5. Performance Measurement Dr Youchang Wu Dr. WS 2007 Asset Management

Youchang Wu

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An overview ƒ Measuring performance with asset pricing models – Jensen’s alpha, Treynor index, Sharpe ratio, M2, APT alpha, alpha Upmarket Upmarket-downmarket downmarket beta

ƒ Measuring performance without asset pricing i i models d l – Grinblatt-Titman measure, Tracking error, Information ratio, Style analysis, Attribution analysis Asset Management

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Performance and CAPM ƒ If the market is in equilibrium and no manager has superior information, then all the funds must be on one line (SML). ƒ The expected return should depend only on the beta factor. Asset Management

E(r) SML B

M A rf

Youchang Wu

0

05 0,5

1

15 1,5

β

3

Jensen‘s Jensen s Alpha (1) ƒ If a fund manager has „superior superior“ performance, then his fund lies above the SML ƒ The Th distance di t to t the th SML is the alpha

E(r) B´

JB' B A´

M

A rf

α = E[rP − rf ] − β P (E[rm − rf ]) Asset Management

SML

Youchang Wu

0

0,5

1

1,5

β

4

Jensen Alpha (2) ƒ Consider Fund A with ith a b beta t off 0.8 and the data f the for th last l t4 quarters

Quarter T-Bill Rate

Fund Return

Excess Return

S&P 500

Excess Return

Q1

2 97% 2,97%

-8,77% 8 77%

-11,74% 11 74%

-5,86% 5 86%

-8,83% 8 83%

Q2

3,06%

-6,03%

-9,09%

-2,94%

-6,00%

Q3

2,85%

14,14%

11,29%

13,77% 10,92%

Q4

1 88% 1,88%

24 99% 24,99%

23 08% 23,08%

14 82% 12,94% 14,82% 12 94%

Mean

2,69%

6,08%

3,39%

4,95%

16,22 %

16,68%

10,87% 11,26%

SD

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2,26%

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Jensen‘s Jensen s Alpha (3) ƒ What Wh t was the th Alpha Al h ffor F Fund d A?

ƒ Suppose Fund B has a beta of 1.2 and an Alpha off 4.5%. % How would you rank Fund A and Fund B? ƒ By how much could one lever Fund A to achieve a beta of 1.2? Asset Management

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Jensen‘s Jensen s Alpha (4) ƒ What is Jensen‘s Alpha for the levered fund?

ƒ How would we now rank Fund A versus F nd B? Fund Asset Management

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Treynor Index (1) ƒ Measures the „excess excess return per unit of systematic risk (beta)“ (beta) . ƒ Considers the l leverage effect. ff t E[rP − rf ] Treynor= βP Asset Management

E(r) B´

JP

M

SML

B

A´ A rf

0

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0,5

1

1,5

β

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Treynor Index (2) ƒ Treynor Index for Fund A: 3.39%/0.8=4.24% ƒ Treynor Index for Fund B: 7.2%/1.2=6% (Note that the excess rate of return of Fund B must have been 4.5%+2.26%*1.2 = 7.2%.) 2% ) ƒ Treynor Index for the S&P 500: 2.26% (=average excess return of the market) Asset Management

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Sharpe Ratio (1) ƒ Divides the excess return by the volatility. ƒ σp is frequently defined as the volatility of the excess rate of return. ƒ Slope of the lines of rf to A or B ƒ Ratio of return to risk

Sharpe= Asset Management

E(r)

B´

M

A

rf

E[rP − rf ]

σP

0

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σ (r)

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Sharpe Ratio (2) ƒ For the previous example, calculate the Sharpe Ratio for Fund A and the S&P 500. ƒ Recall that the estimation of the standard deviation of the rate or return from a sample of T observations is 1 ⎛ ⎞ σ = ⎜∑ ( rt − E [ rt ]) 2 ⎟ ⎝ T −1 ⎠

0,5

ƒ and d the th estimation ti ti off th the expected t d return t is i E[rt ] = Asset Management

1 rt ∑ T Youchang Wu

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Sharpe Ratio (3) ƒ Sharpe ratio for Fund A

ƒ Sharpe ratio for the S&P 500

ƒ Was there „superior“ performance? Asset Management

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Sharpe Ratio (4) ƒ The Sharpe ratio should be expressed on an annual basis. ƒ This requires transforming a quarterly Sharpe ratio in the following way: SharpeRatioAnnual=

