Right, so my statement that the APR and nominal interest rate are the same is not correct (assuming your APR calculation is correct, which it seems to be).
Huh? This is not how you calculate APR. You have to take into account the fact that you are repaying principal, so the size of the loan decreases over its lifetime. In both these loans (with no fees) the APR is simply 5%.
I may have muddied the waters with mentioning potential differences in compounding period or daycount basis. In a mortgage that usually won’t be an issue. @Richard_Pearse was essentially correct - absent fees, the APR equals the nominal interest rate.
According to this page, @Saint_Cad is correct with his calculation.
However, if I plug some numbers into this calculator it agrees with you that the APR = the interest rate.
No he’s not. I assume you are looking at the formula on that page, which corresponds to the simple formula that @Saint_Cad used? But note that “principal” is a variable in that formula. This simple formula would be applicable in the way @Saint_Cad used it only if it were an interest-only mortgage where the principal stays constant through the life of the loan. But in a typical mortgage your monthly mortgage payments are also paying down principal each month, the size of the outstanding loan decreases each month. Both the nominal interest rate and the APR are calculated with respect to this reducing principal in exactly the same way. Absent fees, the APR equals the nominal interest rate.
Thanks. That appears to be where we are both getting confused.
If you buy a $300k house and put down $100k, you need to borrow $200k. If you buy a $500k house and put down $300k, you again need another $200k. The amount of interest you pay would be the same for both. The price of the house is NOT relevant. The only things that matter are the size of the loan (plus fees), the APR, and duration of the mortgage.
There is no way that the APR on a 5% interest loan is 2.65% or 2.67%. There is definitely something wrong with your calculation.
(Nitpicking myself, you could probably construct a case where you had a one year floater that started at 5% but reset to zero after the first year or something. But, for a fixed rate loan, no way)
I explained what’s wrong with that calculation.
Thanks. I’m hoping @Saint_Cad comes in to correct himself.
The problem with the formula “(total interest and fees) ÷ (principal × time)” is that it assumes the amount of the loan is constant throughout the term of the loan. But the amount owed decreases with each payment. To properly determine APR, we need to divide the interest paid each payment by the remaining principal each payment. And then average over all of the payments. But we already know the answer, if there’s no fees: the original 5%.
If you want a crude approximation of this, assume the principal decreases linearly throughout the term of the loan. Then the formula becomes “(total interest and fees) ÷ (1/2 × principal × time)”. For the above example, you’d get the 5-year loan has an approximate APR of 5.29%, and the 6-year, 5.33%. But note that the remaining principal drops exponentially, not linearly, so this approximation is an over-estimation.
(In agreement with Riemann, but thought it might be useful to explain a different way.)
Exactly. To map out all the cash flows and the reducing principal, you need a spreadsheet with 60 rows, updating after each monthly payment. But of course since both the nominal interest rate and the APR are calculated in exactly the same way with respect to the steadily reducing size of the loan, absent any fees we don’t need to actually do the exact same calculation forwards and backwards.
On another note, I think it’s useful to look at the total interest fraction “(total interest and fees) ÷ principal”. This is the total interest as a fraction of the original principal. It excludes the time and is not a rate, but removes the scale of the principal.
In the recent example, for the 5-year loan, the lendee pays interest of 13.23% of the principal. For the 6-year, it’s 16.00%. No hiding behind a yearly rate when you look at the whole loaf.
No, I don’t agree with this. The annual rate is the correct way to analyze the economics of a loan, not something to hide anything behind. Time has value.
Yeah, that’s a huge problem, because non-amortizing loans are really unusual for individual investors.
I agree and that’s why I like to remove the time from formula, because then its impact is directly there. The value of changing from a 5-year loan to a 6-year loan is 2.77% of the principal (the difference of their total fractional interest).
Note that I’m not saying APR is not useful. It’s just that it’s not telling the full story by hiding the additional interest you’re paying over the term of the loan. Some people don’t realize the extra they’re paying in a longer loan.
That’s not agreeing with me. I’m saying that your approach of “removing time” - i.e. looking at the total amount of interest over the lifetime of the loan - is the wrong way to analyze the economics of a loan.
When we think about investments or look at the fixed income yield curve we generally look at annualized yields or returns. That’s because time has real value. A stock market that rises 10% in one year is a much better investment than a stock market that rises 10% in 2 years. Similarly, if don’t have to repay a loan for an extra year, of course I expect to pay more interest - because I have use of that money for an extra year and I can invest that money and generate a return on it. Whether it’s a good or bad loan depends on the rate I’m paying per year.
(underlining added for emphasis.)
You are not innumerate.
I’m trying to explain in simple terms what the cost to the lendee is for agreeing to a longer loan. The innumerate lendee sees the dollar impact of their monthly payment going down, but does not see the dollar impact of the greater total interest they pay. My suggestion is one way to make it clear what that value is.
This is a good point – for my mother’s most recent car purchase, the total sleazeball finance guy convinced her to go from a low rate 3 year loan (that she can easily afford) to a higher rate 6 year loan by showing that the monthly payment is lower. It was like 1.5% to 3% or something, too.
Most people, the vast majority of people, don’t really understand loans and mortgages. The sleazeball finance guy himself said to me that if she paid the loan before the one year point, she won’t pay interest – he thought the interest accrued annually and that’s why it’s called the annual percentage rate. Good sales guy, total moron.
So what? I’m saying that your approach is wrong, so teaching it to someone less financially literate is not helping them. The rational and appropriate way to analyze a loan is NOT to look at the total amount of interest without regard for the term of the loan. That would be teaching a fallacy that time has no value. What’s important is the amount of interest per year.
It’s not as though I’m saying anything controversial here. Why do you think there is strong legislation that specifically forces lenders to be transparent about the APR on their loans?
That supports my point. The rational comparison is between the APRs on the two loans. Noting that the APR doubled from 1.5% to 3.0% showed that the longer term loan was obviously a much worse deal. Whereas if the APR had stayed equal or close to 1.5% for the longer term loan, that would have been an attractive deal, even though the total amount of interest of the lifetime of the loan would have been larger. Because that’s a very attractive annual rate at which to obtain access to funds.
A longer term loan will always have lower periodic payments. But how much lower is a good deal? We look at the APR to judge that, not the total amount of interest payable over the lifetime of the loan.
Yes, again, I agree that APR is necessary when comparing loans. You’re preaching to the choir. But, how do you quantify to an innumerate person what the difference in value is between a 1.5% loan over 3 years and a 1.5% loan over 6 years?