Efficient Point Estimation of the Sharpe Ratio

Transcription

1 Joural of Statitical ad Ecoometric Method, vol., o.4, 013, ISSN: (prit), (olie) Sciepre Ltd, 013 Efficiet Poit Etimatio of the Sharpe Ratio Grat H. Skrepek 1 ad Ahok Sahai Abtract The Sharpe Ratio i defied a the mea exce retur over the tadard deviatio of the exce retur for a give ecurity market portfolio. Due i part to the dyamic ature of thi meaure ad becaue of tatitical iue, the ample etimatio of thi ratio i challegig ad ubject to ubtatial amplig error. A uch, the purpoe of thi reearch wa to develop ad tet a efficiet poit etimator of the Sharpe Ratio utilizig a approach that ought to explicitly reduce it aociated amplig error through the miimizatio of the coefficiet of variatio (CV) ad Mea Squared Error (MSE). A empirical imulatio tudy wa coducted to ae the potetial gai of the ovel method give tochatic variatio preet withi time erie of ecurity price data, with reult offerig improvemet acro all pecificatio of ample ize ad populatio tadard deviatio. Overall, thi work addreed a major limitatio i the exitig poit etimate calculatio of the Sharpe Ratio, particularly ivolvig etimatio error which i preet eve withi large data et. Mathematic Subject Claificatio: 6D05, 60C05 Keyword: Sharpe Ratio; Relative Efficiecy of a Poit Etimator; Coefficiet of Variatio; Miimum Mea Squared Error (MMSE) 1 College of Pharmacy, The Uiverity of Oklahoma Health Sciece Ceter, Oklahoma City, OK, USA. Departmet of Mathematic ad Statitic, The Uiverity of The Wet Idie, Faculty of Sciece ad Agriculture, St. Augutie Campu, Triidad ad Tobago, Wet Idie. Article Ifo: Received : September 15, 013. Revied : October 1, 013. Publihed olie : December 1, 013.

2 130 Efficiet Poit Etimatio of the Sharpe Ratio 1 Itroductio The Sharpe Ratio i a frequetly-ued fiacial portfolio performace meaure that provide a aemet of rik-adjuted performace (i.e., the mea exce retur or rik premium divided by the tadard deviatio of the exce retur), defied mathematically i it ex ate form a a expected value: E( Ra Rb) E( Ra Rb) Sr = = (1.1) var ( Ra Rb) for portfolio i = 1,,..., with R a = aet retur ad R b = rik-free rate of retur or a idex retur.[1,] While utilizig the ame mathematical equatio, the ex-pot form of the Sharpe ratio icorporate realized retur rather tha thoe that are expected.[1,] Withi the mea-variace Markowitz efficiet frotier, by defiitio, the Sharpe Ratio i the lope of the capital market lie.[3,4] Eve though a aalyt may commoly rely upo the Sharpe Ratio to optimize portfolio choice, thee value may be tatitically biaed due to iheret etimatio error, eve withi large data et.[5,6] Chritie (007) directly commeted that A major limitatio of Sharpe Ratio i that the iput, amely expected retur ad tadard deviatio, are meaured with error, beig a iue which the ivetmet commuity practically igore. [5] A uch, umerou author have ought to develop geeralizatio of the Sharpe Ratio that correct for variou tatitical cocer, icludig autocorrelatio, kew ad kurtoi, ad o-ormality.[7-13] Skrepek ad Sahai (011), to illutrate, developed a boottrap reamplig ad Computatioal Itelligece approach to the poit etimate ad cofidece iterval for the Sharpe Ratio that offered improved etimatio error correctio relative to other meaure.[7] Lo (00) derived the tatitical ditributio of the Sharpe Ratio uder umerou retur ditributio, though without explicitly coiderig the impact of amplig error upo tatitical iterpretatio.[11] Furthermore, aumig multivariate ormality, a approach wa alo developed by Jobo ad Korkie (1981) though failig to achieve ufficiet tatitical power.[6] Give the above, the purpoe of thi reearch wa to develop ad ae a efficiet poit etimate of the Sharpe Ratio utilizig a approach that ought to explicitly reduce it aociated amplig error. Baed upo iformatio preet withi the give radom ample, a ovel method wa developed that focue upo miimizig the coefficiet of variatio (CV) ad Mea Squared Error (MSE) to offer improved etimatio for both the umerator ad deomiator of the Sharpe Ratio.

