# Chapter 4

```Chapter 4
The Time Value
of Money
(Part 2)
Learning Objectives
1. Compute the future value of multiple cash flows.
2. Determine the future value of an annuity.
3. Determine the present value of an annuity.
4. Adjust the annuity formula for present value and future
value for an annuity due, and understand the concept of a
perpetuity.
5. Distinguish between the different types of loan repayments:
discount loans, interest-only loans, and amortized loans.
6. Build and analyze amortization schedules.
7. Calculate waiting time and interest rates for an annuity.
4-2
4.1 Future Value of Multiple
Payment Streams
• With unequal periodic cash flows, treat each of
the cash flows as a lump sum and calculate its
future value over the relevant number of periods.
• Sum up the individual future values to get the
future value of the multiple payment streams.
4-3
FIGURE 4.1 The time line of a nest egg
4-4
4.1 Future Value of Multiple
Payment Streams (continued)
Example 1: Future Value of an Uneven
Cash Flow Stream
Jim deposits \$3,000 today into an account
that pays 10% per year, and follows it up
with 3 more deposits at the end of each of the
next three years. Each subsequent deposit is
\$2,000 higher than the previous one. How
much money will Jim have accumulated in his
account by the end of three years?
4-5
4.1 Future Value of Multiple
Payment Streams (Example 1
FV
FV
FV
FV
of
of
of
of
Cash
Cash
Cash
Cash
Flow
Flow
Flow
Flow
at
at
at
at
T0
T1
T2
T3
=
=
=
=
FV = PV x (1+r)n
\$3,000 x (1.10)3 = \$3,000
\$5,000 x (1.10)2 = \$5,000
\$7,000 x (1.10)1 = \$7,000
\$9,000 x (1.10)0 = \$9,000
x
x
x
x
1.331= \$3,993.00
1.210 = \$6,050.00
1.100 = \$7,700.00
1.000 = \$9,000.00
Total = \$26,743.00
4-6
4.2 Future Value of an Annuity
Stream
• Annuities are equal, periodic outflows/inflows., e.g. rent, lease,
mortgage, car loan, and retirement annuity payments.
• An annuity stream can begin at the start of each period (annuity
due) as is true of rent and insurance payments or at the end of
each period, (ordinary annuity) as in the case of mortgage and
loan payments.
• The formula for calculating the future value of an annuity stream
is as follows:
FV = PMT * (1+r)n -1
r
• where PMT is the term used for the equal periodic cash flow, r is
the rate of interest, and n is the number of periods involved.
4-7
4.2 Future Value of an Annuity
Stream (continued)
Example 2: Future Value of an
Ordinary Annuity Stream
Jill has been faithfully depositing \$2,000 at
the end of each year for the past 10 years
into an account that pays 8% per year.
How much money will she have
accumulated in the account?
4-8
4.2 Future Value of an Annuity
Stream (continued)
Example 2
Future Value of Payment One = \$2,000 x 1.089 =
\$3,998.01
Future Value of Payment Two = \$2,000 x 1.088 =
\$3,701.86
Future Value of Payment Three = \$2,000 x 1.087 =
\$3,427.65
Future Value of Payment Four = \$2,000 x 1.086 =
\$3,173.75
Future Value of Payment Five = \$2,000 x 1.085 =
\$2,938.66
Future Value of Payment Six = \$2,000 x 1.084 =
\$2,720.98
Future Value of Payment Seven = \$2,000 x 1.083 =
\$2,519.42
Future Value of Payment Eight = \$2,000 x 1.082 =
\$2,332.80
Future Value of Payment Nine = \$2,000 x 1.081 =
\$2,160.00
Future Value of Payment Ten = \$2,000 x 1.080 =
\$2,000.00
Total Value of Account at the end of 10 years
\$28,973.13
4-9
4.2 Future Value of an Annuity
Stream (continued)
FORMULA METHOD
FV = PMT * (1+r)n -1
r
where, PMT = \$2,000; r = 8%; and n=10.
FVIFA [((1.08)10 - 1)/.08] = 14.486562,
FV = \$2000*14.486562  \$28,973.13
USING A FINANCIAL CALCULATOR
N= 10; PMT = -2,000; I = 8; PV=0; CPT FV = 28,973.13
4-10
4.2 Future Value of an Annuity
Stream (continued)
USING AN EXCEL SPREADSHEET
Enter =FV(8%, 10, -2000, 0, 0); Output = \$28,973.13
Rate, Nper, Pmt, PV,Type
Type is 0 for ordinary annuities and 1 for annuities due.
