Long Maturity = High duration
High Coupn = Low Duration
High YTM = Low Duration
There are some things I am unclear on:
Firstly, I was thinking that a high coupon had higher duration because you receive more payments earlier which you then must reinvest at a different rate. Is the duration relationship entirely different to the reinvestment risk I just described?
Why does a High YTM have a low duration?
The relationships are correct with respect to YTM and coupon. As to Long maturity = High duration, it’s correct in most of the cases (an exception may occur if you have discount bond with coupon rate low relative to YTM and the time to maturity is long).
The duration is the weighted average term to maturity of the present value of the cash flows (cited by Investopedia). If the bond has higher YTM, the present value of the cash flows will be lower, hence the lower duration.
Higher discount rates affect more distant cash flows more than they do the nearer term ones. So, as the YTM (i.e. the discount rate) rises, this effectively decreases the weight on the later cash flows (and since Macauley Duration is the PV weighted time that the cas flows are received), this means that MAcauley Duration decreases when the YTM increases.
Since Modified duration = Macauley Duration / (1 + YTM), an increase in YTM serves to discount Macauley Duration to a greater extent (even if the first effect wasn;t present).
These are all correct for Macaulay duration, modified duration, and effective duration, and their inverses (short maturity = low duration, low coupon = high duration, low YTM = high duration) are correct for Macaulay duration and modified duration, but not necessarily for effective duration; in particular, a callable or prepayable bond will have a low duration when its YTM is low.
I wrote an article on duration that covers all of these: http://financialexamhelp123.com/macaulay-duration-modified-duration-and-effective-duration/.