A $1-$year bond $($face value USD $1,000)$ of firm MFB with significant default risk has a coupon rate of $9\mathrm{.}4\%$ and is selling for USD $970.$ The bond makes its coupon payments semi$-$annually.

A comparable risk$-$free $1-$year government bond $($face value USD $1,000)$ pays a coupon of $5\mathrm{.}4\%$ and is selling for USD $1,000.$

a$)$ What is the risky bond's yield to maturity, and what is the credit spread, that is$,$ the yield in excess of the yield of the risk$-$free bond?

b$)$ Compute the modified duration for both bonds.

c$)$ You believe that credit spreads will narrow, meaning that the difference in yield between risky bonds and risk$-$free government bonds will get smaller. Describe how a portfolio consisting of the $1-$year risky bond and the $1-$year risk$-$free bond can be used to trade on this belief. Explain which bonds you would go long or short.