The Time Value of Money

1. ALAN ANDERSON, Ph.D.
ECI RISK TRAINING
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2. For free problem sets based on this material
along with worked-out solutions, write to
info@ecirisktraining.com. To learn about
training opportunities in finance and risk
management, visit www.ecirisktraining.com
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3. The time value of money is one of the most
fundamental concepts in finance; it is based
on the notion that receiving a sum of money
in the future is less valuable than receiving
that sum today.
This is because a sum received today can be
invested and earn interest.
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4. The four basic time value of
money concepts are:
 future value of a sum
 present value of a sum
 future value of an annuity
 present value of an annuity
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5. If a sum is invested today, it will earn interest
and increase in value over time. The value that
the sum grows to is known as its future value.
Computing the future value of a sum is known
as compounding.
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6. The future value of a sum depends on
the interest rate earned and the time
horizon over which the sum is invested.
This is shown with the following formula:
FVN = PV(1+I)N
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7. where:
FVN = future value of a sum
invested for N periods
I = periodic rate of interest
PV = the present or current
value of the sum invested
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8. Suppose that a sum of $1,000 is invested for
four years at an annual rate of interest of 3%.
What is the future value of this sum?
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9. In this case,
N=4
I=3
PV = $1,000
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10. Using the future value formula,
FVN = PV(1+I)N
FV4 = 1,000(1+.03)4
FV4 = 1,000(1.125509)
FV4 = $1,125.51
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11. The present value of a sum is the amount
that would need to be invested today in
order to be worth that sum in the future.
Computing the present value of a sum is
known as discounting.
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12. The formula for computing the
present value of a sum is:
FVN
PV =
(1 + I ) N
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13. How much must be deposited in a bank
account that pays 5% interest per year in
order to be worth $1,000 in three years?
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14. In this case,
N=3
I=5
FV3 = $1,000
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15. FVN 1, 000
PV = =
(1 + I ) N
(1.05) 3
1, 000
= = $863.84
1.1576
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16. An annuity is a periodic stream of
equally-sized payments.
The two basic types of annuities are:
 ordinary annuity
 annuity due
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17. With an ordinary annuity, the first
payment takes place one period in
the future.
With an annuity due, the first
payment takes place immediately.
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18. The formulas used to compute the
future value and present value of a
sum can be easily extended to the
case of an annuity.
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19. The formula for computing the future
value of an ordinary annuity is:
⎡ (1 + I ) − 1 ⎤ N
FVAN = PMT ⎢ ⎥
⎣ I ⎦
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20. where:
FVAN = future value of an
N-period ordinary annuity
PMT = the value of the
periodic payment
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21. Suppose that a sum of $1,000 is invested at
the end of each of the next four years at an
annual rate of interest of 3%. What is the
future value of this ordinary annuity?
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22. In this case,
N=4
I=3
PMT = $1,000
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23. Using the formula,
⎡ (1 + I ) − 1 ⎤ N
FVAN = PMT ⎢ ⎥
⎣ I ⎦
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24. ⎡ (1 + .03) − 1 ⎤
4
FVA4 = 1,000 ⎢ ⎥ = $4,183.63
⎣ .03 ⎦
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25. The future value of the annuity can
also be obtained by computing the
future value of each term and then
combining the results:
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26. 1,000(1.03)3 + 1,000(1.03)2 +
1,000(1.03)1 + 1,000(1.03)0
= 1,092.73 + 1,060.90 +
1,030.00 + 1,000.00
= $4,183.63
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27. The future value of an annuity due
is computed as follows:
FVAdue = FVAordinary(1+I)
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28. Referring to the previous example, the
future value of an annuity due would be:
4,183.63(1+.03) = $4,309.14
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29. The formula for computing the present
value of an ordinary annuity is:
⎡ 1 ⎤
1−
⎢ (1 + I )N ⎥
PVAN = PMT ⎢ ⎥
⎢ I ⎥
⎢
⎣ ⎥
⎦
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30. where:
PVAN = future value of an
N-period ordinary annuity
PMT = the value of the
periodic payment
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31. How much must be invested today in a bank
account that pays 5% interest per year in order
to generate a stream of payments of $1,000 at
the end of each of the next three years?
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32. In this case,
N=3
I=5
PMT = $1,000
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33. Using the formula,
⎡ 1 ⎤
1−
⎢ (1 + I )N ⎥
PVAN = PMT ⎢ ⎥
⎢ I ⎥
⎢
⎣ ⎥
⎦
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34. ⎡ 1 ⎤
1−
⎢ (1 + .05)3 ⎥
PVA3 = 1, 000 ⎢ ⎥ = $2, 723.25
⎢ .05 ⎥
⎢
⎣ ⎥
⎦
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35. The present value of the annuity can
also be obtained by computing the
present value of each term and then
combining the results:
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36. 1,000(1.05)-3 + 1,000(1.05)-2 + 1,000(1.05)-1
= 863.84 + 907.03 + 952.38
= $2723.25
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37. The present value of an annuity
due is computed as follows:
PVAdue = PVAordinary(1+I)
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38. Referring to the previous example, the
present value of an annuity due would be:
2,723.25(1+.05) = $2,859.41
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