Statistical Inference for Risk-Adjusted Performance Measure. Miranda Lam

Transcription

1 Statstcal Inference for Rsk-Adjusted Performance Measure Mranda Lam Abstract Ths paper examnes the statstcal propertes of and sgnfcance tests for a popular rsk-adjusted performance measure, the M-squared measure. Test statstcs along wth asymptotc moments and probablty dstrbutons developed by pror studes are revewed and ther applcatons to the M- squared measure are dscussed. Ths paper demonstrates through an example that calculaton of the analytc test statstc and the correspondng p-value can be performed easly usng a spreadsheet or mathematcal software. Results of sgnfcance tests usng bootstrappng are smlar to those based on the analytc solutons, suggestng that the analytc soluton approach s a useful method for conductng statstcal nference. The results appear relatvely robust even n small samples. Keywords: Performance Evaluaton, Hypothess Testng, Mutual Funds JEL Classfcaton: C12, G12 Salem State College, voce : , e-mal: mlam@salemstate.edu 27 Electronc copy avalable at:

2 M. Lam Statstcal Inference for Rsk Adjusted Performance Measure Fronters n Fnance and Economcs Vol. 5 No1 Aprl 2008, Statstcal Inference for Rsk-Adjusted Performance Measure 1 - Introducton Measurng the performance of managed mutual funds s an mportant topc of nterest to a wde audence, ncludng economsts, fnancal analysts, nvestors, and legslators. In 1995, the Securty and Exchange Commsson (SEC) receved over 3,500 responses as a result of the Rsk Concept Release (Investment Company Act Release No ) requestng comments on mprovng the descrpton of rsk to nvestors by mutual fund companes. The enthusasm of the responses was matched by ther dversty and the SEC decded not to adopt a quanttatve measure of rsk or rsk-adjusted performance at the current tme. One mportant objecton based on a survey by the Investment Company Insttute (ICI) was that quanttatve rsk measures have a strong potental to confuse or mslead nvestors. (ICI 1996). Modglan and Modglan (1997), motvated by the need to create an easy-to-understand rsk-adjusted performance measure, propose the M-square Measure as an ntutve, rgorous, and flexble alternatve to tradtonal measures based on the Captal Asset Prcng Model. The M-squared measure has ganed steady popularty snce ts ntroducton, and strves as ts goal to nform and not confuse nvestors. In practce, the M-squared measure reles on hstorc returns as nput data and s subject to typcal statstcal nose of hstorc samplng. There are two mportant prncples for evaluatng past performance: the frst s comparng apples wth apples,.e. rskadjustment, and the second s ensurng that the phenomenon s not spurous,.e. statstcal sgnfcance. Whereas the frst prncple, rskadjusted performance, has receved wde-spread attenton, the second prncple, ascertanng statstcal sgnfcance s less understood and often gnored n practce. The purpose of ths paper s to explore the statstcal propertes of and to demonstrate applyng sgnfcance tests for the M-squared measure. We show that under the null hypothess of zero performance, the M-squared measure can be transformed nto a test statstc developed by Jobson and Korke (1981). We present the asymptotc mean and standard devaton for ths test statstc. We also 28 Electronc copy avalable at:

