Theory of finance past paper

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Theory of finance past paper

MSc Examination
M5MF37 Theory of Finance
January 2012
This paper consists of five questions, all carrying equal weight.
Full marks will be awarded for complete solutions to four questions.
Each question is marked out of 20 points.
If you attempt all five questions, marks will be awarded for the best four
questions only.
Notation: Throughout the paper, “standard deviation” is abbreviated to stdev.
c
2014 ?Imperial
College London
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1.
M5MF37
A bond B pays annual coupon at rate c for n years and repays the principal
amount (= 1) at the maturity time n years. It trades at price P .
(i)[4 points] De?ne the yield y of the bond. Explain from a ?nancial perspective the inverse relationship between price and yield: when price goes up,
yield goes down. Show that the bond is at par, i.e. P = 1, if and only if
y = c.
(ii)[5 points] The modified duration of bond B is de?ned by
d(y) = -(?/?y)(log P (y)).
Calculate this and estimate roughly what the modi?ed duration is when n =
3, c = 2% and B is at par.
(iii)[4 points] Consider a portfolio B = a1 B1 + · · · + am Bm , where the
individual bonds may have di?erent coupons and maturities but all trade
at the same yield. Show that the duration D of B can be expressed as
?
D= M
1 wi Di where the Di are the individual durations and wi are certain
weights. What are these weights?
(iv)[7 points] The duration measures the sensitivity of price to a parallel shift
in the yield curve. Explain what the preceding sentence means.
In practice the market yield curve does not move in parallel shifts, but
di?erent parts of it may move by di?erent amounts. Suppose rk is the market
rate for a k-year deposit, so that a deposit of £1 pays £(1 + rk )k after k
years. De?ne what is meant by the forward rate fk for the period from year
k to k + 1. Show that, given the rj , there is a unique arbitrage-free value
for fk . If we value the bond portfolio B in Part (iii) by calculating its NPV
given the market rates rj , what is the sensitivity of the bond value to a shift
in the forward rate fk ?
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2.
M5MF37
This question concerns mean-variance portfolio optimization.
(i)[3 points] Suppose we have n assets whose initial prices are x1 , . . . , xn and
whose ?nal values are, respectively, random variables X1 , . . . , Xn . De?ne the
return ri of asset i and show that if we start with initial capital c and form a
portfolio in which an amount cwi is invested in asset i, i = 1, . . . , n, where wi
?
?
are weights such that ni=1 wi = 1, then the portfolio return is r = n1 wi ri .
Calculate r¯ = E[r] and s
¯ , the stdev of r, when the ri have means and
covariances r¯i = E[ri ], sij = cov(ri , rj ).
(ii)[9 points] Suppose the covariance matrix S = [sij ] is non-singular. Using
Lagrange multipliers, or otherwise, solve the Markowitz problem of minimizing
s
¯ 2 over the choice of weights w1 , . . . , wn subject to the constraints r¯ = r0
?
and i wi = 1, where r0 is a given ‘target’ mean return.
(iii)[3 points] Explain what is meant by the “e?cient frontier”. If, in addition
to the assets described above, we can invest in a risk-free asset with return
rf (and zero variance), the e?cient frontier becomes a straight line. Explain
why this is the case.
(iv)[5 points] Suppose n = 2 and the risky assets A and B in the portfolio
have mean returns and return stdev as given in Table 1. The correlation
coe?cient is 0.3. Find the value of w that minimizes the portfolio stdev
when the weights are w, (1 – w) on assets A, B respectively. Is the minimum
less than 15%? Sketch on the (s, r) plane the locus of points (sw , rw ) of
portfolio stdev and returns as w varies.
Now suppose an investor has a target mean return of 15%, and that a riskless asset with return rf is available in addition to assets A and B. Describe
in qualitative terms the composition of the investor’s mean-variance optimal
portfolio for various values of rf , giving rough estimates of the critical values
involved.
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3.
M5MF37
This is a question about CAPM. We have a market consisting of a riskless
asset with ?xed return rf , and n risky assets with returns ri having expected
value r¯i = E[ri ] and covariance matrix S = [sij ] where sij = cov(ri , rj ).
(i)[3 points] Explain what is meant by the market portfolio and give expressions for the mean r¯M and stdev sM of its return rM , and for the covariances
siM = cov(ri , rM ), i = 1, . . . , n.
