I have this economics problem.
What is the present worth of $500.00$ Rupiah deposited at the end of every three months for $6$ years if the interest rate is $12\%$ compounded semiannually?
According to the solution, the interest rate per quarter must be determined first. And so the solution goes by saying that:
$$(1 + i)^4 - 1 = \left(1 + \frac{0.12}{2}\right)^2 - 1$$
$i=2.96\%$ per quarter
My question is that where did the above equation came from? How was it derived?
Hint: Let i be the quarterly interest rate ( no compounding in other words quoted rate). Let x be the annual interest rate compounded semiannually. 12% is annual interest rate (quoted rate).
6% be the semi-annual interest rate ( quoted rate). Now it is compounded semiannually
Annual interest rate compounded semi annually is $1+x= (1+.06)^2 = 1.06^2 => x = 12.36$% x should be higher than the annual interest rate
Now $(1+i)^4 = 1.1236 =>1+i = 1.02956 => i = 2.956$%