Student Loan Payment Calculation and Effective Interest Rate Analysis

The intention is a bit ambiguous. The question does not explicitly state this. However, the statement "you will pay an extra $100 that you are not required to pay" seems to imply that the borrower is making an extra payment along with the one he is required to pay. If that is the case, then I would go with the first approach. But if the borrower decides to make the extra payment at the end, then the second approach would be correct.
  • #1
issacnewton
998
29

Homework Statement


You have an outstanding student loan with required payments of $500 per month for
the next four years. The interest rate on the loan is 9% APR (monthly). You are
considering making an extra payment of $100 today (i.e., you will pay an extra $100
that you are not required to pay). If you are required to continue to make payments
of $500 per month until the loan is paid off, what is the amount of your final
payment? What effective rate of return (expressed as an APR with monthly
compounding) have you earned on the $100?

Homework Equations


Annuity Formula

The Attempt at a Solution


I will present my solution for the first part. I have question about the second part. Since APR is 9%, the periodic monthly interest rate is ##i = 0.09/12 = 0.0075##. Let ##C= $500## be the monthly payment. There are 48 months in 4 years. So we can find the loan value using PV of annuity formula $$\text{PV} = \frac{500}{0.0075}\left[ 1 - \frac{1}{\left(1+ 0.0075\right)^{48}}\right] $$ Now student makes an extra payment of $100 today and then 47 payments of $500 and one last payment. Let's call this last payment ##x##. The present value of all these payments must equal to ##\text{PV}##. So, we have $$\text{PV} = 100+\frac{500}{0.0075}\left[ 1 - \frac{1}{\left(1+ 0.0075\right)^{47}}\right] + \frac{x}{\left(1+0.0075\right)^{48}} $$ Solving for ##x##, I get the last payment as ##x=$356.86##. Is this correct so far ? Now I don't the last part of the question. How would I get the effective rate of return earned on $100. What exactly is being asked here ?

Thanks
 
Physics news on Phys.org
  • #2
You look at the difference between what the last payment would have been and what it actually will be. If you had put your $100 in a bank, this is what you could have been able to take out in the end. What interest rate would the bank have to pay you for this to be possible?
 
  • #3
Well, in both cases, the payment is $500 except for the last payment and beginning $100. So I am not sure I am following your point.
 
  • #4
The difference between the last payment and 500 (what you would have normally paid at the end) is $143.14
So $100 today, is worth $143.14 in 48 months. What interest rate would do that for you?
 
  • #5
So how many periods should we count here ? Since effective rates are annual, 48 months will be 4 annual periods. Is that right ?
 
  • #6
What did the question state (about compounding)?
 
  • #7
So we would have ##143.14 = 100\left[1+\frac{i}{12}\right]^{48}## And solving for ##i##, we get ##i = 0.0899##, so the effective annual rate of return would be ##8.99\%##. Right ?
 
  • #8
Why do you think that you got slightly less than 9% APR?
 
  • #9
No idea. Why is that ?
 
  • #10
I got 0.08999906 which i would round to 9.00%
The reason it is not exactly 9% is there has alredy been some rounding in the money being rounded to the nearest $0.01
 
  • #11
Yes, that makes sense. I think in financial problems, the rounding should be done at the very end.
 
  • Like
Likes scottdave
  • #12
IssacNewton said:

Homework Statement


You have an outstanding student loan with required payments of $500 per month for
the next four years. The interest rate on the loan is 9% APR (monthly). You are
considering making an extra payment of $100 today (i.e., you will pay an extra $100
that you are not required to pay). If you are required to continue to make payments
of $500 per month until the loan is paid off, what is the amount of your final
payment? What effective rate of return (expressed as an APR with monthly
compounding) have you earned on the $100?

Homework Equations


Annuity Formula

The Attempt at a Solution


I will present my solution for the first part. I have question about the second part. Since APR is 9%, the periodic monthly interest rate is ##i = 0.09/12 = 0.0075##. Let ##C= $500## be the monthly payment. There are 48 months in 4 years. So we can find the loan value using PV of annuity formula $$\text{PV} = \frac{500}{0.0075}\left[ 1 - \frac{1}{\left(1+ 0.0075\right)^{48}}\right] $$ Now student makes an extra payment of $100 today and then 47 payments of $500 and one last payment. Let's call this last payment ##x##. The present value of all these payments must equal to ##\text{PV}##. So, we have $$\text{PV} = 100+\frac{500}{0.0075}\left[ 1 - \frac{1}{\left(1+ 0.0075\right)^{47}}\right] + \frac{x}{\left(1+0.0075\right)^{48}} $$ Solving for ##x##, I get the last payment as ##x=$356.86##. Is this . correct so far ? Now I don't the last part of the question. How would I get the effective rate of return earned on $100. What exactly is being asked here ?

