Chapter 5
Time Value of Money
Slides developed by:
Pamela L. Hall, Western Washington University
Time is Money
$100 in your hand today is worth more
than $100 in one year

Money earns interest
• The higher the interest, the faster your money
grows
Example

Q: How much would $1,000 promised in
one year be worth today if the bank
paid 5% interest?
A:
$952.38. If we deposited $952.38 after one year we
would have earned $47.62 ($952.38 × .05) in
interest. Thus, our future value would be $952.38 +
$47.62 = $1,000.
2
Time is Money

Present Value
The amount that must be deposited today to
have a future sum at a certain interest rate
 The discounted value of a sum is its present
value

3
Outline of Approach

Deal with four different types of problems

Amount
• Present value
• Future value

Annuity
• Present value
• Future value
4
Outline of Approach

Mathematics


For each type of problem an equation will be
presented
Time lines

Graphic portrayal of a time value problem
0
1
•
Helps with complicated problems
2
5
Amount Problems—Future
Value

The future value (FV) of an amount

How much a sum of money placed at interest
(k) will grow into in some period of time
• If the time period is one year
• FV1 = PV + kPV or FV1 = PV(1+k)
• If the time period is two years
• FV2 = FV1 + kFV1 or FV2 = PV(1+k)2
• If the time period is generalized to n years
• FVn = PV(1+k) n
6
Amount Problems—Future
Value

The (1 + k)n depends on

Size of k and n
• Can develop a table depicting different values of n
and k and the proper value of (1 + k)n
• Can then use a more convenient formula
Example
• FVn = PV [FVFk,n]
Q: If we deposited $438 at 6% interest
for five years, how much would we
have?
These values can be
looked up in an interest
factor table.
A: FV5 = $438(1.06)5 = $438(1.3382) = $586.13
7
Table 5-1: The Future Value Factor for k
and n
FVFk,n = (1+k)n
6%
5
1.3382
8
Other Issues

Problem-Solving Techniques

Three of four variables are given
• We solve for the fourth

The Opportunity Cost Rate

The opportunity cost of a resource is the
benefit that would have been available from
its next best use
9
Financial Calculators


Work directly with equations
How to use a typical financial calculator in time
value

Five time value keys
• Use either four or five keys

Some calculators distinguish between inflows and
outflows
• If a PV is entered as positive the computed FV is negative
10
Financial Calculators
Basic Calculator Keys
N
Number of time periods
I/Y
Interest rate (%)
PV
Present Value
FV
Future Value
PMT
Payment
11
Financial Calculators
Q: What is the present value of $5,000 received in one year if interest rates
are 6%?
Example
A: Input the following values on the calculator and compute the PV:
N
1
I/Y
6
FV
5000
PMT 0
PV
4,716.98
Answer
12
The Expression for the
Present Value of an Amount
FVn  PV 1+k 
n
Solve for PV


1
PV = FVn 

n
 1  k  
Interest Factor

The future and present values factors are
reciprocals

Either equation can be used to solve any amount
problems
1
Solving for k or n involves
FVFk,n 
searching a table.
PVFk,n
13
The Expression for the Present
Value of an Amount—Example
Q: What interest rate will grow $850 into $983.96
in three years?
Example
A: Using the FV equation of FV = PV (1 + i)n, the answer
is:
N
3
1.1576 = (1 + i)3
PV
-850
1.1576  1 + i
1.0499 = 1 + i
0.0499 = i or i = 4.99%
FV
983.96
$983.96 = $850 (1 + i)3
3
PMT 0
I/Y
5.0
Or you could
look up the
interest factor of
0.8639 [850 
983.96] in the
proper table with
n=3.
Answer
14
The Expression for the Present
Value of an Amount—Example
Example
Q: How long does it take money invested at 14% to double?
A: The future value must be twice the present value, so make up
some values for present and future values that meet that
criterion, such as $1 doubling to $2.
Or you could
-1
PV
look up the
n
$2 = $1 (1.14)
interest factor
2
FV
of 2.00 [2  1]
$2 = (1.14)n
in the proper
log 2 = n log 1.14
PMT 0
table with
0.3010 = 0.0569n
I/Y
14
k=14%.
5.29 = n
Answer
5.29
N
15
Annuity Problems

