Question

Assume a world has only two assets: a safe asset and a risky asset. There are two rates associated with the safe asset. Specifically, an investor’s borrowing rate of safe asset is 7%, and the lending rate of safe asset is 5%. The risky asset P has rp=13%, σp=22%.

Assume there are 20 investors: investor#1, investor#2, …, investor#20. Investors are ranked by their coefficient of risk aversion (A), which form an arithmetic sequence with a common difference of 0.05, as the following:

Investor#1’s coefficient of risk aversion is 1.05;

Investor#2’s coefficient of risk aversion is 1.10;

Investor#3’s coefficient of risk aversion is 1.15;

… …

Investor#18’s coefficient of risk aversion is 1.90;

Investor#19’s coefficient of risk aversion is 1.95;

Investor#20’s coefficient of risk aversion is 2.00;

Question: Among the 20 investors, identify the investor(s) whose optimal allocation is 100% to risky asset P and 0% to safe asset.

Answer 1

The risk free rate will be 5%

The optimal proportion of the complete portfolio of risky component = (Expected Return on Optimal Risky Portfolio - Risk Free Rate) / (Risk Aversion coeff.*Variance of portfolio)

Numerator = .13 - .05 = .08

Calculation of Weight in Optimal Risky portfolio for Investor 1:

Investor 1

The optimal proportion of the complete portfolio of risky component = (Expected Return on Optimal Risky Portfolio - Risk Free Rate) / (Risk Aversion coeff.*Variance of portfolio)

= (.13-.05) / (1.05 * (.22)^2)

= .08 / .050

= 1.57

Calculating Weights for all investors:

Investor	Investors's Coefficient of Risk Aversion	Denominator = (.22)^2 * Risk Aversion Coefficient	Numerator / Denominator
1	1.05	0.05	1.57
2	1.10	0.05	1.50
3	1.15	0.06	1.44
4	1.20	0.06	1.38
5	1.25	0.06	1.32
6	1.30	0.06	1.27
7	1.35	0.07	1.22
8	1.40	0.07	1.18
9	1.45	0.07	1.14
10	1.50	0.07	1.10
11	1.55	0.08	1.07
12	1.60	0.08	1.03
13	1.65	0.08	1.00
14	1.70	0.08	0.97
15	1.75	0.08	0.94
16	1.80	0.09	0.92
17	1.85	0.09	0.89
18	1.90	0.09	0.87
19	1.95	0.09	0.85
20	2.00	0.10	0.83

For Investor 13, whose risk aversion coefficient is 1.65, the 100% allocation is towards Optimal Risky Portfolio and 0% towards risk free asset.