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E[rp − rf ] ⋅ 4

σp ⋅ 4

= SharpeRatioQuarterly⋅ 4

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Sharpe Ratio (5) ƒ The higher the Sharpe ratio the better is the portfolio performance, because either – the return was higher or – the risk was smaller ƒ Comparison with the Sharpe ratio of the market g g guide: Sharpe p ratio above 1 is very yg good ƒ Rough and above 2 is extraordinary ƒ Note: The Sharpe ratio ignores correlations between the fund and clients’ other investments. Asset Management

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2

Modigliani-Modigliani (M ) Measure (1) ƒ Easier to interpret than Sharpe Ratio ƒ Return of a fund with the same risk as the benchmark, ƒ Includes leverage effekt M = rf + 2 p

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E[rp − rf ]

σp

E(r) ()

A´ A

M A

rf

σm 0

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σ (r) 15

2

Modigliani-Modigliani (M ) Measure (2) ƒ Calculate M for Fund A. 2

ƒ According to M , did Fund A have superior performance? f ? 2

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Summarizing CAPM-based CAPM based PMs ƒ Sharpe p Ratio,, M2: Consider return and total risk (volatility) – Return per unit of total risk – Assumption: entire wealth is invested in the fund

ƒ Alpha Alpha, Treynor: Consider return and systematic risk – Th The idiosyncratic idi ti risk i k off a stock t k is i ignored. i d – Only systematic risk counts. – Assumption: only a small part of the investor‘s wealth is invested in the fund. Asset Management

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Problems of CAPM CAPM-based based PMs ƒ What is the appropriate risk-free rate? ƒ What is the appropriate market portfolio? ƒ Market-timing g abilities may yg give biased results.

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Upmarket and Downmarket Betas ƒ Estimating two regression coefficients βup und βdown

rj

βup

βdown

ƒ Market timing expertise exists when βup-βdown > 0 Asset Management

0

Youchang Wu

rf

rM 19

APT Alpha Al h (1) ƒ Stock returns are related to basic factors: rj ,t = a j + β1, j F1,t + β 2, j F2,t + ... + β n, j Fn,t + ε j ,t

ƒ Expected returns are linearly related to factor exposures: E[rj ] = rf + β1, jγ 1 + β 2, jγ 2 + ... + β n, jγ n

ƒ Performance measurement using APT: Alpha _ APT = E[rp] − [rf + β1, jγ 1 + β2, jγ 2 + ... + βn, jγ n ] Asset Management

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APT Alpha (2) ƒ A popular l model d l used d iin practice ti iis th the F FamaFrench three factor model – Market factor f – Size factor – Book-to-market factor

ƒ The momentum factor is also often considered ƒ Problems with performance measurement using APT: – Factors are not theoretically specified – Factor acto structure st uctu e not ot u uniquely que y dete determined ed Asset Management

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Grinblatt Titman Measure Grinblatt-Titman ƒ Does not require a benchmark portfolio ƒ Assume that portfolioweights are observable ƒ Manager with market timing abilities will increase asset weight in asset classes with increasing returns ƒ Positive P iti correlation l ti b between t – Changes in portfolio weights – Return R ƒ Portfolio Change Measure T N r j ,t w j ,t − w j ,t −1 PCM = ∑∑ T −1 t =2 j =1

(

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Tracking Error and Information Ratio ƒ Relative performance = portfolio return benchmark return ƒ Tracking Error = σ(portfolio return - benchmark return)) RelativePerformance ƒ Information Ratio = TrackingError ƒ Risk relative to benchmark =

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σ(Portfolio) σ(Benchmar ( k))

Tracking Error and Information Ratio ƒ See the data for Fund I and II and the benchmark ƒ Which one has a higher information ratio? ƒ What is the relative risk ratio for each fund? Asset Management

Quar- Bench- Fund I Relative Fund II ter mark Return Perfor- Return mance I

Relative Performance II

Q1

8,77%

11,74% 2,97%

8,95%

0,18%

Q2

-6 03% -9,09% -6,03% -9 09% -3,06% -3 06%

-5 50% -5,50%

0 53% 0,53%

Q3

7,14%

11,29% 4,15%

7,55%

0,41%

Q4

4,99%

23,08% 18,09%

6,00%

1,01%

Mean

3,72%

9,255% 5,54%

4,25%

0,53%

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Style Analysis (1) ƒ Asset returns follow a factor model

rj ,t = α j + β1, j F1,t + β 2, j F2,t + ... + β n, j Fn,t + ε j ,t

ƒ Asset class factor model: each factor is the return of an asset class and Σβi=1 ƒ Choice of basic asset classes

