A famous quarterback just signed a $\$15$ million contract providing $\$3$ million a year for $5$ years. A less famous receiver signed a $\$14$ million $5$ -year contract providing $\$4$ million now and $\$2$ million a year for $5$ years. The interest rate is $10$ $\%$ . $\newline$ a. What is the PV of the quarterback's contract? $\newline$ Note: Do not round intermediate calculations. Enter your answer in millions rounded to $2$ decimal places. $\newline$ Present value = $\square$ million

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Grade 6

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Q. A famous quarterback just signed a

\$15

million contract providing

\$3

million a year for

5

years. A less famous receiver signed a

\$14

million

5

-year contract providing

\$4

million now and

\$2

million a year for

5

years. The interest rate is

10

\%

\newline

a. What is the PV of the quarterback's contract?

\newline

Note: Do not round intermediate calculations. Enter your answer in millions rounded to

2

decimal places.

\newline

Present value =

\square

million

Identify Variables: To calculate the present value (PV) of the quarterback's contract, we need to discount each of the $\$3$ million payments back to the present value using the formula for the present value of an ordinary annuity. The formula for the present value of an ordinary annuity is $PV = PMT \times \left[\frac{1 - (1 + r)^{-n}}{r}\right]$ , where $PMT$ is the annual payment, $r$ is the interest rate per period, and $n$ is the number of periods.
Plug Values: First, we identify the variables needed for the calculation: $\newline$ $PMT$ (annual payment) = $\$3$ million $\newline$ $r$ (interest rate per period) = $10\%$ or $0.10$ $\newline$ $n$ (number of periods) = $5$ years
Calculate Present Value: Now we plug these values into the present value formula for an ordinary annuity: $\newline$ $PV = \$3 \text{ million} \times \left[\frac{1 - (1 + 0.10)^{-5}}{0.10}\right]$
Round to Two Decimals: Next, we calculate the present value: $\newline$ $PV = (\$)3 \text{ million} \times \left[\frac{1 - (1 + 0.10)^{-5}}{0.10}\right]$ $\newline$ $PV = (\$)3 \text{ million} \times \left[\frac{1 - (1.10)^{-5}}{0.10}\right]$ $\newline$ $PV = (\$)3 \text{ million} \times \left[\frac{1 - 0.620921323059155}{0.10}\right]$ $\newline$ $PV = (\$)3 \text{ million} \times \left[\frac{0.379078676940845}{0.10}\right]$ $\newline$ $PV = (\$)3 \text{ million} \times 3.79078676940845$ $\newline$ $PV = (\$)11.37236030822535 \text{ million}$
Round to Two Decimals: Next, we calculate the present value: $\newline$ $PV = \$3 \text{ million} \times \left[\frac{1 - (1 + 0.10)^{-5}}{0.10}\right]$ $\newline$ $PV = \$3 \text{ million} \times \left[\frac{1 - (1.10)^{-5}}{0.10}\right]$ $\newline$ $PV = \$3 \text{ million} \times \left[\frac{1 - 0.620921323059155}{0.10}\right]$ $\newline$ $PV = \$3 \text{ million} \times \left[\frac{0.379078676940845}{0.10}\right]$ $\newline$ $PV = \$3 \text{ million} \times 3.79078676940845$ $\newline$ $PV = \$11.37236030822535 \text{ million}$ Finally, we round the present value to two decimal places as instructed: $\newline$ $PV = \$11.37 \text{ million}$