Chapte
8
Slides Developed by:
Terry Fegarty
Seneca College
Time Value
of Money
Chapter 8 – Outline (1)
• The Time Value of Money
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
Time Value Problems
Amount Problems—Future Value
Other Issues
Financial Calculators
Spreadsheet Solutions
The Present Value of an Amount
Finding the Interest Rate
Finding the Number of Periods
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2
Chapter 8 – Outline (2)
• Annuities
 The Future Value of an Annuity
 The Future Value of an Annuity—Developing a
Formula
 The Future Value of an Annuity—Solving Problems
 The Sinking Fund Problem
 Compound Interest and Non-Annual Compounding
 The Effective Annual Rate
 The Present Value of an Annuity—Developing a
Formula
 The Present Value of an Annuity—Solving Problems
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3
Chapter 8 – Outline (3)






Amortized Loans
Loan Amortization Schedules
Mortgage Loans
The Annuity Due
Perpetuities
Continuous Compounding
• Multipart Problems
 Uneven Streams
 Imbedded Annuities
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4
The Time Value of Money
• $100 in your hand today is worth more
than $100 in one year
 Money earns interest
Example
• The higher the interest, the faster your money
grows
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.
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5
The Time Value of Money
Example
• 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
 In our example, what is the present value?
$952.38
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6
The Time Value of Money
• Future Value
Example
 The amount a present sum will grow into at a
certain interest rate over a specified period of
time
 In our example,
• The
• The
• The
• The
present sum (value) is $952.38
interest rate is 5%
time is 1 year
future value is $1,000
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7
Time Value Problems
• Time value deals with four different types
of problems
 Amount— a single amount that grows at
interest over time
• Future value
• Present value
 Annuity— a stream of equal payments that
grow at interest over time
• Future value
• Present value
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8
Time Value Problems
• 4 methods to solve time value problems




Use
Use
Use
Use
formulas
financial tables
financial calculator
financial functions in spreadsheet
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9
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
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10
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
Example
Example
• Can then use a more convenient formula
• 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
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11
The Future Value Factor
for k and n
FVFk,n = (1+k)n
Table 8-1:
Example
6%
1.3382
1.3382
5
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12
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
• Lost investment income is an opportunity cost
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13
Financial Calculators
• 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
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14
Financial Calculators
Basic Calculator Keys
N
Number of time periods
I/Y
Interest rate (%)
PV
Present Value
FV
Future Value
PMT
Payment
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15
Financial Calculators
Example
Q: What is the present value of $5,000 received in one year if
interest rates are 6%?
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
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16
Spreadsheet Solutions
• Time value problems can be solved on a
spreadsheet such as Microsoft® Excel®
• Click on the fx (function) button
• Select for the category Financial
• Select the function for the unknown variable
• For example, to solve for present value:
 Use PV(k, n, PMT, FV)
 Interest rate (k) is entered as a decimal, not a
percentage
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17
Spreadsheet Solutions
• To solve for:






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.
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)
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 (-)
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The Present Value of an Amount
F Vn  P V 1 + k 
n
S o lv e fo r P V

1
P V = F Vn 
n
1

k
 




In te re s t F a c to r
• Either equation can be used to solve any
amount problems
F V Fk ,n 
1
P V Fk ,n
Solving for k or n involves
searching a table.
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Example
Spreadsheet Solution—PV of Amount
Click on the fx (function) button
Select for the category Financial
In the insert function box, select PV
You will see PV(rate,nper,pmt,fv,type)
Press OK
Select the function
for the unknown
Fill in rate = B4, nper = C1, variable, place the
known variables in
pmt = 0, fv = C2
the proper order
within the
 Enter OK
parentheses and






input 0, for the
unknown variable.
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Example
Spreadsheet Solution—PV of Amount
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21
Example 8.3:
Finding the Interest
Rate
Q: What interest rate will grow $850 into $983.96 in three
years?
A:
Example
Calculator
N
3
PV
-850
FV
983.96
PMT
0
I/Y
5.0
Financial Table
Interest factor is
850.00 / 983.96 = 0.8639
Look up in Table A-2 with
n=3
Interest = 5%
Answer
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22
Spreadsheet Solution
Example
Example 8.3:
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Example
Example 8.3:
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