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Style Analysis (2) ƒ Factor exposures can be estimated by – Regression – Constrained Regression • Sum of factor loadings = 1 – Quadratic programming: minimize the variance of return difference with constraints • Sum of factor loadings = 1 • Individual loading between 0 and 1

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Style Analysis (3) ƒ Estimated style profile: an example – Both unconstrained and constrained regression give problematic results – Qudratic programming is the recommended estimation method Unconstrained Constrained Regression Quadratic

Bills 14.69 42.65 0

Intermedi Long-term Corporate Value Growth ate Bonds Bonds Bonds Mortgages Stocks Stocks -7.86 -69.51 -2.54 16.57 5.19 109.52 -68.64 0

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-2.38 0

15.29 0

4.58 0

110.35 69.81

-8.02 0

Medium Small Stocks Stocks -41.83 45.65 -43.62 0

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47.17 30.04

Foreign Bonds -1.85 -1.38 0

European Japanese Total Stocks Stocks 6.15 -1.46 72.71 5.77 0.15

-1.79 0

100.00 100.00

R-squ 95.20 95.16 92.22

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Style Analysis (4)

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Style Analysis (5) ƒ Consider the rates of return on „factor portfolios“ g p given on the next p page. g ƒ Assume that the fund‘s „style“ is characterized by the regression results estimated above. ƒ What was the relative performance of this fund according to style analysis?

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Style Analysis (6) ƒ Rates of return on „factor portfolios“ V a lu e s to c k s

S m a ll s to c k s

E u ro p e a n F u n d s to c k s R e tu r n I

Q1

8 ,7 7 %

1 1 ,7 4 %

2 ,9 7 %

7 ,4 5 %

Q2

- 6 ,0 3 %

- 9 ,0 9 %

- 3 ,0 6 %

- 6 ,8 7 %

Q3

7 ,1 4 %

1 1 ,2 9 %

4 ,1 5 %

1 3 ,5 4 %

Q4

4 ,9 9 %

2 3 ,0 8 %

1 8 ,0 9 %

1 8 ,6 7 %

M ean

3 ,7 2 %

9 ,2 5 5 %

5 ,5 4 %

8 ,2 0 %

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Style Analysis (7) ƒ What was the average rate of return of the benchmark? ƒ Wh Whatt was the th average rate t off return t off the fund? ƒ What was as the relati relative e performance according to style analyis? Asset Management

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Attribution Analysis (1) ƒ Attribution analysis attemps to identify the performance,, i.e.,, the sources of relative p difference between portfolio return and benchmark return return. ƒ A popula decomposition: Total performance = allocation effect + selection effect Asset Management

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Attribution Analysis (2) Allocation effect

Selection effect

N

N

i =1

i =1

rp − rb = ∑ [( w pi − wbi )( rbi − rb )] + ∑ w pi ( rpi − rbi )

wpi, wbi = weights of the ith market segment (asset class, industry group) in the actual portfolio and the benchmark portfolio respectively portfolio, rpi, rbi = return to the ith market segment in the actual portfolio and the benchmark portfolio portfolio, respectively rb = the total return to the benchmark portfolio Asset Management

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Attribution analysis: example ƒ The following information is given. Compute the allocation effect and selection effect. investment weights Fund

Return

Benchmark Defference

Fund

Benchmark Difference

Stock

0.5

0.6

‐0.1

9.70%

8.60%

1.10%

Bond d

0.38

0.3

0.08

9.10%

9.20%

‐0.10%

Cash

0.12

0.1

0.02

5.60%

5.40%

0.20%

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Summary ƒ R Returns t mustt b be adjusted dj t d b by risk! i k! ƒ Asset pricing models give some guidance on how to adjust for risks ƒ However we do not know exactly what is the right asset pricing p g model and how to implement p it ƒ Measuring performance without asset pricing models is possible if portfolio holding data are available, or if a benchmark is pre pre-specified specified ƒ Style analysis is a tool to backout a suitable benchmark from the data ƒ Attribution analysis identifies the source of relative performance Asset Management

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