From “the Actuary”
I Didn’t Know That The Mathematics of Financial Bonds Was So Interesting
Premium Discount Formula, basically
So, yes. I have been studying a lot for my actuarial exam FM. There are various topics and thus formulae. One interesting aspect about this subject is that you start with using formulae like a tool. But then you realise what they are trying to signify. You get to the core of the matter and then you start to appreciate the “elegance” of the formula.
Anyway, so here comes the stuff you actually came for.
This is how a typical Amortization Schedule looks like. If you have studied about loans then I bet you would be able to notice something.
Let me give you a hint. Notice the book values on the schedule. All of them are essentially “Redemption amount plus some extra cash”. The extra cash, in the first Book value (or the price) of the bond, is what we call the Premium. (In our schedule, the premium is 44.518).
Next, we have the amount of amortization of premium, which basically tells us how much premium is being amortized by the coupon payments (after paying the amount of interest on the book value of the previous term).
I could also have written this schedule in this manner.
Suppose that you are considering that some part of your coupon payment is responsible for making sure that your redemption amount remains 1000 and doesn’t increase. What do I mean?
Consider only the interest that the Redemption Amount Earns:
1000 * 0.04 = 40. This amount is what the redemption earns and has to be paid off by a portion of the coupon(50). So, after paying that off, we have 10 of the coupon left.Now, we check the amount of interest that the Premium(amount above redemption) has earned.
Consider the book value of t=3, i.e. 1018.861. The premium associated with it is 18.861, which earns an effective interest of 18.861 * 0.04 = 0.754.We had 10 of our coupon left, using which we pay this interest off and get left with 10 - 0.754 = 9.246.Now, check the amortization schedule above and see what this value 9.246 represents. (DO IT!)Also notice the value 0.754 is present in the interest paid portion of t = 4.
I encourage you to go through the box again and try to make sense of what is being discussed. It is very rewarding once you do, because then you will understand the Premium Discount formula associated with loans.
Here is the entire schedule if we look at the part amortizing the Premium.
Indeed, it is very beautiful. Because when you get this understanding, you will realise the significance of this formula, instead of just being convinced by the algebra.
So, now if you have the time, then think about how this also works in the case of Discount bond. Truly speaking, you might get the understanding since you realised the case of premium bonds. But it is difficult to put in words. Hence, you are the best person who can teach that to yourself. Hence, try it!
