An investment strategy has an expected return of 18 percent

An investment strategy has an expected return of 18 percent and a standard deviation of 10 percent. Assume investment returns are bell shaped.

a. How likely is it to earn a return between 8 percent and 28 percent? (Enter your response as decimal values (not percentages) rounded to 2 decimal places.)

b. How likely is it to earn a return greater than 28 percent?(Enter your response as decimal values (not percentages) rounded to 2 decimal places.)

c. How likely is it to earn a return below 2 percent?(Enter your response as decimal values (not percentages) rounded to 2 decimal places.)

Solution

Let X be a random variable as invvestment returns.

given that X ~ Normal (mean = 18% , sd = 10%)

mean = 18% = 18/100 = 0.18

sd = 10% = 0.1

Bell shaped distribution is known as normal distribution.

How likely is it to earn a return between 8 percent and 28 percent?

that is here we have to calculate P(8% < X < 28%)

P(0.08 < X < 0.28)

For finding this probability first we have to convert each X into standard normal (z).

z-score = (X - mean) /sd

z-score for X = 0.08

z = (0.08 - 0.18)/0.1 = -1

similarly for X = 0.28,

z = (0.28 - 0.18) / 0.1 = 1

That is now we have to find the P(-1 < Z < 1).

P(-1 < Z < 1) = P(Z 1) - P(Z -1)

This probability we can find by using EXCEL.

EXCEL syntax : normsdist(z)

where z is test statistic value.

P(Z 1) = 0.84

P(Z -1) = 0.16

P(-1 < Z < 1) = 0.84 - 0.16 = 0.68

How likely is it to earn a return greater than 28 percent?

Here we have to find P(X > 28%) = P(X > 0.28)

= 1 - P(X < 0.28)

because EXCEL will always gives us left tail area.

First convert X = 0.28 into z-score.

z = (0.28 - 0.18) / 0.1 = 1

P(Z > 0.28) = 1 - P(Z 0.28) = 1 - 0.84 = 0.16

How likely is it to earn a return below 2 percent?

P(X < -2%) = P(X < - 0.02)

FIrst we have to find z-score for X = -0.02

z = (-0.02 - 0.18) / 0.1 = -2

P(Z < -2) = 0.02