2. (16 points) In this question, we will explore how probabilistic modelling can be used to analyze the risk and return of an investment portfolio. These types of models are also often applied to population dynamics. This problem will be phrased entirely in terms of the course material so no outisde knowledge is required. Throughout this problem, consider normal random variables X and Y such that E[X] = E[Y ] = 0.05, V ar(X) = V ar(Y ) = ^2 and their correlation is 0.4. Each will represent an asset's rate of return after one year. In this problem, we will be considering an idealized model were one dollar is invested into an account over one year, and analyzing how correlation between assets impacts risk and return. We consider two investment strategies: investing our dollar into asset one assest and equally investing our dollar between the two assests. When the dollar invested into only one assest, P1 = eX models the value in the account after one year. Note, eY would have worked too since they follow the same distribution - the statistics and probabilities pertaining to the account will be the same. When the dollar is split equally between the two accounts, P2 = 0.5e^X + 0.5e^Y models the value of the account.

concise version: X N(0.06, ^2) and Y N(0.06, ^2); second parameters are variances. P1 = eX and P2 = 0.5e^X + 0.5e^Y . The correlation between X and Y is = 0.4.

(a) Compute the expected value of P1 and variance of P1. What happens to each as ^2 increases? In financial terms, E[P1] is how much the account is expected to return and Var(P1) is how much the account varies, a measure of how risky the investment is. Riskier strategies have larger variances.