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 $r(t)=\frac{2}{50-t}$.

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    $\begingroup$ Can you post an example ISIN please? $\endgroup$ Jan 19, 2021 at 22:12
  • $\begingroup$ I'm sorry but what is an ISIN? $\endgroup$
    – Podski
    Jan 19, 2021 at 22:19
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    $\begingroup$ 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? $\endgroup$ Jan 20, 2021 at 0:00
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    $\begingroup$ 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. $\endgroup$
    – Podski
    Jan 20, 2021 at 2:01

1 Answer 1


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.

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

Inserting your data, we get

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

The integral on the RHS seems to have no closed form solution (at least thats what Wolfram Alpha tells me).



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