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Lesson 7 Chapter 3 The time value of money Unit1 Core concepts in financial management After reading this lesson you will be able to: Demonstrate the capability to apply the concepts and techniques learnt in the previous lecture. Students I believe you have understood the concept of time value of money now I will test your understanding by doing these exercises. Lets get going EXERCISES SECTION A State whether the following statements are True (T) or false (F) i. Financial analysis requires an explicit consideration of the time value of money because many financial problems involve cash flows occurring at different points of time. ii. A regular annuity in a series of periodic cash flows of equal amounts occurring at the beginning of each period. iv. The inverse of the FVIFA factor is" equal to the PVIFA factor. v. The inverse of the PVIFA factor is called the capital recovery- factor. vi. A bank that pays 10.5 percent interests compounded annually provides a higher effective rate of interest than a bank that pays 10 percent compounded semiannually. vii. The sinking fund factor is used to determine the amount that must be deposited periodically to accumulate a specified sum at the end of a given period at a given rate of interest. viii. The nominal rate of interest is equal to the effective rate of interest when interest is compounded annually. ix. When debt is amortized in equal periodic installments, the total debt-servicing burden (consisting of interest payment and principal repayment) declines over time. x. The present value interest factor for annuity is equal to the product of the future value interest factor for annuity and the present value interest factor. xi. The present value of an uneven cash flow stream can be calculated using the PV1FA tables. xii. The rule of 72 is useful in determining the future value of an annuity for 6 years at an xiii. interest rate of 12% p.a. One of the reasons for attributing time value to money is that individuals prefer future consumption to current consumption. Solution of Section (A) 3.1 T 3.2 F 3.3 F 3.4 T 3.5 T 3.6 T 3.7 T 3.8 F 3.9 T 3.10 F 3.11 F 3.12 F Section (B) Choose the right answer from the' alternatives given. i. Money has time value because a. Money in hand today is more certain than money to be got tomorrow. b. The value of money -gets discounted as time goes by. c. The value of money gets compounded as time goes by. d. Both (a) and (b) above. e. Both (a) and (c) above. ii. Given an investment of Rs 1,000 to be invested for 9 months and interest is credited annually a. It is better to -invest in a scheme, which earns compound interest at 12%. b. It is better to invest in a scheme, which earns simple interest at 12%. c. It is better to invest in a scheme, which earns simple interest at 15%. d. It is better to invest in a scheme, which earns compound interest at 14%. e. The interest rate does not matter. iii. In order to find the value in 1995 of a sum of Rs 100 invested in 1993 at X% interest a. The FVIFA table should be used. b. The PVIFA table should be used. c. The FVIF table should be used. d. The PVIF table should be used. e. Both FVIFA and FVIF tables can be used. iv. The real rate of interest or return is nothing but a) Nominal or market interest rate b) Market interest rate to which expected rate of inflation and risk premium for uncertainty has been added c) Market interest rate, which has been adjusted for inflation d) Nominal interest rate from which expected rate of inflation and risk premium for uncertainty has been deducted. e) None of the above. v. The relationship between effective rate of interest (r) and nominal rate of interest (i) is best represented by a) i = (1 + r m ) −1 m b) r = (1 + r n ) −1 n c) r = (1 + r m ) −1 m d) Both (a) and (c) above f) None of the above. vi. Which of the following statement is /are true? a) Present value interest factor Annuity (PVIFA) is the product of future value interest factor Annuity (FVIFA) and present value interest factor (PVIF) b) PVIFA ( i,n ) = (1 + n) n − 1 i (1 + i ) n c) PVIFA ( i,n) = 1 (1 + i ) n i d) PVIFA is the inverse of FVIFA e) None of above. vii. If P= initial amount, i = interest rate, m = frequency of compounding per year, n= number of years and S = accumulation at the end of year n, which of the following expressions are correct. i a) S = P (1 + ) mn n b) P = S (1 + i mn ) m c) S = [P (1+ i )m]n m d) S= P ( (1 + i mn ) m e) None of the above. viii. The Rule of 72 a) Is used to find the doubling period b) Makes use of the PVIFA tables c) Applies the formula 72 int erest rate d) Both (b) and (c) above e) Both (a) and (c) above Solution to Section (B) 3.1 d 3.8 e 3.2 c 3.3 c 3.4 d 3.5 c 3.6 a 3.7 d Section (C) 1. If you invest Rs. 10,000 today for a period of 5 years, what will be the maturity value if the interest rate is? (a) 8% (b) 10% (c) 12% (d) 15% 2. How many years will it take for Rs. 5,000 invested today at 12% rate of interest to grow to Rs. 1,60,000? Use the rule of 72. 3. Amount invested today = Rs. 1,000; maturity value = Rs. 8,000; time period = 12 years. Use rule of 69 to calculate the implied interest rate. 4. If you invest Rs. 3,000 a year for 3 years and Rs. 5,000 a year for 7 years therefore at a rate of 12%, what will be the maturity value at the end of 10 years? 5. Sunita expects an expenditure of Rs. 2,00,000 after a period of 10 years. How much should she save annually to have the required sum after 10 years, if she invests her savings at a rate of 12%. 6. Annual payment = Rs. 1,500; maturity value = Rs. 12,500, period = 5 years. Find out the implied interest rate. 7. You invest Rs. 3,000 today and get Rs. 10,000 after 6 years. What is the implied interest rate? 8. What will be the present value of Rs. 12,000 receivable after 10 years if the rate of discount is (i) 10% (ii) 12% (iii) 15% 9. What is the present value of an 5 year annuity of Rs. 3,000 at 12% 10. Mr. Srinivas is going to retire after 6 months. He has a choice between (a) an annual pension of Rs. 8,000 as long as he lives, and (b) A lump sum amount of Rs. Amount of Rs. 50,000. If he expects to live for 20 years and the interest rate is 10% which option would you suggest him to go for? 11. Sunil has deposited Rs. 2,00,000 in a bank, which pays interest @8%. How much can he withdraw every year for a period of 25 years, so that there is no balance left at the end? 12. You invest Rs. 1,500 at the end of year one, Rs. 2,000 at the end of the second year, and Rs. 5,000 each year form the third year to the tenth. Calculate the present value of the stream if the discount rate is 10%. 13. You receive Rs. 1,000 a year for the first 8 years, and Rs. 4,000 a year forever therefore. Calculate the present value if the discount rate is 12%. 14. Suman is due to retire 20 years form now. She wants to invest a lump sum now so as to be able to withdraw Rs. 10,000 every year, beginning from the end of the 20th year. How much should she invest now if the deposit earns a return of 12%? 15. A company is offering to pay Rs. 10,000 annually for a period of 10 years if you deposit Rs. 50,000 now. What is the implied interest rate? 16.Using a discount rate of 10%, calculate the present value of the following cash flow streams. End of year Stream A Stream B Stream C 1 1,000 10,000 5,000 2 2,000 9,000 5,000 3 3,000 8,000 5,000 4 4,000 7,000 5,000 5 5,000 6,000 5,000 6 6,000 5,000 5,000 7 7,000 4,000 5,000 8 8,000 3,000 5,000 9 9,000 2,000 5,000 10 10,000 1,000 5,000 17. If you invest Rs.5,000 with a bank which pays 11% interest with quarterly compounding, what amount will it grow to in 7 years? 