Question

Suppose that a floating rate bond with a principal of $100 pays coupons every 6 months. The coupon amount is determined by the 6-month LIBOR quoted in the market 6 months before. That is to say,

- the first coupon is the interest accrued for 6 months using today’s 6-month LIBOR
- the second coupon, one year from now, is the interest accrued for 6 months using the 6-month LIBOR quoted in 6 months
- the third coupon, one and a half years from now, is the interest accrued for 6 months using the 6-month LIBOR quoted in 12 months

The objective of this exercise is to show that the price of such bonds is always the notional
at every coupon date.

(a) Suppose that this is a 6-month bond. What is the price of this bond?
(b) Suppose that this is a 1-year bond. What is the price of this bond? Hint: use (a) to argue that you know the price of the bond in 6 months.
(c) Suppose that this is a N-year bond. What is the price of this bond? Hint: Generalize (a) and (b) to come up with the wanted result.

Edit: I think this is more of a theoretical question as opposed to numerical calculations. As in, how does the price of the 6 month bond relate to the timings of the coupons and notional values? Etc. Such as in question (c) it tells us to generalize the other answers, as in explain the concepts.

Answer 1

As you know, price of the bond at any time is nothing but present value of all the future cash flow it can generate.

(a) Suppose that this is a 6-month bond. What is the price of this bond?

This bond is eligible to get a coupon at the end of 6 month. And this coupon is the interest accrued for 6 months using today’s 6-month LIBOR. On the coupon date, the only pending payment is the redemption payment equal to the face value or the notional. All the accrued interest has been paid. On the coupon date, the price = PV of the notional = Notional. The present value of notional is same as notional, because the notional payment is immediately due and payable on the coupon date.

(b) Suppose that this is a 1-year bond. What is the price of this bond? Hint: use (a) to argue that you know the price of the bond in 6 months.

Using part (a), we know that the price of the bond on the coupon date is the notional value. Next coupon is 6 months away. The situation is similar to what we have in part (a). At the end of 6 months, the one year bond is now a 6-month bond. And we have just proved in part (a) that price of such 6-month bond is the notional on every coupon date. Hence, price of such 1 year bond will be same as notional on any coupon date.

(c) Suppose that this is a N-year bond. What is the price of this bond? Hint: Generalize (a) and (b) to come up with the wanted result.

This can be proven by method of induction. We have seen that in part (a), when N = 1/2 = 6 months, the statement was true. We have seen in part (b) that when N = 1, the statement was true. By making use of these two, we can prove that irrespective of the value of N, the bond will always have a value equal to its notional on any coupon date.