1. MTH 105

2. THE TIME VALUE OF MONEY Which would you prefer? - GH 100 today or GH 100 in 5yrs time. 3/8/20162

3. THE TIME VALUE OF MONEY GH100 to be received in 5yrs time is worth less than the same amount today. Why? - Inflation (decreases purchasing power) - Risk (money might never be received) - Postponed consumption 3/8/20163

4. THE TIME VALUE OF MONEY - The fact that GH100 today is not the same as GH 100 to be received in 5yrs time means money has a time value. (money changes its value over time). 3/8/20164

5. THE TIME VALUE OF MONEY We shall be concerned mainly with two questions: What is the future value of an amount invested or borrowed today? What is the present value of an amount to be paid or received at a certain time in the future? 3/8/20165

6. SIMPLE INTEREST Simple Interest Simple interest refers to interest paid only on the principal. - Suppose that an amount (P) is paid into a bank account, where it is to earn interest. The future value of this investment consists of the initial deposit, called the principal and denoted by P, plus all the interest earned since the money was deposited in the account. (r: interest rate) 3/8/20166

7. SIMPLE INTEREST V(t) after 1yr: = P + rP = P(1+r) V(t) after 2yrs: = P + rP + rP = P(1+2r) V(t) after 3yrs: = P + rP + rP + rP = P(1+3r) 3/8/20167

8. SIMPLE INTEREST 3/8/20168

9. SIMPLE INTEREST 3/8/20169

10. SIMPLE INTEREST Example: A deposit of GH150 paid into a bank account is to attract simple interest at a rate of 8%. Find the future value after: - 3 years - 20 days - 2 months 3/8/201610

11. SIMPLE INTEREST Find the principal to be deposited initially in an account attracting simple interest at a rate of 8% if GH 1,000 is needed after 3 months. 3/8/201611

12. SIMPLE INTEREST If the principal P is invested at time s, rather than at time 0: V (t) = (1 + ( t−s)r)P 3/8/201612

13. SIMPLE INTEREST 3/8/201613

14. SIMPLE INTEREST Example: A sum of GH 9,000 paid into a bank account for two months to attract simple interest will produce GH 9,020 at the and of the term. Find the interest rate r and the return on this investment. 3/8/201614

15. PERIODIC COMPOUNDING Compound Interest Interest is attracted not just by the original deposit, but also by all the interest earned so far. - Supposing an amount P is deposited in a bank account, attracting interest at a constant rate r>0. the future value is calculated as follows:. 3/8/201615

16. PERIODIC COMPOUNDING 3/8/201616

17. PERIODIC COMPOUNDING Periodic Compounding: Unlike simple interest, we assume that the interest earned will be added to the principal periodically, for example: Annually 1 Semi-annually 2 Quarterly 4 Monthly 12 Daily 365 3/8/201617

18. PERIODIC COMPOUNDING 3/8/201618

19. PERIODIC COMPOUNDING 3/8/201619

20. PERIODIC COMPOUNDING Examples: 1. Find the future value after two years of a deposit of GH100 attracting interest at a rate of 10% compounded: a) Annually b) Semi- annually c) Quarterly d) Daily 3/8/201620

21. PERIODIC COMPOUNDING 2. Which will deliver a higher future value after one year, a deposit of GH1,000 attracting interest at 15% compounded daily, or at 15.5% compounded semi-annually? 3. What initial investment subject to annual compounding at 12% is needed to produce GH1,000 after two years? 3/8/201621

22. PERIODIC COMPOUNDING 3/8/201622

23. PERIODIC COMPOUNDING Example: Find the return over one year under monthly compounding with r = 10%. 3/8/201623

24. CONTINUOUS COMPOUNDING 3/8/201624

25. CONTINUOUS COMPOUNDING 3/8/201625

26. CONTINUOUS COMPOUNDING 3/8/201626

27. CONTINUOUS COMPOUNDING Example: What will be the value after one year of GH 100 deposited at 10% compounded: a) Monthly b) Continuously 3/8/201627

28. CONTINUOUS COMPOUNDING Find the present value of GH 3,500 to be received after 20 years assuming continuous compounding at 6%. 3/8/201628

29. COMPARING COMPOUNDING METHODS 3/8/201629

30. COMPARING COMPOUNDING METHODS The compounding method with a higher growth factor is said to be preferable. 3/8/201630

