The owner of Campus Cafe plans to open a second location on a satellite campus in 5yrs. She buys an annuity that pays 10.5% interest compounded annually
A. If the payment is $4000 a year, find the future value of the annuity in 5yrs.
B. How much more interest would be earned if the owner could invest the full amount paid into the annuity in a regular compounded interest account with the same term for 4 years
To find the future value of the annuity in 5 years, we can use the formula for the future value of an ordinary annuity:
Future Value = Payment * [(1 + Interest Rate)^Number of Periods - 1] / Interest Rate
Let's calculate it step by step:
A. Future Value of the Annuity
1. Identify the given values:
- Payment: $4000 per year
- Interest Rate: 10.5% or 0.105 (as a decimal)
- Number of Periods: 5 years
2. Plug the values into the formula:
Future Value = $4000 * [(1 + 0.105)^5 - 1] / 0.105
3. Calculate the formula:
Future Value = $4000 * [(1.105)^5 - 1] / 0.105
Future Value = $4000 * [1.64701092625 - 1] / 0.105
Future Value = $4000 * 0.64701092625 / 0.105
Future Value = $4000 * 6.16105645476
Future Value ≈ $24,644.23
Therefore, the future value of the annuity in 5 years would be approximately $24,644.23.
B. Calculating the additional interest earned
To calculate the additional interest that would be earned if the owner could invest the full amount paid into the annuity in a regular compounded interest account with the same term for 4 years, we need to compare the future value of the annuity with the future value of the regular compounded interest account.
1. Calculate the future value of the annuity as determined in part A, which is approximately $24,644.23.
2. Calculate the future value of the regular compounded interest account using the formula:
Future Value = Principal * (1 + Interest Rate)^Number of Periods
In this case, the principal would be the total amount paid into the annuity over 4 years, which is $4000/year * 4 years = $16,000.
Interest Rate: 10.5% annually or 0.105 (as a decimal)
Number of Periods: 4 years
Future Value = $16,000 * (1 + 0.105)^4
3. Calculate the formula:
Future Value = $16,000 * (1.105)^4
Future Value = $16,000 * 1.47745625
Future Value ≈ $23,639.30
Therefore, the future value of the regular compounded interest account with the same term would be approximately $23,639.30.
To calculate the additional interest earned, we subtract the future value of the regular compounded interest account from the future value of the annuity:
Additional Interest = Future Value of the Annuity - Future Value of the Regular Compounded Interest Account
Additional Interest ≈ $24,644.23 - $23,639.30
Additional Interest ≈ $1,004.93
Therefore, the owner would earn approximately $1,004.93 in additional interest if they could invest the full amount paid into the annuity in a regular compounded interest account with the same term.