Skip to content
All library documents

Why Yield to Maturity Cannot Be Uniquely Allocated Across Cash Flows

Article Quant Q&A · Author: Carl
Read the original

Summary

The document asks whether a bond’s yield to maturity can be broken into separate yields for two component cash-flow streams. It represents the bond’s present value as the discounted value of total cash flows, with each period’s payment split between components A and B. The central issue is whether the total yield determines an individual yield for each component.

The response interprets the task as calculating separate internal rates of return for the two streams and notes that there can be multiple valid solutions. The component present values can be partitioned in different ways while still summing to the bond’s total present value, so the overall bond valuation alone does not identify unique component yields. The discussion is brief and gives no allocation rule or worked numerical example; additional constraints would be needed to select a particular decomposition.

Key ideas

  • A bond’s total cash flow can be expressed as the sum of component cash-flow streams.
  • The bond’s overall yield does not, by itself, uniquely determine separate yields for those streams.
  • Different allocations of present value between components can preserve the total bond present value.
  • A unique component yield decomposition requires additional constraints or a chosen allocation method.

Tags

Fixed incomeStatistics
Full text
# How to break down yield to maturity to different components?


# How to break down yield to maturity to different components?












Suppose we have the PV of a bond, as well as two separate streams of cash flows, say, $C_a$ and $C_b$ that make up the total annual cash flows $C$ (i.e. $C=C_a+C_b$). In other words, suppose we have,

\begin{equation} PV(bond)=\frac{C^{(1)}}{(1+YTM)}+\frac{C^{(2)}}{(1+YTM)^2}=\frac{C^{(1)}_a+C^{(1)}_b}{(1+YTM)}+\frac{C_a^{(2)}+C_b^{(2)}}{(1+YTM)^2}. \end{equation} noting that $C^{(1)}\ne C^{(2)}$. I have been assigned with the task of breaking down the YTM according to the individual $YTMs$, which I cannot figure out. Would appreciate it, if someone could explain this to me.

Thanks.

## Answer by dm63 (score 0, accepted)

https://quant.stackexchange.com/a/70648

I think I understand. You are trying to calculate the IRR of the a-cash flows and the b-cash flows individually ? But there are multiple solutions: you can partition the PV into PV(a) and PV(b) and solve for IRRs of a and b separately with only the constraint that PV= PV(a)+PV(b).

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.