Chapter 3 The Time Value of Money: An Introduction to Financial Mathematics

1. Chapter 3 The Time Value of Money:An Introduction to Financial Mathematics

2. Learning Objectives • Understand and solve problems involving simple interest and compound interest, including accumulating, discounting and making comparisons using the effective interest rate. • Value, as at any date, contracts involving multiple cash flows. • Distinguish between different types of annuity and calculate their present and future values.

3. Learning Objectives (cont.) • Apply knowledge of annuities to solve a range of problems, including problems involving principal-and-interest loan contracts. • Distinguish between ordinary and general annuities and make basic calculations involving general annuities.

4. Fundamental Concepts • Cash flows — fundamental to finance, the funds that flow between parties either now or in the future as a consequence of a financial contract. • Rate of return — relates cash inflows to cash outflows.

5. Interest rate — special case of rate of return (used when the financial agreement is in the form of debt). Time value of money Money received now can be invested to earn additional cash (interest). Relates to opportunity cost of giving up money or resources for a period of time — either forgone investments or consumption, whatever the next best alternative is. Fundamental Concepts (cont.)

6. Time Value of Money • An investment decision will involve an outlay of cash made in one period with the expectation of cash inflows in future periods. • As a significant amount of time may elapse between the outflow of cash and the subsequent inflows, the significance of this time should be considered. • To ignore differences in the timing of cash flows is to ignore the importance of the time value of money. • Cash flows that occur at different points in time cannot simply be added together or subtracted — this is one of the critical issues conveyed in this chapter.

7. Simple Interest • Typically used when there is only a single time period. • Interest is calculated on the original sum invested: • Where S is the lump sum payable:

8. Simple Interest: Present Value • Present cash equivalent of an amount to be paid or received at some future date, calculated using simple interest. • Formula:

9. Compound Interest • Compounding involves accumulating interest on previous interest payments. • This means that, unlike the case of simple interest, previous interest payments will generate further interest. • This earning of interest on interest is one of the key differences between simple interest and compound interest.

10. Compound Interest (cont.) • The backbone of many time-value calculations are the present value (PV) and future value (FV) based on compound interest. • The sum or future value (S ) accumulated after n periods is:

11. Compound Interest (cont.) • The future value formula can be manipulated to provide a formula to determine the present value. • The present value of a future sum is: • It is important to understand that the PV and FV formulas are the inverse of each other — one is derived from the other.

12. Nominal and Effective Interest Rates • Nominal rate • Quoted interest rate where interest is charged or calculated more frequently than the time period specified in the interest rate. • Effective rate • Interest rate where interest is charged at the same frequency as the interest rate quoted. • Used to convert different nominal rates so that they are comparable.

13. Nominal and Effective Interest Rates (cont.) • The distinction is important when interest is compounded over a period different from that expressed by the interest rate, e.g. more than once a year. • The effective interest rate can be calculated as:

14. Example 3.7: Calculate the effective annual interest rates corresponding to 12% p.a., compounding: (a) semi-annually. Solution: Using equation 3.6 Example: Effective Annual Interest Rate

15. Example: Effective Annual Interest Rate (cont.) Example 3.7 (cont.): Calculate the effective annual interest rates corresponding to 12% p.a., compounding: (b) quarterly. Solution: Using equation 3.6

16. Example: Effective Annual Interest Rate (cont.) Example 3.7 (cont.): Calculate the effective annual interest rates corresponding to 12% p.a., compounding: (c) monthly. Solution: Using equation 3.6

17. Example: Effective Annual Interest Rate (cont.) Example 3.7 (cont.): Calculate the effective annual interest rates corresponding to 12% p.a., compounding: (d) daily. Solution: Using equation 3.6

18. Real Interest Rates • The ‘real interest rate’ is the interest rate after taking out the effects of inflation. • The ‘nominal interest rate’ is the interest rate before taking out the effects of inflation. • The real interest rate (i*) can be found as follows:

19. Continuous Interest Rates • ‘Continuous interest’ is a method of calculating interest in which it is charged so frequently that the time period between each charge approaches zero. • Continuous interest is an example of exponential growth:

20. A Generalisation: Geometric Rates of Return • The rate of return between two dates, measured by the change in value divided by the earlier value. • The average of a sequence of geometric rates of return is found by a process that resembles compounding.

21. A Generalisation: Geometric Rates of Return (cont.) • ‘Average geometric rate of return’ is also referred to as the ‘average compound rate of return’.

22. Valuation of Contracts with Multiple Cash Flows • Value additivity • Cash flows occurring at different times cannot be validly added without accounting for timing. • Only cash flows occurring at the same time can be added. • Therefore, it is necessary to convert multiple cash flows into a single equivalent cash flow. • Cash flows can be carried either forward in time (accumulated) or back in time (discounted).

23. Valuation of Contracts with Multiple Cash Flows (cont.) • Where a cash flow of C dollars occurs on a date t, the value of that cash flow at a future valuation date t* is given by: • This formula takes a cash flow of $C and converts it into a future value.

