asset management lecture 6. outline for today treynor black model m2 measure of performance...

Asset Management

Lecture 6

Outline for today

Treynor Black ModelM2 measure of performanceSensitivity to return assumptionTracking error

Treynor Black Model

The optimization of a risky portfolio using a single-index model is know as the Treynor Black model (or diagonal model)

Optimizing procedure

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Table 27.1 active portfolio management with 6 assets

S&P 500 Active Pf A HP DELL WMT TARGET BP SHELL2(e) 0.0705 0.0572 0.0309 0.0392 0.0297 0.0317

0.5505 /2e 0.2126 - 0.1748 - 0.1619 0.1911 0.4045 0.0789

1.0000 W0(i) 0.3863 - 0.3176 - 0.2941 0.3472 0.7349 0.1433

[W0(i)]2 0.1492 0.1009 0.0865 0.1205 0.5400 0.0205

A 0.0222

2(eA) 0.0404

W0 0.1691 Overall

W* 0.8282 0.1718 Portfolio 0.0663 - 0.0546 - 0.0505 0.0596 0.1262 0.0246

Beta 1 1.0922 1.0158 0.0663 - 0.0546 - 0.0505 0.0596 0.1262 0.0246Risk premium 0.06 0.0878 0.0648 0.0750 0.1121 0.0689 0.0447 0.0880 0.0305

SD 0.1358 0.2497 0.1422 0.3817 0.2901 0.1935 0.2611 0.1822 0.1988

Sharpe Ratio 0.44 0.35 0.4556

M-Square 0 - 0.0123 0.0019

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0.5505 /2e 0.2126 - 0.1748 - 0.1619 0.1911 0.4045 0.0789

1.0000 W0(i) 0.3863 - 0.3176 - 0.2941 0.3472 0.7349 0.1433

[W0(i)]2 0.1492 0.1009 0.0865 0.1205 0.5400 0.0205

A 0.0222

2(eA) 0.0404

W0 0.1691 Overall

W* 0.8282 0.1718 Portfolio 0.0663 - 0.0546 - 0.0505 0.0596 0.1262 0.0246


SD 0.1358 0.2497 0.1422 0.3817 0.2901 0.1935 0.2611 0.1822 0.1988

Sharpe Ratio 0.44 0.35 0.4556

M-Square 0 - 0.0123 0.0019

n

ii

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0.5505 /2e 0.2126 - 0.1748 - 0.1619 0.1911 0.4045 0.0789

1.0000 W0(i) 0.3863 - 0.3176 - 0.2941 0.3472 0.7349 0.1433

[W0(i)]2 0.1492 0.1009 0.0865 0.1205 0.5400 0.0205

A 0.0222

2(eA) 0.0404

W0 0.1691 Overall

W* 0.8282 0.1718 Portfolio 0.0663 - 0.0546 - 0.0505 0.0596 0.1262 0.0246


SD 0.1358 0.2497 0.1422 0.3817 0.2901 0.1935 0.2611 0.1822 0.1988

Sharpe Ratio 0.44 0.35 0.4556

M-Square 0 - 0.0123 0.0019

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n

iiA awa

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0.5505 /2e 0.2126 - 0.1748 - 0.1619 0.1911 0.4045 0.0789

1.0000 W0(i) 0.3863 - 0.3176 - 0.2941 0.3472 0.7349 0.1433

[W0(i)]2 0.1492 0.1009 0.0865 0.1205 0.5400 0.0205

A 0.0222

2(eA) 0.0404

W0 0.1691 Overall

W* 0.8282 0.1718 Portfolio 0.0663 - 0.0546 - 0.0505 0.0596 0.1262 0.0246


SD 0.1358 0.2497 0.1422 0.3817 0.2901 0.1935 0.2611 0.1822 0.1988

Sharpe Ratio 0.44 0.35 0.4556

M-Square 0 - 0.0123 0.0019

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0.5505 /2e 0.2126 - 0.1748 - 0.1619 0.1911 0.4045 0.0789

1.0000 W0(i) 0.3863 - 0.3176 - 0.2941 0.3472 0.7349 0.1433

[W0(i)]2 0.1492 0.1009 0.0865 0.1205 0.5400 0.0205

A 0.0222

2(eA) 0.0404

W0 0.1691 Overall

W* 0.8282 0.1718 Portfolio 0.0663 - 0.0546 - 0.0505 0.0596 0.1262 0.0246


SD 0.1358 0.2497 0.1422 0.3817 0.2901 0.1935 0.2611 0.1822 0.1988

Sharpe Ratio 0.44 0.35 0.4556

M-Square 0 - 0.0123 0.0019

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0.5505 /2e 0.2126 - 0.1748 - 0.1619 0.1911 0.4045 0.0789

