Chapter 4, Problem 59QP

Summary Introduction

To determine: The amount of quarterly check.

Annuity:

The annuity refers to the payments of equal amount to be made after certain periods. These payments are made monthly, semi-annually or annually.

Effective Annual Rate:

The effective annual rate is the rate which is incurred or received on the various investment or loans. The effective annual rate is affected by the increase in compounding years.

Explanation of Solution

Given,

The salary structure is offered for 6 years.

The expected increase in the contract amount is $2,700,000.

The amount of signup bonus is $10 million.

The salary is needed in every 3 months.

The annual interest rate is 5.7% that is compounded on daily basis.

Calculation of the value of the quarterly check:

The formula to calculate the quarterly check amount is,

$\text{Quarterly}\text{salary}=\frac{\text{Present}\text{value}}{{\text{PVIFA}}_{r,\text{period}}}$

Substitute $28,069,200 for the present value and 20.19 for the ${\text{PVIFA}}_{1.43\%,24}$ (refer working note) in the above formula.

$\begin{array}{c}\text{Quarterly}\text{salary}=\frac{\$28,069,200}{20.19}\\ =1,390,252.6\end{array}$

The monthly savings are $1,390,252.60.

Working note:

Calculation of the effective annual rate:

$\begin{array}{c}\text{Effective}\text{Annual}\text{Rate}={\left(1+\frac{\text{Annual}\text{percentage}\text{rate}}{\text{Compounding}\text{periods}}\right)}^{\text{Corresponding}\text{periods}}-1\\ ={\left(1+\frac{0.057}{365}\right)}^{365}-1\\ ={\left(1+0.00016\right)}^{365}-1\\ =0.0587\text{or}5.87\%\end{array}$

The effective annual rate is 5.87%.

Calculation of the present value of the contract offer:

The present value of the contract offer is $35,369,200.

Computation of the present value of the new contract,

$\begin{array}{c}\text{Present}\text{value}\text{of}\text{new}\text{contract}=\left(\begin{array}{c}\text{Present}\text{value}\text{of}\text{current}\text{contract}\\ +\text{Increase}\text{in}\text{contract}\text{value}\end{array}\right)\\ =\$35,369,200+\$2,700,000\\ =\$38,069,200\end{array}$

The present value of new contract is $38,069,200.

Calculation of the effective quarterly rate,

$\begin{array}{c}\text{Effective}\text{quarterly}\text{rate}={\left(1+\frac{\text{Annual}\text{percentage}\text{rate}}{\text{Compounding}\text{periods}}\right)}^{\text{Corresponding}\text{periods}}-1\\ ={\left(1+\frac{0.057}{365}\right)}^{\frac{365}{4}}-1\\ ={\left(1+0.00016\right)}^{91.25}-1\\ =0.0143\text{or}1.43\%\end{array}$

The effective quarterly rate is 1.43%.

Calculation of the present value of the quarterly salary,

$\begin{array}{c}\text{Present}\text{value}\text{of}\text{quarterly}\text{salary}=\text{Present}\text{value}\text{of}\text{new}\text{contract}-\text{Signing}\text{bonus}\\ =\$38,069,200-\$10,000,000\\ =\$28,069,200\end{array}$

The present value of quarterly salary is $28,069,200.

Calculation of the ${\text{PVIFA}}_{1.43\%,24}$,

$\begin{array}{c}{\text{PVIFA}}_{1.43\%,24}=\left[\frac{\left\{1-{\left(\frac{1}{1+r}\right)}^n\right\}}{r}\right]\\ =\left[\frac{\left\{1-{\left(\frac{1}{1+0.0143}\right)}^{24}\right\}}{0.0143}\right]\\ =\frac{1-0.7112}{0.0143}\\ =20.19\end{array}$

Conclusion

Thus, the amount of quarterly check will be $1,390,252.60.