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Mccaulay duration and the denominator k


New Member
What is the denominator k in the Mccaulay duration formula?
Also why does the the duration for zero-coupon and deep discount bonds with maturities greater than 20 years fall slightly with increase in maturity date?
A last thing was that you said that reinvestment risk and interest rate risk go hand in hand. Is it because for low coupon bonds or no-coupon bonds if the interest rates go up the price decrease and hence though the reinvestment risk is avoided if YTM increases too much(opposite of reinvestment risk) the no-coupon bond suffers from lack of reinvestment.
Just looked for clarification on this point.

David Harper CFA FRM

David Harper CFA FRM
Staff member
k is the compound frequency. So for Tuckman, where semiannual is the norm, k = 2 periods/year and Mac Duration = Mod Duration * (1 + y/2).

Re duration falling: yea, i've read that in Tuckman, but i frankly don't get it esp for zeros. The Mad duration for zeros is linear with maturity, so i don't see it falling at any maturity. I'm sure it's probably right, but i've never been able to see the calcs. For DV01, different story: DV01 has offsetting price/duration impacts, so DV01 should bend back

reinvestment versus interest rate: they are a tradeoff. The zero coupon bond has no coupons to reinvest; it has high duration to reflect high interest rate. The high coupon bond has lower duration/less interest rate but coupons must be reinvestment. It is an necessary trade-off.