1.0000 W0(i) 0.3863 - 0.3176 - 0.2941 0.3472 0.7349 0.1433

[W0(i)]2 0.1492 0.1009 0.0865 0.1205 0.5400 0.0205

A 0.0222

2(eA) 0.0404

W0 0.1691 Overall

W* 0.8282 0.1718 Portfolio 0.0663 - 0.0546 - 0.0505 0.0596 0.1262 0.0246


SD 0.1358 0.2497 0.1422 0.3817 0.2901 0.1935 0.2611 0.1822 0.1988

Sharpe Ratio 0.44 0.35 0.4556

M-Square 0 - 0.0123 0.0019

i

n

iiA w

1



0.5505 /2e 0.2126 - 0.1748 - 0.1619 0.1911 0.4045 0.0789

1.0000 W0(i) 0.3863 - 0.3176 - 0.2941 0.3472 0.7349 0.1433

[W0(i)]2 0.1492 0.1009 0.0865 0.1205 0.5400 0.0205

A 0.0222

2(eA) 0.0404

W0 0.1691 Overall

W* 0.8282 0.1718 Portfolio 0.0663 - 0.0546 - 0.0505 0.0596 0.1262 0.0246


SD 0.1358 0.2497 0.1422 0.3817 0.2901 0.1935 0.2611 0.1822 0.1988

Sharpe Ratio 0.44 0.35 0.4556

M-Square 0 - 0.0123 0.0019

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0.5505 /2e 0.2126 - 0.1748 - 0.1619 0.1911 0.4045 0.0789

