find payment necessary to amortize a 8% loan of $2000 compounded quarterly with 12 quarterly payments.
i = .08/4 = .02
P( 1 - 1.02^-12)/.02 = 2000
solve for P
To find the payment necessary to amortize a loan, you can use the formula for the present value of an ordinary annuity.
The formula for the present value of an ordinary annuity is:
PV = P * (1 - (1 + r)^(-n)) / r
where:
PV = Present value of the annuity
P = Payment amount
r = Interest rate per period
n = Total number of periods
In this case, the loan has an interest rate of 8% per year, compounded quarterly. So the interest rate per period is 8% divided by 4, or 2%. The total number of periods is 12 (since there are 12 quarterly payments).
Substituting the given values into the formula, we have:
PV = 2000
r = 2% or 0.02
n = 12
Plugging these values into the formula, we get:
2000 = P * (1 - (1 + 0.02)^(-12)) / 0.02
Now, we can solve the equation to find the payment amount (P). Here's the step-by-step calculation:
1. Calculate (1 + 0.02) = 1.02
2. Calculate (1 + 0.02)^(-12) = 0.8872388231
3. Subtract this result from 1 to get (1 - 0.8872388231) = 0.1127611769
4. Divide 0.1127611769 by 0.02 to get 5.638058845
5. Divide 2000 by 5.638058845 to find P, which is approximately $354.55
Therefore, the payment necessary to amortize an 8% loan of $2000 compounded quarterly over 12 quarterly payments is approximately $354.55.