Chapter 4 Time Value of Money. Learning Goals 1. Discuss the role of time value in finance, the use of computational aids, and the basic patterns of cash

Chapter 4Chapter 4

Time Value of Time Value of MoneyMoney

Learning Goals

1. Discuss the role of time value in finance, the use of

computational aids, and the basic patterns of cash

flow.

2. Understand the concept of future value and present

value, their calculation for a single amounts, and the

relationship of present value to future value.

3. Find the future value and the present value of both an

ordinary annuity and an annuity due, and the present

value of a perpetuity.

Learning Goals

4. Calculate both the future value and the present value

of a mixed stream of cash flows.

5. Understand the effect that compounding interest more

frequently than annually has on future value and the

effective annual rate of interest.

6. Describe the procedures involved in (1) determining

deposits to accumulate to a future sum, (2) loan

amortization, (3) finding interest or growth rates, and

(4) finding an unknown number of periods.

The Role of Time Value in Finance

• Most financial decisions involve costs & 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 after one year, or one that would return

$220,000 after two years?

Answer!

It depends on the interest rate!

The Role of Time Value in Finance

• Most financial decisions involve costs & benefits that

are spread out over time.

• Time value of money allows comparison of cash flows

from different periods.

Basic Concepts

• Future Value: compounding or growth over time

• Present Value: discounting to today’s value

• Single cash flows & 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

Computational Aids

Computational Aids

Computational Aids

Advantages of Computers & Spreadsheets

• Spreadsheets go far beyond the computational

abilities of calculators.

• Spreadsheets have the ability to program logical

decisions.

• Spreadsheets display not only the calculated values of

solutions but also the input conditions on which

solutions are based.

• Spreadsheets encourage teamwork.

• Spreadsheets enhance learning.

• Spreadsheets communicate as well as calculate.

Basic Patterns of Cash Flow

• The cash inflows and outflows of a firm can be

described by its general pattern.

• The three basic patterns include a single amount, an

annuity, or a mixed stream:

Simple 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

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

Compound 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

With compound interest, a depositor earns interest on interest!

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 equal payments 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

Future Value Example

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

Algebraically and Using FVIF Tables


You deposit $2,000 today at 6%

interest. How much will you have in 5

years?

Using Excel

PV 2,000$ k 6.00%n 5FV? $2,676

Excel Function

=FV (interest, periods, pmt, PV)

=FV (.06, 5, , 2000)


A Graphic View of Future Value

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.


• 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, an (d) monthly?

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


Annually SemiAnnually Quarterly Monthly

PV 100.00$ 100.00$ 100.00$ 100.00$

k 12.0% 0.06 0.03 0.01

n 5 10 20 60

FV $176.23 $179.08 $180.61 $181.67

On Excel

Continuous Compounding

• With continuous compounding the number of

compounding periods per year approaches infinity.

• Through the use of calculus, the equation thus

becomes:

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 (continuous compounding) = PV x (ekxn)

where “e” has a value of 2.7183.

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

Nominal & Effective Rates

• 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 > nominal rate whenever

compounding occurs more than once per year

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

Nominal & 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?

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

EAR = 19.56%

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.

• Calculating present value is also known as

discounting.

• The discount rate is often also referred to as the

opportunity cost, the discount rate, the required return,

or the cost of capital.

Present Value Example

How much must you deposit today in order to

have $2,000 in 5 years if you can earn 6%

interest on your deposit?

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

$2,000 x 0.74758 = $1,494.52

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?

FV 2,000$ k 6.00%n 5PV? $1,495

Excel Function

=PV (interest, periods, pmt, FV)

=PV (.06, 5, , 2000)

Using Excel


A Graphic View of Present Value

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

Future Value of an Ordinary Annuity

• Annuity = 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

Using the FVIFA Tables

Future Value of an Ordinary Annuity



deposit $100 at the end of each year at 5% interest for

three years.

Using Excel

PMT 100$ k 5.0%n 3FV? 315.25$

Excel Function


=FV (.06, 5,100, )

Future Value of an Annuity Due



deposit $100 at the beginning of each year at 5%

interest for three years.

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

Using the FVIFA Tables

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

Future Value of an Annuity Due



deposit $100 at the beginning of each year at 5%

interest for three years.

Using Excel

Excel Function


=FV (.06, 5,100, )

=315.25*(1.05)

PMT 100.00$ k 5.00%n 3FV $315.25FVA? 331.01$

Present Value of an Ordinary Annuity


• 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?

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

Using PVIFA Tables

Present Value of an Ordinary Annuity


• 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?

Using Excel

PMT 2,000$ I 10.0%n 3PV? $4,973.70

Excel Function

=PV (interest, periods, pmt, FV)

=PV (.10, 3, 2000, )

Present Value of a Mixed Stream

• 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%.

Using Tables

Year Cash Flow PVIF9%,N PV

1 400 0.917 366.80$

2 800 0.842 673.60$

3 500 0.772 386.00$

4 400 0.708 283.20$

5 300 0.650 195.00$

PV 1,904.60$

Present Value of a Mixed Stream

• 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%.

Using EXCEL

Year Cash Flow

1 400

2 800

3 500

4 400

5 300

NPV $1,904.76

Excel Function

=NPV (interest, cells containing CFs)

=NPV (.09,B3:B7)

Future Value of a Mixed Stream

Future Value of a Mixed Stream

Present Value of a Perpetuity

• A perpetuity is a special kind of annuity.

• With a perpetuity, the periodic annuity or cash flow

stream continues forever.

PV = Annuity/k

• 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?

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

Loan Amortization

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?

1994 1,000$ 1995 1,127 1996 1,158 1997 2,345 1998 3,985 1999 4,677 2000 5,525

It is important to notethat although 7 years show, there are only 6 time periods

between the initial deposit and the final value.






1994 which grew as shown in the table below?1994 1,000$ 1995 1,127 1996 1,158 1997 2,345 1998 3,985 1999 4,677 2000 5,525

Thus, $1,000 is the presentvalue, $5,525 is the futurevalue, and 6 is the numberof periods. Using Excel,

we get:







PV 1,000$ FV 5,525$ n 6k? 33.0%







Excel Function

=Rate(periods, pmt, PV, FV)

=Rate(6, ,1000, 5525)

Documents

Chapter 4 Time Value of Money. Learning Goals 1. Discuss the role of time value in finance, the use of computational aids, and the basic patterns of cash