Question

A venture capital company feels that the rate of return (X) on a proposed investment is approximately normally distributed with mean 30% and standard deviation 10%.

(a) Find the probability that the return will exceed 55%.
(b) Find the probability that the return will be less than 25%
(c) What is the expected value of the return?
(d) Find the 75th percentile of returns.

Answer 1

a) 0.0062 = 0.62% probability that the return will exceed 55%.

b) 0.3085 = 30.85% probability that the return will be less than 25%

c) 30%.

d) The 75th percentile of returns is 36.75%.

Explanation:

Normal Probability Distribution

Problems of normal distributions can be solved using the z-score formula.

In a set with mean
`\mu` and standard deviation
`\sigma`, the z-score of a measure X is given by:

`Z = (X - \mu)/(\sigma)`

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the p-value, we get the probability that the value of the measure is greater than X.

Mean 30% and standard deviation 10%.

This means that
`\mu = 30, \sigma = 10`

(a) Find the probability that the return will exceed 55%.

This is 1 subtracted by the p-value of Z when X = 55. So

`Z = (X - \mu)/(\sigma)`

`Z = (55 - 30)/(10)`

`Z = 2.5`

`Z = 2.5` has a p-value of 0.9938

1 - 0.9938 = 0.0062

0.0062 = 0.62% probability that the return will exceed 55%.

(b) Find the probability that the return will be less than 25%

p-value of Z when X = 25. So

`Z = (X - \mu)/(\sigma)`

`Z = (25 - 30)/(10)`

`Z = -0.5`

`Z = -0.5` has a p-value of 0.3085

0.3085 = 30.85% probability that the return will be less than 25%.

(c) What is the expected value of the return?

The mean, that is, 30%.

(d) Find the 75th percentile of returns.

X when Z has a p-value of 0.75, so X when Z = 0.675.

`Z = (X - \mu)/(\sigma)`

`0.675 = (X - 30)/(10)`

`X - 30 = 0.675*10`

`X = 36.75`

The 75th percentile of returns is 36.75%.