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Sample Mathematics and Finance Questions

## Nominal and effective rates of interest

1. Describe briefly what is meant by a nominal rate of interest. Calculate the annual effective rate of interest for a nominal rate of 6% per annum converted 6-monthly.

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2. Discuss briefly any four of the twelve key stages of constructing a model.

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3. If the force of interest is ?(t) = 0.0001+0.02t for all time, calculate the accumulated value at time t = 10 of a continuous payment stream of rate ?(t) = 1+200t between times t = 2 and t = 4.

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4. A non-tax paying investor buys Â£100 nominal of a 5-year fixed-interest bond redeemable at 110% and paying annual coupons of 4% in arrears. Calculate the yield obtained if the n-spot rates are yn = (2.5 + n)% per annum for n = 1,2,...,5. You should assume that the bond is held to redemption. Comment on your answer.

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5. Discuss briefly the limitations of using stochastic models for advising a client on the longevity risk to a pension scheme.

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6. (i) Define what are denoted by the quantities an and sn . Derive closedform expressions for each using standard notation. You may assume a constant force of interest.
(ii) You invest Â£250,000 in an asset that returns Â£15,000 per annum paid on a daily basis. Determine an estimate of the discounted payback period for this investment if the cost of borrowing is 2% per annum effective. State any approximations made.

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7. (i) Derive Makehamâ€™s formula for pricing a fixed-interest bond,
A =
K + (1?t1)I if no capital gain
(1?t2)K+(1?t1)I
1?t2K/C if a capital gain
Define all notation used. [Note: This part of the question is not required.]

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(ii) An investor, subject to income and capital gains tax at rates 40% and 15%, respectively, decides to invest in a 10-year fixed-interest b ond r edeemable a t p ar and bearing annual interest of 7% per annum payable monthly in arrears.
If the investor requires a net yield of 5% per annum, calculate the maximum price she should pay for Â£100 nominal of the bond.

8. (i) Explain the retrospective and prospective methods for calculating the capital outstanding on a loan.
A 25-year repayment mortgage for Â£400,000 was granted on 1 April 2015. The mortgage is to be repaid by monthly instalments in arrears that increase by Â£5 each month.
(ii) Calculate the amount of the first instalment if the underlying interest rate is assumed to be fixed at 2.4% per annum nominal, converted monthly.
(iii) [9 marks] The homeowner decides to renegotiate the repayment terms immediately after the payment on 1 April 2023. Determine the new repayments if the remaining debt is to be repaid at a fixed annual rate for 10 years. You should assume that the prevailing interest rate under the new contract is 3.4% per annum effective.

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9. (i) Derive Redingtonâ€™s conditions for the surplus of a fund to be immunised to small changes in interest rate.
A pension fund has liabilities of Â£1 million in 1 yearâ€™s time, Â£2 million due in 2 yearsâ€™ time, Â£3 million due in 3 yearsâ€™ time, and Â£4 million due in 4 yearsâ€™ time. The fund holds two investments, X and Y. Investment X produces income of Â£1 million payable after 3 years.
Investment Y is a zero coupon bond which pays a lump sum of Â£R at the end of n years.
0.041 ?1 per annum effective.

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(ii) Determine whether particular values of Â£R and n can be found that ensure that the fund is immunised to small changes in the interest rate.

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(iii) The interest rate immediately changes to e 0.045 ? 1 per annum effective.
Use the calculations in part (ii) to estimate the revised present value of both the assets of the fund. Fully explain your reasoning. Would you expect an accurate calculation to demonstrate that the fund is in surplus?