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- 1. Alternative Intelligence QuotientThe Performance of the SFAScore vs Traditional Risk-adjustedPerformance MeasuresPeter Urbani, Infiniti CapitalPeter Urbani is Chief Investment Officer of Infiniti Capital, a Hong-Kong-based hedge fund of funds group. Issue 33, December 2009Ever since the seminal work on Portfolio Theory by Harry Markowitz (1959) and the subsequent work of WilliamSharpe, the measurement of portfolio returns has been inextricably linked to the level of risk associated withachieving those returns.This has led to the introduction of a number of risk-adjusted performance measures (RAPMs), most famouslythe reward-to-variability, or Sharpe Ratio.Typically calculated as the portfolio returns in excess of those of the risk-free rate over the standard deviationof portfolio returns, the Sharpe Ratio embeds the concept of the variance, standard deviation squared, orvolatility as the appropriate measure of ‘risk’ to use.Over time, practitioners and academics alike have realised that this poses a number of problems for theaccurate measurement of ‘risk’. In fact, the use of variance was largely an act of convenience to simplify themath in the days before computers. Markowitz himself has said that for some investors semi-variance mightbe a more appropriate measure to use.The reason for this is simply that variance, or standard deviation as is more commonly used (the square rootof the variance), is not a measure of ‘risk’ at all, but rather a measure of uncertainty. Standard deviationsuffers from a number of well known deficiencies, most particularly the fact that it does not differentiatebetween good (upside) ‘risk’ and bad (downside) ‘risk’. Moreover, it is a symmetric measure that assumesboth upside- and downside-variance are the same.In recognition of these deficiencies, a number of other RAPMs have been developed to better address theseissues. Probably the best known of these is the Sortino Ratio which replaces the standard deviation with thedownside deviation or second lower partial moment (LPM2) as the denominator in the Sharpe Ratio.Still others include the modified Sharpe Ratio where the denominator of risk is represented by the CornishFisher expanded or ‘modified’ Value at Risk (VaR).More recently, Shadwick and Keating pioneered the use of the Omega Functio,n sometimes also used asthe Omega Ratio. In this formula, the area under the probability curve in excess of some threshold return istaken over the area under the curve of the downside part of the distribution. This can be calculated in eithera discrete form using empirical data, or a continuous form by fitting a distribution. 7
- 2. From a practitioner’s perspective, what we care The study referenced below, compares the out- most about is how well these measures predict of-sample performance of a portfolio built using the relative ranking from one period to the next the SFA score as its objective function versus the and whether or not using one particular method performance of portfolios built from the same data produces superior returns to another. Although set using the Sharpe, Sortino and Omega measures. these RAPMs are typically used for calculating the For reference we also include a naive benchmark relative ranking of funds they can also be, and made up of an equally weighted continuously often are, used as the objective function in direct rebalanced portfolio of all of the 36 underlying portfolio optimisations. For instance, maximising hedge funds in the selection universe. The portfolios the value of your portfolio’s Sharpe Ratio is the are re-optimised to the objective function and same as minimising its variance and gives the same rebalanced on a quarterly basis. set of weights as the classical Markowitz mean variance optimisation formulation. Minimising your The results of the study suggest that the SFA score is ‘normal’ VaR will give the same solution. However, as capable of generating annualised rates of returns mentioned previously, measures based on standard (CAGR) of around 15% versus those of around deviation such as Sharpe and the normal VaR 11.5% for the Sharpe Ratio, 13% for the Omega calculation do not consider the asymmetry of returns. Ratio, and just 10% for the Sortino Ratio. In this article, we examine the performance of a More importantly, although the Sharpe Ratio of new risk-adjusted performance measure called the resultant time series remains better for both the Single Fund Analysis (SFA) score, developed by the Sharpe and Omega portfolios, the ratio of the Infiniti Capital. This measure is a weighted average annualised return to the absolute drawdown over of a number of underlying statistics that has also the period, which is arguably a better measure of been standardised to a reference data set making realised