Chapter 4, Problem 59QP

Summary Introduction

To calculate: The monthly savings of person BB.

Explanation of Solution

Time line of the cash flow:

Note: The cash flows from the given information takes place monthly basis, the given interest rate is an effective annual rate. As the cash flows take place on a monthly basis, it is essential to compute the effective monthly rate by finding the annual percentage rate through monthly compounding, and then dividing it by 12. The preretirement annual percentage rate is calculated as follows:

Formula to calculate the effective annual rate:

$\text{Effective annual rate}=\left(1+{\left(\frac{\text{APR}}{12}\right)}^{12}-1\right)$

Compute the effective annual rate:

$\begin{array}{c}\text{Effective annual rate}=\left(1+{\left(\frac{\text{APR}}{12}\right)}^{12}-1\right)\\ 0.11=\left(1+{\left(\frac{\text{APR}}{12}\right)}^{12}-1\right)\end{array}$

Compute the annual percentage rate with the effective annual rate:

$\begin{array}{c}\text{Annual percentage rate}=12\left({\left(1+\text{EAR}\right)}^{\frac{1}{12}}-1\right)\\ =12\left({\left(1+0.11\right)}^{0.083333333}-1\right)\\ =0.1048\end{array}$

Hence, the annual percentage rate is 0.1048 or 10.48%.

The annual percentage rate for the post-retirement is calculated as follows:

Formula to calculate the effective annual rate:

$\text{Effective annual rate}=\left(1+{\left(\frac{\text{APR}}{12}\right)}^{12}-1\right)$

Compute the effective annual rate:

$\begin{array}{c}\text{Effective annual rate}=\left(1+{\left(\frac{\text{APR}}{12}\right)}^{12}-1\right)\\ 0.07=\left(1+{\left(\frac{\text{APR}}{12}\right)}^{12}-1\right)\end{array}$

Compute the annual percentage rate with the effective annual rate:

$\begin{array}{c}\text{Annual percentage rate}=12\left({\left(1+\text{EAR}\right)}^{\frac{1}{12}}-1\right)\\ =12\left({\left(1+0.07\right)}^{0.083333333}-1\right)\\ =12\left(1.005654145-1\right)\\ =0.067849744\end{array}$

Hence, the annual percentage rate is 0.067 or 6.78%.

First, it is essential to compute the retirement needs of Person BB. The amount that is essential for the retirement is the present value of the monthly spending plus the inheritance’s present value. The present value of the two cash flows is as follows:

Formula to calculate the present value annuity:

$\text{Present value annuity}=C\left\{\frac{\left[1-\left(\frac{1}{1+r^t}\right)\right]}{r}\right\}$

Note: C denotes the payments, r denotes the rate of exchange, and t denotes the period.

Compute the present value annuity for without fee:

  $\begin{array}{c}\text{Present value annuity}=C\left\{\frac{\left[1-\left(\frac{1}{{\left(1+r\right)}^t}\right)\right]}{r}\right\}\\ =\$25,000\left\{\frac{\left[1-\left(\frac{1}{{\left(1+\frac{0.067849744}{12}\right)}^{12\left(20\right)}}\right)\right]}{\frac{0.067849744}{12}}\right\}\\ =\$25,000\left\{\frac{\left[1-\left(\frac{1}{{\left(1.005654145\right)}^{240}}\right)\right]}{\frac{0.067849744}{12}}\right\}\\ =\$3,278,926.11\end{array}$

Formula to calculate the present value:

$\text{Present value}=\frac{\text{Cash flow}}{{\left(1+\text{r}\right)}^{\text{t}}}$

Compute the present value:

$\begin{array}{c}\text{Present value}=\frac{\text{Cash flow}}{{\left(1+\text{r}\right)}^{\text{t}}}\\ =\frac{\$1,000,000}{{\left(1+0.07\right)}^{20}}\\ =\frac{\$1,000,000}{{\left(1.07\right)}^{20}}\\ =\$258,419\end{array}$

Hence, the present value is $258,419.

Person BB will be saving $2,200 per month for the upcoming ten years until he buys the cabin. The value of his savings after ten years is calculated as follows:

Formula to calculate the future value annuity:

$\text{Future value annuity}=C\left\{\frac{\left[{\left(1+r\right)}^t-1\right]}{r}\right\}$

Note: C denotes the annual cash flow or the annuity payment, r denotes the rate of interest, and t denotes the number of payments.

Compute the future value annuity of Person BB:

$\begin{array}{c}\text{Future value annuity}=C\left\{\frac{\left[{\left(1+r\right)}^t-1\right]}{r}\right\}\\ =\$2,200\left\{\frac{\left[{\left(1+\frac{0.1048}{12}\right)}^{12\left(10\right)}-1\right]}{\frac{0.1048}{12}}\right\}\\ =\$463,298.72\end{array}$

Hence, the future value of the annuity is $463,298.72.

The amount that remains in the hands of Person BB after the purchase of the cabin is as follows:

$\$463,298.72-\$340,000=\$123,298.72$

Note: The amount that remains in the hands of Person BB is calculated by subtracting the cost of the cabin from the calculated future value of the annuity.

Hence, the amount that person BB has in his hands is $123,298.72

Person BB still has twenty years until the retirement and at the time when he is ready to retire, the amount he would receive is as follows:

Formula to compute the future value:

$\text{Future value}=\text{PV}{\left(1+\text{r}\right)}^{\text{t}}$

Note: C denotes the annual cash flow or the annuity payment, r denotes the rate of interest, and t denotes the number of payments.

Compute the future value:

$\begin{array}{c}\text{Future value}=\text{PV}{\left(1+\text{r}\right)}^{\text{t}}\\ =\$123,298.72{\left(1+\frac{0.1048}{12}\right)}^{12\left(20\right)}\\ =\$994,072.72\end{array}$

Hence, the future value is $994,072.72.

Thus, when Person BB is ready for the retirement based on his present savings, he will be short of the below amount:

$\$3,537,345.11-\$994,072.72=\$2,543,272.39$

Thus, the above calculated amount is the future value of the monthly savings that Person BB has to make between ten to thirty years.

Hence, the future value of an annuity of person BB is $2,543,272.39.

Compute the monthly savings of Person BB using the Formula of the future value of an annuity:

$\begin{array}{c}\text{Future value annuity}=C\left\{\frac{\left[{\left(1+r\right)}^t-1\right]}{r}\right\}\\ \$2,543,272.39=C\left\{\frac{\left[{\left(1+\frac{0.1048}{12}\right)}^{12\left(20\right)}-1\right]}{\frac{0.1048}{12}}\right\}\\ C=\$3,145.49\end{array}$

Hence, the monthly savings of person BB is $3,145.49.