LECTURE 10 APPLICATION OF LINEAR FACTOR MODELS

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

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

Newspaper Comments n The Sunday Times 10. 03. 2002 ‘Nine out of ten funds underperform’ n The Sunday Times 10. 2004 ‘Funds take half your growth in fees’

Introduction n n 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 ? n Assets under management n Number of funds – US mutual fund industry : over $ 5. 5 trillion (2000), with $ 3 trillion in equity 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 Funds Assets under Management

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

Fund Performance : Luck or Skill

Financial Times, Mon 29 th of Nov. 2004

Who Wants to be a Millionaire ? n Suppose £ 500, 000 question : Which of these funds’ performance is not due to luck ? (A. ) Artemis ABN AMRO Equity Income Alpha = 0. 4782 t of alpha = 2. 7771 Alpha = 0. 2840 t of alpha = 2. 6733 Alpha = 0. 3822 t of alpha = 2. 4035 Alpha = 0. 4474 t of alpha = 2. 0235 (B. ) AXA UK Equity Income (C. ) Jupiter Income (D. ) GAM UK Diversified

Measuring Fund Performance : Equilibrium Models 1. ) Unconditional Models CAPM : (ERi – rf)t = ai + bi(ERm – rf)t + eit Fama-French 3 factor model : (ERi – rf)t = ai + b 1 i(ERm–rf)t + b 2 i. SMLt + b 3 i HMLt + eit Carhart (1997) 4 factor model (ERi–rf)t = ai +b 1 i(ERm–rf)t + b 2 i. SMLt + b 3 i. HMLt + b 4 i. PR 1 YRt+ eit 2. ) Conditional (beta) Models Z = {z 1, z 2, z 3, …}, Zt’s are measured as deviations from their mean bi, t = b 0 i + B’(zt-1) CAPM : (ERi – rf)t = ai + bi(ERm – rf)t + B’i(zt-1 [ERm - rf]t) + eit

Measuring Fund Performance : Equilibrium Models (Cont. ) 3. ) Conditional (alpha-beta) Models Z = {z 1, z 2, z 3, …} bi, t = b 0 i + B’(zt-1) and ai, t = a 0 i + A’(zt-1) CAPM : (ERi – rf)t = a 0 i + A’i(zt-1) + bi(ERm – rf)t + B’i(zt-1 [ERm - rf]t) + eit 4. ) Market timing Models (ERi – rf)t = ai + bi(ERm – rf)t + gi(ERm - rf)2 t + eit (ERi – rf)t = ai + bi(ERm – rf)t + gi(ERm - rf)+t + eit

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

UK Mutual Funds / Unit Trusts n Data : – Sample period : n monthly data n 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 n Calculate summary statistics of all the funds regressions : Means n n R 2 n Akaike-Schwartz criteria (SIC) : is adding an extra variable worth losing a degree of freedom n Also want to look at t-statistics of the extra variables

Methodology : Bootstrapping Analysis n When we consider uncertainty across all funds (i. e. L funds) – do funds in the ‘tails’ have skill or luck ? n For each fund we estimate the coefficients (ai, bi) 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). n Simulate the data, under the null hypothesis that each fund has ai = 0.

Alphas : Unconditional FF Model

Residuals of Selected Funds

Methodology : Bootstrapping n Step 1 : Generating the simulated data (ERi – rf)t = 0 + b 1 i(ERm – rf)t + Residit n n n Simulate L time series of the excess return under the null of no outperformance. Bootstrapping on the residuals (ONLY) Step 2 : Estimate the model using the generated data for L funds (ERi – rf)t = a 1 + b 1(ERm – rf)t + eit

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

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

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

Interpretation of the p-Values (Positive Distribution) n Suppose highest alpha is 1. 5 using real data p-value is 0. 20, that means 20% of the a(i)max (i = 1, 2, …, 1000) (under the null of no outperformance) are larger than 1. 5 LUCK n If p-value is 0. 02, that means only 2% of the a(i)max (under the null) are larger than 1. 5 SKILL

Interpretation of the p-Values (Negative Distribution) n Suppose worst alpha is -3. 5 p-value is 0. 30, that means 30% of the a(i)min (i = 1, 2, …, 1000) (under the null of no outperformance) are less than -3. 5 UNLUCKY n If p-value is 0. 01, that means only 1% of the a(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 !) n Different Models – see earlier discussion n Different Bootstrapping – see next slide n

Other Issues (Cont. ) n 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 n 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 0. 7853 4. 0234 0. 056 2 nd best 0. 7239 3. 3891 0. 059 10 th best 0. 5304 2. 5448 0. 022 15 th best 0. 4782 2. 4035 0. 004 Bottom Funds 15 th worst -0. 5220 -3. 6873 0. 000 10 th worst -0. 5899 -4. 1187 0. 000 2 nd worst -0. 7407 -5. 1664 0. 001 Worst -0. 9015 -7. 4176 0. 000

Bootstrap Results : Best Funds

Bootstrapped Results : Worst Funds

UK Mutual Fund Industry

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

References n n 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 http: //www. cass. city. ac. uk/faculty/d. nitzsche/research. html

END OF LECTURE