Chapter 3 Understanding The Time Value of Money

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

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

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

Future Value Equation u FVn – – – = 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 u This equation is used to determine the value of an investment at some point in the future. Prentice-Hall, Inc. 4

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

Reinvesting -- How to Earn Interest on Interest u Future-value interest factor (FVIFi, 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

The Future Value of a Wedding In 1998 the average wedding cost $19, 104. Assuming 4% inflation, what will it cost in 2028? FVn = PV (FVIFi, n) FVn = PV (1 + i)n FV 30 = PV (1 + 0. 04)30 FV 30 = $19, 104 (3. 243) FV 30 = $61, 954. 27 Prentice-Hall, Inc. 7

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

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

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

The Time Value of a Financial Calculator u The – – – Prentice-Hall, Inc. 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 11

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

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

Compounding and the Power of Time u In the long run, money saved now is much more valuable than money saved later. u Don’t ignore the bottom line, but also consider the average annual return. Prentice-Hall, Inc. 14

The Power of Time in Compounding Over 35 Years u u u Prentice-Hall, Inc. 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. 15

The Importance of the Interest Rate in Compounding u From 1926 -1998 the compound growth rate of stocks was approximately 11. 2%, whereas long-term corporate bonds only returned 5. 8%. u The “Daily Double” -- states that you are earning a 100% return compounded on a daily basis. Prentice-Hall, Inc. 16

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

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

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

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

Compound Annuities u 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 u Example -- saving $50 a month to buy a new stereo two years in the future – Prentice-Hall, Inc. By allowing the money to gain interest and compound interest, the first $50, at the end of two years is worth $50 (1 + 0. 08)2 = $58. 32 21

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

Future Value of an Annuity Equation (cont’d) u 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

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? FVn FV 30 Prentice-Hall, Inc. = PMT (FVIFA i, n) = $2000 (FVIFA 9%, 30 yr) = $2000 (136. 305) = $272, 610 24

Present Value of an Annuity Equation u PVn – – – Prentice-Hall, Inc. = PMT (PVIFAi, n) PVn = 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 PVIFAi, n = the present-value interest factor for an annuity 25

Present Value of an Annuity Equation (cont’d) u 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

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 soon-to -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

Amortized Loans u Definition -- loans that are repaid in equal periodic installments u 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. u Examples -- car loans or home mortgages Prentice-Hall, Inc. 28

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, 101. 58 = PMT Prentice-Hall, Inc. 29

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

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

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

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