The Time Value of Money
Suppose market interest rates are 10%/year.
That means, if you invest 1000 today, in 1 year, you will have 1000(1.10) = 1,100 =
1,000 (your original principal) + 100 interest. We could also ask, what is the present
value (PV) (the value NOW) of 1,100 to be received in 1 year, if market interest rates are
10%/year? Answer: PV = X such that X(1.10) = 1100 => X = 1100/1.10 = 1000.
What is the PV of 1000 in 1 year, if the interest rate is 10%? Answer: X such that X(1.10)
= 1000 => X = 1000/1.10 = 909.09 What would you rather have, 909.09 now or 1000 in
1 year?
Example: If interest rates are 12%/year, what does 1500 grow to in 5 years? In 1 year,
1500 => 1500(1.12) = 1680 = 1500 + 180 of interest. In the second year, 1680 =>
1680(1.12) = 1881.60. Note: 1881.60 –1500 = 381.60 and 180 x 2 = 360 and 381.60 –
360 = 21.60. 21.60 is the interest earned in the second year, on the interest earned in the
first year.
Benjamin Franklin: “Money makes money, and the money that money makes, makes
more money.”
In 5 years, 1500 => 1500(1.12)^5 = 1500(1.7623) = 2643.51. We say, 2643.51 is the
future value (FV) of 1500 in 5 years if interest rates are 12% annually with annual
compounding. In general: FV = C
0
(1+r)
T
where C
0
is the amount invested, r = the annual
rate and T = the number of years it is invested for. Similarly, What is the PV of 2643.51
in 5 years if r = 12%? PV = X such that X(1.12)
5
= 2643.51 => X = 2643.51/(1.12)
5
=
1500.
PV of an amount C
1
in 1 year: PV = C
1
/(1+r)
PV of an amount C
T
in T years is PV = C
T
/(1+r)
T
Suppose we invest C
0
now in return for:
C
1
in 1 year
C
2
in 2 years
…
C
T
in T years
First note: If r is the appropriate annual interest rate, the PV of the future payments is
given by: PV= C
1
/(1+r) + C
2
/(1+r)
2
+ … + C
T
/(1+r)
T
And the Net Present Value (NPV) = C
0
+ C
1
/(1+r) + C
2
/(1+r)
2
+ … + C
T
/(1+r)
T
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 Spring '10
 BOOTH
 Interest Rates, Time Value Of Money

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