**The
Geometric Series in
Finance**

© 2007 L.L. Williams

The geometric series is a marvel of mathematics which rules
much
of the natural world. It is in finance, however, that the geometric
series finds perhaps its greatest predictive power. In the 21^{st}
century, our lives are ruled by money. How are mortgages calculated?
How much money will I have at the end of 20 years if I regularly put
away a fixed amount of money and it accrues interest? If you want to
understand money, you need to understand the geometric series. And
the really wonderful thing about the geometric series is, for all its
power in our world, it is one of the few series which lends itself to
a closed form summation. Thanks to a benevolent Creator, the one
series important for us to understand is the one series we can fully
grasp.

**1. Mortgages**

Let's start with a mortgage. You are going to borrow an amount
of
money *P* for the principal. You are going to make a
fixed
monthly payment *m* over a period of *y*
years. You will
pay a fixed annual interest rate *R*, which means
that your
monthly interest is *r = R/*12.

If you wait a month before your first payment, you will owe
interest for that month of *rP *in addition to the
principal *P*.
But when you finally make your first payment, the amount owed,
*P(*1*+r)*, will be reduced by *m*:

After another month, you will now owe interest of *r*[*P(*1*+r)-m*]
on the amount owed above, but you will have made another payment *m*
so that your amount owed is now:

After the third payment,

After *n* payments,

So how big should your monthly payment be? Simple: *m*
is
chosen so that at your final payment, the amount owed is zero. Since
you make 12 payments a year for *y* years, we can put
*n=*12*y*.
Then *P*, *m*, *y*,
and *r *are related by:

As you may have guessed, the term on the right side is the
geometric series. Let us take a moment to consider the series. To
start, let's write the series in a simpler form. For any constant *a*,
the geometric series is:

To get a closed form solution to this sum, a trick is
required. We
multiply both sides of the series by *a*:

Then subtract the first equation from the second. Our trick allows every term on the right sides to cancel out except for the last term in the second equation and the first term in the first equation:

and voila, the series sum *G(a,n)* can be
written very
simply:

Using this expression for the geometric series, the relation
between *P*, *m*, *y*,
and *r *is:

Or, more simply:

Given any three of principal, monthly payment, loan term, and
interest rate, the fourth can be determined. This is the equation
that determines your monthly payment when you borrow an amount *P*
and finance it at an annual rate 12*r *for *y*
years*.*

You can link to a simple mortgage calculator web page which has this equation programmed in Javascript. When you link to the page, click “View Source” from your browser to see this equation doing the calculation for you.

Notice that the power term on the right hand side is usually < 1. Therefore a quick approximation is to write

The interest on the full principal sets the size of the monthly payment, independent of the loan term. As the principal is paid down, more of the payment goes to principal and less to interest, but the monthly payment stays the same.

**2. Investments**

The geometric series also determines how interest-bearing
investments accrue with time. Consider the inverse scenario that we
considered for mortgages: you put away a monthly amount *m*
which accrues interest at an annual rate *R* and
monthly rate
*r=R/*12. Assume you start with an amount *m*
in your
investment account. After a month you have earned *rm*
interest,
and deposited another increment *m*:

After another month, you earn interest of *r*[*m(*1*+r)
+m*] on your balance, and make another deposit *m*:

After *n* months:

Of course, we can write this sum in closed form using the
formula
for *G(a,n)* we derived in the mortgage section:

This equation shows that your investment is proportional to
the
monthly payment. Double the payment yields double the investment at
any given time. While the investment grows linearly with the monthly
payment, it grows exponentially with the rate *r *and
with the
time *n*. The term depending on the rate can be
thought of a
monetary magnification factor *M* which depends on *r
*and
*n*, or more conveniently, on *R=*12*r*
and *n=*12*y:*

M(5%, 5yr) = 68; M(5%, 10yr) = 155; M(5%, 20yr) = 411

M(10%, 5yr) = 77; M(10%, 10yr) = 205; M(10%, 20yr) = 759

So over 5 years you would
pay into your
investment 60*m, *yet the power of compounding
interest would
inflate this to 68*m* at 5%, or 77*m*
at 10%. Over 20 years
you would pay in 240*m*, yet 10% interest would more
than triple
this value to 759*m*. This means that small values of *m*
can work powerfully over long stretches of time. The geometric
magnification over 40 years at 10% would be a staggering 6400, or 13
times the amount payed in. This is how wealthy dynasties are
sustained. Once they start, the power of geometric compounding
ensures they grow exponentially.

Since the geometric magnification factor is independent of the set-aside value, the size of the monthly set-aside is less important than just having geometric compounding at work on as much money as possible.

**3. Exponential Limit**

The geometric growth in the value of an investment can also be properly called exponential, and related to the exponential function. To see this, make use of the simple identity

which is true for *r *<< 1.
Since r = R/12 is indeed
often << 1, the investment value can be approximated:

So, it is mathematically true that geometrically compounding investments grow exponentially.

*4.**Aside: Xeno's
Paradox*

It is difficult to appreciate in this day and age the triumph
represented by the simple formula for *G(a,n)*. An
ancient Greek
named Xeno (or Zeno) came up with a “paradox” that
stumped
thinkers for millenia. Xeno said that you could never get from point
A to point B because first you had to go 1/2 the distance. Then you
had to go 1/2 of the remaining distance, or 1/4. Then half of that,
or 1/8, and so on. Because you could keep taking half steps forever,
reasoned Xeno, you could never get to point B. Of course, Xeon always
made it home at the end of the day, so his argument presented a
paradox.

Resolving Xeno's paradox requires the assistance of the
geometric
series. For Xeno, we set *a=*1/2 and subtract off the
1 so we
start with 1/2, taking the limit of an infinite number of steps.

The only trick is knowing that 1/2 raised to the power of infinity goes to zero. But here the distance sums to one.

Xeno may have been able to sum the series but the key step that eluded him, and that awaited the invention of calculus, was being able to treat infinity as a number.