Converting yield volatility into price volatility

For a bond paying annual coupons, show how to turn a volatility quoted for its yield into a volatility for its price.

Multiply by modified duration: $\sigma_P\approx D_{mod}\,\sigma_y$ (or $D_{mod}\,y\,\sigma_{rel}$ for a relative yield vol). With $P(y)=\sum_i c_i(1+y)^{-t_i}$, a first-order expansion gives $\tfrac{dP}{P}=-D_{mod}\,dy$ where $D_{mod}=\tfrac{1}{P}\sum_it_ic_i(1+y)^{-t_i-1}$. If yield changes have standard deviation $\sigma_y$, percentage price changes have standard deviation about $D_{mod}\sigma_y$.