# Present worth from monthly payment with interest rate compunded quarterly

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MiniG
Present worth from monthly payment with interest rate compunded quarterly

Given the monthly payment for a land loan is 1110. The bank charge 6% interest compunded quarterly for 30 years. What is the price of the land?

*is the concept correct if I first convert 6% to the effective annual rate which it will be 6.14%. Then, dividing it by 12, with the period of 360 (30 x 12).So, the equation will be P=1110(P/A, 0.51%,360). ?

Jhun Vert

Yes, your concept is correct; convert the interest rate of 6% compounded quarterly into equivalent interest rate per payment period which is monthly. We use the concept of effective rates, ER, to do it. Your ER = 6.14% is correct but it is not as simple is dividing it by 12 to find the equivalent monthly interest. The proper way is to calculate the monthly interest is to convert the effective rate of 6% compounded quarterly to an effective rate compounded monthly:

ERcompounded monthly = ER6% compounded quarterly
$\left(1 + \dfrac{r}{12} \right)^{12} - 1 = \left( 1 + \dfrac{0.06}{4} \right)^4 - 1$

$r = 5.97\% ~ \text{compounded monthly}$

Note that for r = 5.97% compounded monthly will yield the same ER of 6.14% annually. The monthly interest that you are going to use is not 6.14% ÷ 12 but 5.97% ÷ 12.

Monthly interest:
$i = \dfrac{5.97}{12} = 0.498\% ~ \text{monthly}$

The direct way to calculate the monthly interest if ER is known is by this:
$ER = (1 + i)^m - 1$

For ER = 6.14% and m = 12 (or monthly)
$0.0614 = (1 + i)^{12} - 1$

$i = 0.498\% ~ \text{monthly}$

The rest of what you did is already correct.

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