Mean and Standard Deviation

Question

I've been struggling to solve this problem for quite some time. The solution states that the mean = 0,4% and SD = 2,5%, but I can't understand why.

"The daily return on a stock has a mean of 0,2% and a standard deviation of 1,25%. We want to find the mean return and the return standard deviation for stock returns measures over a 2 day interval. Under the assumption that stock returns are independent from one day to the next, what are the mean and standard deviation of the two day returns?"

My logic is that since for 1 day the mean is 0,2% and SD is 1,25%, the same will remain true for 2 days, so the answer should be mean = 0,2% and SD=1,25%.

Thank you very much in advance.

Answer 1

I will first give an answer with many symbols. Let $X_1$ be the return on Day $1$, and $X_2$ the return on Day $2$. Let $Y=X_1+X_2$. We want the mean and standard deviation of $Y$.

In general, for any two random variables $U$ and $V$, and any real numbers $a$ and $b$, we have $$E(aU+bV)=aE(U)+bE(V).$$ If $U$ and $V$ ae independent random variables, then $$\text{Var}(aU+bV)=a^2\text{Var}(U)+b^2\text{Var}(V).$$

Let us apply that to our problem. We have $$E(Y)=E(X_1)+E(X_2).$$ In our case, we have $E(X_i)=0.2\%$, so $E(Y)=0.4\%$.

Also, we have $\text{Var}(X_i)=(1.25)^2$. Thus $\text{Var}(Y)=2(1.25\%)^2$, and therefore the standard deviation of $Y$ is $\sqrt{2}(1.25\%)$.

Remark: The problem is somewhat unclear. One could interpret it as asking for the mean and standard deviation of $Y/2$. But then neither of the official answers answer is correct.

You seem to have interpreted the question as asking about $Y/2$. That is very reasonable. Then the mean is indeed $0.2\%$. The variance is then, by the displayed formula for variance above, given by $\frac{1}{4}(1.25\%)^2+ \frac{1}{4}(1.25\%)^2$, giving standard deviation $\frac{1}{\sqrt{2}}(1.25\%)$.

Answer 2

The two-day return is the sum of two one-day returns; if they are each $0,2$% on average, then in two days you get on average a return of $0,4$%. For example, if the investment is £$1000$, the mean one-day return is £2, so the mean two-day return is £4. In more technical language, you’re adding two independent, identically distributed random variables, $X_1$ and $X_2$, to get a new one, $Y=X_1+X_2$; the mean (or expected value) of $Y$ is the sum of the means of $X_1$ and $X_2$.

However, standard deviations aren’t additive, and the standard deviation of $Y$ is neither $1,25$ nor $2,50$; it’s

$$\sqrt{1,25^2+1,25^2}\approx 1,77\;.$$