You make an investment. Assume that returns are normally distributed with a mean return of .20 per year and a standard deviation of .10. Suppose you check on your returns once a week. What is the probability that your return is positive for the week? We define the ratio of noise to performance as the coefficient of variation (the ratio of the standard deviation to the mean). Calculate the number of parts of noise per part performance if you check your returns once a week. In this problem, you can assume that weekly returns are independent of each other.
I've bolded what's throwing me off — the "per year" vs "for the week."
Best Answer
You are expected to assume that the weekly return is also a normal distribution. Leaving aside compounding, the average weekly return is $\frac {0.20}{52}$ We know that the variance of a sum is the sum of the variance, so compute the variance of the annual return from the standard deviation, divide by $52$ to get the weekly variance, then compute the weekly standard deviation. You should find much more noise in weekly data than yearly data.