Time Value of Money

© 2004 Pearson Education Canada Inc.

5-1

Chapter FiveTime Value of Money

Principles of Managerial Finance

First Canadian Edition

Lawrence J. Gitman and Sean Hennessey

© 2004 Pearson Education Canada Inc.

5-2

Learning Goals

LG1 – Discuss role of time value of money in finance, and use of computational aids.

LG2 – Understand concept of future value, its calculation for single amount, and the effect of compounding interest more frequently.

LG3 – Find the future value of an ordinary annuity and an annuity due, and compare these two types of annuities.

© 2004 Pearson Education Canada Inc.

5-3

Learning Goals (continued)

LG4 – Understand concept of present value, its calculation for a single amount, and its relationship to future value.

LG5 – Calculate present value of a mixed stream of cash flows, an annuity, a mixed stream with an embedded annuity, and a perpetuity.

© 2004 Pearson Education Canada Inc.

5-4

Learning Goals (continued)

LG6 – Describe the procedures involved in1) Determining the periodic investments

required to accumulate a future sum,

2) Loan amortization, and

3) Determining growth and interest rates.

© 2004 Pearson Education Canada Inc.

5-5

Role of Time Value in Finance

• The timing of cash flows has important economic consequences that are recognized as the Time Value of Money.

• Time value is based on the belief that a dollar today is worth more than a dollar that will be received at some future date.

© 2004 Pearson Education Canada Inc.

5-6

Future vs. Present Value• Future Value is cash you will receive at a

given future date.• Present Value is the equivalent of cash on

hand today.• A time line can be used to depict the cash

flows associated with a given investment.• Since financial managers make decisions at

time zero, they tend to rely on present value techniques.

© 2004 Pearson Education Canada Inc.

5-7

Figure 5.2 Compounding and Discounting

© 2004 Pearson Education Canada Inc.

5-8

Computational Aids

• Financial Tables are commonly used as quick reference tools for determining present and future values at various interest or discount rates of a range of time periods.

• Modern calculators are programmed to perform the complete computational analysis using the underlying formulas for present and future value.

© 2004 Pearson Education Canada Inc.

5-9

Figure 5.3 Financial Tables

© 2004 Pearson Education Canada Inc.

5-10

Future Value of Single Amount

• Principal is the amount of money on which interest is paid.

• Compound Interest is the interest earned on a given deposit that becomes part of the principal at the end of a specified period.

• Future Value of a present amount is found by applying compound interest over a specified period of time.

© 2004 Pearson Education Canada Inc.

5-11

Equation of Future Value

FVn = future value at the end of period n.

PV = initial principal, or present value.

k = annual rate of return.

n = number of periods the money is left on deposit.

(5.4) FVn = PV (1+k)n

© 2004 Pearson Education Canada Inc.

5-12

Using Tables & Calculators

• Table A-1 provide values for the Future Value Interest Factor (FVIF) which simplifies the process of calculating FV in equation (5.4).

(5.5) FVIFk,n = (1+k)n

(5.6) FVn = PV FVIFk,n

© 2004 Pearson Education Canada Inc.

5-13

Graphic View of Future Value

© 2004 Pearson Education Canada Inc.

5-14

Compounding More Frequently Than Annually

• Semiannual Compounding involves two compounding periods within the year.

• Quarterly Compounding involves four compounding periods within the year.

(5.7) FVn = PV (1+k/m)m*n

Where m is the number of compounding periods within the year.

© 2004 Pearson Education Canada Inc.

5-15

Continuous Compounding

• Continuous Compounding involves compounding over every microsecond.

(5.8) FVn(continuous) = PV (ek*n)

(5.9) FVIFk,n(continuous) = ek*n

© 2004 Pearson Education Canada Inc.

5-16

Nominal and Effective Annual Rates of Interest

• The Nominal, or State, Annual Rate is that charged by a lender or promised by a borrower.

• The Effective Annual Rate (EAR) is the interest actually paid or earned due to compounding.