18. Initial deposit = Rs.10,000 ; Time period = 5 years Interest rate = 10% compounded half-yearly. Maturity value = ? 19. Find out the effective rate of interest in the following cases: Case A: Nominal rate of interest = 8% compounded quarterly. Case B: Nominal rate of interest = 11 % compounded half-yearly. Case C: Nominal rate of interest = 12% compounded 6 times a year. Case D: Nominal rate of interest = 12% compounded 12 times a year. 20. How much do you need to invest now at an interest rate of 10% to have a perpetual annual income of Rs.20,000 from the beginning of the 15th year? 21. If you are given a choice between Rs.4,000 now and Rs.15,000 after a period of 10 years, which one will you choose? What does your choice imply? 22. Initial investment = Rs.10,000; Time period = 7 years Interest rate = 12% compounded half-yearly Maturity value = ? With an inflation rate of 6%, what will be the value of the investment after 7 years in terms of the current rupee? 23. In order to accumulate Rs.25,000 at end of 10 years, how much should you invest at the beginning of each year? The rate of interest is 10%. 24. You require Rs.10,000 at the beginning of each year from the 10th to 14th year. How much should you invest at the end of each year from the 1st to 5th year at an interest rate of 10%? 25. Calculate the present value of an annuity of Rs.5,000 receivable for 35 years, if the first receipt occurs after 15 years. Take the -discount rate as 12%. 26. At the end of 6 years, Mr. X will start receiving a pension of Rs.l,000 per month and will keep receiving it for the next 20 years. How much can he borrow now @10%, so that both the principal and the interest can be paid off with 25% of the pension amount? Assume that the interest payable- will be accumulated till the first pension is received. 27. Akshay takes a bank loan of Rs.10,000 to purchase a scooter. He has to pay an installment of Rs.500 per month for the next two years to the bank. What is the implied interest rate? 28. X Co. needs to redeem Rs.50,00,000 of debentures each at the end of 8,9 and 10 years. What amount should be deposited in a sinking fund account annually for 4 years to meet the requirements? Interest rate is 10%. 3.29. You deposit a sum of Rs.10,000 with a bank at an interest rate of 12%. If you want to withdraw Rs. l ,500 every year, for how many years can you do so? . Solution to Section ( C ) 1. Compound values (S) of the deposit of Rs. 10,000 for 5 years at the given rates of interest will be as follows: i = 8% S = Rs. 10,000 x FVIF(8.5) = Rs. 10,000 x 1.469 = Rs. 14,693 i = 10% S = Rs. 10,000 x FVIF (10.5) = Rs. 10,000 x 1.6105 = Rs. 16,105 i = 12% S = Rs. 10,000 x FVIF(10.5) = Rs. 10,000 x 1.7623 = Rs. 17,623 i = 15% S = Rs. 10,000 x FVIF (15.5) = Rs. 10,000 x 2.0113 = Rs. 20,113. 2. Deposit of Rs. 5,000 at 12% rate of interest will grow to Rs. 5,000 x 2 = Rs. 10,000 in 6 years using rule of 72 (i.e. 72/12). Extending the same principle the deposit will grow to a sum of Rs. 5,000 X 25 = Rs. 1,60,000 in 6 x 5 = 30 years. 