Question

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

Micro Forecasts
Asset	Expected Return (%)			Beta	Residual Standard Deviation (%)
Stock A		25		0.8		52
Stock B		19		1.2		61
Stock C		16		0.6		55
Stock D		12		0.7		47

  

Macro Forecasts
Asset	Expected Return (%)			Standard Deviation (%)
T-bills		8			0
Passive equity portfolio		18			26

a. Calculate expected excess returns, alpha values, and residual variances for these stocks. (Negative values should be indicated by a minus sign. Do not round intermediate calculations. Round "Alpha values" to 1 decimal place.)

b. Compute the proportion in the optimal risky portfolio. (Do not round intermediate calculations. Enter your answer as decimals rounded to 4 places.)

c. What is the Sharpe ratio for the optimal portfolio? (Do not round intermediate calculations. Enter your answers as decimals rounded to 4 places.)

d. By how much did the position in the active portfolio improve the Sharpe ratio compared to a purely passive index strategy? (Do not round intermediate calculations. Enter your answers as decimals rounded to 4 places.)

e. What should be the exact makeup of the complete portfolio (including the risk-free asset) for an investor with a coefficient of risk aversion of 3.0? (Do not round intermediate calculations. Round your answers to 2 decimal places.)

Answer 1

a)

Expected excess return for Stock i = Expected return on Asset i - Risk-free rate of return

In the the question we can see the standard deviation for retun on T-bill is 0, this means that the risk-fee return = Return on T-bill = 8%

Using the above equation to calculate expected excess return of assets A,B,C,D

1)Expected excess return for Stock A = Expected return on asset A - 14%

= 25 -8 = 17%

2) Expected excess return for Stock B = 19 - 8 = 11%

3) Expected excess return for Stock C = 16 - 8 = 8%

4) Expected excess return for Stock D = 12 - 8 = 4%

To calculate Alpha

Alpha = Expected return on an asset - [Beta of that asset * ( Expected return on passive equity portfolio - Risk-free return)]

1) Alpha for asset A = 25 - [0.8*(18 - 8)] = 25 - 8 = 17% = 0.17

2) Alpha for asset B = 19 - [1.2*(18-8)] = 19-12 = 7% = 0.07

3) Alpha for asset C =16 - [0.6*(18 - 8)] = 16 - 6 = 10% = 0.10

4) Alpha for asset D = 12 - [0.7*(18 - 8)] = 12 - 7 = 5% = 0.05

Residual variance for asset i = ( residual standard deviation for stock i )²

using the above equation

1) Residual variance for Asset A = (0.52)² = 0.2704

2) Residual variance for Asset B = (0.61)² = 0.3721

3) Residual variance for Asset C = (0.55)² = 0.3025

4) Residual variance for Asset D = (0.47)² = 0.2209

b)

First we will calculate variance for each asset

variance for asset A = ( Beta of asset A * standard deviation of passive portfolio return )² + Variance of residual of asset A

= (0.8*0.26)² + 0.2704 = 0.313664

Standard deviation for asset A = (variance of asset A)^(1/2) = (0.313664)^(1/2) = 0.560057

Similarly ,

Variance for asset B = (1.2*0.26)² + 0.3721 = 0.469444

Standard deviation for asset B =  variance of asset B)^(1/2) = 0.68516

Variance for asset C = (0.6*0.26)² + 0.3025 = 0.326836

Standard deviation for asset C = 0.571696

Variance for asset D = (0.7*0.26)² + 0.2209 = 0.254024

Standard deviation for asset D = 0.504008

Now we will calculate initial position of each asset in active portfolio

Initial position of asset A = Standard deviation of asset A/variance of residual for asset A =

(0.560057/0.2704) = 2.071217

Initial position of asset B = Standard deviation of asset B/variance of residual for asset B=

(0.68516/0.3721) = 1.841333

Initial position of asset C = Standard deviation of asset C/variance of residual for asset C =

(0.571696/0.3025) = 1.889903

Initial position of asset D = Standard deviation of asset D/variance of residual for asset D =

