chapter 5 time value of money lawrence j. gitman jeff madura introduction to finance

Chapter

5

Time Value of Money

Lawrence J. GitmanJeff Madura

Introduction to Finance

5-2Copyright © 2001 Addison-Wesley

Discuss the role of time value in finance and the use of computational aids to simplify its application.

Understand the concept of future value and its calculation for a single amount; understand the effects on future value and the true rate of interest of compounding more frequently than annually.

Understand the concept of present value, its calculation for a single amount, and the relationship of present to future cash flow.

Learning Goals


Find the future value and present value of an ordinary annuity, the future value of an annuity due, and the present value of a perpetuity.

Calculate the present value of a mixed stream of cash flows, describe the procedures involved in: Determining deposits to accumulate to a future sum

Loan amortization

Finding interest or growth rates

Learning Goals


The Role of Time Value in Finance

Most financial decisions involve costs and benefits that are spread out over time.

Time value of money allows comparison of cash flows from different periods.

Question Would it be better for a company to invest $100,000 in a product

that would return a total of $200,000 in one year, or one that would return $500,000 after two years?

Answer It depends on the interest rate!


Basic Concepts

Future Value Compounding or growth over time

Present Value Discounting to today’s value

Single cash flows and series of cash flows can be considered

Time lines are used to illustrate these relationships


Computational Aids

Use the equations

Use the financial tables

Use financial calculators

Use spreadsheets


Computational Aids

Figure 5.1


Computational Aids

Figure 5.2


Computational Aids

Figure 5.3


Computational Aids

Figure 5.4


Simple Interest

With simple interest, you don’t earn interest on interest.

Year 1: 5% of $100 = $5 + $100 = $105

Year 2: 5% of $100 = $5 + $105 = $110

Year 3: 5% of $100 = $5 + $110 = $115

Year 4: 5% of $100 = $5 + $115 = $120

Year 5: 5% of $100 = $5 + $120 = $125


Compound Interest

With compound interest, a depositor earns interest on interest!

Year 1: 5% of $100.00 = $5.00 + $100.00 = $105.00

Year 2: 5% of $105.00 = $5.25 + $105.00 = $110.25

Year 3: 5% of $110.25 = $5 .51+ $110.25 = $115.76

Year 4: 5% of $115.76 = $5.79 + $115.76 = $121.55

Year 5: 5% of $121.55 = $6.08 + $121.55 = $127.63


Time Value Terms

PV0 = present value or beginning amount

k = interest rate

FVn = future value at end of “n” periods

n = number of compounding periods

A = an annuity (series of equalpayments or receipts)


Four Basic Models

FVn = PV0(1+k)n = PV(FVIFk,n)

PV0 = FVn[1/(1+k)n] = FV(PVIFk,n)

FVAn = A (1+k)n - 1 = A(FVIFAk,n)

k

PVA0 = A 1 - [1/(1+k)n] = A(PVIFAk,n)

k


Algebraically and Using FVIF Tables You deposit $2,000 today at 6% interest.

How much will you have in 5 years?

$2,000 x (1.06)5 = $2,000 x FVIF6%,5

$2,000 x 1.3382 = $2,676.40

Future Value Example


Microsoft® Excel Function

= FV(interest, periods, pmt, PV)

= FV(.06, 5, , 2000)

Future Value Example

Using Microsoft® Excel You deposit $2,000 today at 6% interest.

How much will you have in 5 years?


A Graphic View of Future Value

Figure 5.5


Compounding More Frequently Than Annually

Compounding more frequently than once a year results in a higher effective interest rate because you are earning on interest on interest more frequently.

As a result, the effective interest rate is greater than the nominal (annual) interest rate.

Furthermore, the effective rate of interest will increase the more frequently interest is compounded.


Annually: 100 x (1 + .12)5 = $176.23

Semiannually: 100 x (1 + .06)10 = $179.09

Quarterly: 100 x (1 + .03)20 = $180.61

Monthly: 100 x (1 + .01)60 = $181.67


For example, what would be the difference in future value if I deposit $100 for 5 years and earn 12% annual interest compounded (a) annually, (b) semiannually, (c) quarterly, and (d) monthly?


FVn (continuous compounding) = PV x (ekxn)

where “e” has a value of 2.7183

Continuing with the previous example, find the future value of the $100 deposit after 5 years if interest is compounded continuously.

Continuous Compounding

With continuous compounding the number of compounding periods per year approaches infinity.

Through the use of calculus, the equation thus becomes:

FVn = 100 x (2.7183).12x5 = $182.22


The nominal interest rate is the stated or contractual rate of interest charged by a lender or promised by a borrower.

The effective interest rate is the rate actually paid or earned.

In general, the effective rate is greater than the nominal rate whenever compounding occurs more than once per year.

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

Nominal and Effective Rates


EAR = (1 + .18/12)12 - 1

EAR = 19.56%

Nominal and Effective Rates

For example, what is the effective rate of interest on your credit card if the nominal rate is 18% per year, compounded monthly?


