Question

1. Consider historical data showing that the average annual rate of return on the S&P 500 portfolio over the past 85 years has averaged roughly 8% more than the Treasury bill return and that the S&P 500 standard deviation has been about 20% per year. Assume these values are representative of investors’ expectations for future performance and that the current T-bill rate is 5%.

1. Calculate the expected return and variance of portfolios invested in T-bills and the S&P 500 index with the following weights.                         [5]

Weight in T-Bills	Weight in S& P
0	1.0
0.2	0.8
0.4	0.6
0.6	0.4
0.8	0.2
1	0

2. Calculate the utility levels of each portfolio of Problem in a) for an investor with A =2. What do you conclude? [5]

3. Repeat Problem b) for an investor with A = 3. What do you conclude? [5]

Answer 1

a) S&P Index Premium over Treasury bill = 8% | Current Treasury bill = 5% | Standard Dev of S&P 500 = 20%

Using CAPM equation, where Weights for S&P index is the Beta since the risky part of the portfolio is the market index

E(r) = Risk-free rate + Weight for S&P * Risk premium

i) Weight for T-bills = 0 and Weight for S&P = 1

E(R) = 5% + 1 * 8% = 8%

Since Risk-free asset has zero standard deviation, hence, weighted risk of S&P is the risk of portfolio

Variance of the portfolio = (1*20%)² = 0.04

Utility at (A = 2) = E(R) - 1/2 * A * Variance of the portfolio

Utility at (A = 2) = 8% - 1/2 * 2 * 0.04 = 0.040

Utility at (A = 3) = 8% - 1/2 * 3 * 0.04 = 0.020

ii) W for T-bills = 0.2 and Weight for S&P = 0.8

E(R) = 5% + 0.8 * 8% = 5% + 6.4% = 11.4%

Variance of the portfolio = (0.8 * 20%)² = 0.0256

Utility at (A = 2) = 11.4% - 1/2 * 2 * 0.0256 = 0.088

Utility at (A = 3) = 11.4% - 1/2 * 3 * 0.0256 = 0.0756

iii) Weight for T-bills = 0.4 and Weight for S&P = 0.6

E(R) = 5% + 0.6 * 8% = 9.8%

Variance of the portfolio = (0.6 * 20%)² = 0.0144

Utility at (A = 2) = 9.8% - 1/2 * 2 * 0.0144 = 0.084

Utility at (A = 3) = 9.8% - 1/2 * 3 * 0.0144 = 0.0764

iv) Weight for T-bills = 0.6 and Weight for S&P = 0.4

E(R) = 5% + 0.4 * 8% = 8.2%

Variance of the portfolio = (0.4 * 20%)² = 0.0064

Utility at (A = 2) = 8.2% - 1/2 * 2 * 0.0064 = 0.076

Utility at (A = 3) = 8.2% - 1/2 * 3 * 0.0064 = 0.072

v) Weight for T-bills = 0.8 and Weight for S&P = 0.2

E(R) = 5% + 0.2 * 8% = 6.6%

Variance of the portfolio = (0.2 * 20%)² = 0.0016

Utility at (A = 2) = 6.6% - 1/2 * 2 * 0.0016 = 0.06440

Utility at (A = 3) = 6.6% - 1/2 * 3 * 0.0016 = 0.06360

vi) Weight for T-bills = 1 and Weight for S&P = 0

E(R) = 5% + 0 * 8% = 5%

Variance of the portfolio = (0 * 20%)² = 0

Utility at (A = 2) = 5% - 1/2 * 2 * 0 = 0.050

Utility at (A = 3) = 5% - 1/2 * 3 * 0 = 0.050

Case of Utility at (A = 2)

=> Highest utility of 0.088 is in the portfolio with 0.2 allocation to T-bills and 0.8 allocation to S&P 500 Index

=> Lowest utility of 0.040 is in the portfolio with 100% allocation to S&P index and 0% allocation to T-bills

Case of Utility at (A = 3)

=> Highest utility of 0.07560 is in the portfolio with 0.4 allocation to T-bills and 0.6 allocation to S&P 500 Index

=> Similar to case of A = 2, Lowest utility of 0.020 is in the portfolio with 100% allocation to S&P index and 0% allocation to T-bills