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

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

Prentice-Hall, Inc.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.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 = $61,954.27

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

Prentice-Hall, Inc.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.15 The Power of Time in Compounding Over 35 Years  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.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.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.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.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.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.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.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.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.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.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.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.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 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.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.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.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.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.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.33 Summary (cont’d)  Amortized loans – loans paid in equal periodic installments for a specific length of time  Perpetuities – annuities that continue forever