Spreadsheet Solution
Click on the fx (function) button
Select for the category Financial
In the insert function box, select RATE
You will see RATE(n, PMT, PV, FV)
Press OK
Fill in n = E1, PMT = 0, PV = -B2, FV = E2
Enter OK
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24
Example 8.4:
Finding the Number of
Periods
Q: How long does it take money invested at 14% to double?
Example
A: The future value is twice the present value. If the present
value is $1, the future value is $2
Calculator
PV
-1
FV
2
Interest factor is
2/1=2
Look up in Table A-2 with
k = 14%
n = 5 – 6 years
PMT 0
I/Y
14
N
5.29
Financial Table
Answer
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Example 8.4:
Example




Finding the Number of
Periods
Click on the fx (function) button
Select for the category Financial
In the insert function box, select NPER
You will see NPER(k, PMT, PV, FV)
 Enter OK
 Fill in the cell references for
• k (= .14)
• PMT = 0
• PV (= -1)
• FV = 2
 Enter OK
 NPER = 5.29
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Spreadsheet Solution
Example
Example 8.4:
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27
Annuities
• Annuity
 A finite series of equal payments separated
by equal time intervals
• Ordinary annuity
• Payments occur at the end of the time periods
• Monthly lease, pension, and car payments are annuities
• Annuity due
• Payments occur at the beginning of the time periods
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Figures 8.1 and 8.2:
Ordinary Annuity
and Annuity Due
Ordinary Annuity
Annuity Due
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Figure 8.3:
Timeline Portrayal of an
Ordinary Annuity
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Future Value of an Annuity
• Future value of an annuity
 The sum, at its end, of all payments and all
interest if each payment is deposited when
received
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Figure 8.4:
FV of a Three-Year
Ordinary Annuity
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The Future Value of an Annuity—
Developing a Formula
• Thus, for a 3-year annuity, the formula is
F V A = P M T  1 + k   P M T 1 + k   P M T 1 + k 
0
1
2
G e n e ra lizin g th e E xp re ssio n :
F V A n = P M T  1 + k   P M T 1 + k   P M T 1 + k  
0
1
2
 P M T 1 + k 
n -1
w h ich ca n b e w ritte n m o re co n v e n ie n tly a s:
n
FVA n 
 P M T 1 + k 
ni
i= 1
F a cto rin g P M T o u tsid e th e su m m a tio n , w e o b ta in :
n
FVA n  PM T

1 + k 
i= 1
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ni
FVFAk,n
33
The Future Value of an Annuity—
Solving Problems
• There are four variables in the future
value of an annuity equation




The
The
The
The
future value of the annuity itself
payment
interest rate
number of periods
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34
Example
Example 8.5:
The Future Value of an
Annuity
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?
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Example
Example 8.5:
The Future Value of an
Annuity
A: Use the future value of an annuity equation:
FVAn = PMT[FVFAk,n]
In Table A-3, look up the interest factor at an n of 10 and
a k of 7
Interest factor = 13.8164
Future value = $100,000[13.8164] = $1,381,640
N
I/Y
PMT
PV
FV
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10
7
100000
0
1,381,645
Answer
36
Example
Example 8.5:






Spreadsheet Solution
Click on the fx (function) button
Select for the category Financial
In the insert function box, select FV
You will see FV(k, n, PMT, PV)
Enter OK
Fill in the cell references for
•
•
•
•
k (=.07)
n (= 10)
PMT (=100000)
PV (=0)
 Enter OK
 FV = 1,381,644.80
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Spreadsheet Solution
Example
Example 8.5:
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38
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
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39
Example
Example 8.6:
The Sinking Fund
Problem
Q: The Greenville Company issued bonds totaling $15 million for 30
years. A sinking fund must be maintained after 10 years, which will
retire the bonds at maturity. The estimated yield on deposited funds
will be 6%.
How much should Greenville deposit each year to be able to retire the
bonds?
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Example 8.6:
The Sinking Fund
Problem
Example
A: The time period of the annuity is the last 20 years of the bond
issue’s life. On your calculator:
N
I/Y
FV
PV
20
6
15000000
0
PMT 407,768.35
Answer
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41
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
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The Effective Annual Rate
• Effective annual rate (EAR)
 The annually compounded rate that pays the
same interest as a lower rate compounded
more frequently
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43
The Effective Annual Rate—
Example
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.
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The Effective Annual Rate
• EAR can be calculated for any
compounding period using the following
formula:
EAR 

