Duration Question

Why duration increases with decrease in interest rates??

consider this very simple example in layman’s terms you invest $100 today at 10% pa to get $110 at the end of the year. Now, if you invest $100 at 5%, how long will it take to get $110? 1.95 years ! So you see, duration has increased with decrease in interest rate.

Oh man…

The way i think about it is: a change from 5 - 6% is a 20% increase. A change from 9 - 10% is an 11% increase. An equal change when dealing with lower rates will have a larger impact than with higher rates. I’m sure there is a better way to explain, but that seems to work for me. I believe the post above is the way you’re definitely not supposed to think about duration.

There are a few ways to think about this - First, duration is the (first-order) sensitivity of a bond to interest rate fluctuations. If you’re a calculus type, a really easy way to see this is to take the derivative wrt r and see that it is decreasing when r increases. But really heuristically, suppose you have a 30 year semi-annual pay coupon bond paying a 10% coupon tomorrow. Alas, it’s 1923 Weimar and interest rates are 1200%. How much do you care if interest rates change to 1300% and how much do you care about that bond except the coupon payment tomorrow? (Not at all and not at all) By the formula, D = 1/V *Sum (T(i) * PV(i)) where T(i) is the time until the ith cash flow and PV(i) is the PV of that cash flow. V is the value of the bond or the sum of the PV(i)'s. Duration is big when the early times have low weights compared to the later ones. If interest rates are high, later cash payments are unimportant compared to earlier ones because their PV is low. Hence, duration is lower when interest rates are higher.

haha… i should apologise here. i had no intent to be misleading. just shows how much i have to gear up for the exam. anyway, i just started preparing two weeks back and am still on the quant section. need to get a lot more intricate. apologies cfalevel_1. took the question totally out of context.

Draw a Price-Yield graph and notice that duration is = the slope of any given price/yield point on that convex graph. See how the slope changes when you go from very low YTMs -> higher YTMs?

^ That’s a good way to look at it if you know that graph, but it’s a little circular.

Nonetheless, I find that my students understand the concept better when illustrated as such in conjunction with the mathematical explanation.

Right but the question is how do you know that graph is concave up without knowing the answer to the original question? (Graph it with Mathematica or something I suppose).