24. Valuation of Contracts with Multiple Cash Flows (cont.) • Measuring the rate of return • Where there are n cash inflows Ct (t = 1, ..., n), following an initial outflow of C0 , the internal rate of return is that value of r that solves the equation:

25. Example: Internal Rate of Return • Consider three cash flows: –$1000 today, +$1120 in 1 year, +$25 in 2 years • What is the average rate of return on the initial investment of $1000, taking into account compounding, that is, the IRR? • The IRR is the r that satisfies the following equation:

26. Example: Internal Rate of Return (cont.) • The answer can be solved for precisely, as the equation is a quadratic equation. • Alternatively, and more generally, trial and error can be used, substituting different values for r. • In practice, this would be done with a computer, using a program such as Excel or Lotus.

27. Example: Internal Rate of Return (cont.) • The solution is, IRR = 14.19%. • This can be confirmed by substituting r = 0.1419. • The result is zero, confirming that the IRR = 14.19%

28. Annuities • An annuity is a stream of equal cash flows, equally spaced in time. • We consider four types of annuities: • Ordinary annuity • Annuity due • Deferred annuity • Ordinary perpetuity

29. Ordinary Annuities • Annuities in which the time period from the date of valuation to the date of the first cash flow is equal to the time period between each subsequent cash flow. • Assume that the first cash flow occurs at the end of the first time period:

30. Valuing Ordinary Annuities • Present value of an ordinary annuity: • Using the present value of annuity tables, values of A(n,i ) for different values of n and i can be found.

31. Example: Ordinary Annuities Example 3.16: • Find the present value of an ordinary annuity of $5000 p.a. for 4 years if the interest rate is 8% p.a. by: • (a) Discounting each individual cash flow.

32. Example: Ordinary Annuities (cont.) Example 3.16 (cont.): • Find the present value of an ordinary annuity of $5000 p.a. for 4 years if the interest rate is 8% p.a. by: • (b) Using equation 3.19.

33. Example: Ordinary Annuities (cont.) Example 3.16 (cont): • Find the present value of an ordinary annuity of $5000 p.a. for 4 years if the interest rate is 8% p.a. by: • (c) Using Table 4, Appendix A and equation 3.20.

34. Annuity Due • An annuity where the first cash flow is to occur immediately: • An annuity due of n cash flows is simply an ordinary annuity of (n – 1) cash flows, plus an immediate cash flow of C.

35. Annuity Due (cont.) • The present value of an annuity due:

36. Deferred Annuity • Annuity in which the first cash flow is to occur after a time period that exceeds the time period between each subsequent cash flow:

37. Deferred Annuity (cont.) • Present value of a deferred annuity:

38. Deferred Annuity (cont.) • The present value of a deferred annuity involves taking the present value of an ordinary annuity. • This figure is a present value but, as the annuity is deferred, we need to discount the PV further. • If the first cash flow is k periods into the future, we discount the PV by (k – 1) periods.

39. Ordinary Perpetuity • An ordinary annuity where the cash flows are to continue forever: • The present value of an ordinary perpetuity:

40. Valuing Ordinary Annuities • Future value of an ordinary annuity: • Using the future value of annuity tables, values of S(n,i) for different values of n and i can be found.

41. Example: Ordinary Annuities Example 3.20: • Starting with his next monthly salary, Harold intends to save $200 each month. • If the interest rate is 8.4% p.a., payable monthly, how much will Harold have saved after 2 years? • Solution: Monthly interest rate is 0.4/12 = 0.7%. Using equation 3.28, Harold’s savings will amount to:

42. Example: Ordinary Annuities (cont.) • Substituting the values we have: • Thus, at the end of 2 years, Harold will have saved $5206.99.

43. Principal-and-Interest Loans • An important application of annuities is to loans involving a sequence of equal cash flows, each of which is sufficient to cover the interest accrued since the previous payment and to reduce the current balance owing. • Such loans can be referred to as: • Principal-and-interest loans • Credit foncier loans • Amortised loans

44. Principal-and-Interest Loans (cont.) Example 3.22: • Borrow $100 000. • Make 5 years of annual repayments at a fixed interest rate of 11.5% p.a. • What is the annual repayment? • Use the PV of annuity formula:

45. Principal-and-Interest Loans (cont.) Example 3.22 (cont): • Substituting values: • Thus, annual repayments on this loan are $27 398.18.

46. Principal-and-Interest Loans (cont.) • Balance owing at a given date • Equals the present value of the then-remaining repayments • Loan term required • Solving for the required loan term n:

47. Principal-and-Interest Loans (cont.) • Changing the interest rate: • In some loans (usually called variable interest rate loans), the interest rate can be changed at any time by the lender. • Two alternative adjustments can be made: • The lender may set a new required payment which will be calculated as if the new interest rate is fixed for the remaining loan term. • The lender may allow the borrower to continue making the same repayment and, instead, alter the loan term to reflect the new interest rate.

48. General Annuities • Annuity in which the frequency of charging interest does not match the frequency of payment; thus, repayments may be made either more frequently or less frequently than interest is charged. • Link between short period interest rate (iS) and long period interest rate (iL)

49. Summary • Fundamental concepts in financial mathematics include rates of return, simple and compound interest. • Valuation of cash flows: • Present value of a future cash flow • Future value of a current payment/deposit. • Annuities are a special class of regularly spaced fixed cash flows.