LECTURE 10 : APPLICATION OF LINEAR FACTOR MODELS (Asset Pricing and Portfolio Theory)

Contents Mutual fund industry Mutual fund industry Measuring performance of mutual funds (risk adjusted rate of return) Measuring performance of mutual funds (risk adjusted rate of return)  Jensen’s alpha Using factor models to measure fund performance due luck or skill Active vs passive fund management Active vs passive fund management

Newspaper Comments The Sunday Times The Sunday Times ‘Nine out of ten funds underperform’ The Sunday Times The Sunday Times ‘Funds take half your growth in fees’

Introduction Diversification in practice : invest in different mutual funds, with different asset classes (e.g. bonds, equity), different investment objectives (e.g. income, growth funds) and different geographic regions. Diversification in practice : invest in different mutual funds, with different asset classes (e.g. bonds, equity), different investment objectives (e.g. income, growth funds) and different geographic regions. Should we buy actively managed funds or index trackers ? Should we buy actively managed funds or index trackers ? Assets under management Assets under management –US mutual fund industry : over $ 5.5 trillion (2000), with $ 3 trillion in equity funds Number of funds Number of funds –US : 393 funds in 1975, 2424 in 1995 (main US database) –UK : 1167 funds in 1996, 2222 in 2001 (yearbook)

UK Unit Trust Industry Number of FundsAssets under Management

Classification of Unit Trusts - UK Income Funds (7 subgroups) Income Funds (7 subgroups) UK Corporate Bonds (74 funds) UK Corporate Bonds (74 funds) Global Bonds (52 funds) Global Bonds (52 funds) UK Equity and Bond Income (46 funds) UK Equity and Bond Income (46 funds) UK Equity Income (85 funds) UK Equity Income (85 funds) Global Equity Income (4 funds) Global Equity Income (4 funds) … Growth Funds (21 subgroups) Growth Funds (21 subgroups) UK All Companies (290 funds) UK All Companies (290 funds) UK Smaller Companies (73 funds) UK Smaller Companies (73 funds) Japan (75 funds) Japan (75 funds) North America (84 funds) North America (84 funds) Global Emerging Markets (23 funds) Global Emerging Markets (23 funds) Properties (2 funds) Properties (2 funds) … Specialist Funds (3 subgroups) Specialist Funds (3 subgroups)

Fund Performance : Luck or Skill

Financial Times, Mon 29 th of Nov. 2004

Who Wants to be a Millionaire ? Suppose £ 500,000 question : Suppose £ 500,000 question : Which of these funds’ performance is not due to luck ? (A.) Artemis ABN AMRO Equity Income Alpha = t of alpha = (B.) AXA UK Equity Income Alpha = t of alpha = (C.) Jupiter Income Alpha = t of alpha = (D.) GAM UK Diversified Alpha = t of alpha =

Measuring Fund Performance : Equilibrium Models 1.) Unconditional Models CAPM : (ER i – r f ) t =  i +  i (ER m – r f ) t +  it Fama-French 3 factor model : (ER i – r f ) t =  i +  1i (ER m –r f ) t +  2i SML t +  3i HML t +  it Carhart (1997) 4 factor model (ER i –r f ) t =  i +  1i (ER m –r f ) t +  2i SML t +  3i HML t +  4i PR1YR t +  it 2.) Conditional (beta) Models Z = {z1, z2, z3, …}, Z t ’s are measured as deviations from their mean  i,t = b 0i + B’(z t-1 ) CAPM : (ER i – r f ) t =  i +  i (ER m – r f ) t + B’ i (z t-1 [ER m - r f ] t ) +  it

Measuring Fund Performance : Equilibrium Models (Cont.) 3.) Conditional (alpha-beta) Models Z = {z1, z2, z3, …}  i,t = b 0i + B’(z t-1 ) and  i,t = a 0i + A’(z t-1 ) CAPM : (ER i – r f ) t = a 0i + A’ i (z t-1 ) +  i (ER m – r f ) t + B’ i (z t-1 [ER m - r f ] t ) +  it 4.) Market timing Models (ER i – r f ) t =  i +  i (ER m – r f ) t +  i (ER m - r f ) 2 t +  it (ER i – r f ) t =  i +  i (ER m – r f ) t +  i (ER m - r f ) + t +  it

Case Study : Cuthbertson, Nitzsche and O’Sullivan (2004)

UK Mutual Funds / Unit Trusts Data : Data : –Sample period : monthly data monthly data April 1975 – December 2002 April 1975 – December 2002 –Number of funds : 1596 (‘Live’ and ‘dead’ funds) –Subgroups : equity growth, equity income, general equity, smaller companies

Model Selection : Assessing Goodness of Fit Say, if we have 800 funds, have to estimate each model for each fund Say, if we have 800 funds, have to estimate each model for each fund Calculate summary statistics of all the funds regressions : Means Calculate summary statistics of all the funds regressions : Means R 2 R 2 Akaike-Schwartz criteria (SIC) : is adding an extra variable worth losing a degree of freedom Akaike-Schwartz criteria (SIC) : is adding an extra variable worth losing a degree of freedom Also want to look at t-statistics of the extra variables Also want to look at t-statistics of the extra variables

Methodology : Bootstrapping Analysis When we consider uncertainty across all funds (i.e. L funds) – do funds in the ‘tails’ have skill or luck ? When we consider uncertainty across all funds (i.e. L funds) – do funds in the ‘tails’ have skill or luck ? For each fund we estimate the coefficients (  i,  i ) and collect the residuals based on all the data available for the fund (only funds with at least 60 observations are included in the analysis). For each fund we estimate the coefficients (  i,  i ) and collect the residuals based on all the data available for the fund (only funds with at least 60 observations are included in the analysis). Simulate the data, under the null hypothesis that each fund has  i = 0. Simulate the data, under the null hypothesis that each fund has  i = 0.