USING FVIFA TABLE (Appendix 3 in text)
Find the FVIFA in the 8% column and the 10 period row:
FVIFA = 14.486
FV = 2000*14.4865 = \$28.973.13
4-11
FIGURE 4.3 Interest and principal
growth with different interest rates for
\$100-annual payments.
4-12
4.3 Present Value of an Annuity
To calculate the value of a series of equal
periodic cash flows at the current point in time,
we can use the following simplified formula:



1
1  
n 
 1  r   

PV  PMT 
r
The last portion of the equation is the
Present Value Interest Factor of an Annuity (PVIFA).
Practical applications include figuring out the nest egg needed
prior to retirement or the lump sum needed for college expenses.
4-13
FIGURE 4.4 Time line of present value of
annuity stream.
4-14
4.3 Present Value of an Annuity
(continued)
Example 3: Present Value of an Annuity.
John wants to make sure that he has saved up
enough money prior to the year in which his
daughter begins college. Based on current
estimates, he figures that college expenses will
amount to \$40,000 per year for 4 years (ignoring
any inflation or tuition increases during the 4 years
of college). How much money will John need to
have accumulated in an account that earns 7% per
year, just prior to the year that his daughter starts
college?
4-15
4.3 Present Value of an Annuity
(continued)
USING THE EQUATION
  1 
1 
n
  1 r   
PV  PMT 
r
1. Calculate the PVIFA value for n=4 and r=7%3.387211.
2. Then, multiply the annuity payment by this factor to get the PV:
PV = \$40,000 x 3.387211 = \$135,488.45
4-16
4.3 Present Value of an Annuity
(continued)
USING A FINANCIAL CALCULATOR:
Set the calculator for an ordinary annuity (END mode) and
then enter:
N= 4; PMT = 40,000; I = 7; FV=0; CPT PV = 135,488.45
USING AN EXCEL SPREADSHEET:
Enter =PV(7%, 4, 40,000, 0, 0); Output = \$135,488.45
Rate, Nper, Pmt, FV, Type
4-17
4.3 Present Value of an Annuity
(continued)
USING A PVIFA TABLE (APPENDIX 4)
For r =7% and n = 4, PVIFA =3.3872
PVA = PMT*PVIFA = 40,000*3.3872
= \$135,488 (Notice the slight rounding error!)
4-18
4.4 Annuity Due and Perpetuity
A cash flow stream such as rent, lease, and
insurance payments, which involves equal
periodic cash flows that begin right away or at
the beginning of each time interval, is known
as an annuity due.
Figure 4.5
4-19
4.4 Annuity Due and Perpetuity
PV annuity due = PV ordinary annuity x (1+r)
FV annuity due = FV ordinary annuity x (1+r)
PV annuity due > PV ordinary annuity
FV annuity due > FV ordinary annuity
Can you see why?
Financial calculator
Mode  BGN for annuity due
Mode  END for an ordinary annuity
Type = 0 or omitted for an ordinary annuity
Type = 1 for an annuity due.
4-20
4.4 Annuity Due and Perpetuity
(continued)
Example 4: Annuity Due versus Ordinary Annuity
Let’s say that you are saving up for retirement and
decide to deposit \$3,000 each year for the next 20
years into an account that pays a rate of interest of
8% per year. By how much will your accumulated nest
egg vary if you make each of the 20 deposits at the
beginning of the year, starting right away, rather than
at the end of each of the next twenty years?
4-21
4.4 Annuity Due and Perpetuity
(continued)
Given information: PMT = -\$3,000; n=20; i= 8%; PV=0;
1 r n  1

FV  PMT  
r
FV ordinary annuity = \$3,000 * [((1.08)20 - 1)/.08]
= \$3,000 * 45.76196
= \$137,285.89
FV of annuity due = FV of ordinary annuity * (1+r)
FV of annuity due = \$137,285.89*(1.08) = \$148,268.76
4-22
4.4 Annuity Due and Perpetuity
(continued)
Perpetuity
A perpetuity is an equal periodic cash flow
stream that will never cease.
The PV of a perpetuity is calculated by
using the following equation:
PMT
PV 
r
4-23
4.4 Annuity Due and Perpetuity
(continued)
Example 5: PV of a Perpetuity
If you are considering the purchase of a
consol that pays \$60 per year forever, and the
rate of interest you want to earn is 10% per
year, how much money should
you pay
for the consol?
r=10%, PMT = \$60, and PV = (\$60/.1) = \$600.
