=========================preview======================
(MATH4512)[2013](s)midterm~tkzhuang^_50976.pdf
Back to MATH4512 Login to download
======================================================
MATH 4512
Fundamentals of Mathematical Finance
C Spring 2013
Time allowed: 80 minutes Course instructor: Prof. Y. K. Kwok
[points]
1. Suppose there are only 2 fully negatively correlated risky assets in the portfolio, whose expected rate of return and variance are ri and i 2, respectively, i =1, 2.
(a)
Show that it is possible to construct a portfolio that is riskfree. Find this riskfree portfolio and the corresponding expected rate of return. [2]
(b)
Next, we assume that these two risky assets now become correlated and let be the coe.cient of correlation between their random rates of return, where .1 << 1. Find the corresponding minimum variance portfolio. [3]
2. Suppose there are N risky assets (no riskfree asset) whose covariance matrix of their random rates of return is denoted by .. Let denote the vector of expected rates of return of these N risky assets.
(a)
Find the global minimum variance portfolio, g. [2]
(b)
Let P be any portfolio of these N risky assets. Show that
cov(rg,rP ) = var(rg),
where rg and rP are the random rates of return of g and P , respectively. [3]
(c) Suppose . is singular (..1 does not exist). Show that the minimum variance port-folio has zero variance. Find the corresponding riskfree rate of return of this zero-variance portfolio. [3]
3. Consider the universe of N risky assets and one riskfree asset whose riskfree rate of return is r. Let . and be the covariance matrix and expected rate of return vector of the N risky assets.
(a)
Explain why the Market Portfolio does not exist when r = b/a, where a = 1T ..11 and b = T ..11. State another case where the Market Portfolio does not exist. [3]
(b)
Under the scenario of r = b/a, show that the e.cient frontier in the P -P diagram
is given by
.
P = r + P ,
a
T ..1
where . = ac . b2 , c = . [5]
4. The formulation of the risk tolerance model is given by
P
maximize P . 2 , with . 0, subject to 1T w =1.
2
Here, is the risk tolerance factor and w is the portfolio weight vector. 1
(a) Show that the optimal portfolio weight vector is given by
w . = wg + z . ,
where wg is the portfolio weight vector of the global minimum variance portfolio and z. is a vector whose sum of components is zero. Find z. explicitly. [4]
(b) Let P be the expected rate of return of the optimal portfolio. Show that
()
ab
= P . ,
. a
where a = 1T ..11, b = T ..11, c = T ..1 and .= ac . b2 . [3]
5. Consider the zero-beta Capital Asset Pricing Model (CAPM):
rQ = rZM + QM (rM . rZM ),
where Q is any portfolio and ZM is the uncorrelated counterpart of the Market Portfolio
M.
(a)
Illustrate the method of constructing ZM on a graphical plot of the P -P diagram. Explain the rationale behind the construction. [3]
(b)
We may choose any e.cient fund P other than the Market Portfolio M in the zero-beta version of the CAPM. Explain th