Time Value of Money
Chapter 5
© 2003 South-Western/Thomson Learning
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
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
• FVn = PV [FVFk,n]
These values can be
looked up in an interest
factor table.
7
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
8
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
9
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
10
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
11
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
12
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
13
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
14
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
15
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
16
The Effective Annual Rate

Effective annual rate (EAR)

The annually compounded rate that pays the
same interest as a lower rate compounded
more frequently
17
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
18
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
19
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
PMT
PMT
PMT
PVA =


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
20
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
21
Spreadsheet Solutions


Time value problems can be solved on a
spreadsheet such as Microsoft Excel™ or
Lotus 1-2-3™
Select the function
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)
for the unknown
variable, place the
known variables in
the proper order
within the
parentheses, and
input 0 for the
unknown variable.
22
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 (-)
23
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
24
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)

25
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
26
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
27
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 
28
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
29
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
PMT
PVp 
k
30
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
31
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
32
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
33
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

34
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
35