Question

(20 points) Suppose that the return of stock A is normally distributed with mean 4% and standard deviation 5%, the return of stock B is normally distributed with mean 8% and standard deviation 10%, and the covariance between the returns of stock A and stock B is -30(%)2. Now you have an endowment of 1 dollar, and you decide to İnvset w dollar in stock A and 1 - w dollar in stock B. Let rp be the overall return of your portfolio, then Answer the following questions using w 0.5 (a) Compute the expectation and variance of your portfolio. (i.e., Elrp and Var(rp)) (b) Find the 5th percentile of rp. (This quantity is often called the 5% Value at Risk.) (c) If Cov(rA,TB) = 301%)2 rather than-30(%)2, find EPPI and Var(rp) agian. (d) Would Cov(rA, TB) have an impact on E[rp? (e) In order to more efficiently reduce the risk (i.e., variance) of the investment portfolio through diversification, should one look for financial assets that are positively correlated or negatively correlated?

Answer 1

	A	B
E	4%	8%
V	$(5\%)^{2}$	(10%)2

Cor( ra , rb)--30%
$\omega =0.5$

on ITb

a.

Taking expectation

$E(r_{p})=\omega E(r_{a})+(1-\omega)E(r_{b})$

0.5 * 4%+ 0.5 * 8%

E(rp) 5%.

Taking Variance

Using formula V(ax + by) = $a^{2}V(x)+b^{2}V(y)+2Cov(x,y)S_{a}S_{b}$

$V(r_{p})=\omega^{2} V(r_{a})+(1-\omega)^{2}V(r_{b})-2Cov(r_{a},r_{b})*\omega *(1-\omega )$

0.52 * (5%)-+ 0.52 * (10%), + (-30%" * 5 * 10)

Var(rp) = 29.75%% (5.454% )

Final Ans:
E(rp) 5%.
and
Var(rp) = 29.75%% (5.454% )

b.

Since returns on asset a and asset b both follow normal dist our portfolio return will follow normal dist as well

$r_{p}\sim N(6\%, 5.454\%)$

$z=\frac{x-0.06}{0.05454}$

VaR is the value of loss that will not exceed $\alpha$ level in (1 - $\alpha$) confidence level for a given period of time. For normal distribution$P(x<t)=\alpha$

Where t = - VaR and $\alpha =5\%$

$P(x<-VaR)=0.05$

$P(z<\frac{-VaR-0.06}{0.05454})=0.05$

$\frac{-VaR-0.06}{0.05454}=1.6449$

VaR = - 0.1497 (A loss of 14.97% of value invested)

Final Ans: For $1 invested VaR is $0.1497 or 14.97p.

c.

$Cov(r_{a},r_{b})=30\%^{2}$

$V(r_{p})=\omega^{2} V(r_{a})+(1-\omega)^{2}V(r_{b})-2Cov(r_{a},r_{b})*\omega *(1-\omega )$

$0.5^{2}*(5\%)^{2}+0.5^{2}*(10\%)^{2}+30\%^{2}*5*10$

$Var(r_{p})=32.75\%\%=(5.722\%)^{2}$

d.

$Cov(r_{a},r_{b})$ does not have any impact on $E(r_{p})$. Since expected returns are irrespective of the covariance.

e.

In order to diversify portfolio, one should invest in negatively related assets. This is because the movements in the assets returns will be different directions due to the changes. This is not beneficial if we were going to get profits on both assets. But we assume that the investors are risk averse. Therefore we look from the point of view of losses, so if we are facing losses on one asset we will have profit on the other asset. Otherwise, we will have losses on both assets.

From b. and c. we can see that for -vely correlated assets Variance was lesser than that of +vely correlated. Smaller variance means less riskier portfolio.