Amount of amortization

This is a little note after the thread A useful weapon for annuities. It is just a simple idea which I have used to simplify my life a little bit.

In Exam FM, we would find amortization problems (with level payments) in which we are asked to calculate the amount of principal/amount of interest in the k-th payment, or, similarly, the write-up/write-down in a coupon payment. Often one resorts to formulae which are tricky to remember and easy to get wrong (e.g. "X v^(n+1-t)" stuff).

I find that the most intuitive way to such problems is to write out in a series the Present Value

```(i)  P = X (v+v^2+v^3 + ... + v^n)
```

The counter part for bonds is

```(ii) P= C+ (Fr-Ci)(v+v^2+...+v^n).
```

## Examples

Take the simplest example,

• What are the amount of principal and the amount of interest in the first payment?

A moment of thought around what the sum (i) above means, one can simply look at (or remove) the last term, and see that

• X v^n is the difference of principal for times 0 and 1, hence the principal paid at the first payment;
• Therefore X - X.v^n = X(1-v^n) is the amount of interest

The amount of interest payment at the m-th payment is equally easy. Just look for the m-th term, counting from the back. So the amount of amortization is X.v^(n-m+1), amount of interest is X.(1-v^(n-m+1)). You don't need to remember any formula.

The same idea works for bond write-ups. Just use the Price as the Present Value, and the Redemption C as the final value (instead of 0). Or, if you like, subtract C from both sides of (ii).