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Math  /  Calculus

QuestionBasketball player Patrick Ewing received a contract from the New York Knicks in which he was offered $3\$ 3 million a year for 10 years. Determine the present value of the contract on the date of the first payment if the interest rate is 4%4 \% per year, compounded continuously.
Round your answer to two decimal places.
Present value == i □\square millions of dollars

Studdy Solution

STEP 1

What is this asking? How much is Patrick Ewing's 10-year, $3\$3 million/year contract *really* worth today, considering a continuous 4% yearly interest rate? Watch out! Don't forget that the interest is compounded *continuously*, not annually!
This changes the formula we'll use.

STEP 2

1. Define the present value formula for continuous compounding.
2. Calculate the present value of each payment.
3. Sum the present values to find the total present value.

STEP 3

Alright, so we're dealing with continuous compounding, which means our present value formula looks a little different than the usual one.
The present value PVPV of a single payment PP made tt years from now, with a continuous interest rate rr, is given by: PV=P⋅e−rt PV = P \cdot e^{-rt} Where ee is the magical mathematical constant approximately equal to 2.71828.
It's like the superhero of continuous growth and decay!

STEP 4

Ewing's contract pays him $3\$3 million each year for 10 years.
So, we need to calculate the present value of each of these $3\$3 million payments.

STEP 5

For the **first payment**, t=0t = 0, since it's received today: PV0=3⋅e−0.04⋅0=3⋅e0=3⋅1=$3 million PV_0 = 3 \cdot e^{-0.04 \cdot 0} = 3 \cdot e^0 = 3 \cdot 1 = \$3 \text{ million}

STEP 6

For the **second payment**, t=1t = 1: PV1=3⋅e−0.04⋅1≈3⋅0.9608≈$2.8824 million PV_1 = 3 \cdot e^{-0.04 \cdot 1} \approx 3 \cdot 0.9608 \approx \$2.8824 \text{ million}

STEP 7

We continue this for all 10 payments, up to the **tenth payment** at t=9t = 9: PV9=3⋅e−0.04⋅9≈3⋅0.7047≈$2.1141 million PV_9 = 3 \cdot e^{-0.04 \cdot 9} \approx 3 \cdot 0.7047 \approx \$2.1141 \text{ million}

STEP 8

To find the total present value of the contract, we add up the present values of all 10 payments: Total PV=PV0+PV1+⋯+PV9 \text{Total } PV = PV_0 + PV_1 + \dots + PV_9 Total PV=∑t=093⋅e−0.04t \text{Total } PV = \sum_{t=0}^{9} 3 \cdot e^{-0.04t}

STEP 9

Calculating this sum, we get: Total PV≈3+2.8824+⋯+2.1141≈$24.0159 million \text{Total } PV \approx 3 + 2.8824 + \dots + 2.1141 \approx \$24.0159 \text{ million}

STEP 10

The present value of Patrick Ewing's contract is approximately $24.02\$24.02 million.

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