a. If you borrow $1,900 and agree to repay the loan in five equal annual payments at an interest rate of 12%, what will your payment be?
b. What will your payment be if you make the first payment on the loan immediately instead of at the end of the first year?
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a. To calculate the annual payment, we can use the formula for the annuity payment:
P = PV * [r * (1 + r)^n] / [(1 + r)^n - 1]
where:
P = annual payment
PV = present value of the loan = $1,900
r = annual interest rate = 12% = 0.12
n = number of years = 5
P = $1,900 * [0.12 * (1 + 0.12)^5] / [(1 + 0.12)^5 - 1]
P = $1,900 * [0.12 * 1.76234] / [1.76234 - 1]
P = $1,900 * 0.21148 / 0.76234
P = $528.94
So, the annual payment will be approximately $528.94.
b. If you make the first payment on the loan immediately instead of at the end of the first year, the loan amount will be reduced by the first payment. Therefore, the new loan amount will be $1,900 - $528.94 = $1,371.06.
Then, we can calculate the new annual payment for the remaining 4 years:
P = $1,371.06 * [0.12 * (1 + 0.12)^4] / [(1 + 0.12)^4 - 1]
P = $1,371.06 * [0.12 * 1.57352] / [1.57352 - 1]
P = $1,371.06 * 0.18882 / 0.57352
P = $458.15
So, if you make the first payment on the loan immediately, the remaining annual payments will be approximately $458.15.
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