Abstract
An intersection–union test for supporting the hypothesis that a given investment strategy is optimal among a set of alternatives is presented. It compares the Sharpe ratio of the benchmark with that of each other strategy. The intersection–union test takes serial dependence into account and does not presume that asset returns are multivariate normally distributed. An empirical study based on the G–7 countries demonstrates that it is hard to find significant results due to the lack of data, which confirms a general observation in empirical finance.
22 L 44 (cid:33) . Schmid and Schmidt (2009) obtain the same large-sample variance of {n} under the assumption that the processes are strongly mixing (Bradley 2005), but that assumption seems to be more restrictive than Gordins condition. To the best of my knowledge, Lo (2002) is the rst who analyzes the potential impact of serial dependence when estimating the Sharpe ratio. Mertens (2002) points out that the formula for independent and identically distributed asset returns presented by Lo (2002) is based, implicitly, 5 of 13 L vs. 2, Risks 2018, 6, 40 on the normal-distribution hypothesis. More precisely, he shows that the large-sample variance of {n} is 1 + 2 2 3 + 4 3 4 2 if the components of {Rt} are independent and identically distributed, where 3 := E(cid:0)(Rt )3(cid:1) 3
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Lo (2002) presumes that 3 = 0 and 4 = 3, in which case the large-sample variance of {n} is 1 + 2/2. Some of those results can be found also in Opdyke (2007). However, Ledoit and Wolf (2008) mention that the formula for serially dependent asset returns presented by Opdyke (2007) is wrong because it does not distinguish between large-sample and stationary (co-)variances. One purpose of this work is to clarify the aforementioned misunderstandings. Suppose, without loss of generality, that we want to compare the Sharpe ratio of Strategy 1 with that of Strategy 2. In Appendix A, the reader can verify that n (cid:32)(cid:34) 1n 1 2n 2 (cid:35)(cid:33) (cid:32) N (cid:32) 0, (cid:34) 11 12 21 22 (cid:35)(cid:33) with 11 = 2 L1 2 1 1L1 3 1 + 1 2 2 L1 44 1 , 22 = 2 L2 2 2 2L2 3 2 + 2 2 2 L2 44 2 , and 12 = 21 = 11 12 2112 + 1221 1 2 22 2 + 1222 1 2 42 2 ,
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We conclude that n (cid:0)n (cid:1) (cid:32) N (cid:0)0, 11 + 22 212 with n := 1n 2n and := 1 2. It is worth emphasizing that the benchmark must be chosen before examining the Sharpe ratios. Otherwise, the entire procedure would suffer from a selection bias and then the results derived so far are no longer valid. However, this is not a serious drawback: If our choice of the benchmark is based on historical data, we can simply apply the test out of sample. (cid:1) As already mentioned at the end of Section 1, the given result represents a nonparametric generalization of the JobsonKorkie test (Jobson and Korkie 1981), which is frequently used in nance. The latter is based on the assumption that asset returns are serially independent and multivariate normally distributed. In this special case, it follows that
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This expression for the large-sample variance of {n} corrects a typographical error made by Jobson and Korkie (1981), which is observed by Memmel (2003). 6 of 13 Risks 2018, 6, 40
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