Closed-form formula

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A single arithmetic formula obtained to simplify an infinite sum in a general formula. The general formula of bond duration and bond convexity cannot be said closed-form as there is an infinite sum over the different time periods. Using a closed-form formula, a bond’s duration or convexity can be calculated at any point in its life time.

Bond duration closed-form formula:

Dur=\frac{C}{P}\frac{(1+ai)(1+i)^m-(1+i)-(m-1+a)i}{i^2(1+i)^{(m-1+a)}}+\frac{100(m-1+a)}{P(1+i)^{(m-1+a)}}


C = coupon payment per period (half-year)
i = discount rate per period (half-year)
a = fraction of a period remaining until next coupon payment
m = number of coupon dates until maturity

Bond convexity closed-form formula (Blake and Orszag):

Conv=-\frac{Dur}{P}\begin{Bmatrix}\frac{(m-1+a+1)(m-1+a+2)(1/(1+i))^{(m-1+a+2)}}{i}+\\2\frac{(m-1+a+2)(1/(1+i))^{(m-1+a+2)}-(1/(1+i))}{i^2}+\\2\frac{(1/(1+i))^{(m-1+a+2)}-(1/(1+i)}{i^3}\end{Bmatrix}+\frac{B}{P}\frac{(m-1+a)(m-1+a+1)}{(1+i)^{(m-1+a+2)}}


Dur = duration
P = present value (price)
B = face value
i = discount rate per period (half-year)
a = fraction of a period remaining until next coupon payment
m = number of coupon dates until maturity

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