Finding the expected rate of return

Question

Given the problem:

The risk-free rate (per annum with semi-annual compounding) is 5%.

You have a portfolio worth 10000 dollars today. Let X be the return of your portfolio in 6 months.

Suppose the p.d.f. of X is given by

$fX(x)$ = $\frac{x + 50} {80000}$ if − 50 ≤ x ≤ 350 and 0 otherwise

Find the expected rate of return.

To calculate this rate, is it just finding $\mu$ in the formula:

E[$(S_\frac{6}{12}$)]= $S_0$$e^{\mu(.05)}$

Also, is the pdf relative in this instance?

Answer 1

Your question says that $X$ is the return and its pdf $f(x)$ has support $[-50,350]$. So the expected return is the expected value of $X$, that is $$E(X) = \int_{-50}^{350} x f(x) dx = \int_{-50}^{350} x \frac{x + 50} {80000} dx = \frac{650}{3} \approx 216.67$$ Then the expected rate of return is $$\frac{216.67}{10000} = 0.02167$$ or approximately $2.17\%$.