Fast Formulas #1 (Follow-up): New Insight into an Old Formula

Ugarte (Peter Lorre):              You despise me, don’t you?
Rick (Humphrey Bogart):     If I gave you any thought I probably would.

Casablanca

If you work in finance, I bet there is something you use all the time, but give no more thought than Rick gives Peter Lorre’s slimy character in Casablanca.

I refer to the level payment formula. See? You don’t want to think about it, do you?

The level payment formula is used millions of times every day, by lenders, analysts, bankers, traders, and actuaries. Not that they need to think about it. The formula is usually buried out of sight, deep inside their software tools.

This is the formula:

{L=\dfrac {r\times P}{1-\dfrac{1}{(1+r)^n}} }

Unlike Peter Lorre’s character, however, the level payment formula merits our appreciation.  More like Cinderella than Ugarte, this industrious formula has its own disguised beauty.

I found what might be a new way to make sense of this old formula. Once you see its financial meaning, you might never forget it again.

The Usual Explanation

Let’s begin by reviewing a standard derivation of the formula. Every derivation I have seen is equivalent to what follows, and uses the same trick.

Here are the symbols used by the formula, with some sample values for a 30-year loan with level monthly payments:

Symbol Description

Sample Value

L The level payment amount.

$599.55

r The periodic interest rate.

.005 (6%/12)

P The principal amount.

$100,000

n The number of payments.

360    (30 x 12)

L is the combined payment of principal and interest each period.  The principal and interest components of L change from one period to the next, but the total is constant.  The formula gives the value of L that completely amortizes the loan over n periods. This means that L is exactly sufficient to repay the loan principal and to pay all of the interest on the loan over its life.

We could also describe L as the level annuity that could be generated by an investment of P earning r for n periods.

Note that r is the rate per period, not the annual rate. An annual rate of 6% for a monthly mortgage corresponds to a periodic rate of 0.5%.

We assume that the interest rate r is exactly the market rate for this sort of loan. Thus we can say that P must equal the present value of all future payments of L, discounted at r. So, with d=\dfrac {1}{1+r}, the discount factor for one period, we have

P=dL+d^{2}L+d^{3}L+{...}+d^{n}L.

This is a geometric series. The geometric series has a distinguished history. It was suggested by one of Zeno’s paradoxes before 400 B.C., and a solution was known to Euclid. The geometric series is even connected to music and fractals.

While the geometric series looks unworkable, its regular pattern allows us to cancel out most of its terms to get a compact expression. I explained this in Fast Formulas #4: Geometric Series (One of the Tricks).  The shortcut is given by Formula 4.4, and we can use it to find that

P=\dfrac {d^{n+1}-d}{d-1}\times L.

Now divide the numerator and denominator by d, and then replace \frac {1}{d} with 1+r. In a few more steps we can see that

L=rPD _{n},                                                                                                       (Formula 1.3)

where D_{n}=\dfrac {1}{1-d^n}.

This is Formula 1.3 from Fast Formulas #1: Average Life of Mortgage (As Scheduled). It is just another way to write the level payment formula.

Plugging our sample values into the formula, we calculate that L={\$599.55}.

Financial Meaning

Sybil (Alice Krige):  That sounds clever.  What does it mean?

Chariots of Fire

We just saw how to derive the level payment formula by using an old algebra trick. But does the formula tell us anything, or is it just an undigested bit of math?

Working on this blog led me to a simple financial meaning for the formula, one that is hidden in plain sight.  Once I saw how the formula makes financial sense, it has been easy to remember.

Consider another instrument with the same interest rate and repayment term as our amortizing loan. The principal amount is only \$1, and no principal is repaid until the final payment.  There are n-1 payments of \$r in interest and a final payment of \$(1+r) that includes both principal and interest.   We could call this a bullet loan, or a bond.

The principal amount of the bond is $1, and it also has a present value of $1, because we assume (as before) that the interest rate r is also the appropriate rate at which to discount the future cash flows (i.e. we have a par bond, where the yield equals the coupon).

Now, if the bond is worth $1 today, that means that the combined present value of the bond payments must equal $1:

\$1=PV({All\;Pmts}), or

\$1=d\times \$r+d^{2}\times \$r+d^{3}\times \$r+{...}+d^{n-1}\times \$r+d^{n}\times \$(1+r).

We can break down the present values like this:

PV({All\;Pmts})=PV({All\;Int\;Pmts})+ PV({Prin\;Pmt}), or

\$1=(d\times \$r+d^{2}\times \$r+d^{3}\times \$r+{...}+d^{n}\times \$r)+(d^{n}\times \$1).

The present value of the principal payment is

PV({Prin\;Pmt})=d^{n}\times \$1=\dfrac {\$1}{(1+r)^{n}}.

Because the present value of the combined bond cash flows is $1, the present value of the interest payments is simply

PV({All\;Int\;Pmts})={\$1}- PV({Prin\;Pmt})={\$1}-\dfrac {\$1}{(1+r)^{n}}.

Now consider the ratio

\dfrac {\$r}{PV({All\;Int\;Pmts})}=\dfrac {r}{1-\dfrac{1}{(1+r)^n}}.

Since all of the interest payments on the bond are equal to \$r, we now have the ratio of the amount \$r to the present value of a stream of cash flows at that same amount.  This is very useful information.  The same ratio holds for any (non-zero) cash flow amount.  Therefore, we can use this ratio to find the level payment or annuity that corresponds with any present value.

So if the present value is P, then

L=\dfrac {rP}{1-\dfrac{1}{(1+r)^n}},

which is the same level payment formula we saw above.

Let’s explore this bond-based interpretation with an example. We need the separate present values of the principal and interest on our $1 bond:

PV({Prin\;Pmt})=\dfrac {\$1}{(1.005)^{360}}={\$0.166042}

and

PV({Int\;Pmts})={\$1}-{\$0.166042}={\$.833958}.

Most of the bond’s value is due to the interest payments, which is not unusual for a bond with a long maturity.

Now let’s find the ratio of one interest payment to the present value of all the interest payments:

\dfrac {\$r}{PV({All\;Int\;Pmts})}=\dfrac{{\$0.005}}{{\$.833958}}=.0059955.

The ratio then tells us the level payment for any principal amount:

L=.0059955\times {\$100,000}={\$599.55},

as before.

If you can remember that

L=\dfrac {\$r}{ PV({All\;Int\;Pmts})}\times P,

there is a good chance you will remember the level payment formula.

I hope you agree that the level payment formula deserves neither to be forgotten nor to be despised.

Copyright 2013.  All Rights Reserved.

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