2.2a. Annuities
Annuities (Future Value Annuities)
When periodic payments are made into an account in order to increase the value of the account, we call this a future value annuity.
When periodic payments are paid from an account (or paid on a loan) in order to decrease the value of the account, we call this a present value annuity.
We will first discuss future value annuities.
For most of us, we aren’t able to put a large sum of money in the bank today. Instead, we save for the future by depositing a smaller amount of money from each paycheck into the bank. This idea is called a
savings annuity. Most retirement plans like 401k plans or IRA plans are examples of savings annuities.
An annuity can be described recursively in a fairly simple way. Recall that basic compound interest follows from the relationship
P
m
=
(
1
+
r
k
)
P
m
−
1
P
m
=
(
1
+
r
k
)
P
m
−
1
For a savings annuity, we simply need to add a deposit, d, to the account with each compounding period:
P
m
=
(
1
+
r
k
)
P
m
−
1
+
d
P
m
=
(
1
+
r
k
)
P
m
−
1
+
d
Taking this equation from recursive form to explicit form is a bit trickier than with compound interest. It will be easiest to see by working with an example rather than working in general.
Suppose we will deposit $100 each month into an account paying 6% interest. We assume that the account is compounded with the same frequency as we make deposits unless stated otherwise. In this example:
r = 0.06 (6%)
k = 12 (12 compounds/deposits per year)
d = $100 (our deposit per month)
Writing out the recursive equation gives
P
m
=
(
1
+
0.06
12
)
P
m
−
1
+
100
=
(
1.005
)
P
m
−
1
+
100
P
m
=
(
1
+
0.06
12
)
P
m
−
1
+
100
=
(
1.005
)
P
m
−
1
+
100
Assuming we start with an empty account, we can begin using this relationship:
P0 = 0
P1 = (1.005)P0 + 100 = 100
P2 = (1.005)P1 + 100 = (1.005)(100) + 100 = 100(1.005) + 100
P3 = (1.005)P2 + 100 = (1.005)(100(1.005) + 100) + 100 = 100(1.005)2 + 100(1.005) + 100
Continuing this pattern, after
m deposits, we’d have saved:
Pm = 100(1.005)m – 1 + 100(1.005)m – 2 + L + 100(1.005) + 100
In other words, after
m months, the first deposit will have earned compound interest for m – 1 months. The second deposit will have earned interest for m – 2 months. Last months deposit would have earned only one month worth of interest. The most recent deposit will have earned no interest yet.
This equation leaves a lot to be desired, though—it doesn’t make calculating the ending balance any easier! To simplify things, multiply both sides of the equation by 1.005:
1.005
Pm = 1.005(100(1.005)m – 1 + 100(1.005)m – 1 + L + 100(1.005) +100)
Distributing on the right side of the equation gives
1.005
Pm = 100(1.005)m + 100(1.005)m – 1 + L + 100(1.005)2 + 100(1.005)
Now we’ll line this up with like terms from our original equation, and subtract each side
1.005
P
m
1.005
P
m
=
100
(
1.005
)
m
100
(
1.005
)
m
+
100
(
1.005
)
m
−
1
100
(
1.005
)
m
−
1
+
L
L
+
100
(
1.005
)
100
(
1.005
)
P
m
P
m
=
100
(
1.005
)
m
−
1
100
(
1.005
)
m
−
1
+
L
L
+
100
(
1.005
)
100
(
1.005
)
+
Almost all the terms cancel on the right hand side when we subtract, leaving
1.005Pm – Pm = 100(1.005)m – 100
Solving for Pm
0.005
P
m
=
100
(
(
1.005
)
m
−
1
)
0.005
P
m
=
100
(
(
1.005
)
m
−
1
)
P
m
=
100
(
(
1.005
)
m
−
1
)
0.005
P
m
=
100
(
(
1.005
)
m
−
1
)
0.005
Replacing
m months with 12N, where N is measured in years, gives
P
N
=
100
(
(
1.005
)
12
N
−
1
)
0.005
P
N
=
100
(
(
1.005
)
12
N
−
1
)
0.005
Recall 0.005 was
r/k and 100 was the deposit d. 12 was k, the number of deposit each year. Generalizing this result, we get the saving annuity formula.
Annuity Formula
P
N
=
d
(
(
1
+
r
k
)
N
k
−
1
)
(
r
k
)
P
N
=
d
(
(
1
+
r
k
)
N
k
−
1
)
(
r
k
)
PN is the balance in the account after N years.
d is the regular deposit (the amount you deposit each year, each month, etc.)
r is the annual interest rate in decimal form.
k is the number of compounding periods in one year.
Insert By Professor P: The above formula actually describes the future value (FV) of an ordinary annuity. I typically use this formula for the Future Value of an ordinary annuity.
F
V
=
P
M
T
(
(
1
+
r
m
)
n
−
1
)
(
r
/
m
)
F
V
=
P
M
T
(
(
1
+
r
m
)
n
−
1
)
(
r
/
m
)
FV= future value of the annuity
PMT= amount of the periodic payment
r= annual interest rate written in decimal form
m=number of compounding periods per year.
n=total number of compounding periods. {n=mt, where t=number of years}
Note: Some texts prefer to replace the rate per period (r/m) with i. The above formula would look like this.
F
V
=
P
M
T
(
(
1
+
i
)
n
−
1
)
(
i
)
F
V
=
P
M
T
(
(
1
+
i
)
n
−
1
)
(
i
)
If the compounding frequency is not explicitly stated, assume there are the same number of
