PRM Study Guide : Exam III Question 1

[Anonymous]

The question is :
Under the standard parametric VaR methodology, which of the following assumptions is true?
a) Returns follow a log normal distribution
b) Log returns follow a normal distribution
c) Mean log return is zero for daily VaR
d) All of the above

The correct answer is : c)

I don’t understand why b) is not a correct answer ?

[kzg]

PRM Study Guide : Exam III Question 1

[Anonymous]

I think that (c) is true, but the question is whether (b) is also true. I think that it’s the log of “price relatives” that is Normal. Say you have stock prices on successive days of 100,110,104.5 which is returns of 10% and -5%. The price relatives are 110/100 = 1.1 and 104.5/110 = 0.95. It is the log of the numbers 1.1 and 0.95 (the price relatives) that are normally distributed, not the returns themselves (0.10 and -0.05). Suppose the price was 100 every day, meaning the price relative is 1, meaning ln(1) = 0….i.e., the mean return is zero which is what you want to see. But if return is zero, ln(0) = not defined.

[Anonymous]

Hi Jim,
What you call “log of price relatives” is by definition “log returns”.
So we agree that answer b is correct.

[Anonymous]

“log of price relatives” is the continuously compounded return as I see it. So it is correct to say that “continuously compounded returns are normal”. Asset prices are lognormal and continuously compounded asset returns are normal. I don’t believe that it is correct to say that log of returns is normal. Suppose that an asset grows from S(0) to S(T) in time T. If mu is the continuously compounded return per unit of time we have S(T) = S(0)exp(mu * T). This gives S(T)/S(0) = exp(mu * T) or ln[S(T)/S(0)] = mu * T. In the Black-Scholes analysis
ln[S(T)/S(0)]is normally distributed. So mu * T is the continuously compounded return over time T and is normally distributed.

[Anonymous]

Jim,

Well, what you call “continuously compounded returns” are the same as “log returns”
http://en.wikipedia.org/wiki/Rate_of_return#Logarithmic_or_continuously_compounded_return

So b) and c) are both correct answer.

[Anonymous]

Yes, I can see now from the Wikipedia page that the term “logarithmic returns” is synonymous with “continuously compounded returns”. What’s becoming clearer to me now is the issue of how you define returns. If you defined returns in a non-continuously compounded way, e.g., use a simple definition of return such as a rise from 100 to 110 being 10% return, then the log of these simple returns does not conform to a normal distribution as I see it. Or if the returns were defined from the outset as continuously compounded returns then the log of these continuously compounded returns would not be normal (because the continuously compounded returns themselves are normal). So I guess the term “log returns” is a bit loose. If it means logarithmic or continuously compounded returns, then Normal for sure, meaning (b) is definitely correct as you say. But I was taking it to possibly mean the log of simple returns, so I didn’t take it to be generally true.

[Author]

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