31. COMPARING COMPOUNDING METHODS Example: 1. Which of the two compounding methods will you prefer? Semi-annual compounding at 10% or annual compounding at 10.25%. 2. Which of the two compounding methods will you prefer? Semi-annual compounding at 10% or monthly compounding at 9%. 3/8/201631

32. COMPARING COMPOUNDING METHODS Assignment: Find the rate for continuous compounding equivalent to monthly compounding at 12%. NOTE: We can switch from one compounding method to another equivalent method. 3/8/201632

33. COMPARING COMPOUNDING METHODS 3/8/201633

34. COMPARING COMPOUNDING METHODS 3/8/201634

35. COMPARING COMPOUNDING METHODS 3/8/201635

36. COMPARING COMPOUNDING METHODS Example: 1. Which of the two compounding methods will you prefer? Semi-annual compounding at 10% or annual compounding at 10.25%. 2. Which of the two compounding methods will you prefer? Semi-annual compounding at 10% or monthly compounding at 9%. 3/8/201636

37. COMPARING COMPOUNDING METHODS TRY: Which compounding method would you prefer? Daily compounding at 15% or semi-annual compounding at 15.5%. 3/8/201637

38. SERIES OF PAYMENTS Annuity: An annuity is a sequence of finitely many payments of a fixed amount due at equal time intervals. Suppose that payments of an amount C are to be made once a year for n years. Year 0 Year 1 Year 2 Year 3… Year n C C C C 3/8/201638

39. SERIES OF PAYMENTS Assuming that annual compounding applies, how do we find the present value of such a stream of payments? We compute the present values of all payments and add them up to get. 3/8/201639

40. SERIES OF PAYMENTS 3/8/201640

41. SERIES OF PAYMENTS 3/8/201641

42. SERIES OF PAYMENTS 3/8/201642

43. SERIES OF PAYMENTS Example of Annuity: Payment of a Mortgage gives an example of an annuity. Mortgage: A mortgage is a loan given purposely to finance the purchase of a house. The house bought is used as a collateral for the loan. This loan is repaid in equal installments over a specified period of time. (Amortized) 3/8/201643

44. SERIES OF PAYMENTS Amortization: The gradual elimination of a loan by equal installment payments is called amortization over a period of time. Amortization Schedule: The repayment schedule of a loan represented in tabula form. 3/8/201644

45. SERIES OF PAYMENTS Example: - How much can you borrow if the interest rate is 18%, you can a ff ord to pay GH 10,000 at the end of each year, and you want to clear the loan in 10 years? - Suppose you borrowed GH 22,000 at 12% to be repaid over the next 6yrs. How much will you pay each year if equal installments payments are required? 3/8/201645

46. SERIES OF PAYMENTS Perpetuity: Perpetuity is a sequence of payments of a fixed amount to be made at equal time intervals and continuing indefinitely into the future. Year 0 Year 1 Year 2 Year 3..... C C C........ 3/8/201646

47. SERIES OF PAYMENTS 3/8/201647

48. SERIES OF PAYMENTS Coupon Bonds: Bonds promising a sequence of payments are called coupon bonds. These payments consist of the face value due at maturity, and coupons paid regularly, the last coupon due at maturity. The price of a coupon bond is computed by discounting all the future payments. 3/8/201648

49. SERIES OF PAYMENTS Terms: - Coupons: Interest payments - Coupon rate: Interest rate of the bond - Face value or Par value: An amount of money the bondholders receive at maturity - Maturity: the date after which bond ceases to bear interest. Bondholder receives Face value and last interest payment 3/8/201649

50. SERIES OF PAYMENTS Example: Consider a bond with face value GH 100 maturing in 5 years, with coupons of C = GH 10 paid annually. Given the annual compounding rate of 12%, find the price of the bond. 3/8/201650

51. SERIES OF PAYMENTS Find the price of a bond with face value GH100 and GH 5 annual coupons that matures in four years, given that the continuous compounding rate is 5%. 3/8/201651

52. SERIES OF PAYMENTS 3/8/201652

53. Example: Find the price of a zero coupon bond with face value GH 100 maturing in 2 years if the annual compounding rate is 12% 3/8/201653