SD(X)=SD(log(X))? (if X is lognormal)

[ajsa]

Hi David,

Could yuo take a look of this one? Does it mean SD(X)=SD(log(X)) if X is lognormal?

Thanks.

EXAMPLE 11.1: FRM EXAM 1997–QUESTION 14
What is the implied correlation between JPY/EUR and EUR/USD when given
the following volatilities for foreign exchange rates?
JPY/USD at 8%
JPY/EUR at 10%
EUR/USD at 6%.
a. 60%
b. 30%
c. −30%
d. −60%

 

[David Harper CFA FRM]

This looks like this: http://forum.bionicturtle.com/viewthread/1684/
...which confused me...but I did not consider the lognormal distributional assumption...
...is the answer given as -0.6?

If so, then *apparently* you are right and it assumes lognormal, such that
LN(JPY/USD) ~ N(.)
and as JPY/EUR * EUR/USD = JPY/USD
(this is why it confused me, I thought it needed the variance of a product, which is nontrivial)

but LN(JPY/EUR) + LN(EUR/USD) = LN(EUR/USD)
i.e., LN (x*y) = LN(X) + LN(Y)

so we can use variance of sum:
8%^2 = 10%^2 + 6%^2 + 2*10%*6%*correlation(), and
correlation() = (8%^2 - 10%^2 - 6%^2) / (2*10%*6%) = -0.60

...although i am still a bit puzzled, i wouldn't have thought of LN() unless you mentioned it...must refer to an old reading (?) ...further note the functional difference given by FullofQuestion's answer (his answer applies a different variance formula?)

Thanks, David

 

[ajsa]

Hi David,

you are correct. it is -0.6. but what puzzled me is it seems it assumes SD(LN(JPY/USD))=SD(JPY/USD), right?

Thanks.

 

[David Harper CFA FRM]

Hi ajsa,

i totally agree ... i think we have the same confusion from different angles. When they say "JPY/USD at 8%," it appears they mean LN(JPY/USD) is normal and, just as you say, SD(LN(JPY/USD) = 8% ... but i don't see why you can make that assumption, nor is it correct: if LN(X) is normal with variance Y, that does not imply X has variance Y (see variance of lognormal @ http://en.wikipedia.org/wiki/Log-normal_distribution) ... so the question makes hidden assumptions I think ...David

 

[ajsa]

Thanks David,, The more I do those old practice questions, I more I hope GARP has already improved to make their questions more clear (with least assumptions or guessworks). Do you think if it is the case? (I have not done 2009 practice yet..)

 

[David Harper CFA FRM]

Hi ajsa,

I really don't know ... it's a major concern of mine ... I've written long briefs detailing the issues (e.g., 5 pages of definitional problems) ... I don't think you should have to "hope." I think you've spent a lot of time and a lot of money and I think you have an absolute *right,* as a paying customer, to an exam that has (i) accurate, fair, technically proper, *vetted* questions that (ii) are aligned with the assignments.

Thanks, David

(i edited my original reaction...)