Attributing Portfolio Yield Changes Across Bond Purchases
Summary
The document asks how to assign a portfolio’s yield change to separate bond purchases when the order of those purchases should not affect their attributed contributions. It describes a sequential approach: calculate the yield change after each purchase and assign that step’s change to the purchase. Because reversing the purchase order can produce different contributions, the questioner suggests averaging the results across the two possible orders.
The accepted response instead derives sensitivities of the portfolio’s weighted-average yield to each bond weight. These derivatives indicate how a marginal change in a position affects portfolio yield at a given composition; multiplying a derivative by a weight change gives an approximate contribution. The same approach can be applied to yield change relative to the initial yield. This is a local approximation, not an exact order-independent allocation for large trades, and the document offers it as a possible indicator rather than a definitive attribution method. It provides equations but no empirical comparison or validation.
Key ideas
- Sequential yield attribution can depend on the order in which bond purchases are evaluated.
- The portfolio yield is the weighted average of the component bond yields.
- Derivatives with respect to position weights measure marginal effects on portfolio yield.
- Multiplying a derivative by a position change approximates that trade’s yield contribution.
- The proposed sensitivity method is approximate and is not empirically validated in the document.
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# Attributing change in yield as a result of structural change
# Attributing change in yield as a result of structural change
Suppose your portfolio has $w_0$ amount of bonds with yield $r_0$. Now you buy additional $w_1$ amount of bonds with yield $r_1$, then buy additional $w_2$ amount of bonds with yield $r_2$. Eventually your portfolio yield is $$r_f = \frac{\sum_0^2 w_i \cdot r_i}{\sum _0 ^2 w_i }$$. And change in yield is $r_f - r_0$
The question is, how do you attribute this difference to each of the two purchases?
The obvious answer is, first find the yield after you bought $w_1$, compare that to the original yield $r_0$, and let this difference be the contribution of $w_1$. Then find the yield after buying $w_2$, compare to the yield after buying $w_1$, and set the difference as the contribution of $w_2$.
The problem is that there is no natural order which the purchase decisions are made. We could have bought $w_2$ first and then bought $w_1$. If we use the same technique then we will calculate a difference level of attribution for the two purchasing decisions.
I am thinking that I should use the average of the two techniques above. But I am suspecting that there is a better measurement out there?
## Answer by Probilitator (score 2, accepted)
https://quant.stackexchange.com/a/10497
This is not a perfect solution but perhaps the following approach could also serve you well as an indicator.
Assuming you are only using a finite number (e.g. $n$) of bonds with fixed yields $r_i$ you can write $r_f(w_1, \dots,w_n)=\frac{\sum_0^n w_ir_i}{\sum_0^n w_i}$ with most of the weights being zero. Using the quotient rule you can now calculate derivatives $r^j_f(w_1,\dots,w_n)=\frac{\sum_{i=0}^n w_i(r_j-r_i)}{(\sum_{i=0}^n w_i)^2}$
Thus given a concrete portfolio-composition $\vec{w}=(w_1,\dots,w_n)$ you can calculate the vector of derivatives $(r^0_f(\vec{w}),\dots,r^n_f(\vec{w}))$.
Intepretation: Having a portfolio $\vec w$ how will the porfolio yield be affected by marginal changes in the weights $w_i$. You could also do something like $r^j_f(\vec w)(w^{new}_j-w^{old}_j$) to approxiate the change in yield.
One can also do the above for $\tilde r_f(\vec w)=r_f(\vec w)-r_0=\frac{\sum_0^n w_i(r_i-r_0)}{\sum_0^n w_i}$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.