[Question 1] (20 marks)

a)    Consider a bond which has 4 years to maturity and pays coupon payments of £100 annually. The bond’s par value  is £1200. Moreover, the yield to maturity 2.5% is compounded annually. The bond is currently trading at around £1463.

Derive the new price of the bond using the modified duration when the yield to maturity increases 1% from the current yield.

(10 marks)

b)    Marry wishes to buy a new electric car and is planning to apply for a loan of £20000. She would like to repay the loan over 10 years in equal monthly payments, each of which includes an interest and principal. She assumes that the current interest rate of 7.5% per annum remains the same over this period and compounds monthly.

What is the monthly repayment?

(10 marks)

[Question 2] (30 marks)

a)   A manufacturing company is currently considering five risky stocks (labelled as i = 1, 2, 3, 4, 5) to create an investment portfolio. They have estimated the first two moments (i.e., expected value and covariances) of rate of return of these stocks using the historical data. Let μi represent the expected rate of return  of asset i. The covariance between stocks i and j is denoted by σij. They would like to develop the Markowitz  portfolio allocation model to determine the optimal investment strategy. The transaction cost for buying and selling any asset is 1.5% and the current portfolio position is given as 0.1 for each asset. The manager thinks that they should invest in no more than four different stocks. Moreover, at most 30% of capital should be invested on any asset to be included into the portfolio. No short sale is allowed.

Formulate (but do not solve) the portfolio allocation problem that maximises the expected portfolio return to achieve at most 15% expected portfolio risk. Briefly describe decision variables, constraints, and the objective function.

(15 marks)

b)   The manager of aconsulting company would like to create the least expensive portfolio of two assets (labelled by i  = 1,  2) to meet their short-term obligations. Let  x  = (x1, x2 )T   ∈ R2   denote  a  vector  of   decision  variables   representing  amount  of  assets   ( i  =  1, 2)  purchased.  The  feasible  set X  consists  of  all  linear  constraints.  Let p1   and p2   denote market    prices   of   assets   i  = 1, 2,    respectively.    They    assume    that   asset    prices ̃(p) = ( p1, p2) ∈ R2  are  uncertain.  A  stochastic  linear  programming  model  (SLP)  of  the

portfolio allocation problem is formulated in a compact form as,

SLP:    minx∈x    ̃(p) x .

For a fixed parameter β  ∈ [0, 1), an uncertainty set U (that asset prices belong to) is given as follows

U  = {̃(p) = (p1, p2) ∈ R2  | 2 − 2β ≤ p1  ≤ 2 + 2β,  4 − 4β  ≤ p2  ≤ 4 + 4β,  p1  + p2   = 6}

Derive (but do not solve) the robust counterpart of the portfolio allocation problem in view of the given uncertainty set.

(15 marks)

[Question 3] (25 marks)

Henry, the manager of a technology company, aims to develop a financial plan such that the firm’s expected final wealth at the end of the planning horizon is to be maximised by meeting their liabilities over three years. Currently, the firm has a capital of (£). The firm's liabilities are given as L1, L2, and L3  for the first, second and third years, respectively. Henry considers three different assets (labelled as A, B and C) and generates the following scenario tree to model uncertain asset returns over three years. The planning horizon of 3 years is represented by discrete time periods (labelled as  t  = 0, 1,2,3) where investment decisions are made. Time period t = 0 represents today.

The scenario tree consists of nodes representing different realisations of asset returns (with certain probabilities) at each time period. Each node of the scenario tree is labelled in terms of time period and node number as (time_period, node_number). For instance, (2,3) at the top of a node of the scenario tree shows the third scenario realised in year 2 with branching probability of 0.4. The gross returns of each asset and the probability of occurrence of each scenario arepresented in the following table.