Chapter 6 - Time Value of Money
Time Value of Money
A sum of money in hand today is worth more than the
same sum promised with certainty in the future.
Think in terms of money in the bank
The value today of a sum promised in a year is the amount
you'd have to put in the bank today to have that sum in a
year.
Example:
Future Value (FV) = $1,000
k = 5%
Then Present Value (PV) = $952.38 because
$952.38 x .05 = $47.62
and $952.38 + $47.62 = $1,000.00
Time Value of Money
Present Value
– The amount that must be deposited today to
have a future sum at a certain interest rate
Terminology
– The discounted value of a future sum is its
present value
3
Outline of Approach
Four different types of problem
– Amounts
Present value
Future value
– Annuities
Present value
Future value
4
Outline of Approach
Develop an equation for each
Time lines - Graphic portrayals
Place information on the time line
5
The Future Value of an Amount
How much will a sum deposited at interest rate
k grow into over some period of time
If the time period is one year:
FV1 = PV(1 + k)
If leave in bank for a second year:
FV2 = PV(1 + k)(1 ─ k)
FV2 = PV(1 + k)2
Generalized:
FVn = PV(1 + k)n
6
The Future Value of an Amount
(1 + k)n depends only on k and n
Define Future Value Factor for k,n as:
FVFk,n = (1 + k)n
Substitute for:
FVn = PV[FVFk,n]
7
The Future Value of an Amount
Problem-Solving Techniques
– All time value equations contain four
variables
In this case PV, FVn, k, and n
Every problem will give you three and
ask for the fourth.
8
Concept Connection Example 6-1
Future Value of an Amount
How much will $850 be worth in three years at 5% interest?
Write Equation 6.4 and substitute the amounts given.
FVn = PV [FVFk,n ]
FV3 = $850 [FVF5,3]
Concept Connection Example 6-1
Future Value of an Amount
Look up FVF5,3 in the three-year row under the 5%
column of Table 6-1, getting 1.1576
Concept Connection Example 6-1
Future Value of an Amount
Substitute the future value factor of 1.1576 for FVF5,3
FV3 = $850 [FVF5,3]
FV3 = $850 [1.1576]
= $983.96
Financial Calculators
Work directly with equations
How to use a typical financial calculator
– Five time value keys
Use either four or five keys
– Some calculators require inflows and
outflows to be of different signs
If PV is entered as positive the computed FV is
negative
12
Financial Calculators
Basic Calculator functions
Financial Calculators
What is the present value of $5,000 to be received in one year
if the interest rate is 6%?
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
14
The Present Value of an Amount
FVn  PV 1+k 
n
Solve for PV


1
PV = FVn 

n
 1  k  
Interest Factor
1
FVFk,n 
PVFk,n
PV= FVn [PVFk,n ]
Future and present value factors are reciprocals
– Use either equation to solve any amount problems
15
Concept Connection Example 6-3
Finding the Interest Rate
Finding the Interest Rate
what interest rate will grow $850 into $983.96 in three
years. Here we have FV3, PV, and n, but not k.
Use Equation 6.7
PV= FVn [PVFk,n ]
16
Concept Connector Example 6-3
PV= FVn [PVFk,n ]
Substitute for what’s known
$850= $983.96 [PVFk,n ]
Solve for [PVFk,n ]
[PVFk,n ] = $850/ $983.96
[PVFk,n ] = .8639
Find .8639 in Appendix A (Table A-2). Since n=3 search only row 3,
and find the answer to the problem is (5% ) at top of column.
Concept Connection Example 6-3
Finding the Interest Rate
Annuity Problems
Annuities
– A finite series of equal payments separated
by equal time intervals
Ordinary annuities
Annuities due
19
Figure 6-1 Future Value: Ordinary
Annuity
20
Figure 6-2 Future Value: Annuity Due
21
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
– Figure 6-3 Time Line Portrayal of an
Ordinary Annuity
22
Figure 6-4 Future Value of a Three-Year
Ordinary Annuity
23
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 
i=1
n i
FVFAk,n
The Future Value of an Annuity—
Solving Problems
Four variables in the future value of an
annuity equation
– FVAn
– PMT
–k
–n
future value of the annuity
payment
interest rate
number of periods
Helps to draw a time line
25
Concept Connection Example 6-5 The
Future Value of an Annuity
Brock Corp. will receive $100K per year for 10 years
and will invest each payment at 7% until the end of
the last year.
How much will Brock have after the last payment is
received?
26
Concept Connection Example 6-5 The
Future Value of an Annuity
FVAn = PMT[FVFAk,n]
FVFA 7,10 = 13.8164
– FVA10 = $100,000[13.8164] = $1,381,640
The Sinking Fund Problem
Companies borrow money by issuing
bonds
– No repayment of principal is made during
the bond’s life
– Principal is repaid at maturity in a lump sum
A sinking fund provides cash to pay off principal
at maturity
See Concept Connection Example 6-6
28
Compound Interest and
Non-Annual Compounding
Compounding
– Earning interest on interest
Compounding periods
– Interest is usually compounded annually,
semiannually, quarterly or monthly
29
Figure 6-5
The Effect of Compound Interest
30
The Effective Annual Rate
Effective annual rate (EAR)
– The annually compounded rate that pays
the same interest as a lower rate
compounded more frequently
31
Year-end Balances at Various Compounding
Periods for $100 Initial Deposit and knom = 12%
Table 6.2
32
The Effective Annual Rate
EAR can be calculated for any compounding period
using the formula
 knom 
EAR  1 

