VALUE AT RISK FOR LONG/SHORT POSITIONS. VaR = E[Portfolio] - q α (1) - PDF

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VALUE AT RISK FOR LOG/SHORT POSITIOS Sorn R. Straa, Ph.D., FRM Montgomery Investment Technology, Inc. 200 Federal Street Camden, J 0803 Phone: (60) The obectve

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VALUE AT RISK FOR LOG/SHORT POSITIOS Sorn R. Straa, Ph.D., FRM Montgomery Investment Technology, Inc. 200 Federal Street Camden, J 0803 Phone: (60) The obectve of ths paper s to provde a methodology for the computaton of the value at rsk (VaR) of a portfolo contanng both long and short postons. The frst sectons present the defnton of VaR and the usual approxmatons used for VaR estmates. The followng sectons provde VaR estmates for portfolo havng only long (or short) postons, and VaR estmates for portfolo havng both long and short postons. VALUE AT RISK DEFIITIO Accordng to Joron (997; page 87), for a gven (-α)% confdence level (e.g., 95%) the defnton of VaR relatve to the mean s: where: VaR = E[Portfolo] - q α () E[Portfolo] = expected value of the portfolo on the date of nterest q α = the α quantle (e.g., 5%) of the portfolo value on the date of nterest The tme horzon (.e., the tme nterval between today and the future date of nterest) and the confdence level (e.g., 95%) are the two parameters used n the defnton of VaR. Based on the defnton of VaR t results that, wth (-α)% confdence level (e.g., 95%), the losses for the selected tme horzon wll not exceed the value of VaR. Ths s the reason why VaR s commonly related to a confdence level. However, t s more adequate to refer to VaR as a quantle estmate. Because VaR s an estmate, t s possble and useful to provde a confdence nterval for VaR. Ths confdence nterval bult around the estmated VaR has ts own confdence level. Labelng VaR as a quantle removes the confuson of a confdence nterval bult around a confdence level. It s less confusng to deal wth a confdence nterval bult for a quantle. However, the usage of confdence level as a parameter for VaR s wdely accepted by the ndustry and academa. Value at Rsk for Long/Short Postons Montgomery Investment Technology, Inc. / Sorn R. Straa, Ph.D., FRM Page of 6 APPROXIMATIOS USED FOR VaR ESTIMATES Longerstaey and Spencer (996) defne the contnuously compounded returns (r, ) and the percent returns (R, ) as follows: r, = S, ln ( ) S 2, R, = S S S,,, ( 3) where S, s the value of asset at tme t. It should be noted that the return rates are equal to the correspondng returns dvded by the tme nterval t = t t -. The correct temporal aggregaton of these returns s as follows (Longerstaey and Spencer 996): r = r, ( 4) = R = ( + R, ) (5) = Smlarly, the correct cross-secton aggregaton s as follows (Longerstaey and Spencer 996):, r = ln( w e ) ( 6) = R = w R, ( 7) = r where w s the fracton of asset wth respect to the total portfolo. The contnuously compounded returns are used n RskMetrcs as the bass for all computatons (Longerstaey and Spencer 996). In practce, RskMetrcs assumes that a portfolo return s a weghted average of contnuously compounded returns (Longerstaey and Spencer 996): r = w r, (8) = Value at Rsk for Long/Short Postons Montgomery Investment Technology, Inc. / Sorn R. Straa, Ph.D., FRM Page 2 of 6 It should be noted that ths weghted average of contnuously compounded returns - used as an approxmaton by RskMetrcs (Longerstaey and Spencer 996) - s dfferent wth respect to the correct cross-secton aggregaton mentoned on the same page 49 of Longerstaey and Spencer (996). Another approxmaton largely used for VaR computatons s (Longerstaey and Spencer 996; page 8): e x + x ( 9) For the purpose of VaR computatons, some authors (Longerstaey and Spencer 996; Joron 997) assume that the expected rate of return of the portfolo s zero. The man reason s the relatve hgh value of the portfolo volatlty. The dffculty of obtanng a good qualty estmate for the rate of return of the portfolo s another argument. VaR ESTIMATES FOR PORTFOLIO WITH LOG (OR SHORT) POSITIOS OLY For a gven (-α)% confdence level (e.g., 95%), assumng a normal probablty dstrbuton functon for the rates of return, the VaR of a gven portfolo s computed as (Longerstaey and Spencer 996; page 8): where: VaR = Portfolo [ exp( µ t z σ t )] Portfolo ( µ t + z σ t ) Portfolo z σ t ( 0) Portfolo = portfolo value µ = portfolo growth rate t = tme horzon σ = portfolo volatlty z = the (-α) quantle of the standard normal dstrbuton (0,) It should be noted that usng the z value s correct only when the portfolo volatlty s known exactly. If an estmate of the portfolo volatlty s used nstead of the true value of the portfolo volatlty, than a correct estmate of VaR requres the use of the Student t value. The Student t value s larger than the correspondng z value. The VaR based on the Student t value s larger than the VaR based on the correspondng z value. Ths fact accounts for the uncertanty generated as a result of usng an estmate of the portfolo volatlty nstead of the true value of the portfolo volatlty. However, when the portfolo volatlty s estmated usng a large number of data (e.g., greater than 30), the Student t value s practcally equal to the z value. Value at Rsk for Long/Short Postons Montgomery Investment Technology, Inc. / Sorn R. Straa, Ph.D., FRM Page 3 of 6 Based on the volatltes of the ndvdual assets (σ ), ther weghts n the portfolo value (w ), and ther correlaton coeffcents (ρ ), the portfolo volatlty s computed as (Joron 997; page 50): = = σ = ρ w w σ σ ( ) It should be noted that the weghts w cannot be computed for a portfolo wth both long and short postons havng a total net value of zero. Moreover, a portfolo wth both long and short postons havng a total value close to zero may have a relatvely hgh volatlty (McCarthy 999). VaR ESTIMATES FOR PORTFOLIO WITH BOTH LOG AD SHORT POSITIOS Based on the RskMetrcs approxmatons for contnuously compounded returns and VaR, t results that: VaR = Asset µ t + = + z t ρ Asset Asset σ σ = = z t ρ Asset Asset σ σ ( 2) = = where: Asset = portfolo value nvested n asset µ = the growth rate for asset. A postve value for Asset denotes a long poston, whle a negatve value denotes a short poston. Ths approxmaton (equaton 2) s smlar to equaton (.39) of Deutsch (2003). It should be noted that equaton (.39) of Deutsch (2003) s for portfolos havng ether long or short postons. If t = equaton (2) becomes dentcal wth equaton (0) of Chapados and Bengo (2000). TotalSum (2003) provdes a numercal example for a portfolo havng both long and short postons usng equaton (2) as a VaR approxmaton wth µ = 0. Value at Rsk for Long/Short Postons Montgomery Investment Technology, Inc. / Sorn R. Straa, Ph.D., FRM Page 4 of 6 The correspondng un-dversfed VaR s: VaR = Asset µ t + + z t Asset σ undversfed = = z t Asset σ ( 3) The VaR approxmaton (equaton 2) handles n a smooth manner the case of portfolos havng an aggregate value close to zero. As long as the drect effect of the rates of return s neglected for VaR estmates (.e., µ = 0), the long and short holders of a gven portfolo have the same rsk. If the total amount (.e., long and short postons) nvested n each asset (or assets havng exactly the same volatlty and correlaton coeffcent +.0) s zero, then the VaR s zero. Smlarly, f all postons n each asset are matched by equal postons n another asset that has exactly the same volatlty and the correlaton coeffcent of these two assets s exactly.0, then the VaR s zero. When the hedgng s done usng assets that () are not perfectly (postvely or negatvely) correlated and (2) have dfferent volatltes, then the VaR s greater than zero, even when the total portfolo value s zero. If the growth rates are taken nto account, ths VaR approxmaton accounts for an ncreased rsk due to () long postons wth negatve growth rates or (2) short postons wth postve growth rates. Smlarly, t accounts for a reduced rsk due to () long postons wth postve growth rates or (2) short postons wth negatve growth rates. MOTE CARLO VALUE AT RISK ESTIMATES FOR PORTFOLIO WITH BOTH LOG AD SHORT POSITIOS All the long postons can be lumped together nto an ndex. Based on hstorcal data for the ndvdual stocks, the hstory of ths ndex can be establshed. Smlarly, all short postons can be lumped together nto another ndex and ts hstory can be establshed from the hstorcal data avalable for the ndvdual stocks. For each one of these ndexes we can compute ts growth rate and volatlty. Addtonally, we can compute the correlaton coeffcent between the rates of return of the two ndexes. The Monte Carlo smulaton s requred only for two postons,.e. the two ndexes. The actual return rates for these ndexes are generated based on correlated random numbers. To be compatble wth the VaR estmates provded by equatons (0) and (2), the Monte Carlo smulaton must assume the normalty of the return rates for these ndexes. (Usng a hstorcal approach for Monte Carlo smulatons may be requred when we are faced wth sgnfcant departures from normalty. However, ths case s outsde the scope of our analyss.) When the portfolo contans only long (or short) postons, the Monte Carlo VaR estmate s close to the VaR estmate provded by equaton (0) (wthout applyng the approxmaton for the exponental functon provded by equaton (9)). When the portfolo contans both long and short = Value at Rsk for Long/Short Postons Montgomery Investment Technology, Inc. / Sorn R. Straa, Ph.D., FRM Page 5 of 6 postons resultng n an overall portfolo value close to zero, the Monte Carlo VaR estmate s close to the VaR estmate provded by equaton (2). Therefore, for portfolos contanng both long and short postons, n order to check the numercal effects of dfferent approxmatons, t s recommended to use both the Monte Carlo method and equaton (2). For portfolos contanng only long (or short) postons, the VaR estmate provded by equaton (0) s preferable to the estmates provded by equaton (2) or Monte Carlo smulaton. REFERECES Chapados,.; Bengo, Y. Cost functons and model combnaton for VaR-based asst allocaton usng neural networks. Montreal: Computer Scence and Operatons Research Department Unversty of Montreal; (May ) Accessed on August 22, 2003 at Deutsch, H. P. Value at Rsk. Eschborn: d-fne; Accessed on August, 2003 at Joron, P. Value at rsk. Chcago, IL: Irwn; 997. Longerstaey, J.; Spencer, M. RskMetrcs Techncal Document. 4 th Edton. ew York, Y: J. P. Morgan/Reuters; (December 7) 996. McCarthy, M. How Rsk Management Can Beneft Portfolo Managers. In: Rsk Management: Prncples and Practces. Charlottesvlle, VA: Assocaton for Investment Management and Research; 999. TotalSum. Calculatng Accessed on August 2, 2003 at Value at Rsk for Long/Short Postons Montgomery Investment Technology, Inc. / Sorn R. Straa, Ph.D., FRM Page 6 of 6
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