1. 6-1 CHAPTER 3 Time Value of Money Future value Present value Annuities Amortization

2. 6-2 Time Value of Money Definition: Value of money changes as time changes. This is because of the positive rate of interest in all the markets. If the market interest rate is 10%, then Tk.100 today has the same value as Tk.110 after 1 year from now and Tk.121 after 2 years from now. So the present value of Tk.110 of the next year is Tk.100, or the future value of Tk.100 now is Tk.110 in the next year. FV n =PV(1+i) n PV=FV n /(1+i) n

3. 6-3 Solving for PV: The arithmetic method Problem 1: How much should you set aside now to get Tk.100 after 3 years from now? Solve the general FV equation for PV: PV = FV n / ( 1 + i ) n PV = FV 3 / ( 1 + i ) 3 = Tk.100 / ( 1.10 ) 3 = Tk.75.13

4. 6-4 Finding the interest rate and time period Problem 2. What is the rate of interest by what Tk.100 becomes Tk.200 in 4 years? 200=100(1+i) 4 (1+i) 4 =2, 1+i=2 1/4 =2.25 =1.1892, i=18.92% Problem 3. How long time it takes to double an amount if the interest rate is 15% per annum? 200=100(1+.15) n (1.15) n =2, n log(1.15)=log(2) n=log(2)/log(1.15)=4.96 years

5. 6-5 Compounding more than once in a year For round year case: Step 1: “i” (interest rate) should be divided by “m” (how many times to be compounded in a year) Step 2: “n” (number of years) should be multiplied by “m” (how many times to be compounded in a year)

6. 6-6 Compounding more than once in a year (Contd.) For broken year case: Step 1: “i” (interest rate) should be divided by “m” (how many times to be compounded in a year) Step 2: Look at the interest rate in step 1. Now the power would be determined on the basis of how many times such interest rate gets compounded throughout the whole life.

7. 6-7 Compounding more than once in year (Example) Problem 4: You like to set aside an amount of money so that you get Tk.50,000 after 5 years from now. Bank One offers you 10% annual interest rate and Bank Two offers you 9.5% interest rate compounded monthly. Where should you put the money? Bank One: PV=50,000/(1.1) 5 =Tk.31046.07 Bank Two: PV=50,000/(1+.095/12)) 60 =Tk.31152.46 Bank One is a better choice

8. 6-8 Classifications of interest rates EFF% for 10% semiannual investment EFF%= ( 1 + i NOM / m ) m - 1 = ( 1 + 0.10 / 2 ) 2 – 1 = 10.25% An investor would be indifferent between an investment offering a 10.25% annual return and one offering a 10% annual return, compounded semiannually.

9. 6-9 Effective Annual Rate EFF%= ( 1 + i NOM / m ) m - 1 Problem 5: A Credit card charges 2% interest rate per month. What is the effective interest rate? EAR=(1+.24/12) 12 -1 =(1.02) 12 -1 =26.82%

10. 6-10 Why is it important to consider effective rates of return? An investment with monthly payments is different from one with quarterly payments. Must put each return on an EFF% basis to compare rates of return. Must use EFF% for comparisons. See following values of EFF% rates at various compounding levels. EAR ANNUAL 10.00% EAR QUARTERLY 10.38% EAR MONTHLY 10.47% EAR DAILY (365) 10.52%

11. 6-11 Annuity Definition: A series of equal payments is made against what an accumulated sum can be received either at the beginning or at the end of the period of annuity. If the accumulated sum takes place at the beginning then it is a Present Value Annuity, and if the accumulated sum takes place at the end then it is a Future Value Annuity.

12. 6-12 Annuity 100 0123 i% 3 year $100 ordinary annuity. PV?

13. 6-13 Present Value Annuity All kinds of consumers’ credit schemes follow present value annuity. A lump sum amount is borrowed now against what payments would be made in equal installments at a regular interval for a definite period of time. For example, at 10% interest rate, you can borrow Tk.173.55 in a 2 year annuity of Tk.100 installment. The amount of Tk.173.55 is composed of (the PV of FV 1 of Tk.100 or) Tk.90.91 and (FV 2 of Tk.100) or Tk.82.64.

14. 6-14 PVIFA= 1- 1 (1+i) n i Formulae for Present Value Interest Factor of Annuity (PVIFA)

15. 6-15 Present Value Annuity Problem 6: At 10% interest rate, How much can you borrow now against the repayment 3 equal annual installments of Tk.1000? PV Annuity=C*(PVIFA) =C{[1-(1/(1+i) n )]/i} =1000{[1-(1/(1.1) 3 ]/.1} =1000*2.4869 =2486.90

16. 6-16 Present Value Annuity Problem 7: You have a plan to deposit Tk.1,000 per month in a bank for next 20 years. If the interest rate is 8.5% per annum then how much can you borrow from the bank against that?

