Omega as a Performance Measure

Transcription

1 Omeg s Performnce Mesure Hossein Kzemi Thoms Schneeweis Rj Gupt c CISDM University of Msschusetts, Amherst Isenerg School of Mngement Amherst, MA Preliminry. Corresponding Author. Associte Professor of Finnce nd Associte Director of Center for Interntionl Securities nd Derivtives Mrkets kzemi@som.umss.edu.. Professor of Finnce nd Director of Center for Interntionl Securities nd Derivtives Mrkets schneeweis@som.umss.edu. c. Reserch Associte, Center for Interntionl Securities nd Derivtives Mrkets ; hswr@som.umss.edu.

2 1. Introduction In recent pper, Shdwick nd Keting (2002) present new mesure of performnce clled Omeg. According to the uthors, Omeg ws developed to overcome the indequcy of mny trditionl performnce mesures when pplied to investments tht do not hve normlly distriuted return distriutions. Unlike other mesures of performnce, Omeg ws developed with the intention to tke the entire return distriution into ccount. In this pper we show tht Omeg is essentilly the rtio of cll price to put price. This result hs severl implictions s fr s this mesure of performnce is concerned. First, it provides very intuitive explntion for wht Omeg is nd wht its limittions re. Second, it communictes to investment professionls tht Omeg is not significntly new mesure of risk since its uilding locks (i.e., cll nd put options) hve long een used to mesure potentil risk nd return of investments. In sense, Omeg tkes these well known uilding locks nd rrnges them in mnner tht provides n intuitive mesure of risk/return. Third, one could use the vst ody of reserch in the re of option pricing to otin ccurte estimtes of Omeg nd to gin etter understnding of its properties. In prticulr, since most investments do no stisfy the sic ssumptions of Blck-Scholes option pricing model, 1 non-prmetric s well s pproximte option pricing models (e.g., see Jrrow nd Rudd (1982), Ruinstein (1998), Stutzer (1996)) cn e used to otin estimtes of Omeg. Finlly, this pper presents new version of Omeg clled Shrpe-Omeg. This mesure provides exctly the sme informtion tht Omeg provides, ut the formultion is very similr to the well known Shrpe rtio. 2 In prticulr, Shrpe- Omeg of n investment is given y: Expected Return Threshold Shrpe-Omeg = Put Option Price This mesure will lwys give the sme rnking s Omeg. The pln of the pper is s follows. The next section discusses the results in greter detil nd Section 3 provides some numericl exmples. All mthemticl proofs pper in the Appendix. 2. Results Shdwick nd Keting (2002) define Omeg s: [ 1 F( x) ] dx Ω ( ) =, F( x) dx (1) where x is the rndom one-period rte of return on n investment, F( y) Pr{ x y} = is cumultive distriution of the one-period return, is threshold selected y the investor, nd (, ) represent the upper nd lower ounds of the return distriution respectively. We show tht Omeg cn e written s: C ( ) Ω ( ) =, (2) P( ) where C ( ) is essentilly the price of Europen cll option written on the investment nd 1

