Mid Term Examination Semester A 17/18 MTH5124 Actuarial Mathematics I SOLUTIONS Duration: 2 hours 1. A saver invests £1,000 in a bank account earning interest at an AER of 6% per annum. What is the balance on the account after 6.5 years? A. £1,460.45 CORRECT B. £1,459.14 C. £1,418.52 D. £1,390.00 2. A loan shark charges simple interest of 10% per week. What is the effective annual rate of interest if a customer repays the loan plus interest after 1 year? A. 10.00% B. 10.51 % C. 520.00% CORRECT D. 1410.43% 3. What is the present value of a payment of £1,000 due in 9 years time if the force of interest is 0.075 per year? A. £509.16 CORRECT B. £598.69 C. £637.76 D. £1,952.08 4. Which of the following interest rates has the highest equivalent Annual Effective Rate? A. An interest rate of 6% every 6 months. B. i(2) = 7% per annum. C. i = 13% per annum. CORRECT D. i(52) = 7% per annum. c Queen Mary, University of London (Semester A 17/18) Turn Over MTH5124 Actuarial Mathematics I SOLUTIONS (Semester A 17/18) Page 2 5. An individual wins a lottery prize of £4,000 payable every 3 months in advance for 10 years. Which of the following expressions is an incorrect expression for the present value of the cashflows at an interest rate of 10% per annum. (4) A. Present value = 16000ä10 at i = 10% per unit of time B. Present value = 4000ä40 at i = 2.411% per unit of time (4) C. Present value = 16000a10 at i = 10% per unit of time INCORRECT (4) D. Present value = 16000(0.25 + a9.75 ) at i = 10% per unit of time 6. What is s̈12 at an effective annual interest rate of 14.0%? A. 9.8 B. 13.9 C. 17.9 D. 31.1 CORRECT 7. What is the correct symbol for the present value of a continuously payable increasing annuity payable, for n units of time, at a rate of t per unit of time in year t? A. (I¯ā)n 0 0 B. an at i = i− j 1+ j C. (Ia)n D. (I ā)n CORRECT 8. What is the effective annual rate of discount if the nominal rate of interest is 8% per annum convertible 4 times per year. A. 8.24% per annum. B. 8.00% per annum. C. 7.62% per annum. CORRECT D. 7.41% per annum. 9. A loan of £20,000 is repayable by payments of £C annually in arrears for 20 years. Given an expression for the capital outstanding after 8 years if the effective annual rate of interest is i%. c Queen Mary, University of London (Semester A 17/18) Turn Over MTH5124 Actuarial Mathematics I SOLUTIONS (Semester A 17/18) Page 3 The most common errors were: • Confusing the loan and annual repayment amounts (a very serious problem); and • calculating answers at the beginning of the loan rather than after 8 years. There are a variety of correct solutions including: • Capital outstanding = Ca12 12 • Capital outstanding = 20, 000 1−v 1−v20 • Capital outstanding = 20, 000(1 + i)8 −Cs8 • Capital outstanding = 20, 000 −Ca8 (1 + i)8 (2) 10. Explain in words the meaning of the symbol a12 . This a good student solution: Present value of an annuity of 1 per unit of time: • for 12 units of time • payments are made in arrears • payments(of 12 ) are made 2 times per unit of time 11. A purchaser buys £10,000 nominal of a corporate bond on 1st January 2018. The bond is repayable on the 1st November 2028 at 120% of nominal. Interest is paid on the bond at a rate of 5% per year quarterly in arrears. Write an expression for the price of the bond on 1st January 2018 at a gross redemption yield of i% per annum. c Queen Mary, University of London (Semester A 17/18) Turn Over MTH5124 Actuarial Mathematics I SOLUTIONS (Semester A 17/18) Page 4 Most students could reproduce the standard formula Price per N nominal (p) = CNat + RNvn and recognised from the question that: • N = 10000 • C = .05 • p=4 • R = 1.20 1 • v = 1+i ; ie the calculation is made at an interest rate equal to the gross redemption yield. Many students did not recognise that the standard formula must be adjusted for the non integer term (10 years and 10 months) and the fact that the number of years worth of coupons is different to the outstanding term. The investor receives 11 years worth of coupons ( every 3 months from 1st February 2018 until 1st November 2028). A fully i be price per N nominal = h correct 2 answer would 10 (4) 10, 000 0.05(1 + i) 12 a11 + 1.2v10 12 . c Queen Mary, University of London (Semester A 17/18) Turn Over MTH5124 Actuarial Mathematics I SOLUTIONS (Semester A 17/18) Page 5 12. A charity receives the income from a perpetuity of £400 per annum payable annually in advance. Determine the value of the perpetuity at an effective interest rate of 8% per annum. The answer is £54,000. The solution is 400ä∞ = 400(1+i) . i 13. A newsagent buys a news round for £1200. He estimates that he will make additional profits of £55 per month, payable monthly in arrears. Assuming an effective interest rate of 2% per month, calculate the discounted payback period to the nearer month. The answer was 29 months. It is found by finding the smallest value of t such that 55at ≥ 1200 at an interest rate of 2% per unit of time. Natural logs are used to find t. c Queen Mary, University of London (Semester A 17/18) Turn Over MTH5124 Actuarial Mathematics I SOLUTIONS (Semester A 17/18) Page 6 14. The table below shows the cashflows and investment performance of a fund over 2016. All non-investment cashflows occur immediately at the start of each six month period. Year Jan - June July - December Fund at start of Period 10000 10750 New Cashflow 0 1250 Investment Return (£) 750 1200 Investment Return 7.5% 10% Fund at end of Period 10750 13200 Determine the time weighted rate of return for the fund in 2016. If i1 is the investment return for the first 6 month period and i2 is the investment return for the second period, then the time weighted rate of return is found from (1 + i1 )(1 + i2 ) − 1. But i1 = 0.075 and i2 = 0.10, which leads to the TWRR of 18.25%. End of Paper. c Queen Mary, University of London (Semester A 17/18)