LECTURE 10 :LECTURE 10 :

APPLICATION OF LINEAR APPLICATION OF LINEAR FACTOR MODELSFACTOR MODELS

(Asset Pricing and Portfolio (Asset Pricing and Portfolio Theory)Theory)

ContentsContents

Mutual fund industry Mutual fund industry Measuring performance of mutual Measuring performance of mutual

funds (risk adjusted rate of return)funds (risk adjusted rate of return)

Jensen’s alphaJensen’s alpha

Using factor models to measure Using factor models to measure fund performance due luck or skillfund performance due luck or skill

Active vs passive fund management Active vs passive fund management

Newspaper Comments Newspaper Comments

The Sunday Times 10.03.2002The Sunday Times 10.03.2002

‘‘Nine out of ten funds Nine out of ten funds underperform’underperform’

The Sunday Times 10.10.2004The Sunday Times 10.10.2004

‘‘Funds take half your growth in Funds take half your growth in fees’fees’

IntroductionIntroduction

Diversification in practice : invest in different Diversification in practice : invest in different mutual funds, with different asset classes (e.g. mutual funds, with different asset classes (e.g. bonds, equity), different investment objectives bonds, equity), different investment objectives (e.g. income, growth funds) and different (e.g. income, growth funds) and different geographic regions. geographic regions.

Should we buy actively managed funds or index Should we buy actively managed funds or index trackers ? trackers ?

Assets under management Assets under management – US mutual fund industry : over $ 5.5 trillion (2000), with US mutual fund industry : over $ 5.5 trillion (2000), with

$ 3 trillion in equity funds$ 3 trillion in equity funds Number of funds Number of funds

– US : 393 funds in 1975, 2424 in 1995 (main US US : 393 funds in 1975, 2424 in 1995 (main US database)database)

– UK : 1167 funds in 1996, 2222 in 2001 (yearbook)UK : 1167 funds in 1996, 2222 in 2001 (yearbook)

UK Unit Trust IndustryUK Unit Trust Industry

UK Unit Trusts : Assets under Management

0

50000

100000

150000

200000

250000

300000

350000

1996 1997 1998 1999 2000 2001

UK Unit Trusts : Number of Funds

0

500

1000

1500

2000

2500

1996 1997 1998 1999 2000 2001

Number of Funds Assets under Management

Classification of Unit Classification of Unit Trusts - UKTrusts - 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 : Fund Performance : Luck or SkillLuck or Skill

Financial Times, Mon Financial Times, Mon 2929thth of Nov. 2004 of Nov. 2004

Who Wants to be a Who Wants to be a Millionaire ? Millionaire ? Suppose £ 500,000 question : Suppose £ 500,000 question :

Which of these funds’ performance is not Which of these funds’ performance is not due to luck ?due to luck ?

(A.) Artemis ABN AMRO Equity Income (A.) Artemis ABN AMRO Equity Income Alpha = 0.4782Alpha = 0.4782 t of alpha = 2.7771t of alpha = 2.7771

(B.) AXA UK Equity Income(B.) AXA UK Equity IncomeAlpha = 0.2840Alpha = 0.2840 t of alpha = 2.6733t of alpha = 2.6733

(C.) Jupiter Income (C.) Jupiter Income Alpha = 0.3822Alpha = 0.3822 t of alpha = 2.4035t of alpha = 2.4035

(D.) GAM UK Diversified(D.) GAM UK DiversifiedAlpha = 0.4474Alpha = 0.4474 t of alpha = 2.0235t of alpha = 2.0235

Measuring Fund Measuring Fund Performance : Equilibrium Performance : Equilibrium Models Models 1.) Unconditional Models 1.) Unconditional Models

CAPM : (ERCAPM : (ERii – r – rff))tt = = ii + + ii(ER(ERmm – r – rff))tt + + itit

Fama-French 3 factor model : Fama-French 3 factor model : (ER(ERii – r – rff))tt = = ii + + 1i1i(ER(ERmm–r–rff))tt + + 2i2iSMLSMLtt + + 3i3i HML HMLtt + + itit

Carhart (1997) 4 factor model Carhart (1997) 4 factor model (ER(ERii–r–rff))tt = = ii + +1i1i(ER(ERmm–r–rff))tt + + 2i2iSMLSMLtt + + 3i3iHMLHMLtt + + 4i4iPR1YRPR1YRtt+ + itit

2.) Conditional (beta) Models 2.) Conditional (beta) Models Z = {z1, z2, z3, …}, ZZ = {z1, z2, z3, …}, Ztt’s are measured as deviations ’s are measured as deviations from their mean from their mean

i,ti,t = b = b0i0i + B’(z + B’(zt-1t-1) ) CAPM : (ERCAPM : (ERii – r – rff))tt = = ii + + ii(ER(ERmm – r – rff))tt + B’ + B’ii(z(zt-1t-1 [ER [ERmm - r - rff]]tt) + ) + itit