3 P( ) is essentilly the price of Europen put option written on the investment (See the Appendix for the proof nd discussion of the difference etween C ( ) nd P( ) nd trditionl option prices). The mturity of oth options is one period (e.g., 1month) nd strike price of oth options is siclly. In prticulr, we show tht the numertor nd the denomintor of Omeg cn e expressed s (see Appendix) [ 1 F( x) ] dx ( x ) f ( x) dx E mx( x 0) = =,, (3) ( ) F( x) dx = ( x) f ( x) dx = E mx x, 0, (4) where f ( x ) is the density function of the one-period rte of return on the investment. Equtions (3) nd (4) represent essentilly undiscounted cll nd put prices. To clculte the present vlues of these two vlues, we multiply oth sides y exp r, where r f is the per-period riskless rte. The results will e C ( ) nd P( ). In other words, ( ) mx ( 0) C = e E x,, P ( ) = e E mx( x, 0). f Shrpe-Omeg: A Better Omeg In this section first we present vrition in Omeg tht preserves ll of its fetures nd t the sme time provides mesure of risk tht is similr to the Shrpe rtio nd thus more intuitive. We show in the Appendix tht ( Ω 1) is proportionl to the expected excess return on the investment divided y the price of put option written on the sset. We cll this Shrpe-Omeg rtio nd it is formlly defined s x Shrpe-Omeg =, (5) P where x the expected rte of return on the investment. Shrpe-Omeg is proportionl to Ω 1 nd thus provides the sme informtion s Omeg mesure nd it lwys gives the sme rnking of investments tht Omeg offers. On the other hnd, we cn see tht Shrpe- Omeg represents mesure of return/risk tht is more intuitive thn Omeg. Since the price of the put option is the cost of protecting n investment s return elow the trget rtio, it is resonle mesure of the investment s riskiness, nd similr to Omeg, this mesure tkes the shpe of the distriution elow the threshold into ccount. In the next section of the pper we present numericl results concerning the ehvior of Omeg nd Shrpe-Omeg. We cn see from eqution (5) tht when the threshold is set equl to the expected return on investment, the Shrpe-Omeg is zero. We now exmine two different cses: (1) The expected return is less thn the threshold nd (2) the most likely cse tht the expected return is greter thn threshold. 1. When x <, Shrpe-Omeg mesure will e negtive. In this cse the higher the put price the etter the investment. For instnce, higher voltility will increse the put price nd thus incresing the vlue of Shrpe-Omeg (note tht its solute vlue ( ) 2

4 declines). This is similr to the cse of Omeg eing less thn one. As shown in Shdwick nd Keting (2002), when Omeg is less thn one, higher voltility increses the vlue of Omeg mesure ecuse it increses the likelihood of erning return ove the men. In prctice this is not likely to e scenrio of interest ecuse the threshold is usully set elow the men. 2. When x >, Shrpe-Omeg rtio will e positive. Unlike the previous cse, higher voltility will increse the put price nd thus reducing the Shrpe-Omeg rtio. This is similr to the cse of Omeg eing greter thn 1. Of course, the prime ojective of Shdwick nd Keting in the development of Omeg ws to provide universl performnce mesure. Given the ove results, the crucil question is how universl is this mesure nd wht re its properties? First, s mesure of risk or performnce, Omeg nd Shrpe-Omeg re not tht new in the sense tht investors hve for yers worked with cll nd put options s instruments for cpturing return nd reducing risk. In this frmework, the numertor of the Omeg (see eqution (2)) represents the cost of cquiring the return ove threshold nd denomintor represents the cost of protecting the return elow the threshold. Therefore, is seems quite sensile to use the rtio s mesure of return/risk of n investment ( similr rgument pplies to Shrpe- Omeg). Second, given wht ppers in eqution (5), we cn see tht investments with the sme expected excess return will hve different Omegs (or Shrpe-Omegs) to the degree tht prices of puts written on these investments re different. Even though this definition of risk (i.e., price of put) is fr more generl thn stndrd devition, some investors my still consider it to e rther nrrow definition of risk. Third, n ttrctive feture of Omeg nd Shrp-Omeg is tht under some circumstnces the denomintor of eqution (5) cn e pproximted y the mrket price of put option written on the investment (This is shown through numericl exmples in the next section). Fourth, in the originl presenttion of Omeg it ws uncler how this mesure is ffected y chnges in higher moments unless the distriution function ws prmetriclly specified. The results presented here cn e used to provide pproximte nswers to such questions. In prticulr, there is vst literture on pproximte option pricing for ritrry distriutions. For instnce Corrdo nd Su (1996 nd 1997) Posner nd Milevsky (1998) hve provided nlyticl solutions for option prices in terms of the first 5 nd 4 moments of the distriution of n investment s return. These solutions cn e used to otin ccurte estimtes of Omeg or Shrpe-Omeg for ritrry distriution functions. 3. Numericl Exmples In this section we provide some numericl exmples. If n investment s return is lognormlly distriuted, then one could use the Blck-Scholes option pricing formul to clculte Omeg quickly. The only djustment tht needs to e done is to use the men return on the investment s the riskless rte. Of course, most investments, nd in prticulr, lterntive investments, do not hve lognormlly distriuted returns. For this reson we use generliztion of Blck-Scholes formul to otin numericl vlues for Omeg nd Shrpe- Omeg (see Jurczenko et. l. (2002)). Also, we use the Blck-Scholes formul to exmine the differences etween mrket prices of puts nd put prices tht pper in Omeg nd Shrp- Omeg mesures. In Blck-Scholes formul option prices re effectively determined in risk neutrl world. Thus, the expected rte of return on the sset is set equl to the riskless rte. The put price tht ppers in Omeg nd Shrpe-Omeg hs to e clculted in the risk-verse world nd thus the true men return of the investment must e used. This will crete difference etween the mrket price of the put nd the price tht should pper in performnce mesures. Of course, given enough informtion, the pproprite put price tht ppers in performnce mesures cn e clculted, ut it will e eneficil to study the mrket prices of these put 3