3 M. Lam Statstcal Inference for Rsk Adjusted Performance Measure Fronters n Fnance and Economcs Vol. 5 No1 Aprl 2008, llustrate how to perform statstcal test usng the test statstc for the M-square measure through an example contanng seven mutual funds. Lastly, we conduct the same statstcal tests usng the bootstrappng method. Results from the bootstrappng method are very smlar to those based on the asymptotc analytc solutons of the test statstc. For robustness check, we mplement the same tests usng sub-samples of varyng sample szes and fnd that the analytc approach s farly robust. Ths s an mportant fndng because the analytc approach can be easly appled usng spreadsheet or mathematcal software. Ths paper s organzed as follows. Secton 2 presents the M- squared measure and ts test statstc wth asymptotc frst and second moments. Secton 3 presents an applcaton of sgnfcance tests for the M-squared measure and compares results of sgnfcance tests based on the asymptotc dstrbuton to the bootstrappng method. Selected results from sub-samples of dfferent sample szes are also presented. Secton 4 provdes a bref summary and conclusons. 2 - Statstcal Propertes of Rsk-adjusted Performance Measures Modglan and Modglan (1997) ntroduced the Rsk-adjusted performance measure (RAP) for mutual funds, usng a benchmark portfolo as the reference pont for rsk. The term benchmark wll be used n the remander of the paper to denote the unmanaged dversfed portfolo aganst whch the performance of the mutual funds wll be evaluated. Most mportantly, the rsk and return of the benchmark portfolo serve as a reference for the market opportunty cost of rsk. The RAP s defned as M M RAP = rf σ + σ μ σ 1 (1) σ where σ M = standard devaton of return on the benchmark portfolo; σ = standard devaton of return on the mutual fund; μ = mean return on the mutual fund; r F = rsk-free nterest rate; The RAP s, n effect, a complete portfolo contanng the mutual fund and the rsk-free asset and ths complete portfolo has the same standard 29

4 M. Lam Statstcal Inference for Rsk Adjusted Performance Measure Fronters n Fnance and Economcs Vol. 5 No1 Aprl 2008, devaton as the benchmark portfolo. A popular transformaton of the RAP, dubbed the M-squared measure, M, s expressed as the dfference between the RAP and return on the benchmark portfolo (Bode, Kane, Marcus, 2001): M = RAP - μ M (2) where RAP s the rsk-adjusted performance defned n (1) and μ M s the mean return on the benchmark portfolo. The Sharpe rato (Sharpe 1966) s closely related to the M-squared measure and the RAP n that they all use standard devaton as the measure of rsk. The advantage of the M-squared measure over the Sharpe rato s that the average nvestor s more famlar wth bass ponts than wth ratos. Graham and Harvey (1997) ntroduced a performance measure (GH2) that s also based on adjustng the standard devaton of a complete portfolo of T-bll and the rsky asset beng evaluated to match the S&P500 (benchmark portfolo). The GH2 measure allows returns on the T-bll to have non-zero varance and covarance wth other rsky assets. The GH2 measure s a more generalzed form that ncludes the M-square measure as a specal case. Graham and Harvey (1997) noted that for well-dversfed mutual funds where substantal leverage s not needed to match the benchmark portfolo s volatlty, the assumpton of a rsk-free asset by the M-squared measure s not an mportant ssue. Thus, the M-square measure remans a useful performance evaluaton tool for stock mutual funds. The followng secton explans how statstcal nference based on parametrc testng can be conducted for the M-square measure. The statstcal property of the GH2 measure s beyond the scope of ths paper. In practce, the true values of the mean and standard devaton of returns on the mutual fund and on the benchmark portfolo are unknown. When sample means and sample standard devatons are used n place of the true values, the results are estmators of the RAP and the M-squared measure: s RAP = r + 1- s M M rf (3) s s and M = RAP r M (4) 30

5 M. Lam Statstcal Inference for Rsk Adjusted Performance Measure Fronters n Fnance and Economcs Vol. 5 No1 Aprl 2008, where 1 r = T t 1 r M = T s = T = 1 T r t= 1 t r 1 T 1 Mt = sample mean of returns on mutual fund ; = sample mean of returns on the benchmark; T ( r r ) t t= 1 2 = sample standard devaton of returns on mutual fund ; T 1 2 sm = ( rmt rm ) = sample standard devaton of T 1 t= 1 returns on the benchmark; For sgnfcance testng, the null hypothess for the M-squared measure s 1 H O : M = 0. (5) A prme canddate for the test statstc s the sample estmator for the M-squared measure, M. The asymptotc statstcal propertes, ncludng the frst and second moments, of the estmator, M, can be derved drectly usng Taylor seres expansons. Expandng the terms of RAP, equaton (4) becomes sm M = ( r rf ) ( rm rf ) (6) s Snce the rsk-free rate s a constant, the statstcal propertes of M depend only on the sample means and sample standard devatons of the mutual fund and the benchmark. Under the null hypothess that M = 0, equaton (6) can be rewrtten as ( ) ( ) ' M F M F M = s r r s r r (7) 1 The null hypothess for testng the RAP s H 0 : RAP = μ M. Snce ths equaton s a lnear transformaton of (6) and (6) s a more famlar format n hypothess testng, the dscusson n the paper focuses on dervng statstcal tests for (6). Dervatons of the test statstc for RAP and emprcal results for RAP are avalable from the author. 31