3 G.H. Skrepek ad A. Sahai 131 A Propoed Efficiet Poit Etimator of the Sharpe Ratio µ The Sharpe Ratio for a give portfolio, Sr =, i typically etimated a x Sr = = Er ( r). I more detail, give that the populatio value of the Sharpe µ i z µ i R f Ratio for portfolio i may be expreed a Sri = = for i = 1,,...,, i i thee populatio parameter are etimated by the ample couterpart icludig the ample mea ad tadard deviatio baed upo a radom ample X1, X,..., X a: i= x i= 1 i ample mea, x = i= 1 ample tadard deviatio, ; with ( x ) i i x = = = (.1) ( 1) Therefore, the typically-ued poit etimator for the Sharpe Ratio i x Sr = = Er ( r). Notably, 1 may ot ecearily be a efficiet etimate of the 1 populatio value, irrepective of large ample ize, or may x be a efficiet etimate of µ i term of a optimal coefficiet of variatio.[5,7] I the forthcomig, a propoed efficiet etimator pecifically focuig upo the umerator of thi ratio (i.e., the populatio mea, µ) i offered. Followig, the developmet of a efficiet etimator of the deomiator (i.e., the populatio tadard deviatio, ) i udertake. Fially, the proceed are collectively icorporated ito a ovel, propoed poit etimate of the Sharpe Ratio..1 Efficiet Etimatio of Sharpe Ratio Numerator, µ The propoed efficiet etimator of the umerator of the Sharpe Ratio, µ, eek to utilize iformatio withi the data more fully via icorporatio of the ample coefficiet of variatio (CV), which i a ormalized diperio of a probability or frequecy ditributio that repreet the extet of variability from a populatio mea, defied a a radom variable ratio of tadard deviatio to it expected value.[14] A lower CV reflect a maller reidual veru predicted value i a give model, uggetig improved goode of fit. Quatitatively, the potetial beefit of more efficiet etimator alo iclude, for example, a lower ample ize requiremet to achieve robut reult.

4 13 Efficiet Poit Etimatio of the Sharpe Ratio I quatitative fiacial or ecoometric aalye, it may be commo to ecouter a aalytic ceario wherei the ample etimate of the coefficiet of variatio of the ample mea (i.e., a more table radom variable tha the origial variable) will ot be large. I thoe ituatio, the aalyt may prefer a coefficiet of variatio a CV ( x ) < 1.0, ad more likely eve markedly below 1.0. Hece, i uch cae, the propoed alterative etimator of µ, deoted t, veru the uual etimator which i deoted x, i preeted a: x t = x+ (.) ( x ) 1 I percetage term, the relative efficiecy of thi propoed etimator t with repect to the uual etimator x would be: 100 Ex ( µ ) η = 100 Et ( µ ) (.3) A ubiaed etimator of the efficiecy ratio, η, a a fuctio of ( x, ) aloe i alo required, a ( x, ) i joitly a complete ufficiet tatitic for ( µ, ). Beig a fuctio of a complete ufficiet tatitic, the ubiaed etimator would be a Uiform Miimum Variace Ubiaed Etimator (UMVUE), take that: E x η = E t ( µ ) ( µ ) x = E + ( x µ ) ( x ) 1 = 1+ A+ B (.4) The followig ca be developed from (.4), pecifically cocerig A : A E x x µ = ( x ) 1 x ( x µ ) = E Ex ( x) 1

5 G.H. Skrepek ad A. Sahai c x x µ = E ( x µ ) exp d x ( x ) 1 + x x µ c E d = exp d x ( x dx ) 1 a: 1/ c = π (.5) Additioally, by applyig itegratio by part, the followig proceedig are offered: A = E Ex[ a] = E [ a] with: ( x ) ( x ) a = = ( u+ 1) ( u 1) a: ( x) u = (.6) Pertaiig pecifically to B withi (.4), it hould be recogized that idepedetly of x, ( 1) i ~ χ ditributed with ( 1) degree of freedom, yieldig: B E Ex ( x) = ( x ) 1

6 134 Efficiet Poit Etimatio of the Sharpe Ratio ( x) ( x) + 1 = where: c 0 Ex ( 1 ) ( 1) exp d ( 1 ) 1 1 c = Γ ( 1) (.7) Importatly, the followig i alo oted: ( 1 ) d ( 1) exp d 1 1 = ( 1 ) ( 1) ( 3 ) exp By applyig (.7) ad (.8): + ( x ) ( x ) 1 c 0 B= Ex ( 3 ) ( 1) exp d c ( 1) ( x) ( x ) 1 d + d x 0 ( 1 ) ( 1) exp d Through the applicatio of itegratio by part: (.8) (.9)