m 

m
m is number of compounding periods per year
Effect of more frequent compounding is greater at
higher interest rates
33
The APR and the EAR
The annual percentage rate (APR)
associated with credit cards is actually
the nominal rate and is less than the
EAR
34
Compounding Periods and the
Time Value Formulas
n must be compounding periods
k must be the rate for a single
– E.g. for quarterly compounding
k = knom divided by 4, and
n = years multiplied by 4
35
Concept Connection Example 6-7
Compounding periods and Time Value Formulas
Save up to buy a $15,000 car in 2½ years.
Make equal monthly deposits in a bank account which
pays 12% compounded monthly
How much must be deposited each month?
A “Save Up” problem
Payments plus interest accumulates to a known amount
Save ups are always FVA problems
36
Concept Connection Example 6-7
Compounding periods and Time Value Formulas
Calculate k and n for monthly compounding,
k
k nom
12
12%

 1%
12
and
n = 2.5 years x 12 months/year = 30 months.
Concept Connection Example 6-7
Compounding periods and Time Value Formulas
Write the future value of an annuity expression and substitute.
FVAN = PMT [FVFAk,n ]
$15,000 = PMT [FVFA1,30 ]
From Appendix A (Table A-3) FVFA1,30 = 34.7849 substituting
$15,000 = PMT [34.7849]
Solve for PMT
PMT = $431.22
Concept Connection Example 6-7
Compounding periods and Time Value Formulas
Figure 6-6 Present Value of a Threeperiod Ordinary Annuity
40
The Present Value of an Annuity
Developing a Formula
Present value of an annuity
– Sum of the present values of all of the annuity’s
payments
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
41
The Present Value of an Annuity—
Solving Problems
There are four variables
– PVA
present value of the annuity
– PMT
payment
–k
interest rate
–n
number of periods
– Problems give 3 and ask for the fourth
42
Concept Connection Example 6.9
PVA - Discounting a Note
Shipson Co. will receive $5,000 every six months (semiannually) for 10
years. The firm needs cash now and asks its bank to discount the
contract and pay Shipson the present value of the expected annuity.
This is a common banking service called discounting.
If the payer has good credit, the bank will discount the contract at the
current rate of interest, 14% compounded semiannually and pay
Shipson the present value of the annuity of the expected payments.
How much should Shipson receive?
Solution:
43
Amortized Loans
An amortized loan’s principal is paid off over
its life along with interest
Constant Payments are made up of a varying
mix of principal and interest
The loan amount is the present value of the
annuity of the payments
44
Concept Connection Example 6-11
Amortized Loan – Finding PMT
What are the monthly payments on a $10,000, four year
loan at 18% compounded monthly?
Solution:
k= kNom /12 = 18%/12 =1.5%
n = 4 years x 12 months/ year = 48 months
PVA = PMT [PVFAk,n ]
$10,000 = PMT [PVFAk,n ]
PVFA15,48 = 34.0426
$10,000 = PMT [34.0426]
PMT = $293.75
45
Concept Connection Example 6-12
Amortized Loan – Finding Amount Borrowed
A car buyer and can make monthly payments of $500. How
much can she borrow with a three-year loan at 12%
compounded monthly.
Solution:
k= kNom /12 = 12%/12 =1%
n=3 years x 12 months/year = 36 months
PVA= PMT [PVFAk,n ]
PVA = PMT [PVFA1,36 ]
Appendix A (Table A-4) gives PVFA1,36 = 30.1075
PVA= 500 (30.1075)
PVA =$15,053.75
46
Loan Amortization Schedules
Shows interest and principal in each loan payment
Also shows beginning and ending balances of unpaid
principal for each period
To construct we need to know
– Loan amount (PVA)
– Payment (PMT)
– Periodic interest rate (k)
47
Table 6-4
Partial Amortization Schedule
Develop an amortization schedule for
the loan in Example 6 -12
Note that the Interest portion of the
payment is decreasing while the
Principal portion is increasing.
48
Mortgage Loans
Used to buy real estate
Often the largest financial transaction in
a person’s life
– Mortgages are typically amortized loans,
compounded monthly over 30 years
Early years most of payment is interest
Later on principal is reduced quickly
49
Concept Connection 6-13
Interest Content of Early Loan Payment
Calculate interest in the first payment on a 30-year,
$100,000 mortgage at 6%, compounded monthly.
Solution:
n= 30 years x 12 months/year = 360
k=6%/12 months/year = .5%
PVA= PMT [PVFAk,n ]
$100,000 = PMT [PVFA.5,360 ]
$100,000 = PMT [166.792 ]
PMT = $599.55
First month’s interest = $100,000 x .005 = $500
leaving $99.55 to reduce principal.
First payment is 83.4%
50
Concept Connection 6-13
Interest Content of Early Loan Payment
Next, solve for the monthly payment
PVA= PMT [PVFAk,n ]
$100,000 = PMT [PVFA.5,360 ]
$100,000 = PMT [166.792 ]
PMT = $599.55
The first month’s interest is .5% of $100,000
$100,000 x .005 = $500
The $500 of the first payment goes to interest, leaving $99.55
to reduce principal. The first payment is 83.4%
Concept Connection 6-13
Interest Content of Early Loan Payment
Mortgage Loans
Implications of mortgage payment pattern
– Early mortgage payments provide a large tax
savings, reducing the effective cost of borrowing
– Halfway through a mortgage’s life, half of the loan is
not yet paid off
Long-term loans result in large total interest amounts
over the life of the loan
Adjustable rate mortgage (ARM)
53
The Annuity Due
Payments occur at beginning of periods
The future value of an annuity due
– Each PMT earns interest one period longer
– Formulas adjusted by multiplying by(1+k)
– FVAdn = PMT [FVFAk,n](1+k)
– PVAdn = PMT [PVFAk,n](1+k)
54
Figure 6-7 Future Value of a
Three-Period Annuity Due
55
Concept Connection Example 6-17
Annuity Due
Baxter Corp started 10 years of $50,000 quarterly
sinking fund deposits today at 8% compounded
quarterly. What will the fund be worth in 10 years?
Solution:
k = 8%/4 = 2%
n = 10 years x 4 quarters/year x 40 quarters
FVAdn = PMT [FVFAk,n](1+k)
FVAd40 = $50,000[FVFA2,40](1+.02)
FVAd40 = 60.4020 from Appendix A (Table A-3).
FVAd40 = $50,000[60.4020](1.02)
=$3,080,502
Recognizing Types of Annuity Problems
Annuity problems always involve a stream of equal
payments with a transaction at either the end or
the beginning
– End
— future value of an annuity
– Beginning — present value of an annuity
57
Perpetuities
A stream of regular payments goes on forever
– An infinite annuity
Future value of a perpetuity
– Makes no sense because there is no end point
Present value of a perpetuity
– The present value of payments is a diminishing
series
– Results in a very simple formula
PMT
PVp 
k
58
Example 6-18 Perpetuities – Preferred Stock
Longhorn Corp issues a security that pays $5 per quarter
indefinitely. Similar issues earn 8% compounded. How
much can Longhorn sell this security for?
Solution: Longhorn’s security pays a quarterly perpetuity.
It is worth the perpetuity’s present value calculated using
the current quarterly interest rate.
k = .08 / 4 = .02
PVP = PMT / k = $5.00/.02 = $250
Continuous Compounding
Compounding periods can be any length
– As the time periods become infinitesimally
short, interest is compounded continuously
To determine the future value of a
continuously compounded value:
 
