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Using Continuous Coupon Cash Flows to Express Bond Value as an Integral

Article Quant Q&A · Author: findingmyway
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Summary

The document considers how to connect an integral to internal rate of return in a cash-flow example with an initial outflow, a single terminal inflow, and no intervening payments. It notes that integrating an isolated IRR expression is not meaningful because IRR is not itself a cumulative quantity. The proposed way to make integration relevant is to change the cash-flow model to a bond that pays coupons continuously.

Under that assumption, the bond's value is represented by integrating discounted coupon payments over the bond's term and adding the discounted principal repayment at maturity. Continuous coupon payments are presented as an approximation to very frequent discrete payments, such as daily coupons. The explanation is brief and does not derive an IRR solution or discuss how the result changes with compounding conventions, day counts, or nonuniform cash flows. It serves as a modeling example for applying calculus to discounted cash flows.

Key ideas

  • An integral is not useful when applied directly to a standalone IRR value that is not cumulative.
  • A continuously paying coupon bond provides a cash-flow stream that can be integrated over time.
  • Bond value in this setup combines the integral of discounted coupons with discounted principal at maturity.
  • Continuous coupon payments can approximate sufficiently frequent discrete payments.

Tags

Fixed incomeStatistics
Full text
# Using Integrals With Internal Rate of Return?


# Using Integrals With Internal Rate of Return?












I'm taking a Calculus 2 course this Fall, and for my honors project, I will be using the IRR function. My professor is requesting that I figure out a way to use an integral with the IRR.

The cash flow scenario being modeled has first period/outgoing/negative cash flow (NPV variable), and one final terminal positive cash flow (CFx), with each period between the first period and final period being 0.

I've taken the NPV function: $NPV=\sum _{n=0}^N\:\frac{CFx}{\left(1+IRR\right)^x}$ , and as we just have one positive terminal cash flow, I've removed the summation and solved for the IRR (with the terminal year=x), with the function becoming $IRR=\left(\frac{CFx}{NPV}\right)^{\frac{1}{x}}-1$

As the IRR function is not cumulative, taking an integral of the IRR function as defined would be meaningless.

Does anyone have an idea as to how an integral could be used for the project?

I really appreciate any assistance that's provided.

## Answer by Adam N. (score 2)

https://quant.stackexchange.com/a/57199

The simplest approach would probably be to consider a bond that pays coupon continuously. This can be justified as a useful approximation for very frequent (ex. daily) coupon payments, and would allow you to use an integral instead of a summation. The value of such a bond would be: $NPV = \int_0^T \frac{100c}{(1+IRR)^t} \text{d}t + \frac{100}{(1+IRR)^T}$

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

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