II. Portfolios Risk and Return

You are given the following distributions of returns for assets (single stock or portfolio) A, B, and C.

Economic Return on Asset

Conditions Probability A B C

Boom .30 60% 50% 10%

Normal .40 40 30 50

Bust .30 20 10 90

1. Find the expected return (think of this as the mean return) and standard deviation of B and C. Answers for A: Expected return is 40% and its standard deviation is 15.49%. The formulas:

• Exp. Return (ER) = Sum of {Probability * Return} for each state of the economy
• STD = Square root of Variance; Variance = Sum of [Prob. * (Return ER)², for each of the three econ conditions]}.

That is, for asset A, ER = .30 * 60% + .40 * 40% + .30 * 20% = 40%; and the variance is given by = .30 * (60% - 40%)² + .40 * (40% - 40%)² + .30 * (20% - 40%)²

A similar example is given in Ch. 11, Section 11.2, but the example is for a unique case that involves equally-likely states of the economy. With 4 equally likely states in the example, the probability of each state = 25%. So, rather than weigh each return figure by 25%, the authors divide the results by 4, t getting the same answer. Their footnote (Page 329) explains this.

2. Portfolio AB is formed by investing 50% of the funds in each of the assets A and B. A similar (equally weighted) portfolio has also been created from A and C, called AC. Find the rates of return not given for AB and AC. (Partial answers under Boom are given below.)

Econ AB= AC=

Conditions Probability .50A+.50B .50A+.50C

Boom .30 55% 35%

Normal .40 35 ?

Bust .30 15 ?

(For example, the highlighted return of 55% for Portfolio AB under Boom is found as follows: .50*60% + .50*50%. That is, 50% is invested in A, with A having a return of 60% and 50% invested in B, with B having a return of 50%, together they give a return of 55% for the portfolio.)

1. Find the expected return and std. deviation of AB and AC. (The formulas are given in Part 1 above.) Do you see any evidence of risk reduction from what you obtain? What do you think caused this?