Final answer:
The quarterly payment necessary to amortize an 8% loan of $2000 compounded quarterly with 12 quarterly payments is $189.24. This calculation utilizes the formula for the present value of an ordinary annuity, factoring in the principal, interest rate per period, and total number of payments.
Step-by-step explanation:
To find the payment necessary to amortize an 8% loan of $2000 compounded quarterly with 12 quarterly payments, one can use the formula for the present value (PV) of an annuity. The calculation requires the loan amount (also known as the principal), the rate per period, and the total number of payments to determine the periodic payment amount.
The formula for the PV of an ordinary annuity is:
PV = R * [1 - (1 + i)^(-n)] / i
Where:
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- PV is the loan amount (present value)
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- R is the periodic payment
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- i is the interest rate per period
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- n is the total number of payments
In this case:
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- PV = $2000
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- i = 8% annual interest rate compounded quarterly (0.08/4 = 0.02 or 2% per quarter)
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- n = 12 quarterly payments
The periodic payment R is the unknown variable we seek in order to amortize the loan.
Rearranging the formula to solve for R:
R = PV * i / [1 - (1 + i)^(-n)]
Inserting the given values:
R = $2000 * 0.02 / [1 - (1 + 0.02)^(-12)]
Performing the calculation:
R = $2000 * 0.02 / [1 - (1 + 0.02)^(-12)] = $189.24
The quarterly payment necessary to amortize the loan is $189.24.
Note that this is a simplified example and does not take into account other potential factors such as fees, insurance, or taxes that might affect the actual payment amount. Additionally, rounding can affect the precision of the payment value.