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

a$.$ Calculate duration and convexity for all three

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

c$.$ What is the $convexity for LONG $100$ Million dollars of $20$ Year zero? What is $convexity for short $100$ Million dollar portfolio of $10$ Year and $30$ Year as constructed in Question b $?$

d$.$ For the long$-$short duration$-$neutral portfolio above in Question c$,$ if yields moved up or down by $10$ basis points for ALL THREE bonds in a single day, what is the long$-$short portfolio's duration and convexity Profit and Loss $($PnL$)?$

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

duration$-$neutral portfolio above in Question 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$)$