6 M. Lam Statstcal Inference for Rsk Adjusted Performance Measure Fronters n Fnance and Economcs Vol. 5 No1 Aprl 2008, Jobson and Korke (1981) derve a test statstc, whch s dentcal n form to (7), to test the dfference between two Sharpe ratos. 2 Followng Jobson and Korke (1981), the asymptotc dstrbuton of ' M s normal wth mean M ' and varance ϖ 2,.e. where and M ' ~ N(M ', ϖ 2 ) (8) M ' = σ M (μ r F ) - σ (μ M r F ) (9) μμ M ϖ = 2 σσm 2 σσmσm + μσm + μmσ T σσ M ( σ σ σ ) M M 2 (10) To test the null hypothess, H O : M = 0, a standardzed test statstc, MT, s formed by substtutng sample estmators for the means, varances, and covarances n (8), (9) and (10): ' M MT = ϖ. (11) The test statstc, MT, has an asymptotc standard normal dstrbuton. ' The sample estmator for M s gven by (7) and the sample estmator 2 for ϖ s 2 1 M ϖ M M M M M ( M M ) T ss 2 2 ss s 1 rs r 2 s 2 1 rr 2 = s s s ss M (12) where r, r M,s, and s M are defned under (4) and s M s the sample covarance between returns on mutual fund and returns on the benchmark: 3 T 1 s = ( )( r r r r ) (13) T 1 1 M t Mt M t= 2 Jobson and Korke (1981) derve test statstcs to compare two or more Sharpe ratos. They explore the statstcal propertes of several transformatons of the dfference between two Sharpe ratos. They present n ther paper the most successful transformaton, whch s dentcal to (4)'. 3 ' ' Note that M s a based estmator of M, E( M ) = M (1 ¼ T /32 T -2 ), and the bas to O(T -2 ) s M ( ¼ T /32 T -2 ). See Jobson and Korke (1981) for a dscusson on the behavor of the estmator for the varance, ϖ 2,based on smulaton experments. 32

7 M. Lam Statstcal Inference for Rsk Adjusted Performance Measure Fronters n Fnance and Economcs Vol. 5 No1 Aprl 2008, The next secton presents an example to demonstrate the applcaton of sgnfcance tests for the M-squared measure. Results obtaned usng the bootstrappng method are compared to results from the analytc test statstcs presented n ths secton. 3 Estmaton and Sgnfcance Tests for the M-Squared Measure Data The mutual fund sample conssts of the seven funds from the Modglann and Modglann (1990) study. The sample perod s from January 1988 through Aprl 2002, resultng n 172 monthly observatons. 4 These funds represent sx dfferent categores, ncludng Large Growth, Large Value, Large Blend, Md-cap Growth, Small Growth, and Domestc Hybrd. Two benchmarks, the S&P 500 Index, the most popular benchmark for performance evaluaton, and the Russell 3000 Index, whch ncludes the 3000 largest U.S. companes by market captalzaton, are studed. Returns on 3-month Treasury Blls are used as proxy for rsk-free returns. In practce, standard devaton of returns on a 3-month Treasury Bll s not zero. However, for the purpose of evaluatng performance, snce the same rsk-free return s subtracted from all returns, mutual funds and benchmarks, we follow conventonal treatment and use return on a 3-month Treasury Bll as proxy for the rsk-free rate. Returns on the mutual funds, benchmarks, and Treasury Blls are computed as: r,t = (P,t P,t-1 ) / P,t-1 (14) where P,t s the closng prce for month t. Monthly prces, adjusted for dvdends and captal gans dstrbuton, for mutual funds and closng values for benchmarks are obtaned from Prces for 3-month treasury blls are computed as: P t = x (1 T-Bll Rate t x 90 / 360) (15) where T-Bll Rate t s the dscount rate of 3-month Treasury Bll traded n the secondary market n month t. Data on T-Bll Rates are from the Federal Reserve Board of Governors. Excess monthly return, defned as the dfference between total monthly return and return on the T-bll, s used n the followng emprcal calculatons. 4 Data for the Amercan Income Fund of Amerca s only avalable begnnng May Hence the sample perod for ths fund s from May 1996 through Aprl

