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 38% 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 = 3. Assume the utility function is U = E(r) − 0.5 × Aσ2. (Negative amounts should be indicated by a minus sign. Do not round intermediate calculations. Round your answers to 4 decimal places.)

WBills	WIndex	U(A=3)
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

Computing utility from the above formula U=E(r)-0.5*Aσ²  by using calculations in below table

(a)	(b)	(c)	(d)	(e)= E(r)	(f)	(g) = (f)2	U(A=3)
WB	rB	WI	rI=8%+5%	rP=a*b+c*d	σp=(WI*0.38)	σ2p	using given formula
0.0	5%	1.0	13%	13% or 0.130	0.38	0.144	-0.086
0.2	5%	0.8	13%	0.1140	0.304	0.092	-0.024
0.4	5%	0.6	13%	0.098	0.228	0.052	0.02
0.6	5%	0.4	13%	0.082	0.152	0.023	0.0475
0.8	5%	0.2	13%	0.066	0.076	0.006	0.057
1.0	5%	0.0	13%	0.050	0	0	0.050