Valuing a Bond with Continuous Coupons and Time-Varying Rates

Summary

The document outlines how to value a bond that pays a continuous coupon stream and returns principal at maturity when the stated zero rate changes with time. It assumes continuous compounding, so each payment is discounted using a factor derived from the zero rate and its payment time.

The present value is expressed as the discounted redemption amount plus an integral of discounted coupon payments over the bond’s life. The example substitutes the specified coupon and rate functions into that formula and notes that the resulting coupon integral appears not to have an elementary closed form, so numerical integration may be needed. The answer flags that the bond structure is unusual and its calculation rests on interpreting the given rate as a continuously compounded zero rate. It provides no numerical evaluation or alternative treatment if the rate instead represents an instantaneous forward rate.

Key ideas

• Bond value combines discounted principal repayment with the present value of coupon payments.
• A continuous coupon stream is valued by integrating each payment after discounting it to the present.
• Under continuous compounding, the discount factor depends on the zero rate and the payment horizon.
• The example’s coupon integral is reported to lack a convenient closed form and may require numerical evaluation.

Tags

# Valuing Bonds With Continuous Coupon Yields


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

## Answer by Kermittfrog (score 2)

https://quant.stackexchange.com/a/60581

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).

HTH?

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.