Present Value

Present value is the current dollar value of a future amount of money.

It is based on the idea that a dollar today is worth more than a dollar tomorrow.

It is the amount today that must be invested at a given rate to reach a future amount.

It is also known as discounting, the reverse of compounding.

The discount rate is often also referred to as the opportunity cost, the discount rate, the required return, and the cost of capital.


$2,000 x [1/(1.06)5] = $2,000 x PVIF6%,5

$2,000 x 0.74758 = $1,494.52

Present Value Example

Algebraically and Using PVIF Tables How much must you deposit today in order to have

$2,000 in 5 years if you can earn 6% interest on your deposit?



=PV(interest, periods, pmt, FV)

=PV(.06, 5, , 2000)

Present Value Example

Using Microsoft® Excel How much must you deposit today in order

to have $2,000 in 5 years if you can earn 6% interest on your deposit?


A Graphic View of Present Value

Figure 5.6


Annuities

Annuities are equally-spaced cash flows of equal size.

Annuities can be either inflows or outflows.

An ordinary (deferred) annuity has cash flows that occur at the end of each period.

An annuity due has cash flows that occur at the beginning of each period.

An annuity due will always be greater than an otherwise equivalent ordinary annuity because interest will compound for an additional period.


Annuities

Table 5.1


Using the FVIFA Tables An annuity is an equal annual series of cash flows.

Example

• How much will your deposits grow to if you deposit $100 at the end of each year at 5% interest for three years?

FVA = 100(FVIFA,5%,3) = $315.25

Year 1 $100 deposited at end of year = $100.00

Year 2 $100 x .05 = $5.00 + $100 + $100 = $205.00

Year 3 $205 x .05 = $10.25 + $205 + $100 = $315.25

Future Value of an Ordinary Annuity



=FV(interest, periods, pmt, PV)

=FV(.06,5,100, )

Future Value of an Ordinary Annuity

Using Microsoft® Excel An annuity is an equal annual series of cash flows.

Example

• How much will your deposits grow to if you deposit $100 at the end of each year at 5% interest for three years?


FVA = 100(FVIFA,5%,3)(1+k) = $330.96

FVA = 100(3.152)(1.05) = $330.96

Future Value of an Annuity Due

Using the FVIFA Tables An annuity is an equal annual series of cash flows.

Example

• How much will your deposits grow to if you deposit $100 at the beginning of each year at 5% interest for three years.



=FV(interest, periods, pmt, PV)

=FV(.06, 5,100, )

=315.25*(1.05)

Future Value of an Annuity Due

Using Microsoft® Excel An annuity is an equal annual series of cash flows. Example

• How much will your deposits grow to if you deposit $100 at the beginning of each year at 5% interest for three years.


Present Value of an Ordinary Annuity

PVA = 2,000(PVIFA,10%,3) = $4,973.70

Using PVIFA Tables An annuity is an equal annual series of cash flows.

Example

• How much could you borrow if you could afford annual payments of $2,000 (which includes both principal and interest) at the end of each year for three years at 10% interest?



=PV(interest, periods, pmt, FV)

=PV(.10, 3, 2000, )

Present Value of an Ordinary Annuity

Using Microsoft® Excel An annuity is an equal annual series of cash flows. Example

• How much could you borrow if you could afford annual payments of $2,000 (which includes both principal and interest) at the end of each year for three years at 10% interest?



=NPV(interest, cells containing CFs)

=NPV(.09,B3:B7)

Present Value of a Mixed Stream

Using Microsoft® Excel A mixed stream of cash flows reflects no particular pattern

Find the present value of the following mixed stream assuming a required return of 9%.


A perpetuity is a special kind of annuity.

With a perpetuity, the periodic annuity or cash flow stream continues forever.

PV = Annuity/k

PV = $1,000/.08 = $12,500

Present Value of a Perpetuity

For example, how much would I have to deposit today in order to withdraw $1,000 each year forever if I can earn 8% on my deposit?


Loan Amortization

Table 5.7


It is important to note that although there are

7 years shown, there are only 6 time periods

between the initial deposit and the final value.

Determining Interest or Growth Rates

At times, it may be desirable to determine the compound interest rate or growth rate implied by a series of cash flows.

For example, you invested $1,000 in a mutual fund in 1994 which grew as shown in the table below?





Thus, $1,000 is the present value, $5,525 is the future value,

and 6 is the number of periods.



=Rate(periods, pmt, PV, FV)

=Rate(6, ,1000, 5525)





Using Microsoft® Excel

The Microsoft® Excel Spreadsheets used in the this presentation can be downloaded from the Introduction to Finance companion web site: http://www.awl.com/gitman_madura

Chapter

5

End of Chapter

Lawrence J. GitmanJeff Madura

Introduction to Finance

chapter 5 time value of money lawrence j. gitman jeff madura introduction to finance

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