Final answer:
To find the nominal annual interest rate for an investment compounded semi-annually, we use the compound interest formula, rearrange it to solve for the interest rate, and then apply logarithms to isolate the rate before converting it to a percentage.
Step-by-step explanation:
To find the nominal annual interest rate of an investment that is compounded semi-annually, we need to use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A = the future value of the investment/loan, including interest
P = the principal investment amount (the initial deposit or loan amount)
r = the annual nominal interest rate (as a decimal)
n = the number of times that interest is compounded per year
t = the number of years the money is invested or borrowed for
Given that:
- A = $938.56
- P = $500
- n = 2 (since the interest is compounded semi-annually)
- t = 5 years
We can rearrange the formula to solve for r:
(1 + r/2)^(2*5) = 938.56 / 500
(1 + r/2)^10 = 1.87712
Now we can use logarithms to solve for r:
10 * log(1 + r/2) = log(1.87712)
log(1 + r/2) = log(1.87712) / 10
1 + r/2 = 10^(log(1.87712) / 10)
r/2 = 10^(log(1.87712) / 10) - 1
r = 2 * (10^(log(1.87712) / 10) - 1)
The calculation will give us the nominal annual interest rate as a decimal, which we then convert to a percentage by multiplying by 100.