So \$600 is the most you should pay for the
consol.
4-24
4.5 Three Payment Methods
Loan payments can be structured in one of
3 ways:
1) Discount loan
•
Principal and interest is paid in lump sum at end
2) Interest-only loan
•
Periodic interest-only payments, with principal due
at end
3) Amortized loan
•
Equal periodic payments of principal and interest
4-25
4.5 Three Payment Methods
(continued)
Example 6: Discount loan versus Interest-only loan versus
Amortized loans
Roseanne wants to borrow \$40,000 for a period of 5 years.
The lender offers her a choice of three payment structures:
1) Pay all of the interest (10% per year) and principal in one lump
sum at the end of 5 years.
2) Pay interest at the rate of 10% per year for 4 years and then a final
payment of interest and principal at the end of the 5th year.
3) Pay 5 equal payments at the end of each year inclusive of interest
and part of the principal.
Under which of the three options will Roseanne pay the least interest
and why? Calculate the total amount of the payments and the
amount of interest paid under each alternative.
4-26
4.5 Three Payment Methods
(continued)
Option 1: Discount Loan
Since all the interest and the principal is paid at the
end of 5 years, we can use the FV of a lump sum
equation to calculate the payment required:
FV
= PV x (1 + r)n
FV5 = \$40,000 x (1+0.10)5
= \$40,000 x 1.61051
= \$64, 420.40
Interest paid = Total payment - Loan amount
Interest paid = \$64,420.40 - \$40,000 = \$24,420.40
4-27
4.5 Three Payment Methods
(continued)
Option 2: Interest-Only Loan
Annual Interest Payment (Years 1-4)
= \$40,000 x 0.10 = \$4,000
Year 5 payment
= Annual interest payment + Principal payment
= \$4,000 + \$40,000 = \$44,000
Total payment = \$16,000 + \$44,000 = \$60,000
Interest paid = \$60,000 - \$40,000 = \$20,000
4-28
4.5 Three Payment Methods
(continued)
Option 3: Amortized Loan
n = 5; I = 10%; PV=\$40,000; FV = 0;CPT PMT=\$10,551.86
Total payments = 5*\$10,551.8 = \$52,759.31
Interest paid = Total Payments - Loan Amount
= \$52,759.31-\$40,000
Interest paid = \$12,759.31
Loan Type
Discount Loan
Interest-only Loan
Amortized Loan
Total Payment
\$64,420.40
\$60,000.00
\$52,759.31
Interest Paid
\$24,420.40
\$20,000.00
\$12,759.31
4-29
4.6 Amortization Schedules
An amortization schedule is a tabular listing of the
allocation of each loan payment towards interest and
principal reduction.
It can help borrowers and lenders figure out the payoff
balance on an outstanding loan.
Procedure
1) Compute the amount of each equal periodic payment
(PMT).
2) Calculate interest on unpaid balance at the end of each
period, subtract it from the PMT, and reduce the loan
balance by the remaining amount.
3) Continue the process for each payment period, until
you get a zero loan balance.
4-30
4.6 Amortization Schedules
(continued)
Example 7: Loan Amortization Schedule.
Prepare a loan amortization schedule for the
amortized loan option given in Example 6. What
is the loan payoff amount at the end of 2 years?
PV = \$40,000; n=5; i=10%; FV=0;
CPT PMT = \$10,551.89
4-31
4.6 Amortization Schedules
(continued)
Year
Beg. Bal
Payment Interest
Prin. Red End. Bal
1
40,000.0 10,551.8 4,000.0
0
9
0
33,448.1
6,551.89 1
2
33,448.1 10,551.8 3,344.8
1
9
1
26,241.0
7,207.08 3
3
26,241.0 10,551.8 2,264.1
3
9
0
18,313.2
7,927.79 4
4
18,313.2 10,551.8 1,831.3
4
9
2
8,720.57 9,592.67
5
10,551.8
9,592.67 9
959.27
9,592.67
0
The loan payoff amount at the end of 2 years is \$26,241.03
4-32
4.7 Waiting Time and Interest Rates
for Annuities
Problems involving annuities typically have 4 variables: i.e.
PV or FV, PMT, r, n.
If any 3 of the 4 variables are given, we can easily solve for
the fourth one.
This section deals with the procedure of solving problems
where either n or r is not given.
