lower tail error rate (LE): the proportion of the true difference of the KS Sharpe ratios falls below the lower limit of the 95% confidence interval (3) e upper tail error rate (UE): the proportion of the true difference of the KS Sharpe ratios falls above the upper limit of the 95% confidence interval e nominal values for these three criteria are 0.95, 0.025, and 0.025, respectively. e nominal value for the CP criteria is 0.95. Moreover, since the three methods considered in this paper have the same limiting standard normal distribution, the nominal values for the LE and UE criteria are 0.025 and 0.025, respectively. Tables 3-5 record the simulation results. e central coverage of our proposed method is superior to both the Wald and signed log-likelihood ratio methods, even for small sample sizes. Furthermore, our proposed method has extremely accurate and symmetric tail error rates. e tail error probabilities produced by the other two methods are wildly asymmetric but perform better as the sample size increases, indicating that these two methods are converging to the normal distribution. From all our additional simulation studies, we can conclude that our proposed method is indisputably superior and thus recommended for all empirical applications.

Conclusion
We considered lognormally distributed returns and used the KS Sharpe ratio as a measure of an asset's expected return relative to its volatility. We proposed a third-order likelihood-based method for the difference between two such ratios. Numerical studies verified the extreme accuracy obtained by the proposed method. When returns are assumed to be lognormally distributed, we advocate the use of our proposed method. It is both easy to implement and extremely accurate regardless of the size of the samples.

Data Availability
e dataset used in the example is from a public domain (downloaded from Yahoo Finance). e other numerical examples in the submitted paper are based on simulation studies, which are available from the corresponding author upon request.    Journal of Probability and Statistics 5