91Ó°ÊÓ

$\begin{array}{c}\mathbf{Presentvalue}\mathbf{}\mathbf{=}\mathbf{Coupon}\mathbf{×}\frac{\mathbf{1}\mathbf{-}{(\mathbf{1}\mathbf{+}\mathbf{r})}^{-n}}{r}\mathbf{+}\frac{\mathrm{Facevalue}}{{(\mathbf{1}\mathbf{+}\mathbf{r})}^{-n}}\\ \mathbf{=}\mathbf{150}\mathbf{×}\frac{\mathbf{1}\mathbf{-}\mathbf{(}\mathbf{1}\mathbf{+}\mathbf{0}\mathbf{.}\mathbf{12}{\mathbf{)}}^{-20}}{\mathbf{0}\mathbf{.}\mathbf{12}}\mathbf{+}\frac{1,000}{{\mathbf{(}\mathbf{1}\mathbf{+}\mathbf{0}\mathbf{.}\mathbf{12}\mathbf{)}}^{-20}}\\ \mathbf{=}\mathbf{\$}\mathbf{1}\mathbf{,}\mathbf{224}\mathbf{.}\mathbf{35}\end{array}$

$\begin{array}{c}\mathbf{Presentvalue}\mathbf{=}\mathbf{Annualpayment}\mathbf{×}\frac{\mathbf{1}\mathbf{-}{(\mathbf{1}\mathbf{+}\mathbf{r})}^{-n}}{r}\mathbf{+}\frac{\mathrm{Facevalue}}{{(\mathbf{1}\mathbf{+}\mathbf{r})}^{-n}}\\ \mathbf{=}\mathbf{\$}\mathbf{30}\mathbf{×}\frac{\mathbf{1}\mathbf{-}{(\mathbf{1}\mathbf{+}\mathbf{0}\mathbf{.}\mathbf{12})}^{-20}}{\mathbf{0}\mathbf{.}\mathbf{12}}\mathbf{+}\frac{\mathbf{1},\mathbf{000}}{{\mathbf{(}\mathbf{1}\mathbf{+}\mathbf{0}\mathbf{.}\mathbf{12}\mathbf{)}}^{-20}}\\ \mathbf{=}\mathbf{\$}\mathbf{1},\mathbf{224}\mathbf{.}\mathbf{07}\end{array}$

The part a $1,224.35 and part b $1,224.07 are actually the same because in both cases the present value computed of a $30 differential between actual return and required return for 20 years.

In part a, the goal is accomplish by valuing all future benefits at a 3% differential between stated return (coupon = 12%) and required return (10%)to arrive at $1,224.35.

In part b, the present value of the $30 differential to arrive at $224.07. Add this value to the $1,000 par value that is exactly equal to its market value because the remaining 12% coupon ($150 – $30 = $120 coupon) equals the 12%market rate.When the coupon rate equals the required rate, the market value equals the par value.