# Duration Measures

A zero-coupon bond with a maturity of 10 years has an annual effective yield of 10%. What is the closest value for its modified duration? a. 9 b. 10 c. 100 d. Insufficient Information Answer: a You must first recall that the Macauley duration of a zero-coupon bond is equal to its maturity. Then, the modified duration of a zero-coupon bond is: Macauley duration / 1+ i = 10 / 1.10 = 9.09. Okay so my question is: Do we not use semi-annual yield in the formula? As per my understanding, Modified Duration = Macauley Duration / (1+semi-annual yield) So should the answer not be 1/1.05 = 0.95238?

Sorry I meant 10/1.05 = 9.5238 (Also, when you a Duration calculation assuming annual payout, the duration comes out to be 9.09 and when you do it for semi-annual payout, it gives 9.5238. I remember reading that we use semi-annual for a zero.)

divide by 1.05^2

Now this is even more confusing. AFA I can understand, JDV means that my interpretation is correct and the answer is 9.5238 and hence closer to 10 which implies B. However, wyantjs says that it should be (1/1.05^2) which is 9.07 and A. The original answer uses 1/1.1 = 9.09 and hence A. Could someone please clarify? Many thanks.

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That is strange. Investopedia also uses MD/((1+(ytm/number of coupons per periods). So, since there are no coupons per period, you would think that the answer would be 1/1. But, I guess one could argue that the theoritical number of coupons would be 1 per period, which would work to: 1/1.1= .9090909090909090…

I believe I made a mistake, and Joey is correct. You should not square the denominator. Answer is B

If Joey is right, then Investopedia is wrong… Which isn’t impossible…

I checked three different Fixed Income books written by Fabozzi. I trust that Frank is right on this one…

I just checked in Fixed Income by Fabozzi. The formula is the same as Investopedia’s. So here is the answer: .10/(1+(.10/1)= .0909090909090909…

This is assuming that we value zero coupon bonds with 1 period per year… Valuing it with zero periods per year would be: .1/1 = .1

Point being, the answer definetely isn’t 9.52…

I think it is 10/1.1 = 9.0909?

In computing modified duration, you use the periodic yield. That is, if you had a 10% coupon bond paying coupons semiannually, you’d divide by 1.05, but in this case, of course, the zero pays no coupons, so you divide using the effective annual yield rate provided, that is, D_Mod = D_Macauley/1.10 = 10/1.10 = 9.0909. Example: Let’s calculate the impact of a change in yield of 1 bp and compare it to the change implied using the Macauley duration. Actual Change: The price of the 10% zero should be 100/1.10^10 = \$38.554, the price of the zero with a +1bp change in yield, to 10.01%, is 100/1.1001^10 = \$38.519297. The change in price is -0.035032/bp. The Macauley duration provides an indication of the relative rate of change in price of the bond per dollar invested as a function of change in yield. Then, P(i+h)-P(i) is about equal to -h*D_Macauley*P, or -.0001*38.554*9.0909 = -0.035049, which is pretty close to what we had. If we’d used the semi-annual yield, we would have computed a change in price of -0.0001*10/1.05*38.554 = -0.036718, which is off by about 4.8%.