# Valuing Bonds With Continuous Coupon Yields

How do I find the value of bonds with continuous coupon yields and interest rates that are both a function of time?

The bond has a redemption of 2000 at time $$t=2$$ and pays continuous coupon payments of $$K(t)=100e^{-t}$$. The spot interest rate is $$r(t)=\frac{2}{50-t}$$.

• Can you post an example ISIN please? Jan 19, 2021 at 22:12
• I'm sorry but what is an ISIN? Jan 19, 2021 at 22:19
• ISIN is a kind of an identifier for a particular bond issue. Can you please point to an example of such a bond that exists in practice? Jan 20, 2021 at 0:00
• I don't know if it's a real life example but the problem i'm trying to solve is A bond V(t) has a redemption of Z = £1000 at time t = 2. The interest rate is r = 2/(50−t) and the bond pays a continuous coupon yield of K(t) = 100 exp(−t). Determine the value of the bond at time t = 0. But I'm getting to a point where I have an integral that I can't solve using methods I've been taught, so I'm inclined to think that i'm wrong. Jan 20, 2021 at 2:01

To echo on @Dimitri Vulis comment on your question, this kind of bond structure is rather contrived, but nevertheless let me try to give you some starting pointers.

I am assuming that your interest rate is a continuously compounded zero rate, i.e. the discount factor for time t equals $$D(t)=e^{-r(t)t}$$. Then the present value fo the bond equals discounted (one-time) redemption payment and discounted coupon income stream, i.e.

$$V(0)=B\times D(T) + \int_{s=0}^{T}K(s)D(s)\,\mathrm{d}s$$

$$V(0)=2000e^{-\frac{1}{12}} + 100\int_{s=0}^{T}e^{\frac{2}{50-s}s^2}\,\mathrm{d}s$$