8 M. Lam Statstcal Inference for Rsk Adjusted Performance Measure Fronters n Fnance and Economcs Vol. 5 No1 Aprl 2008, Summary statstcs n Exhbt 1 show that average excess monthly returns on the mutual funds range from % to % and average excess returns on the S&P 500 Index s % and on the Russell 3000 Index s %. Sample standard devatons for the mutual funds range between % to %, and sample standard devaton for the S&P 500 Index s % and for the Russell 3000 Index s %. Returns on all mutual funds are postvely correlated wth returns on the benchmarks. 5 Wth the excepton of 2 funds, hgher sample standard devatons are assocated wth hgher average monthly returns. An mportant concern to nvestors s whether the hgher average return represents approprate compensaton for the hgher rsk. Currently, the SEC requres mutual funds to present rsk/return data n a bar chart format. The followng except from the SEC s Fnal Release explans the ratonale for adoptng the Bar Chart n the presentaton standard for mutual funds. The proposed rsk/return summary would requre a fund s profle to nclude a bar chart showng the fund's annual returns for each of the last 10 calendar years and a table comparng the fund's average annual returns for the last 1-, 5-, and 10-fscal years to those of a broadbased securtes market ndex. The bar chart reflects the Commsson s determnaton that nvestors need mproved dsclosure about the rsks of nvestng n a fund. The bar chart s ntended to llustrate graphcally the varablty of a fund s returns (e.g., whether a fund's annual returns for a 10-year perod have vared sgnfcantly from year to year or were relatvely even over the perod). (SEC 1998) The SEC mandated bar chart apples only to hstorc returns on the mutual fund and does not requre the bar chart to contan returns on the benchmark. Fgure 1 ncludes returns on both the mutual fund and the 5 The two benchmarks, the S&P 500 Index and the Russell 3000 Index, are hghly correlated wth a correlaton coeffcent of Emprcal results for performance and statstcal tests are very smlar usng both benchmarks. In the nterest of brevty, only results usng the S&P500 Index are reported n the paper. Results usng the Russell 3000 Index are avalable from the author. 34

9 M. Lam Statstcal Inference for Rsk Adjusted Performance Measure Fronters n Fnance and Economcs Vol. 5 No1 Aprl 2008, benchmark. Exhbt 2 contans such a bar chart usng annual returns for 10 years from the Am Constellaton fund. Whle a bar chart shows the varablty of annual returns from year to year, t does not address the queston of whether the reward, n the form of average returns, s adequate compensaton for the rsk. Ths queston s partcularly mportant for nvestors who desre nvestments wth dfferent rsk level than the benchmark. A bar chart, currently requred by the SEC, does not convey explctly the drect trade-off between rsk and return. Rsk-Adjusted Performance Exhbt 3 shows the estmated values for several rsk-adjusted performance measures, ncludng the Sharpe rato, the Rsk-Adjusted Performance and the M-squared Measure for the sample mutual funds aganst the S&P 500 Index as the benchmark. The estmated Sharpe rato for the S&P 500 Index durng the sample perod s and ts average excess return s %. The Rsk-adjusted Performance (RAP) measure of a mutual fund represents an nvestment strategy combnng the mutual fund and a 3-month Treasury Bll to acheve the same standard devaton as the S&P 500 Index. A mutual fund wth a hgher Sharpe rato or a hgher RAP relatve to those of the benchmark s consdered to have superor performance. Note that the RAP and the M-squared measure are consstent wth the Sharpe rato,.e. a fund demonstratng superor performance usng the RAP or the M-squared measure wll also demonstrate superor performance usng the Sharpe rato as the evaluaton crtera. The M-squared measure s the dfference between a mutual fund s RAP and the average excess return on the benchmark. The M-squared measure gves the absolute advantage, n bass ponts, of the nvestment strategy of holdng a portfolo of the mutual fund and a 3-month Treasury Bll over nvestng n the benchmark alone. It s a rsk-adjusted representaton of performance because both strateges have the same standard devaton. In the sample, 4 funds have negatve M-squared measures and 3 funds have postve M-squared measures and the values range from % to %. Snce these values are computed usng sample statstcs, sgnfcance tests are needed to ascertan whether the fndngs represent true measure of performance or the results are spurous due to 35