(ii)[3 points] Explain the basic principle of the CAPM by drawing a (s, r¯)
diagram showing the relationship between the capital market line (what is
that?) and the set G of all (s, r¯) points achievable by forming portfolios of
the n risky assets.
(iii)[8 points] For the jth risky asset, consider the curve {sj (w), r¯j (w)), w ?
[0, 1]} where r¯j (w) and sj (w) are the mean and stdev of the portfolio
wrj + (1 – w)rM . In view of your answer to Part (ii), what geometric
property holds for this curve as w ? 0? Use this property to show that the
mean return for asset j must be given by
r¯j = rf + ßj (¯
rM – rf )
(1)
2
where ßj = sjM /sM
.
(iv)[6 points] Now interpret the CAPM (1) as a pricing formula. Write the
return rj as rj = (Xj – xj )/xj where xj is the price of the jth stock at the
beginning of the period and Xj is its random value at the end of the period
(with mean mj ). Now express r¯j and ßj in terms of xj and the statistical
properties of Xj and use (1) to derive a formula for xj . Show that this
formula gives a linear pricing rule.
Asset
A
B
r¯
s
0.10 0.15
0.18 0.30
Fraction
w
1-w
Table 1: Data for Question 2(iv)
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4.
M5MF37
(i)[5 points] Describe the properties of a utility function and give two examples. De?ne the Arrow-Pratt index of risk aversion and compute it for your
examples. How are utility functions used to formalize a concept of optimal
investment, and what is the economic signi?cance of the Arrow-Pratt index
in terms of investors’ attitude to risk?
(ii)[5 points] Suppose we have initial capital x and n investment opportunities:
we pay a ?xed amount pk (at time 0) to enter one unit of the kth investment
and it returns (at time 1) a random amount Xk , modelled as a random
variable on some probability space (?, F, P). One of these investments is a
bank account where a deposit of 1 at time zero yields a sure return R > 1
at time 1. Now consider the asset allocation problem of choosing a portfolio
?
X = n1 ak Xk so as to maximize the expected utility E[U (X)]. Formulate
this as a constrained, or unconstrained, optimization problem and write down
the necessary conditions for optimality of an asset allocation vector a =
(a1 , . . . , an ).
(iii)[5 points] Use the result of Part (ii) to show that the price pk of asset k
can be expressed as
[ ]
Xk
pk = EQ
,
R
where EQ is the expectation with respect to a measure Q obtained by transforming the original measure P in a way that depends on the solution of the
optimal allocation problem.
(iv)[5 points] Your utility function is U (y) = log y and your initial capital
is x > 0. You have the opportunity to bet on the home team winning at a
football match with odds of 4 to 1, meaning that with a £1 stake (paid before
the match) you are paid after the match £5 (= 4×the stake + a refund of
the stake) if the home team wins and nothing if it loses.
(a) Show by using Jensen’s inequality that if you assess the probability of a
win as 0.2 (i.e. ‘fair odds’) then you will not bet.
(b) Calculate the fraction of your initial wealth you will bet if you think the
win probability is 0.25.
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5.
M5MF37
(i)[7 points] State the Fundamental Theorem of Asset Pricing for a singleperiod market model in which n assets have known prices at time 0 and, at
time 1, there are N possible ‘states of the world’; the value of asset i at time
1 is dij if state j occurs. Show the relationship with “risk-neutral pricing”
when asset 1 is a zero-coupon bond, i.e. d1j = 1 for all j.
The ?gure shows a single-period trinomial tree modelling an asset whose
initial price is S0 = 90 and ?nal price ST ? {S1 , S2 , S3 } where S1 = 120, S2 =
100, S3 = 80. There is also a riskless asset paying zero interest rate.
q
S0
1
q2
q
3
S1
S2
S3
Figure 1: Single-period trinomial tree.
(ii)[6 points] By considering ‘martingale’ probability distributions (q1 , q2 , q3 ),
determine the range of arbitrage-free prices for an at-the-money (ATM) call
option, whose exercise value is max{ST – S0 , 0}. Show in particular that 6
is an arbitrage-free price.
(iii)[7 points] Now suppose the ATM option is traded at a market price of
6, and that we want to price a digital option whose exercise value is 100 if
ST = 110 and zero otherwise. Show that this payo? can be replicated by a
portfolio containing the riskless asset, the underlying asset and the ATM call.
Hence show that the unique arbitrage-free value of the digital option is 10.
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