For example:
– Finding out how many deposits (n) it would take to reach a
retirement or investment goal
– Figuring out the rate of return (r) required to reach a
retirement goal, given fixed monthly deposits
4-33
4.7 Waiting Time and Interest Rates
for Annuities (continued)
Example 8: Solving for the Number of
Annuities Involved
Martha wants to save up \$100,000 as soon as
possible so that she can use it as a down
payment on her dream house. She figures
that she can easily set aside \$8,000 per year
and earn 8% annually on her deposits. How
many years will Martha have to wait before
she can buy that dream house?
4-34
4.7 Waiting Time and Interest Rates
for Annuities (continued)
USING A FINANCIAL CALCULATOR
INPUT
?
8.0
0
-8000
100000
TVM KEYS
N
I/Y PV
PMT
FV
Compute
9.00647
USING AN EXCEL SPREADSHEET
Using the “=NPER” function we enter the following:
Rate = 8%; Pmt = -8000; PV = 0; FV = 100000;
Type = 0 or omitted;
=NPER(8%,-8000,0,100000,0).
The cell displays 9.006467.
4-35
4.8 Solving a Lottery Problem
In the case of lottery winnings, there are
two choices:
1) Annual lottery payment for fixed number of
years
2) Lump sum payout
How do we make an informed judgment?
We need to figure out the implied rate of
return of both options using TVM functions.
4-36
4.8 Solving a Lottery Problem
(continued)
Example 9: Calculating an Implied
Rate of Return, Given an Annuity
Let’s say that you have just won the state
lottery. The authorities have given you a
choice of either taking a lump sum of
\$26,000,000 or a 30-year annuity of
\$1,500,000. Both payments are assumed
to be after-tax. What will you do?
4-37
4.8 Solving a Lottery Problem
(continued)
Using the TVM keys of a financial calculator, enter:
PV=26,000,000; FV=0; N=30; PMT = -\$1,625,000;
CPT I = 4.65283%
4.65283% = rate of interest used to determine the
30-year annuity of \$1,625,000 versus the
\$26,000,000 lump sum payout.
Choice: If you can earn an annual after-tax
rate of return higher than 4.65% over the next
30 years, go with the lump sum.
Otherwise, take the annuity option.
4-38
4.9 Ten Important Points about the
TVM Equation
1. Amounts of money can be added or
subtracted only if they are at the same point
in time.
2. The timing and the amount of the cash flow
are what matters.
3. It is very helpful to lay out the timing and
amount of the cash flow with a time line.
4. Present value calculations discount all future
cash flow back to current time.
5. Future value calculations value cash flows at
a single point in time in the future.
4-39
4.9 Ten Important Points about the
TVM Equation (continued)
6. An annuity is a series of equal cash payments at regular
intervals across time.
7. The time value of money equation has four variables
but only one basic equation, and so you must know
three of the four variables before you can solve for the
missing or unknown variable.
8. There are three basic methods to solve for an unknown
time value of money variable:
(1) Using equations and calculating the answer;
(2) Using the TVM keys on a calculator;
(3) Using financial functions from a spreadsheet.
4-40
4.9 Ten Important Points about the
TVM Equation (continued)
9. There are 3 basic ways to repay a loan:
(1) Discount loans
(2) Interest-only loans
(3) Amortized loans
10. Despite the seemingly accurate answers from the time
value of money equation, in many situations not all the
important data can be classified into the variables of
present value, time, interest rate, payment, or future
value.
4-41
Problem 1
Present Value of an Annuity Due.
Julie has just been accepted into Harvard and
her father is debating whether he should make
monthly lease payments of \$5,000 at the
beginning of each month on her flashy
apartment or to prepay the rent with a one-time
payment of \$56,662. If Julie’s father earns 1%
per month on his savings, should he pay by
month or take the discount by making the
single annual payment?
4-42
P/Y = 12; C/Y = 12; MODE = BGN
INPUT
12
-56,662
TVM KEYS
N
I/Y
PV
OUTPUT
12.70%
5,000
PMT
0
FV
Monthly rate = 12.7%/12 = 1.0583%
If he can get 1% interest per month then his annual rate is 12%
and he can generate \$4,984.51 per month with the \$56,662 it
would take to pay off the rent. He is ahead \$15.49 per month by
making the one-time payment.
INPUT
TVM KEYS
OUTPUT
12
N
12
I/Y
-56,662
PV
PMT
4,984.51
0
FV
4-43
Problem 2
Future Value of Uneven Cash Flows. If
Mary deposits \$4000 a year for three
years, starting a year from today, followed
by 3 annual deposits of \$5000 into an
account that earns 8% per year, how much
money will she have accumulated in her
account at the end of 10 years?