Measuring Fund Measuring Fund Performance : Equilibrium Performance : Equilibrium Models (Cont.)Models (Cont.)3.) Conditional (alpha-beta) Models 3.) Conditional (alpha-beta) Models

Z = {z1, z2, z3, …} Z = {z1, z2, z3, …}

i,ti,t = b = b0i0i + B’(z + B’(zt-1t-1) ) and and i,ti,t = a = a0i0i + A’(z + A’(zt-1t-1))

CAPM : (ERCAPM : (ERii – r – rff))tt = a = a0i0i + A’ + A’ii(z(zt-1t-1) + ) + ii(ER(ERmm – r – rff))tt

+ B’+ B’ii(z(zt-1t-1 [ER [ERmm - r - rff]]tt) + ) + itit

4.) Market timing Models4.) Market timing Models

(ER(ERii – r – rff))tt = = ii + + ii(ER(ERmm – r – rff))tt + + ii(ER(ERmm - r - rff))22tt + + itit

(ER(ERii – r – rff))tt = = ii + + ii(ER(ERmm – r – rff))tt + + ii(ER(ERmm - r - rff))++tt + + itit

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

UK Mutual Funds / Unit UK Mutual Funds / Unit TrustsTrusts Data : Data :

– Sample period : Sample period : monthly data monthly data April 1975 – December 2002April 1975 – December 2002

– Number of funds : 1596 (‘Live’ and Number of funds : 1596 (‘Live’ and ‘dead’ funds)‘dead’ funds)

– Subgroups : equity growth, equity Subgroups : equity growth, equity income, general equity, smaller income, general equity, smaller companiescompanies

Model Selection : Model Selection : Assessing Goodness of Assessing Goodness of Fit Fit Say, if we have 800 funds, have to Say, if we have 800 funds, have to

estimate each model for each fund estimate each model for each fund Calculate summary statistics of all Calculate summary statistics of all

the funds regressions : Meansthe funds regressions : Means RR22

Akaike-Schwartz criteria (SIC) : is adding Akaike-Schwartz criteria (SIC) : is adding an extra variable worth losing a degree of an extra variable worth losing a degree of freedomfreedom

Also want to look at t-statistics of the Also want to look at t-statistics of the extra variablesextra variables

Methodology : Methodology : Bootstrapping Analysis Bootstrapping Analysis When we consider uncertainty across all funds When we consider uncertainty across all funds

(i.e. L funds) – do funds in the ‘tails’ have skill or (i.e. L funds) – do funds in the ‘tails’ have skill or luck ? luck ?

For each fund we estimate the coefficients (For each fund we estimate the coefficients (ii, , ii) ) and collect the residuals based on all the data and collect the residuals based on all the data available for the fund (only funds with at least 60 available for the fund (only funds with at least 60 observations are included in the analysis). observations are included in the analysis).

Simulate the data, under the null hypothesis that Simulate the data, under the null hypothesis that each fund has each fund has ii = 0. = 0.

Alphas : Unconditional Alphas : Unconditional FF ModelFF Model

Residuals of Selected Residuals of Selected FundsFunds

Methodology : Methodology : BootstrappingBootstrapping Step 1 : Generating the simulated data Step 1 : Generating the simulated data

(ER(ERi i – r– rff))tt = 0 + = 0 + ii(ER(ERm m – r– rff))tt + Resid + Residitit

Simulate L time series of the excess return under Simulate L time series of the excess return under the null of no outperformance.the null of no outperformance.

Bootstrapping on the residuals (ONLY)Bootstrapping on the residuals (ONLY)

Step 2 : Estimate the model using the Step 2 : Estimate the model using the generated data for L fundsgenerated data for L funds

(ER(ERii – r – rff))tt = = 11 + + 11(ER(ERmm – r – rff))tt + + itit

Methodology : Methodology : Bootstrapping (Cont.)Bootstrapping (Cont.) Step 3 : Sort the alphas from the L - OLS Step 3 : Sort the alphas from the L - OLS

regressions from step 2regressions from step 2

{{11(1)(1), , 22

(1)(1), …, , …, LL(1)(1)} } maxmax

(1)(1)

Repeat steps 1, 2 and 3 1,000 timesRepeat steps 1, 2 and 3 1,000 times Now we have 1,000 highest alphas all Now we have 1,000 highest alphas all

under the null of no outperformance. under the null of no outperformance. Calculate the p-values of Calculate the p-values of maxmax (real data) (real data)

using the distribution of using the distribution of maxmax from the from the bootstrap (see below)bootstrap (see below)