10 M. Lam Statstcal Inference for Rsk Adjusted Performance Measure Fronters n Fnance and Economcs Vol. 5 No1 Aprl 2008, nose n the sample. The next secton dscusses a smple approach to determne statstcal sgnfcance for the estmated M-squared measure. Statstcal Test for Rsk-adjusted Performance Snce the M-squared measure s computed usng hstorc data, t s a sample estmator and proper nterpretaton of performance must nclude statstcal nference. Equaton (11) presented a test statstc, MT, for testng the null hypothess of zero rsk-adjusted performance. In ths secton, two methods are used to compute the p-value for MT. The frst method evaluates the asymptotc mean and varance of MT usng sample estmators as outlned n equatons (7) and (12). We refer to ths method as the analytcal method and present ts results n Panel A of Exhbt 4. The p-values range from to Out of the seven funds n the example, only the performance of the Magellan Fund s statstcally sgnfcant. Performance measures for the remanng sx funds are statstcally ndstngushable from zero. The second method to compute p-value for the test statstc employs a bootstrappng procedure and does not requre explct evaluaton of the mean and varance of the test statstc, MT. The bootstrap procedure generates 1000 repettons of random samplng, wth replacement, from the orgnal sample of 172 monthly observatons. In each repetton, 172 data ponts are selected randomly for each mutual fund and the benchmark. Sample mean and standard devaton, computed n each teraton, are used to calculate the test statstc MT. The bootstrappng method produces 1000 estmates of MT and the p-value for the mean value of these 1000 estmates s presented n Panel B of Exhbt 4. Results from the bootstrappng procedures are vrtually dentcal to those obtaned usng the asymptotc moments. Ths s good news to practtoners and nvestors because computng the test statstcs, MT, usng the asymptotc moments does not requre sophstcated software and snce MT has a standard normal dstrbuton, ts p-value s readly avalable. 6 6 For example, the Sharpe ratos, RAPs, and the M-squared measures n Table 2 and the asymptotc moments of the test statstcs, MT, and the p-values n Panel A of Table 3 are computed usng MS Excel. The bootstrappng procedure, whch produces the results n Panel B of Table 3, s conducted usng SAS and requres consderably more work. 36

