Chapter 12, Problem 8QP

a.

Summary Introduction

To determine: Choosing between appropriate risk-averse person prefer to invest.

Introduction:

Arbitrage Pricing Theory (APT) is a substitute form of CAPM (Capital Asset Pricing Model). This hypothesis, as CAPM gives financial specialists or investors assessed required rate of return for the risky securities. APT reflects on risk premium premise indicated set of elements notwithstanding the correlation of the cost of the asset with expected surplus return on the portfolio.

Answer to Problem 8QP

The investor should prefer second market.

Explanation of Solution

In order to prefer the suitable investment it is required to calculate the variance of the portfolios formed by the various stocks from market. The diversification is effective here and it is rational that each investor prefers their own market to invest and buy from the market. We know the variables,

$\begin{array}{l}{E}_F=0\\ \sigma =0.10and\\ E_{\epsilon }=0\\ S_{\epsilon i}=0.20foralli\end{array}$

Suppose if we believe that the stocks in portfolio are weighted equally, the weight of each stock is $\left(\frac{1}{N}\right)$ which means,

$\begin{array}{l}{X}_i=\left(\frac{1}{N}\right)\\ foralli\end{array}$

Suppose if a portfolio is formulated with many stocks with a percentage of $\frac{1}{N}$ of total portfolio, the total return on the portfolio is $\frac{1}{N}$ times the total returns on N stocks. If we need to calculate the variance of each portfolio from the 2 markets it is required to implement the meaning of variance from statistics as below,

$Var\left(X\right)=\left[E\times {\left(X-E\left(X\right)\right)}^2\right]$

In our scenario,

$Var\left(R_p\right)=\left[E\times {\left(R_P-E\left(R_P\right)\right)}^2\right]$

To implement the above we need to calculate R_P and E(R_P). By using the supposition on equal weights and substituting the equation for R_i,

$\begin{array}{l}{R}_P=\left[\frac{1}{N}{\displaystyle \sum R_i}\right]\\ R_P=\left[\frac{1}{N}{\displaystyle \sum \left(0.10+\beta F+{\epsilon }_i\right)}\right]\\ R_P=0.10+\beta F+\frac{1}{N}{\displaystyle \sum {\epsilon }_i}\end{array}$

We evoke from statistics a property of expected value which is if,

$\tilde{Z}=a\tilde{X}+\tilde{Y}$

Here “a” is constant and $\tilde{Z}$, $\tilde{X}$ and $\tilde{Y}$ are stated as unsystematic variables and,

$\begin{array}{l}E\left(\tilde{Z}\right)=E\left(a\right)E\left(\tilde{X}\right)+E\left(\tilde{Y}\right)\\ and\\ E\left(a\right)=a\end{array}$

We use the above equation to find $E\left(R_P\right)$

$\begin{array}{l}E\left(R_P\right)=E\times \left(0.10+\beta F+\left(\frac{1}{N}\right){\displaystyle \sum {\epsilon }_i}\right)\\ E\left(R_P\right)=0.10+\beta E\left(F\right)+\left(\frac{1}{N}\right){\displaystyle \sum E\left({\epsilon }_i\right)}\\ E\left(R_P\right)=0.10+\beta \left(0\right)+\left(\frac{1}{N}\right){\displaystyle \sum 0}\\ E\left(R_P\right)=0.10\end{array}$

Now we combine and substituting both the results into a equation for variance,

$\begin{array}{l}Var\left(R_P\right)=E\left[{\left(R_P-E\left(R_P\right)\right)}^2\right]\\ Var\left(R_P\right)=E\left[{\left(0.10+\beta F+\left(\frac{1}{N}\right){\displaystyle \sum {\epsilon }_i-0.10}\right)}^2\right]\\ Var\left(R_P\right)=E\left[{\left(\beta F+\left(\frac{1}{N}\right){\displaystyle \sum \epsilon }\right)}^2\right]\\ Var\left(R_P\right)=E\left[{\left({\beta }^2{F}^2+2\beta F\times \left(\frac{1}{N}\right){\displaystyle \sum \epsilon +}\left(\frac{1}{N^2}\right){\left({\displaystyle \sum \epsilon }\right)}^2\right)}^2\right]\\ Var\left(R_P\right)=\left[{\left({\beta }^2{\sigma }^2+\left(\frac{1}{N}\right){\sigma }^2\epsilon +\left(1-\left(\frac{1}{N}\right)\right)Cov\left({\epsilon }_i,{\epsilon }_j\right)\right)}^2\right]\end{array}$

The last point is since there is many stocks in each market as we require the limit is $N\to \infty ,\left(\frac{1}{N}\right)\to 0$ and we get

$Var\left(R_P\right)={\beta }^2{\sigma }^2+Cov\left({\epsilon }_i,{\epsilon }_j\right)$

Because

$Cov\left({\epsilon }_i,{\epsilon }_j\right)=\left[{\sigma }_i{\sigma }_j\rho \left({\epsilon }_i,{\epsilon }_j\right)\right]$

