$6\mathrm{.}1\mathrm{.}4$

$($a$)$ You are given the following information about two $10-$year

bonds. Both bonds have face amount $100$ and coupons

payable semi$-$annually, with next coupon due in $\frac{1}{2}-$year.

Bond $1$: Coupon rate $4\%$ per year, price $85\mathrm{.}12$

Bond $2$: Coupon rate $10\%$ per year, price $133\mathrm{.}34.$

Find the yield rate for a $10-$year zero coupon bond.

$($b$)$ You are given the following term structure $($effective annual

interest rates$)$ for zero coupon bond maturities up to $n$ years:

$nnr>0j{r}_{n-1}.r_1=r_2=$cdots$=r_{n-1}(flat$ term structure except for $n-$year

maturity$).Ann-$year bond has annual coupon rate $r>0$ and

annual coupons. Show that the yield $to$ maturity for the bond

$j$ must satisfy $r_{n-1}.$