2.1a.3 Price Sensitivity

Duration & Convexity

The duration of a bond measures the sensitivity of its price to changes in yield (interest). There are two types of duration:

Macaulay duration is the weighted average time until all cashflows are received, where the weights are the present values of each period’s cashflow. This is formally defined as
$D = \frac{1}{n P} [\sum_{t = 1}^{n T} \frac{C _{t}}{( 1 + y / n ) ^{t}} t + \frac{F}{( 1 + y / n ) ^{n T}} n T] .$

Modified duration adjusts Macaulay duration to approximate the percentage change in the price of a bond for a 1% change in yield, such that
$D^{*} = - \frac{1}{P} \cdot \frac{d P}{d y} = \frac{D}{1 + y / n},$
where the negative sign arises from the inverse relationship between bond price and yield.

Generally, price-yield curves are convex. Higher price-yield convexity implies a higher price regardless of which way prices or directions move:

The convexity of a price-yield curve is represented as the second derivative of its price with respect to yield, such that
$Γ = \frac{1}{P} \cdot \frac{d ^{2} P}{d y ^{2}} = \frac{1}{n ^{2} P ( 1 + y / n ) ^{2}} [\sum_{t = 1}^{n T} \frac{C _{t}}{( 1 + y / n ) ^{t}} (t^{2} + t) + \frac{F}{( 1 + y / n ) ^{n T}} (n^{2} T^{2} + n T)] .$

Convexity can be used in tandem with modified duration to create a better approximation for price:
$\frac{d P}{P} = - D^{*} \cdot d R + \frac{1}{2} Γ \cdot d R^{2} .$

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2.1a.3 Price Sensitivity

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