Exam May 2014, questions PDF

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MATH1510 MATH151001 This question paper consists of 3 printed pages, each of which is identified by the reference MATH1510.

All calculators must carry an approval sticker issued by the School of Mathematics.

Examination for the Module MATH1510 (May/June 2014)  c University of Leeds Financial Mathematics 1 Time allowed: 2 hours 30 minutes Answer all questions in Section A, and no more than three from Section B. Each question in Section A carries 4 marks, and each question in Section B carries 20 marks. SECTION A — Attempt all questions in this section

A1. Compute the present value of a payment of £350 due in four months on the basis of a compound interest rate of 3 41 % per annum. A2. The interest rate is 5% p.a. Compute the nominal rate of discount compounded weekly. A3. State the formula using the nominal and effective interest rate, assuming that interest is paid p times per year. A4. Given that a20 at 5% equals 27.537790, and a20 at 10% equals 22.972873, use linear interpolation to approximate the rate at which a20 = 25. (2) A5. Express a¨n in terms of only the interest rate.

A6. Compute future value of level annuity for ten years, with monthly payments at the end of each month. The interest rate is 7%. A7. A loan of £2,000 is repaid by a level annuity due over ten years. Compute the annual payment on the basis of 643% interest. A8. The money rate of interest is 5% and the rate of inflation is 3.5%. Compute the real rate of interest.

CONTINUED...

MATH1510 A9. An investor bought some property for £1000, and is selling it four years later. The investor, who is liable for capital gains tax at the rate of 18%, wishes to obtain a net yield of at least 8% per annum. What is the minimum price the investor can sell the property for? A10. What is an index-linked bond?

SECTION B — Attempt three questions in this section

B1. (a) How many days does it take for £1200 to accumulate to £1250 under 3% p.a. simple interest? (b) How many days does it take for £1200 to accumulate to £1250 under an interest rate of 3% p.a. convertible quarterly? (c) A capital of £100 at time t = 0 grows to £105 at t = 2, while a capital of £100 at t = 2 grows to £120 at t = 5. Assume that the principle of consistency holds. Compute the accumulated value at t = 5 of a payment of £60 at t = 0, showing clearly where the principle of consistency is used. (d) Assume that the force of interest is given by δ(t) = 0.03 + 0.02t. Compute the present value at t = 0 of a payment of £3000 due at t = 5. (e) Under what constant rate of interest would the present value of a payment of £3000 due at t = 5 be the same as the one computed in question (d)?

B2. A loan of £60,000 is to be repaid in 10 years by equal monthly payments, done in arrears. The interest rate is 5%. (a) Show that the monthly payment is £633.20. (b) Compute the outstanding balance after 6 years. (c) After n years, the interest rate decreases to 421%. The bank recomputes the required monthly payment under the assumption that the length of the loan does not change and find that the monthly payment decreases by £7.733. Write down an equation for n (you do not need to solve for n). (d) Use linear interpolation with trial values of n = 2 and n = 4 to estimate n.

CONTINUED...

MATH1510 B3. Mrs Jones invests a sum of money for her retirement which is expected to be in 20 year’s time. The money is invested in a zero coupon bond which provides a return of 5% per annum effective. At retirement, she requires sufficient money to purchase an annuity certain of £10,000 per annum for 25 years. The annuity will be paid in arrears and the purchase price will be calculated at the rate of interest of 4% per annum convertible half-yearly. (a) Calculate the present value of annuity certain at retirement. (b) Calculate the sum of money she needs to invest at the beginning of the 20-year period. The index of retail prices has a value of 143 at the beginning of the 20-year period and 340 at the end of 20-year period. (c) Calculate the annual effective real return she would obtain from zero coupon bond. The government introduces a capital gains tax on zero coupon bonds of 25 per cent of the nominal capital gains. (d) Calculate the amount of capital gain before and after tax.

B4. A bank makes a loan to be repaid by instalments paid annually in arrears. The first instalment is £400, the second is £380 with payments reducing to £20 per annum until the end of 15th year, after which there are no further repayments. the rate of interest charged is 4% per annum effective. For this question, you may use that the present value (at t = 0) of the standard increasing annuity, which pays 1 at t = 1, 2 at t = 2, 3 at t = 3, . . . , n at t = n, is given by a¨n − nv n (Ia)n = . i (a) Calculate the amount of loan. (b) Calculate the capital and interest component of first payment. At the beginning of the ninth year, the borrower can no longer make the scheduled repayment. The bank agrees to reduce the capital by 50 per cent of the loan outstanding after the eighth payment. The bank requires that the remaining capital is repaid by a 10-year annuity paid annually in arrear, increasing by £2 per annum. The bank changes the rate of interest to 8% per annum effective. (c) Calculate the first repayment under the revised loan.

END

IMPORTANT NOTE The attached check-sheet contains the final answers to some, not necessarily all questions on the exam. Answers to questions requiring longer answers, for example proofs, are not given.

Please note. In the exam, students are expected to show their full work on the exam script, not just final answers.

Advice. Use this check-sheet to check your answers AFTER you have worked through the exam as if you were in an exam situation, i.e. without access to notes, books, answers to exercises, etc. This way you will test whether you can tackle the problems without any help as in the exam.

Check-sheet for May/June 2014 Examination for MATH1510: Financial Mathematics 1 N.B.: For some reason, the author of Question A7 assumed that the payments were made in arrears and the results to A7 given below is for an annuity immediate, see the comment on the VLE. A1. A2. A3.

£346.29 d (52) = 0.0488

A4. A5.

Approximately 7.78% −n (2) a ¨ n = 2(11−(1+i) −(1+i)−1/2 )

A6. A7. A8. A9. A10.

s 10 = 14.254311 £281.47325. Real rate of interest is 1.4492%. Minimum selling price is 1439.62 No numerical answer

1+i= 1+

i(p) p

p

(12)

B1(a) B1(b) B1(c) B1(d) B1(e)

1 year and 142 days. 1 year and 133 days. 60A(0, 5) = 75.60 £2010.9610 Rate of 8.328%

B2(a) B2(b)

Monthly payment of £633.14. (Instead of £633.20.) £27,553.0232.

B2(c) B2(d)

a

(12)

10−n 5% a 10−n 4.5% (12)

=

625.467 . 633.20

Linear interpolation suggests n ≈ 4.8846.

N.B.: The solutions below to question B3 are valid if one understands that at her retirement Mrs Jones will receive monthly payments (even if the rate of 4% the nominal rate p.a. convertible semi-annual ly ), see the comment on the VLE. B3(a) £158,422. B3(b) £59,708. B3(c) 0.550%. B3(d) £98,717 (before tax); £74,039 (after tax). B4(a) B4(b) B4(c)

£3,052.65. £277.84. £74.16....