k n o m in a l 

1 

m


m
- 1
Effect of more frequent compounding is
greater at higher interest rates
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45
Impact of Compounding
Frequency
$1,000 Invested at
10% Nominal Rate for One Year
$1,106
$1,105
$1,104
$1,103
$1,102
$1,101
$1,100
$1,099
$1,098
$1,097
Annual
SemiAnnual
Quarterly Monthly
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Daily
46
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
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47
Example
Example 8.7: The
Effective Annual
Rate
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. On your calculator:
N
30
I/Y
1
FV
15000
PV
0
PMT
431.22
Answer
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48
The Present Value of an Annuity—
Developing a Formula
• Present value of an annuity
 Sum of all of the annuity’s payments
PVA =
PMT
1 + k 

PMT
1 + k 
2

PMT
1 + k 
3
w h ic h c a n a ls o b e w ritte n a s :
P V A = P M T 1 + k 
1
 P M T 1 + k 
2
 P M T 1 + k 
3
G e n e ra lize d fo r a n y n u m b e r o f p e rio d s :
P V A = P M T 1 + k 
1
 P M T 1 + k 
2

 P M T 1 + k 
n
F a c to rin g P M T a n d u s in g s u m m a tio n , w e o b ta in :
 n
i 
P V A  P M T   1 + k  
 i= 1

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PVFAk,n
49
Figure 8.6:
PV of a Three-Payment
Ordinary Annuity
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50
The Present Value of an Annuity—
Solving Problems
• There are four variables in the present
value of an annuity equation




The
The
The
The
present value of the annuity itself
payment
interest rate
number of periods
• Problem usually presents 3 of the 4 variables
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51
Example
Example 8.9:
The Present Value of
an Annuity
Q: The Shipson Company has just sold a large machine on an
installment contract. The contract calls for payments of $5,000 every
six months (semiannually) for 10 years. Shipson would like its cash
now and asks its bank to it the present (discounted) value. The bank is
willing to discount the contract at 14% compounded semiannually.
How much should Shipson receive?
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52
Example 8.9:
The Present Value of
an Annuity
Example
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 Table A4.
Look up n = 20 and k = 7%.
PVA = $5,000[10.594]
= $52,970
Answer
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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
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54
Example
Example 8.10:
Amortized Loans
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. On your calculator:
N
48
I/Y
1.5
PV
10000
FV
0
PMT 293.75
This can also be calculated
using the PVA formula of
PVA = PMT[PVFAk, n]
Answer
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n = 48 and k = 1.5%
$10,000 = PMT[34.0426]
= $293.75.
55
Example 8.11:
Amortized Loans
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. On your calculator:
N
36
I/Y
1
FV
0
PMT
500
PV
15,053.75
This can also be calculated
using the PVA Table A4.
Look up n = 36 and k = 1%.
PVA = $500[30.1075]
= $15,053.75
Answer
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56
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)
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57
Example 8.11:
Loan Amortization
Schedule
Example
Q: Develop an amortization schedule for the loan demonstrated
in Example 8.11
Note that the Interest portion
of the payment is decreasing
while the Principal portion is
increasing.
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58
Mortgage Loans
• Mortgage loans (AKA: mortgages)
 Loans used to buy real estate
• Often the largest single financial
transaction in a 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
 Halfway through a mortgage’s life half of the
loan has not been paid off
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59
Mortgage Loans
• Implications of mortgage payment pattern
 Long-term loans like mortgages result in large total
interest amounts over the life of the loan
• At 6% interest, compounded monthly, over 25 years,
borrower pays almost the amount of the loan just in
interest!
• Canadian banks compound semi-annually, thus lowering
interest charges
 Early mortgage payments and more frequent
mortgage payments provide a large interest saving
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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.
Monthly payment
$1,015.65
X # of payments
360
N
360
I/Y
0.5979
Total payments
$365,634
FV
0
- Original Loan
$150,000
PV
150000
Total Interest
$215,634
Interest / Principal
143.76%
PMT
1,015.65
Answer
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61
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
F V A d n = P M T + P M T 1 + k  