5 prices s well. Exhiit 1 displys the rtio of put prices tht pper in the performnce mesures to the mrket prices of puts under vrious conditions. We cn see tht in generl the put price tht ppers in Omeg nd Shrpe-Omeg will e less thn the mrket price of the put. The difference is generlly lower when voltility nd/or threshold re high. Exhiit 1 As shown in the Appendix, there exists fmily of distriution functions known s Johnson fmily (see Johnson (1949)). This fmily of distriutions cn mtch the first 4 moments (i.e., men, vrince, skewness, nd kurtosis) of ny well defined proility distriution (see Exhiit 2). We use this fmily of distriutions to clculte Omeg nd then exmine its sensitivity to chnges in these moments. Further, for illustrtive purposes we use this fmily of distriutions to disply estimtes of Shrpe-Omeg for S&P500 index s well s CSFB convertile ritrge nd CSFB equity mrket neutrl index. We use monthly dt covering Jnury 1994-My Exhiits 2 nd 3 We cn see tht the CSFB-Tremont equity mrket neutrl nd the convertile ritrge hedge fund indices re eqully good low levels of the thresholds while the S&P500 index hs higher Shrpe-Omeg mesure for high vlues of the threshold. Interestingly, the equity mrket neutrl index is etter thn the convertile ritrge index when the threshold is high though they re eqully good for low levels of thresholds. For illustrtive purposes we present our estimtes of Omeg for these investments in Exhiit 4. As expected, Omeg gives the sme rnking s Shrpe-Omeg. Exhiit 4 Finlly, in exhiits 5-8 we present results regrding the sensitivity of the Shrpe- Omeg mesure to chnges in the distriution of n investment s return. We use the sme Johnson fmily of distriutions to present estimtes Shrpe-Omeg under vrious scenrios. Exhiits 5-8 Similr to results reported in Shdwick nd Keting (2002) for Omeg mesure, these grphs show tht the Shrpe-Omeg is most sensitive to chnges in men nd vrince. Higher skewness nd kurtosis hve reltively smll effect on Omeg nd Shrpe-Omeg unless the threshold is sustntilly less thn the men return. 4

