Final answer:
To calculate the final payment that would make the bank indifferent to receiving three annual payments or a single payment after four years at a 7% interest rate, you need to find the future value of each skipped payment and sum them up. The total comes to $14,152.34 as the needed final payment.
Step-by-step explanation:
The question is asking to calculate what the final payment would be if you choose to skip making the next three annual payments of $3,000 and instead make one large payment after four years, considering a 7% interest rate. The calculation required is to determine the future value of an annuity due because the normal payments would have been made at the end of each year. The future value of each $3,000 payment needs to be calculated and then summed to find the equivalent large payment at the end of the fourth year.
To calculate the future value (FV) of each payment, we use the formula:
FV = P × (1 + r)^n
Where:
- P is the payment amount
- r is the interest rate per period
- n is the number of periods
Calculating the future value for each of the four $3,000 payments at 7% interest:
- FV of 1st payment after 1 year: $3,000 × (1 + 0.07)^1 = $3,210
- FV of 2nd payment after 2 years: $3,000 × (1 + 0.07)^2 = $3,429
- FV of 3rd payment after 3 years: $3,000 × (1 + 0.07)^3 = $3,649.03
- FV of 4th payment after 4 years: $3,000 × (1 + 0.07)^4 = $3,864.31
Adding all these together gives the final payment amount the bank would require to be indifferent to receiving this single payment or the three annual payments:
$3,210 + $3,429 + $3,649.03 + $3,864.31 = $14,152.34
Therefore, the bank would require a final payment of $14,152.34 at the end of the fourth year.