4-44
Future Value in Year 10 = \$4000*(1.08)9 +
\$4000*(1.08)8 + \$4000*(1.08)7 + \$5000*(1.08)6 +
\$5000*(1.08)5 + \$5000*(1.08)4
=\$4000*1.999+\$4000*1.8509+
\$4000*1.7138+\$5000*1.5868+
\$5000*1.4693+\$5000*1.3605
=\$7,996+\$7,403.6+\$6,855.2+
\$7,934+ \$7,346.5+6,802.5
=\$44,337.8
4-45
Problem 3
Present Value of Uneven Cash Flows:
Jane Bryant has just purchased some
equipment for her beauty salon. She
plans to pay the following amounts at the
end of the next five years: \$8,250;
\$8,500; \$8,750; \$9,000; and \$10,500. If
she uses a discount rate of 10 percent,
what is the cost of the equipment that she
purchased today?
4-46
\$8,250 \$8,500 \$8,750 \$9,000 \$10,500
PV 




2
3
4
(1.10) (1.10) (1.10) (1.10)
(1.10)5
 \$7,500  \$7,024.79  \$6,574  \$6,147.12  \$6,519.67
 \$33,765.58
4-47
Problem 4
Computing an Annuity Payment
The Corner Bar & Grill is in the process of taking a five-year loan of
\$50,000 with First Community Bank. The bank offers the
restaurant owner his choice of three payment options:
1)
Pay all of the interest (8% per year) and principal in one lump sum
at the end of 5 years;
2)
Pay interest at the rate of 8% per year for 4 years and then a final
payment of interest and principal at the end of the 5th year;
3)
Pay 5 equal payments at the end of each year inclusive of interest
and part of the principal.
Under which of the three options will the owner pay the least
interest and why?
4-48
Under Option 1: Principal and Interest Due at end
Payment at the end of year 5 = FVn = PV x (1 + r)n
FV5 = \$50,000 x (1+0.08)5
= \$50,000 x 1.46933
= \$73,466.5
Interest paid = Total payment - Loan amount
Interest paid = \$73,466.5 - \$50,000 = \$23,466.50
4-49
Problem 4 (ANSWER continued)
Under Option 2: Interest-only Loan
Annual Interest Payment (Years 1-4)
= \$50,000 x 0.08 = \$4,000
Year 5 payment = Annual interest payment +
Principal payment
= \$4,000 + \$50,000 = \$54,000
Total payment = \$16,000 + \$54,000 = \$70,000
Interest paid = \$20,000
4-50
Problem 4 (ANSWER continued)
Under Option 3: Amortized Loan
To calculate the annual payment of principal and
interest, we can use the PV of an ordinary annuity
equation and solve for the PMT value using n = 5; I =
8%; PV=\$50,000; and FV = 0.
PMT  \$12,522.82
Total payments = 5*\$12,522.82 = \$62,614.11
Interest paid = Total Payments - Loan Amount
= \$62,614.11-\$50,000
Interest paid = \$12,614.11
4-51
Problem 4 (ANSWER continued)
Comparison of total payments and interest paid under
each method:
Loan Type
Total Payment
Discount Loan
\$73,466.50
Interest-only Loan \$70,000.00
Amortized Loan
\$62,614.11
Interest Paid
\$23,466.50
\$20,000.00
\$12,614.11
So, the amortized loan is the one with the lowest
interest expense, since it requires a higher annual
payment, part of which reduces the unpaid balance
on the loan and thus results in less interest being
charged over the 5-year term.
4-52
Problem 5
Loan amortization.
Let’s say that the restaurant owner in
Problem 4 above decides to go with the
amortized loan option, and after having
paid 2 payments decides to pay off the
balance. Using an amortization schedule,
calculate his payoff amount.
Amount of loan = \$50,000; Interest rate = 8%;
Term = 5 years; Annual payment = \$12,522.82
4-53
AMORTIZATION SCHEDULE
Year Beg. Bal. PaymentInterest Prin. Red.End Bal.
1 50,000.00 12,522.82 4,000.00
8,522.82
2 41,477.18 12,522.82 3,318.17
9,204.65
3 32,272.53 12,522.82 2,581.80
9,941.02
4 22,331.51 12,522.82 1,786.52 10,736.30
5 11,595.21 12,522.82
927.62 11,595.21
41,477.18
32,272.53
22,331.51
11,595.21
0
The loan payoff amount at the end of 2 years is \$32,272.53
4-54
FIGURE 4.2 The time line of a \$1,000
per year nest egg