3. In a period of 12 years a deposit of Rs. 1,000 will grow to Rs. 8,000 i.e. 23 times of the initial deposit. Therefore the initial deposit double in 12/3 = 4 years. Using rule of 69, the approximate rate of interest at which the amount will double will be 0.35 + 69 =4 int erest rate Interest rate = 18.9% 4. Savings at the end of 10th year will be S10 = Rs. [3,000 X FVIFA (12,.3) x FVIF (12,7) + 5,000 X FVIFA(12.7)] = Rs. [3,000 X 3.374 X 2.211 + 5,000 x 10.089] = Rs. 72,824 Alternatively, the problem can be solved as follows: S10 = Rs. [ 3,000 x FVIFA (12,10)+ 2,000 x FVIFA (12,7)] = Rs. 72,824 5. Let the annual savings required = S Then S x FVIFA (12,10) = Rs. 2,00,000 S = 2,00,000 = Rs.11,397 17.548 6. Given Rs. 1,500 x FVIFA(K,5) = 12,500 FVIF (k,5) = 8.33 From tables, FVIFA(24,5) = 8.048 FVIF (28,5) = 8.699 Applying linear approximation in this interval, we get = 24% + (28% - 24%) x 8.33 − 8.048 8.699 − 8.048 Implied interest rate = 25.73% 7. The equation will be Rs. 3,000 x FVIF (k,6) = 10,000 = FVIF(k,6) = 3,333 From tables, we get FVIF(20,6) FVIF(24.6) = 2.986 = 3.635 Applying a linear approximation in this interval, we get = 20% + (24 – 20)% x 3.333 − 2,986 3,635 − 2,986 = 22.22% Therefore, the implied interest rate is 22.22% 8. If P is present value of Rs. 12,000 receivable after 10 years, at the following discount rates. P will be a. k =10% p = Rs. 12,000 x PVIF (10,10) =Rs. 12,000 x 0.386 = Rs. 4,6325 b. k = 12% = Rs. 12,000 x PVIF(12,10) =Rs. 12,000 x 0.322 = Rs. 3,864 c. k =15% p = Rs. 12,000 x PVIF(15,10) =Rs. 2,964 9. Present value of the 5-year annuity of Rs. 3,000 at a discount rate of 12% will be = Rs. 3,000 X PVIF(12,5) = Rs. 3,000 X 3.605 =Rs. 10,815 10. Present value of annual pension = Rs. 8,000 X PVIFA (10,20) = Rs. 8,000 x 8.514 =Rs. 68,112 The second choice is Rs. 50,000 lump sum now Hence he should opt for the first choice. 11. If x be the annual amount, Sunil will withdraw x. PVIFA(8.25) = Rs. 2,00,000 x = = Rs.2,00,000 PVIFA(8, 25) Rs.2,00,000 = Rs.18,737 10.674 12. The present value of this income stream can be obtained as follows: Rs. [ 1,500 X PVIF(10,1) + Rs. 2,000 X PVIF(10,2) + 5,000 X PVIF(10,8) x PVIF (10,2)] Rs. [ 1363.6 + 1652.9 + 13,773.5] Rs. 16,790 13. The present value of the given income stream =Rs. 1,000 x PVIFA(12,8) + Rs.4,000 PVIF(12,8) 0.12 =Rs. (4,968 + 13,467) =Rs. 18,435 14. Discounted value of the annual income of Rs. 10,000, beginning from the end of the 20th year, evaluated at the end of 19th year = Rs.10,000 = Rs.83.333 0.12 If X be the amount to be deposited today to get a sum of Rs. 83,333 at the end of 19th year, then, x FVIF(12,19) x =Rs. 83,333 = 83,333 = Rs.9,675 8.613 Thus Suman has to invest Rs. 9,675 to withdraw Rs. 10,000 every year after 20 years. 15. Given Rs. 10,000 X PVIFA(k,10) = 50,000 PVIFA(K,10) =5 From tables, we get PVIFA(15,10) = 5.019 PVIFA(18,10) = 4.494 Applying a linear approximation in this interval, the value of ‘k’ can be determined as follows: K = 15% + (18 – 15)% x 5.019 − 5 5.019 − 4.494 =15.1% 16. The present values of cash flow streams at a discount rate of 10% are as follows. PV of (Stream A) =Rs. 1000 PVIF(10,1) + 2000 PVIF(10,2) + 3000 PVIF(10.3) + … + 10,000 PVIF(10,10) =Rs. 1,000 ( 1 x 0.909+ 2 x 0.826 + 3 x 0.751 + 4 x 0.683 + 5x 0.621 + 6 x 0.564 + 7 x 0.513 + 8 x 0.467 + 9 + 0.424 + 10 x 0.386) = Rs. 1000(0.909 + 1.652 + 2.253 + 2.753 + 2.732 + 3.105 + 3.384 + 3.591 + 3.736 + 3.816 + 3.86 = 100 x 26.306 = Rs.26,306. PV of (Stream B) = Rs. 1000 x 0.909 + 9000 x 0.826 + 8000 x 0.751 +… + 1000 x 0.386) = Rs. 1000 (10 x 0.909 + 9 x 0.826 + 8 x 0.751 + … + 1 x 0.386) = 1000 x 38.546 = Rs. 38,546. PV of (Stream C) = Rs. 5000 x PVIFA (10,10) = Rs. 5000 x 6.145 = Rs. 30,725. 