Chapter 6, Problem 6.14P

(1)

Present Value:

The value of today’s amount to be paid or received in the future at a compound interest rate is called as present value. The following formula is used to calculate the present value of an amount:

$\text{Present value of an amount = }\frac{\text{Future value}}{{\left(\text{1 + interest rate}\right)}^{\text{number}\text{of}\text{periods}}}$

Future Value: The future value is value of present amount compounded at an interest rate until a particular future date. The future value of an amount is calculated by using the following formula:

$\text{Future value of an amount = Present value}\times {\left(\text{1+ Interest rate}\right)}^{\text{Number}\text{of}\text{periods}}$

To determine

The present value of the pension obligation to three employees as of December 31, 2018.

Explanation of Solution

• For Employee T:

Determine the present value of an ordinary annuity.

$\begin{array}{c}\text{Present value annuity=}\left(\begin{array}{l}\text{Annuity amount×Present value}\\ \text{of an oridinary annuity of \$1}\end{array}\right)\\ \text{=\$20,000×7}\text{.19087}\\ \text{=\$143,817}\end{array}$ (1)

PV factor (Present value of an ordinary annuity of $1: n = 15, i = 11%) is taken from the table value (Table 4 in Appendix from textbook).

Determine the present value.

$\begin{array}{c}\text{Present value=Present value annuity×Present value of \$1 }\\ \text{=\$143,817}\left(\text{1}\right)\text{×}\text{.81162}\\ \text{=\$116,725}\end{array}$

PV factor (Present value of $1: n = 2, i = 10%) is taken from the table value (Table 2 in Appendix from textbook).

• For Employee E:

Determine the present value of an ordinary annuity.   

$\begin{array}{c}\text{Present value annuity=}\left(\begin{array}{l}\text{Annuity amount×Present value}\\ \text{of an oridinary annuity of \$1}\end{array}\right)\\ \text{=\$25,000×7}\text{.19087}\\ \text{=\$179,772}\end{array}$

PV factor (Present value of an ordinary annuity of $1: n = 15, i = 11%) is taken from the table value (Table 4 in Appendix from textbook).

Determine the present value.

$\begin{array}{c}\text{Present value=Present value annuity×Present value of \$1 }\\ \text{=\$179,772}\left(\text{1}\right)\text{×}\text{.73119}\\ \text{=\$131,447}\end{array}$

PV factor (Present value of $1: n = 3, i = 11%) is taken from the table value (Table 2 in Appendix from textbook).

• For Employee C:

Determine the present value of an ordinary annuity.   

$\begin{array}{c}\text{Present value annuity=}\left(\begin{array}{l}\text{Annuity amount×Present value}\\ \text{of an oridinary annuity of \$1}\end{array}\right)\\ \text{=\$30,000×7}\text{.19087}\\ \text{=\$215,726}\end{array}$

PV factor (Present value of an ordinary annuity of $1: n = 15, i = 11%) is taken from the table value (Table 4 in Appendix from textbook).

Determine the present value.

$\begin{array}{c}\text{Present value=Present value annuity×Present value of \$1 }\\ \text{=\$215,726}\left(\text{1}\right)\text{×}\text{.65873}\\ \text{=\$142,105}\end{array}$

PV factor (Present value of $1: n = 4, i = 11%) is taken from the table value (Table 2 in Appendix from textbook).

Conclusion

Thus, following are the present values of the pension obligation of the three employees.  

Employees	Present values ($)
T	116,725
E	131,447
C	142,105

Table (1)

(2)

To determine

To compute: The annual contribution.

Explanation of Solution

Determine the present value of pension obligation as of December 31, 2021.

Employee	PV as of December 31, 2018	x	FV of $1 factor,	=	FV as of December 31, 2018
			n = 3, i = 11%
T	$116,725	x	1.36763	=	$159,637
E	131,448	x	1.36763	=	179,772
C	142,105	x	1.36763	=	194,347
Total present value, December 31, 2021					$533,756

Table (2)

Compute the annual contribution using future value of annuity due.

$\begin{array}{c}\begin{array}{l}\text{Future value}\\ \text{ annuity due}\end{array}\}=\text{Annuity amount}\times \text{Future value of an annuity due of \$1 }\\ \$533,756=\text{Annuity amount}\times 3.7097\end{array}$

$\begin{array}{l}\Rightarrow \text{Annuity amount = }\frac{\$533,756}{3.7097}\\ \Rightarrow \text{Annuity amount = \$143,881}\end{array}$

FV factor (Future value of an annuity due of $1: n =3, i =11%) is taken from the table value (Table 5 in Appendix from textbook).

Conclusion

Hence, the first contribution that will be made on December 31, 2016 is $143,881.