A simple example shows that losing all money is compatible with a very high
Sharpe ratio (as computed after losing all money). However, the only way that
the Sharpe ratio can be high while losing money is that there is a period in
which all or almost all money is lost. This note explores the best achievable
Sharpe and Sortino ratios for investors who lose money but whose one-period
returns are bounded below (or both below and above) by a known constant.Comment: 6 page

Vovk, Vladimir

English

arXiv

A simple example shows that losing all money is compatible with a very high Sharpe ratio (as computed after losing all money). However, the only way that the Sharpe ratio can be high while losing money is that there is a period inwhich all or almost all money is lost. This note explores the best achievable Sharpe and Sortino ratios for investors who lose money but whose one-period returns are bounded below (or both below and above) by a known constant

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Losing money with a high Sharpe ratio

Royal Holloway - Pure

arXiv:1109.0706v1  [q-fin.RM]  4 Sep 2011Losing money with a high Sharpe ratioVladimir VovkSeptember 6, 2011AbstractA simple example shows that losing all money is compatible with avery high Sharpe ratio (as computed after losing all money). However,the only way that the Sharpe ratio can be high while losing money isthat there is a period in which all or almost all money is lost. This noteexplores the best achievable Sharpe and Sortino ratios for investors wholose money but whose one-period returns are bounded below (or bothbelow and above) by a known constant.1 IntroductionSharpe ratio [2, 3] has become a “gold standard” for measuring performance ofhedge funds and other institutional investors (this note uses the generic term“portfolio”). It is sometimes argued that it is applicable only to i.i.d. Gaussianreturns, but we will follow a common practice of ignoring such assumptions. Forsimplicity we assume that the benchmark return (such as the risk-free rate) iszero.The (ex post) Sharpe ratio of a sequence of returns x1, . . . , xN ∈ [−1,∞) isdefined as ShN = Sh(x1, . . . , xN ) := µN/σN , whereµN :=1NN∑n=1xn, σN :=√√√√ 1NN∑n=1(xn − µ)2.(None of our results will be affected if we replace, assuming N ≥ 2, 1Nby 1N−1,as in [3], (6).) Intuitively, the Sharpe ratio is the return per unit of risk.Another way of measuring the performance of a portfolio whose sequence ofreturns is x1, . . . , xN is to see how this sequence of returns would have affected aninitial investment of 1 assuming no capital inflows and outflows after the initialinvestment. The final capital resulting from this sequence of returns is∏Nn=1(1+xn). We are interested in conditions under which the following anomaly ispossible: the Sharpe ratio Sh(x1, . . . , xN ) is large while∏Nn=1(1 + xn) < 1.(More generally, if we did not assume zero benchmark returns, we would replace∏Nn=1(1 + xn) < 1 by the condition that in the absence of capital inflows andoutflows the returns x1, . . . , xN underperform the benchmark portfolio.)1Suppose the return is 5% over k−1 periods, and then it is −100% in the kthperiod. As k →∞, µk → 0.05 and σk → 0. Therefore, making k large enough,we can make the Sharpe ratio Shk as large as we want, despite losing all themoney over the k periods.If we want the sequence of returns to be i.i.d., let the return in each periodn = 1, 2, . . . be 5% with probability (k− 1)/k and −100% with probability 1/k,for a large enough k. With probability one the Sharpe ratio ShN will tend to alarge number as N → ∞, despite all money being regularly lost. Of course, inthis example the returns are far from being Gaussian (strictly speaking, returnscannot be Gaussian unless they are constant, since they are bounded from belowby −1).It is easy to see that our examples lead to the same conclusions when theSharpe ration is replaced by the Sortino ratio [5, 4] SoN = So(x1, . . . , xN ) :=µN/σ′N, whereσ′N:=√√√√ 1NN∑n=1((xn − µ)−)2 ≤ σN .2 Upper bound on the Sharpe ratio for losersThe examples of the previous section are somewhat unrealistic in that there is aperiod in which the portfolio loses almost all its money. In this section we showthat only in this way a high Sharpe ratio can become compatible with losingmoney.For each B ∈ (0, 1], defineF1(B) := sup{Sh(x1, . . . , xN ) |N∏n=1(1 + xn) < 1}, (1)where N ranges over the positive integers and (x1, . . . , xN ) over [−B,∞)N . Inother words, F1(B) is the best achievable Sharpe ratio for sequences of returnsthat lose money, assuming that none of the returns falls below −B.It is not difficult to show that F1(0+) = 0, and in the previous section wesaw that F1(1) =∞. In this section we are interested in the behaviour