A Note on a New Weighted Idiosyncratic Risk Measure

Transcription

1 Inernaional Journal of Financial Research Vol. 5, No. 3; 14 A Noe on a New Weighed Idiosyncraic Risk Measure Yin-Ching Jan 1 1 Deparmen of Disribuion Managemen, Naional Chin-Yi Universiy of echnology, aiwan Correspondence: Yin-Ching Jan, Deparmen of Disribuion Managemen, Naional Chin-Yi Universiy of echnology, aiwan. jan511@mail.ncu.edu.w Received: May 15, 14 Acceped: June 9, 14 Online Published: June 3, 14 doi:1.543/ijfr.v5n3p194 URL: hp://dx.doi.org/1.543/ijfr.v5n3p194 Absrac his noe remedies a risk measure, which was proposed by he work of Jan and Wang (1). hey used propery of maringale o measure idiosyncraic risk, and illusraed ha i is beer han he measuremens of variance and semivariance. However, heir risk measure can disinguish beween he asses whose reurn rising firsly and hen declining, and he asses whose reurn declining firsly and hen rising. In his noe, I propose a remedied mehod, which pus more weigh o he recen reurn s variaion, and demonsrae ha he new weighing risk measure is more close o he invesor risk concepion. Keywords: risk measure, maringale, idiosyncraic risk 1. Inroducion Only sysemaic risk is priced in he radiional asse pricing model. ha is because he idiosyncraic risk can be diversified away and herefore should no play any role in he risk premium. However, Meron (1987) saes ha idiosyncraic risk should be priced when invesors hold undiversified porfolios. When invesors don hold he diversified porfolios, he invesors porfolios reain he idiosyncraic risk. As a resul, risk premium should be compensaed for he idiosyncraic risk when invesors are risk averer. Many sudies have discovered ha idiosyncraic risk premium can help o explain expeced asse reurn. See, for example, he sudies of Malkiel and Xu s (1997), Malkiel and Xu (3), Goyal and Sana-Clara (3), Ang, e al. (6), and Eiling (13). Mos measuremens of he idiosyncraic risk go o he radiional variance and semivariance. Balzer (1995) and Nawrocko (1999) provide reviews of he risk measuremen and sugges he use of semivariance. However, he calculaion of semivariance needs benchmark o compue he downside deviaion. Differen benchmark conceived by differen invesor would resul in differen risk measures. Jan and Wang (1) propose an objecive benchmark, which is named as maringale variance, o obain a single objecive risk measure. hey sugges he use of previous reurn as benchmark insead of mean reurn o calculae he idiosyncraic risk. he use of previous reurn is based on he propery of maringale. Jan, e al. (13) apply he new risk measure o examine wheher idiosyncraic risk can play an imporan role in explaining he expeced reurn in aiwan sock marke. hey find ha idiosyncraic risk has a posiive relaion o expeced reurns for socks wih smaller bea porfolio. However, he new measure can disinguish beween he asses whose reurn rising firsly and hen declining, and he asses whose reurn declining firsly and hen rising. In his noe, I propose a remedied mehod, which pus more weigh o he recen reurn s variaion and demonsrae ha he new weighing risk measure is more close o he invesor risk concepion.. Weighed Maringale Variance he expecaion of nex period reurn is oday reurn by he propery of maringale. herefore, Jan and Wang (1) use previous reurn as benchmark o measure he maringale variance ( MVAR ). Specifically, i is calculaed as follows: Published by Sciedu Press 194 ISSN E-ISSN

2 Inernaional Journal of Financial Research Vol. 5, No. 3; 14 MVAR ( R) 1 ( R R 1 {( R E ( R) ) ( 1 { VAR ( R) COV ( R, R ) 1 1 R R )} 1 E ( R) where R denoes an asse reurn a ime, is sample size, and E () represens expecaion operaor. VAR and COV are convenional variance and covariance operaors, respecively. We see ha he maringale variance is compued by he variance minus auocorrelaion, and is differen from he work of Goyal and Sana-Clara (3) in he second erm. Jan and Wang (1) have provided an example o show he superioriy of he maringale variance han he radiional variance. he comparison is shown in Appendix A. he maringale variance can capure he ime series auocorrelaion paern, bu can disinguish he asse whose reurn are rising, and he reurn whose reurn are declining. See also he example presened in Appendix A. he problem can be solved by maringale semivariance ( MSAVR ), which can be wrien as: { Min (, R R )} 1 1 MSVAR ( R ) () where Min (, ) is minimum operaor. he maringale semivariance can be hough as relaive semivariance inroduced by Balzer (1995). Neverheless, he maringale uses pas reurn as he ime-varying reurn benchmark. Jan, e al. (13) have shown ha he superioriy of using maringale semivariance in explaining he expeced reurn in aiwan sock marke. Neverheless, he maringale semivariance can disinguish beween he asses whose reurn rising firsly and hen declining, and he asses whose reurn declining firsly and hen rising. ake wo ficiious asses E and F for example. (Noe 1) See Figure 1 as follows. )} (1) wo hypoheical asses E and F Rae of reurn E F Figure 1. wo hypoheical asses E and F We can see ha he asses E and F have he same measures of variance,semivariance, maringale variance, and maringale seimivariance. However, mos invesors would consider he asse E be riskier han he asse F, because he reurn of E is growing and hen declining, while he reurn of F is declining and hen growing. he separaion of hese wo asses risk measure can be reached by he combinaion of he maringale semivariance and he use of Published by Sciedu Press 195 ISSN E-ISSN

