Question

The historical returns on a balanced portfolio have had an average return of 9% and a standard deviation of 11%. Assume that returns on this portfolio follow a normal distribution. [You may find it useful to reference the z table.]

a. What percentage of returns were greater than 20%?(Round your answer to 2 decimal places.)

b. What percentage of returns were below −13%? (Round your answer to 2 decimal places.)

Answer 1

a)

Here, μ = 9, σ = 11 and x = 20. We need to compute P(X >= 20). The corresponding z-value is calculated using Central Limit Theorem

z = (x - μ)/σ
z = (20 - 9)/11 = 1

Therefore,
P(X >= 20) = P(z <= (20 - 9)/11)
= P(z >= 1)
= 1 - 0.8413 = 0.1587

= 15.87%

b)

Here, μ = 9, σ = 11 and x = -13. We need to compute P(X <= -13). The corresponding z-value is calculated using Central Limit Theorem

z = (x - μ)/σ
z = (-13 - 9)/11 = -2

Therefore,
P(X <= -13) = P(z <= (-13 - 9)/11)
= P(z <= -2)
= 0.0228

= 2.28%

b)