1. ## Lognormally distributed

The rate of return on Your portfolio, R, has a mean value of 1% and a standard deviation of 5%. Suppose that (1 + R) is lognormally distributed.

1) Calculate at what level of rate of return y, the probability that R is less than or equal to y is equal to 10%.

2) Calculate the probability that R is greater than 10%.

P.S. I undarstand that the shape of a lognormall distribution is skeewed to the left. But do not understand how to solve this. Is there a trick to simplify this to a normal distribution? If so please help and explain it to me. Thank you.

2. ## Re: Lognormally distributed

Several things you may need to know:

1. If ,
then

http://en.wikipedia.org/wiki/Log-nor...dard_deviation

2. If ,
then

3. Since you know the mean and variance of the log normal distribution,
you may solve the parameters and thus
you can convert any problem about the probability of a lognormal random
variable into a problem about the normal random variable.

3. ## Re: Lognormally distributed

Originally Posted by BGM
Several things you may need to know:

1. If ,
then

http://en.wikipedia.org/wiki/Log-nor...dard_deviation

2. If ,
then

3. Since you know the mean and variance of the log normal distribution,
you may solve the parameters and thus
you can convert any problem about the probability of a lognormal random
variable into a problem about the normal random variable.
I solved the equestion for E[1+R] and Var[1+R], by inputing mu=0.01 and sigma=0.05.

So I got
E[1+R]=1.012578452
Var[1+R]=0.002560086

But I don't understand how to go from here. I am asked to find (R<y)=10%

But I do not have E[R] and Var[R] values, and I am not sure how I can convert

E[1+R] in to E[R], and Var[1+R] into Var[R].

Maybe I do not full understand somthing, if u see where I went wrong in my thought process, please point it out. Thank you for the help. This question is bugging me very much.

4. ## Re: Lognormally distributed

The parameters I typed are not the mean
and variance of .
They are the mean and variance of

So the first thing you should know

Next, solve for the which is a two equations
with two unknowns. It is easy. You can either look for the solution at the
wikipedia, or consider

After obtaining the parameters, note that function is a
strictly increasing function, and the inequality will preserve. So the required
probability and is easy once you have access to the normal CDF.