(5.10) EAR = (1+k/m)m - 1

© 2004 Pearson Education Canada Inc.

5-17

Future Value of An Annuity

• An Annuity is a stream of equal annual cash flows, either inflows or outflows.

• There are two basic types of annuities:– Ordinary Annuity where the cash flow occurs at

the end of each period, and– Annual Due Annuity where the cash flow

occurs at the beginning of each period.

© 2004 Pearson Education Canada Inc.

5-18

FV of Ordinary Annuity

• The Future Value Interest Factor for an Annuity (FVIFA) is:

(5.14)

(5.15) FVAn = PMT (FVIFAk,n)

Where PMT is the amount of each cash flow payment.

n

t

tFVIFA knk

1

1)1(,

© 2004 Pearson Education Canada Inc.

5-19

FV of Annuity Due

• Since an Annuity Due requires the cash flow at the beginning of the period only a simple adjustment to the FVIFA is needed.

(5.16) FVIFAk,n(Annuity Due)=FVIFAk,n (1+k)

© 2004 Pearson Education Canada Inc.

5-20

Present Value of Single Amount

• Present Value is the current dollar value of a future amount; the amount of money that would have to be invested today at a given rate of return over a specified period to equal the future amount.

• The process of finding Present Value is often referred to as Discounting Cash Flows.

• The annual rate of return used is referred to as the discount rate, required return, cost of capital, or opportunity cost.

© 2004 Pearson Education Canada Inc.

5-21

Equation for Present Value

(5.19) PV = FVn = FVn ( 1 )(1+k)n (1+k)n

• Tables may also be used to look up the Present Value Interest Factor (PVIF).

(5.21) PVIFk,n = ( 1 ) (1+k)n

© 2004 Pearson Education Canada Inc.

5-22

Graphic View of Present Value

© 2004 Pearson Education Canada Inc.

5-23

PV of Mixed Stream

• A Mixed Stream is cash flows of different amounts during the future periods.

• To determine the Present Value of a Mixed Stream we must calculate the present value of each future amount, then sum the total of the present value calculations.

© 2004 Pearson Education Canada Inc.

5-24

PV of an Annuity• An Annuity is a series of uniform future cash

flows. We may determine the Present Value of an Annuity using a Present Value Interest Factor for an Annuity (PVIFA).

(5.26)

(5.27) PVAn = PMT (PVIFAk,n)

n

tt

nkk

PVIFA1

,)1(

1

© 2004 Pearson Education Canada Inc.

5-25

PV of Mixed Stream with Embedded Annuity

• Three steps to determine the Present Value of a Mixed Stream with an Embedded Annuity.– Find the present value of the annuity at specified

discount rate.– Add the present value calculated to any other cash

flow occurring in the period just before the start of the annuity to determine a revised cash flows.

– Discount the revised cash flows back to time zero in the normal fashion at specified discount rate.

© 2004 Pearson Education Canada Inc.

5-26

PV of a Perpetuity

• A Perpetuity is an annuity with an infinite life.

• Adjusting the PVIFA where n= we have:

(5.28) PVIFAk, = 1k

© 2004 Pearson Education Canada Inc.

5-27

Investments Required to Accumulate a Future Sum

• To determine the Payments necessary to accumulate a Future Sum, we simply rearrange the formula for the future value of an annuity (5.15):

(5.30) PMT = FVAn

FVIFAk,n

© 2004 Pearson Education Canada Inc.

5-28

Loan Amortization• Loan Amortization is the determination of

the equal annual loan payments necessary to provide a lender with a specified interest return and to repay the loan principal over a specified period.

• Rearranging the formula for PVA (5.27):

(5.32) PMT = PVAn

PVIFAk,n

© 2004 Pearson Education Canada Inc.

5-29

Growth or Interest Rates

• It is often necessary to calculate the compound annual growth rate of a series of cash flows.

• Either future value or present value interest factors can be used depending on the situation.

• Financial Calculators can determine the precise annual interest rate.

• This rate is called the Internal Rate of Return (IRR).