Question

suppose you are going to invest $1000 in some stocks and holdthem for one year. suppose safe corp. stock currently costs $100.its price a year from now is a random variable x with mean $105and standard deviation of 10. risky business stock also currentlycosts $100. but its price a year from now is a random variable ywith mean $110 and standard deviation of 20. suppose the correla-tion coefficient of x and y is 0.2.(a) how much should you invest in each stock if you want to max-imize the expected value of what your stocks are worth one yearfrom now?(b) how much should you invest in each stock if you want to mini-mize the variance of what your stocks are worth one year from now?note: you are allowed to buy part of a stock. so for example, youcould invest $333.33 in safe corp. and $666.67 in risky busi-ness.

Answer 1

To maximize expected value, invest the entire $1000 in Risky Business stock, with an expected value of $110. To minimize variance, we need to solve an optimization problem that considers the variances of the stocks and their correlation coefficient of 0.2.

To answer the student's questions on how much to invest in each stock to maximize expected value and minimize variance while considering stock costs and variables, we can follow these steps:

Maximize Expected Value:

To maximize the expected value of the investment after one year, the entire amount should be invested in Risky Business stock. The expected value for Risky Business is $110, which is higher than Safe Corp.'s $105. Thus, you should invest $1000 in Risky Business.

Minimize Variance:

To minimize variance, we can use the formula for the variance of a portfolio which takes into account the weights of the investments, the variances of the individual investments, and the correlation between them. To solve for the weights that minimize variance, we need to find the portfolio that offers the lowest possible variance, considering the correlation coefficient of 0.2 between the variables. This involves solving an optimization problem, which might require the use of calculus or numerical methods.