Chapter 4, Problem 61QP

Summary Introduction

To calculate: The quarterly salary of the player.

Explanation of Solution

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:

$\begin{array}{c}\text{Effective annual rate}=\left(1+{\left(\frac{\text{APR}}{12}\right)}^{12}-1\right)\\ =\left(1+{\left(\frac{0.05}{365}\right)}^{365}-1\right)\\ =\left(1.05126-1\right)\\ =0.513\end{array}$

Hence, the effective annual rate is 0.513 or 5.13%.

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:

$\begin{array}{c}\text{Present value }=\frac{\text{Cash flow}}{{\left(1+\text{r}\right)}^{\text{t}}}\\ =\left(\begin{array}{l}\$7,000,000+\frac{\$6,100,000}{\left(1+0.0513\right)}+\frac{\$6,900,000}{{\left(1+0.0513\right)}^2}+\frac{\$7,600,000}{{\left(1+0.0513\right)}^3}+\\ \frac{\$8,200,000}{{\left(1+0.0513\right)}^4}+\frac{\$9,500,000}{{\left(1+0.0513\right)}^5}+\frac{\$8,400,000}{{\left(1+0.0513\right)}^6}\end{array}\right)\\ =\$45,922,807.29\end{array}$

Hence, the present value is $45,922,807.29.

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:

$\begin{array}{c}\text{Present value}=\$45,922,807.29+\$2,500,000\\ =\$48,422,807.29\end{array}$

Hence, the present value of the new contract is $48,422,807.29.

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.

$\begin{array}{c}\text{Remaining amount}=\$48,422,807.29-\$9,000,000\\ =\$39,422,807.29\end{array}$

Hence, the remaining amount is $39,422,807.29.

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:

$\begin{array}{c}\text{Effective annual rate}=\left(1+{\left(\frac{\text{APR}}{365}\right)}^{365}-1\right)\\ =\left(1+{\left(\frac{0.05}{365}\right)}^{91.25}-1\right)\\ =\left(1.01258-1\right)\\ =0.01258\end{array}$

Hence, the effective annual rate is 0.01258 or 1.258%.

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:

  $\begin{array}{c}\text{Present value annuity}=C\left\{\frac{\left[1-\left(\frac{1}{{\left(1+r\right)}^t}\right)\right]}{r}\right\}\\ \$39,422,807.29=C\left\{\frac{\left[1-\left(\frac{1}{{\left(1+0.01258\right)}^{24}}\right)\right]}{0.01258}\right\}\\ \$39,422,807.29=C\left\{\frac{\left[1-\left(\frac{1}{1.349908}\right)\right]}{0.01258}\right\}\\ C=\$1,913,224.07\end{array}$

Hence, the quarterly salary of the player is $1,913,224.07.