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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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    $\begingroup$ Homework problem? I don't think quant finance se knows your the proposition numbers for your textbook. Can you edit to clarify a single part of this problem that you are having issues with? $\endgroup$ – rhaskett Nov 7 '14 at 23:18
  • $\begingroup$ A more descriptive title would be nice as well. $\endgroup$ – rhaskett Nov 7 '14 at 23:18
  • $\begingroup$ Is a consol bond the same thing as an annuity? $\endgroup$ – barrycarter Nov 8 '14 at 16:18

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