Question: User: When Justin had 5 years left in college, he took out a student loan for $19,246. The loan has an annual interest rate of 4.2%. Justin graduated 5 years after acquiring the loan and began repaying the loan immediately upon graduation.
According to the terms of the loan, Justin will make monthly payments for 5 years after graduation. During the 5 years he was in school and not making payments, the loan accrued simple interest.
Answer each part. Do not round intermediate computations, and round your answers to the nearest cent. If necessary, refer to the list of financial formulas.
(a)If Justin's loan is subsidized, find his monthly payment.
Subsidized loan monthly payment: $
(b)If Justin's loan is unsubsidized, find his monthly payment.
Unsubsidized loan monthly payment: $
1. **Problem statement:** Find Justin’s monthly loan payment for a subsidized loan and an unsubsidized loan.
2. **Formula used:** For an amortized loan, the monthly payment is
$$M=P\frac{r}{1-(1+r)^{-n}}$$
where $P$ is the principal, $r$ is the monthly interest rate, and $n$ is the total number of monthly payments.
3. **Important rules:**
- Convert the annual rate to a monthly rate by dividing by $12$.
- Use $n=5\cdot 12=60$ monthly payments.
- For a subsidized loan, the amount owed when repayment begins is still $P=19246$ because no interest accrues during school.
- For an unsubsidized loan, simple interest accrues during the $5$ years in school, so first find the balance at graduation.
4. **Subsidized loan:**
The monthly rate is
$$r=\frac{0.042}{12}=0.0035$$
and the number of payments is
$$n=60.$$
Now substitute into the payment formula:
$$M=19246\frac{0.0035}{1-(1.0035)^{-60}}$$
Evaluate the denominator and numerator:
$$M\approx 19246\frac{0.0035}{1-0.8117362782}$$
$$M\approx 19246\frac{0.0035}{0.1882637218}$$
$$M\approx 357.84$$
So the subsidized loan monthly payment is **357.84**.
5. **Unsubsidized loan:**
First find the balance after $5$ years of simple interest.
Use
$$A=P(1+rt)$$
with $P=19246$, $r=0.042$, and $t=5$.
$$A=19246(1+0.042\cdot 5)$$
$$A=19246(1+0.21)$$
$$A=19246(1.21)$$
$$A=23287.66$$
Now use this as the new principal in the payment formula:
$$M=23287.66\frac{0.0035}{1-(1.0035)^{-60}}$$
$$M\approx 23287.66\frac{0.0035}{0.1882637218}$$
$$M\approx 432.82$$
So the unsubsidized loan monthly payment is **432.82**.
6. **Final answers:**
(a) **357.84**
(b) **432.82**