Not sure how to solve the following problem.
Suppose two athletes sign 10-year contracts for $80 million. In one case, we're told that the $80 million will be paid in 10 equal installments. In the other case, we're told that the $80 million will be paid in 10 installments, but the installments will increase by 5 percent per year. Who got the better deal?
To determine who got the better deal, we need to compare the total amount received by each athlete over the 10-year period. Let's calculate this.
For the first case, where the $80 million is paid in 10 equal installments, we can divide $80 million by 10 to find each installment's amount. Each installment will be $8 million.
For the second case, where the installments increase by 5 percent per year, we need to calculate the installment amount for each year. The initial installment will still be $8 million, but it will increase by 5 percent each year.
To find the installment amount for each year, we can use the formula:
Installment_amount = Initial_installment_amount * (1 + Annual_increase_rate)^Number_of_years.
For the second case, the installment amounts for each year will be as follows:
Year 1: $8 million * (1 + 0.05)^1 = $8.4 million
Year 2: $8 million * (1 + 0.05)^2 = $8.82 million
Year 3: $8 million * (1 + 0.05)^3 = $9.26 million
and so on, until Year 10.
Now, we can sum up the installment amounts for each athlete to see who received more money over the 10-year period.
For the first case: 10 installments * $8 million = $80 million
For the second case, we need to calculate the sum of the increasing installments. Let's calculate this using the installment amounts we found earlier:
Sum = $8 million + $8.4 million + $8.82 million + $9.26 million + ... (continue summing until Year 10)
Now, you can evaluate the sum, and the athlete who received the higher total amount over the 10-year period will have gotten the better deal.
To determine which athlete got the better deal, we need to compare the total value of the contracts for both cases.
Case 1: $80 million paid in 10 equal installments.
In this case, each installment will be $80 million divided by 10, which equals $8 million.
Case 2: $80 million paid in 10 installments increasing by 5 percent per year.
To calculate the increasing installments, we'll need to use the formula for compound interest:
A = P(1 + r)^n
Where:
A = the future value of the investment
P = the initial principal amount
r = annual interest rate
n = number of times the interest is compounded per year
In this case, we'll treat each installment as a separate investment and compound the interest yearly. Therefore, the formula becomes:
A = P(1 + r)^n
Using this formula, we can calculate the value of the installments as follows:
Year 1: $80 million
Year 2: $80 million * (1 + 0.05)^1 = $84 million
Year 3: $80 million * (1 + 0.05)^2 = $88.2 million
Year 4: $80 million * (1 + 0.05)^3 = $92.61 million
Year 5: $80 million * (1 + 0.05)^4 = $97.23 million
Year 6: $80 million * (1 + 0.05)^5 = $102.09 million
Year 7: $80 million * (1 + 0.05)^6 = $107.19 million
Year 8: $80 million * (1 + 0.05)^7 = $112.55 million
Year 9: $80 million * (1 + 0.05)^8 = $118.18 million
Year 10: $80 million * (1 + 0.05)^9 = $124.09 million
Now, let's calculate the total value of the contract under Case 2:
Total Value = Year 1 + Year 2 + Year 3 + Year 4 + Year 5 + Year 6 + Year 7 + Year 8 + Year 9 + Year 10
Total Value = $80 million + $84 million + $88.2 million + $92.61 million + $97.23 million + $102.09 million + $107.19 million + $112.55 million + $118.18 million + $124.09 million
Adding up all the installments, the total value of the contract under Case 2 is:
Total Value = $944.14 million
Comparing the total value of the contracts, we can see that the athlete in Case 2, with increasing installments, got the better deal as the total value of the contract is higher.