Performance Evaluation Fan Longzhen Discussion Points ? How to measure returns? ? How to choose benchmark? ? How to adjust for risk? ? Performance attribution. ? Active return and risk. Why evaluate performance? ? If you manage your own portfolio, then you may possibly want to make alternations to your investment procedures. ? If someone else is managing your portfolio, you want to know if they are adding value. ? If not, perhaps you may have them manage a smaller amount of your money (or even replace them), or add constraints to what they want can do. How to measure returns ? The portfolio value is affected by cash inflow and outflows. How do we compute the portfolio return when portfolio size changes over time? ? In the following case, the portfolio has an ending value of $135, 000, and then an additional $50000 cash investment was made to the portfolio on January 5. What is the portfolio return in January. asset 31-Dec Before After 31-Jan. cash equivalents $5,000 4000 54000 7000 Bonds 10000 11000 11000 60650 stocks 120000 122700 122700 130000 total 135000 137700 187700 197650 January 5 Investment market value of portfolio on Time-weighted returns ? Compute the return for two periods (Dec 31-Jan 5 and Jan 5-Jan 31) ? Return on Dec 31-Jan 5: – ($137700-135000)/135000=2.0% ? Return on Jan 5-Jan 31 – ($197650-187700)/$187700=5.3% ? Time-weighted returns: – (1+2%)(1+5.3%)-1=7.41% Dollar-weighted return ? Dollar-weighted return: – Compute the internal rate of return that will equate the future value of the beginning value and cash inflows with the ending value Dollar-weighted return( r) =7.16% 31/26 )1(50000)1(135000197650 rr +++= ? Which return measure is better ? Dollar weighted return: – Affected by any cash investments or withdrawals to the portfolio during the period over which the return is calculated. – Measure the returns to the owner of the portfolio allowing the benefits or losses associated with cash distributions and withdrawals during the measurement period. ? Time-weighted return – Not affected by any cash investment or withdrawals to the portfolio during the period over which the return is calculated. – Measure returns on the securities held in a portfolio (or performance of the manager) How to choose Benchmark? ? Purpose of benchmark – Evaluate performance – Provide value weightings – Provide constraints to portfolio manager ? Benchmark portfolio – Usually a passive index or portfolio – Sometimes no appropriate single benchmark exists, so you build your own. Performance measure of your portfolio ? Any abnormal returns – Compare the return of the portfolio with a comparable market index and adjust for risk. ? Measure of abnormal returns – Jensen’s alpha – Treynor’s measure – Sharpe’s measure – Information ratio Performance measure of your portfolio ? Fix a particular time interval for which you are interested in assessing the performance of your portfolio; ? Find a market index as market portfolio; ? Determine the portfolio beta ? compute the average return of your portfolio (denoted by ), the average risk-free return , the average benchmark return over the interval you are assessing the performance of your portfolio. p β p ar f ar M ar Jensen’s alpha ? Obtain the benchmark return based on SML ? Determine the portfolio’s ex-post alpha, denoted by ? If then your portfolio beat the market; ? If , then your portfolio lost to the market. ? You may also compare with alpha of other portfolios. ? Alternatively, the benchmark for your portfolio can be based on the APT by estimating the sensitivities of your portfolio. pb ar , pfmfpb arararar β)( , ?+= bppp arar ?=α 0> p α 0< p α p α Jensen’s alpha ? During the 1994-1998 period, the average annual return on the S&P 500 was 24.1%, the average annual return on Fidelity Magllan was 20.5%, and the average annual riskfree return in 1994-1998 was 5%. The betas of Fidelity Magllan and Fidelity asset Manager are 1.0 and 0.6, respectively. Jensen’s alpha are as follows %6.3 0.1%)5%1.24(%5[%5.20 )([ ?= ×?+?= ?+?= MagellanfmfMagllanMagllean arararar βα %66.4 6.0%)5%1.24(%5[%8.11 )([ ?= ×?+?= ?+?= ManagerAssetfmfmanagerAssetManagerAsset arararar βα Treynor’s Reward-to-Beta Ratio ? Treynor’s reward-to-beta ratio (RVOLp)is given by ? Compare RVOLp with the reward-to-beta ratio of the market ( ), i,.e. the slope of the ex-post SML: – If , then your portfolio beat the market. – If , then your portfolio lost to the market pfpp ararRVOL β/)( ?= fMM ararRVOL ?= Mp RVOLRVOL > mp RVOLRVOL > mp RVOLRVOL < Treynor’s Reward-to-Beta Ratio ? Treynor’s reward-to-beta ration for the two mutual funds: %5.150.1/%)5%5.20( /)( =?= ?= MagellanfMagellanmagellan ararRVOL β %3.136.0/%)5%8.11( /)( =?= ?= ManagerAssetfManagerAssetMangerAsset ararRVOL β %1.190.1/%)5%1.24( )( =?= ?= fMM ararRVOL Sharpe’s Reward-to-Variability Ratio ? Sharpe’s ratio is based on ex-post capital market line ? Sharpe’s reward-to-variability ratio is given by ? Compare with the reward-to-variability ratio of the market , or the slope of the ex-post CML, that is, – If , then your portfolio beat the market; – If , then your portfolio lost to the market. pfpp ararRVAR σ/)( ?