Annuity

A finite series of equal payments separated
by equal time intervals
• Ordinary annuities
• Payments occur at the end of the time periods
• Annuity due
• Payments occur at the beginning of the time periods
16
Figure 5.1: Ordinary Annuity
17
Figure 5.2: Annuity Due
18
The Future Value of an Annuity—
Developing a Formula

Future value of an annuity

The sum, at its end, of all payments and all
interest if each payment is deposited when
received
19
Figure 5.4: FV of a Three-Year
Ordinary Annuity
20
The Future Value of an Annuity—
Developing a Formula

Thus, for a 3-year annuity, the formula is
FVA = PMT 1+k   PMT 1+k   PMT 1+k 
0
1
2
Generalizing the Expression:
FVA n = PMT 1+k   PMT 1+k   PMT 1+k  
0
1
2
 PMT 1+k 
n -1
which can be written more conveniently as:
n
FVA n   PMT 1+k 
n i
i=1
Factoring PMT outside the summation, we obtain:
FVA n  PMT
n
 1+k 
n i
FVFAk,n
i=1
21
The Future Value of an
Annuity—Solving Problems

There are four variables in the future value
of an annuity equation
The future value of the annuity itself
 The payment
 The interest rate
 The number of periods

• Helps to draw a time line
22
Example
The Future Value of an Annuity—
Solving Problems—Example
Q: The Brock Corporation owns the patent to an industrial process and receives
license fees of $100,000 a year on a 10-year contract for its use. Management
plans to invest each payment until the end of the contract to provide funds for
development of a new process at that time. If the invested money is expected
to earn 7%, how much will Brock have after the last payment is received?
A: Using the FVAn = PMT[FVFAk,n] equation and looking up the interest factor of
the annuity at an n of 10 and a k of 7, we obtain an interest factor of 13.8164.
This gives up a FVA10 of $100,000[13.8164] = $1,381,640.
N
10
I/Y
7
PMT
100000
PV
0
FV
1,381,645
Answer
23
The Sinking Fund Problem

Companies borrow money by issuing
bonds for lengthy time periods

No repayment of principal is made during the
bonds’ lives
• Principal is repaid at maturity in a lump sum
• A sinking fund provides cash to pay off a bond’s principal
at maturity
• Problem is to determine the periodic deposit to have
the needed amount at the bond’s maturity—a future
value of an annuity problem
24
Example
The Sinking Fund Problem –
Example
Q: The Greenville Company issued bonds totaling $15 million for 30 years. The
bond agreement specifies that a sinking fund must be maintained after 10
years, which will retire the bonds at maturity. Although no one can accurately
predict interest rates, Greenville’s bank has estimated that a yield of 6% on
deposited funds is realistic for long-term planning. How much should Greenville
plan to deposit each year to be able to retire the bonds with the money put
aside?
A: The time period of the annuity is the last 20 years of the bond issue’s life. Input
the following keystrokes into your calculator.
N
20
I/Y
6
FV
15,000,000
PV
0
PMT
407,768.35
Answer
25
Compound Interest and NonAnnual Compounding

Compounding


Earning interest on interest
Compounding periods

Interest is usually compounded annually,
semiannually, quarterly or monthly
• Interest rates are quoted by stating the nominal
rate followed by the compounding period
26
Figure 5.5:
The Effect of Compound Interest
27
The Effective Annual Rate

Effective annual rate (EAR)

The annually compounded rate that pays the
same interest as a lower rate compounded
more frequently
28
Example
The Effective Annual Rate—
Example
Q: If 12% is compounded monthly, what annually
compounded interest rate will get a depositor the
same interest?
A: If your initial deposit were $100, you would have
$112.68 after one year of 12% interest compounded
monthly. Thus, an annually compounded rate of
12.68% [($112.68  $100) – 1] would have to be
earned.
29
The Effective Annual Rate

EAR can be calculated for any
compounding period using the following
formula:
EAR 

knominal 

1



m


m
-1
Effect of more frequent compounding is
greater at higher interest rates
30
Table 5.3
31
The Effective Annual Rate

