Question

You are a creating a portfolio of investments for a retiring person. The portfolio consists of ten securities. Three of them must be stocks, and seven of them must be bonds. You can choose among ten stocks, and twenty bonds. Answer the following-

1. If you randomly choose ten securities, what is the probability that the portfolio specification will be met, i.e., you would have three stocks and seven bonds in the portfolio?

2. Suppose that you are told that two stocks, GM and GE must be in the portfolio. Also, you are told that bonds of ATT and Verizon must be in the portfolio. These are already in the specified set of ten stocks and twenty bonds. How likely is it that a randomly formulated portfolio of ten securities will contain these stocks and bonds, and also satisfy the requirements for containing three stocks and seven bonds?

   (Note: the values in this problem are rather large to compute manually. You will be better off using Excel functions such as =permut or =combin for your analysis. If you do use them, write down the exact expressions you used as part of your analysis).

Answer 1

Given,

Total number of stocks (s) = 10 ...(1)

Total number of bonds (b) = 20 ...(2)

Total number of securities (n) = s + b = 10 + 20 = 30 ...(3)

In the question, mentioned portfolio specification 3 of them should be stocks and 7 of the them should be bonds

Hence, if 10 securities are selected at random, the probability that the portfolio specification is met is given in the below formula

... (4)

where

; ;

Substituting (1), (2), and (3) in (4)

.... (A)

Using formula (A)

Hence, the probability that the portfolio specification is met p = 0.3096

Given the condition that

(i) Of the 3 stocks to be chosen, two stocks GM and GE are to be in the portfolio, i.e., we have to choose 1 (r1)  stock out of the remaining 8 (s) stocks

(ii) Of the 7 bonds to be chosen, two bonds ATT and VERIZON are to be in the portfolio, i.e., we have to choose 5 (r2) bonds out of the remaining 18 (b) bonds

(iii) Finally, we have to choose 6 (r) securities from the total of 26 securities ( n = 18 + 8 = 26)

Hence substituting (i), (ii) and (iii) in (4) and using formula (A), the probability that the portfolio specification is met

Hence, with the conditions mentioned in (i) and (ii), probability that the portfolio specification is met p = 0.2977