Chapter 8, Problem 18PS

a.

Summary Introduction

To determine: The cost of restriction in terms of Sharpe’s measure

Introduction: The Sharpe ratio is used to measure the accumulated performance of an aggregate investment portfolio or an individual stock. It evaluates the performance of equity investment to the rate of return.

Answer to Problem 18PS

The cost of restriction in terms of Sharpe’s measure is in the cost

Explanation of Solution

Given Information:

Forecast returns, standard deviations and the beta values are given.

Sharpe ratio is mostly used to measure the risk return. For this, first compute the expected return on investment or individual stock and then subtract it from the risk free rate of return. Generally, when the ratio is greater than 1, it is considered as acceptable by the investors.

If the manager would not sell the securities he would consider only the stock A and C

	$a$	${\sigma }^2(e)$	$\frac{a}{{\sigma }^2(e)}$	$\frac{\frac{a}{{\sigma }^2}(e)}{\frac{S_a}{{\sigma }^2}(e)}$
A	1.6	3,364	0.00048	0.3352
C	3.4	3,600	0.00094	0.6648
Total			0.001420	1

The alpha of active portfolios,

  $\begin{array}{l}a=[0.3352\times 1.6]+[0.6648\times 3.4]\\ =2.80\%\end{array}$

The beta of active portfolio

  $\begin{array}{l}\beta =[0.3352\times 1.3]+[0.6648\times 0.7]\\ =0.90\end{array}$

So, the standard deviation of portfolio is,

  $\begin{array}{l}$ {\sigma }^2(e)=[{(0.3352)}^2\times {58}^2]+[{(0.6648)}^2\times {36}^2]$=1,969.03\end{array}$

  $\sigma (e)=\sqrt{1,969.03}=44.37\%$

The active risky portfolio,

  $W_0=\frac{\frac{a}{{\sigma }^2}(e)}{\left[\frac{E(r_M)-r_f}{{\sigma }_M{}^2}\right]}$

  $=\frac{\frac{2.80}{1,969.03}}{\frac{8}{{23}^2}}=0.0940$

The adjustment of beta,

  $w*=\frac{w_0}{1+(1-\beta )w_0}$

  $\begin{array}{l}=\frac{0.094}{1+[(1-0.90)\times 0.094]}\\ =0.0931\end{array}$

The information ratio for the active portfolio,

  $\begin{array}{l}A=\frac{a}{\sigma (e)}\\ =\frac{2.80}{44.37}=0.0631\end{array}$

The Sharpe ratio for the optimal portfolio,

  $\begin{array}{l}{S}^2=S_M{}^2+A^2\\ ={\left(\frac{8}{23}\right)}^2+{0.0631}^2=0.1250\end{array}$

  $\Rightarrow S=\sqrt{0.1250}=0.3535$

  $S_M=0.3478$

Calculation of Beta,

  $\begin{array}{l}{\beta }_P=W_M+(W_A\times {\beta }_A)\\ =(1-0.0931)+[(0.0931)\times 0.9]\\ =0.99\end{array}$

The expected return,

  $\begin{array}{l}(R_P)=a_P+{\beta }_{P}E(R_M)\\ =[(0.0931)\times (2.80\%)]+(0.99\times 8\%)]\\ =8.18\%\end{array}$

The variance,

  ${\sigma }_P{}^2=535.54$

  $\begin{array}{l}{\sigma }_P=\sqrt{535.54}\\ =23.14\%\end{array}$

As, A = 2.8, the optimal position is calculated as,

  $y=\frac{8.18}{0.01\times 2.8\times 535.54}=0.5455$

So, the share per each asset is,

BILLS	1-0.5455	45.45%
MARKET	0.5455x(1-0.0931)	49.47%
A	0.5455x (0.0931)x (0.3352)	1.70%
C	0.5455 x (0.0931)x (0.6648)	3.38%
		100.00%

b.

Summary Introduction

To determine: The utility loss to the investor in new complete portfolio

Introduction: The Sharpe ratio is used to measure the accumulated performance of an aggregate investment portfolio or an individual stock. It evaluates the performance of equity investment to the rate of return.

Answer to Problem 18PS

The utility levels for the unconstrained, constrained and passive strategy are 10.40%, 10.23% and 10.16% respectively.

Explanation of Solution

Given Information:

Forecast returns, standard deviations and the beta values are given.

Sharpe ratio is mostly used to measure the risk return. For this, first compute the expected return on investment or individual stock, then subtract it from the risk free rate of return. Generally, when the ratio is greater than 1, it is considered as acceptable by the investors.

The variance and for passive strategy, the calculations are

	E ( R)	${\sigma }^2$
UNCONSRAINED STRATEGY	4.79	170.95
CONSTRAINED STRATEGY	4.46	159.36
PASSIVE STRATEGY	4.32	154.31

By using the utility level formula, $E(r)-0.005A{\sigma }^2$

  $UnconstrainedStrategy=8\%+4.79\%-(0.005\times 2.8\times 170.95)=10.40\%$

  $ConstrainedStrategy=8\%+4.46\%-(0.005\times 2.8\times 159.36)=10.23\%$

  $PassiveStrategy=8\%+4.32\%-(0.005\times 2.8\times 154.31)=10.16\%$