Topic $12$ Portfolio Theory

Exercise $1$: One Risk free and One risky asset

The simplest example is a portfolio containing two assets: a risk$-$free Treasury bill

and a stock. Assume that the expected return of the Treasury bill is $3\%$ and its risk is

$0\%.$ Further, assume that the expected return of the stock is $10\%$ and its standard

deviation is $20\%.$

If the investor allocated $25\%$ to the risk$-$free asset and $75\%$ to the risky asset:

a$)$ Calculate the expected return of the portfolio.

b$)$ Calculate the risk of the portfolio.

Exercise $2$: Capital Allocation Line $($CAL$)$

Assume that the average rate of return on the S&P$500$ portfolio is $8\%$ more than the

T$-$Bill return and that the S&P$500$ standard deviation is $22\%$ per year. Assume that

these values are representative of the expectations of the representative investor

for future performance and that the current T$-$Bill rate is $3\%.$

Portfolio Weights in T$-$Bills Weights in S&P$500$

$10\%100\%$

$250\%50\%$

$3100\%0\%$

a$)$ Calculate the expected return and variance of the following portfolios

invested in T$-$bills and the S&P$500$ index with the following weights.

b$)$ Given a level of risk aversion, if the investor wanted to reduce his risk level,

what would he have to do in his portfolio composition?

c$)$ Calculate the level of risk associated with each portfolio.

d$)$ Sketch the CAL.

e$)$ If the investor wanted to reduce his risk to $5\%,$ how would his portfolio look like?

Exercise $3$: Security Market Line $($SML$)$

Consider the following table which gives a security analyst$$s expected return on two

stocks for two particular market returns.

Market Return Aggressive Stock $($A$)$ Defensive Stock $($B$)$

$5\%-2\%6\%$

$25\%38\%12\%$

a$)$ Calculate the betas of these two stocks.

b$)$ What is the expected rate of return on each stock if the market return is

equally likely to be $5\%$ or $25\%?$

c$)$ If the T$-$Bill rate is $6\%$ and the market return is equally likely to be $5\%$ or $25\%,$

define the CAPM equation.