The APR and EAR

Annual percentage rate (APR)
• Is actually the nominal rate and is less than the EAR

Compounding Periods and the Time Value
Formulas


Time periods must be compounding periods
Interest rate must be the rate for a single
compounding period
• For instance, with a quarterly compounding period the knominal
must be divided by 4 and the n must be multiplied by 4
32
The Effective Annual Rate –
Example
Example
Q: You want to buy a car costing $15,000 in 2½ years. You plan to save
the money by making equal monthly deposits in your bank account,
which pays 12% compounded monthly. How much must you deposit
each month?
A: This is a future value of an annuity problem with a 1% monthly interest rate and
a 30-month time period. Input the following keystrokes into your calculator.
N
30
I/Y
1
FV
15,000
PV
0
PMT
431.22
Answer
33
The Present Value of an Annuity—
Developing a Formula

Present value of an annuity

Sum of all of the annuity’s payments
• Easier to develop a formula than to do all the calculations
individually
PVA =
PMT
PMT
PMT


1+k  1+k 2 1+k 3
which can also be written as:
PVA = PMT 1+k   PMT 1+k   PMT 1+k 
1
2
3
Generalized for any number of periods:
PVA = PMT 1+k   PMT 1+k  
1
2
 PMT 1+k 
n
Factoring PMT and using summation, we o btain:
 n
i 
PVA  PMT   1+k  
 i=1

PVFAk,n
34
The Present Value of an
Annuity—Solving Problems

There are four variables in the present
value of an annuity equation
The present value of the annuity itself
 The payment
 The interest rate
 The number of periods

• Problem usually presents 3 of the 4 variables
35
The Present Value of an Annuity—
Solving Problems–Example
Example
Q: The Shipson Company has just sold a large machine to Baltimore Inc. on an
installment contract. The contract calls for Baltimore to make payments of $5,000
every six months (semiannually) for 10 years. Shipson would like its cash now and
asks its bank to discount the contract and pay it the present (discounted) value.
Baltimore is a good credit risk, so the bank is willing to discount the contract at 14%
compounded semiannually. How much should Shipson receive?
A: The contract represents an annuity with payments of $5,000. Adjust the interest rate and
number of periods for semiannual compounding and solve for the present value of the
annuity.
N
20
I/Y
7
FV
0
PMT
5000
PV
52,970.07
This can also be calculated
using the PVA formula of
PVA = PMT[PVFAk, n] with
an n of 20 and a k of 7%,
resulting in PVA =
$5,000[10.594] = $52,970.
Answer
36
Spreadsheet Solutions


Time value problems can be solved on a
spreadsheet such as Microsoft Excel™ or Lotus
1-2-3™
To solve for:





FV use =FV(k, n, PMT, PV)
PV use =PV(k, n, PMT, FV)
K use =RATE(n, PMT, PV, FV)
N use =NPER(k, PMT, PV, FV)
PMT use =PMT(k, n, PV, FV)
Select the function
for the unknown
variable, place the
known variables in
the proper order
within the
parentheses and
input 0, for the
unknown variable.
37
Spreadsheet Solutions

Complications
Interest rates in entered as decimals, not
percentages
 Of the three cash variables (FV, PMT or PV)

• One is always zero
• The other two must be of the opposite sign
• Reflects inflows (+) versus outflows (-)
38
Amortized Loans

