Stock& Watson question 206.2

[orit]

Hi David,
Can you please help me to understand the rational behind the calculation, in particular:
Why do you divide the annual volatility 10% by five to get the five year horizon? what do you mean by the volatility of the average? (I don't recall from our study notes that we simply divide the volatility by the number of years)

Thanks,
Orit

 

[David Harper CFA FRM]

Hi Orit,

There is discussion on the source Q&A here at http://forum.bionicturtle.com/threads/p1-t2-206-variance-of-sample-average.5271/

I think it helps to keep in mind that we are referring to random variables, here.
The question assumes a random variable, we can call it R, that characterizes a normal distribution, where R ~N(9%, 10%^2), where we are assuming R represents an annual (one-year) return.

Note that a five-year period can thusly be viewed as a sequence of five of these random variables, R.

We can ask at least two questions, among others, about this five-year sequence:

1. What is the five-year volatility; i.e., the volatility over a five year period instead of one year?
2. What is the average return over the five-year period; i.e., what is [R(1) + R(2) + R(3) + R(4) + R(5)]/5 ?

The first question wants to scale volatility, such that if the variables are i.i.d., we can infer the 5-year volatility = 10%*SQRT(5).
The second question, however, invokes the central limit theorem (CLT), such that the AVERAGE of (n) returns tends toward zero as (n) increases. CLT tells us that if R ~N(9%, 10%^2), then the the sample average has variance of 10%^2/n. Because this question is concerned with the AVERAGE of five years as the random variable.

If it helps, we can think of rolling a large number (n) of six-sided die. As (n) increases:

1. the variance of the sum of the dice naturally increases; in fact, the variance of the sum of n dice is n*2.92 since 2.92 is the variance of one die.
2. the variance of the (sample) average of the dice decreases per CLT; in fact, although the individual die has a uniform distribution, CLT tells us the variance of the average of (n) dice is 2.92/n

 

[orit]

Thank you David!! it is really challenging to understand that this question relates to a sample of 5 stocks..

 

[David Harper CFA FRM]

Hi orit, 206.2 refers only to a single stock: "A stock has an expected (i.i.d.) return of 9.0% per annum and volatility of 10% per annum."
It's not five stocks, but rather a single random variable (annual return), drawn five times (five years). Thanks,

 

[orit]

Thanks David, now it's clear