CHAPTER FIVE. Time Value of Money. J.D. Han

Transcription

1 CHAPTER FIVE Time Value of Money J.D. Han

2 Learning Objectives. Explain what is meant by the time value of money. 2. Define discounting and compare it to compounding. 3. Explain the difference between the nominal and the effective rate of interest. 4. Discuss how discounting and compounding affect effective yields and payment levels of term loans.

3 5. Introduction Time value of money refers to the fact that $ received today is worth more than $ received tomorrow, or vice versa Compounding and discounting form the basis for the valuation process used in finance.

4 5.2 Basic Compounding The table shows the ending wealth that an investor could have accumulated by the end of 998 had he invested $000 in 938 Cumulative Wealth ($000s) Stocks,09 2,03 0,28 27,639 5,038 93, ,068 Bonds,056,434,623 2,084 3,69 0,480 37,720 T-Bills,006,058,244,893 3,678,489 23,253 U.S. stocks,344 2,65 5,602 44,804 66,85 303,322 2,260,43

5 Simple versus Compound Interest Rate Investing $000 at 3% per year for 4 years According to Simple Interest: 4 years later The total future cash flow is F = $000x ( + 0.8x4)=$,320 According to Compound Interest: F = $,000 (+0.08)(+0.08)(.08)(.08) =$,000(.08) 4 =$,360.49

6 5.2 Compound and Discounting Variables P = current cash flow F = future cash flow PV = present value of a future cash flow(s) FV = future value of a cash flow(s) i = the stated (or nominal) interest rate per period r = the effective rate of return per period n = # of periods under consideration A = the amount of annuity

7 Compounding and Discounting Compounding : For now i = r with an annual compounding and annual payment F n = P( + r) n OR FV = PV( + r) n The equations represent the compounding relationship that is the basis for determining equivalent future and present values of cash flows

8 Discounting PV = FV ( + r) n Discounting the process of converting future values of cash flows into their present value equivalents

9 Annuities Annuity series of payments over a specific period that are of the same amount and are paid at the same interval where one discount rate is applied to all cash flows Examples of annuities: interest payments on debt and mortgages

10 Future Value of an Annuity ) Numerical Illustration Year $ ,33 $ ,20 $000->,00 $,000 Total FV = 4,64

11 2) Formula FV = A(+r) n- +A(+r) n A(+r)+ A FV = A ( + r ) r n

12 Present Value of an Annuity Year Total PV = $3,

13 2) Formula PV = A+ A/(+r)+A(+r) A(+r) n PV = A ( r + r ) n

14 *Annuity Due Annuity due - payments are made at the beginning of each period (Example: leasing arrangements) Formula: multiply the future or present value annuities factors by ( +r) PV Ø ø Œ - n ( + r ) œ = A Œ œ ( Œ r œ Œº œß + r )

15 Perpetuities Exist when an annuity is to be paid in perpetuity Present value of Perpetuity Example: Equities PV = A r

16 5.3 Varying Compound Periods More than one compounding a year: m=, 2, 4, 2, or 365 times compounding By using the annual, semiannual, quarterly, monthly and daily interest rates ( = i, i/2, i/4, i/2, i/365) More than one payment a year f=,2,4,2 Effective interest rate actual interest rate earned after adjusting the nominal interest rate for the number of compounding periods

17 Varying Compound Periods Effective annual rate formula m = # of compounding periods per year r annual = m i + m Effective period rate formula f= # of payments in a year;, 2, 4, or 2 r effective = m i f +

18 Application: Amortization of Term Loans Compounding and discounting are found in debt financing Installments consist of two portions:. Interest 2. Principle

19 Numerical Examples Savings at (the annual interest rate of) 2% compounded quarterly Effective annual interest rate r annual: r annual = (+0.03) 4 -= 0,255 or 2.55% Effective montly interest rate r monthly : (+ r monthly ) 2 =.255 r monthly = (.255) /2 - = or 0.990%

20 5.4 Amortization of Term Loans Common computational problems with term loans or mortgages include:. What effective interest rate is being charged? 2. Given the effective interest rate, what amount of regular payments have to be made over a given time period, or what is the duration over which payments have to take place given the amount? 3. Given a set of repayments over time, what portion represents interest on principle? represents repayment of principle?

21 Repayment Schedules for Term Loan and Mortgages Most loans are not repaid on an annual basis Loans can have monthly, bi-monthly or weekly repayment schedules In Canada, interest on mortgages is compounded semi-annually posing a problem in calculating the effective period interest rate

22 Numerical Example Question: What is the Canadian monthly payment of a $ 00,000 mortgage with an amortization period of 25 years, a quoted rate of 2 %? Answer: m=?; f=2 Effective annual interest rate= Effective monthly interest rate= PV=?; new n =old n times 2 Which formula to use?

23 Answer: m=2; f=2 Numerical Example Effective annual interest rate= (+0.2/2) 2 - Effective monthly interest rate= [(+0.2/2) 2 ] /2 - PV=00,000; n = 300 PV 00, = 000 A Ø Œ Œ Œ Œº = - A Ø Œ Œ Œ ( r - + r ( ) n + 0. ø œ œ œ œß ) 300 ø œ œ œ