Thanks

Careful!

Your formula ##PV = (500/.0075) [ 1- 1/(1.0075)^{48}]## expresses ##\sum_{n=1}^{48} 500/(1.0075)^n##, so reckons the first payment at time ##t = 1## (month). However, when you put ##100 + (500/.0075) [1-1/(1.0075)^{47}] + x/(1.00750^{48}## you are putting the first $100 payment at time ##t=0##, not at ##t=1## the way all the other payments are timed. If that is your intention, then OK, but be sure to clarify it in your writeup. Otherwise, you should use ##\$100/1.0075## instead of ##\$100## when reckoning the first extra payment. Either that, or else start all payments at the start of each month, so you would use instead the formula ##PV = \sum_{n=0}^{47} 500/(1.0075)^n,## which gives a slightly different annuity expression.
 
  • #13
Ray Vickson said:
Careful!

Your formula ##PV = (500/.0075) [ 1- 1/(1.0075)^{48}]## expresses ##\sum_{n=1}^{48} 500/(1.0075)^n##, so reckons the first payment at time ##t = 1## (month). However, when you put ##100 + (500/.0075) [1-1/(1.0075)^{47}] + x/(1.00750^{48}## you are putting the first $100 payment at time ##t=0##, not at ##t=1## the way all the other payments are timed. If that is your intention, then OK, but be sure to clarify it in your writeup. Otherwise, you should use ##\$100/1.0075## instead of ##\$100## when reckoning the first extra payment. Either that, or else start all payments at the start of each month, so you would use instead the formula ##PV = \sum_{n=0}^{47} 500/(1.0075)^n,## which gives a slightly different annuity expression.

Interesting. Re-reading the question, I think the question implies that the borrower decides to send in an extra $100 along with the first payment. Making a payment at time 0 is the same as when you make a down payment, at the time of purchasing a car, and financing the rest.
 
  • #14
Ray, in the statement of the problem it states that $100 are paid today, so that is ##t=0##. So no need to discount here.
 
  • #15
So is my answer correct then ? Scott, Ray can you please respond ?
 

1. What is a student loan word problem?

A student loan word problem is a mathematical question or scenario that involves the concept of student loans. It typically requires the use of equations and calculations to solve for unknown variables related to student loan payments, interest rates, and loan amounts.

2. Why are student loan word problems important?

Student loan word problems are important because they help students understand the real-world application of mathematical concepts and the impact of student loans on their finances. They also help students develop critical thinking and problem-solving skills.

3. What are common elements in a student loan word problem?

Common elements in a student loan word problem include the initial loan amount, interest rate, loan term, monthly payments, and any additional fees or charges. These elements may vary depending on the specific problem.

4. How can I solve a student loan word problem?

To solve a student loan word problem, you can follow a few simple steps. First, identify and list all the given information. Then, determine which equations or formulas are needed to solve the problem. Next, substitute the given values into the equations and solve for the unknown variables. Finally, check your solution to ensure it makes sense in the context of the problem.

5. How can I apply student loan word problems to my own finances?

You can apply student loan word problems to your own finances by using them to understand the impact of different interest rates, loan terms, and payment amounts on your student loans. This can help you make informed decisions about managing your student loan debt and planning for repayment.

Similar threads

  • Precalculus Mathematics Homework Help
Replies
1
Views
874
  • Precalculus Mathematics Homework Help
Replies
10
Views
5K
  • Precalculus Mathematics Homework Help
Replies
2
Views
1K
  • Precalculus Mathematics Homework Help
Replies
8
Views
2K
  • General Math
Replies
8
Views
2K
  • Precalculus Mathematics Homework Help
Replies
4
Views
1K
  • Calculus and Beyond Homework Help
Replies
12
Views
1K
  • Engineering and Comp Sci Homework Help
Replies
2
Views
1K
  • General Math
Replies
4
Views
2K
Back
Top