Deriving Duration and Convexity for a Perpetual Bond

Summary

The document presents an exercise about a consol bond that pays one unit of cash at every future period while the market yield is constant across maturities. It asks for the bond’s price, duration as an infinite series and in analytical form, and convexity. The central connection is between bond price sensitivity to yield and derivative sensitivity: duration corresponds to a first derivative measure, while convexity is introduced as a second derivative measure.

The supplied material gives a price derivative relationship in which the yield derivative is negative duration times price, and states a definition of convexity. It does not show the requested price summation or derive the closed-form duration and convexity expressions. Thus, it offers a useful setup and sensitivity interpretation, but leaves most of the exercise for the reader. The setup assumes an indefinitely paying bond and a constant yield; it does not address credit risk, changing rates, or practical bond conventions.

Key ideas

• A consol pays the same cash amount at every future period and is priced using a constant yield assumption.
• The exercise asks for price, duration, and convexity, including analytical sensitivity formulas.
• Duration measures the first-order response of bond price to yield changes.
• Convexity is presented as the second derivative of price with respect to the relevant variable.

Tags

# How to prove following order?


# How to prove following order?












Consider a consol bond, i.e. a bond which will forever pay one unit of cash at $t = 1, 2, . . ..$ Suppose that the market yield $ y$ is constant for all maturities.

(a) Compute the price, at $t = 0$, of the consol.

(b) Derive a formula (in terms of an infinite series) for the duration of the consol.

(c) Use (a) and Proposition 22.11 in order to compute an analytical formula for the duration.

(d) Compute the convexity of the consol.

Proposition 22.11 we have $$\frac{dp}{dy}=\frac{d}{dy} \{\sum_{1}^n c_ie^{-yT_i}\}=-D.p.$$

Thus we see that duration is essentially for bonds (w.r.t. yield) what delta is for derivatives (w.r.t. the underlying price). The bond equivalent of the gamma is convexity, which is defined as $$C=\frac{\partial^2p}{\partial u^2}$$

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.