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TIM E VALUE OF M ONEY TUTORIAL QUESTION AN D SOLUTION Question 1 Find the following present and future values: a. An initial £ 500 compounded for 1 year at 6 percent. Present and future b. An initial £ 500 compounded for 2 years at 6 percent. values c. The present value of £ 500 due in 1 year at a discount rate of 6 percent. d. The present value of £ 500 due in 2 years at a discount rate of 6 percent. S O L U T IO N a. Given, Present value (PV) = £. 500 Interest rate (i) = 6% 0 6% 1 - 500 FV = ? FVn = PV(1 + i)n FV1 = PV (1 + i)1 = £ 500 (1 + 0.06)1 = £. 530 b. Present value (PV) = £. 500 Interest rate (i) = 6% 0 6% 1 2 - 500 FV = ? FVn = PV(1 + i)n FV2 = PV (1 + i)2 = £. 500 (1 + 0.06)2 = £. 561.80 c. Answer: = £. 471.70 d. Future value (FV) = £. 500 Interest rate (i) = 6% No. of periods (n) = 2 Present value (PV) = ? 0 6% 1 2 PV = ? FV = 500 FVn FV2 PV = (1 + i)n = (1 + i)2 = £ 445 Question 2 Suppose John deposits £ 10,000 in a bank account that pays 10 percent interest annually. Future value How much money will be in his account after 5 years? SOLUTION Here, Present value (P) = £ 10,000, Interest rate (k) = 10% Number of years (n) = 5 years, Future value (FV5) = ? 0 1 2 3 5 10% £10,000 FV = ? We have, FV5 = PV × (1 + k)n = £ 10,000 × (1.10)5 = £ 10,000 × 1.6105 = £ 16,105.10 John will have £ 16,105.10 at the end of year 5 in his account. Question 3 What is the present value of a security that promises to pay you £ 5,000 in 20 years? Present value Assume that you can earn 7 percent if you were to invest in other securities of equal risk? SOLUTION Here, Future value (FV) = £ 5,000 Number of years (n) = 20 years Interest rate (k) = 7% Present value (PV) = ? 0 1 2 3 20 7% PV = ? £5,000 We have, FV20 PV = (1 + k)n = 5000/ (1.07)20 =5000/3.8697 = £ 1,292.09 Question 4 If you deposit money today into an account that pays 6.5 percent interest, how long will it Time for a lump take for you to double your money? sum to double SOLUTION Here, Interest rate (i) = 6.5% Number of period (n) = ? Present value (PV) = £ 1000 (assume) Future value (FV) = £ 2000 0 1 2 3 n=? 6.5% PV = £1000 FV = £ 2,000 We have, FV Present value (PV) = (1 + i)n or, £ 1000 = 2000 / (1+0.065)n or, (1 + 0.065)n = 2000/1000 or, (1.065)n = 2 .... (i) Trying at n = 11 We get, If n = 11, the left hand side in above equation (i) is approximately equal to 2. Hence the required no. of years to double the sum of money is 11 years. Question 5 Your broker offers to sell a note for £ 13250 that will pay £ 2345.05 per year for 10 years. If Effective rate of you buy the note, what rate of interest will you be earning? Calculate to the closest interest percentage. S O L U T IO N Here, Present value of annuity (PVA) = £. 13,250 Periodic equal payment (PMT) = £. 2345.05 No. of periods (n) = 10 years Interest rate (i) = ? Time Line 0 1 2 3 4 5 6 7 8 9 10 PVA = 2345.05 2345.05 2345.05 2345.05 2345.05 2345.05 2345.05 2345.05 2345.05 2345.05 13250 We have, PVA = PMT × PVIFA i × n yrs. or, £. 13,250 = £. 2345.05 × PVIFAi% 10 yrs or, PVIFAi%, 10 yrs = 5.6502 From the PVIFA table, the value of 5.6502 in 10 years lies at 12%. The required interest rate is 12%. Question 6 Your parents are planning to retire in 18 years. They currently have £ 250,000, and they would like to have £ 1,000,000 when they retire. What annual rate of interest would they Effective rate of interest have to earn on their £ 250,000 in order to reach their goal, assuming they save no more money? SOLUTION Here, Future value (FV) = £ 1,000,000 Present value (PV) = £ 250,000 Time period (n) = 18 years Interest rate (i) = ? 