An amortized loan’s principal is paid off
regularly over its life

Generally structured so that a constant
payment is made periodically
• Represents the present value of an annuity
39
Amortized Loans—Example
Example
Q: Suppose you borrow $10,000 over four years at 18% compounded
monthly repayable in monthly installments. How much is your loan
payment?
A: Adjust your interest rate and number of periods for monthly
compounding and input the following keystrokes into your calculator.
N
48
I/Y
1.5
PV
10,000
FV
0
PMT
293.75
Answer
This can also be calculated
using the PVA formula of
PVA = PMT[PVFAk, n] with
an n of 48 and a k of 1.5%,
resulting in $10,000 =
PMT[34.0426] = $293.75.
40
Amortized Loans—Example
Example
Q: Suppose you want to buy a car and can afford to make payments of
$500 a month. The bank makes three-year car loans at 12%
compounded monthly. How much can you borrow toward a new car?
A: Adjust your k and n for monthly compounding and input the following
calculator keystrokes.
N
36
I/Y
1
FV
0
PMT
500
PV
15,053.75
Answer
This can also be calculated
using the PVA formula of
PVA = PMT[PVFAk, n] with
an n of 36 and a k of 1%,
resulting in PVA =
$500[30.1075] = $15,053.75.
41
Loan Amortization Schedules
Detail the interest and principal in each
loan payment
 Show the beginning and ending balances
of unpaid principal for each period
 Need to know

Loan amount (PVA)
 Payment (PMT)
 Periodic interest rate (k)

42
Loan Amortization Schedules—
Example
Example
Q: Develop an amortization schedule for the
loan demonstrated in Example 5.12.
Note that the Interest portion
of the payment is decreasing
while the Principal portion is
increasing.
43
Mortgage Loans

Mortgage loans (AKA: mortgages)


Loans used to buy real estate
Often the largest single financial
transaction in an average person’s life

Typically an amortized loan over 30 years
• During the early years of the mortgage nearly all
the payment goes toward paying interest
• This reverses toward the end of the mortgage
44
Mortgage Loans

Implications of mortgage payment pattern
Early mortgage payments provide a large tax
savings which reduces the effective cost of a
loan
 Halfway through a mortgage’s life half of the
loan has not been paid off


Long-term loans like mortgages result in
large total interest amounts over the life of
the loan
45
Mortgage Loans—Example
Example
Q: Calculate the monthly payment for a 30-year 7.175% mortgage
of $150,000. Also calculate the total interest paid over the life of
the loan.
A: Adjust the n and k for monthly compounding and input the
following calculator keystrokes.
N
Monthly payment
360
X number of payments
I/Y
.5979
FV
0
PV
150,000
1,015.65
Answer
360
Total payments
$365,634
- Original Loan
$150,000
Total Interest
$215,634
Tax Savings @ 30%
PMT
$1,015.65
Net interest cost
$64,690
$150,944
46
The Annuity Due
In an annuity due payments occur at the
beginning of each period
 The future value of an annuity due


Because each payment is received one
period earlier
• It spends one period longer in the bank earning
interest
n -1
FVAdn = PMT + PMT 1+k    PMT 1+k   1  k 


which written with the interest factor becomes:
FVAdn  PMT FVFA k,n  1  k 
47
Example
The Annuity Due—Example
Q: The Baxter Corporation started making sinking fund deposits of
$50,000 per quarter today. Baxter’s bank pays 8% compounded
quarterly, and the payments will be made for 10 years. What will the
fund be worth at the end of that time?
A: Adjust the k and n for quarterly compounding and input the following
calculator keystrokes.
NOTE: Advanced
calculators allow you to
switch from END (ordinary
annuity) to BEGIN (annuity
due) mode.
N
40
I/Y
2
PMT
50,000
PV
0
FV
3,020,099 x 1.02 = 3,080,501
Answer
48
The Annuity Due

The present value of an annuity due

Formula
PVAd  PMT PVFAk,n  1 k 

Recognizing types of annuity problems


Always represent a stream of equal payments
Always involve some kind of a transaction at one
end of the stream of payments
•
•
End of stream—future value of an annuity
Beginning of stream—present value of an annuity
49
Perpetuities

A perpetuity is a stream of regular payments
that goes on forever


Future value of a perpetuity


An infinite annuity
Makes no sense because there is no end point
Present value of a perpetuity

A diminishing series of numbers
• Each payment’s present value is smaller than the one before
PVp 
PMT
k
50
Perpetuities—Example
Example
Q: The Longhorn Corporation issues a security that promises to pay its holder $5
per quarter indefinitely. Money markets are such that investors can earn about
8% compounded quarterly on their money. How much can Longhorn sell this
special security for?
A: Convert the k to a quarterly k and plug the values into the equation.
PVp 
PMT
$5

