f = p * (1 + r) ^ n
f is the future value
p is the present value
r is the interest rate per time period.
(1 + r) is the growth rate per time period.
n is the number of time periods.
your interest rate is 7% per year / 2 semi-annual periods per year = 3.5% per semi-annual time period.
that's the percent.
the rate is that / 100 = .035 per semi-annual time period.
the growth rate is 1.035 per semi-annual time period.
the number of time periods is equal to 2 times the number of years.
3 * 2 = 6 time periods.
the formula of f = p * (1 + r) ^ n becomes:
13,000 = p * 1.035 ^ 6
solve for p to get p = 13,000 / (1.035 ^ 6) = 10575.50838.
you could also have used a financial calculator, such as the one found at https://arachnoid.com/finance/
here are the results from using that calculator.
with the calculator, you use the rate percent rather than the rate.
money that you spend is shown as negative.
money you receive is shown as posiive.
since you receive the future value, the preset value is shown as negative because that's what you invested to get back the future value.
the same rules apply.
the interest rate per time period is equal to the interest rate per year divided by the number of compounding periods per year (7% / 2 = 3.5%).
the number of time periods is equal to the number of year times the number of compounding compounding periods per year (3 * 2 = 6).
7% is your nominal interest rate.
1.035 ^ 2 = 1.071225 -1 = .071225 * 100 = 7.1225% = the effective annual interest rate.
13000 / 1.035 ^ 6 is the same result as 13000 / 1.071225 ^ 3.
the effective annual interest rate take into account compounding.
the nominal annual interest rate doesn't.