Problem $1$ You must invest $$100,000,$ and the bonds listed below from A to E are the only investments available today $($assume that it is possible to buy a fraction of a bond in order to invest the full $$100,000).$ Assume all these bonds have the same yield$-$to$-$maturity $(4\%$; APR, compounded semi$-$annually$)$: A$.$ A bond with $6$ years to maturity and a zero coupon rate $($discount bond$).$ B$.$ A bond with $6$ years to maturity and a $4\%$ coupon rate $($coupons are paid semi$-$annually$).$ C$.$ A bond with $6$ years to maturity and a $4\%$ coupon rate $($coupons are paid annually$).$ D$.$ A bond with $4$ years to maturity and a $7\%$ coupon rate $($coupons are paid semi$-$annually$).$ E$.$ A bond with $4$ years to maturity and a $5\%$ coupon rate $($coupons are paid semi$-$annually$).$ a$)$ Rank these bonds according to their interest rate sensitivities$,$ from the most interest rate sensitive to the least interest rate sensitive$.$ b$)$ If you want to profit from an unexpected decrease in market interest rates, which bond should you purchase? c$)$ What is the duration of the bond you chose in part b$)?$ d$)$ If you are worried that market interest rates might increase unexpectedly, which bond should you purchase?