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(MATH4512)[2013](s)final~=u_cildkj^_67321.pdf
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MATH 4512
Fundamentals of Mathematical Finance


C Spring 2013
Time allowed: 120 minutes Course instructor: Prof. Y. K. Kwok
[points]
1. Consider the following one-factor model, where the rate of return ri of the risky asset
i, i =1, 2, is given by
ri = ai + bif + ei.

Here, f is the random factor, ai and bi are constant parameters and ei is the random
residual. We assume E[e1]= E[e2] = 0.

(a) Under the assumption of
cov(e1,f) = cov(e2,f) = cov(e1,e2)=0
for asset 1 and asset 2, .nd cov(r1,r2) in terms of b1,b2 and .f 2, where r1 and r2 are the random rates of return of asset 1 and asset 2, .f 2 is the variance of f. [3]
(b) Given a1 =0.10,b1 =2,a2 =0.08 and b2 = 1, and assuming E[f]= e1 = e2 = 0, .nd the factor risk premium . and the return of the zero-beta portfolio .0. How to construct the zero-beta portfolio from these two risky assets? [3]
Hint: ri = E[ri]= .0 + bi., i =1, 2.
2. Consider the choice set
B = {(x, y): x [0, ) and y [0, )}
and de.ne the following preference solution .:
For any (x1,y1) and (x2,y2) B,
(x1,y1) . (x2,y2) if and only if [x1 >x2] or [x1 = x2 and y1 y2].
Consider the Order Preserving Axiom, Intermediate Value Axiom and Boundedness Ax-iom, does the above preference relation satisfy each of the above axioms? Give detailed explanation to your answer. [4]
3. Consider the investment wheel problem with m sectors whose return vector is given by R(i) = (0 ai 0)T , with ai > 0 and i corresponds to the event where the pointer falls on the ith sector, i =1, 2, ,m. Let pi,i =1, 2, ,m, denote the probability that the pointer lands on the ith sector. Let wi,i =1, 2, ,m, be the proportion of wealth
m

placed on the ith sector, with wi = 1.
i=1

(a)
Find an optimal betting strategy that maximizes the long-term growth of the capital. Show details of your calculations. [4]

(b)
How does the strategy for the long-term growth di.er from that of the single-period growth? How would you modify the strategy obtained in part(a) when maximization of the single-period growth is considered? [3]


1

Hint: Use the logarithm utility for maximizing the long-term growth.
4. (a) Suppose all security returns are normal random variables so that for a utility function U, the corresponding expected utility value E[U(y)] is a function of M and .. Here, M and . are the mean and standard deviation of the random wealth variable y. We write

1
.(y.M)2/2.2
E[U(y)] = U(y) e dy = f(M, .). . 2..2
f f
Explain why > 0 and < 0. Here, U(y) is concave and an increasing function
M .
of y. [3]
(b) Suppose we choose the quadratic concave utility function
b
u(x)= ax . x 2 , 0 x a/b, a> 0 and b> 0.
2
Show that
E[u(z)] = aE[z] . b (E[z])2 . b var(z),

22where z is the random wealth value. [2] Explain why
(i) for a given value of E[z], [1]
maximizing E[u(z)] . minimizing var(z);
(ii) For a given