The best way to understand modified (or effective) duration – interest rate sensitivity – is to start with Macaulay duration; whatever is true for Macaulay is true for the others.
Macaulay duration is the weighted-average time to receipt of cash flows, where the weighting on the time to a given payment is the present value of that payment divided by the sum of all present values (i.e., the price).
The Macaulay duration of a 10-year zero-coupon bond is 10 years: 100% of the cash flow comes at maturity.
The Macaulay duration of a 20-year zero is 20 years. Thus (compared to the 10-year zero), we see that greater the longer the maturity, the longer the duration.
The Macaulay duration of a 10-year, 6% coupon bond is less than 10 years, because some of the cash flows come earlier than 10 years. Thus (compared to the 10-year zero), we see that the higher the coupon, the shorter the duration: the more cash we get earlier.
The higher the YTM, the lower the PV of more distant cash flows, so the less weight there is to the longer time to those cash flows. For example, for a 10-year, 6% coupon, semi-annual pay bond, if YTM = 4%, the PV of the first coupon is 98% of the FV, while the PV of the last payment is 67% of the FV; if the YTM were 8%, the PV of the first coupon is 96% of the FV (98% of the 4% YTM value), while the PV of the last payment is 46% of the FV (68% of the 4% YTM value). Thus, as YTM increases, the near-term cash flows represent a greater percentage of the price than the far-term cash flows; the duration becomes shorter.
Longer maturity means longer duration.
Higher coupon means shorter duration.
Higher YTM means shorter duration.
This is true for Macaulay duration, modified duration , and effective duration (everything else being equal).
Unlike Macaulay and modified duration (which assume that the cash flows never change), effective duration allows for changes in the cash flows (e.g., for bonds with embedded options, or floating-rate bonds); like modified duration, effective duration measures the percent price change in a bond per percent change in YTM.
Because callable and prepayable bonds have a lower cap on their price than an option-free bond would, there is less price change when YTM is low; thus, duration is shorter for these bonds when YTM is low.
Their Macaulay duration is longest because 100% of the cash flow is at maturity. As Macaualy duration goes, so go modified and effective duration: they’re longest for zeros.
I could show you all of the math, but I doubt that that would give you the intuitive feel for it.
Try this: draw a price vs. YTM graph for an option-free bond. Notice that the slope is quite steep for low YTM, and nearly flat for high TYM. The implication is that as YTM increases, the price change decreases.
(Yes, I know that (modified) duration is not the slope of this curve; it’s the slope of the ln(price) curve. Nevertheless, the curves look similar, and it gives you the intuition you need without complicating things with a stupid natural logarithm.)