0 1 2 3 18 i=? £ 250,000 £ 1,000,000 We have, FV = PV (1 + i)n or, £ 1,000,000 = £ 250,000 (1 + i)18 or, (1 + i)18 = 1,000,000/ 250,000 or, (1 + i)18 = 4 or, 1 + i = (4)1/18 or, i = 1.08 - 1 = 0.08 or 8% The required rate of interest to reach the goal is 8%. Question 7 What is the future value of a 5-year ordinary annuity that promises to pay you £ 300 each Future value of an year? The rate of interest is 7 percent. annuity SOLUTION Here, Future value of annuity (FVA) = ? Payment (PMT) = £ 300 Number of period (n) = 5 years Interest rate (i) = 7% 0 1 2 3 5 7% £300 £300 £300 £300 We have, FVA = ? (1 + i) - 1 FVA = PMT n i (1 + 0.07)5 - 1 = £ 300 0.07 = £ 300 × 5.7507 = £ 1,725.21 Question 8 What is the future value of a 5-year annuity due that promises to pay out £ 300 each year? Assume that all payments are reinvested at 7% a year, until year 5. Future value of an annuity due SOLUTI ON Here, Future value of annuity due (FVAdue) = ? Payment (PMT) = £ 300 Number of period (n) = 5 years Interest rate (i) = 7% 0 1 2 3 5 7% £300 £300 £300 £300 FVA (due) = ? We have, (1 + i)n - 1 FVAdue = PMT i (1 + i) (1 + 0.07)5 - 1 = £ 300 0.07 (1 + 0.07) = £ 300 × 5.7507 × 1.07 = £ 1,845.97 Question 9 A company invests £ 4 million to clear a tract of land and to set out some young pine trees. The trees will mature in 10 years, at which time the company plans to sell the forest Expected rate of at an expected price of £ 8 million. What is company's expected rate of return? return SOLUTION Here, Future value (FV) = £ 8,000,000 Present value (PV) = £ 4,000,000 Time period (n) = 10 years Expected rate of return (i) = ? First set up time line as follows: 0 1 2 3 10 i=? We have, £ 4 million £ 8 million FV = PV (1 + i)n or, £ 8,000,000 = £ 4,000,000 (1 + i)10 or, (1 + i)10 =8,000,000/ 4,000,000 or, (1 + i)10 =2 or, 1 + i = (2)1/10 i = 1.0718 - 1 = 0.0718 or 7.18% Question 10 Rachel wants a refrigerator that costs £ 12000. She has arranged to borrow the total purchase price of refrigerator from a finance company at a simple interest rate equal to 12 Solving for payment percent. The loan requires quarterly payments for a period of three years. If the first payment is due three months after purchasing the refrigerator, what will be the amount of her quarterly payments on the loan? SOLUTION Given, Present value of an annuity (PVA) = £. 12,000 Simple interest rate per annum (i) = 12% Loan requires quarterly payment i.e. m = 4 Number of years (n) = 3 year Periodic equal payment (PMT) = ? Time line: 0 12% 1 2 3 4 12 -12000 PMT PMT PMT PMT PMT We have, 1 - 1 n × m PVA = PMT (1 + i/m) i m 1 - 1 3 × 4 or, 12,000 = PMT (1 + 0.12/4) 0.12 4 or, 12,000 = PMT × 9.9540 PMT = 12,000/ 9.9540 = £ 1,205.55 Question 11 You need to accumulate £ 10,000. To do so, you plan to make deposits of £ 1,250 per year, with the first payment being made a year from today, in a bank account which pays 12 Reaching a financial percent annual interest compounded annually. Your last deposit will be less than £ 1,250 goal if less is needed to round out to £ 10,000. How many years will it take you to reach your £ 10,000 goal, and how large will the last deposit be? SOLUTION Here, Annual payment (PMT) = £ 1,250 Future value of annuity (FVAn) = £ 10,000 Interest rate (i) = 12% Time to maturity (n) = ? Last deposit = ? First, we determine the 1 0 number of periods of the financialngoal. This is calculated using 2 3 =? 