This SEC document claims that increasing the ocupon on a bond decreases the interest rate risk (bottom of page 3):
And the Finra SIE exam states the same also.
I cannot understand the logic behind this statement, it just seems wrong to me. If we consider a simple example where we have a flat interest rate of $r$, and a bond that pays semiannually, then the value of the bond can be written as:
$$ B = \frac{1}{(1+r)^{t_n}} + c \sum_{i=0 \ldots n} \frac{1}{(1+r)^{t_i}}$$
Where if we're just comparing two bonds to each other then, for the sake of comparison, the principal repayment can be ignored, and we can then look at the interest rate risk of the coupons. Here, we can happily say that the rate risk is linear in the coupons, and if we have larger coupons then we must have more risk.
So how is it that the SEC can say that a lower coupon bond has more interest rate risk? What am i missing?