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LECTURE 10 : APPLICATION OF LINEAR FACTOR MODELS

LECTURE 10 : APPLICATION OF LINEAR FACTOR MODELS. (Asset Pricing and Portfolio Theory). Contents. Mutual fund industry Measuring performance of mutual funds (risk adjusted rate of return)  Jensen’s alpha Using factor models to measure fund performance due luck or skill

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LECTURE 10 : APPLICATION OF LINEAR FACTOR MODELS

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

  2. Contents • Mutual fund industry • 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

  3. Newspaper Comments • The Sunday Times 10.03.2002 ‘Nine out of ten funds underperform’ • The Sunday Times 10.10.2004 ‘Funds take half your growth in fees’

  4. 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. • Should we buy actively managed funds or index trackers ? • Assets under management • US mutual fund industry : over $ 5.5 trillion (2000), with $ 3 trillion in equity funds • Number of funds • US : 393 funds in 1975, 2424 in 1995 (main US database) • UK : 1167 funds in 1996, 2222 in 2001 (yearbook)

  5. UK Unit Trust Industry Number of Funds Assets under Management

  6. Classification of Unit Trusts - UK • Income Funds (7 subgroups) • 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) • … • Growth Funds (21 subgroups) • 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)

  7. Fund Performance : Luck or Skill

  8. Financial Times, Mon 29th of Nov. 2004

  9. Who Wants to be a Millionaire ? • 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 (B.) AXA UK Equity Income Alpha = 0.2840 t of alpha = 2.6733 (C.) Jupiter Income Alpha = 0.3822 t of alpha = 2.4035 (D.) GAM UK Diversified Alpha = 0.4474 t of alpha = 2.0235

  10. 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 + b1i(ERm–rf)t + b2iSMLt + b3i HMLt + eit Carhart (1997) 4 factor model (ERi–rf)t = ai +b1i(ERm–rf)t + b2iSMLt + b3iHMLt + b4iPR1YRt+ eit 2.) Conditional (beta) Models Z = {z1, z2, z3, …}, Zt’s are measured as deviations from their mean bi,t = b0i + B’(zt-1) CAPM : (ERi – rf)t = ai + bi(ERm – rf)t + B’i(zt-1 [ERm - rf]t) + eit

  11. Measuring Fund Performance : Equilibrium Models (Cont.) 3.) Conditional (alpha-beta) Models Z = {z1, z2, z3, …} bi,t = b0i + B’(zt-1) and ai,t = a0i + A’(zt-1) CAPM : (ERi – rf)t = a0i + 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)2t + eit (ERi – rf)t = ai + bi(ERm – rf)t + gi(ERm - rf)+t + eit

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

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

  14. Model Selection : Assessing Goodness of Fit • Say, if we have 800 funds, have to estimate each model for each fund • Calculate summary statistics of all the funds regressions : Means • R2 • 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

  15. Methodology : Bootstrapping Analysis • 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 (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). • Simulate the data, under the null hypothesis that each fund has ai = 0.

  16. Alphas : Unconditional FF Model

  17. Residuals of Selected Funds

  18. Methodology : Bootstrapping • Step 1 : Generating the simulated data (ERi – rf)t = 0 + b1i(ERm – rf)t + Residit • 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 = a1 + b1(ERm – rf)t + eit

  19. Methodology : Bootstrapping (Cont.) • Step 3 : Sort the alphas from the L - OLS regressions from step 2 {a1(1), a2(1), …, aL(1)}  amax(1) • 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)

  20. The Bootstrap Alpha Matrix (or t-of Alpha)

  21. The Bootstrap Matrix – Sorted from high to low

  22. Interpretation of the p-Values (Positive Distribution) • Suppose highest alpha is 1.5 using real data • If 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 • If p-value is 0.02, that means only 2% of the a(i)max (under the null) are larger than 1.5  SKILL

  23. Interpretation of the p-Values (Negative Distribution) • Suppose worst alpha is -3.5 • If 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 • 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

  24. Other Issues • 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 Bootstrapping – see next slide

  25. Other Issues (Cont.) • 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

  26. UK Results : Unconditional Model

  27. Bootstrap Results : Best Funds

  28. Bootstrapped Results : Worst Funds

  29. UK Mutual Fund Industry

  30. Summary • Asset returns are not normally distributed  Hence should not use t-stats • 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’

  31. References • 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

  32. END OF LECTURE

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