 P M T 1 + k 
n -1
 1  k 

w h ich w ritte n w ith th e in te re st fa cto r b e co m e s:
F V A d n  P M T  F V F A k ,n   1  k 
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62
The Future Value of a
Three-Period Annuity Due
Figure 8.7:
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63
Example
Example 8.12:
The Annuity Due
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
50000
PV
0
FV
3,020,099 x 1.02 = 3,080,501 Answer
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64
The Annuity Due
• The present value of an annuity due
 Formula
P V A d  P M T  P V F A k ,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
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65
Perpetuities
• A perpetuity is a stream of regular payments
that 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
 A diminishing series of numbers
• Each payment’s present value is smaller than the one before
P Vp 
PMT
k
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66
Example 8.13:
Perpetuities
Example
Q: The Longhorn Corporation issues a security that promises to
pay its holder $5 per quarter indefinitely. Investors can earn
8% compounded quarterly on their money. How much can
Longhorn sell this security for?
A: Convert the k to a quarterly k and plug the values into the
equation.
P Vp 
PMT
k

$5
0 .0 2
 $250
You may also work this by inputting a
large n into your calculator (to
simulate infinity), as shown below.
N
I/Y
PMT
FV
PV
© 2006 by Nelson, a division of Thomson Canada Limited
999
2
5
0
250
Answer
67
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:

F Vn  P V e
kn

Where k = nominal rate, n = number of years, e =
2.71828
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68
Continuous
Compounding
Example
Example 8.15:
Q: The First National Bank of Cardston is offering continuously
compounded interest on savings deposits.
If you deposit $5,000 at 6½% compounded continuously and leave it
in the bank for 3½ years, how much will you have?
What is the equivalent annual rate (EAR) of 12% compounded
continuously?
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69
Continuous
Compounding
Example 8.15:
A: To determine the future value of $5,000, plug the appropriate values
into the equation
FVn
=
P V e kn 
=
Example
FV 3.5= $5,000(2.71828)
 .065  3.5  =
$6.277.29
To determine the EAR of 12% compounded continuously, find the
future value of $100 compounded continuously in one year, then
calculate the annual return
F V 1 = $ 1 0 0  e  .1 2 1  
EAR=
=
$ 1 0 0  2 .7 1 8 2 8  .1 2  
$112.75-$100
= 12.75%
$100
© 2006 by Nelson, a division of Thomson Canada Limited
=
$112.75
NOTE: Some advanced
calculators have a function
to solve for the EAR with
continuous compounding
70
Table 8.5:
Time Value Formulas
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71
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
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72
Example
Example 8.16:
Multipart Problems
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.
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 account will pay 12% compounded
monthly.
How much should Exeter deposit with the bank each month ?
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73
Example
Example 8.16:
Multipart Problems
A: Two things are happening in this problem
 Exeter is saving money every month (an annuity) and
 The money invested in securities (an amount) is growing
independently at interest
We have three steps
 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 solve a future value of an annuity problem for the
payment
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74
Example 8.16:
Multipart Problems
Example
To find the future value of
the $75,000…
N
I/Y
PMT
FV
PV
To find the savings
annuity value
N
I/Y
PV
FV
PMT
24
1
0
8
4
0
75000
102,645 Answer
$500,000 - $102,645
= $397,355
397355
14,731 Answer
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Uneven Streams
• 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
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76
Example 8.18:
Uneven Streams
Q: Calculate the interest rate at which the present value of the stream of
payments shown below is $500.
Example
$100
$200
$300
A: Start with a guess of 12% and discount each amount separately at
that rate.
P V = F V1  P V Fk ,1   F V 2  P V Fk ,2   F V 3  P V Fk ,3 
 $ 1 0 0  P V F1 2 ,1   $ 2 0 0  P V F1 2 ,2   $ 3 0 0  P V F1 2 ,3 
 $ 1 0 0  .8 9 2 9   $ 2 0 0  .7 9 7 2   $ 3 0 0  .7 1 1 8 
 $ 4 6 2 .2 7
This value is too low; select a lower interest rate.
Using 11% gives us $471.77. The answer is
between 8% and 9%.
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Imbedded Annuities
• Sometimes uneven streams of cash flows
will have annuities imbedded within
them
 We can use the annuity formula to calculate
the present or future value of that portion of
the problem
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78