Question about calculating the price of a coupon bond at different times: FM question.

Question

Let's say we are working with a coupon bond with face value of $F = 100€$, maturity of $T = 5$ years and with $10€$ coupons paid anually. Also, consider we're dealing with a continuously compounded interest rate of $r = 12\%.$ I tried to compute the price of this bond at the actual time and after $1$ year of its emission.

For the actual time, we have that

$$ B_c(0) = \sum_{i=1}^5 10e^{-i \cdot r}+ 100e^{-5r} = \sum_{i=1}^5 10e^{-i \cdot 0.12} + 100e^{-5 \cdot 0.12} \approx 90.27€ $$

and I believe that, for this case, my calculations are correct. My main problem comes when I try to compute the price of this bond after $1$ year.

My thought process was the following: after $1$ year, the coupon bound in question becomes a coupon bond with maturity in $4$ years, with annualy coupons of $10€$ and with the same interest rate and face value of the initial text. With this in mind, we directly have that

$$ B_c(1) = \sum_{i=1}^4 10e^{-i\cdot r} + 100e^{-4r} = \sum_{i=1}^4 10e^{-i\cdot 0.12} + 100e^{-4\cdot 0.12} \approx 91.78€$$

After these calculations, I started to wonder: does it really makes sense that a bond increases its price after $1$ year? And my intuition tells me that this shouldn't be the case since after $1$ year, we have lost one payment of coupons. So I think that I am doing something wrong in my calculations, but I can't figure what it is.

Thanks for any help in advance.

Answer 1

The interest rate is higher than the coupon rate.

Consider investing from year 0 to year 1. The person with $90.27€$ can

• buy this bond, or
• invest elsewhere with the $12\%$ continuous compounding rate instead.

After one year, the second option gives $90.27€\cdot e^{0.12} = 101.78€$, which is a $11.51€$ interest on top of the initial investment. This total is the same as in the first option, which gives only $10€$ coupon.

In a slightly different interpretation, while paying $90.27€$ for the bond at emission is a lower price than paying $91.78€$ one year later, the first is in fact a higher cost, because $90.27€$ at $t=0$ should worth $90.27€\cdot e^{0.12} = 101.78€$ at $t=1$. The difference is exactly $101.78€ - 91.78€ = 10€$ in $t=1$ value: the missed coupon.

For the extreme case at the maturity, after the 5th coupon is paid, the bond would have a price $B_c(5)$ equal to the face value $100€$. This is a further increase from your $B_c(0)$ and $B_c(1)$.