Consider three zero coupon bonds: $10Y$ yielding $4\mathrm{.}40\%,20Y$ yielding $4\mathrm{.}60\%,$ and $30Y$ yielding $4\mathrm{.}50\%.$

a$.$ Calculate duration and convexity for all three

b$.$ If you want to construct a $100M$ portfolio of $10Y$ and $30Y$ to match the dollar duration of $100M$ position in $20Y,$ what is the weight in $5Y$ and $30Y$ respectively?

c$.$ What is the $convexity for LONG $100M$ of $20$Y zero? What is $convexity for short $100M$ portfolio of $10Y$ and $30Y$ as constructed in Question $2b?$

d$.$ For the long$-$short duration$-$neutral portfolio above in Q$2$c$,$ if yields moved up or down by $10$ bps for ALL THREE bonds in a single day, what is the long$-$short portfolio's duration and convexity PnL$?$

e$.$ What is the $1-$day net interest payment $($ie carry$)$ for the long$-$short

duration$-$neutral portfolio above in Q$2$c$?$

f$.$ What is the break$-$even amount of parallel movement in yield curve in a single day $($breakeven means the convexity $p$nl will offset daily interest payment$)$