Question

A portfolio manager summarizes the input from the macro and micro forecasts in the following table:

a. Calculate expected excess returns, alpha values, and residual variances for these stocks.

b. Construct the optimal risky portfolio.

c. What is Sharpe’s measure for the optimal portfolio and how much of it is contributed by the active portfolio? What is the M² ?

Short answer

a. Residual variance is 0.3364

b. The position in index portfolio = 1.0486

c. S = .3662;M² =.423%

Step-by-step solution

Calculation of expected excess returns, alpha values, and residual variances

Alpha (a):α_i = E(r_i) – {r_f + β_i [E(r_M) –r_f ]} Expected excess return: E(r_i) – r_f

α_A = .20 – [ .08 + 1.3 ´ ( .16 – .08)] = 1.6% .20 – .08 = 12%

α_B = .18 – [ .08 + 1.8 ´ ( .16 – .08)] = -4.4% .18 - .08 = 10%

α_C = .17 – [ .08 + 0.7 ´ ( .16 – .08)] = 3.4% .17 - .08 = 9%

α_D = .12 – [ .08 + 1.0 ´ ( .16 – .08)] = - 4.0% .12 - .08 = 4%

Hence Stock A and C have positive alpha while stock B and D have negative alpha..

Their residual variances=

s²(e_A) = .58² = .3364

s²(e_B) = .71² = .3364

s²(e_C) = .60² = .3364

s²(e_D) = .55² = .3364

Calculation of optimal risk portfolio

For this let’s first create optimal active portfolio using Trteynor Black technique.

Portfolio return	Portfolio Weight (w0 = a/2(e) [E( rM ) － rf ]/ 2M)
.0476	-0.6136
-.0873	1.1261
.0944	-1.2185
-.1322	1.7060
-.0775	1.0000

Hence the forecast of the active portfolio =

α= [– .6136´.016] + [1.1261´(– .044)] – [1.2185´.034]

α= [– .6136´.016] + [1.1261´(– .044)] – [1.2185´.034]+ [1.7060´(– .04)] = –16.90%

b = [– .6136 ´ 1.3] + [1.1261 ´ 1.8] – [1.2185 ´ 0.70] + [1.7060 ´ 1.0] = 2.08

s²(e) = [(– .6136)² ´ .3364] + [1.12612 ´ .5041] + [(–1.2185)² ´ .36] + [1.70602 ´ .3025]

=2.18082

s(e) = 147.68

Since optimal portfolio has a proportion w* in the active portfolio, it is therefore:

W_{0 =}a/2 (e) / [E( r_M )－r_f ]/ 2M

= －. 1690/2.18082 / . 08/ .232

=-.05124

Adjustment beta = W* = W₀ / 1+1－b) w₀

= － . 05124 / 1 + (1－2.08 )´(－ . 05124 )

= -.0486

The position in index portfolio = 1 – (-0.0486)

= 1.0486

Calculation of Sharpe measure for optimal risk portfolio

For this, let’s first calculate appraisal ratio A.

A =a/(se)= – .1690/1.4768 = – .1144

A² = .0131

The square of Sharpe measure of optimized risk portfolio=

S²= S²_M+ A²

=(8/23)² + 0.0131

=0.1341

S = .3662

Comparison with S_M=8 / 23 = .3478

Difference = .3662 - .3478 = 0.184

M²is “Modigliani Squared” measure. This is calculated as follows:

E(r_P*) = r_f + S_P s_M = .08 + ( .3662´.23) = 16.423%

M² = E(r_P*) – E(r_M) = .16423 – .16 = .423%