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Derivation of modified duration formula for par bond

Jose V

New Member
Thread starter #1
In which study guide or video can i find the derivation of the formula of modified duration of par bond?
I understand MD formula for par bond is: MD=[1-(1+y)˄-maturity] / y
But still searching for derivation..

David Harper CFA FRM

David Harper CFA FRM (test)
Staff member
Hi @Jose V I'm not sure it's explicitly in a study note (we should add to the formula sheet revision, however ....). I think I derived it somewhere in the forum almost a decade ago (ha), but it's also based on Tuckman 4.45 (which in turn makes use of his own formula 4.37), please see (4.45) below where D(c = y) and notice his formula is for a semi-annual compound frequency; of course, (c=y) refers to a par bond. So in my XLS etc, I just used the annual compound frequency equivalent (Jorion assumes annual):

And recent note here:
https://www.bionicturtle.com/forum/threads/l2-t5-63-fixed-income-mapping-jorion.3617/post-72820 i.e.,
"Hi @Stuart D Moncrieff That is not a general equation for modified duration; to my knowledge, there are no super simple (efficient) formulas for the modified duration of coupon-bearing bonds, with the exception of par bonds (of course, zero coupon bonds are easiest, hence their popularity in exam question assumptions!).

In Chapter 4, Tuckman shows the simple formulas for modified duration in three special cases: zero-coupon, par bonds (i.e., when the coupon rate equals the yield to maturity) and perpetuities. So we have a "simple" formula when the bond happens to be priced at par (or as approximations when the price is near to par; Tuckman: "The yield-based DV01 and duration of par bonds are useful formulae as relatively simple approximations for bonds with prices close to par.")

For the modified duration of a par bond (i.e., "c = y") with a semi-annual coupon, Tuckman gives formula 4.45:
D(c=y) = 1/y*[1 - 1/(1+y/2)^2T], which is the semi-annual equivalent to the annual (compound frequency) formula that you quoted. I hope that's helpful, thanks!"
I hope that's helpful!