MAS362/MAS462/MAS6053 Financial Mathematics Problem Sheet 1

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MAS362/MAS462/MAS6053 Financial Mathematics Problem Sheet 1 1. Consider a £100, 000 mortgage paying a fixed annual rate of 5% which is compounded monthly. What are the monthly repayments if the mortgage is to be repaid in 25 years? 2. You want to deposit your savings for a year and you have the choice of three accounts. The first pays 5% interest per annum, compounded yearly. The second pays 4.8% interest per annum, compounded twice a year. The third pays 4.9% interest per annum, compounded continuously. Which one should you choose? 3. Consider any traded asset whose value vT in T years is known with certainty. −rT Prove that the current value of the asset must be vT e where r is the continuously- compounded T -years interest rate. 4. A perpetual bond is a bond with no maturity, i.e., it pays a fixed payment at fixed intervals forever.1 Under the assumption of constant (continuously compounded) interest rates of 5%, what is the price of a perpetual bond with face value £100 paying annual coupon of 2.50% once a year and whose first coupon payment is due in three months? 5. The prices of zero coupon bonds with face value of £100 and maturity dates in 6, 12 and 18 months are £98, £96 and £93.5 respectively. The price of a 8% bond maturing in 2 years with semiannual coupons is £107. Use the bootstrap method to find the two-year discount factor and spot rates. 6. Consider the following zero-coupon bonds with face value of £100: Time to maturity Annual interest Bond price (in years) (in £) 0.5 0 97.8 1. 0 95.5 Suppose that you are offered by a risk free institution the opportunity to deposit or borrow £1,000,000 in six months for a period of six months earning an interest rate of 5%. Describe in detail an arbitrage opportunity available to you. 1An example of such a bond is the consol bond, which was issued in 1749 by the British gov- ernment to convert a number of outstanding debt issues into a promise to pay 3% interest to the note holders forever. In 1888, the consols were converted from 3% into 2.5% notes so the holder of a £100 consol today receives £2.50 a year in interest. 1.

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