11 M. Lam Statstcal Inference for Rsk Adjusted Performance Measure Fronters n Fnance and Economcs Vol. 5 No1 Aprl 2008, Exhbt 5 presents the M-squared measure and ther p-values computed usng the analytcal approach and the bootstrap procedure. Smlar to fndngs for the test statstc, MT, estmates for the M- squared measures usng both methods are very smlar. P-values, n parenthess, are ndcated next to the M-squared measures to gve a complete pcture of both economc and statstcal sgnfcance of performance. Calculaton of the p-values for the M-squared measures usng the analytcal approach s dentcal to the calculaton of p-value for the test statstc MT because under the null hypothess, MT s a lnear transformaton of the M-squared measure. The bootstrap estmaton for the M-squared p-value also employs 1000 repettons of random samplng, wth replacement, from the orgnal sample of 172 monthly observatons. In each repetton, 172 data ponts are selected randomly for each mutual fund and the benchmark. Sample mean and standard devaton are computed n each teraton and are used to calculate the M-squared measure. The bootstrappng method produces 1000 estmates of M-squared measure for each mutual and the p-value for the mean value of these 1000 estmates s presented n Panel B of Exhbt 5. Out of the sample of 7 funds, only one fund, Fdelty Magellan, has sgnfcant postve performance wth a M-squared measure of %. Snce Magellan has a hgher standard devaton, %, than the S&P 500 Index, %, the M-squared measure represents the gan of a strategy that nvests 93% n the Magellan fund and 7% n a 3-month T-Bll. The M-squared measures of the other 6 funds are not statstcally sgnfcant even though the estmated values can be qute large. Ths example demonstrates the mportance of ncludng sgnfcance test when presentng rsk-adjusted performance nformaton. Sample Sze and Robustness The analytcal approach s based on the asymptotc statstcal propertes of the test statstc and t s mportant to examne robustness, especally n smaller samples. 7 We create fve sub-samples from the orgnal sample of 172 monthly observatons by random selecton 7 We thank partcpants at the 2002 FMA conference for suggestng the examnaton of robustness n smaller samples. 37

12 M. Lam Statstcal Inference for Rsk Adjusted Performance Measure Fronters n Fnance and Economcs Vol. 5 No1 Aprl 2008, stratfed across years. The samples szes are 24, 36, 48, 60, and 120 monthly observatons. For each sample sze, we repeated the statstcal tests descrbed n the prevous secton. 8 Results n Exhbt 6 show that nference based on p-values computed usng the analytcal approach s smlar to those based on bootstrap estmaton. Panels A and B present the results for the M-squared measure and Panels C and D present the results for the test statstc, MT. For example, n Panels A and B, the analytcal approach generated a p-value of whle the bootstrap estmaton generated a p-value of for the sub-sample wth 36 observatons. Both methods wll lead to not rejectng the null hypothess and conclude that the M-squared measure s not statstcally dfferent from zero. In fact, the two methods lead to the same concluson for four of the fve sub-samples examned, suggestng that the analytcal approach s qute robust even n small samples. 4 Summary and Conclusons Ths paper llustrates that the two mportant prncples of performance evaluaton, rsk-adjustment and sgnfcance testng, can be easly mplemented. We present a test statstc developed by Jobson and Korke (1981) and demonstrate that ths test statstc can be used to perform sgnfcance test for the M-squared measure. Two approaches are used to compute the p-value for the test statstc. The frst approach uses the analytcal solutons to the asymptotc moments of the test statstcs wth values from sample estmators. The second approach uses bootstrappng. P-values of the M-squared measures for the sample funds obtaned usng both approaches are very smlar and the results are relatvely robust even n small samples. Ths s good news for the nvestors and fnancal advsors because the analytcal approach can be appled easly wthout sophstcated software or mathematcal programmng. 8 In the nterest of brevty, only results for the Magellan Fund are reported for the sub-sample robustness check. 38

