Chapter 3. Understanding The Time Value of Money. Prentice-Hall, Inc. 1

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1 Chapter 3 Understanding The Time Value of Money Prentice-Hall, Inc. 1

2 Time Value of Money A dollar received today is worth more than a dollar received in the future. The sooner your money can earn interest, the faster the interest can earn interest. Prentice-Hall, Inc. 2

3 Interest and Compound Interest Interest -- is the return you receive for investing your money. Compound interest -- is the interest that your investment earns on the interest that your investment previously earned. Prentice-Hall, Inc. 3

4 Future Value Equation FV n = PV(1 + i) n FV = the future value of the investment at the end of n year i = the annual interest (or discount) rate PV = the present value, in today s dollars, of a sum of money This equation is used to determine the value of an investment at some point in the future. Prentice-Hall, Inc. 4

5 Compounding Period Definition -- is the frequency that interest is applied to the investment Examples -- daily, monthly, or annually Prentice-Hall, Inc. 5

6 Reinvesting -- How to Earn Interest on Interest Future-value interest factor (FVIF i,n ) is a value used as a multiplier to calculate an amount s future value, and substitutes for the (1 + i) n part of the equation. Prentice-Hall, Inc. 6

7 The Future Value of a Wedding In 1998 the average wedding cost $19,104. Assuming 4% inflation, what will it cost in 2028? FV n = PV (FVIF i, n ) FV n = PV (1 + i) n FV 30 = PV ( ) 30 FV 30 = $19,104 (3.243) FV 30 = $61, Prentice-Hall, Inc. 7

8 The Rule of 72 Estimates how many years an investment will take to double in value Number of years to double = 72 / annual compound growth rate Example / 8 = 9 therefore, it will take 9 years for an investment to double in value if it earns 8% annually Prentice-Hall, Inc. 8

9 Compound Interest With Nonannual Periods The length of the compounding period and the effective annual interest rate are inversely related; therefore, the shorter the compounding period, the quicker the investment grows. Prentice-Hall, Inc. 9

10 Compound Interest With Nonannual Periods (cont d) Effective annual interest rate = amount of annual interest earned amount of money invested Examples -- daily, weekly, monthly, and semi-annually Prentice-Hall, Inc. 10

11 The Time Value of a Financial Calculator The TI BAII Plus financial calculator keys N = stores the total number of payments I/Y = stores the interest or discount rate PV = stores the present value FV = stores the future value PMT = stores the dollar amount of each annuity payment CPT = is the compute key Prentice-Hall, Inc. 11

12 The Time Value of a Financial Calculator (cont d) Step 1 -- input the values of the known variables. Step 2 -- calculate the value of the remaining unknown variable. Note: be sure to set your calculator to end of year and one payment per year modes unless otherwise directed. Prentice-Hall, Inc. 12

13 Tables Versus Calculator REMEMBER -- The tables have a discrepancy due to rounding error; therefore, the calculator is more accurate. Prentice-Hall, Inc. 13

14 Compounding and the Power of Time In the long run, money saved now is much more valuable than money saved later. Don t ignore the bottom line, but also consider the average annual return. Prentice-Hall, Inc. 14

15 The Power of Time in Compounding Over 35 Years $200,000 $150,000 $100,000 $50,000 $0 $198,422 Selma $146,212 Patty Selma contributed $2,000 per year in years 1 10, or 10 years. Patty contributed $2,000 per year in years 11 35, or 25 years. Both earned 8% average annual return. Prentice-Hall, Inc. 15

16 The Importance of the Interest Rate in Compounding From the compound growth rate of stocks was approximately 11.2%, whereas long-term corporate bonds only returned 5.8%. The Daily Double -- states that you are earning a 100% return compounded on a daily basis. Prentice-Hall, Inc. 16

17 Present Value Is also know as the discount rate, or the interest rate used to bring future dollars back to the present. Present-value interest factor (PVIF i,n ) is a value used to calculate the present value of a given amount. Prentice-Hall, Inc. 17

