C422: Computational Finance
Coursework
Exercise 1. Dividend Discount Model. The dividend discount model is a way of
valuing a company based on the theory that a stock is worth the discounted sum of
all of its future dividend payments. In other words, it is used to value stocks based
on the net present value of the future dividends.
Assume that each stock of the company pays a dividend Dt at the end of every year,
t = 1, 2, 3, . . ., and denote by r > 0 the constant annual interest rate (use yearly
compounding).
a
What is the price of the stock if the dividends have constant growth rate g > 0
(i.e., Dt+1 = (1 + g)Dt for each t ∈ N)? Your result should only depend on D1,
r and g and no longer contain an infinite sum. Hint: Calculate the net present
value of the dividend stream.
b
What is the price of the stock if the dividends increase by a constant amount I
every year (i.e., Dt+1 = Dt + I for each t ∈ N)? Your result should only depend
on D1, r and I and no longer contain an infinite sum. Hint: Derive first a
formula for the expression
∑∞
t=1 tx
t for −1 < x < +1. Note that this expression
can be reduced to a geometric series by using the identity txt = x d
dx
xt.
20 marks and the two parts carry equal weight
1
Exercise 2. Fixed-Income Securities.
Consider a 10% coupon bond with a face value of £100 that matures in 2 years
from now. Assume that there is one coupon payment per year and that the bond is
absolutely risk free. The current spot rate curve is given in the following table (yearly
compounding).
s1 s2 s3
4% 6% 8%
a
What is the price of this bond?
b
Find an analytical formula for the yield of this bond and use your formula to
compute the bond’s yield numerically.
c
Compute the bond’s Macauley duration.
d
Compute the bond’s price and yield in one year from now assuming that the
spot rates will evolve according to expectation dynamics. Assume further that
the first coupon payment is still to be received.
20 marks and the four parts carry equal weight
2
Exercise 3. Portfolio Optimisation. Consider a market that contains n risky
assets and one risk-free asset. Denote by r¯ = (r¯1, . . . , r¯n) the vector of expected
returns and by
Σ =
 cov(r1, r1) · · · cov(r1, rn)... . . . ...
cov(rn, r1) · · · cov(rn, rn)

the covariance matrix of the risky assets, respectively. Moreover, let rf be the risk-free
rate of return. Assume that Σ is strictly positive definite and therefore invertible.
a
What is the expected return r¯P and the variance σ2P of a portfolio that assigns
weights w = (w1, . . . , wn) to the risky assets and weight w0 to the risk-free
asset?
b
Consider an investor who wishes to maximise r¯P − a2σ2P , where the constant
a > 0 is called the risk-aversion parameter. Assume that short-selling is allowed.
Formulate this investment problem as a quadratic program without constraints.
Hint: Use the budget constraint to eliminate w0.
c
Derive the optimality conditions for the optimisation problem formulated in
part b and solve it analytically, that is, find an expression for the problem’s
optimal value and the optimal portfolio weights in terms of rf , r¯ and Σ.
d
Characterise the optimal objective value and the optimal portfolio when the
risk-aversion parameter a tends to +∞.
e
Assume now that r¯ cannot be estimated exactly. Instead, only an unbiased
estimator ˆ¯r is known, which satisfies E(ˆ¯r) = r¯. Show that an investor who
uses ˆ¯r to approximate r¯ will—on average—overestimate the optimal value of
the investment problem from part b. Assume that rf and Σ are still known
exactly. Hint: use Jensen’s inequality, which states that E(f(X)) ≥ f(E(X))
for any convex function f and random vector X.
The five parts carry, respectively, 2, 6, 6, 2, and 4 marks
3

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