Question

John has a utility function given by the expression U(x) = E(r) -½A(s²). Where E(r) is the expected return on an asset and s is the standard deviation of returns on that asset.John has the opportunity to purchase the XJKsecurity that returns 25.9% with 23% probability and returns 8.6% the remainder of the time.

The security has a price of $33 and A=11

a) What is the risk-neutral valuation of the XJK security? Recall the risk-neutral value is simply the expected value.

b) Using the utility function above, find John's risk-averse valuation of XJK security. Hint: Find John's certainty equivalent (CEQ) for this security's payoff.

c) If the expected annual return on the market is 6.575%, the standard deviation of the market return is 8.9% and the risk-free rate for the next year is 1.22% then what is John's optimal percent of funds that he'll invest in the market?

d) Use the rates given in part c to answer this question. If a stock had a Beta of 2.92 what would be the expected return for that stock in the coming year?

Answer 1

Part a)

$U=E(r)-\frac{1}{2}As^{2}\\$

John has the opportunity to purchase the XJKsecurity that returns 25.9% with 23% probability and returns 8.6% the remainder of the time. The security has a price of $33 and A=11

expected return is 12.579%

Part b)

$\mu=\frac{0.259+0.086}{2}=0.1725$

$x_{i}$	$P(x_{i})$	$(x_{i}-\mu)^{2}P(x_{i})$
0.259	0.23	0.00172
0.086	0.77	0.005761

$s=\sqrt{\sum{(x_{i}-\mu)^{2}P(x_{i})}}=\sqrt{0.005761+0.00172}=0.08649\\$

$expected\ utility\ of\ john=0.23(0.259-\frac{11*0.007481}{2})+0.77(0.086-\frac{11*0.007481}{2})=0.23*0.21785+0.77*0.04485=0.050105+0.03453=0.084635\\$

$Expected\ utility=0.084635\\ E(r)-\frac{1}{2}As^{2}=0.084635\\ E(r)=0.084635+\frac{11*0.08649}{2}=0.56033\\$

Part c)

If an investor in 2 assets and one of them is risk free and risk free asset has zero variance

then using slope of capital allocation line that is the sharpe ratio is

$sharpe-ratio=\frac{R_{m}-R_{f}}{\sigma_{m}}=\frac{0.06575-0.0122}{0.089}=0.6016$

John would invest 60.16 % of his funds in market

Part d)

$expected\ return\ on\ XJK\ security,E(R_{xjk})=R_{m}+\beta_{xjk}(R_{m}-R_{f})\\ E(R_{xjk})=0.06575+2.92(0.06575-0.122)=0.222116$