Question

[Investment Planning] Temoc is a high tech company that has recently gone public, and trades using the symbol TEMO. Comet is a rival high tech company that has also recently gone public, and trades using the symbol COME. Currently, the price of one share of each company is $10. Based on extensive market research and financial analysis, you have forecast that the price of one share of TEMO one year from now can be modeled as a normal distribution, with mean $12 and standard deviation $3. To put this formally, let ???? (1) represents the price of one share of TEMO one year from now. Then, ???? (1) ~ N(12, 3). Similarly, you have forecast that ???? (1) ~ ?(12, 4). Also, assume that the two stocks are independent of each other.

Investor B’s portfolio consists of 300 shares of TEMO and 400 shares of COME. What is the probability that Investor B’s portfolio is worth more than $8,000 one year from now? (a) 0.414 (b) 0.436 (c) 0.564 (d) 0.586

Answer 1

Let TEMO (1) be represented as X and COME(1) be Y. We have

  

Variable X and Y Represent 1 share of each company. We have a portfolio of a number of shares. There we have to apply central limit theorem for a sum of variables.

Central limit theorem states for n independent and identically distributed distribution X random variable

Proof:

First we solve for sum of means. Due independence between the shares and identical distribution  

   

Similarly for Y

For sum of variances and standard deviation

  

Similarly

Given information

300shares of TEMO and 400 shares of COME

Therefore We already have

We now have sum of distributions as

  

Substituting values we get

Portfolio consists of both the shares. Using the independence assumption we can add the distribution and name it Z. Such that Z=X+Y

The mean of the distributions is the sum of the means. But for S.D. we cannot simply add them. We add the variances and then take its square root.

Proof of variance

Var(X+Y)=Var(X) + Var(Y) - Cov(X,Y)

since both are independent Cov(X,Y) = 0. Therefore Var(X+Y)=Var(X)+Var(Y)

  

Probability that portfolio is worth more than 8000

Using this approach I am getting 0.9999 as the probability. which is neither of the options. i am unable find my mistake but I feel that the approach is correct. My apologies for not giving full solution.