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

Funds

1 2 3 4 … 849 850  

Bootstraps 1 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 … 849,999 850,999  

1000 1,1000 2,1000 3,1000 4,1000 … 849,1000 850,1000  

The Bootstrap Matrix – The Bootstrap Matrix – Sorted from high to Sorted from high to lowlow

Highest 2nd highest

3rd highest

4th highest

… 2nd Worst

Worst

Bootstraps 1 151,1 200,1 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 11,999

1000 17,1000 9,1000 233,1000 47,1000 … 12,1000 696,1000

Interpretation of the p-Interpretation of the p-Values (Positive Values (Positive Distribution)Distribution) Suppose highest alpha is 1.5 using real dataSuppose highest alpha is 1.5 using real data

If p-value is 0.20, that means 20% of the If p-value is 0.20, that means 20% of the (i)(i)maxmax

(i = 1, 2, …, 1000) (under the null of no (i = 1, 2, …, 1000) (under the null of no outperformance) are larger than 1.5outperformance) are larger than 1.5

LUCKLUCK

If p-value is 0.02, that means only 2% of the If p-value is 0.02, that means only 2% of the (i)(i)

maxmax (under the null) are larger than 1.5 (under the null) are larger than 1.5

SKILLSKILL

Interpretation of the p-Interpretation of the p-Values (Negative Values (Negative Distribution)Distribution) Suppose worst alpha is -3.5 Suppose worst alpha is -3.5

If p-value is 0.30, that means 30% of the If p-value is 0.30, that means 30% of the (i)(i)

minmin (i = 1, 2, …, 1000) (under the null of (i = 1, 2, …, 1000) (under the null of no outperformance) are less than -3.5no outperformance) are less than -3.5

UNLUCKY UNLUCKY

If p-value is 0.01, that means only 1% of If p-value is 0.01, that means only 1% of the the (i)(i)

minmin (under the null) are less than -3.5 (under the null) are less than -3.5

BAD SKILLBAD SKILL

Other IssuesOther Issues

Instead of using sorting according Instead of using sorting according to the alphas, we can sort the to the alphas, we can sort the funds by the t of alphas (or funds by the t of alphas (or anything else !)anything else !)

Different Models – see earlier Different Models – see earlier discussiondiscussion

Different Bootstrapping – see next Different Bootstrapping – see next slideslide

Other Issues (Cont.)Other Issues (Cont.)

A few questions to address : A few questions to address : – Minimum length of fund performance Minimum length of fund performance

date required for fund being considereddate required for fund being considered– Bootstrapping on the ‘x’ variable(s) and Bootstrapping on the ‘x’ variable(s) and

the residuals or only on the residuals the residuals or only on the residuals – Block bootstrapBlock bootstrap

Residuals of equilibrium models are often Residuals of equilibrium models are often serially correlatedserially correlated

UK Results : UK Results : Unconditional Model Unconditional Model Fund Position Fund Position Actual alphaActual alpha Actual t-alphaActual t-alpha Bootstr. P-Bootstr. P-

valuevalue

Top Funds Top Funds

Best Best 0.78530.7853 4.02344.0234 0.0560.056

22ndnd best best 0.72390.7239 3.38913.3891 0.0590.059

1010thth best best 0.53040.5304 2.54482.5448 0.0220.022

1515thth best best 0.47820.4782 2.40352.4035 0.0040.004

Bottom Funds Bottom Funds

1515thth worst worst -0.5220-0.5220 -3.6873-3.6873 0.0000.000

1010thth worst worst -0.5899-0.5899 -4.1187-4.1187 0.0000.000

22ndnd worst worst -0.7407-0.7407 -5.1664-5.1664 0.0010.001

WorstWorst -0.9015-0.9015 -7.4176-7.4176 0.0000.000

Bootstrap Results : Bootstrap Results : Best Funds Best Funds

Bootstrapped Results : Bootstrapped Results : Worst FundsWorst Funds

UK Mutual Fund UK Mutual Fund IndustryIndustry

SummarySummary

Asset returns are not normally Asset returns are not normally distributed distributed

Hence should not use t-stats Hence should not use t-stats Skill or luck : Evidence for UKSkill or luck : Evidence for UK

– Some top funds have ‘good skill’, good Some top funds have ‘good skill’, good performance is luck for most fundsperformance is luck for most funds

– All bottom funds have ‘bad skill’All bottom funds have ‘bad skill’

References References

Cuthbertson, K. and Nitzsche, D. (2004) Cuthbertson, K. and Nitzsche, D. (2004) ‘Quantitative Financial Economics’, Chapters ‘Quantitative Financial Economics’, Chapters 99

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

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

END OF LECTUREEND OF LECTURE