12% future value of annuity formula as follows: We have, FVAn PMT 1,250 1,250 =1,250 × FVIFAi, n Last deposit = ? £ 10,000 = £ 1,250 × PViFA12, n FVA = £ 10,000 10‚000 FVIFA12, n = 1‚250 = 8 Looking FVIFA table the value 8 at 12 percent interest rate lies approximately in 6 years. Therefore the number of years to reach the financial goal is 6 years. Now we calculate the future value of £ 1,250 for 5 years at 12%, it is £ 7,941.06 FV = £ 1,250 × FVIFA12, 5 = £ 1,250 × 6.3528 = £ 7,941 Compounding this value after 6 years and before the last payment is made, it is £ 7,941 (1.12) = £ 8,893.92. Thus, we will have to make a payment of £ 10,000 - £ 8,893.92 = £ 1,106.08 at year 6, therefore it will take 6 years, and £ 1,106.08 must be paid in the last installment. Question 12 Your Company is planning to borrow £ 1,000,000 on a 5-year, 15 percent, annual Loan amortization payments, fully amortized term loan. What fraction of the payment made at the end of the second year will represent repayment of principal? SOLUTION Here, Loan amount (PVA) = £ 1,000,000 Number of years (n) = 5 years Interest rate (i) = 15% First we determine the annual installment or payment (PMT) We have, PVA PMT = PVIFA = 1,000,000/PVIFA15,5 =1,000,000/ 3.3522 = £ 293,311.55 i‚ n Preparation of Amortization Schedule, Amortization schedule Year Payments Payment of Principal Interest Ending Balance 1 £ 298,311.5566 £ 150,000 £ 148,311.5566 £ 851,688.4434 2 298,311.5566 127,753.2665 170,558.2901 681,130.1533 Principal payment in 2nd year % principal in 2nd year = Payments = 57.17% That is 57.17% of the payment in second year represents the principal. Question 13 You are branch manager of town centre Natwest Bank, Manchester. A borrower Loan amortization approaches you for a term loan of £ 500,000. You agreed to give loan to be fully amortized in a period of 5 year at 10 percent, annual payment. What will be the size of each installment? What fraction of the payment made at the end of second year represents repayment of interest? SOLUTION Here, Loan amount (PVA) = £ 500,000 Number of years (n) = 5 years Interest rate (i) = 10% First we determine the annual installment or payment (PMT) We have, PVA PMT = PVIFA = £ 500,000/ PVIFA10,5 = 500,000/3.7908 i‚ n = £ 131898.28 Preparation of Amortization Schedule, Amortization schedule Beginning Repayment of Ending Year PMT Interest balance principal balance 1 £500,000 £131,898.28 £50,000 £81898.28 £418,101.72 2 418,101.72 131,898.28 41,810.17 90,088.11 328,013.61 Interest payment in 2nd year % interest in 2nd year = Payments = 31.7% That is 31.7% of the payment in second year represents the interest. Question 14 a. It is now January 1, 2007. You plan to make 5 deposits of £ 100 each, on every 6 months, with the first payment being made today. If the bank pays a nominal Non annual compounding interest rate of 12 percent, but uses semiannual compounding, how much will be in your account after 10 years? b. Ten years from today you must make a payment of £ 1,432.02. To prepare for this payment, you will make 5 equal deposits, beginning today and for the next 4 quarters, in a bank that pays a nominal interest rate of 12 percent, quarterly compounding. How large must each of the 5 payments be? SOLUTION a. Here, Number of deposits (n) = 5 deposits; Semiannual (every 6 months), payment = £ 100; Nominal interest rate (i) = 12%, Present value of annuity (PVA) = ? 0 1 2 10 6% £100 £100 £100 £100 £100 FV = ? We have, (1 + i)n - 1 (1 + 0.06)5 - 1 FVA = PMT × (1 + i) = £ 100 i 0.06 (1 + 0.06) = £ 100 × 5.6371 × 1.06 = £ 597.5326 Now remaining period is 15 periods (20 periods - 5 periods), so we calculate the future value of this £ 597.5326 for remaining periods. We have, FV = PV (1 + i)n = £ 597.5326 (1 + 0.06)15 = £ 1,432.02 b. Here, Future value at the end of 10 years = £ 1,432.02; n = 35 periods because quarterly compounding (in 10 years there are 40 quarters); Quarterly interest rate = 3%, PMT = ?, PV = ? 