of F1(B)for the intermediate values of B, B ∈ (0, 1).Figure 1 shows the graph of F1 over B ∈ (0, 0.9] and over B ∈ [0.9, 0.999].Over the interval B ∈ (0, 0.9] the slope of F1 is roughly 1. We can see that evenfor a relatively large value of B = 0.5, the Sharpe ratio of a losing portfolionever exceeds 0.5; according to Table 1, F1(0.5) = 0.424 (much less than theconventional threshold of 1 for a good Sharpe ratio [1]).Table 1 gives approximate numerical values of F1(B) for selected B. Theseapproximate values are attained for sequences of returns involving only twolevels of returns: −B and another level c. The table also lists the value of cand the fraction α of returns equal to c at which the given approximate valueof F1(B) is attained.2Figure 1: The function F1(B) in the ranges B ∈ (0, 0.9] (left) and B ∈[0.9, 0.999] (right).B F1(B) c α B F1(B) c α0.1 0.160 10.49 0.041 0.8 0.673 6.37 0.4460.2 0.235 10.00 0.085 0.9 0.826 5.42 0.5530.3 0.299 9.50 0.132 0.99 1.239 3.90 0.7430.4 0.361 8.97 0.182 0.999 1.564 3.35 0.8240.5 0.424 8.40 0.236 0.9999 1.836 3.10 0.8670.6 0.493 7.80 0.296 0.99999 2.075 2.96 0.8930.7 0.572 7.13 0.365 0.999999 2.289 2.88 0.911Table 1: The approximate values of F1(B), c, and α for selected B.B F2(B) α B F2(B) α0.1 0.050 0.525 0.4 0.210 0.6030.2 0.101 0.550 0.5 0.271 0.6310.3 0.154 0.576 0.6 0.340 0.661Table 2: The approximate values of F2(B) and α for selected B.3Figure 2: The function F2(B) in the ranges B ∈ (0, 0.9] (left) and B ∈[0.9, 0.999] (right).A striking feature of Table 1 is the values of c: they exceed 1 even forB = 0.999999. The value of c = 10.00 corresponding to B = 0.2 in Table 1means that the portfolio increases its value 11-fold in one period. Figure 2 isanalogous to Figure 1 but imposes the upper bound of B on the absolute valuesof one-period returns. Namely, it plots the graph of the function F2 which isdefined by the same formula (1) as F1 but with (x1, . . . , xN ) now ranging over[−B,B]N . An abridged analogue of Table 1 for F2 is given as Table 2; the latterdoes not give the value of c as it is equal to B in all the entries. Not surprisinglyα are not so different from 1/2 in Table 2, especially for smaller B.The analogues of Figures 1 and 2 for the Sortino ratio are Figures 3 (one-sided) and 4 (two-sided). The function G1 of Figure 3 is defined, in analogywith (1), byG1(B) := sup{So(x1, . . . , xN ) |N∏n=1(1 + xn) < 1}, (2)where N ranges over the positive integers and (x1, . . . , xN ) over [−B,∞)N .The function G2 of Figure 4 is defined as the right-hand side of (2) but with(x1, . . . , xN ) ranging over [−B,B]N .The values of G1(B) and G2(B) for selected B are shown in Table 3, G1on the left and G2 on the right. The meaning of α is the same as in Tables 1and 2. We do not give the values of c; they are huge on the left-hand side of thetable and equal to B on the right-hand side. The left-hand side suggests thatG1(0+) = 1, and this can be verified analytically.4Figure 3: The function G1(B) in the ranges B ∈ (0, 0.9] (left) and B ∈[0.9, 0.999] (right).Figure 4: The function G2(B) in the ranges B ∈ (0, 0.9] (left) and B ∈[0.9, 0.999] (right).5B G1(B) α B G2(B) α0.1 1.005 0.011 0.1 0.069 0.5250.2 1.011 0.026 0.2 0.136 0.5500.3 1.020 0.045 0.3 0.203 0.5760.4 1.030 0.069 0.4 0.271 0.6030.5 1.044 0.099 0.5 0.341 0.6310.6 1.062 0.137 0.6 0.418 0.661Table 3: The approximate values of G1(B) and α (left) and G2(B) and α (right)for selected B.3 DiscussionFigures 1–4 can be regarded as a sanity check for the Sharpe and Sortino ratio.Not surprisingly, they survive it, despite the theoretical possibility of havinga high Sharpe and, a fortiori, Sortino ratio while losing money. In the caseof the Sharpe ratio, such an abnormal behaviour can happen only when someone-period returns are very close to −1. In the case of the Sortino ratio, suchan abnormal behaviour can happen only when some one-period returns are veryclose to −1 or when some one-period returns are huge.AcknowledgementsI am grateful to Boris Afanasiev for his advice and to Wouter Koolen for usefulcomments. All computations for this note were done in Maple™.References[1] Investopedia. Understanding the Sharpe ratio, July 2010.[2] William F. Sharpe. Mutual fund performance. Journal of Business, 39:119–138, 1966.[3] William F. Sharpe. The Sharpe ratio. Journal of Portfolio Management,49:49–58, 1994.[4] Frank A. Sortino and Lee N. Price. Performance measurement in a downsiderisk framework. Journal of Investing, 3:59–64, 1994.[5] Frank A. Sortino and Robert van der Meer. Downside risk. Journal ofPortfolio Management, 17:27–31, 1991.6

Vladimir Vovk

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Losing money with a high Sharpe ratio

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