13 M. Lam Statstcal Inference for Rsk Adjusted Performance Measure Fronters n Fnance and Economcs Vol. 5 No1 Aprl 2008, References Bode, Kane, and Marcus, 2001, Essentals of Investments, 4 th Edton, New York: McGraw Hll, pp Graham, J. and Harvey C., 1997, Gradng the Performance of Market Tmng Newsletters, wth John Graham, Fnancal Analysts Journal 53, 6, 1997, pp Gbbons, M., Ross, S., and Shanken, J., 1989, A Test of the Effcency of a Gven Portfolo, Econometrca, September 1989, pp Investment Company Act Release No (Mar. 29, 1995) [60 FR 17172], Rsk Concept Release. Investment Company Insttute, 1996, Shareholder Assessment of Rsk Dsclosure Methods, Washngton DC: Investment Company Insttute. Jensen, M., 1968, The Performance of Mutual Funds n the Perod , Journal of Fnance 23, May 1968, pp Jobson, J., and Korke B., 1981, Performance Hypothess Testng wth the Sharpe and Treynor Measures, The Journal of Fnance, September 1981, pp Jobson, J., and Korke B., 1989, A Performance Interpretaton of Multvarate Tests of Asset Set Intersecton, Spannng, and Mean-varance Effcency, Journal of Fnancal and Quanttatve Analyss 24/2, June 1989, pp Judge, G., Hll, R., Grffths, W., Lutkepohl H, and Lee, T., 1988, Introducton to the Theory and Practce of Econometrcs, New York: John Wley & Sons. Kandel, S. and Stambaugh R., 1989, A Mean-varance Framework for Tests of Asset Prcng Models, The Revew of Fnancal Studes, 2/2 1989, pp Kennedy, P., 1998, A Gud to Econometrcs 4 th Edton, Cambrdge, MA: the MIT Press. Modglan, F., and L. Modglan, 1997, Rsk-Adjusted Performance, The Journal of Portfolo Management, Wnter 1997, pp Sharpe, W., 1966, Mutual Fund Perfomance, Journal of Busness A Supplement, January 1966, pp Treynor, J., 1965, How to Rate Management of Investment Funds, Harvard Busness Revew 43, January-February 1965, pp

14 M. Lam Statstcal Inference for Rsk Adjusted Performance Measure Fronters n Fnance and Economcs Vol. 5 No1 Aprl 2008, Exhbt 1 Summary Statstcs The sample perod s from January 1988 through Aprl Monthly NAVs, dvdends and captal gan dstrbutons for the mutual funds and closng values for the S&P 500 ndex are obtaned from fnance.yahoo.com. Mutual Fund categores, expense rato and total assets as of December 2001 are obtaned from Monthly return s computed as r,t = (NAV,t + D,t NAV,t-1 ) / NAV,t-1 ; excess return s the dfference between monthly return and return on a 3-month Treasury Bll, R,t = r,t r f,t. Average Monthly Standard Mutual Expense Total Assets Excess Return (%) Devaton (%) Fund Category Rato (Mls) 3-month U.S. Treasury Blls N/A N/A N/A S&P 500 Index N/A N/A N/A Russell 3000 Index N/A N/A N/A Am Constellaton A Large Growth ,000 Amercan Century Vsta Investors Md-cap Growth ,207 T. Rowe Prce New Horzons Small Growth ,622 Fdelty Magellan Large Blend ,893 Vanguard Wndsor Large Value ,592 Fdelty Purtan Domestc Hybrd ,363 Amercan Income Fund of Amerca A Domestc Hybrd ,949 Correlaton between the S&P 500 and the Russell 3000 Index Correlaton Coeffcent Matrx S&P500 CSGTX TWCVX PRNHX FMAGX VWNDX FPURX AMECX S&P 500 Index Am Constellaton A (CSGTX) Amercan Century Vsta Investors (TWCVX) T. Rowe Prce New Horzons (PRNHX) Fdelty Magellan (FMAGX) Vanguard Wndsor (VWNDX) Fdelty Purtan (FPURX) Amercan Income Fund of Amerca (AMECX)

15 M. Lam Statstcal Inference for Rsk Adjusted Performance Measure Fronters n Fnance and Economcs Vol. 5 No1 Aprl 2008, Exhbt 2 Bar Chart for Am Constellaton 40% 30% 20% Annual Return 10% 0% % -20% -30% Year 41

16 Exhbt 3 Rsk-adjusted Performance The sample perod s from January 1988 through Aprl Monthly NAVs, dvdends and captal gan dstrbutons for the mutual funds and closng values for the S&P 500 ndex are obtaned from fnance.yahoo.com. Monthly return s computed as r,t = (NAV,t + D,t NAV,t-1 ) / NAV,t-1 ; excess return s the dfference between monthly return and return on a 3-month Treasury Bll, R,t = r,t r f,t ; sample mean of monthly excess return s R = T -1 Σ R t ; sample standard devaton of monthly excess return s s = ((T-1) - 1 Σ(R t R ) 2 ) ½ ; sample Sharpe rato s SH = RAP = (s M /s ) R. M-squared measures computed usng sample statstcs s R /s ; sample Rsk-Adjusted Performance n excess return s M = (s M /s ) R - R M. Sharpe Ratos RAP (n %) M-square measures (n %) Benchmark: S&P 500 Index Mutual Funds Am Constellaton A Amercan Century Vsta Investors T. Rowe Prce New Horzons Fdelty Magellan Vanguard Wndsor Fdelty Purtan Amercan Income Fund of Amerca A Sample perod for the Amercan Income Fund of Amerca s from May 1996 through Aprl The Sharpe rato for the S&P 500 Index durng the same perod was and the RAP (n %) for the S&P 500 Index n the same perod was

