Perpetuity and zero

(a) Using (11.4) and (11.8) with annual compounding, derive the Macaulay duration, modified duration and convexity of a perpetuity paying $c$ a year at yield $y$. Check the convexity against $P''(y)/P$ for $P=c/y$. (b) Derive the convexity of a $T$-year zero with $m$ compounding periods per year.