 $250
k
0.02
You may also work this by inputting a
large n into your calculator (to
simulate infinity), as shown below.
N
999
I/Y
2
PMT
5
FV
0
PV
250
Answer
51
Continuous Compounding

Compounding periods can be shorter than
a day


As the time periods become infinitesimally
short, interest is said to be compounded
continuously
To determine the future value of a
continuously compounded value:
 
FVn  PV ekn
52
Continuous Compounding—
Example
Example
Q: The First National Bank of Charleston is offering continuously
compounded interest on savings deposits. Such an offering is generally
more of a promotional feature than anything else. If you deposit $5,000
at 6½% compounded continuously and leave it in the bank for 3½ years,
how much will you have? Also, what is the equivalent annual rate (EAR)
of 12% compounded continuously?
A: To determine the future value of $5,000, plug the appropriate values into the
equation.
 

FVn  PV ekn  $5,000 e
.0653.5
  $5,000 1.2255457  $6,277.29
A: To determine the EAR of 12% compounded continuously, find the future value of
$100 compounded continuously in one year, then calculate the annual return.
 
 
FVn  PV ekn  $100 e.12  $100 1.1275   $112.75
NOTE: Some advanced calculators
$112.75  $100
Annual Return =
 12.75% have a function to solve for the EAR
$100
with continuous compounding.
53
Table 5.5:
Time Value Formulas
54
Multipart Problems

Time value problems are often combined
due to complex nature of real situations

A time line portrayal can be critical to keeping
things straight
55
Example
Multipart Problems—Example
Q:Exeter Inc. has $75,000 invested in securities that
earn a return of 16% compounded quarterly. The
company is developing a new product that it plans to
launch in two years at a cost of $500,000. Exeter’s
cash flow is good now but may not be later, so
management would like to bank money from now until
the launch to be sure of having the $500,000 in hand
at that time. The money currently invested in
securities can be used to provide part of the launch
fund. Exeter’s bank has offered an account that will
pay 12% compounded monthly. How much should
Exeter deposit with the bank each month to have
enough reserved for the product launch?
56
Multipart Problems—Example
A: Two things are happening in this problem

Example

Exeter is saving money every month (an annuity) and
The money invested in securities (an amount) is growing
independently at interest
We have two problems that must be handled
sequentially.



First, we need to find the future value of $75,000
Then, we subtract that future value from $500,000 to
determine how much extra Exeter needs to save via the
annuity
Then, we’ll solve a future value of an annuity problem for
the payment
57
Multipart Problems—Example
Example
To find the future value of
the $75,000…
N
8
To find the savings
annuity value
N
24
I/Y
1
PV
0
FV
397,355
PMT
14,731
I/Y
4
PMT
0
FV
75,000
PV
102,643
Answer
$500,000 - $102,645
= $397,355
Answer
58
Uneven Streams and
Imbedded Annuities

Many real world problems have sequences of
uneven cash flows

These are NOT annuities
• For example, if you were asked to determine the present
value of the following stream of cash flows
$100

$200
$300
Must discount each cash flow individually

Not really a problem when attempting to determine either a
present or future value
•
Becomes a problem when attempting to determine an interest rate
59
Uneven Streams and
Imbedded Annuities
Q:Calculate the interest rate at which the present value
of the stream of payments shown below is $500.
Example
$100
$200
$300
A: We’ll start with a guess of 12% and discount each amount
separately at that rate.
PV = FV1 PVFk,1   FV2 PVFk,2   FV3 PVFk,3 
 $100 PVF12,1   $200 PVF12,2   $300 PVF12,3 
 $100 .8929   $200 .7972   $300 .7118 
 $462.27
This value is too low, so we need to select a
lower interest rate. Using 11% gives us
$471.77. The answer is between 8% and 9%.
60
Calculator Solutions for Uneven
Streams
Financial calculators and spreadsheets
have the ability to handle uneven streams
with a limited number of payments
 Generally programmed to find the present
value of the streams or the k that will
equate a present value to the stream

61
Imbedded Annuities

Sometimes uneven streams of cash flows
will have annuities embedded within them

We can use the annuity formula to calculate
the present or future value of that portion of
the problem
62