First-principles and perturbational attribution

FIA offers two approaches to attribution of returns generated by curve movements.

First-principles pricing: by ‘first-principles’, we mean valuing each cash flow generated by the security at the appropriate maturity on the yield curve, and summing the total value of all these discounted cash flows to produce a market price. The change in price caused by pricing with two different curves provides the return due to risk. This approach requires a zero coupon curve for each security, and sufficient information about each security to calculate its cash flows, such as maturity date, coupon and coupon frequency.
Perturbational attribution, using perturbational returns and risk numbers. This approach uses risk numbers, such as modified duration and convexity, as a proxy for a pricing model.

To run this type of attribution, perform a Taylor expansion on the price of a security $P\left( {y,t} \right)$ and remove higher order terms, to give

\delta P = \frac{\partial P}{\partial t} \delta t + \frac{\partial P}{\partial y}\delta y + \frac{1}{2}\frac{\partial ^2 P}{\partial y^2}\delta y^2 + O\left( {\delta t^2 ,\delta y^3 } \right)

Writing the return of the security as

\delta r = \frac{\delta P}{P}

leads to the perturbation equation

\delta r = y \cdot \delta t - MD \cdot \delta y + \frac{1}{2}C \cdot \delta y^2 + O\left( {\delta t^2 ,\delta y^3 } \right)

where the last term denotes higher-order corrections that may be ignored, and

MD = - \frac{1}{P}\frac{\partial P}{\partial y}

C = \frac{1}{P}\frac{\partial ^2 P}{\partial y^2}

The terms $MD$ and $C$ measure first- and second-order interest rate sensitivity. These are conventionally referred to as the modified duration and convexity of the security. Together with the yield to maturity $y$ , they are often called risk numbers.

The modified duration of a security measures its price sensitivity to parallel changes in the level of the yield curve; more specifically,

r_{duration} = -MD \cdot \delta y

where $r_{duration}$ is return, $MD$ is modified duration, and $\delta y$ is the change in yield of the security.

Perturbational attribution implicitly assumes that the risks of a security can be modelled by representing its cash flows as a single bullet payment, rather than as a stream of individual cash flows over time. For some securities such as coupon-paying bonds, this gives accurate results, since the bulk of the security’s cash flows are concentrated at maturity. For other securities such as mortgage-backed securities, where cash flows are more widely spread over the security’s lifetime, the approach is less ideal, although lack of information about a complex securitised security may compel its use.

At what point on the yield curve should we measure changes in the yield curve for perturbational attribution? The maturity is not suitable, since this assumes that changes in price are only affected by the cash flow at maturity. Our preferred measure is the security’s Macauley duration, which is related to the modified duration by:

MD = \frac {D}{(1+ y/n)}

where $MD$ is modified duration, $D$ is Macauley duration, $y$ is yield to maturity, and $n$ is the coupon frequency for the security. Macauley duration is a cash-weighted average of the term to maturity of a bond, and provides a suitable average at which yield changes should be measured. This measure is also suitable for instruments such as floating rate notes, which may have a maturity many years in the future but a very short modified duration due to frequent coupon resets.