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MATH0094 Market Risk and Portfolio Theory

创建日期：2024-05-08 17:33:52

所属分类：数学mathematics

标签： MATH0094

Market Risk and Portfolio Theory

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MATH0094 Market Risk and Portfolio Theory

Mock Exam
TIME: 2.5 HOURS
Each question is worth 25 marks.
MATH0094 1 TURN OVER
Question 1. For a constant c ∈ R, let u : R→ R be given by
u(x) =
{√
x− c x ≥ c
−√c− x x < c ,
where
√
x is the non-negative square root for x ≥ 0.
(a) Calculate the coefficient of absolute risk aversion and the coefficient of relative
risk aversion as a function of wealth for an investor with this utility function.
[ 8 marks]
(b) Two investment opportunities are available to an investor with the above util-
ity function and initial wealth c: i) An investment with (gross) return R
distributed like U [1/2, 3/2]; or ii) An investment with return R˜ distributed
like U [2/3, 4/3].
According to utility theory, which of these investments will the investor prefer?
Generalise your answer to the comparison of two investments with distribution
U [1− p, 1 + p].
[9 marks]
(c) Consider the risk measure ρu defined by
ρ(X) := −u−1(E[u(X)]), for X such that E|u(X)| <∞.
Show that u is not a monetary risk measure.
[8 marks]
[Total: 25 marks]
MATH0094 2 CONTINUED
Question 2.
Set a one-period market model with a risk-free asset with rate of return r01 and n
risky assets with normal rate of return vector r1 ∼ N (µ,Σ), where µ ∈ Rn and
Σ ∈ Rn×n is definite positive and invertible. Recall that
ri1 :=
Si1 − Si0
Si0
; i = 0, . . . , n.
Assume that 0 < r01 < µ
i for i = 1, . . . n.
(a) Show that if M is an SDF in this market, then M is negatively correlated with
each rate of return ri1 (that is, show that corr(M, r
i
1) < 0).
[7 marks]
(b) Find the optimal investment solving (a) when the investor has CARA utility
with risk aversion α.
[10 marks]
Consider now a two-period version of the above market, assuming that the rate of
returns in time are i.i.d. so that r2 ∼ r1, r2 ⊥ r1 and r02 = r01.
(c) The same investor as in (b) wants now to maximise
E[u(W2)]
where W2 is their wealth at time 2. Assume they have initial wealth w0, no
consumption and no endowments. Find the optimal investment strategy for
this investor.
Hint: Take advantage of the independence assumption and use dynamic pro-
gramming.
[8 marks]
[Total: 25 marks]
MATH0094 3 TURN OVER
Question 3.
(a) Show that for α ∈ (0, 1) and any two random variables X, Y such that (X, Y )
is jointly Gaussian we have that V@Rα(X + Y ) ≤ V@Rα(X) + V@Rα(Y ).
[9 marks]
(b) Show via a counterexample that Value at Risk is, however, not subadditive in
general. Verify on the same example the subadditivity of expected shortfall.
[9 marks]
(c) Assume that you are calculating capital on the daily returns of a portfolio
using as risk measure value at risk at 97.5%.
Use a Z-test (Gaussian test) to find the minimal number of excess losses in a
trading year (252 days) that would put in doubt the coverage property of your
calculation, if the accepted I-type error is 5%.
[7 marks]
[Total: 25 marks]
MATH0094 4 CONTINUED
Question 4. Consider a financial market composed by one risk-free asset with
return R0, and n risky assets with returns Rˆ = (R1, . . . , Rn)>. We define µ =
(µ1, . . . , µn)> where µi = E[Ri], and the matrix Σ where Σij = cov[Ri, Rj], for
i, j = 1, . . . , n. Assume that Σ is invertible and that
R0 6= µ
>Σ−11
1>Σ−11
,
Furthermore, let pi = (pi0, . . . , pin)> where pii denotes the proportion of investment
in the i-th risky asset, and pi is the sub-vector corresponding to the risky assets.
(a) We showed that all portfolios in the mean-variance frontier can be written as
pi∗ = δpΣ−1(µ −R01); pi∗0 = 1− 1>pi∗
Show that the value of the constant δp needed to obtain a portfolio in the
frontier with mean µp is
δp =
µp −R0
(µ −R01)>Σ−1(µ −R01) .
[8 marks]
(b) For a portfolio pi let Rpi, µpi and σpi be respectively its return, expected return
and standard deviation of its return. Recall that the Sharpe ratio is given by
S(pi) = µpi −R
0
σpi
.
Show that
|S(pi)| ≤
√
(µ −R01)>Σ−1(µ −R01)
with equality if and only if pi is in the mean-variance frontier.
[9 marks]
(c) Show that if there is a beta pricing model having as factor the return of a
market portfolio pi∗, i.e., if for any market portfolio pi we have that
µpi −R0 = (µpi∗ −R0)cov(Rpi, Rpi∗)
σ2pi∗
,
then pi∗ is a portfolio in the mean-variance frontier. Hint: Use (b).

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