Question

A bank estimates that its profit next year is normally distributed with a mean of 0.8% of assets and the standard deviation of 2% of assets. How much equity (as a percentage of assets) does the company need to be (a) 99% sure that it will have a positive equity at the end of the year and (b) 99.9% sure that it will have positive equity at the end of the year

Answer 1

Answer:

a) 5.45%

b) 6.98%

Explanation:

We are given the following information in the question:

Mean, μ = 0.8%

Standard Deviation, σ = 2%

We are given that the distribution of profit is a bell shaped distribution that is a normal distribution.

Formula:

$z_{score} = \displaystyle\frac{x-\mu}{\sigma}$

a) We have to find the value of x such that the probability is 0.99

P(X < x)  

$P( X < x) = P( z < \displaystyle\frac{x - 0.8}{2})=0.99$  

Calculation the value from standard normal z table, we have,  

$P(z < 2.326) = 0.99$

$\displaystyle\frac{x - 0.8}{2} = 2.326\\\\x = 5.452 \approx 5.45$

Thus, 5.45% of assets does the company need to be 99% sure that it will have a positive equity at the end of the year.

b) We have to find the value of x such that the probability is 0.999

P(X < x)  

$P( X < x) = P( z < \displaystyle\frac{x - 0.8}{2})=0.999$  

Calculation the value from standard normal z table, we have,  

$P(z < 3.090) = 0.999$

$\displaystyle\frac{x - 0.8}{2} = 3.090\\\\x = 6.98$

Thus, 6.98% of assets does the company need to be 99% sure that it will have a positive equity at the end of the year.