Chapter 6, Problem 11PS

Summary Introduction

Introduction: Portfolio consists of assets which have different rates as well as different risk associated with the assets, a risk free asset and asset that contains risk, both have a certain percentage of return.

To calculate: the utility levels of each portfolio for an investor with A=2.

Utility of portfolio can be calculated by the formula provided below:

  $\begin{array}{c}U=E\left(r\right)-\frac{1}{2}A{\sigma }^2\\ U=\text{ Utility}\\ E\left(r\right)=\text{Expected Return}\\ A=\text{Risk Aversion Coefficient}\\ \sigma \text{= Standard Deviation}\end{array}$

Calculating utility for each investment taking A=2

Return of portfolio (Return $\times$ Weight of bills $$	Standard deviation of portfolio (Standard deviation $\times$ weight of bills)	Variance of Portfolio	Weight of bills	Weight of Index	U(A=2)
0.130	0.20	0.0400	0.0	1.0	$U=0.13-\frac{1}{2}\times 2\times {\left(0.20\right)}^2=0.090$
0.114	0.16	0.0256	0.2	0.8	$U=0.114-\frac{1}{2}\times 2\times {\left(0.16\right)}^2=0.0884$
0.098	0.12	0.0144	0.4	0.6	$U=0.098-\frac{1}{2}\times 2\times {\left(0.12\right)}^2=0.0836$
0.082	0.08	0.0064	0.6	0.4	$U=0.082-\frac{1}{2}\times 2\times {\left(0.08\right)}^2=0.0756$
0.066	0.04	0.0016	0.8	0.2	$U=0.066-\frac{1}{2}\times 2\times {\left(0.04\right)}^2=0.0644$
0.050	0.00	0.0000	1.0	0.0	$U=0.050-\frac{1}{2}\times 2\times {\left(0.0\right)}^2=0.050$

Explanation of Solution

Utility of portfolio can be calculated by the formula provided below:

  $\begin{array}{c}U=E\left(r\right)-\frac{1}{2}A{\sigma }^2\\ U=\text{ Utility}\\ E\left(r\right)=\text{Expected Return}\\ A=\text{Risk Aversion Coefficient}\\ \sigma \text{= Standard Deviation}\end{array}$

Calculating utility for each investment taking A=2

Return of portfolio (Return $\times$ Weight of bills $$	Standard deviation of portfolio (Standard deviation $\times$ weight of bills)	Variance of Portfolio	Weight of bills	Weight of Index	U(A=2)
0.130	0.20	0.0400	0.0	1.0	$U=0.13-\frac{1}{2}\times 2\times {\left(0.20\right)}^2=0.090$
0.114	0.16	0.0256	0.2	0.8	$U=0.114-\frac{1}{2}\times 2\times {\left(0.16\right)}^2=0.0884$
0.098	0.12	0.0144	0.4	0.6	$U=0.098-\frac{1}{2}\times 2\times {\left(0.12\right)}^2=0.0836$
0.082	0.08	0.0064	0.6	0.4	$U=0.082-\frac{1}{2}\times 2\times {\left(0.08\right)}^2=0.0756$
0.066	0.04	0.0016	0.8	0.2	$U=0.066-\frac{1}{2}\times 2\times {\left(0.04\right)}^2=0.0644$
0.050	0.00	0.0000	1.0	0.0	$U=0.050-\frac{1}{2}\times 2\times {\left(0.0\right)}^2=0.050$