Consider a coupon bond with maturity in $2$ years, with a coupon rate of $4.375\%$ (coupons are paid twice a year) and with a face value of $100€$. Let's say this coupon bond has a varying continuously compounded interest rate, in the sense that for each half year we have a different continuously compounded interest rate. Furthermore, assume you also know the following discount factors:
$$ B(t,0.5) = 0.978 \quad B(t,1) = 0.957 \quad B(t,1.5) = 0.937 \quad \text{ and } \quad B(t,2) = 0.917. $$
Compute the price of this bond.
My interpretation of the problem. According to the coupon rate and to the face value, I was able to determine the value of each coupon, which came out to be $c = 2.1875€$. On the other hand, I believe that I understood the idea behind the exercise. Essentialy, the bond in question has different interest rates $r_1,r_2,r_3$ and $r_4$ in the following way: in the first half year, interest rate $r_1$ is applied, in the second half of the first year the interest rate $r_2$ is applied and so on. What I really can't understand is how to use the discount factors provided in order the determine the price of the bond. I think that, mainly, I am having a hard time understanding exactly what is a discount factor because in my mind the values provided don't make much sense.
So, in essence, I am looking for hints that can lead me to solve this exercise and I think my main problem is in understanding what a discount factor is, for this particular bond.
Thanks for any help in advance.
The notation $B(t,T)$ means the price at time $t < T$ of a zero-coupon bond that is redeemable for the face value (principal) of 1€ at maturity time $T$. The buyer of this pure-discount bond is receiving a continuously compounded interest rate of $r_T$ given by
$$B(t,T) =e^{-r_T(T-t)},$$
when the bond is held to maturity and the principal payment is received.
The coupon bond that makes coupon payments of 2.19€ every $6$ months and repays the principal of 100€ along with the final coupon payment in two years is equivalent to a package of zero-coupon bonds. Given the current market price of those zero-coupon bonds, the fair value of the coupon bond (precluding any arbitrage opportunities) must be
$$P(t) = 2.19B(t,0.5) + 2.19B(t,1) + 2.19B(t,1.5) + 102.19B(t,2)= 99.99€$$
Using the prices of the zero-coupon bonds in this way, you are finding the present value of the coupon bond with the different interest rates that are appropriate for each future payment date.