1.0000 W0(i) 0.3863 - 0.3176 - 0.2941 0.3472 0.7349 0.1433

[W0(i)]2 0.1492 0.1009 0.0865 0.1205 0.5400 0.0205

A 0.0222

2(eA) 0.0404

W0 0.1691 Overall

W* 0.8282 0.1718 Portfolio 0.0663 - 0.0546 - 0.0505 0.0596 0.1262 0.0246


SD 0.1358 0.2497 0.1422 0.3817 0.2901 0.1935 0.2611 0.1822 0.1988

Sharpe Ratio 0.44 0.35 0.4556

M-Square 0 - 0.0123 0.0019

** 1 AM ww



0.5505 /2e 0.2126 - 0.1748 - 0.1619 0.1911 0.4045 0.0789

1.0000 W0(i) 0.3863 - 0.3176 - 0.2941 0.3472 0.7349 0.1433

[W0(i)]2 0.1492 0.1009 0.0865 0.1205 0.5400 0.0205

A 0.0222

2(eA) 0.0404

W0 0.1691 Overall

W* 0.8282 0.1718 Portfolio 0.0663 - 0.0546 - 0.0505 0.0596 0.1262 0.0246


SD 0.1358 0.2497 0.1422 0.3817 0.2901 0.1935 0.2611 0.1822 0.1988

Sharpe Ratio 0.44 0.35 0.4556

M-Square 0 - 0.0123 0.0019

AAMAAMp wREwwRE *** )()()(



0.5505 /2e 0.2126 - 0.1748 - 0.1619 0.1911 0.4045 0.0789

1.0000 W0(i) 0.3863 - 0.3176 - 0.2941 0.3472 0.7349 0.1433

[W0(i)]2 0.1492 0.1009 0.0865 0.1205 0.5400 0.0205

A 0.0222

2(eA) 0.0404

W0 0.1691 Overall

W* 0.8282 0.1718 Portfolio 0.0663 - 0.0546 - 0.0505 0.0596 0.1262 0.0246


SD 0.1358 0.2497 0.1422 0.3817 0.2901 0.1935 0.2611 0.1822 0.1988

Sharpe Ratio 0.44 0.35 0.4556

M-Square 0 - 0.0123 0.0019

2*22** )( AAMAAMP ewww



0.5505 /2e 0.2126 - 0.1748 - 0.1619 0.1911 0.4045 0.0789

1.0000 W0(i) 0.3863 - 0.3176 - 0.2941 0.3472 0.7349 0.1433

[W0(i)]2 0.1492 0.1009 0.0865 0.1205 0.5400 0.0205

A 0.0222

2(eA) 0.0404

W0 0.1691 Overall

W* 0.8282 0.1718 Portfolio 0.0663 - 0.0546 - 0.0505 0.0596 0.1262 0.0246


SD 0.1358 0.2497 0.1422 0.3817 0.2901 0.1935 0.2611 0.1822 0.1988

Sharpe Ratio 0.44 0.35 0.4556

M-Square 0 - 0.0123 0.0019

2*22**

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AAMAAM

AAMAAM

P

pp

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wREwwRES



0.5505 /2e 0.2126 - 0.1748 - 0.1619 0.1911 0.4045 0.0789

1.0000 W0(i) 0.3863 - 0.3176 - 0.2941 0.3472 0.7349 0.1433

[W0(i)]2 0.1492 0.1009 0.0865 0.1205 0.5400 0.0205

A 0.0222

2(eA) 0.0404

W0 0.1691 Overall

W* 0.8282 0.1718 Portfolio 0.0663 - 0.0546 - 0.0505 0.0596 0.1262 0.0246


SD 0.1358 0.2497 0.1422 0.3817 0.2901 0.1935 0.2611 0.1822 0.1988

Sharpe Ratio 0.44 0.35 0.4556

M-Square 0 - 0.0123 0.0019

M2 Measure

Developed by Modigliani and Modigliani Create an adjusted portfolio P* with T-bills

and the managed portfolio P so that SD[r(P*)]= SD[r(M)]

Example: Volatility of r(P)=1.5*volatility of r(M) P*=2/3P+1/3T

With the same SD, you can now compare the performance

2*P MM r r

M2 Measure: Example

Managed Portfolio: return = 35% standard deviation = 42%

Market Portfolio: return = 28% standard deviation = 30% T-bill return = 6%

Hypothetical Portfolio:

30/42 = .714 in P

(1-.714) or .286 in T-bills

r(P*)=(.714) (.35) + (.286) (.06) = 26.7%

Since this return is less than the market, the managed portfolio underperformed

M2 Measure: Example

σ

E(r)

PM

T

σ(P)σ(M)

P*M2

M2 Measure: Example

σ

E(r)

P M

T

σ(P) σ(M)

P*

M2

M2 Measure

Simplification for calculation

MP rrM *2

fP

MP

P

MP rrr

1*P

MT

P

MP ww

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P

fMfPPMP

rrrr

* fPMP rSr *

)()(2MPMfMPM SSrrSM



0.5505 /2e 0.2126 - 0.1748 - 0.1619 0.1911 0.4045 0.0789

1.0000 W0(i) 0.3863 - 0.3176 - 0.2941 0.3472 0.7349 0.1433

[W0(i)]2 0.1492 0.1009 0.0865 0.1205 0.5400 0.0205

A 0.0222

2(eA) 0.0404

W0 0.1691 Overall

W* 0.8282 0.1718 Portfolio 0.0663 - 0.0546 - 0.0505 0.0596 0.1262 0.0246


SD 0.1358 0.2497 0.1422 0.3817 0.2901 0.1935 0.2611 0.1822 0.1988

Sharpe Ratio 0.44 0.35 0.4556

M-Square 0 - 0.0123 0.0019

Target price and alpha on June 1, 2006

The Optimal Risky Portfolio with the Analysts’ New Forecasts

The Optimal Risky Portfolio (WA < 1)

Drawback of the model

Extreme sensitivity to expected return assumptions

The results often run against investor intuition

Such quantitative optimization processes are rarely employed by managers

What about putting some constraints to this model?

Tracking error

Portfolios are often compared against a benchmark

Tracking error

Benchmark Risk: SD of Tracking error

MPE RRT

AAMAAAAP ewRwawR *** )]1(1[

AAMAAAAE ewRwawT *** )1(

2*22* )]([)]1([)( AAMAAE ewwTVar

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The Optimal Risky Portfolio with the Analysts’ New Forecasts

)()1()( 222*AMAAE ewT

Tracking error

Set weight in the active portfolio to meet the desired benchmark risk

For a unit investment in the active portfolio

For our example:

For a desired benchmark risk

Assume that the desired benchmark risk is 0.0385 Wa(Te)=0.0385/0.0885 Wa(Te)=0.43

)()]1()1;( 22*AMAAE ewT

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*0

AE

EEA wT

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)(0 ET0885.0)1;( * AE wT

Constrained benchmark risk

asset management lecture 6. outline for today treynor black model m2 measure of performance...

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