6 4. References Corrdo C. nd T. Su. S&P 500 Index Option Tests of Jrrow nd Rudd s Approximte Option Vlution Formul. Journl of Futures Mrkets, 16(6), pp , Corrdo C. nd T. Su Implied Voltility Skews nd Stock Index Skewness nd Kurtosis Implied y S&P 500 Index Option Prices. Journl of Derivtives, 4, pp. 8-19, Jrrow R., nd A. Rudd. Approximte Option Vlution for Aritrry Stochstic Processes. Journl of Finncil Economics, 10, pp , Johnson N.. Systems of Frequency Curves Generted y Methods of Trnsltion. Biometrik, 36, pp , Jurczenko E., B. Millet nd B. Negre. Revisited Multi-moment Approximte Option Pricing Models: A Generl Comprison (Prt 1). Working Pper, Posner S., nd M. Milevsky. Vluing Exotic Options y Approximting the SPD with Higher Moments. Journl of Finncil Engineering, 7(2), pp , Ruinstein M. Edgeworth Binomil Trees. Journl of Derivtives, 5(3), pp , Shdwick W., nd C. Keting. A Universl Performnce Mesure. Working Pper, The Finnce Development Centre, ondon, Shdwick W., nd C. Keting. A Universl Performnce Mesure. Journl of Performnce Mesurement, Spring 2002, pp , Stutzer M. A Simple Nonprmetric Approch to Derivtive Security Vlution. Journl of Finnce, 51, pp ,

7 5. Appendix y f x nd { } et ( ) F( y) = f( x) dx = Pr x y denote the density function nd the cumultive distriution function of x, the one-period rte of return on n sset. We now show tht Omeg, defined s (see Keting nd Shdwick (2002)) ( 1 F( x) ) Ω =, ( ) F x dx dx (6) where is the threshold, is essentilly equl to rtio of cll price to put price. Tht is, C ( ) Ω =, P( ) where C ( ) nd P( ) re prices of one period cll nd put prices with the men return of the underlying sset sustituted for the riskless rte in the option pricing formul (i.e., in pricing the options the expecttion is tken under true mesure rther thn the risk neutrl mesure). et (, ) denote the domin of x, which mens tht lim F( x) x = 0, x ( F x ) lim 1 ( ) x = 0. x First, we look t the numertor of Omeg. To understnd this term, we expnd the term d x( 1 F( x) ) nd then integrte it over (, ). ( 1 ( )) ( 1 ( )) ( 1 ( )) d x F x = F x dx+ xd F x, ( 1 F( x) ) dx xdf( x) =. (7) Note tht ( 1 F ( x )) d x cn lso e written s ( 1 ( )) lim ( 1 ( )) lim x x ( 1 ( )) = 0 ( 1 F( ) ), = Pr{ x } x F x = x F x x F x, (8) Thus, comining equtions (7) nd (8), we hve 6

8 ( 1 F( ) ) ( 1 F( x) ) dx ( ) xdf x ( 1 F ( x) ) dx = ( 1 F ( ) ) + xdf ( x) = ( x ) = f( x) dx ( ) = mx x, 0 f( x) dx The lst line is essentilly the undiscounted price of cll option. To understnd the denomintor, we expnd the term d[ xf( x )] nd then integrte it over (, ). d xf( x) = F( x) dx+ xdf( x) (9) Note tht ( ) d xf x cn lso e written s xf( x) = lim xf( x) lim xf( x), x x ( ) 0, { } = F = Pr x. (10) Thus, comining eqution (9) nd (10), we hve ( ) ( ) F( ) = F x dx+ xdf x, ( ) = ( ) ( ) F x dx F xdf x ( ) = x f( x) dx, ( ) = mx x, 0 f( x) dx. The lst line is essentilly the undiscounted price of put option. Given these results, we hve, (11) Ω= r ( 1 F( x) ) ( ) F x dx dx ( ) ( ) E mx x,0 = E mx x,0 ( ) ( ) f e E mx x,0 C ( ) = =, e E mx x,0 P( ) (12) where C ( ) nd P( ) denote cll nd put prices where the expecttion is tken under the true mesure rther thn the risk neutrl mesure. 7

