Question

You are a financial analyst facing the task of selecting bond mutual funds to purchase for a​ client's portfolio. You have narrowed the funds to be selected to twelve different funds. In order to diversify your​ client's portfolio, you will recommend the purchase of four different funds. Five of the funds are​ short-term corporate bond funds. Complete parts​ (a) through​ (d) below.

Include Excel Formulas to solve A-C.

a. What is the probability that of the four funds selected exactly 1 is a​ short-term corporate bond​ fund?

​ P(X=​1)=___________(Round to two decimal places as needed.)

b. at least 1 is a short-term corporate bond fund?

P(X ≥ 1)= ___________(Round to two decimal places as needed.)

c. two are short-term corporate bond funds?

P(X=22 )= ____________(Round to two decimal places as needed.)

d. Another financial analyst narrowed their own funds to be selected to ten different funds. The probability that the three of the recommended funds were exactly two of the six short-term corporate bond funds is 50.00 %. How does this compare to the results of c. ?

Answer 1

Solution:

We are given: out of 12 funds, 5 are short and 7 are long.

Let X be the number of short among randomly selected 4 funds.

Here X follows the hypergeometric distribution with parameters (N = 12, K = 5, n = 4)

a. What is the probability that of the four funds selected exactly 1 is a short-term corporate bond fund?

Answer: We have to find $\dpi{100} \large P(X=1)$

Using the hypergeometric probability model, we have:

$\dpi{100} \large P(X=1)=\frac{\binom{5}{1}\binom{12-5}{4-1}}{\binom{12}{4}}$

$\dpi{100} \large \boldsymbol{=\frac{5\times 35}{495}=0.3535}$

Using the excel, we need to use the below function:

$\dpi{100} \large =HYPGEOMDIST(1,4,5,12)$

Where:

1 is X

4 is n

5 is K

12 is N

b. at least 1 is a short-term corporate bond fund?

Answer:

$\dpi{100} \large P(X\geq 1)=1-P(X<1)$

Now using the excel we have:

$\dpi{100} \large =1-HYPGEOMDIST(0,4,5,12)=0.9293$

$\dpi{100} \large \boldsymbol{P(X\geq 1)=0.9293}$

c. two are short-term corporate bond funds?

Answer: $\dpi{100} \large \boldsymbol{P(X=2)=HYPGEOMDIST(2,4,5,12)=0.4242 }$

d. Another financial analyst narrowed their own funds to be selected to ten different funds. The probability that the three of the recommended funds were exactly two of the six short-term corporate bond funds is 50.00 %. How does this compare to the results of c. ?

Answer: The other financial's results are greater than yours because 50% is greater than 42.42%