[Question 1]  (20 marks)

a)   Consider a bond which has a maturity of 2 years and pays coupon payments of £75 annually. The bond’s par value is £425, and it is currently trading at £500.

i)       Find the yield to maturity and compute the Macaulay duration and convexity.  (6 marks)   

ii)       Derive the new price of the bond using the modified duration and convexity if the yield to maturity increases 1% from the current yield. Verify your answer by deriving the actual price of the bond using the new yield.  (7 marks)

b)   Suppose that you want to buy a £180,000 flat in Coventry. You get a loan with terms that require a £30,000 deposit and  repayment of the  rest in monthly instalments  in 25 years.

Assume that the interest rate of 8% per annum remains the same for the life of the loan and compounds monthly. What is the monthly payment? (7 marks) 

[Question 2]  (30 marks)

Consider a portfolio with two risky assets A and B . Let wA  and wB  denote the weights of assets A and B, respectively.

a)   Assume that the mean rates of returns are estimated as 0.015 and 0.012 for assets A and B,

0.007    0.001

Answer the following questions using the Markowitz portfolio allocation model.

i) Compute the expected rate of return and expected risk of the portfolio constructed by assets A and B . Write the mean-variance portfolio allocation model. (6 marks)

ii) Formulate the  pure  portfolio  risk  minimization  problem . Compute the  investment strategy that minimizes the expected portfolio risk. What is the expected portfolio return and risk for the optimal investment strategy? (12 marks)

b)   Consider the worst-case analysis for the stochastic portfolio allocation problem that aims to maximize portfolio return subject to the sum of weights is equal to one . Let  A  and B  denote

uncertain rates of returns of assets A and B, respectively. Assume that asset returns belong to the following uncertainty set:

U = {(A, B )|A  = 0.015 +  A, B  = 0.012 + B ,  A  ∈ [−0.01, 0.25],  B  ∈ [−0.02, 0.35] }

where A  and B  are random parameters defined within different intervals. Formulate the robust portfolio management problem in view of the uncertainty set. Derive (but do not solve) the robust counterpart of the portfolio allocation problem using the duality theory. (12 marks)

[Question 3]  (22 marks)

A large technology company would like to determine the optimal combination of different financial instruments so that their short-term cash requirements during April - September 2021 will be met. They consider three different sources of funding opportunities as bank credit, issue of bonds, and cash investment. They can receive a bank credit up to £200k at the beginning of each month with an interest rate of 1. 1% per month. In each one of the first four months, they plan to issue 60-day commercial paper requiring a total interest of 2.5% for two-month period. Finally, they wish to invest any amount of excess funds remaining at the end of each month with an interest rate of 1.5% per month.

Because of other commitments to large technology projects they run, they forecast two potential scenarios (labelled as Scenarios 1 and 2) with probabilities of 0.35 and 0.65, respectively, for the net cash  flow  requirements.  The  following  table  presents  net  cash  flow  requirements  (in  terms  of thousands of pounds) under both scenarios where positive entries represent the cash needs while negative entries correspond to the cash surpluses.

a)    Formulate (but do not solve) the company’s short term financing problem as a linear optimization model that maximizes the average excess funds to be received at the beginning of October 2021. Briefly describe decision variables, constraints and the objective function. (17 marks)

b)   Now they would like to ensure that the average excess cash fund must be at least £75k at each month during April - August 2021. Briefly explain how to modify the optimization model developed in part (a) to take into account this condition. (5 marks)

[Question 4] (28 marks)

An investment company would like to construct a portfolio from two risky assets (i = 1, 2) and one risk-free asset (i = 3) to meet their cash requirement at the end of the investment horizon. The investment decisions are made at discrete time periods t = 0,1, … , T where t = 0 represents today. The  manager  is  concerned  with  restructuring  of the  portfolio  at  discrete  time  periods  over  the investment  horizon.  He  assumes  that the  initial  capital 0  is to  be  invested  at t = 0 . The  cash requirement  at  t = T  is  C .  In  addition,  they  consider  borrowing  (as  any  amount  required)  at intermediate time periods. In this case, the amount borrowed must be paid off with the same rate of risk-free asset at the next time period. The cost of borrowing (denoted by t ) is proportional to the amount received at time t, and to be paid at the end of investment horizon. The short sale is not allowed at any time period. Moreover, there is no trading and no borrowing at the final stage.