You make an investment. Assume that returns are normally distributed with a mean

Question

You make an investment. Assume that returns are normally distributed with a mean annual 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. Please write out all work.

Explanation / Answer

I am assuming that trading happens for 365 days or 52 weeks in a year. (In reality, trading only happens for 252 days in an year)

Now, I am also assuming that weekly returns are normally distributed.

Let us call E(W) be the mean weekly return

since every week return is independent of each other,

52*E(W) = E(Y)

E(W) = 0.2/52 = 0.002

Simlarly

52*Var(W) = Var(Y)

Var(W) = Var(Y) / 52

thus, std. deviation (W) = std. deviation(Y)/sqare root(52) = 0.1/sqare root(52) = 0.028

P (W>0) = P(W-mean/sd > -mean/sd ) = P (Z> -0.07) = 0.527

Noise to performance = std. dev / mean = 14

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