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# Calculation of macaulay duration and Modified duration

#### nickkl

##### New Member
Subscriber
Hi,

I am confused on the difference in how John Hull (John Hull Example 4.6) and Tuckman (Tuckman Table 4.6) calculates their Macaulay duration in order to determine the modified duration.

In John Hull's example, he uses continuous compounding to determine present value of the cash flows and calculates the Mac duration. He then uses the continously compounded Mac duration value 2.653 (which is calculated under continuous compounding) and calculates the modified duration under semi-annual compounding frequency.

In Tuckman's Table 4.6, he uses semi-annual compounding to determine the present value of the cash flows and calculates the Mac duration. He then uses its value calculate the modified duration under semiannual compounding frequency.

Since the mac duration is needed in order to calculate the modified duration under semi-annual frequency, which method is correct? Should the present value of the cash flows be discounted by continously compounding or semi-annual compounding to first determine the mac duration?

#### Nicole Seaman

Staff member
Subscriber
Hi,

I am confused on the difference in how John Hull (John Hull Example 4.6) and Tuckman (Tuckman Table 4.6) calculates their Macaulay duration in order to determine the modified duration.

In John Hull's example, he uses continuous compounding to determine present value of the cash flows and calculates the Mac duration. He then uses the continously compounded Mac duration value 2.653 (which is calculated under continuous compounding) and calculates the modified duration under semi-annual compounding frequency.

In Tuckman's Table 4.6, he uses semi-annual compounding to determine the present value of the cash flows and calculates the Mac duration. He then uses its value calculate the modified duration under semiannual compounding frequency.

Since the mac duration is needed in order to calculate the modified duration under semi-annual frequency, which method is correct? Should the present value of the cash flows be discounted by continously compounding or semi-annual compounding to first determine the mac duration?
Hello @nickkl

There are a great number of posts in our forum already regarding these concepts. If you use our search function, it will bring up many threads related to Macaulay duration and modified duration. We also have tags (which you can see that I've added at the top of your post). When you click on those tags, it will bring up a ton of threads that discuss these.

During this VERY busy time before the exam, we ask that all of our members search the forum for answers before they post a question. This not only prevents us from repeating the same answers to duplicate questions, but it keeps our forum organized so all members can find the information that they are looking for. Here are links for our search function and tag functions for reference:
This is also an informational thread about posting in the forum: https://www.bionicturtle.com/forum/...d-before-posting-questions-in-the-forum.9071/

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi @nickkl Since we do have so much reference material already on Mac vs. modified duratoin, per Nicole's help, I don't want to repeat too much now. But briefly this may help:
• Macaulay duration is the bond's weighted average maturity, where the weights are PV cash flows as a % of price (itself sum of PV cash flows). Hull's weights are based on spot rates given with continuous compounding; Tuckman's are based on spot rates given with s.a. compounding. The spot rates could be translated into their discrete/continuous equivalents, and you'd get the same discount factors (discount factors "never lie" as they embed whatever compound frequency assumption). So the answer to your question, " Should the present value of the cash flows be discounted by ...." is: the Mac duration is weighting the maturity, those weights are based on (accurate) present values, which is a function of the specified zero rates (Hull's are continuous, Tuckman's are s.a.. Neither is wrong, either could itself be translated).
• Please note about the relationship Mod Duration = Mac Duration/(1 + y/k), that (y) is not any particular spot rate but rather a yield (i.e., yield to maturity). Just in case this wasn't noticed. Hence the meaning of the fact we are generally (in the FRM) dealing with "yield-based duration" anb "yield-based DV01" where yield is just one type of (convenient) interest rate. I hope that's helpful,

#### nickkl

##### New Member
Subscriber
Hi David / Nicole,

Thank you for the explanation and for highlighting the search function! The youtube video was also particularly helpful.