(0.504008/0.2209) = 2.281611

Sum of these initial positions = 2.071217 +1.841333 + 1.889903 +2.281611 = 8.084064

Now we will scale these initial positions such that portfolio weights of active portfolio to sum to 1

Wa = weight of asset A in active portfolio

Wa = Initial position of asset A in active portfolio / sum of the initial positions of assets A,B,C,D

= 2.071217/8.084064 = 0.25621

Wb = 1.841333 /8.084064 = 0.227773

Wc = 1.889903 /8.084064 = 0.233781

wd = 2.281611 /8.084064= 0.282236

Now we will calculate the alpha of this active portfolio of asset A,B,C,D

alpha = weighted average of the alpha of individual assets

alpha = wa*(alpha a) + wb*(aplha b ) + wc*(alpha c) + wd*(alpha d)

= (0.25621* 0.17) + (0.227773 *0.07) + (0.233781*0.1) +( 0.282236*0.05)

= 0.09699

Residual variance of active portfolio = weighted average sum of the individua residual variances for each asset3

= (wa*residual variance for A) + (Wb*residual variance for asset B) + (Wc* residual variance for asset C) + (Wd* residual variance for C)

= (0.25621*0.2704) +(0.227773 *0.3721)+ (0.233781*0.3025) + (0.282236*0.2209)

= 0.287098

Initial position in the active portfolio , IA = [(alpha of active portfolio/residual variance of active portfolio)]/(excess return of market portfolio/variance of passive equity portfolio)

= [(0.09699/0.287098)]/(0.1/0.26*0.26) = 0.337828/1.47929 = 0.228371

Beta of active portfolio = (Wa*Beta of A) +(Wb*Beta of B) +(Wc*beta of C) + (Wd*Beta of D)

=( 0.25621*0.8) + (0.227773 *1.2) + (0.233781*0.6)+( 0.282236*0.7)

= 0.816129

adjusted initial position in active portfolio,WA = (IA/(1+((1-beta of active portfolio)*IA))

= 0.228371/(1+((1-0.816129)* 0.228371))

= 0.228371/1.041991 = 0.219168

Hence, Optimal risky portfolio now has weights:

Weight of passive portfolio = 1- adjusted initial position in active portfolio = 1-0.219168 = 0.780832 or 0.7808   (after rounding to 4 decimal places)

Weight of asset A = WA*Wa = 0.219168*0.25621 = 0.056153 or 0.0562 (after rounding to 4 decimal places)

Weight of asset B = WA*Wb = 0.219168*0.227773 = 0.049921 or 0.0499 (after rounding to 4 decimal places)

Weight of asset C = WA*Wc = 0.219168*0.233781 = 0.051237 or 0.0512 (after rounding to 4 decimal places)

Weight of asset D = WA*Wd = 0.219168*0.282236 = 0.061857 or 0.0619(after rounding to 4 decimal places)

c)

(Sharpe ratio of optimal risky portfolio)^2 =(sharpe ratio of passive portfolio)^2 + (alpha of active portfolio/standard deviation of residual of active portfolio)^2

sharpe ratio of passive portfolio = (Expected return of passive portfolio – risk-free return)/standard deviation of passive portfolio

= (18-8)/26 = 0.38461538

Standard deviation of residual of active portfolio = (residual variance of active portfolio)^(1/2) = (0.287098)^(1/2) = 0.535815

(alpha of active portfolio/standard deviation of residual of active portfolio)= 0.09699/0.535815

= 0.181013

Substituting the calculated values in the main equation

(Sharpe ratio of optimal risky portfolio)^2 = (0.38461538)^2 + (0.181013)^2 = 0.14792899 + 0.0327657 = 0.1806946

Sharpe ratio of optimal risky portfolio = (0.1806946)^(1/2) = 0.42508187 or 0.4251 ( rounding to 4 decimal places

d)

( Sharpe ratio of optimal risky portfolio - sharpe ratio of passive portfolio)/ sharpe ratio of passive portfolio

=( 0.42508187 – 0.38461538)/0.38461538 = 0.1052128 or 0.1052   (after rounding to 4 decimal places)