# 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

1. 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.

2. 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 $\Delta y_i = \Delta y_j.$

• So I take it the assumption really is that there are equal changes in yield to maturity: $\Delta y_i = \Delta y_j?.$ Or more precisely $\frac{\partial y_i}{\partial y_j} = 1$ so that way the above definition of modified duration makes sense since by the chain rule $\frac{\partial B_i}{\partial y_j} = \frac{\partial B_i}{\partial y_i}\frac{\partial y_i}{\partial y_j} = \frac{\partial B_i}{\partial y_i}$ so that the derivatives can be thought of as being taken with respect to a single quantity? – moquant Aug 9 '18 at 13:16