18 Present Value Equation PV = FV n (PVIF i,n ) PV = the present value, in today s dollars, of a sum of money FV n = the future value of the investment at the end of n years PVIF i,n = the present value interest factor This equation is used to determine today s value of some future sum of money. Prentice-Hall, Inc. 18

19 Calculating Present Value for the Prodigal Son If promised $500,000 in 40 years, assuming 6% interest, what is the value today? PV = FV n (PVIF i, n ) PV = $500,000 (PVIF 6%, 40 yr ) PV = $500,000 (.097) PV = $48,500 Prentice-Hall, Inc. 19

20 Annuities Definition -- a series of equal dollar payments coming at the end of a certain time period for a specified number of time periods. Examples -- life insurance benefits, lottery payments, retirement payments. Prentice-Hall, Inc. 20

21 Compound Annuities Definition -- depositing an equal sum of money at the end of each time period for a certain number of periods and allowing the money to grow Example -- saving $50 a month to buy a new stereo two years in the future By allowing the money to gain interest and compound interest, the first $50, at the end of two years is worth $50 ( ) 2 = $58.32 Prentice-Hall, Inc. 21

22 Future Value of an Annuity Equation FV n = PMT (FVIFA i,n ) FV n = the future value, in today s dollars, of a sum of money PMT = the payment made at the end of each time period FVIFA i,n = the future-value interest factor for an annuity Prentice-Hall, Inc. 22

23 Future Value of an Annuity Equation (cont d) This equation is used to determine the future value of a stream of payments invested in the present, such as the value of your 401(k) contributions. Prentice-Hall, Inc. 23

24 Calculating the Future Value of an Annuity: An IRA Assuming $2000 annual contributions with 9% return, how much will an IRA be worth in 30 years? FV n = PMT (FVIFA i, n ) FV 30 = $2000 (FVIFA 9%,30 yr ) FV 30 = $2000 ( ) FV 30 = $272,610 Prentice-Hall, Inc. 24

25 Present Value of an Annuity Equation PV n = PMT (PVIFA i,n ) PV n = the present value, in today s dollars, of a sum of money PMT = the payment to be made at the end of each time period PVIFA i,n = the present-value interest factor for an annuity Prentice-Hall, Inc. 25

26 Present Value of an Annuity Equation (cont d) This equation is used to determine the present value of a future stream of payments, such as your pension fund or insurance benefits. Prentice-Hall, Inc. 26

27 Calculating Present Value of an Annuity: Now or Wait? What is the present value of the 25 annual payments of $50,000 offered to the soonto-be ex-wife, assuming a 5% discount rate? PV = PMT (PVIFA i,n ) PV = $50,000 (PVIFA 5%, 25 ) PV = $50,000 (14.094) PV = $704,700 Prentice-Hall, Inc. 27

28 Amortized Loans Definition -- loans that are repaid in equal periodic installments With an amortized loan the interest payment declines as your outstanding principal declines; therefore, with each payment you will be paying an increasing amount towards the principal of the loan. Examples -- car loans or home mortgages Prentice-Hall, Inc. 28

29 Buying a Car With Four Easy Annual Installments What are the annual payments to repay $6,000 at 15% interest? PV = PMT(PVIFA i%,n yr ) $6,000 = PMT (PVIFA 15%, 4 yr ) $6,000 = PMT (2.855) $2, = PMT Prentice-Hall, Inc. 29

30 Perpetuities Definition an annuity that lasts forever PV = PP / i PV = the present value of the perpetuity PP = the annual dollar amount provided by the perpetuity i = the annual interest (or discount) rate Prentice-Hall, Inc. 30

31 Summary Future value the value, in the future, of a current investment Rule of 72 estimates how long your investment will take to double at a given rate of return Present value today s value of an investment received in the future Prentice-Hall, Inc. 31

32 Summary (cont d) Annuity a periodic series of equal payments for a specific length of time Future value of an annuity the value, in the future, of a current stream of investments Present value of an annuity today s value of a stream of investments received in the future Prentice-Hall, Inc. 32

33 Summary (cont d) Amortized loans loans paid in equal periodic installments for a specific length of time Perpetuities annuities that continue forever Prentice-Hall, Inc. 33