Another way to write a duration$-$based hedge ratio is as follows:

Hedge Ratio $=\frac{C{F}_{cotd}x(P_{b}x{D}_b)}{(P_{f}x{D}_f)}(1+YT{M}_{ctd}),$ where

$CF=$ conversion factor for CTD bond

$P_b=$ price of bond portfolio as percentage of par

$D_b=$ duration of bond portfolio

$P_f=$ price of futures contract as percentage of $100\%$

$D_f=$ duration of CTD bond for futures contract

YTd ${?}_{ctd}=$ Yield to Maturity of CTD bond

$YT{M}_b=$ Yield to Maturity of the portfolio $($average$)$

A bond portfolio manager holds a government bond portfolio with a face value of $$10$

million that is currently worth a market value of $$9\mathrm{.}7$ million. The manager is concerned

about future rising interest rates and so decides to hedge with a T$-$Bond futures contract.

The cheapest to deliver bonds have a projected duration at maturity of $11\mathrm{.}14$ years. Their

conversion factor is $1\mathrm{.}1529$ and at their current price the futures price is $90-22.$ The

projected average duration of the bond portfolio is $9\mathrm{.}0$ years. Current Yield to Maturity is

$7\mathrm{.}8\%$ on the portfolio and $7\mathrm{.}1\%$ on the CTD bond.

a$.$ Based on the above data, compute the optimal hedge ratio.

b$.$ Based on the interest rate expectations, should they take a short or long position?

c$.$ The optimal number of contracts to hedge with is given by:

Number of contracts $=$ HR X $($Portfolio par value$/$value of futures contract$)$

Where each futures contract is for $$100,000$ of bonds.

Based on this, compute the optimal number of futures contracts to hold.

d$.$ The closing futures contract price is $89-16.$ Based on this, how did the futures

position perform?