FVn  PV e
kn
60
Example 6-20 Continuous Compounding
First Bank is offering 6½% compounded continuously on
savings deposits. If $5,000 is deposited and left for 3½
years, how much will it grow into?
Solution:
 

FVn  PV ekn  $5,000 e
.0653.5
  $5,000 1.2255457  $6,277.29
61
Multipart Problems
Time value problems are often combined
due to the complexity of real situations
– A time line portrayal can be critical to
keeping things straight
62
Concept Connection Example 6-21
Simple Multipart
Exeter Inc. has $75,000 in securities earning 16%
compounded quarterly. The company needs $500,000
in two years.
Management will deposit money monthly at 12%
compounded monthly to be sure of having the cash.
How much should Exeter deposit each month.
Solution: Calculate the future value of the $75,000 and
subtract it from $500,000 to get the contribution
required from the deposit annuity.
Then solve a save up problem (future value of an
annuity) for the payment required to get that amount.
63
Concept Connection Example 6-21
Simple Multipart
Concept Connection Example 6-21
Simple Multipart
Find the future value of $75,000 with Equation 6.4.
FVn = PV [PVFk,n ]
FV8 = $75,000 [FVF4,8]
= $75,000 [1.3686]
= $102,645
Then the savings annuity must provide:
$500,000 - $102,645 = $397,355
Concept Connection Example 6-21
Simple Multipart
Use Equation 6.13 to solve for the required payment.
FVAn = PMT [FVFA k,n ]
$397,355 = PMT [FVFA1,24]
$397,355 = PMT [26.9735]
PMT = $14,731
Uneven Streams and Imbedded
Annuities
Many real problems have uneven cash flows
– These are NOT annuities
For example, determine the present value of the
following stream of cash flows
Must discount each cash flow individually
67
Example 6-23 Present Value of an
Uneven Stream of Payments
Calculate the interest rate at which the present value
of the stream of payments shown below is $500.
$100
$200
$300
We’ll start with a guess of 12% and discount each amount separately
at that rate.
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%.
68
Imbedded Annuities
Sometimes uneven streams cash have
annuities embedded within them
– Use the annuity formula to calculate the
present or future value of that portion of the
problem
69
Present Value of an Uneven Stream
70