risk to return, remains highest for the SFA it both a relative and conditional measure. The SFA score portfolio. score can further be decomposed into risk, return and persistence sub-scores. Equally- Key statistics SFA Total Omega Sharpe Sortino weighted fund CAGR 15.09% 12.93% 11.44% 10.07% 10.36% Annual return 14.34% 12.35% 10.97% 9.80% 10.08% Annual SD 6.50% 5.07% 4.35% 5.72% 6.12% Skew 0.78 -0.13 0.4 -0.21 -0.67 Kurtosis 0.00 -0.39 0.67 0.29 0.52 Normal VaR 95% -1.89% -1.38% -1.15% -1.90% -2.07% Infiniti VaR 95% -1.25% -1.43% -1.15% -1.99% -2.30% Sharpe Ratio 1.77 1.86 1.85 1.20 1.17 CAGR/ABS (Drawdown) 3.41 2.39 3.27 1.05 0.93 Max drawdown -4.43% -5.41% -3.50% -9.56% -11.10% Johnson Mixture of Modified Log normal Best-fit distribution Gumbel (Max) (Lognormal) normals normal (max) Portfolio’s relationship to benchmark (equally-weighted fund ) Correlation 0.79 0.87 0.78 0.91 1.00 Beta 0.84 0.72 0.56 0.85 Alpha (monthly) 0.49% 0.42% 0.45% 0.10% Information Ratio 1.03 0.76 0.23 -0.118
- 3. Alternative Intelligence QuotientThe reason for the superior performance of the SFA We were somewhat surprised by the poorscores is due to it not being a simple point estimate, performance of the Sortino Ratio portfolio relativebut being calibrated relative to a reference to that of the Sharpe Ratio portfolio. We believedata set of other hedge funds. This improves the this may have been due to the fact that the qualitypredictive power of the method because it responds of the 36 underlying hedge funds used was verydynamically to market conditions. In order to ensure good. This enabled the computer to select portfoliothe availability of data for the SFA reference scores, weights that gave a portfolio with zero downsidethe portfolios are also optimised with a one-month deviation in the in-sample optimisation periods.data lag. This means that January SFA scores which The low variance of these portfolios did not persistonly become available in February are used to out-of-sample in the subsequent periods causing thisobtain the March opening portfolio weights. portfolio to underperform.Unlike traditional performance measures, the This is a classical problem of over-fitting yourSFA score is both conditional on the time period data which results in there being little relationshipbeing used and relative to a large reference data between the in-sample period and the performanceset of other hedge funds. Where other methods in the next period. The Sharpe Ratio suffers from atypically standardise everything back to a normal similar problem, but more because it is capturing Issue 33, December 2009or Gaussian distribution, the IAS uses the best only the linear effects of the portfolio whereas wefitting distributions throughout. This has the effect know there are significant non-linear effects presentof calibrating the range of scores more closely to in hedge funds.real-world data. The SFA score is able to capture some of theseOf course, the method is not perfect. The SFA non-linear artifacts because of the statistics usedscores will not provide the best returns over each in its calculation and the best-fitting non-normaland every single time period, however, over any distributions it uses. These have the effect both ofmeaningful length of time they will tend to out- improving the predictive power of the method andperform. ensuring the resultant pay-off is positively skewed, or as close to positively skewed as possible. This translates into more upside risk than downside risk. 9
- 4. Subsequent studies, where we have used less well-performing hedge fund indices, have confirmed our beliefs and the intuitive expectation that the Sortino Ratio should out-perform the Sharpe Ratio. The SFA score has been used by Infiniti to manage real portfolios over the past two years. All of the calculations used to obtain these results, the dataset used, and a trial version of the software used, are freely downloadable from www.infiniti-analytics.com for third parties to evaluate. Peter Urbani Infiniti Capital peter.urbani@infiniti-capital.com Tel: 64 3 977 8811 References: The Infiniti SFA score as a RAPM, Peter Urbani (2009), www.infiniti-analytics.com/kb/kb/article/ infinitisfascorearapm. Portfolio Selection: Efficient Diversification of Investments, Harry Markowitz (1959), http://cowles. econ.yale.edu/P/cm/m16/index.htm. Sharpe Ratio, William Sharpe, http://en.wikipedia.org/wiki/Sharpe_ratio. Sortino Ratio, Frank Sortino, http://en.wikipedia.org/wiki/Sortino_ratio. Omega Ratio — A Universal Performance Measure, Keating and Shadwick (2002), www.performance- measurement.org/KeatingShadwick2002a.pdf. Modified Sharpe Ratio, www.andreassteiner.net/performanceanalysis/?External_Performance_ Analysis:Risk-Adjusted_Performance_Measures:Modified_Sharpe_Ratio.10

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