IB3K20 Financial Optimisation

IB3K20 Financial Optimisation
Exam Paper

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 princ 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.

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)

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 aregiven 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 scenari 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 n 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 ass