Alphas : Unconditional FF Model

Residuals of Selected Funds

Methodology : Bootstrapping Step 1 : Generating the simulated data Step 1 : Generating the simulated data (ER i – r f ) t = 0 +   i (ER m – r f ) t + Resid it Simulate L time series of the excess return under the null of no outperformance. Simulate L time series of the excess return under the null of no outperformance. Bootstrapping on the residuals (ONLY) Bootstrapping on the residuals (ONLY) Step 2 : Estimate the model using the generated data for L funds Step 2 : Estimate the model using the generated data for L funds (ER i – r f ) t =  1 +  1 (ER m – r f ) t +  it

Methodology : Bootstrapping (Cont.) Step 3 : Sort the alphas from the L - OLS regressions from step 2 Step 3 : Sort the alphas from the L - OLS regressions from step 2 {  1 (1),  2 (1), …,  L (1) }   max (1) Repeat steps 1, 2 and 3 1,000 times Repeat steps 1, 2 and 3 1,000 times Now we have 1,000 highest alphas all under the null of no outperformance. Now we have 1,000 highest alphas all under the null of no outperformance. Calculate the p-values of  max (real data) using the distribution of  max from the bootstrap (see below) Calculate the p-values of  max (real data) using the distribution of  max from the bootstrap (see below)

The Bootstrap Alpha Matrix (or t-of Alpha) Funds 1234… Bootstraps1  1,1  2,1  3,1  4,1 …  849,1  850,1 2  1,2  2,2  3,2  4,2 …  849,2  850,2 3  1,3  2,3  3,3  4,3 …  849,3  850,3 4  1,4  2,4  3,4  4,4 …  849,4  850,4 …………………… 999  1,999  2,999  3,999  4,999 …  849,999  850,  1,1000  2,1000  3,1000  4,1000 …  849,1000  850,1000

The Bootstrap Matrix – Sorted from high to low Highest2 nd highest 3 rd highest 4 th highest …2 nd Worst Bootstraps1  151,1  200,1  23,1  45,1 …  800,1  50,1 2  23,2  65,2  99,2  743,2 …  50,2  505,2 3  55,3  151,3  78,3  95,3 …  11,3  799,3 4  68,4  242,4  476,4  465,4 …  352,4  444,4 …………………… 999  76,999  12,999  371,999  444,999 …  31,999  11,  17,1000  9,1000  233,100 0  47,1000 …  12,1000  696,1000

Interpretation of the p-Values (Positive Distribution) Suppose highest alpha is 1.5 using real data Suppose highest alpha is 1.5 using real data If p-value is 0.20, that means 20% of the  (i) max (i = 1, 2, …, 1000) (under the null of no outperformance) are larger than 1.5 If p-value is 0.20, that means 20% of the  (i) max (i = 1, 2, …, 1000) (under the null of no outperformance) are larger than 1.5  LUCK If p-value is 0.02, that means only 2% of the  (i) max (under the null) are larger than 1.5 If p-value is 0.02, that means only 2% of the  (i) max (under the null) are larger than 1.5  SKILL

Interpretation of the p-Values (Negative Distribution) Suppose worst alpha is -3.5 Suppose worst alpha is -3.5 If p-value is 0.30, that means 30% of the  (i) min (i = 1, 2, …, 1000) (under the null of no outperformance) are less than -3.5 If p-value is 0.30, that means 30% of the  (i) min (i = 1, 2, …, 1000) (under the null of no outperformance) are less than -3.5  UNLUCKY If p-value is 0.01, that means only 1% of the  (i) min (under the null) are less than -3.5 If p-value is 0.01, that means only 1% of the  (i) min (under the null) are less than -3.5  BAD SKILL

Other Issues Instead of using sorting according to the alphas, we can sort the funds by the t of alphas (or anything else !) Instead of using sorting according to the alphas, we can sort the funds by the t of alphas (or anything else !) Different Models – see earlier discussion Different Models – see earlier discussion Different Bootstrapping – see next slide Different Bootstrapping – see next slide

Other Issues (Cont.) A few questions to address : A few questions to address : –Minimum length of fund performance date required for fund being considered –Bootstrapping on the ‘x’ variable(s) and the residuals or only on the residuals –Block bootstrap Residuals of equilibrium models are often serially correlated Residuals of equilibrium models are often serially correlated

UK Results : Unconditional Model Fund Position Actual alpha Actual t-alpha Bootstr. P-value Top Funds Best nd best th best th best Bottom Funds 15 th worst th worst nd worst Worst

Bootstrap Results : Best Funds

Bootstrapped Results : Worst Funds

UK Mutual Fund Industry

Summary Asset returns are not normally distributed Asset returns are not normally distributed  Hence should not use t-stats Skill or luck : Evidence for UK Skill or luck : Evidence for UK –Some top funds have ‘good skill’, good performance is luck for most funds –All bottom funds have ‘bad skill’

References Cuthbertson, K. and Nitzsche, D. (2004) ‘Quantitative Financial Economics’, Chapters 9 Cuthbertson, K. and Nitzsche, D. (2004) ‘Quantitative Financial Economics’, Chapters 9 Cuthbertson, K., Nitzsche, D. and O’Sullivan, N. (2004) ‘Mutual Fund Performance : Skill or Luck’, available on Cuthbertson, K., Nitzsche, D. and O’Sullivan, N. (2004) ‘Mutual Fund Performance : Skill or Luck’, available on

END OF LECTURE