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
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?
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.
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 & 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
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 equal payments or
receipts)
Four Basic Models
• FVn
=
PV0(1+k)n
=
PV(FVIFk,n)
• PV0
=
FVn[1/(1+k)n]
=
FV(PVIFk,n)
A (1+k)n - 1
=
A(FVIFAk,n)
• PVA0 = A 1 - [1/(1+k)n] =
A(PVIFAk,n)
• FVAn =
k
k
Future Value Example
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
Using Excel
You deposit $2,000 today at 6%
interest. How much will you have in 5
years?
PV
k
n
FV?
$
2,000
6.00%
5
$2,676
Excel Function
=FV (interest, periods, pmt, PV)
=FV (.06, 5, , 2000)
Future Value Example
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.
Compounding More Frequently
than Annually
• 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
Compounding More Frequently
than Annually
On Excel
Annually
PV
$
Sem iAnnually Quarterly
100.00
k
12.0%
n
5
FV
$176.23
$
100.00
0.06
10
$179.08
$
100.00
Monthly
$
100.00
0.03
0.01
20
60
$180.61
$181.67
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
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?
$2,000 x [1/(1.06)5] = $2,000 x PVIF6%,5
$2,000 x 0.74758 = $1,494.52
Present Value Example
Using 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?
FV
k
n
PV?
$
2,000
6.00%
5
$1,495
Excel Function
=PV (interest, periods, pmt, FV)
=PV (.06, 5, , 2000)
Present Value Example
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
Using the FVIFA Tables
• 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
Future Value of an Ordinary Annuity
Using Excel
• 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.
PMT
k
n
FV?
$
100
5.0%
3
$ 315.25
Excel Function
=FV (interest, periods, pmt, PV)
=FV (.06, 5,100, )
Future Value of an Annuity Due
Using the FVIFA Tables
• Annuity = 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.
FVA = 100(FVIFA,5%,3)(1+k) = $330.96
FVA = 100(3.152)(1.05) =
$330.96
Future Value of an Annuity Due
Using Excel
• Annuity = 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.
PMT $ 100.00
k
5.00%
n
3
FV
$315.25
FVA? $ 331.01
Excel Function
=FV (interest, periods, pmt, PV)
=FV (.06, 5,100, )
=315.25*(1.05)
Present Value of an Ordinary Annuity
Using PVIFA Tables
• Annuity = 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?
PVA = 2,000(PVIFA,10%,3) = $4,973.70
Present Value of an Ordinary Annuity
Using Excel
• Annuity = 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?
PMT
I
n
PV?
$
2,000
10.0%
3
$4,973.70
Excel Function
=PV (interest, periods, pmt, FV)
=PV (.10, 3, 2000, )
Present Value of a Mixed Stream
Using Tables
• 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%.
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
Using 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%.
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 note
that although 7 years show,
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?
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 present
value, $5,525 is the future
value, and 6 is the number
of periods. Using Excel,
we get:
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
PV
FV
n
k?
$
$
1,000
5,525
6
33.0%
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
Excel Function
=Rate(periods, pmt, PV, FV)
=Rate(6, ,1000, 5525)