Post on 20-Dec-2015
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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Treynor Black Model
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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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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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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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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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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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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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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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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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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
AAMAAMp wREwwRE *** )()()(
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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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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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
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
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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
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
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The Optimal Risky Portfolio with the Analysts’ New Forecasts
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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
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Constrained benchmark risk