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 21st 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.
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=12y. 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 12r for y years.
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.
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=12r and n=12y:
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 60m, yet the power of compounding interest would inflate this to 68m at 5%, or 77m at 10%. Over 20 years you would pay in 240m, yet 10% interest would more than triple this value to 759m. 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.