Student Loan Payment Calculation and Effective Interest Rate Analysis

[issacnewton]

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

 

[Chestermiller]

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?

 

[issacnewton]

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.

 

[scottdave]

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?

 

[issacnewton]

So how many periods should we count here ? Since effective rates are annual, 48 months will be 4 annual periods. Is that right ?

 

[scottdave]

What did the question state (about compounding)?

 

[issacnewton]

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 ?

 

[scottdave]

Why do you think that you got slightly less than 9% APR?

 

[issacnewton]

No idea. Why is that ?

 

[scottdave]

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

 

[issacnewton]

Yes, that makes sense. I think in financial problems, the rounding should be done at the very end.

 

[Ray Vickson]

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.

 

[scottdave]

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.

 

[issacnewton]

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.

 

[issacnewton]

So is my answer correct then ? Scott, Ray can you please respond ?