17. D7 7 x4 0.11 = Rs. 5000 x1 + 4 = Rs. 5000 x (1.027528 = Rs. 5000 x 2.137 =Rs. 10,685 18. D5 5x2 0.1 = 10,000 x1 + 2 = Rs. 16,289 19. Case A: K = 8% Compounded quarterly 4 0.08 Effective rate r = 1 + −1 4 = 0.0824 = 8.24% Case B: K = 11%, Compounded half- yearly 2 0.11 r = 1 + −1 2 = 0.113 = 11.3% Case C: k = 12% Compounded 6 times a year 6 0.12 r = 1 + −1 6 = 0.126 = 12.6% Case D: K = 12%, Compounded 12 times a year 12 0.12 = 1 + −1 12 = 0.1268 = 12.68% 20. Investment needed at the beginning of 14 the year to generate an income of Rs 20,000 from the beginning of 15th year = Rs. 20,000 = Rs.2,00,000. 0.1 If A be the amount to be deposited now Then A x FVIF (10,13) = Rs. 2,00,000 A = Rs. = Rs. 2,00,000 FVIF(10,13) 2,00,000 = Rs.57,937 3.452 21. The interest rate I earn if I opt for Rs. 15,000 after 10 years, vis-à-vis Rs. 4,000 now, will be Rs. 4,000 x FVIF (k,10) = Rs. 15,000 FVIF(k,10) = 3.75 From the tables, it is found that this value is slightly higher than @ 14%. Thus, if I opt Rs. 4,000 now, it implies that I can productively invest it in such a way that I earn an interest rate of more than 14%. 22. Maturity value =Rs. 10,000 x FVIF(6%,14) = Rs. 10,000 x 2.261 = Rs. 22,610 If the inflation rate were 6% per year, the real interest rate would be 6% per annum. [(12 – 6)%] Thus the value of the deposit in terms of current rupee 0.06 = Rs. 10,000 1 + 2 = Rs. 10,000 x 1.5125 = Rs. 15,125 7 x2 23. If y is the annuity due and y10 indicates the future value of the annuity at the end of 10 years, then y (1 + k) FVIFA(k,n) = y10 it is given that y10 = Rs. 25,000 k = 10% n = 10 years we get Y 25,000 = Rs. (1 + 0.1) FVIFA(10,10 ) 25,000 (1.1) (15.94) = Rs. =Rs. 25,000 Rs.1,425.80 17.534 24. The discounted value of annuity due of Rs. 10,000 receivable at the beginning of each year form the 10th to 14th year, calculated at the beginning of 10th year is = Rs. 10,000 (1 + k) PVIFA(k,5) = 10,000 (1 + 0.1) PVIFA(10,5) = 10,000 x 1.1 x 3.790 = Rs. 41,690…(1) Discounted value of the amount in equation (1), calculated at the end of 5th year = 41,690 x PVIF(10,4) = Rs. 41,690 x 0.683 = Rs. 28,474. If y be amount to be deposited at the end of each year form 1st to 5th year, then =y x FVIFA(10,5) = Rs. 28,474 y = Rs. 28,474 = Rs.4,664. 6.1051 25. The discounted value of the annuity receivable for 35 years calculated as at the end of 14th year = Rs. 5000 PVIFA(10,5) = Rs. 5000 x 8.176 = Rs. 40,880. PV of this amount = 40,880 x PVIF(12,14) = 40,880 x 0.205 = Rs. 8,380.4 26. The discounted value of the pension receivable for 20 years at the Rs. 1,000 per month at the end of 6th year =Rs. 1,000 PVIFA(0.833%., 240 months) = Rs. 1,000 x 103.65 = Rs. 1,03,650. If A be the amount Mr. X. can borrow now so that he can repay the principal plus interest @ 10% with 25% of the pension amount, then A x FVIF(10,6) = Rs. (0.25 x 1,03,650) = Rs. 25,912.5 A = Rs. 25,912.5 1.772 = Rs. 14,623. 27. Given Rs. 500 x PVIFA(k,24months) =10,000 =20 PVIFA(k,24months) From tables, we get PVIFA(1%,24) = 21.244 PVIFA(2%,24) = 18.914 Using a linear approximation, in this interval, we calculate the value of ‘k’ as follows: K =1% + (2 – 1)% x (21.244 − 20) (21.244 − 18.914) = 1.53% (per month) Thus the implicit interest rate is (1.0153)12 - 1 = 0.1998 i.e. 19.98% 28. Discounted value of the debentures to be retired between 8 to 10 years evaluated at the end of 4th year =Rs. 50 lakhs [PVIF (10,4)+ PVIF (10,5) + PVIF (10,6)] = Rs. 50 lakhs [0.683 + 0.621 + 0.564] = Rs. 93.4 lakhs. If y is the annual deposit to be made to the sinking fund between years 1 and 4, then Y x FVIFA (10,4) = Rs. 93.4 lakhs Y = Rs. 93.4 = Rs.20.125 lakhs 4.641 29. Let ‘n’ be the number of years for which a sum of Rs. 1,500 can be withdrawn annually, then Rs. 1,500 x PVIFA(12,n) =10,000 PVIFA (12,n) = 6.667 From the tables, we find PVIFA (12,10) = 6.814 PVIFA (12,10) = 6.492 By a linear approximation, we calculate n = 10 + (11 – 10) x =10.5 years. 6.814 − 6.667 6.814 − 6.492