Question

Consider historical data showing that the average annual rate of return on the S&P 500 portfolio over the past 90 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%.
Calculate the utility levels of each portfolio for an investor with A = 2. Assume the utility function is U = E(r) − 0.5 × Aσ2. (Do not round intermediate calculations. Round your answers to 4 decimal places.)
WBills WIndex U(A=2)
0.0 1.0
0.2 0.8
0.4 0.6
0.6 0.4
0.8 0.2
1.0 0.0

Answer 1

Answer:

1) 0.0900

2) 0.0884

3) 0.0836

4) 0.0756

5) 0.0644  

6) 0.0500

Explanation:

WBills   Rbils   Windex   Rindex   R-portfolio   α-portfolio   α²-portfolio   (A=2)

 (A)        (B)         (C)     D(8%+5%)   AB+CD          C20%                  

0.0        5%        1.0           13%          0.13              0.20          0.04          0.0900

0.2        5%        0.8          13%          0.114             0.16           0.0256     0.0884

0.4        5%        0.6          13%          0.098           0.12           0.0144      0.0836

0.6        5%        0.4          13%          0.082           0.08          0.0064     0.0756

0.8        5%        0.2          13%          0.066           0.04          0.0016      0.0644  

1.0         5%        0.0          13%          0.050           0.00         0.0000     0.0500

Utility level values of each portfolio for an investor with A=2

1) 0.13 - (0.5 ×2×0.04) = 0.0900

2) 0.114 - (0.5×2×0.0256) = 0.0884

3) 0.098 - (0.5×2×0.0144) = 0.0836

4) 0.082 - ( 0.5×2×0.0064) = 0.0756

5) 0.066 - ( 0.5×2×0.0016) = 0.0644

6) 0.050 - ( 0.5×2×0.000) = 0.0500

Answer 2

First is 0.0900

Second is 0.0884

Third is 0.0836

Fourth is 0.0756

Fifth is 0.0644  

Sixth is 0.0500

What is Utility Function?

W-Bills   R-bils   Windex   R-index   R-portfolio   α-portfolio   α²-portfolio   (A=2)

(A)        (B)         (C)     D(8%+5%)   AB+CD          C20%                  

0.0        5%        1.0           13%          0.13             0.20          0.04          0.0900

                                                                                                               

0.2        5%        0.8          13%          0.114             0.16           0.0256     0.0884

                                                                                                                 

0.4        5%        0.6          13%          0.098           0.12           0.0144      0.0836

                                                                                                                     

0.6        5%        0.4          13%          0.082          0.08          0.0064     0.0756

                                                                                                                   

0.8        5%        0.2          13%          0.066          0.04          0.0016      0.0644  

                                                                                                                   

1.0         5%        0.0          13%          0.050           0.00        0.0000     0.0500

Then the Utility level values of each portfolio for an investor with A is = 2

First is 0.13 - (0.5 ×2×0.04) is = 0.0900

2) Then 0.114 - (0.5×2×0.0256) is = 0.0884

3) After that 0.098 - (0.5×2×0.0144) is = 0.0836

4) Then 0.082 - ( 0.5×2×0.0064) is = 0.0756

5) Now 0.066 - ( 0.5×2×0.0016) is = 0.0644

6) Then 0.050 - ( 0.5×2×0.000) is = 0.0500

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