Suppose two athletes sign 10-year contracts for €80 million. In one case, we are told that the €80 million will be paid in 10 equal installments. In the other case, we are told that the Euros €80 million will be paid in 10 installments, but the installments will increase by 5% per year. Who got the better deal?
As I read the problem, the first athlete who will be paid in 10 equal installments got the better deal.
I assume that the second athlete also gets a total of 80 million Euros, but that his initial yearly payments are less than 8 million.
To determine who got the better deal, we need to compare the total amount received by each athlete over the 10-year period.
First, let's calculate the total amount received by the athlete who signed the contract with 10 equal installments. Since the €80 million will be paid in equal installments, each installment will be €80 million divided by 10, which equals €8 million. Since there are 10 equal installments, the total amount received will be €8 million multiplied by 10, which equals €80 million.
Now, let's calculate the total amount received by the athlete who signed the contract with increasing installments. In this case, the first installment will be €80 million divided by 10, which is €8 million. However, each subsequent installment will increase by 5% compared to the previous installment. To calculate the increasing installments, we will use the following formula:
Installment_n = Installment_1 * (1 + Increase_rate)^(n-1)
Where:
- Installment_n represents the nth installment
- Installment_1 represents the first installment
- Increase_rate represents the percentage increase rate
- n represents the installment number
Using this formula, we can calculate the increasing installments for each year. Let's do the calculations:
Installment_1 = €8 million (as calculated earlier)
Increase_rate = 5% = 0.05
Installment_2 = €8 million * (1 + 0.05)^(2-1) = €8 million * 1.05 = €8.4 million
Installment_3 = €8 million * (1 + 0.05)^(3-1) = €8 million * 1.1025 = €8.82 million
Installment_4 = €8 million * (1 + 0.05)^(4-1) = €8 million * 1.157625 = €9.26 million
Installment_5 = €8 million * (1 + 0.05)^(5-1) = €8 million * 1.21550625 = €9.72 million
Installment_6 = €8 million * (1 + 0.05)^(6-1) = €8 million * 1.2762815625 = €10.21 million
Installment_7 = €8 million * (1 + 0.05)^(7-1) = €8 million * 1.34009564063 = €10.71 million
Installment_8 = €8 million * (1 + 0.05)^(8-1) = €8 million * 1.40710042266 = €11.28 million
Installment_9 = €8 million * (1 + 0.05)^(9-1) = €8 million * 1.4774554438 = €11.91 million
Installment_10 = €8 million * (1 + 0.05)^(10-1) = €8 million * 1.55132821699 = €12.41 million
To get the total amount received, we add up all the installments:
Total amount received = Installment_1 + Installment_2 + ... + Installment_10
Total amount received = €8 million + €8.4 million + €8.82 million + €9.26 million + €9.72 million + €10.21 million + €10.71 million + €11.28 million + €11.91 million + €12.41 million
Total amount received ≈ €108.41 million
Now, comparing the total amount received by each athlete over the 10-year period, we find that the athlete who signed the contract with increasing installments received approximately €108.41 million, while the athlete who signed the contract with equal installments received €80 million.
Therefore, the athlete who signed the contract with increasing installments got the better deal, receiving a higher total amount over the 10-year period.
To determine which athlete got the better deal, we need to calculate the total amount of money they will receive in each case. Let's analyze each case step by step:
Case 1: €80 million paid in 10 equal installments
In this case, the athlete will receive €80 million divided into 10 equal installments. We can calculate the amount of each installment by dividing €80 million by 10:
€80 million / 10 = €8 million
Therefore, the athlete will receive €8 million per year for 10 years.
Case 2: €80 million paid in 10 increasing installments
In this case, the athlete will receive €80 million divided into 10 installments, but the installments will increase by 5% per year. To calculate the amount of each installment, we can use the formula for compound interest:
Principal (P) = €80 million
Interest rate (r) = 5%
The formula to calculate compound interest is:
A = P(1 + r)^n
Where A is the final amount, P is the principal, r is the interest rate, and n is the number of years.
Using this formula, we can calculate the amount of each installment for the 10 years:
Year 1: P = €80 million
Installment = P = €80 million
Year 2: P = €80 million + (€80 million * 5%) = €80 million + €4 million = €84 million
Installment = €84 million
Year 3: P = €84 million + (€84 million * 5%) = €84 million + €4.2 million = €88.2 million
Installment = €88.2 million
And so on, until we reach the 10th year.
By calculating the installments for all 10 years, we can add them up to find the total amount the athlete will receive over the course of the contract.
Now that we have all the necessary information, we can compare the total amount received by each athlete in both cases.