From Actuarial Outpost Wiki
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).
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).