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# Thread: Intuitive Explanation of Duration

1. ## Intuitive Explanation of Duration

Hi guys

As if my 3 page thread on Macaulay Duration wasn't enough, I am starting another! I think (through the help of Dr. Ostaszewski, many people here including ctperng, and Prof Zara at UMass) I am finally able to understand what Duration means.

I wrote a big PDF file which I would like to share with all of you! I tried to make it as easy to read as possible. In it I offer 2 common interpretations of the Macaulay Duration, namely

1) It can be thought of as the time T in the future such that the yield of a security will be independent (ideally) of the interest rate.
2) It can be thought of as the expected time till maturity of cash flows of a security.

I then proceed to demonstrate where these interpretations come from.

The PDF can be found at:

http://evo-games.com/music/Macaulay_...n_05-15-27.pdf

I hope anybody struggling with Duration can use this to help themselves.

Thanks again

- junk

2. Thanks for sharing this! Although I have not reached this topic yet, I am sure i will find it helpful in the near future!

3. You are welcome

I do have to make one tiny update to the intuitive interpretation of Duration as the "expected time till maturity of cash flows in a security." This intuitive interpretation really only "makes sense" if you have either all positive cash flows or all negative cash flows. If you have cash flows where some are positive and some are negative, you will not be able to set up a proper probability distribution based on the mixing weights w_{t} = PV(A_{t})/P(i) where

PV(A_{t}) = present value of cash flow at time t
P(i) = price of security

since some weights will be "negative" and thus the discrete probability distribution given by Pr(T=t) whose distribution equals w_{t} really is no longer a distribution (we can't have "negative probabilities"). Anyway, it's a detail but one worth noting.

- junk

4. I found a lot of the PDF confusing and I realize I have no clue what I'm talking about, but others could find this helpful.

You can think of Macaulay duration like you would center of mass (if anyone took physics...) It's has the same final result: [as he stated a weight average of payments]

CM = sum(m.i*r)/M [M = sum(m.i)]
DM = sum(pv.i(0)*t)/PV(0) [PV(0) = sum(pv.i(0))]

If payments have equal present value it will be half of the midpoint of the payments (if discount rate is small and payments are all equal this is the case as well). If the bond is a lump sum payment at the end its Macaulay Duration equals the life of the bond.

Here's an example:

You can see that T=5 has 4 payments on the left and 5 on the right (and one at 5), due to the fact that payments 1-4 have a higher present value.

The PDF mathematical definition is slightly wrong, I'll show this with an example [from Mathematics of Investment and Credit]:
n-year 0-coupon bond:
P = K*(1+i)^-n
-dP/di = K*n*(1+i)^(-n-1)
(-dP/di)/P = n/(1+i) this of course is modified duration that is to say
Macaulay = Modified/(1+i) [By definition]
(-dP/di)*(1+i)/P

5. Originally Posted by JavaGeek
You can think of Macaulay duration like you would center of mass (if anyone took physics...) It's has the same final result: [as he stated a weight average of payments]
Right! This is exactly how one can think of the expected value of a discrete probability distribution (hence the name "probability mass function"). However, be warned that intuitively it becomes a little less clear when you have opposite valued cash flows as this might potentially lead to "expected values" of time to maturity of cash flows that are beyond the maturity date of the entire security! Or even more confusing, negative expected values

Originally Posted by JavaGeek
The PDF mathematical definition is slightly wrong, I'll show this with an example [from Mathematics of Investment and Credit]:
n-year 0-coupon bond:
P = K*(1+i)^-n
-dP/di = K*n*(1+i)^(-n-1)
(-dP/di)/P = n/(1+i) this of course is modified duration that is to say
Macaulay = Modified/(1+i) [By definition]
(-dP/di)*(1+i)/P
Not quite. If you look at the mathematical definition given for the Macaulay Duration, you will see that it is correct. Recall that Macaulay Duration is the measure of sensitivity with respect to the force of interest. As such, we use e^d instead of (1+i) (where d is delta).

If you do out the derivation, you will see that it makes sense. Check out the derivation here: http://evo-games.com/music/Macaulay_...Derivation.pdf

Does this make sense now?

- junk

6. Originally Posted by Junkmonkey
Not quite. If you look at the mathematical definition given for the Macaulay Duration, you will see that it is correct. Recall that Macaulay Duration is the measure of sensitivity with respect to the force of interest. As such, we use e^d instead of (1+i) (where d is delta).
Thanks, I found it confusing in your explanation that you'd use force of interest and call it interest rate. But It makes sense now, it should have been obvious though as everything was written as e^d...

7. Originally Posted by JavaGeek
Thanks, I found it confusing in your explanation that you'd use force of interest and call it interest rate. But It makes sense now, it should have been obvious though as everything was written as e^d...
I suppose it might have been a bit unclear, but you are right. Whether it is a nominal interest rate or an effective interest rate or even a made up interest rate (like for instance "cardboard box percent interest rate per year"), you could call them all "interest rates" and be correct - although you wouldn't be very specific.

Hrmm.. I'd have to double check that "cardboard box percent interest rate per year" ... Anyway, sorry for the confusion. Hope it all makes sense.

- junk