- Closed-form formula
A single
arithmetic formula obtained to simplify an infinite sum in a general formula. The general formula ofbond duration andbond 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 [http://www.google.com/url?sa=t&ct=res&cd=1&url=http%3A%2F%2Fpages.stern.nyu.edu%2F~msiegel%2FUnderstandingDuration.doc&ei=usfLRp65Lon2hQPq8PT6CQ&usg=AFQjCNEBCNz1CXJOO8VOVCMfYvyPysnxUA&sig2=Wk2Bb7Db8_jUeZSN4baGcg (Richard Klotz)] :
Dur=frac{Cfrac{(1+ai)(1+i)^m-(1+i)-(m-1+a)i}{i^2(1+i)^{(m-1+a)+frac{100(m-1+a)}{(1+i)^{(m-1+a)}{P}
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 maturityBond convexity closed-form formula [http://scholar.google.com/scholar?hl=en&lr=&q=cache:MGf3icGTSYsJ:ideas.repec.org/p/bbk/bbkpip/9602.html+ (Blake and Orszag)] : Conv=-frac{D}{P}egin{Bmatrix}frac{(m-1+a+1)(m-1+a+2)(1/(1+i))^{(m-1+a+2){i}+\2frac{(m-1+a+2)(1/(1+i))^{(m-1+a+2)}-(1/(1+i))}{i^2}+\2frac{(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)
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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