Deriving the formula for PV of finite income streams
In class we proved that the present value of a finite stream of payments can be written
as
PV = F
xn+1
x
−
1−x 1−x
1
where x = 1+i
, i is the period interest rate, F is the period payment, and n is the number
of periods. Of course, this doesn’t do us much good, as we don’t want things in terms of x,
we want them in terms of i.
Plugging in for x gives
!
n+1
1
1+i
F
1
1+i
1
1+i
1−
1
1+i
1
1+i
1
1+i
which looks evil. Multiplying by
F
1−
−
1
1+i
1−
1
1+i
−
1 n+1
1+i
1
1 − 1+i
!
∗
1
1+i
1
1+i
n+1
F
1
(1 + i) ∗ 1+i
1
−
1+i−1
1+i−1
!
and simplifying the denominators
F
1 (1 + i) ∗
−
i
i
1 n+1
1+i
!
Now take the right hand portion of this equation and move the
the denominator
1
1+i
F
−
i
i(1 + i)n+1
1 n+1
1+i
= (1 + i)−(n+1) to
The (1 + i) in the numerator of the right hand portion means we can subtract 1 from the
exponent in the denominator, and this becomes
1
1
−
F
i
i(1 + i)n
Prepared by Nick Sanders, UC Davis Graduate Department of Economics 2006
Pulling
1
i
to the outside
1
1
1−
PV = F
i
(1 + i)n
(1)
We now have a handy formula for expressing a finite payment stream in terms of the
period interest rate i, the period payment F , and the number of periods n. This has also
popped up in class in the following form
c 1 − (1 + i )−2m sa
Pm =
2
isa
when discussing the coupon payment portion of a coupon bond. Here, 2c is the period
payment and isa represents the semi-annual interest rate (or the interest rate divided by
two, a good approximation of the interest rate if i is small enough).
Returning to our first form in equation (1), consider the situation where the stream of
payments is endless (e.g. a consol). As n → ∞, the denominator of the right had side
becomes infinitely large, which means the equation becomes
1
(1 − 0)
PV = F
i
PV =
F
i
(2)
giving us the formula for an infinite payment stream.
2