X6 Q5 (ii) query

[Mohammed]

Hi, I'd appreciate if someone could help with this query I have regarding X6 Q5 (ii).
From the solutions:

I would have thought it should be like this:
Let R(t) is the random variable denoting portfolio returns over a time period t. Then
R(30 days) = (R(annual)/365) * 30
Hence Var(R(30 days)) = (30/365)^2 * Var(R(annual))
Hence s.d. (30 days) = (30/365) * s.d.(R(annual))
= (30/365) * 10.82
= 0.8893151

Why does the solution have sqrt(30/365) rather than 30/365?

 

[Colin McKee]

Standard deviation of a time series increases with the root of time. So if a variable has a 5% standard deviation over 30 days, it will have a 5% *(2^0.5) over 60 days. Variance of a time series increases with t. So if you are given volatility or standard deviation over 365 days, and you want the volatility over 30 days, then multiply by root 30, and divide by root 365. If you are given the variance, you would multiply it by 30/365.

 

[Mohammed]

Standard deviation of a time series increases with the root of time. So if a variable has a 5% standard deviation over 30 days, it will have a 5% *(2^0.5) over 60 days. Variance of a time series increases with t. So if you are given volatility or standard deviation over 365 days, and you want the volatility over 30 days, then multiply by root 30, and divide by root 365. If you are given the variance, you would multiply it by 30/365.

Thanks for your response. Please could you provide a proof or reference to the course notes (CMP) for "Standard deviation of a time series increases with the root of time"?

 

[Colin McKee]

Not sure if there is a proof anywhere. If X(i) are independent variables, such as the return of an asset over a period of time such as a day, then the variance of the time series over N periods would be Var(X(1) + X(2) + ... X(N) ) and if we assume that all time periods are the same and independent, then that would be Var ( X(1) )+Var( (X(2)) + ... + Var (X(N) ) which is N * Var (X(1)). So variance increases with time, and standard deviation increases with root time.

 

[Mohammed]

Ok, thanks. I think I see why the solution is the way it is.
However, you did assume that the one-day returns are iid random variables, which we don't necessarily know from the question.
Just to illustrate my point, we could've said

R(annual) = R(30 days)/30 * 365, leading to s.d. (30 days) = (30/365) * s.d.(R(annual)). Or even an obscure example

R(annual) = 36 * R(first 30 days in year) - 24 * R(last 30 days in year), giving
Var(R(annual)) = 36^2 * Var(R(first 30 days)) + 24^2 * Var(R(last 30 days)) - 2*36*24 * Cov(R(first 30 days), R(last 30 days))
Hence, even assuming the first and last 30 days in a year have the same return distribution and are uncorrelated, we could get
s.d.(R(Annual)) = sqrt(36^2 + 24^2) * s.d.(R(first/last 30 days)) = 43.26662 * s.d.(R(first/last 30 days))

I'm happy with your response, though. I was just concerned that I had missed out some information (possibly from earlier subjects), but I don't think this is the case.