17 Exhbt 4 Results for the Test Statstc, MT, Calculated Usng the Analytcal Approach and the Bootstrap Approach The sample perod s from January 1988 through Aprl Monthly NAVs, dvdends and captal gan dstrbutons for the mutual funds and closng values for the S&P 500 ndex are obtaned from fnance.yahoo.com. Monthly return s computed as r,t = (NAV,t + D,t NAV,t-1 ) / NAV,t-1 ; excess return s the dfference between monthly return and return on a 3-month Treasury Bll, R,t = r,t r f,t. All values are n 10-3 except for p-value. Panel A: Analytcal Approach Usng Sample Moments 1 Panel B: Bootstrap Estmaton 2 Sample Bas (to Standard Sample Standard Mutual Funds Mean order T -2 ) Error p-value Mean Bas Errorp- value Am Constellaton A Amercan Century Vsta Investors T. Rowe Prce New Horzons Fdelty Magellan * * Vanguard Wndsor Fdelty Purtan Amercan Income Fund of Amerca A The asymptotc mean and varance of MT are gven n equatons (9) and (10) respectvely and the estmators for the asymptotc moments usng samples moments are gven n equatons (7) and (12). 2 Bootstrap estmaton s performed based on 1000 repettons of random samplng wth replacement from the orgnal sample of 174 monthly observatons. 3 Sample perod for the Amercan Income Fund of Amerca s from May 1996 through Aprl * Statstcally Sgnfcant at 5%. 43

18 Exhbt 5 M-squared Measures and Statstcal Sgnfcance Tests The sample perod s from January 1988 through Aprl Monthly NAVs, dvdends and captal gan dstrbutons for the mutual funds and closng values for the S&P 500 ndex are obtaned from fnance.yahoo.com. Monthly return s computed as r,t = (NAV,t + D,t NAV,t-1 ) / NAV,t-1 ; excess return s the dfference between monthly return and return on a 3-month Treasury Bll, R,t = r,t r f,t. Panel A: Analytcal Approach Usng Sample Moments Panel B: Bootstrap Estmaton M-square M-square Mutual Funds Measure (%) p-value Measure (%) p-value Am Constellaton A Amercan Century Vsta Investors T. Rowe Prce New Horzons Fdelty Magellan * * Vanguard Wndsor Fdelty Purtan Amercan Income Fund of Amerca A Sample perod for the Amercan Income Fund of Amerca s from May 1996 through Aprl * Statstcally Sgnfcant at 5%. 44

19 Exhbt 6 Robustness n Small Samples (Fdelty Magellan Fund) The sample perod s from January 1988 through Aprl Sub-samples are created by random selecton stratfed across years. Monthly NAVs, dvdends and captal gan dstrbutons for the mutual funds and closng values for the S&P 500 ndex are obtaned from fnance.yahoo.com. Monthly return s computed as r,t = (NAV,t + D,t NAV,t-1 ) / NAV,t-1 ; excess return s the dfference between monthly return and return on a 3- month Treasury Bll, R,t = r,t r f,t. Panel A: Analytcal Approach Panel B: Bootstrap Estmaton M-square M-square Sample Sze Measure (%) p-value Measure (%) p-value Panel C: Analytcal Approach Panel D: Bootstrap Estmaton Sample Bas (to Standard Sample Standard Sample Sze Mean of MT order T -2 ) Error p-value Mean of MT Bas Error p-value