Chapter 4
The Time Value of Money
(Part 2)

Learning Objectives
1.
2.
3.
4.

5.
6.
7.
8.
9.

Compute the future value of multiple cash flows.
Determine the future value of an annuity.
Determine the present value of an annuity.
Adjust the annuity equation for present value and future value for an annuity due (not so much) and understand the concept of a perpetuity.
Distinguish between the different types of loan repayments: discount loans, interest-only loans and amortized loans.
Build (not so much) and analyze amortization schedules.
Calculate waiting time and interest rates for an annuity.
Apply the time value of money concepts to evaluate the lottery cash flow choice.
Summarize the ten essential points about the time value of money. Copyright ©2016 Pearson Education, Inc. All rights reserved.

4-2

4.1 Future Value of Multiple Payment
Streams
1. 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.
2. Sum up the individual future values to get the future value of the multiple payment streams. –
–
–

You can add and subtract cash flows if they are in the same time period.
“Cash flows are additive.”
Picture next.

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4-3

Figure 4.1 The time line of a nest egg
This can be 1415, 1972, 2015, 2115, or any other time.

You can add them because they are all at the same time (T3)
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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 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? Note: Remember how you used to hate story problems in elementary school?
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4-5

Draw Cash Flows!

Now
3000

Now+1 Now+2
5000
10%

7000

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Now+3
9000

4-6

4.1 Future Value of Multiple Payment
Streams (Example 1 Answer)

FV = PV x (1+r)n
FV
FV
FV
FV

of of of of Cash Flow at T0 = $3,000
Cash Flow at T1 = $5,000
Cash Flow at T2 = $7,000
Cash Flow at T3 = $9,000
Total = $26,743.00

x x x x (1.10)3
(1.10)2
(1.10)1
(1.10)0

=
=
=
=

$3,000
$5,000
$7,000
$9,000

x x x x 1.331
1.210
1.100
1.000

=
=
=
=

$3,993.00
$6,050.00
$7,700.00
$9,000.00

Why does this increase by $2000 each year?

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4-7

4.1 Future Value of Multiple Payment Streams
(Example 1 Answer) (We skip in class.)

ALTERNATIVE METHOD: Using the Cash Flow (CF) key of the calculator, enter the respective cash flows.
CF0=-$3000;CF1=-$5000;CF2=-$7000;
CF3=-$9000;
Next calculate the NPV using I=10%; NPV=$20,092.41;
Finally, using PV=-$20,092.41; n=3; i=10%;PMT=0;
CPT FV=$26,743.00

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4-8

4.2 Future Value of an Annuity Stream (Same as previous example, but same size payments)
• Annuities are equal, periodic outflows/inflows.
• Rent, lease, mortgage, car loan, alimony, 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 is:
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.

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4-9

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