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# Calculating interest using the effective interest rate?

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# Calculating interest using the effective interest rate?

August 22, 2014 7:02 PM Subscribe

I need to calculate the interest to be paid on a loan of $300,000 with a term of 4 years where the effective interest rate is 10%, payable monthly in arrears. How do I do this?

This is for a grad assignment - unfortunately I have no accounting background and the professor was unwilling to explain. I've Googled this to death but to no avail as most pages explain how to calculate effective interest rate and not how to calculate the amount of interest itself. There are two other interest payment options as well: 9% payable annually in advance and 7% where the entire 4 years of interest is paid in advance.

The question doesn't state whether the effective interest rate is annual - this is also confusing me.

I've seen how useful MetaFilter is in other situations so hoping someone out there will be able to help, thank you in advance!

This is for a grad assignment - unfortunately I have no accounting background and the professor was unwilling to explain. I've Googled this to death but to no avail as most pages explain how to calculate effective interest rate and not how to calculate the amount of interest itself. There are two other interest payment options as well: 9% payable annually in advance and 7% where the entire 4 years of interest is paid in advance.

The question doesn't state whether the effective interest rate is annual - this is also confusing me.

I've seen how useful MetaFilter is in other situations so hoping someone out there will be able to help, thank you in advance!

Excel comes with a template, called "Loan Calculator" in my version, that does this. If you look in the cells you can see how it's calculated and it also splits out the interest.

posted by mewsic at 8:36 PM on August 22, 2014

posted by mewsic at 8:36 PM on August 22, 2014

If this is simple interest...300,000 times 10% is 30,000 times four(years) is $120,000. If you need to know the monthly interest just divide the yearly interest by 12.

If it's compounded other than yearly hopefully someone else will chime in but this is just simple real estate math otherwise. Every problem I had to work assumed interest compounded yearly, if that helps.

If you needed to know the P&I (principal and interest) you would need to know what the loan factor would be. Simplest way to do that is get a chart from a mortgage person. My real estate book chart starts at ten years, so that's no help.

posted by St. Alia of the Bunnies at 9:02 PM on August 22, 2014

If it's compounded other than yearly hopefully someone else will chime in but this is just simple real estate math otherwise. Every problem I had to work assumed interest compounded yearly, if that helps.

If you needed to know the P&I (principal and interest) you would need to know what the loan factor would be. Simplest way to do that is get a chart from a mortgage person. My real estate book chart starts at ten years, so that's no help.

posted by St. Alia of the Bunnies at 9:02 PM on August 22, 2014

Really, this is middle school math. Don't let it scare you. If I could do it, anybody can!

posted by St. Alia of the Bunnies at 9:11 PM on August 22, 2014

posted by St. Alia of the Bunnies at 9:11 PM on August 22, 2014

If you know the effective annual rate, you don't need to know anything about the monthly rate because that is already compounded to give you the effective annual rate.

So your total interest is just 10% compounded annually.

1.10^4 = 1.4641 So your total interest is 46.4%

46.41% of $300,000 = $139,230

----------

Now you could also work backwards to find the nominal interest rate and compound that for 48 months instead. Let's do that just for fun.

First you need to compute the nominal interest rate given the effective interest rate. Find i given r:

r = ((1 + (i/n))^n) - 1

0.1 = ((1 + (i/12))^12) - 1

1.1 = ((1 + (i/12))^12) Added 1 to both sides

1.00797 = (1 + i/12) Took the 1/12 root of both sides

0.00797 = i/12 Subtracted 1 from both sides

0.0957 = i Multiplied both sides by 12

So the nominal annual rate is 9.57% and the monthly interest is 0.797% or 0.00797

To find the compound interest over 4 years you have 48 months at 0.00797

1.00797^48 = 1.464 Raise to the 48th power.

So the compounded interest is 46.4%.

46.4% of 300,000 = $139,223

You get the same result either way.

posted by JackFlash at 9:21 PM on August 22, 2014 [2 favorites]

So your total interest is just 10% compounded annually.

1.10^4 = 1.4641 So your total interest is 46.4%

46.41% of $300,000 = $139,230

----------

Now you could also work backwards to find the nominal interest rate and compound that for 48 months instead. Let's do that just for fun.

First you need to compute the nominal interest rate given the effective interest rate. Find i given r:

r = ((1 + (i/n))^n) - 1

0.1 = ((1 + (i/12))^12) - 1

1.1 = ((1 + (i/12))^12) Added 1 to both sides

1.00797 = (1 + i/12) Took the 1/12 root of both sides

0.00797 = i/12 Subtracted 1 from both sides

0.0957 = i Multiplied both sides by 12

So the nominal annual rate is 9.57% and the monthly interest is 0.797% or 0.00797

To find the compound interest over 4 years you have 48 months at 0.00797

1.00797^48 = 1.464 Raise to the 48th power.

So the compounded interest is 46.4%.

46.4% of 300,000 = $139,223

You get the same result either way.

posted by JackFlash at 9:21 PM on August 22, 2014 [2 favorites]

Wait, scratch that. I think what they are asking is how much is each monthly interest payment. So for the first part they want to know how much you pay each month in interest if the effective annual rate is 10%.

So if you look at the second part of my calculation above, you get an annual nominal rate of 9.57% and therefore a monthly rate of 0.797%. So the monthly interest payment would be 0.797% of $300,000 = $2393.

The total over 48 months is 114,840. This is slightly less than the 120,000 if you paid 10% annually because by paying monthly, you are paying incrementally in advance.

posted by JackFlash at 10:03 PM on August 22, 2014

So if you look at the second part of my calculation above, you get an annual nominal rate of 9.57% and therefore a monthly rate of 0.797%. So the monthly interest payment would be 0.797% of $300,000 = $2393.