9 Since x is the rte of return on the underlying sset, we cn think of C( ) nd P ( ) s cll nd put prices where the initil price of the sset is 1 nd the exercise price is given y exp( ) (plese note tht we re ssuming continuous compounding). For exmple, when the vlue of the underlying sset is lognormlly distriuted, P ( ) will result in the Blck nd Scholes option pricing formul with men return on the investment sustituted for the riskless rte in the formul. Tht is, ( ) exp ( ) exp ( ) P = r N d x r N d, d d f 2 f 1 2 x σ =, σ = d σ, 1 z 1 2 y 2 N( z) = e dy 2π, where x, is the expected continuously compounded per period rte of return on the investment. We now discuss Shrpe-Omeg rtio nd its reltionship to Omeg. As shown in eqution (12), Omeg is given y e E mx ( x,0) Ω ( ) =. (13) e E mx ( x,0) If we sutrct 1 from oth sides nd then rerrnge the terms, we will hve the Shrpe-Omeg rtio. e e rf rf ( ( ) 1) ( ( ) 1) r ( ) ( ) ( ) ( ) e E mx ( x,0) f e E mx x,0 Ω( ) 1= e E mx x,0 1 Ω E mx = x,0 E mx x,0 Ω x = e E mx ( x,0) r We cn see tht Shrpe-Omeg is given y e f ( ( ) 1) sme rnking s Omeg does. The Johnson fmily of distriutions is given y Ω nd therefore it lwys gives the 2 z exp 2 s k 3 q z z z z z 2π ( ) = 1+ ( 3 ) + ( 6 + 3) This distriution hs 0 men, unit vrince, skewness s nd kurtosis k. By defining x = σ z + µ, we cn chnge the men nd the stndrd devition s well. 8

10 6. Exhiits Exhiit 1 Rtio of P ( ) to Mrket Put Prices for Vrious nd σ Monthly Figures Men = 1%, Riskless Rte = 0.4% P H Mrket Price σ=2% σ=5% σ=7% Threshold 0.65 Exhiit 2 Density Function of the Four Prmeter Johnson Distriution for Vrious evels of Skewness nd Kurtosis Men = 0, Vrince = 1 Density s=0,k=1 s=0,k=5 s=1,k=3 s= 1,k= Vlue 9

11 Exhiit 3 Shrpe-Omeg Mesure for S&P500, CSFB-Tremont Convertile Aritrge nd Equity Mrket Neutrl Indices Shrpe Omeg S&P500 CA EMN Threshold -5 Exhiit 4 Omeg Mesure for S&P500, CSFB-Tremont Convertile Aritrge nd Equity Mrket Neutrl Indices Omeg S&P500 CA EMN Threshold

12 Exhiit 5 Shrpe-Omeg Mesure for Vrious evels of Monthly Men (Stndrd Devition = 5%; Skewness = 0; Kurtosis = 3) Shrpe Omeg µ=0.5 % µ=1% µ=2% Threshold -0.5 Exhiit 6 Shrpe-Omeg Mesure Vrious evels of Monthly Stndrd Devition (Men = 1%; Skewness = 0; Kurtosis = 3) Shrpe Omeg 4 σ=0.1 % σ=0.2 % 2 σ=0.5 % Threshold -2 11

13 Exhiit 7 Shrpe-Omeg Mesure Vrious evels of Monthly Skewness (Men = 1%; Stndrd Devition = 5%; Kurtosis = 3) Shrpe Omeg s= 2 s=0 0.2 s= Threshold Exhiit 8 Shrpe-Omeg Mesure Vrious evels of Monthly Kurtosis (Men = 1%; Stndrd Devition = 5%; Skewness = 0) Shrpe Omeg k=3 k=5 k= Threshold

14 7. Notes 1 Among others, the Blck-Scholes option pricing model ssumes tht sset vlues re lognormlly distriuted. 2 The Shrpe rtio is defined s: Expected Return Riskless Rte Stndrd Devition of Return 13