Mean and Variance of Lognormal Distributions

1. Oct 13, 2011

royalstatus

1. The problem statement, all variables and given/known data
Given that the natural log of the growth consumption rate is conditionally normally distributed. I am trying to convert it to a lognormal distribution, but I keep getting a variance that is different from what is in the solution manual. The problem is #3 in the document below
http://www.wiwi.uni-frankfurt.de/profs/faia/welcome_files/ch2ans.pdf [Broken]

2. Relevant equations
Log Ct+1/Log Ct ~ N[Et(logCt+1/logCt), Var (logCt+1/logCt)]

3. The attempt at a solution

I re-wrote Log Ct+1/Log Ct as an exponential function since log and e cancel out, and the mean is simply the expected value of function, but the variance I got was -1/2σ(Var Ct+1/Ct) which is different from what the solution manual says.

My problem is I dont understand how they got 1/2σ^2(Var Ct+1/Ct).
If anyone can explain that to me, I would be highly grateful.

For a clearer understanding of the question, you can look at #3 from the Solution Manual I posted above. It is Chapter 2, # 3 from Obstfeld and Rogoff's Foundation of International Economics. Thanks

Last edited by a moderator: May 5, 2017
2. Oct 14, 2011

Ray Vickson

What do you get? For now, just forget the attached document and answer the question for X = exp(Y), where Y is normal with mean m and variance s^2.

RGV

Last edited by a moderator: May 5, 2017
3. Oct 14, 2011

royalstatus

The variance I got was -1/2σ(Var Ct+1/Ct)

I think the true variance might have something to do with the moment generating function for the lognormal distribution.

My answer is different from what is in the document, so i'm guessing i'm not thinking of this the right way

4. Oct 14, 2011

Ray Vickson

Your variance formula cannot possibly be right: it is < 0, while variance is always >= 0 if it exists at all. Anyway, I asked you to forget the attachment and just answer a *simple question* for X = exp(Y) with Y~N(m,s^2). Using Ct and 1/Ct, etc., just confuses the issue. Possibly the reason you are having trouble is that you are so focused on the model and notation in the attachment that you are neglecting some basic probability results and methods that apply to any model, not just the one you are researching.

RGV

5. Oct 14, 2011

royalstatus

RGV,

Will it be Y~N(ln(m),ln(s^2))?

I'm not particularly sure of the question you are asking.

6. Oct 14, 2011

Ray Vickson

I am asking you to find EX and Var(X)---that was the question you originally posed to this forum. You claim your formula for Var(X) is wrong, because it disagrees with something in some paper. Well, maybe your formula is wrong, or maybe it is correct and the paper is wrong. Let us see your formula, so we can tell which is which.

RGV