17. 6-17 Solution of Problem 5 PVIFA={1-1/(1+.085/12) 12*20 ]}/(.085/12) =115.2308 PV Annuity=C*PVIFA =1000*115.2308=1,15,230.80

18. 6-18 Present Value Annuity Problem 8: Find the amount of installment of a loan of Tk.5,000 to be repaid in 4 equal monthly installment at 12% interest. Make an amortization schedule. 5000=C(PVIFA, i=.12, m=12, n=4) =C(3.901966) C=5000/3.901966=1281.405

19. 6-19 Amortization Schedule

20. 6-20 Present Value Annuity Problem 9: You need Tk.12 lakh now to buy a car, under the terms and condition of monthly installments for 10 year. Interest rate is 15% per annum. (a) What would be the amount of installments? (b) How much would be the accumulated liability of interest?

21. 6-21 Solution: Problem 9 (a) Installment =PV Annuity/PVIFA =12,00,000/61.98285=Tk.19,360.19 (b) Accumulated Interest=Total payments – Present value of annuity =(19,360.19*120)-12,00,000 =23,23,223-12,00,000=11,23,223

22. 6-22 Problem 9a In 1992, a 60 year old nurse bought a $12 dollar lottery ticket and won the biggest jackpot to that date of $9.3 million. Later it turned up that she would be paid in 20 annual installments of $465,000 each. If the interest rate was 8%, then what was the amount she was deprived of in present value?

23. 6-23 Answer to problem 9a PV = $465,000*PVIFA i=.08, n=20 = $ 465,000 * $ 9.818147 = $4,565,417 So, she was paid less than $9.3 million by an amount of $4,734,583. 4734583

24. 6-24 Future Value Annuity Definition: FV Annuity is different from PV Annuity in that the accumulated sum takes place at the end of the period of the annuity. In a savings scheme if you deposit equal installment regularly and at the maturity of the annuity receive the accumulated sum then it is an example of future value annuity. It is composed of the principal amounts and the interest thereof. FVIFA=[(1+i) n -1]/i FV of Annuity=C*FVIFA

25. 6-25 Composition of Future Value of Annuity Suppose, there is a 2 year annuity of $100 installments at 10% interest. The future value is FV Annuity= C*FVIFA= =100*[(1.1) 2 -1]/0.1=$210 This is composed of $110 and $100.

26. 6-26 Future Value Annuity (Contd.) Problem 10: You like to deposit Tk.1000 per month for a period of 15 years. Assuming an interest of 10% how much would you get at the end? FV Annuity=C*(FVIFA) =1000*{[(1+.1/12) 15*12 ]-1}/(.1/12) =1000*414.4703 =Tk.4,14,470.30

27. 6-27 Future Value Annuity (Contd.) Problem 11: You need to have Tk.1 million after 20 years from now. Assuming the market interest rate of 13% per annum if you like to deposit equal quarterly installments during the period in a bank then how much would be the amount of each installment? What is the interest accumulation in the annuity? Given, FV=Tk.1,000,000, i=.13/4, n=20*4, C=?

28. 6-28 Solution: Problem 11 C=FV/FVIFA. C=1,000,000/366.7164=Tk.2,726.90 Interest accumulation=FV Annuity-Total payments =1,000,000-(C*n)=1,000,000- (2726.90*80) =Tk.781,847.80 (This is 78.18% of face value)

29. 6-29 Ordinary Annuity versus Annuity Due The installments of an annuity can be paid either at the beginning or at the end of the period. If it is paid at the end of the period then it is called ordinary annuity. If it is paid at the beginning of the period then it is called annuity due. Both present value annuity and future value annuity can be an ordinary annuity or annuity due. To convert ordinary annuity into annuity due multiply the value by (1+i).

30. 6-30 What is the difference between an ordinary annuity and an annuity due? Ordinary Annuity PMT 0123 i% PMT 0123 i% PMT Annuity Due

31. 6-31 Annuity Due Problem 12: You need to receive Tk.10,000 monthly for a period of 2 years to pursue your MBA program. You make an arrangement with a Bank that says the interest rate is 15%. (a) How much will you have to return back to the bank at the end? (b) How much should you deposit to the bank now to get the same monthly installments throughout the MBA program?

32. 6-32 Solution: Problem 12(a) (a) FV Annuity=C*FVIFA =10000*[(1+.15/12) 24 -1]/(.15/12) =10000*27.78808=Tk.2,77,880.80 Since you need the money at the beginning of the month so it is an annuity due. In that case, FV Annuity Due=2,77,880.80*(1+.15/12)=Tk.2,81,354.40

33. 6-33 Solution: Problem 12(b) (b) This is the present value annuity due. PV Annuity due=C*PVIFA*(1+i) =10,000*20.62423*(1+.15/12) =2,08,820.4 Also notice: you can get answer to (b) by dividing answer to (a) by (1+i) n or [(1+.15/12) 2*12 ] Or, you can get (a) through multiplying (b) by (1+i) n factor For example, 208820.4[(1+.15/12) 2*12 ] =208820.4*[(1.0125) 24 ]=281354.40