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UNIVERSITY OF WARWICK
Department	Warwick Business School
Level	3
Module Code	IB3K20
Module Title	Financial Optimisation
Exam Paper Code	IB3K20

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[Question 1-35% of total marks]
a) Robert, as the fund manager of a telecommunication company, is currently considering two types of investments stated as Case 1 and Case 2.
Case 1: They can deposit £80000 for 5 years in a saving account that pays simple interest at rate of 2.5% per annum. There is no withdrawal during the investment period.
Case 2: They can deposit £80000 for three years in a saving account that pays simple interest at rate of 2.5% per annum. They can then withdraw the money with interest and deposit into another account paying simple interest at 2.5% for another two years.
Which case should Robert choose as a profitable investment?Justify your answer. (9 marks)

b) Robert is not satisfied with the investment opportunity stated in part (a).Thus, he considers three securities (labelled as A, B, C) whose market prices and pay-off values estimated in two different states of the economy are presented in the following table.

Security ID	Market Price	Payoff in State 1	Payoffin State 2
A B C	128 120 124	35 45 41	50 35 41

Robert seeks for a riskless profit if possible. Thus, he receives a capital of £10000 by short-selling security Cand then invests 25% of the capital in security A and the remaining capital in security B. Compute value of each investment on securities A, B, and C today as well as under states 1 and 2. (10 marks)

c)Robert also investigates another investment opportunity on a fixed income security. He considers a bond with 4 years maturity and £4000 par value. The bond pays a coupon of £125 annually. Moreover, the yield to maturity is 5.5% and compounded annually. They assume that the yield to maturity increases from the current rate by 1.25%.Calculate the new price of the bond approximately by using the modified duration. Show whether this is a good approximation or not. (16 marks)

[Question 2-30% of total marks]
a) David, the portfolio manager of an investment bank, is now willing to create a portfolio using the Markowitz portfolio allocation model. He considers two assets (labelled as A and B) from automotive and biotechnologysectors. They consider a historical monthly data ofthe past 10 years and estimate average monthly returnsof assets A and B as 1.55% and 1.25%, respectively,whereas stan dard deviations of assets A and B are computed as 12.5% and 6.5%, respectively. Given the correlation of both assets, they would like to analyse how a portfolio of the two assets would perform. They assume that the correlation between assets A and B is either -1 or +1. Under this assumption, compute the expected portfolio return and standar d deviation of portfolio return. Fill in missing cells in the following table. Give at least one observation on how the asset correlation and the portfolio risk (i.e., standard deviation of portfolio return)are related. (12 marks)

Portfolio Weights		Expected Portfolio Return (%)	Standard Deviation of Portfolio Return
Asset A	Asset B		Correlation =1	Correlation =-1
0	1
0.25	0.75
0.75	0.25
1	0

b) David now ignores the model developed in part (a) and would like to construct a diversified portfolio. He extends the universe of assets from different sectors. He considers five different assets (labelled as A1, A2, A3, A4, and As,respectively) from automotive, tourism, construction, IT technology and biotechnology sectors as they have an initial amount of 0.15 for each asset. He estimates the first two moments as the average rate of returns and the covariance matrix of these asset returns and develops the mean-variance portfolio allocation model by considering specific conditions. David thinks that the portfolio must consist of at most four different assets and one of which much be an asset from IT technology.The transaction cost for buying and selling per share of any asset is 0.25%. Moreover, at most 40% of the capital should be invested on each asset and the short sale is not allowed. Formulate (but do not solve) the portfolio optimisation model that minimises the portfolio risk to achieve the expected portfolio return to be at least 13.5%. Clearly define decision variables and briefly explain constraints, and the objective function. (18 marks)
[Question 3-35% of total marks]
Helen, as the fund manager of Tesco Ltd., is aiming to develop a portfolio dedication strategy.She currently considers investing in 8 different bonds to pay off their liabilities over the next 3 years.The features of these bonds are presented in Table 1 in terms of maturity, coupon payments, face values as well as the current market prices. They assume that all bonds are widely available and can be purchased in any quantities at given prices. The investment horizon of 3 years is represented by d iscrete time periods as t = 0, 1, 2, 3 and the investment decisions are made today (i.e.,t = 0). The total cash obligations are denoted as Lt at year t for t =1, 2, 3. The remaining cash surplus, after paying off liabilities from the return received, is reinvested at each year into their savings account which applies the Bank of England's base rate of 5.25%.They currently have £1000 in the savings account. Helen also considers having a 1-year loan in the second year up to £15000 if needed. The amount of loan must be paid off at a rate of 1% higher than the Bank of England interest rate.

Table 1: Features of different bonds

Bond IDs	1	2	3	4	5	6	7	8
Price(£)	106	114	105	117	98	104	112	109
Coupon rates (£)	12.5	15	12	14	13	10.5	9.5	13
Maturity (year)	2	1	2	3	1	3	3	2
Face Value (£)	108.5	105	110	111	125	116.5	105.5	107

a) They assume that the Bank of England's interest rate remains the same over the three-year investment horizon. They wish to develop a deterministic linear programming model such that the firm's final wealth at the end of the planning horizon is maximised while the total investment cost and amount of borrowing are to be minimised. Formulate (but do not solve) the financial planning problem and briefly describe decision variables,constraints, and objective function. (12 marks)
b) Helen now ignores the optimisation model developed in part (a) and would like to consider uncertain interest rates over the next 3 years. She expects that the Bank of England's base rate may vary over the planning horizon and generates a scenario tree.Each node of the scenario tree represents different realisations of the Bank of England's base rate with certain branching probabilities. The scenario tree has a special structure with two-branching at each node over the first two years and then one-branching in year 3 (i.e., the base rate remains at the same rate of the previous time-period). For two scenarios realised (at each node of the first 2 years), they assume that the interest rate may increase by 0.25% (with probability of 0.6) or decrease by 0.15% (with probability of 0.4) from the current base rate. Draw the scenario tree and clearly present stages, branching probabilities, and realisations at each node. Formulate (but do not solve) the financial planning problem using a scenario-based linear program that minimises the total investment cost and expected amount of loan to borrow while the expected cash surplus at the final year is maximised. (23 marks)

End of Paper