Consider historical data showing that the average annual rate of return on the S

Question

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 35% per year. Assume these values are representative of investors' expectations for future performance and that the current T-bill rate is 3%. Calculate the expected return and variance of portfolios invested in T-bills and the S&P 500 index with weights as shown below. (Enter your answers as decimals rounded to 4 places. Leave no cells blank-be certain to enter "" wherever required.) WBills WIndex Expected Return Variance 0.1100 0.1225 Example 0.0 0.2 0.4 0.6 0.8 1.0 0.8 0.6 0.4 0.2 0.0

Explanation / Answer

The answer is required in decimals and not percentage. So, to convert percentage to decimal, just divide it by 100 such as 8% / 100 = 0.08.

Expected return of portfolio = Wbills x Expected return on bills + WIndex x Expected return on Index

Expected return on Index = 11% or 0.11 (See Windex 1.0 which means only Index is included in portfolio)

Return of index is 8% more than T-bills. Therefore, expected return on T-bills = 11% - 8% = 3% or 0.03

Variance of portfolio = (Wbill x standard deviation of Bills)2 + (WIndex x standard deviation of Index)2 + 2 x Wbills x standard deviation of bills x WIndex x standard deviation of Index x Correlation

Since, T-bills have zero standard deviation (as they provide fixed return), the variance formula would become -

Variance of portfolio = (WIndex x standard deviation of Index)2

standard deviation of Index = (Variance of Index)1/2 = (0.1225)1/2 = 0.35

WBills WIndex Expected return Variance 0.0 1.0 0.1100 0.1225 0.2 0.8 0.2 x 0.03 + 0.8 x 0.11 = 0.0940 (0.8 x 0.35)2 = 0.0784 0.4 0.6 0.4 x 0.03 + 0.6 x 0.11 = 0.0780 (0.6 x 0.35)2 = 0.0441 0.6 0.4 0.6 x 0.03 + 0.4 x 0.11 = 0.0620 (0.4 x 0.35)2 = 0.0196 0.8 0.2 0.8 x 0.03 + 0.2 x 0.11 = 0.0460 (0.2 x 0.35)2 = 0.0049 1.0 0.0 1 x 0.03 + 0 x 0.11 = 0.0300 0

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