Downside Risk-Adjusted Performance Measurement

Transcription

1 Downside Risk-Adjusted Performance Measurement Paul D. Kaplan, Ph.D., CFA Chief Investment Officer Morningstar Associates, LLC 2005 Morningstar, Associates, LLC. All rights reserved. <#>

2 Agenda Omega, Sharpe-Omega, & the Sortino Ratio Kappa as a generalization The Lottery Test Skewness & kurtosis Johnson distributions Estimating Kappa Impact of Kappa variant choice on performance rankings Summary 2

3 Shadwick & Keating s Omega ( τ) Ω = τ τ ( ) 1 F R d R ( ) F R dr τ = threshold return F(.) = cumulative density function of returns 3

4 Example of Omega Ω ( 0) = = Area = Cumulative Density Function Area =

5 Kazemi, Schneeweis, & Gupta s Sharpe-Omega Target-Based Sharpe Ratio = µ τ σ Sharpe-Omega = µ τ ( ) P 1+τ µ = mean of return distribution σ = standard deviation of return distribution P(1+τ) = price of put option (with strike price = 1+τ) 5

6 Equivalence of Omega & Sharpe-Omega r f ( ) Sharpe-Omega = e Ω τ 1 r f = continuous risk-free rate 6

7 Sortino Ratio µ τ ( ) S τ = τ 2 ( τ R) df( R) 7

8 Lower Partial Moments (Downside Risk Measures) τ ( ) ( ) ( ) LPM τ = τ R df R n n 8

9 Kappa: A Generalized Downside Risk-Adjusted Measure (Kaplan & Knowles) µ τ Κ n ( τ) = n LPM n ( τ) 9

10 Omega & the Sortino Ratio as Special Cases of Kappa n=1 Omega: Ω τ = Κ τ + 1 ( ) ( ) 1 n=2 Sortino Ratio: S τ = Κ τ ( ) ( ) 2 10

11 Equivalence of Kappa & Target-Based Sharpe Ratio Rankings when Returns are Normally Distributed (Di Pierro & Mosevich) Kappa 6 4 Kappa 1 Kappa Kappa Target-Based Sharpe Ratio 11

12 Shadwick s Lottery Test Consider a lottery ticket Price = $1 1 out of 1,000,000 chance of paying $1,000,000 Is it better to buy or sell this lottery ticket? 12

13 Characteristics & Ranking of Sides of Lottery Ticket Characteristics Statistic Mean Standard Deviation Skewness Kurtosis Buy Sell Rankings Κ 1 (Omega) ranks Buy better than Sell for all thresholds. For all other values of n, ranking depends on threshold. So Sortino Ratio (n=2) fails lottery test. 13

14 Differences in Kappa between Sides of Lottery Ticket Kappa(Buy) - Kappa(Sell) Kappa Kappa Threshold 14

15 Skewness 0.5 s= s=0 s=

16 Kurtosis 0.5 k=5 0.4 k=3 (Normal) 0.3 k=

17 Johnson Family of Distributions Unbounded Kurtosis Bounded Normal 3-Param Lognormal Bounded 5 Inadmissible Inadmissible Skewness 17

18 Estimating Kappa with Johnson Distributions Estimate mean, standard deviation, skewness, & kurtosis of returns. From skewness & kurtosis, select appropriate Johnson distribution: s 0 & k 3 Normal (s, k) near boundary 3-Parameter Lognormal (s,k) above boundary Unbounded (s,k) below boundary Bounded Calculate parameters of selected distribution. Calculate Kappa using numerical integration. 18

19 Example: HFR Monthly Hedge Fund Indices, 1/ /2004 Returns in excess of T-bill Kurtosis Skewness 19

20 Estimates of Kappa(0) for HFR Monthly Hedge Fund Indices Index 1. Distressed Securities 2. Convertible Arbitrage 3. Equity Market Neutral 4. Event-Driven 5. Equity Hedge 6. Merger Arbitrage 7. Macro 8. Sector 9. Fund of Funds 10. Fixed Income Arbitrage 11. Emerging Markets 12. Short Selling Johnson Dist. Unbounded Bounded Unbounded Unbounded Unbounded Bounded Unbounded Unbounded Unbounded Unbounded Unbounded Unbounded Kappa Kappa Kappa Sharpe

21 Kappa-Based Ranks on HFR Monthly Hedge Fund Indices Index Distressed Securities Convertible Arbitrage Equity Market Neutral Event-Driven Equity Hedge Merger Arbitrage Macro Sector Fund of Funds Fixed Income Arbitrage Emerging Markets Short Selling Kappa Kappa Kappa Sharpe

22 Summary Omega & the Sortino Ratio are special cases of Kappa. Sharpe-Omega is just a restatement of Omega. All Kappa variant give same rankings as the Sharpe Ratio when returns are normal. Omega is the only version of Kappa that passes the lottery test. Skewness & kurtosis can be modeled using Johnson distributions. Johnson distributions can be used to estimate Kappa from the first four moments. The 3-parameter lognormal, used by Sortino & Forsey to estimate downside risk, is a special case of the Johnson distributions. Choice of Kappa variant affects risk-adjusted performance rankings. 22

23 References Forsey Hal, The Mathematician s View: Modelling Uncertainty with the Three Parameter Lognormal, in Managing Downside Risk in Financial Markets, Frank A. Sortino and Stephen E. Satchell, eds., Reed Educational and Professional Publishing Ltd., Kaplan, Paul D. and James A. Knowles, Kappa: A Generalized Downside Risk-Adjusted Performance Measure, Journal of Performance Measurement, Spring Kazemi, Hossein, Thomas Schneeweis, and Bhaswar Gupta, Omega as a Performance Measure, Journal of Performance Measurement, Spring Di Pierro, Massimo and Jack Mosevich, On Ranking Schemes and Portfolio Selection, DePaul University working paper, January Shadwick, William F., The Sortino Ratio and Darsinos and Satchell s Generalised Sharpe Ratios Fail the Lottery Test, Finance Development Centre Limited, February Shadwick, William F. and Con Keating, A Universal Performance Measure, Journal of Performance Measurement, Spring Sortino, Frank A., From Alpha to Omega, in Managing Downside Risk in Financial Markets, Frank A. Sortino and Stephen E. Satchell, eds., Reed Educational and Professional Publishing Ltd.,