Additionally the question says that ${\sigma }_1={\sigma }_2=0.10$ therefore,

$\begin{array}{l}Var\left(R_P\right)=\left[{\beta }^2{\sigma }^2+{\sigma }_1{\sigma }_2\rho \left({\epsilon }_{i,}{\epsilon }_j\right)\right]\\ Var\left(R_P\right)=\left[{\beta }^2\left(0.01\right)+0.04\rho \left({\epsilon }_{i,}{\epsilon }_j\right)\right]\end{array}$

Summary of Equations

$\begin{array}{l}{R}_{1i}=\left[0.10+1.5F+{\epsilon }_{1i}\right]\\ R_{2i}=\left[0.10+0.5F+{\epsilon }_{2i}\right]\\ E\left(R_{1P}\right)=E\left(R_{1P}\right)=0.10\\ Var\left(R_{1P}\right)=\left[0.0225+0.04\rho \left({\epsilon }_{1i},{\epsilon }_{1j}\right)\right]\\ Var\left(R_{2P}\right)=\left[0.0025+0.04\rho \left({\epsilon }_{2i},{\epsilon }_{2j}\right)\right]\end{array}$

Now we choose between appropriate risk-averse person prefer to invest,

By substituting $\rho \left({\epsilon }_{1i},{\epsilon }_{1j}\right)=\rho \left({\epsilon }_{2i},{\epsilon }_{2j}\right)=0$ into the particular variance formula,

$\begin{array}{l}Var\left(R_{1P}\right)=0.0225\\ Var\left(R_{2P}\right)=0.0025\end{array}$

Since the $Var\left(R_{1P}\right)>Var\left(R_{2P}\right)$ expected returns are equal a risk averse, the variance of second market is less than the first market.

Therefore investor should prefer the second market

b.

Summary Introduction

To determine: Choosing between appropriate risk-averse person prefer to invest.

Answer to Problem 8QP

The investor should prefer second market.

Explanation of Solution

Determine the Variance for each Portfolio

Assuming that $\rho \left({\epsilon }_{1i},{\epsilon }_{1i}\right)=9$ and $\rho \left({\epsilon }_{2j},{\epsilon }_{2j}\right)=0$ we calculate the variance of each portfolio as,

$\begin{array}{c}Var\left(R_{1P}\right)=\left[0.0225+0.04\rho \left({\epsilon }_{1i},{\epsilon }_{1j}\right)\right]\\ =\left[0.0225+0.04\times 9\right]\\ =0.0585\end{array}$

$\begin{array}{c}Var\left(R_{2P}\right)=\left[0.0025+0.04\rho \left({\epsilon }_{2i},{\epsilon }_{2j}\right)\right]\\ =\left[0.0025+0.04\times 0\right]\\ =0.0025\end{array}$

Since the $Var\left(R_{1P}\right)>Var\left(R_{2P}\right)$ expected returns are equal a risk averse, the variance of second market is less than the first market.

Therefore investor should prefer the second market

c.

Summary Introduction

To determine: Choosing between appropriate risk-averse person prefer to invest.

Answer to Problem 8QP

The investor should be indifferent between the two markets.

Explanation of Solution

Determine the Variance for each Portfolio

Assuming that $\rho \left({\epsilon }_{1i},{\epsilon }_{1i}\right)=0$ and $\rho \left({\epsilon }_{2j},{\epsilon }_{2j}\right)=0.5$ we calculate the variance of each portfolio as,

$\begin{array}{c}Var\left(R_{1P}\right)=\left[0.0225+0.04\rho \left({\epsilon }_{1i},{\epsilon }_{1j}\right)\right]\\ =\left[0.0225+0.04\times 0\right]\\ =0.0225\end{array}$

$\begin{array}{c}Var\left(R_{2P}\right)=\left[0.0025+0.04\rho \left({\epsilon }_{2i},{\epsilon }_{2j}\right)\right]\\ =\left[0.0025+0.04\times 0.5\right]\\ =0.0225\end{array}$

Since the $Var\left(R_{1P}\right)=Var\left(R_{2P}\right)$ expected returns are equal a risk averse, indifferent between the two markets.

Therefore investor should be indifferent between the two markets

d.

Summary Introduction

To determine: The Relationship between the Correlations of Disturbances in two markets.

Explanation of Solution

It is found that the expected return are equal the indifference involves that the variance of portfolio of the two markets are identical. As a result we set the variance formula identical and calculate for the correlation of a market with the other market.

$\begin{array}{l}Var\left(R_{1P}\right)=Var\left(R_{2P}\right)\\ 0.0225+0.04\rho \left({\epsilon }_{1i},{\epsilon }_{1j}\right)=0.0025+0.04\rho \left({\epsilon }_{2i},{\epsilon }_{2j}\right)\\ \rho \left({\epsilon }_{2i},{\epsilon }_{2j}\right)=\rho \left({\epsilon }_{1i},{\epsilon }_{1j}\right)+0.50\end{array}$

Hence for each set of correlations that have a relationship like in part c, a risk averse investor is entitled as indifferent between the two markets.