# Interpretation of Macaulay Duration

I am having a difficulty conceptualizing the meaning of "Macaulay duration" - I want to note I completely understand the math, this isn't the issue. Modified duration & Efficitive Duration make total sense to me as they are refer to a first order approximation of a change in yield on the price of a bond (eg, a 100 bp change in yield causes price to increase/decrease 110 bp). But Macaulay duration is usually quoted in years/time. How does one interpret this? What does it mean for a bond to have a duration of 6 yrs? Can someone help enlighten me? Thank you!

Maccauly Duration means nothing else than that after the given amount of years, you will have your capital investment back as nominal amount.

If you have \$100 invested, and you have a duration of two years, after two years you will have gotten \$100 repaid, not directly dependent of interest rate or payment scheduling (indirectly they are of course!).

In this example, the bond will be valued at 130.45, which is the sum of PV's of all CF's. After the MD - 1.78 years - you will have received exactly your capital investment, which was the nominal amount of the bond. You are investing the nominal amount and this series of cashflows is valued higher (which is economically reasonable, due to risk exposure!). So while the value of the bond is 130.45, after 1.78 years you will have exactly 100 in your hands, and receive the remainder - 30.45 - in the remaining lifetime of the bond (here thats 0.22 years).

You can see how the small payments sum up to the invested amount, and why the Macaulay Duration is always shorter than the period of payments of the bond.

Of course this number is not exact! You will not have 100 on your account after 1.78 years, but less. You will have to wait for the coupon payment after the MD to actually surpass the 100\$(in the example thats the final payment.) • Hi Phi, thank you very much for the response!! This helps. Though I still have a couple of question. I actually priced this bond in the example diagram you posted. The bond costs 1305 dollar, so its trading at a premium to par. Lets say you buy the bond for 1305 dollars. the NPV all the cash flows 1.78 yrs into the bond total 288 dollars (CF1:98,CF2:96,CF3:94) . This is of course well before you receive the last coupon payment and the principal invested. How is 1.78 the point your capital is returned? – gabriel Dec 9 '12 at 6:30 • incorrect, the pv (price of the bond) is definitely not 1305 dollars, how could it be on a$100 par bond? – Matt Dec 9 '12 at 10:25
• You made a mistake with the bond pricing. I've added and explained the example from the picture. Thanks to chrisaycock for the edit! – zuiqo Dec 9 '12 at 13:17
• who is "you"? You are saying that you had your post edited and adjusted and then tell someone else they got it wrong? Confused here... – Matt Dec 9 '12 at 14:08
• and by the way, your definition is incorrect, nowhere does it say that Macaulay duration is defined as the amount of years by which you have your initial investment back, that is simply incorrect, which is proven by your example above. – Matt Dec 9 '12 at 14:11

The simple but accurate answer should be that Macaulay Duration is the weighted average maturity of cash flows (in years). That is how it is defined in almost every text book and looked at by most market practitioners. That is why its quoted in years and it gives an indication of when, on a weighted basis, cash flows are paid out (mature). For example, in the image of phi, the pv of cash flow at t1 (9.61) is paid out/matures at t1. In his example MD is 1.78 meaning the bulk of the maturity of cash flows occurs close to t2, simply because the last coupon is paid at t2 plus the par value is returned to the investor. I would not make it more complicated than it really is.

Edit: The following link may make it clearer in case there is still confusion out there: http://www.econ.ohio-state.edu/jhm/ts/duration.htm

Just keep in mind the MD of a zero-coupon bond equals the maturity of the bond.

• Where is the example you refer to? – SRKX Dec 9 '12 at 17:38
• referring to phi's example who posted an answer first – Matt Dec 9 '12 at 18:21

Perhaps there is another way to arrive at the "weighted average maturity of cash flows". Suppose that we have a coupon paying bond with a continuously-compounded yield $y$ which pays a coupon of value $C_i$ at time $t_i$ for $1 \leq i \leq n$. What would be the maturity of a zero-coupon bond with the same yield $y$ which has the same present value as the coupon-paying bond?

Let $X$ be the face-value of such a zero-coupon bond and let $t$ be its maturity date so that the zero-coupon bond has present value $Xe^{-yt}$. It follows that $Xe^{-yt} = \sum_{i=1}^n C_ie^{-y t_i}$. If one differentiates with respect to $y$ one sees that $t = \frac{\sum_{i=1}^n t_iC_ie^{-rt_i}}{\sum_{i=1}^n C_ie^{-y t_i}} = \sum_{i=1}^n \omega_i t_i$ which is the weighted weighted average maturity of cash flows with $\omega_i$ equal to the proportion of the present value of the coupon-paying bond associated to the cash flow at time $t_i$.

• This helps! But why differentiate with respect to $y$? – andre de boer Jul 5 '17 at 9:02