0 1 10 3% PMT = ? PMT = ? PMT = ? PMT = ? PMT = ? FV = £1,432.02 We have, FV PV = (1 + i)n = 1,432.02/(1+0.03)35 = £ 508.91 Now we calculate the payment (PMP) Here, n = 5 periods, i = 3%, PV = ?; FV = £ 508.91, FVA = £ 508.91 PMT = ? We have, (1 + i)n - 1 FVA = PMT i (1 + i) (1 + 0.03)5 - 1 or, £ 508.91 = PMT 0.03 (1 + 0.03) or, £ 508.91 = PMT × 5.3091 × 1.03 PMT = £ 508.91/ 5.4684 = £ 93.06 Question 15 The prize in last week's Lottery was estimated to be worth £ 35 million. If you were lucky enough to win, then it will pay you £ 1.75 million per year over the next 20 years. Assume Value of an annuity that the first installment is received immediately. a. If interest rates are 8 percent, what is the present value of the prize? b. If interest rates are 8 percent, what is the future value after 20 years? c. How would your answers change if the payments were received at the end of each year? SOLUTION Here, Payment (MPT) = £ 1.75 million Number of periods (n) = 20 years, a. Present value of annuity (PVA) = ? interest rate (i) = 8% 1 - 1 n 1 - 1 PVA = PMT × (1 + i) (1 + 0.08)20 i (1 + i) = £ 1.75 0.08 (1 + 0.08) = £ 1.75 × 9.8181 × 1.08 = £ 18.56 million b. Future value of annuity (FVA) = ?, Interest rate (i) = 8% (1 + i)n - 1 FVA = PMT i (1 + i) (1 + 0.08)20 - 1 = £ 1.75 0.08 (1 + 0.08) = £ 1.75 × 45.7620 × 1.08 = £ 86.49 million c. PVA and FVA assuming payments received at the end of year. Present value of annuity (PVA) = ?, Interest rate (i) = 8% We have, 1 - 1 n PVA = PMT × (1 + i) i 1 - 1 = £ 1.75 (1 + 0.08)20 0.08 = £ 1.75 × 9.8181 = £ 17.18 million Future value of annuity (FV) = ?, Interest rate (i) 8% (1 + i)n - 1 (1 + 0.08)20 - 1 FVA = PMT = £ 1.75 i 0.08 = £ 1.75 × 45.7620 = £ 80.08 million Question 16 Ashley has £ 42,180.53 in brokerage account, and plans to contribute an additional £ 5,000 Solving for time every year at an annual interest rate of 12 percent. If Ashley has to accumulate £ 250,000, how many years will it take for him to reach his goal? SOLUTION Here, Present value (PV) = £ 42,180.53 Payment (PMT) = £ 5,000 Annual return (i) = 12% Future value (FV) = £ 250,000 Number of years to reach goal (n) = ? 0 1 2 3 n 12% 42,180.53 5000 5000 5000 2,50,000 We have, (1 + 0.12)n - 1 £ 42,180.53 (1 + 0.12)n + £ 5,000 0.12 = £ 250,000 or, 5,061.66 (1.12)n + £ 5,000 (1.12)n - £ 5,000 = £ 30,000 or, 10,061.66 (1.12)n = £ 35,000 or, (1.12)n = £ 35,000/£ 10,061.66 or, (1.12)n = 3.4786 ... (i) In above eqn. (i) if we go for trying several values of 'n', the left hand side is exactly equal to right hand side at n = 11. The required no. of years to reach the goal is 11 years. Question 17 Your client is 40 years old and wants to begin saving for retirement. You advise the client to put £ 5,000 a year into the stock market. You estimate that the market's return will be, Future value of an on average, 12 percent a year. Assume the investment will be made at the end of the year. annuity a. If the client follows your advice, how much money will she have by age 65? b. How much will she have by age 70? SOLUTION Here, Your client is 40 years old, Payment (PMT) = £ 5,000, Interest rate (i) = 12% Investment will be made at the end of the year a. Future value of annuity (FVA) at the age of 65? Number of periods (n) = 65 - 40 = 25 years (1 + i)n - 1 (1 + 0.12)25 - 1 FVA = PMT = £ 5,000 i 0.12 = £ 5,000 × 133.3338 = £ 666,669 b. Future value of annuity (FVA) at the age of 70? Number of periods (n) = 70 - 40 = 30 years (1 + i)n - 1 (1 + 0.12)25 - 1 FVA = PMT = £ , 5,000 i 0.12 = £ 5,000 × 241.3327 = £ 1,206,66 Question 18 Jason has inherited £ 25,000 and wishes to purchase an