How much more money would I have if I invested $14,000.00 for 8 years at 6.4% in an account that compounded annually rather than in an account that just earned simple interest? Why is this?
With annual compounding you would have
14,000*(1.064)^8 = 22,996.46
With simple interest you end up with
14,000*[1+(.064)*8] = 21,168.00
With simple interest, you do not earn interest on the interest that you (the investor) should have collected during the term of the deposit, but which the bank kept.
To calculate the difference in money between compounding interest and simple interest, we can use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A is the final amount (including interest)
P is the principal amount (initial investment)
r is the annual interest rate (in decimal form)
n is the number of times the interest is compounded per year
t is the number of years
For the case of simple interest, we can use the formula:
A = P(1 + rt)
In this scenario, you want to compare the result of 8 years of compounding interest with an annual interest rate of 6.4% to the result of simple interest with the same principal amount.
First, let's calculate the compound interest:
P = $14,000.00
r = 6.4% = 0.064 (in decimal)
n = 1 (compounded annually)
t = 8 years
A_compound = P(1 + r/n)^(nt)
A_compound = $14,000.00(1 + 0.064/1)^(1 * 8)
A_compound = $14,000.00(1 + 0.064)^8
A_compound = $14,000.00(1.064)^8
A_compound ≈ $21,157.91
Now, let's calculate the result of simple interest:
P = $14,000.00
r = 6.4% = 0.064 (in decimal)
t = 8 years
A_simple = P(1 + rt)
A_simple = $14,000.00(1 + 0.064 * 8)
A_simple = $14,000.00(1 + 0.512)
A_simple = $14,000.00(1.512)
A_simple ≈ $21,168.00
The difference in money between compound interest and simple interest is:
Difference = A_compound - A_simple
Difference ≈ $21,157.91 - $21,168.00
Difference ≈ -$10.09
Therefore, you would have approximately $10.09 less if you invested $14,000.00 for 8 years at 6.4% in an account that earned simple interest rather than compounding interest.
The reason for this difference is that compound interest applies the interest earned on the initial principal and any previously earned interest, leading to exponential growth. Simple interest, however, only calculates interest based on the initial principal amount, resulting in a linear growth pattern. Hence, with compound interest, the accumulated interest over time is higher compared to simple interest.