The total over 48 months is 114,840. This is slightly less than the 120,000 if you paid 10% annually because by paying monthly, you are paying incrementally in advance.

posted by JackFlash at 10:03 PM on August 22, 2014

Thanks so much for the help so far, everyone! Especially JackFlash - that answer was really helpful. Just to answer essexjan's query, it's not amortising - the repayments are for interest only. The question is worded very vaguely which is why I think I'm struggling so much; I don't even know if it's compounded annually or if it is a flat 10% of the $300,000 principal charged per year.

posted by herschellie at 10:34 PM on August 22, 2014

posted by herschellie at 10:34 PM on August 22, 2014

I don't understand your answers JackFlash.

If it's an interest only loan, then the amount due at the end of the first month - 10% of the principal divided by 12 - is $2500. Given that the principal remains unchanged, then the remaining 47 monthly payments will be the same and that's $120,000.

Can you explain what's wrong with this reckoning please?

posted by mewsic at 4:21 AM on August 23, 2014

If it's an interest only loan, then the amount due at the end of the first month - 10% of the principal divided by 12 - is $2500. Given that the principal remains unchanged, then the remaining 47 monthly payments will be the same and that's $120,000.

Can you explain what's wrong with this reckoning please?

posted by mewsic at 4:21 AM on August 23, 2014

I'm not positive what the professor is asking but I am interpreting it as the actual monthly interest given an effective rate of 10% annually. The effective rate is not the actual rate paid monthly. If you pay monthly, the actual (or nominal) rate is slightly less than the annual effective rate because you are pre-paying some of your annual interest each month.

The formula for effective rate is:

r = ((1 + i/n)^n) - 1) where r is the effective rate and i is the nominal rate. You are given r and must solve for i as I showed above in a series of steps.

The result is a nominal rate of 9.57% which is slightly less than the effective rate, as expected because you are pre-paying annual interest each month. Divide by 12 and the nominal rate is 0.797% per month. This works out to $2393 per month, which is a little less than the $2500 a month you would pay if you paid it all at once at the end of the year.

Forget the parts above where I was compounding the interest over 4 years. That is an answer to a different question and is clearly wrong.

posted by JackFlash at 9:35 AM on August 23, 2014

The formula for effective rate is:

r = ((1 + i/n)^n) - 1) where r is the effective rate and i is the nominal rate. You are given r and must solve for i as I showed above in a series of steps.

The result is a nominal rate of 9.57% which is slightly less than the effective rate, as expected because you are pre-paying annual interest each month. Divide by 12 and the nominal rate is 0.797% per month. This works out to $2393 per month, which is a little less than the $2500 a month you would pay if you paid it all at once at the end of the year.

Forget the parts above where I was compounding the interest over 4 years. That is an answer to a different question and is clearly wrong.

posted by JackFlash at 9:35 AM on August 23, 2014

Thanks for the explanation. :) I've never seen it calculated like that before. The rate calculation on my mortgage and the loans I've had is just a straight twelfth of the annual interest rate.

posted by mewsic at 5:55 PM on August 23, 2014

posted by mewsic at 5:55 PM on August 23, 2014

That's because the rate they are quoting you already is the nominal rate, not the effective rate. In this problem, they are starting with the effective rate and you have to work backwards to get the nominal rate that you are more accustomed to. The effective rate is akin to and part of the APR. The APR rate is always higher than the nominal rate you use to calculate each month's interest.

In fact you can see an example of the identical calculation in Wikipedia here. It just happens to be the same numbers -- 10% effective rate and 9.57% nominal rate.

posted by JackFlash at 6:37 PM on August 23, 2014 [1 favorite]

In fact you can see an example of the identical calculation in Wikipedia here. It just happens to be the same numbers -- 10% effective rate and 9.57% nominal rate.

posted by JackFlash at 6:37 PM on August 23, 2014 [1 favorite]

Now that I look back on this whole problem, I can see that professor just formed it poorly.

It turns out that all of the options have the same effective interest rate, 10%. So it would have been more enlightening if the professor had asked you to compute the effective interest rate for all of the options and you would have seen they were all identical. This would have illustrated something about the different ways you can formulate the same loan. The timing of interest dollars is different, but they are all equivalent loans, financially.

10% paid once a year in arrears

9.57% annual nominal rate divided into 12 monthly payments (0.797% a month)

9% paid each year in advance

7% annual rate with all four years paid in advance

It turns out these are different ways of describing the same loan and all have the same 10% effective interest rate. Each has a slightly lower nominal rate than the previous one because you are paying more money in advance in each case.

posted by JackFlash at 11:29 PM on August 23, 2014

It turns out that all of the options have the same effective interest rate, 10%. So it would have been more enlightening if the professor had asked you to compute the effective interest rate for all of the options and you would have seen they were all identical. This would have illustrated something about the different ways you can formulate the same loan. The timing of interest dollars is different, but they are all equivalent loans, financially.

10% paid once a year in arrears

9.57% annual nominal rate divided into 12 monthly payments (0.797% a month)

9% paid each year in advance

7% annual rate with all four years paid in advance

It turns out these are different ways of describing the same loan and all have the same 10% effective interest rate. Each has a slightly lower nominal rate than the previous one because you are paying more money in advance in each case.

posted by JackFlash at 11:29 PM on August 23, 2014

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This loan calculator allows you to put in the amount of the loan, the term of years, the interest rate and whether it's to be amortizing (repayment method) or interest-only.

posted by essexjan at 7:11 PM on August 22, 2014