On February $24$ a company decides to issue $$5$ million in new bonds on May $24.$ They

desire to issue them at their current coupon rate of $13\mathrm{.}76\%.$ They will be priced at par value

with a $20-$year maturity and duration of $7\mathrm{.}22$ years. However, if rates rise while due diligence

is occurring, the market will factor that into the bonds' value, resulting in less funds being

raised. To deal with this, they decide to hedge the issue.

a$.$ June futures contracts are trading at $68-11.$ The CTD bond underlying the contract has a

yield of $13\mathrm{.}60\%$ and a projected duration of $7\mathrm{.}83$ years.

b$.$ The optimal number of contracts is given by:

$N^{**}=\frac{(P_{b}x{D}_b)}{(P_{f}x{D}_{\text{h}\text{a}\text{t}(f)})}\frac{(1+YT{M}_{\text{c}\text{t}\text{a}})}{(1+YT{M}_b)}$ where

$P_b=$ dollar value of bond portfolio at par

$D_b=$ duration of bond portfolio

$P_f=$ dollar value of one futures contract at current price

$D_f=$ duration of CTD bond for futures contract

$YT{M}_{ctd}=$ Yield to Maturity of CTD bond

$YT{M}_h=$ Yield to Maturity of the portfolio

c$.$ Should they take a short or long position and why?

d$.$ On May $24$ the bonds are issued and the futures position closed out. The yield on

comparable bonds is now $15\mathrm{.}25\%,$ so the bonds are issued at a $13\mathrm{.}76$ coupon but at a price of

$90\mathrm{.}74638/100$ face. Compute the new value of the portfolio and how much it lost in value

because of the rate change.

e$.$ The futures price at close is now $60-25.$ Compute the gain on the futures position based on

this and $N^{**}.$

f$.$ Compute the performance of the hedge. Did the hedged portfolio gain or lose value?