annuity that will provide him with a steady income over the next 12 years. He has heard that the local savings and loan Solving for payment association is currently paying 6 percent compound interest on an annual basis. If he were to deposit his funds, what year-end equal pound amount (to the nearest pound) would he be able to withdraw annually such that he would have a zero balance after his last withdrawal 12 years from now? SOLUTION Here, Present value of annuity (PVA) = £ 25,000 Number of years (n) = 12 years Interest rate (i) = 6% Equal annual withdraw (PMT) = ? 1 - 1 n PVA = PMT × (1 + i) i 1 - 1 or, £ 25,000 = PMT (1 + 0.06)12 0.06 or, £ 25,000 = PMT × 8.3838 PMT = £ 2,981.9414 Question 19 You need to have £ 50,000 at the end of 10 years. To accumulate this sum, you have decided to save a certain amount at the end of each of the next 10 years and deposit it in Solving for payment the bank. The bank pays 8 percent interest compounded annually for long term deposits. How much will you have to save each year (to the nearest Pound)? SOLUTION (1 + i)n - 1 FVA = PMT i (1 + 0.08)10 - 1 or, £ 50,000 = PMT 0.08 or, £ 50,000 = PTM × 14.4866 PMT = 50,000/14.4866 = £ 3,451.46 Question 20 Louise wishes to borrow £ 10,000 for three years. A group of individuals agrees to lend Annual interest rate her this amount if she contracts to pay them £ 16,000 at the end of the three years. What is the implicit compound annual interest rate you receive (to the nearest whole percent)? SOLUTION Here, Present value (PV) = £ 10,000 Number of year (n) = 3 years Future value (FV) = £ 16,000 End payment, interest rate (i) = ? We have, FV = PV (1 + i)n or, £ 16,000 = £ 10,000 (1 + i)3 or, 1.6 = (1 + i)3 or, (1.6)1/3 - 1 = 1 or, i = 0.1695 or 16.95% Question 21 Calculate the present value of the following cash flow stream. Assume that the stated rate Uneven cash flow of interest is 14 percent per annum discounted semiannually. stream Cash flow 1600 1500 850 1000 SOLUTION End of year 3 1 If stated annual rate is014 percent, discounted semiannually, first we calculate the effective 2 annual rate as follows: Effective interest rate (EAR) = (1 + 0.14/2)2 - 1 = 14.49% Now present value of given cash flow stream discounted at 14.49 percent effective annual rate is calculated as follows: Year Cash flow 14.49% PVIF PV 0 £ 1,000 1.0000 £ 1,000.00 1 1,600 0.8734 1,397.44 2 1,500 0.7629 1,144.35 3 850 0.6663 566.36 Total present value £ 4,108.15 Question 22 National Lottery has offered you the choice of the following alternative payments. Alternative 1: £ 10,000 one year from now Uneven cash flow stream Alternative 2: £ 20,000 five years from now. a. Which should you choose if the discount rate is 0 percent? 20 percent? b. What rate makes the options equally attractive? SOLUTION a. Calculation of present value if discount rate is 0 percent Alternative 1: FV 10000 PV = (1+i)n = (1+0)1 = £ 10,000 Alternative 2: FV 20000 PV = (1+i)n = (1+0)5 = £ 20,000 If discount rate is 0 percent, Alternative 2 is preferable because of higher present value. Calculation of present value if discount rate is 20 percent Alternative 1: FV 10000 PV = (1+i)n = (1+0.20)1 = £ 8,333.33 Alternative 2: FV 20000 PV = (1+i)n = (1+0.20)5 = £ 8,037.55 If discount rate is 20 percent, Alternative 1 is preferable because of higher present value. b. Calculation of rate of interest (i) at which both the options are equally likely PV of Alternative 1 = PV of Alternative 2 10000 20000 (1+i)1 = (1+i)5 (1+i)4 = 2 i = (2)1/4 – 1 = 0.1892 or 18.92% That is, if the discount rate is 18.92 percent both the alternative would produce equal present value so that both are equally likely.