Question

Here are the probability distributions for three investment project returns: Up (prob = Down (prob = 0.4) 0.6) -2% -2.5% 3% 14% -8% 5% Calculate the expected return and standard deviation of each project and explain which one a rational investor would choose. b)Given your answer in part a., if you can only split your investment by half, which two projects would you invest in, and what becomes the expected return and standard deviation of your portfolio?

Answer 1

There are two events. Up and Down.

Probability of Up event = P (up event) = 0.4, Probability of Down event = P (down event) = 0.6

Expected return will be calculated using the formula: P (up event) * return for up event + P (down event) * return for down event

Standard deviation will be calculated using the formula:

square root of [P(up event) * (Expected return - return for up event)² + P(down event) * (Expected return - return for down event)² ]

For Project X: Expected return = 0.4 * 5% + 0.6 * -2% = 0.8%

Standard deviation = sqrt [ (0.8 - 5)² *0.4 + (0.8 - (-2))² *0.6 ] = 2.99%

For Project Y: Expected return = 0.4 * -2.5% + 0.6 * 3% = 0.8%

Standard deviation = sqrt [ (0.8 - (-2.5))² *0.4 + (0.8 - 3)² *0.6 ] = 2.35%

For Project Z: Expected return = 0.4 * 14% + 0.6 * -8% = 0.8%

Standard deviation = sqrt [ (0.8 - 14)² *0.4 + (0.8 - (-8))² *0.6 ] = 9.40%

All the three projects have same expected returns but Project Y has the least standard deviation, hence a rational investor will chose Project Y because the investor prefers less risk for same returns.

Answer: Rational investor will choose Project Y

Part b:

If investor has to choose two projects, then he/she will chose Project X and Project Y.

Expected return = 50% * 0.8 + 50% * 0.8 = 0.8%

Let us assume that the two projects are not correlated that means that changes in project x doesn't impact changes in project Y.

standard deviation = $\sqrt{w_1^2*\sigma_1^2 + w_2^2*\sigma_2^2 + Cov (X,Y)}$

w1 = w2 = w = 50% = 0.5

Standard deviation = sqrt [ 0.5² * 2.99² + 0.5² * 2.35² ] = 1.901%

Answer:

Expected return of the portfolio = 0.8%

Standard deviation of the portfolio = 1.901%