7 G.H. Skrepek ad A. Sahai 135 = B E Ex ( x) ( x) 1 ( x) ( x) 1 d c + d Ex ( 1) ( 1 ) ( 1) 0 exp d = E ( x) ( x) 1 ( x) ( x) ( x) 3 E ( 1) ( x) ( x) 1 = E ( 1) 3 ( x ) ( x ) ( x ) (.10) Alteratively, (.10) may be expreed a B =E(b), where: 3 b= u ( u 1) u ( u+ 1) ( u 1) ( 1) ( + 1) ( 1) 3 u = u ( u 1) ( 3) by ettig the followig from (.6): ( x) u = (.11)

8 136 Efficiet Poit Etimatio of the Sharpe Ratio Overall, the UMVUE of the relative efficiecy of t with repect to the uual etimator of x may be more appropriately expreed a 1+ a+ b rather tha 1+ A+ B. Per (.6) ad (.8), thi i: 3 u ( u 1) ( 3) u ( 1) + 1+ a+ b= 1 ( u+ 1) ( u 1 ) + 1 { } 3 ( u 1) u ( 3) + u ( 3) ( 1) ( 1) = 1 with the followig: 100 Ex ( µ ) = 100 > 100% η Et ( µ ) if 0 η 1, 0 1 a b 1 (.1), a 1 { } < + + <, or 0 u ( 3) u ( 3) ( 1) u > for all ( x) x < 1 per (.11), or if u > (.1) (.13) < + per ad give that the coefficiet of variatio of Importatly, the aforemetioed i coitet for all oberved coefficiet of variatio for x i practice ad, a uch, the propoed alterative etimator t defied i (.) i a more efficiet etimator of the ormal mea µ rather tha the uual etimator x. Additioally, the propoed etimator t may alo be expreed a a fuctio of the quare of the ample coefficiet of variatio, deoted v, a: 1 t x + v with v = ( x) (.14). Efficiet Etimatio of Sharpe Ratio Deomiator, I etablihig the propoed approach for a improved Sharpe Ratio poit etimatio ad focuig o the deomiator,, a efficiet etimator of the ivere of the ormal tadard deviatio, 1, build toward a proof of the followig lemma.

9 G.H. Skrepek ad A. Sahai 137 Lemma. For a radom ample populatio (, ) 1 X1, X, X3,..., X from a ormal N µ, K i the Miimum Mea Squared Error (MMSE) of 1 the ivere of the ormal populatio tadard deviatio,, wherei: ( ) Γ ( 1) K =. (.15) ( 3) Γ Proof. K 1 i the MMSE of give that: 1 1 E E K = 1 E with: ( 1 ) ~ χ 1.3 The Propoed Efficiet Poit Etimate of the Sharpe Ratio, Sr µ = A ofte appear withi the literature, the uual applied poit etimate for x Er r =, depite ackowledgig that 1 doe ot provide the Sharpe Ratio i a good tatitical etimate of 1 or doe x etimated from yield a optimal coefficiet of variatio.[5,7] Buildig upo (.14), the cla of etimator K r i coidered, wherei the MMSE etimator i developed a a more efficiet poit etimate of the Sharpe Ratio a: Er r = K (1+ v) r = K (.16) t

10 138 Efficiet Poit Etimatio of the Sharpe Ratio Give the aforemetioed, K would be defied by miimizig the Mea Squared Error (MSE) of the etimator i the cla of etimator K r. Notably, the aemet of MSE withi quatitative fiacial aalye i of importace primarily becaue the MSE i a predomiat tatitical approach ued to ae the differece betwee value oberved by a etimator veru the true value of the quatity beig etimated.[14,15] The MSE i expreed mathematically a: MSE ( θ) = E ( θ θ) (.17) Marked differece reflected i the MSE may occur becaue a etimator poorly capture relevat iformatio from the ample, therei producig a iaccurate etimate of the true value. 3 Empirical Simulatio Study 3.1 Methodology To ae the efficiecy of the propoed poit etimator of the Sharpe Ratio from (.16), a empirical imulatio tudy wa developed utilizig Matlab 010b [The Mathwork Ic., Natick, Maachuett]. Compario were draw for the propoed poit etimator, deoted Er (r), veru the curretly-utilized etimator for the Sharpe Ratio, deoted Er(r), acro illutrative ample ize of = 6, 11, 1, 31, 41, 57, 71, 101, 0, ad 303. The paret populatio wa defied a ormal with a populatio Sharpe Ratio, Sr = 0.5, ad with varyig populatio tadard deviatio of = 0.0, 0.5, 0.30, 0.35, 0.40, 0.45, ad The umber of replicatio wa 51,000. The actual Mea Squared Error (MSE) of the uual etimator, Er(r), ad the propoed etimator, Er (r), were calculated by averagig the quared deviatio of the etimator value from the populatio Sharpe Ratio (i.e., 0.5). A uch, the Relative Efficiecy, RelEff, of the propoed etimator compared to the uual etimator wa calculated accordigly: MSE{ Er ( r )} RelEff % { Er ( r) veru Er ( r) } = 100 (3.1) MSE Er r { } 3. Reult Preeted i Table 1, the relative efficiecy of the propoed poit etimator, Er (r), raged from a miimum of percet at the lower ample ize ad tadard deviatio ( = 0.0, = 11) to a maximum of percet with icreaig tadard deviatio ad ample ize ( = 0.35, = 303). Variatio i the Relative Efficiecy of the propoed etimator wa, however, predomiatly