3 Inernaional Journal of Financial Research Vol. 5, No. 3; 14 weighed average mehod, which is called weighed maringale semivariance (WMSVAR ). ha is. WMSVAR ( R ) w { Min (, R R )} 1 1 (3) where w / x. x 1 (Noe ) he weighed average mehod can be adjused o reflec more closely more recen negaive reurn flucuaions. herefore, more weighs are assigned o he mos recen daa. he calculaions of hese wo asses weighed maringale semivariance are presened in Figure. I se 36 o calculae he risk measure in he oal 58 reurn daa. As a resul, we have moving measures of weighed maringale semivariance. Jus as shown in he Figure, we can see ha he risk measure of F is larger han hose of E a he ouse. However, he risk measures of F are declining, while hose of E are escalaing. he weighed maringale semivariance measures of E exceed hose of F evenually. When he invesors care more o he recen reurn flucuaion, he weighed mehod would be closed o he invesors risk measure han oher measures. Weighed maringale semivariance of wo hypoheical asses E and F.5 Weighed maringale semivariance E F Figure. Weighed maringale semivariance of wo hypoheical asses E and F 3. Conclusion Idiosyncraic risk has been proved is role in helping o explain expeced asse reurn. However, we can ge an objecive idiosyncraic risk measure wihou a common benchmark. Jan and Wang (1) provide an objecive risk measure, which is based on he propery of maringale. Neverheless, heir risk measure can disinguish beween he asses whose reurn rising firsly and hen declining, and he asses whose reurn declining firsly and hen rising. In his noe, I propose a remedied mehod, which pus more weigh o he recen reurn s variaion, and demonsrae ha he new weighing risk measure is more close o he invesor risk concepion. he new risk measure can be applied o improve he performance of porfolio selecion and evaluaion. I leave hese for furher research. Acknowledgemens I am graeful for commens o wo referees and edior. his research is funded by Naional Chin-Yi Universiy of echnology, NCU 14-R-MD-3. References Ang, A., Hodrick, R.J., Xing, Y., & Zhang, Y. (6). he cross-secion of volailiy and expeced reurns. Journal of Finance, 61(1), hp://dx.doi.org/1.1111/j x Balzer, L. (1994). Measuring Invesmen Risk: A Review. Journal of Invesing, 3(3), hp://dx.doi.org/1.395/joi Drew, M. E., Naughon., & Veeraraghavan, M. (4). Is idiosyncraic volailiy priced? Evidence from he Published by Sciedu Press 196 ISSN E-ISSN

4 Inernaional Journal of Financial Research Vol. 5, No. 3; 14 Shanghai Sock Exchange. Inernaional Review of Financial Analysis, 13, hp://dx.doi.org/1.116/j.irfa.4..7 Drew, M. E., & Veeraraghavan, M. (). Idiosyncraic volailiy and securiy reurns: Evidence from he Asian region. Inernaional Quarerly Journal of Finance,, Eiling E. (13). Indusry-specific human capial, idiosyncraic risk, and he cross-secion of expeced sock reurn. Journal of Finance, 68(1), hp://dx.doi.org/1.1111/j x Fu, F. (9). Idiosyncraic Risk and he Cross-Secion of Expeced Sock Reurns. Journal of Financial Economics, 91(1), hp://dx.doi.org/1.116/j.jfineco.8..3 Goyal, A., & Sana-Clara, P. (3). Idiosyncraic risk maers! Journal of Finance, 58(3), hp://dx.doi.org/1.1111/ Jan, Y-C., & Wang, J.M.C. (1). A Noe on Idiosyncraic Risk Measure. Journal of Money, Invesmen and Banking, 5, Jan, Y-C., Chiu, S-L., & Wang, J.M.C. (13). New Risk Measure and Idiosyncraic Risk in aiwan Sock Marke. Inernaional Journal of Financial Research, 4(), hp://dx.doi.org/1.543/ijfr.v4np77 Malkiel, B., & Xu, Y. (1997). Risk and reurn revisied. Journal of Porfolio Managemen, 3(3), hp://dx.doi.org/1.395/jpm Nawrocko, D. (1999). A brief hisory of downside risk measures. Journal of Invesing, 8(3), 9-5. hp://dx.doi.org/1.395/joi Xu, Y., & Malkiel, B. (4). Idiosyncraic risk and securiy reurns, AFA 1 New Orleans Meeings. Rerieved from SSRN:hp://ssrn.com/absrac=5533 Noes Noe 1. hese wo asses series reurn can be found in Appendix B. Noe. Oher weighed mehod can be used insead, such as mehod of exponenial smoohing. Appendix A In his Appendix, I follow he work of Jan and Wang (1) o illusrae he comparison of some risk measures. Firsly, consider wo hypoheical asses whose reurns (A and B) are shown in Figure A1. We can see ha he variance and semivariance of A and B are equal. However, mos invesors would hink he asse B riskier han he asse A, because he reurn paern of B is more volaile han A. he more risk of asse A han asse B can be capured by he use of maringale variance. We have seen ha he auocorrelaion feaure has been included in he calculaion of maringale variance. wo hypoheical asses A and B.1 Rae of reurn A B -.1 Figure A1. wo hypoheical asses A and B Published by Sciedu Press 197 ISSN E-ISSN

5 Inernaional Journal of Financial Research Vol. 5, No. 3; 14 However, he maringale variance may no be an adequae risk measure. ake wo ficiious asses C and D for example. See Figure A as follows. wo hypoheical asses C and D.1 Rae of reurn C D -.1 Figure A. wo hypoheical asses C and D he reurn of asse C is growing, while asse D is declining. As a resul, mos invesors would consider D more risky han C. However, he variance, semivariance, and maringale variance of C are he same as hose of D. he problem in he comparison of his wo asses C and D can be disinguished by he use of maringale semivariance. Appendix B he ime-series reurns of wo hypoheical asses E and F are presened as follows. E F Published by Sciedu Press 198 ISSN E-ISSN