证明题代写-MATH 11158-Assignment 1
时间:2022-02-23
MATH 11158 : Optimization Methods in Finance
Assignment 1
14 February 2022 Due: 22 February, 2022, 4pm
Question 1 (Markowtiz Efficient Frontier, (15 marks)). The minimum variance portfolio problem
min
x
1
2
x>Σx s.t. e>x = 1, µ>x= R
where the target return must be achieved exactly, short-selling is allowed, and the covariance matrix
is assumed to be positive definite. Here, e is the n-dimensional vector of all ones. Assume that there
are at least two assets whose mean returns are not equal to each other.
Denote three scalars A = e>Σ−1e, B = µ>Σ−1e and C = µ>Σ−1µ.
1. Prove that AC −B2 > 0. (6 marks)
2. Prove that for every R, the point (σR, R) lies on the hyperbola
σ2R
1/A − (R−B/A)
2
(C/A−B2/A2) = 1 in the
(σ,R)-plane. (6 marks)
3. Explain why the efficient frontier is produced by all portfolios x∗R that have the value of R
being R ∈
[
B
A ,∞
)
. (3 marks)
Question 2 (Convexity, (20 marks)). Prove convexity of the following:
1. an ellipsoid in Rn. (5 marks)
2. the risk function of portfolio x
Risk(x) := E
[
R(x)
]
+ δE
[∣∣∣R(x)−E [R(x)]∣∣∣]
where R(x) is the portfolio return and δ is any positive scalar. (10 marks)
3. the Lagrangian function of the nonlinear optimization problem
max f(x) s.t. gi(x) ≤ bi, i = 1, . . . ,m, x ∈ P,
where P is a polyhedron. (5 marks)
Question 3 (Type-B arbitrage, (5 marks)). Consider the sports betting example from lecture
slides. Determine whether there exists type-B arbitrage by solving the appropriate liner program.
Question 4 (Portfolio Optimization, (20 marks)). Download the file indices.csv from Learn. It
has mostly closing values for 7 leading stock market indices, Dow-Jones (USA), FTSE (UK), Dax
(Germany), CAC (France), Nikkei, HSI (Hongkong), BOVESPA (Brazil) as well as monthly prices
for Gold for the 8 years from January 2008 to January 2016.
1. Compute the covariance matrix and state the variance for the Nikkei index. (3 marks)
2. What are the smallest and largest values of the target returns which seem sensible, i.e., beyond
which the minimum variance portfolio model will not change its answer ? (2 marks)
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MATH 11158 : Optimization Methods in Finance
3. Plot the efficient frontier and composition of optimal portfolio for the above range of target
return values. (5 marks)
4. Replace variance with the risk measure from the second question in this sheet and solve your
model for a target return rate of 0.24%. (7 marks)
5. State the optimal portfolio and its volatility after solving the model in part (4). (4 marks)
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