= MfMM ararRVAR σ/)( ?= p RVAR M RVAR Mp RVARRVAR > Mp RVARRVAR < Sharpe’s Reward-to-Variability Ratio ? During the 1994-1998 period, the standard deviation of the S&P 500 was 14%, the standard deviation of Magellan was 15%, and the standard deviation of the Asset Manager was 10%. The Sharpe’s reward-to-volatility ratio for the two mutual funds and the market portfolio as follows: 033.1 15.0/)05.0205.0( /)( = ?= ?= MagellanfMagellanMagellan ararRVAR σ 364.1 14.0/)05.0241.0( /)( = ?= ?= MfMM ararRVAR σ 68.0 10.0/)05.0118.0( /)( = ?= ?= managerassetfmanagerassetManagerAsset ararRVAR σ Information Ratio ? The information ratio adjusts for the amount of residual risk in the portfolio; ? The information ratio is to divide the Jensen’s alpha by the standard deviation of residual risk ? The information ratio is especially useful in the context of active portfolio, where measures the active return and ? measure active risk. p IR )(/ ppp IR εσα= p α )( p εσ Information Ratio ? Using the two mutual funds again, recall that standard deviation of the S&P 500 was 14%, the standard deviation of Magellan was 15%, and the standard deviation of Asset Manager was 10%. Also, the betas of the Fidelity Magellan and Fidelity Asset Manager are 1.0 and 0.6, respectively. The alphas of Fidelity Magellan and Fidelity Asset manager are –3.6% and –4.66%. ? To obtain the residual risk ? The information ratios for the two funds are: 222 )( mppp σβσεσ ?= %38.5 14.00.115.0)( 222 = ×?= Magellan εσ %43.5 14.06.010.0)( 222 . = ×?= MA εσ 86.0%43.5/%66.4 67.0%38.5/%6.3 .. ?=?= ?=?= MA Magellan IR IR Performance Attribution ? Possible explanation of superior performance – Allocation effect: allocation among different economic sectors – Selection effect: allocation between bonds and equities ? Why performance attribution – If you manage the portfolio yourself – If someone manage the portfolio for you Allocation VS selection effect ? Attribute abnormal returns to allocation effect and selection effect ? Allocation effect: – Over- or under weighting of the sectors within the portfolio – Selection effect: – Stock selection within the sectors ? weight of the ith sector in the actual portfolio; ? weight of the ith sector in the benchmark portfolio; ? return of the ith sector in the actual portfolio; ? return of the ith sector in the benchmark portfolio; ? total return to the benchmark portfolio; ∑ = ?×? N i bbibipi RRww 1 )]()[( )]([ 1 bipi N i pi RRw ?× ∑ = = pi w = bi w = pi R = bi R = b R Allocation vs. Selection effect ? The following are the returns of the benchmark and managed portfolio, with a breakdown by three economic sectors sector weight sector return product sector weight sector return product industrial 20% -10% -2% industrial 20% -10% -1% financial 50% 5% 2.50% financial 60% 8% 4.80% telecom 30% 10% 3.00% telecom 30% 12% 3.60% total 100% Bechmark re tur 3.50% total 100% Portfolio return 7.40% Benchmark portfolio Managed portfolio Allocation vs. Selection effect ? Allocation effect: AB CDE F=C*E Sector weight of portfolio weight of benchmark Deviation Sector return sector return-market return product Industrial 10% 20% -10% -10% /=-10%-3.5%=-13.5% 1.35% Financial 60% 50% 10% 5% 5%-3.5%=1.5% 0.15% Telecom 30% 30% 0% 10% 10.5%-3.5%=6.5% 0.00% Allocation vs. Selection effect ? Selection effect: AB CDE=A*D Sector weight of portfolio portfolio return Benchmark returnDifference product Industrial 10% -10% -10% 0% 0.00% Financial 60% 8% 5% 3% 1.80% Telecom 30% 12% 10% 2% 0.60% 2.40%selction effect ?Portfolio return= Benchmark return+allocation effect ? + selection effect Market Timing Skills ? Conventional measures of performance assess selectivity skill of managers ? If manager can time market: – Shifting mix across asset class – Sensitivity of portfolio returns to market returns, or betas, higher (lower) during up (down) markets ? Tests designed to capture nonlinearity of the relationship between portfolio and benchmark returns: – Treynor &Mazuy (1966) “quadratic model” – Hendrikkson & Merton (1981) “Swith-point regression model” Treynor-Mazuy Quadratic model ? TM extends the usual excess returns market model regression with a quadratic term: ? Timing skill if itftMtiftMtiiftit rRrRrR εγβα +?+?+=? 2 )()( 0> i γ Hendrikkson & Merton Swithpoint ? HM segment sample by excess market returns and for each estimate market model regressions ? Timing skill if itft down Mt down ift UP Mt UP iiftit rRrRrR εββα +?+?+=? )()( downUP ββ > Empirical Evidence ? Mutual fund managers do not seem to be able to beat a passively managed index; ? Also little empirical support in favor of managers having market timing ability. Active Risk under multiple factor Model (BARRA) ? Active return: Ractive=Rmanaged-Rbenchmark ?Where ? Ractive=active return; ? Rmanaged=return on management portfolio ? Suppose the active return follows a multiple factor model ? =active exposure of portfolio to factor I ? =factor return returnspecificfxfxfxR nnbbaaactive ++++= ... i x i f