Chapter 4, Problem 54QP

Summary Introduction

To determine: The each years amount of deposits for person X’s children.

Explanation of Solution

The computation of present value of college expenses is as follows:

$\begin{array}{l}\begin{array}{l}\text{PVA =C}\left(\frac{\left\{\text{1 – }{\left[\frac{\text{1}}{\left(\text{1 +r}\right)}\right]}^{\text{t}}\right\}}{\text{r}}\right)\hfill \\ =\$55,000\left(\frac{\left\{1–\left[\frac{1}{{\left(1+.092\right)}^4}\right]\right\}}{.092}\right)\hfill \end{array} \\ =\$177,405.18\end{array}$

Hence, the present value of college expense is $177,405.18.

The computation of oldest child’s expenses at present is as follows:

$\begin{array}{l}\text{PV }=\frac{\text{FV}}{{\left(\text{1 +r}\right)}^{\text{t}}}\hfill \\ =\frac{\$177,405.18}{{\left(1+.092\right)}^{14}}\hfill \\ =\$51,742.73\hfill \end{array}$

Hence, the oldest child’s expenses at present is $51,742.73

The computation of expenses of youngest child at present is as follows:

$\begin{array}{l}\text{PV = }\frac{\text{FV}}{{\left(\text{1 +r}\right)}^{\text{t}}}\hfill \\ =\frac{\$177,405.18}{{\left(1+.092\right)}^{16}}\hfill \\ =\$43,391.44\hfill \end{array}$

$\begin{array}{l}\text{PV = }\frac{\text{FV}}{{\left(\text{1 +r}\right)}^{\text{t}}}\hfill \\ =\frac{\$177,405.18}{{\left(1+.092\right)}^{16}}\hfill \\ =\$43,391.44\hfill \end{array}$

Hence, the youngest child’s expenses at present is $43,391.44.

The computation of total college expenses is as follows:

$\begin{array}{l}\text{Cost today = \$51,742}\text{.73 + 43,391}\text{.44}\hfill \\ =\$95,134.17\hfill \end{array}$ $\begin{array}{l}\text{Cost today = \$51,742}\text{.73 + 43,391}\text{.44}\hfill \\ =\$95,134.17\hfill \end{array}$

Hence, the total college expense is $95,134.17.

The computation of current value of annual savings is as follows:

$\begin{array}{c}\text{PVA =C}\left(\frac{\left\{\text{1 – }{\left[\frac{\text{1}}{\left(\text{1 +r}\right)}\right]}^{\text{t}}\right\}}{\text{r}}\right)\\ \text{\$95,134}\text{.17}=\text{C}\left(\frac{\left\{\text{1 – }\left[{\frac{\text{1}}{\left(\text{1 + }\text{.092}\right)}}^{\text{15}}\right]\right\}}{\text{.092}}\right)\\ \text{C}=\text{\$11,941}\text{.94}\end{array}$

Hence, the current value of annual savings is $11,941.94.