Chapter 6, Problem 59QP

Summary Introduction

To calculate: The quarterly salary of the player.

Introduction:

The series of payments that are made at equal intervals is an annuity payment. The amount of annuity payments is mainly calculated based on the particular situation.

Answer to Problem 59QP

The quarterly salary of the player is $1,597,735.62.

Explanation of Solution

Given information:

The defensive lineman of the AP Team is in a contract of negotiations. The salary structure of the team is as follows:

• At time zero, the salary is $6,500,000
• At time one, the salary is $5,100,000
• At time two, the salary is $5,600,000
• A time three, the salary is $6,100,000
• At time four, the salary is $7,500,000
• At time five, the salary is $8,200,000
• At time six, the salary is $9,000,000

The payment of the salaries is made in lump sum. The player has requested his agent Person X to negotiate again on the terms. The player needs a bonus of $10 million that must be paid today and the value of the contract rises to $2,000,000. The player also needs an equivalent salary that must be paid every 3 months from the present. The rate of interest is 4.8% that is compounded daily. It is assumed that there are 365 days in a year.

Note: To determine the player’s quarterly salary, it is essential to find the present value of the current contract. The contract’s cash flow is on the annual basis and the given information is the daily interest rate. Thus, it is essential to determine the effective annual rate so that the compounding interest would be similar to the cash flows timing.

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:

$\begin{array}{c}\text{Effective annual rate}=\left(1+{\left(\frac{\text{APR}}{12}\right)}^{12}-1\right)\\ =\left(1+{\left(\frac{0.048}{365}\right)}^{365}-1\right)\\ =\left(1.049164721-1\right)\\ =0.4916\end{array}$

Hence, the effective annual rate is 0.4912 or 4.92%.

Formula to calculate the present value:

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

Note: r denotes the rate of discount and t denotes the number of years. The present value of the current contract is the sum of the cash flows’ present value.

Compute the present value:

$\begin{array}{c}\text{Present value }=\frac{\text{Cash flow}}{{\left(1+\text{r}\right)}^{\text{t}}}\\ =\left(\begin{array}{l}\$6,500,000+\frac{\$5,100,000}{\left(1+0.0492\right)}+\frac{\$5,600,000}{{\left(1+0.0492\right)}^2}+\frac{\$6,100,000}{{\left(1+0.0492\right)}^3}+\\ \frac{\$7,500,000}{{\left(1+0.0492\right)}^4}+\frac{\$8,200,000}{{\left(1+0.0492\right)}^5}+\frac{\$9,000,000}{{\left(1+0.0492\right)}^6}\end{array}\right)\\ =\left(\begin{array}{l}\$6,500,000+\$4,860,846.359+\$5,087,113.919+\$5,281,472.092+\\ \$6,189,109.662+\$6,449,446.475+\$6,746,722.04\end{array}\right)\\ =\$41,118,705.53\end{array}$

Hence, the present value is $41,118,705.53.

As the player wishes to increase the contract value by $2,000,000, then the present value of the contract can be calculated as follows:

$\begin{array}{c}\text{Present value}=\$41,118,705.53+\$2,000,000\\ =\$43,118,705.53\end{array}$

Hence, the present value of the new contract is $43,118,705.53.

Note: The player also requested to sign the bonus that is payable at present with the amount of $10 million. To compute the remaining amount, the bonus amount should be subtracted from the new contract’s present value. The remaining amount is the present value of the quarterly payments.

$\begin{array}{c}\text{Remaining amount}=\$43,118,705.53-\$10,000,000\\ =\$33,118,705.53\end{array}$

Hence, the remaining amount is $33,118,705.53.

Note: To determine the quarterly payments first, it is essential to understand that the interest rate that is needed is the effective annual rate. The quarterly interest rate can be found by the effective annual rate equation with the daily interest rate. The number of days in a quarter is 91.25 days $\left(\frac{365}{4}\right)$.

Formula to calculate the effective quarterly rate:

$\text{Effective annual rate}=\left(1+{\left(\frac{\text{APR}}{365}\right)}^{365}-1\right)$

Compute the effective annual rate:

$\begin{array}{c}\text{Effective annual rate}=\left(1+{\left(\frac{\text{APR}}{365}\right)}^{365}-1\right)\\ =\left(1+{\left(\frac{0.048}{365}\right)}^{91.25}-1\right)\\ =\left(1.01207149-1\right)\\ =0.01207\end{array}$

Hence, the effective annual rate is 0.01207 or 1.207%.

Now the rate of interest, the annuity length, and the present value is known. Utilizing the Formula of the present value of annuity, the quarterly salary of the player can be determined.

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:

$\begin{array}{c}\text{Present value annuity}=C\left\{\frac{\left[1-\left(\frac{1}{{\left(1+r\right)}^t}\right)\right]}{r}\right\}\\ \$33,118,705.53=C\left\{\frac{\left[1-\left(\frac{1}{{\left(1+0.01207\right)}^{24}}\right)\right]}{0.01207}\right\}\\ \$33,118,705.53=C\left\{\frac{\left[1-\left(\frac{1}{1.333732051}\right)\right]}{0.01207}\right\}\\ \$33,118,705.53=C\left\{\frac{0.250224211}{0.01207}\right\}\end{array}$

$\begin{array}{c}\$33,118,705.53=C\left\{20.72852744\right\}\\ C=\$1,597,735.62\end{array}$

Hence, the quarterly salary of the player is $1,592,735.62.