What Parallel Yield-Curve Shifts Mean for Portfolio Duration
Summary
The document examines the parallel-shift assumption behind aggregating bond durations into a portfolio duration. It distinguishes shifts in the zero-rate curve from equal changes in the yields to maturity of individual coupon bonds, noting that coupon-bond yields are nonlinear functions of zero rates and can differ across bonds with the same maturity. It also asks whether perfect correlation among yield changes implies equal-sized changes.
The response agrees that a one-basis-point move in each bond’s yield to maturity is not equivalent to a one-basis-point shift in the zero curve. Summing sensitivities to different yield measures can therefore mix unlike quantities, though it may be acceptable as a first-order approximation. It also distinguishes perfect correlation from equal volatility: correlation of one alone does not require yield changes to have the same magnitude. The exchange is brief and does not develop a full curve-based duration or key-rate framework.
Key ideas
- Portfolio duration as a weighted average assumes a specified way of moving yields across bonds.
- A parallel shift in zero rates does not generally create equal changes in the yields to maturity of coupon bonds.
- Summing yield-to-maturity sensitivities across different bonds can mix measures, though it may work as a first-order approximation.
- Perfect correlation between yield changes does not imply equal-sized changes because their volatilities may differ.
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Full text
# Duration: Parallel shift in yield curve assumption
# Duration: Parallel shift in yield curve assumption
### General Intro
I'm trying to really understand the assumptions of dollar duration for a portfolio of bonds. In particular I don't fully understand that the assumption that there are parallel shifts in the yield curve when computing the duration of a portfolio of bonds.
### Background
Assume continuous compounding so that the duration of a bond $B$ is $-\frac{1}{B}\frac{\partial B}{\partial y}$ where $y$ is the yield to maturity of the bond. The duration of a portfolio of bonds $\{B_1, \ldots, B_p\}$ is defined as $$\frac{-\sum_{i=1}^p \frac{\partial B_i}{\partial y_i}}{\sum_{i=1}^t B_i}$$ which is the weighted average duration (weighted by value, assuming for simplicity all bonds have par value of 1).
The above definition of weighted average duration always comes with the caveat that the definition depends on the assumption that there are parallel shifts in the yield curve.
### Question
- What curve are the shifts talking about? Parallel shifts in the yield curve means (to me) parallel shifts in the zero-rate yield curve, aka the yield to maturity against maturity for zero-coupon bonds. If the assumption is talking about parallel shifts in this zero-rate curve then I don't understand the assumption at all since the yield (to maturity) of the coupon bond is a non-linear function of zero-rates. I don't see how parallel shifts in the zero-rate curve translate into a easy statement about moves in yield (to maturity) of bonds with different maturities. If the shifts are referring to parallel shifts in the yield to maturity curve of the bonds themselves, then this also does not make much sense to me. In this case the bonds with the same maturity may have different yields depending on their coupon rates and price so the 'yield curve' in this sense is not well defined.
- In "The Handbook of Fixed Income Securities" by Frank Fabozzi (2005) on page 208 he claims the the assumption that each yield has moved the same amount is equivalent to the correlation between the change in the yields is 1. Is this correct? Such an assumption would say $\Delta y_i = a\Delta y_j + b$ but it seems we need the much stronger assumption that $\Delta y_i = \Delta y_j.$
## Answer by dm63 (score 2, accepted)
https://quant.stackexchange.com/a/41165
I agree with you on both points. Changing the yield to maturity of coupon bonds by 1bp is not consistent with changing the zero curve by 1bp. Hence , adding the dP/dY of different bonds is apples and oranges, although it's probably ok to first order.
The second point is true as well. Lots of non quants confuse "100pct correlation "with "equal volatility."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.