You have $1 million to invest. You can only invest in two stocks, A and B, with the following annualized expected returns and volatilities (i.e. standard deviation of return):
Stock Expected Return Volatility
A 10 % 10 %
B 15 % 20 %
Assume that interest rates are zero, and that the stocks’ returns are independent. Find the fully invested portfolios that maximize the Sharpe ratio, which is defined as the ratio between expected return and volatility. What is the maximum Sharpe ratio you can achieve by combining investments in A and B in this way?
Attempt: I believe the expected returns are
$E[R] = (1-w)*.10 + w*.15 = .10 + .05w$
However, the I am not sure how to set up the volatility.
I would appreciate a thorough solution and explanation of this problem.
Hints:
Let $1-w$ be the proportion of stock A. Therefor $w$ is the proportion of stock B. w is a constant.
The expected value of the portfolio is $\mathbb E(R_p)=(1-w)\cdot 0.1+w\cdot 0.15$
And the variance of the portfolio (independent random variables) is
$Var(R_p)=(1-w)^2\cdot 0.1^2+w^2\cdot 0.2^2$
Consequently the sharpe ratio (with a risk free rate of $0$) is
$$S_p(w)=\frac{\mathbb E(R_p)}{\sqrt{Var(R_p)}}=\frac{(1-w)\cdot 0.1+w\cdot 0.15}{\sqrt{(1-w)^2\cdot 0.1^2+w^2\cdot 0.2^2}}$$
Then calculate $\frac{dS_p}{dw}$ by using the quotient rule. At the next step you take the numerator of $\frac{dS_p}{dw}$ and set it equal to $0$ and solve this equation for $w$. Finally you insert the value $w^*$ into $S_p(w)$
$\small{\color{grey}{\texttt{The maximum of the sharpe ratio in this case is 1.25}}}$