Chapter 6, Problem 32QP

Summary Introduction

To calculate: The amount that Person X can withdraw every month.

Introduction:An annuity is a fixed flow of cash for a particular period. They often appear in financial arrangements and it is a shortcut and a useful method for finding the present and future value of the cash flows.

Answer to Problem 32QP

Person X can withdraw $15,621.21 per month.

Explanation of Solution

Given information:

Person X is planning to maintain a balance for retirement for upcoming 30 years. Thus, Person X will invest $800 and $400 in stock and bond account respectively. The return from the stock and bond account is 10% and 6% respectively. However, Person X will combine amount of the two accounts into one account with 7% return when he retires. The period of withdrawal is assumed for 25 years.

Note:

• We must compute the annuity payment at the time of retirement.
• The savings for retirement ends and the withdrawals of retirement begins. Hence, the present value of the retirement withdrawals will be the future value of the savings of retirement.
• Thus, we must compute the future value of the stock and bond account and then we must take the sum of the two future values.

Equation 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 annuity payment, r denotes the rate of interest, and t denotes the period.

Compute the future value annuity for stock account:

$\begin{array}{c}\text{Future value annuity}=C\left\{\frac{\left[{\left(1+r\right)}^t-1\right]}{r}\right\}\\ =\$800\left[\frac{{\left(1+\frac{0.10}{12}\right)}^{30\times 12}-1}{\frac{0.10}{12}}\right]\\ =\$800\left[\frac{{\left(1+0.008333333333\right)}^{360}-1}{0.008333333333}\right]\\ =\$800\left[\frac{{\left(1.008333333\right)}^{360}-1}{0.008333333333}\right]\end{array}$

$\begin{array}{c}=\$800\left[\frac{19.83739937-1}{0.008333333333}\right]\\ =\$800\left[\frac{18.83739937}{0.008333333333}\right]\\ =\$800\times 2,260.487924\\ =\$1,808,390.34\end{array}$

Note: The investment amount and the rate of return for stock account are $800 and 10% respectively. To compute the monthly return, divide the rate by 12 and multiply the number of periods by 12.

Hence, the future value annuity for stock account is $1,808,390.34.

Compute the future value annuity for bond account:

$\begin{array}{c}\text{Future value annuity}=C\left\{\frac{\left[{\left(1+r\right)}^t-1\right]}{r}\right\}\\ =\$400\left[\frac{{\left(1+\frac{0.06}{12}\right)}^{30\times 12}-1}{\frac{0.10}{12}}\right]\\ =\$400\left[\frac{{\left(1+0.005\right)}^{360}-1}{0.005}\right]\\ =\$400\left[\frac{{\left(1.005\right)}^{360}-1}{0.005}\right]\end{array}$

$\begin{array}{c}=\$400\left[\frac{6.022575212-1}{0.005}\right]\\ =\$400\left[\frac{5.022575212}{0.005}\right]\\ =\$400\times 1,004.515042\\ =\$401,806.02\end{array}$

Note: The investment amount and the rate of return for stock account are $400 and 6% respectively. To compute the monthly return, divide the rate by 12 and multiply the number of periods by 12.

Hence, the future value annuity for stock account is $401,806.02.

Formula to calculate the overall amount saved for retirement:

$\text{Total amount for retirement}=\text{Future value of stock account}+\text{Future value of bond account}$

Compute the overall amount saved for retirement:

$\begin{array}{c}\text{Total amount for retirement}=\text{Future value of stock account}+\text{Future value of bond account}\\ =\$1,808,390.3392+\$401,806.017\\ =\$2,210,196.36\end{array}$

Note: This total amount is the present value annuity.

Hence, the total amount that is saved for retirement is $2,210,196.36.

Formula of present value annuity to calculate the cash withdrawal per month:

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

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

Compute the cash withdrawal per month:

$\begin{array}{c}\text{Present value annuity}=C\left\{\frac{\left[1-\left(\frac{1}{{\left(1+r\right)}^t}\right)\right]}{r}\right\}\\ \$2,210,196.356=C\left\{\frac{\left[1-\frac{1}{{\left(1+\frac{0.07}{12}\right)}^{25\times 12}}\right]}{\frac{0.07}{12}}\right\}\\ \$2,210,196.356=C\left\{\frac{\left[1-\frac{1}{{\left(1.005833333\right)}^{300}}\right]}{0.005833333333}\right\}\\ \$2,210,196.356=C\left\{\frac{\left[1-\frac{1}{5.725418209}\right]}{0.005833333333}\right\}\end{array}$

$\begin{array}{c}\$2,210,196.356=C\left\{\frac{1-0.17465973}{0.005833333333}\right\}\\ \$2,210,196.356=C\left(\frac{0.82534027}{0.005833333333}\right)\\ \$2,210,196.356=C\times 141.4869034\end{array}$

$\begin{array}{c}C=\frac{\$2,210,196.356}{141.8469034}\\ C=\$15,621.21\text{ per month}\end{array}$

Note:

• As the period of withdrawal is assumed for 25 years, t must be computed by multiplying the 25 by the number of months in a year.
• The rate of return after combining the amount of the two accounts is 7%. Hence, to compute the monthly return, divide it by 12.

Hence, the cash withdrawal per month is $15,621.21.