11 G.H. Skrepek ad A. Sahai 139 oberved to be a fuctio of ample ize rather tha populatio tadard deviatio, with abolute percetage icreae beig approximately percet from = 11 to 1, percet from = 1 to 31, percet from = 31 to 41, percet from = 41 to 51, percet from = 51 to 71, percet from = 71 to 101, percet from = 101 to 0, ad percet from = 0 to 303. Acro every pecificatio of ample ize ad populatio tadard deviatio, Er (r) yielded improvemet i efficiecy regardig MSE vi-à-vi the approach commoly utilized i practice.

12 140 Efficiet Poit Etimatio of the Sharpe Ratio Table 1: Relative Efficiecy of the Propoed Efficiet Poit Etimator of the Sharpe Ratio, Er (r), relative to the Uual Etimator, Er(r), i Percetage Term Populatio Stadard Deviatio () Sample Size () = 0.0 = 0.5 = 0.30 = 0.35 = 0.40 = 0.45 = 0.50 = = = = = = = = =

13 G.H. Skrepek ad A. Sahai Cocluio By focuig upo the ample coefficiet of variatio (CV) ad Miimum Mea Squared Error (MMSE), the propoed poit etimator of the Sharpe Ratio yielded improvemet acro all pecificatio of ample ize ad populatio tadard deviatio to the exitig method, with miimum relative efficiecy icreae begiig with 137 percet at lower ample ize ad ultimately exceedig 00 percet at ample ize above 71. Overall, thi work addreed a major limitatio i the exitig poit etimate calculatio of the Sharpe Ratio, particularly ivolvig etimatio error which i preet eve withi large data et. Referece [1] W.F. Sharpe, Mutual Fud Performace, Joural of Buie, 36, (1966), [] W.F. Sharpe, The Sharpe Ratio, Joural of Portfolio Maagemet, 1, (1994), [3] H.M. Markowitz, Portfolio Selectio, Joural of Fiace, 7, (195), [4] H.M. Markowitz, Portfolio Selectio: Efficiet Diverificatio of Ivetmet, Joh Wiley ad So, [5] S. Chritie, Beware the Sharpe Ratio, Macquarie Uiverity Applied Fiace Cetre, (007). [6] J.D. Jobo ad B.M. Korkie, Performace Hypothei Tetig with the Sharpe ad Treyor Meaure, Joural of Fiace, 36, (1981), [7] G.H. Skrepek ad A. Sahai, A Etimatio Error Corrected Sharpe Ratio Uig Boottrap Reamplig, Joural of Applied Fiace ad Bakig, 1, (001), [8] J.P. Pézier, Maximum Certai Equivalet Retur ad Equivalet Performace Criteria, ICMA Ceter Dicuio Paper i Fiace, Readig Uiverity, (010). [9] S.D. Hodge, A Geeralizatio of the Sharpe ratio ad it Applicatio to Valuatio Boud ad Rik Meaure, Fiacial Optio Reearch Cetre, Uiverity of Warwick, (1998). [10] S. Koekebakker ad V. Zakamoulie, Geeralized Sharpe ratio ad Portfolio Performace Evaluatio, Agder Uiverity College, (007). [11] A.W. Lo, Statitic of Sharpe Ratio, Fiacial Aalyt Joural, 58, (00), 6-5. [1] H.D. Viod ad M.R. Morey, A Double Sharpe Ratio, Fordham Uiverity, (1999). [13] H.D. Viod ad M.R. Morey, A Double Sharpe Ratio, Advace i Ivetmet Aalyi ad Portfolio Maagemet, 8, (001),

14 14 Efficiet Poit Etimatio of the Sharpe Ratio [14] D. Ruppert, Statitic ad Data Aalyi for Fiacial Egieerig, Spriger, 011. [15] E.L. Lehma ad G. Caella, Theory of Poit Etimatio, Secod Editio, Spriger, 1998.