Copyright ©2003 South-Western/Thomson Learning Chapter 4 The Time Value Of Money.

Copyright ©2003 South-Western/Thomson Learning

Chapter 4The Time Value Of Money

Introduction

• This chapter introduces the concepts and skills necessary to understand the time value of money and its applications.

Simple and Compound Interest

• Simple Interest – Interest paid on the principal sum only

• Compound Interest – Interest paid on the principal and on prior

interest that has not been paid or withdrawn

t to denote time

PV0 = principal amount at time 0

FVn = future value n time periods from time 0

PMT to denote cash payment

PV to denote the present value dollar amount

T to denote the tax rate

I to denote simple interest

i to denote the interest rate per period

n to denote the number of periods

Notation

Future Value of a Cash Flow

• At the end of year n for a sum compounded at interest rate i is FVn = PV0 (1 + i)n Formula

• In Table I in the text, (FVIFi,n) shows the future value of $1 invested for n years at interest rate i: FVIFi,n = (1 + i)n Table I

• When using the table, FVn = PV0 (FVIFi,n)

Tables Have Three Variables

• Interest factors (IF)

• Time periods (n)

• Interest rates per period (i)

• If you know any two, you can solve algebraically for the third variable.

Present Value of a Cash Flow

• PV0 = FVn [ ] Formula

• PVIFi, n = Table II

• PV0 = FVn(PVIFi, n) Table II

1 (1 + i)n

1 (1 + i)n

Example Using Formula

• What is the PV of $100 one year from now with 12 percent interest compounded monthly?

PV0 = $100 1/(1 + .12/12)(12 1)

= $100 1/(1.126825)

= $100 (.88744923)

= $ 88.74

Example Using Table II

PV0 = FVn(PVIFi, n)

= $100(.887) From Table II

= $ 88.70

Annuity

• A series of equal dollar CFs for a specified number of periods

• Ordinary annuity is where the CFs occur at the end of each period.

• Annuity due is where the CFs occur at the beginning of each period.

FVIFAi, n = Formula for IF

FVANn = PMT(FVIFAi, n) Table III

Future Value of an Ordinary Annuity

(1 + i)n – 1i

Derivation of the FVAN formula(1)

nn-1 n-2 n-n

FVAN =PMT 1+i +PMT 1+i + +PMT 1+i

The FVAN formula is a geometric series because each term on the right side is equal to the previous term multiplied by a common factor: 1/(1+i).

Multiply both sides of the equation above by the common factor to create a second equation.

Derivation of the FVAN formula(2)

1 n-1 1 n-2 1 n-n 1FVAN =PMT 1+i +PMT 1+i + +PMT 1+in1+i 1+i 1+i 1+i

1 n-2 n-3 -1FVAN =PMT 1+i +PMT 1+i + +PMT 1+in1+i

Subtract this new equation from the original equation on the previous slide. The result:

n-1 -1n n

1FVAN - FVAN =PMT 1+i -PMT 1+i

1+i

Solve for FVAN.

n

n

1+i -1FVAN =PMT

i

Present Value of an Ordinary Annuity

PVIFAi, n = Formula

PVAN0 = PMT( PVIFAi, n) Table IV

1 (1 + i)n

1 –

i

Annuity Due

• Future Value of an Annuity Due– FVANDn = PMT(FVIFAi, n)(1 + i) Table III

• Present Value of an Annuity Due – PVAND0 = PMT(PVIFAi, n)(1 + i) Table IV

Other Important Formulas

• Sinking Fund– PMT = FVANn/(FVIFAi, n) Table III

• Payments on a Loan– PMT = PVAN0/(PVIFAi, n) Table IV

• Present Value of a Perpetuity– PVPER0 = PMT/i

Interest Compounded More Frequently Than Once Per Year

Future Valuenm

nom0n

m

i1PVFV )( +=

Present Value

)nm

minom(1 +

FVnPV0 =

m = # of times interest is compoundedn = # of years

Interest Compounded More Frequently than Once Per Year

• Texas Instruments BA II Plus Calculator – set the number of compounding periods to 12 per year:

• 2nd, P/Y, , 12, ENTER, CE/C, CE/C

• When finished: 2nd, CLR TVM

• And, reset compounding to once per year: 2nd, P/Y, , 1, ENTER, CE/C, CE/C

Effective Annual Rates

A nominal rate of interest (or Annualized Percentage Rate) is found by multiplying the rate charged or paid per period by the number of periods during the year.

#periodsRatei =APR=nom yearperiod

This rate does not include the effect of compounding of interest at the end of each period of the year.

Effective Annual Rates

For comparison purpose, we need an effective annual rate that includes the effect of compounding.

1

m

inom1+i 1+ meff

Solve for the rate that gives the same effect with once per year compounding as the APR gives with more frequent compounding than annual.

minomi =1+ -1meff

Effective Annual Rates

If compounding is done continuously,

minomi = lim 1+ -1 mmeff

inom i = e - 1eff

Compounding and Effective Rates

• Rate of interest per compounding period

im = (1 + ieff)1/m – 1

• Effective annual rate of interest ieff = (1 + inom/m)m – 1