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# Bond convexity closed-form formula

Bond convexity closed-form formula (Blake and Orszag): $Conv=-frac\left\{D\right\}\left\{P\right\}begin\left\{Bmatrix\right\}frac\left\{\left(m-1+a+1\right)\left(m-1+a+2\right)\left(1/\left(1+i\right)\right)^\left\{\left(m-1+a+2\right)\right\}\right\}\left\{i\right\}+2frac\left\{\left(m-1+a+2\right)\left(1/\left(1+i\right)\right)^\left\{\left(m-1+a+2\right)\right\}-\left(1/\left(1+i\right)\right)\right\}\left\{i^2\right\}+2frac\left\{\left(1/\left(1+i\right)\right)^\left\{\left(m-1+a+2\right)\right\}-\left(1/\left(1+i\right)\right\}\left\{i^3\right\}end\left\{Bmatrix\right\}+frac\left\{B\right\}\left\{P\right\}frac\left\{\left(m-1+a\right)\left(m-1+a+1\right)\right\}\left\{\left(1+i\right)^\left\{\left(m-1+a+2\right)\right\}\right\}$

D = coupon payment per period
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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