Chapter 24, Problem 8P

a)

Summary Introduction

To determine: The present value of $10million if the GNMA is fully amortized.

Explanation of Solution

In this case 180(15 years x 12 months) monthly payments are there. The coupon rate at which each month in GNMA is 8.5% ‑ 0.5% = 8 percent per year, and the monthly GNMA pass‑through payment is:

 $\begin{array}{l}\text{\$10Million}=\left\{\frac{\left[\text{1-}\left(\frac{\text{1}}{{\left(1+\frac{\text{coupon rate}}{12}\right)}^{\text{Number of monthly payments}}}\right)\right]}{\left(\frac{\text{coupon rate}}{12}\right)}\right\}\times \text{PMT }\\ \text{10Million}=\left\{\frac{\left[\text{1-}\left(\frac{\text{1}}{{\left(1+\frac{0.08}{12}\right)}^{\text{180}}}\right)\right]}{\left(\frac{0.08}{12}\right)}\right\}\times \text{PMT }\\ \text{ PMT = \$95,565}\text{.21}\end{array}$

Hence, the PMT is $95,565.21

The computation of current value of GNMA at a market rate of 5% is as follows:

$\begin{array}{c}\text{Present value = }\times \text{ PMT}\times \left\{\frac{\left[\text{1-}\left(\frac{\text{1}}{{\left(1+\frac{\text{Market rate}}{12}\right)}^{\text{Number of monthly payments}}}\right)\right]}{\left(\frac{\text{Market rate}}{12}\right)}\right\}\\ =\text{ \$95,565}\text{.21}\times \left\{\frac{\left[\text{1-}\left(\frac{\text{1}}{{\left(1+\frac{0.05}{12}\right)}^{180}}\right)\right]}{\left(\frac{0.05}{12}\right)}\right\}\\ \text{= \$81,115}\text{.94}\end{array}$

b)

Summary Introduction

To compute: The present value of $10million if the GNMA is only half amortized and pass through 50% face value of mortgage pools.

Explanation of Solution

While considering the GNMA is only half amortized. At the maturity of the GNMA the payment is lump sum that is same as 50 % face value of mortgage pools.

In case of amortization at which 50% the monthly payments of GNMA are:

The computation is as follows:

 $\begin{array}{l}\text{\$10Million}=\left(\begin{array}{l}\text{PMT}\times \left\{\frac{\left[\text{1-}\left(\frac{\text{1}}{{\left(1+\frac{\text{coupon rate}}{12}\right)}^{\text{Number of monthly payments}}}\right)\right]}{\left(\frac{\text{coupon rate}}{12}\right)}\right\}\\ +\frac{50\%\text{10million}}{{\left(1+\frac{\text{coupon rate}}{12}\right)}^{\text{Number of monthly payments}}}=\text{R}\end{array}\right)\\ \text{R}=\left\{\frac{\left[\text{1-}\left(\frac{\text{1}}{{\left(1+\frac{0.08}{12}\right)}^{\text{180}}}\right)\right]}{\left(\frac{0.08}{12}\right)}\right\}\text{+ }\frac{5\text{Million}}{{\left(1+\frac{0.08}{12}\right)}^{\text{180}}}\\ =\$81,115.94\end{array}$

The computation of current value of GNMA at a market rate of 5% is as follow:

$\begin{array}{c}\text{Present value = }\left(\begin{array}{l}\text{ R}\times \left\{\frac{\left[\text{1-}\left(\frac{\text{1}}{{\left(1+\frac{\text{Market rate}}{12}\right)}^{\text{Number of monthly payments}}}\right)\right]}{\left(\frac{\text{Market rate}}{12}\right)}\right\}\\ +\frac{50\%\text{10million}}{{\left(1+\frac{\text{Market rate}}{12}\right)}^{\text{Number of monthly payments}}}\end{array}\right)\\ =\text{ \$81,115}\text{.94}\times \left\{\frac{\left[\text{1-}\left(\frac{\text{1}}{{\left(1+\frac{0.05}{12}\right)}^{180}}\right)\right]}{\left(\frac{0.05}{12}\right)}\right\}+\frac{50\%\text{10million}}{{\left(1+\frac{0.05}{12}\right)}^{\text{180}}}\\ =\$12,623,051.35\\ \end{array}$

Hence, the present value is $12,623,051.35