The Interest Rate And Monthly Payment

A friend of mine was thinking about buying couple of properties for cash flow purposes. He asked me to have a cup of coffee with him and go over his numbers to make sure that he did not overlook anything. Since I love “math”, I even offered to buy the coffee :).

My buddy understood the process of buying like no one else. He knew the ins and outs and all the hidden expenses. But when it came to simple math, I was shocked to know that he did not get it at all. Without the bank excel sheets he was totally lost. He had no idea how his monthly payment is getting calculated and even worse, did not have any feel or intuition on how the interest rate might impact his payment. So if you are like my buddy, let me show you how these magic numbers work together. It is relatively simple.

Let us say that you take out a loan of $P_o$ at interest rate of $\alpha$ and for $N$ years. These are the only 3 variables in this whole process.

Let us look what happens to the principle $P_o$ after one month (at the end of the month).

$P_1 = (1+\frac{\alpha}{12})*P_o-x$ ,

where $\alpha/12$ is your monthly interest rate and $x$ is your monthly payment. By the way we do not know what $x$ is yet. That is what we are trying to calculate.

The above equation says that at the end of the first month you owe the “bank” the principle + the principle multiplied by the interest rate (monthly), minus your monthly payment. We can rewrite the above equation as

$P_1 = \beta*P_o-x$ ,

where we renamed $\beta = 1+\alpha/12$ .

Note that now $P_1$ becomes the new principle. If you got this, the rest is a piece of cake. Let us see what happens at the end of the second month. It is the same exact process.

$P_2 = \beta*P_1-x$

which we can write as

$P_2 = \beta*(\beta*P_o-x)-x$ , or

$P_2 = \beta^2*P_o-x*\beta-x$ , or

$P_2 = \beta^2*P_o-x*(1+\beta)$

And if we do this again for the third month we can write

$P_3 = \beta*P_2-x$

Which can be written as

$P_3 = \beta^3*P_o-x*(1+\beta+\beta^2)$

And if we keep on going till month $n$ we are write

$P_n = \beta^n*P_o-x*(1+\beta+\beta^2+\ldots+\beta^{n-1})$

So not what happens at the last month when $n = N$ . That is when you are “even Steven” with the bank. So the principle $P_N = 0$ . This allows us to write the payment equation as

$0 = \beta^N*P_o-x*(1+\beta+\beta^2+\ldots+\beta^{N-1})$

Note that the only unknown in the above equation is $x$ the monthly payment. We can solve the above equation to get $x$ . But let us simplify the above equation even further. Let us call

$Q_{n-1} = 1+\beta+\beta^2+\ldots+\beta^{n-1}$

This is the expression that is showing in the payment equation above. To calculate $x$ we need to calculate the above sum. Although we can do it term by term, but that is a pretty lengthy process. So let us see if there is a smarter way around it. If we multiply both side by $\beta$ then we get

$\beta*Q_{n-1} = \beta+\beta^2+\ldots+\beta^{n-1}+\beta^n$

Then let us subtract the last equations to come up with

$Q_{n-1}-\beta*Q_{n-1} = 1+\beta+\beta^2+\ldots+\beta^{n-1}-(\beta+\beta^2+\ldots+\beta^{n-1}+\beta^{n})$

Which we can rewrite as

$Q_{n-1}*(1-\beta) = 1+\beta^n$

Or we can write

$Q_{n-1} = \frac{1+\beta^n}{1-\beta}$

Now that we have figured out the sum of $Q_{n-1}$ we can solve the payment equation to get $x$

$0 = \beta^N*P_o-x*(1+\beta+\beta^2+\ldots+\beta^{N-1})$

$\beta^N*P_o = x*Q_{N-1}$

$\beta^N*P_o = x*\frac{1+\beta^N}{1-\beta}$

And the final expression for the monthly payment becomes:

$x = \frac{1-\beta}{1-\beta^N}*\beta^N*P_o$

In the last expression, $N$ is the duration of the loan in months. So if you took out a 30 year mortgage, then $N = 12*30 = 360$ , $P_o$ is the principle, $\beta = 1+\alpha$ , and $\alpha$ is the monthly interest rate. All are known quantities.

As you see, it is not really that complicated at the end. Now you can calculate your own payment based on any term that you like. Have fun.

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