# Index

**Topics:**Variance, Sharpe ratio, Modern portfolio theory

**Pages:**4 (410 words)

**Published:**April 13, 2014

Alpha (α)

Expected excess return

αi = ri – [rf + βi × (rM – rf ) ]

E(ri ) – rf

αA = 20% – [8% + 1.3 × (16% – 8%)] = 1.6%

20% – 8% = 12%

αB = 18% – [8% + 1.8 × (16% – 8%)] = – 4.4%

18% – 8% = 10%

αC = 17% – [8% + 0.7 × (16% – 8%)] = 3.4%

17% – 8% = 9%

αD = 12% – [8% + 1.0 × (16% – 8%)] = – 4.0%

12% – 8% = 4%

Stocks A and C have positive alphas, whereas stocks B and D have negative alphas. The residual variances are:

2(eA ) = 582 = 3,364

2(eB) = 712 = 5,041

2(eC) = 602 = 3,600

2(eD) = 552 = 3,025

b.To construct the optimal risky portfolio, we first determine the optimal active portfolio. Using the Treynor-Black technique, we construct the active portfolio:

A

0.000476

–0.6142

B

–0.000873

1.1265

C

0.000944

–1.2181

D

–0.001322

1.7058

Total

–0.000775

1.0000

Be unconcerned with the negative weights of the positive α stocks—the entire active position will be negative, returning everything to good order.

With these weights, the forecast for the active portfolio is: α = [–0.6142 × 1.6] + [1.1265 × (– 4.4)] – [1.2181 × 3.4] + [1.7058 × (– 4.0)] = –16.90%

β = [–0.6142 × 1.3] + [1.1265 × 1.8] – [1.2181 × 0.70] + [1.7058 × 1] = 2.08 The high beta (higher than any individual beta) results from the short positions in the relatively low beta stocks and the long positions in the relatively high beta stocks. 2(e) = [(–0.6142)2×3364] + [1.12652×5041] + [(–1.2181)2×3600] + [1.70582×3025] = 21,809.6

(e) = 147.68%

The levered position in B [with high 2(e)] overcomes the diversification effect, and results in a high residual standard deviation. The optimal risky portfolio has a proportion w* in the active portfolio, computed as follows:

The negative position is justified for the reason stated earlier. The adjustment for beta is:

Since w* is negative, the result is a positive position in stocks with positive alphas and a negative position in stocks with negative alphas. The position in the index...

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