What is the value of P(Periodic payment)?

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