Miller Chapt. 4 - End of chapter question12

[FlorenceCC]

Hi,

I am working on Chapt.4 end of chapter Q&A (which I now understand are not written by David! ). I am slightly confused by the second portion of Q12 (p96 of study notes) -"what are the mean and standard deviation of a portfolio where the return is a 50/50 mixture distribution of A and B"

• first of all, wasn't 50% of A and 50% of B a certain type of mixture distribution as well?
• second of all, if for the first part, my pdf is f(x)= 0.5A +0.5B, then what is my pdf for the second part of the question, i.e. I am not sure I understand "the return is a 50/50 mixture distribution of A and B" -> what mixture pdf does this translate into? I understand the integration part itself but I need to establish my pdf to then calculate my mean/standard deviation.

Thank you in advance for your feedback.

Florence

 

[David Harper CFA FRM]

HI @FlorenceCC Miller's question formulations are often not stellar, but this question 4.12 is instructive. We are given two density (pdf) functions, call them f(x) and f(y), such that f(x) = uniform (2,1) and f(y) = uniform (+1, +2). The random variable is produced by the function; e.g., X = f(x).

• The first question is the more familiar sum of two random variables, so is asking about σ^2(X+Y). This is found with 50%^2*(1/12)^2 + 50%^2*(1/12)^2 + 0, because their correlation is zero.
• The second question asks about a mixture distribution, which is a new density, call this new density f(w) and it is given by f(w) = 50%*f(x) + 50%*f(y). A mixture is not a sum of random variables but rather the probability-weighted sum of other distribution functions. You can see this in the solution where he is solving for mean and variance literally of a distribution that "mixes" two other distributions. I hope that's helpful!

 

[FlorenceCC]

Thank you @David Harper CFA FRM
It was actually a great question to underline the difference between sum of variable and weighted sum of distribution functions. I hadn't even realized I didn't really understand the difference until now.