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Chapter 7

The Time Value of Money

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Annuities - Future Sum

A. An annuity is a series of equal payments or receipts that occur at evenly spaced intervals. Leases and rental payments are examples.  The payments or receipts occur at the end of each period for an ordinary annuity   while they occur at the beginning of each period for an annuity due.

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FV Time Lines

Ordinary annuity

Year Payment - $100 100 100

0 1 2 3

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FV Time Lines

Annuity due

Year Payment $100 100 100 -

0 1 2 3

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Future Value of an Ordinary Annuity Illustrated

A. Future Value of an Ordinary Annuity

1. FVoa = PMT [((1 + i)n - 1) / i]

a) FVoa = Future Value of an Ordinary Annuity

b) PMT = Amount of each payment

c) i = Interest Rate per Period

d) n = Number of Periods

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Future Value of an Ordinary Annuity Illustrated

PV = 0PMT = -100I = 5N = 3FV = ?

= 315.25

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Future Value of an Annuity Due Illustrated

A. Future Value of an Annuity Due

1. FVad = FVoa (1+i)

a) FVad = Future Value of an Annuity Due

b) FVoa = Future Value of an Ordinary Annuity

c) i = Interest Rate per Period

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Greater Terminal Values

Higher interest rates

Longer time periods

Result in greater terminal values

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Greater Terminal Values

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Present Value of an Annuity

The present value of an annuity brings a series of payments in the future back to the present.

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Present Value of an Ordinary Annuity Illustrated

1. PVoa = PMT [(1 - (1 / (1 + i)n)) / i]

a) PVoa = Present Value of an Ordinary Annuity

b) PMT = Amount of each payment

c) i = Interest Rate per Period

d) n = Number of Periods

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Present Value of an Ordinary Annuity Illustrated

FV = 0PMT = 100I = 6N = 3PV = ?

= -267.30

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Present Value of anAnnuity Due

C. Present Value of an Annuity Due

1. PVad = PVoa (1+i)

a) PV-ad = Present Value of an Annuity Due

b) PV-oa = Present Value of an Ordinary Annuity

c) i = Interest Rate per Period

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Annuities - Present Value

Higher interest rates result in lower present values.

But longer time periods increases the present value (because more payments are received).

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Annuities - Present Value

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Illustration 1

You deposit $1,000 in an account at the end of each year for twenty years. What is the total amount in the account if you earn 6 percent annually?

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Future Value of an Ordinary Annuity

The unknown: FVThe givens:

–PV = 0

–PMT = -1,000

–N = 20

–I = 6

The answer:$36,786

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Interpretation

For an annual cash payment of $1,000, you will have $36,786 after twenty years.

Of the $36,786

–$20,000 is the total cash outflow

–$16,786 is the earned interest.

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Illustration 2

What is the present value of (or required cash outflow to purchase) an ordinary annuity of $1,000 for twenty years, if the rate of interest is 6 percent?

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Present Value of an Annuity

The unknown: PVThe givens:

–FV = 0

–PMT = 1,000

–N = 20

–I = 6

The answer:$11,470

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Interpretation

For a present payment of $11,470, the individual will annually receive $1,000 for the next twenty years.

The $11,470 is an immediate cash outflow.

The $1,000 annual payment to be received is a cash inflow.

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Illustration 6

If an investment pays $50 a year for 10 years and repays $1,000 after 10 years, what is this investment worth today if you can earn 6 percent? (Like a bond)

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Determination of Present Value

The unknown: PVThe givens:

–FV = 1,000

–PMT = 50

–I = 6

–N = 10

The answer: $926

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Interpretation

If you collect $50 a year for 10 years and receive $1,000 after 10 years, those cash inflows are currently worth $926 at 6 percent.

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Illustration 7

Time value is used to determine a loan repayment schedule, such as a mortgage.

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Loan Repayment Schedule

Amount borrowed (PV) = $80,000Interest rate (I) = 8%Term of the loan (N) = 25 yearsNo future value since loan is repaidAmount of the annual payment =

$7,494.30

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Loan Repayment Schedule

Principal Balance

Pmnt Interest Repayment Owed

1 $6,400.00 $1,094.15 $78,905.85

2 6,312.47 1,181.68 77,724.17

. . . .

. . . .

. . . .

25 555.13 6,939.17 .00

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Illustration 8

You have $170,000 and spend $36,000 a year. If you earn 8% annually, how long will your funds last?

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Determination of Number of Years

The unknown: NThe givens:

–PV = 170,000

–I = 8

–FV = 0

–PMT = -36,000

The answer: 6.2 years

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Interpretation

If you have $170,000 and earn 8 percent annually, you can spend $36,000 per year for approximately 6 years and 2 months.