The math of this kind of finance is simple. What is confusing to someone with math skills is just the unfamiliar terminology. Be clear on definitions, and don’t use the word “interest” ambiguously.
Is this right? If it were, I would expect APRs on 30 year mortgages to be substantially higher than 15 year mortgages, leaving aside the yield curve question. I thought that, with no fees, interest rate and APR are exactly the same, so the term wouldn’t affect it.
Finally.
The very first post says “APR is (total interest and fees) ÷ (principal x time)”. This seems to be the first time someone has mentioned time. Without that, you can’t calculate APR.
APR is basically “what proportion extra will this loan cost me over its lifetime, over and above the principle?” It makes it harder to hide exploding rates, fancy extra fees with other names, etc.
A loan for $10,000 with $5,000 interest (plus fees) over 10 years, and a loan for $1,000,000 with $500,000 interest over 10 years, have the same APR. But - one loan I can afford, one I can’t.
So essentially, APR is a valuable tool but not the only one necessary to judge a loan. A person who buys their house with a down payment of $10,000 or $50,000 will have the same APR but their monthly payments will be smaller with the larger down payment. Or, with the same monthly payment amount, pay off the smaller loan faster, and so less interest, smaller APR.
Finally what?
I’m not sure how OP could have mentioned time prior to his first post. And I have discussed several times the importance of distinguishing between a rate and an absolute or total amount. What do you think “rate” means?
Because, ironically, you have omitted part of the concept of “rate” from this definition of APR. “Rate” does not mean the proportion extra over the lifetime, it means the proportion extra per unit time.
Lesson 4 should show how the total cost of the loan (and hence the house) goes down the more you pay up front/the less you borrow. If I put 50% down on the house and borrow less, the total cost I eventually pay will be less than if I put no money down and borrowed more.
Not to mention that in my home town, the difference in interest rates between a 15- and 30-year fixed rate is almost a full percentage point, so Lessons 1 and 3 would end up showing students the same thing.
As most loan fees are a points. That is fees based on the size of the loan. Larger down payment would mean less fees therefore the ratio of fees to interest would be very close.
NO ! . The loan amount is the amount loaned initially . Many many mortgages are for only 70%,50%,30% of the value of the property from the start. You can get a second mortgage, say you have 70% equity in a property, ie you owe 30% still, so you can get a second mortgage of up to 70%, rather than close the first mortgage, like in the olden days that wasn’t cheap if possible ??, and get a larger new mortgage. Every mortgage is only about the LOAN AMOUNT not the full value of the property. APR is about the initial balance , after the “down payment” … down payment is jargon anyway,it can mean deposit, so as to lock in the vendor to selling to you, or the minimum amount of equity to get the mortgage, or to get the mortgage at a cheaper interest rate , due to the lower risk to the lender, and avoiding mortgage insurance…
The APR is a function of the loan amount plus the fees charged. That is why you can get a loan at say 2.5% with an APR of 2.6% or maybe and APR of 2.8%. The APR verses the interest rate will give you an idea of the amount of fees charged. Most of the fees are based on the amount of the loan, but some fees are fixed.
What does any of this have to do with my scenario?
Everyone else, once again, assume no fees.
I think this is over complicated and is diluting what you are trying to teach. (What are you actually trying to teach?)
The APR on a loan with no fees is just the basic interest rate. It is an input to the interest calculation and will only change if the interest rate is different between different loans. A short term car loan vs a long term car loan might the same interest rate or it might not. You can’t say a longer term loan will have a higher APR because it should actually be the same provided there are no fixed fees and the basic interest rate is the same.
I think with lesson one and two you are leading your students astray by using loan figures that result in a different interest rate. That then causes confusion in lesson three and four when you discover that there is no need for the APR to change at all.
Depending on the age of your students I’d be doing the following:
Lesson 1. Given an interest rate of 7%. A long term loan vs a short term loan for the same amount. Which term results in the higher total interest paid? Answer. The longer term.
Lesson 2. Given an interest rate of 7%. A bigger downpayment on a fixed price item results in a smaller loan. Do you pay more interest on a smaller loan or bigger loan over the same term? Answer, bigger loan equals more interest. (Seems blindingly obvious to me but I guess I learned this stuff somewhere.)
Conclusion. Given a certain interest rate, borrowing more money and/or borrowing money for a longer time results in paying more in interest. To compare loans you need to compare both the interest rate and the term of the loan to work out which is more beneficial. It may be most beneficial to go for a long term loan with no penalty to make early payments.
As stated above, I’m already doing this.
No, you are not doing this.
The way you describe it you are deriving an APR from loan data. Then you wonder why in one case the APR changes but in another case it doesn’t. The reason for this is that one scenario uses different interest rates and the other one doesn’t. The students are getting confused because you have inconsistent variables.
It sounds like the puzzle with three men renting a hotel room for $30. They pay $10 each and go upstairs to the room. A few minutes after they leave, the clerk discovers the room rate should be $25 so he calls a bellhop over and gives him $5. “Give this to the guys in room 210.”
In the elevator the bellhop thinks, They won’t know what the rate is supposed to be, and pockets $2, giving the room-renters $1 each.
So, now the men in the room have paid $9 apiece. 3x$9 = $27. This plus the $2 the bellhop kept = $29. Where did the extra dollar go?
Nope. Try again. For the car the nominal rate stays the same and the term changes thus the APR changes. For the house the nominal rate stays the same and the principal changes but the APR stays the same. I have already explained repeatedly part of the curriculum already is to look at how the total interest changes. The part I was missing was what Reimann filled in which basically is: what does it mean to be a different loan. That was already part of my instruction (in another lesson) and Reimann gave me idea on how to integrate that into this particular set of lessons.
So thank you Riemann, you answered my question.
Ok. Help me out here then. An APR on a loan with no fees IS the nominal rate isn’t it? In which case if the nominal rate hasn’t changed, neither has the APR.
You are correct that the main difference is that (by law) the APR exposes hidden fees. But there may also be a slight difference attributable to the compounding period or the daycount basis for the interest rate calculation. If such a difference exists then unlike one-time fees it should be consistent across different terms.
Way too complicated for a normal high school class, but if you ever teach this to an AP class you can point out that, all other things being equal, paying $50,000 down on a $250,000 home will result in paying more dollars in interest than paying $25,000 down. But, unless you keep that extra cash in a jar buried in the backyard, there’s a cost to using that additional $25,000 for the down payment.
Although you very likely (like almost 100%) can’t guarantee a rate of return on your investment that exceeds the interest rate on the mortgage, it is possible that if you put that $25k in an index fund and the market does well, you could end up better off financially at the end of the mortgage term by increasing the loan amount.
Reimann answered this but I will point out that with the investment equivalent of rate of return (not CAGR) if you use simple interest then the RoR is the nominal rate.
Reimann’s answer is not consistent with your contention that the APR changes when the term changes given a static nominal interest rate and no fees.
Edit. In case it’s not clear, you are acting like Reimann is agreeing with you but I don’t think he is and I think you have a misconception about how changing the term of a loan affects the APR for a no fees loan. It doesn’t.
APR changes when the term changes on an amortized loan. That’s a fact.
$50,000 loan 5% interest over 5 years => total interest = $6,613.70. APR = 6613.70 ÷ (50000 x 5) = 2.6455%
$50,000 loan 5% interest over 6 years => total interest = $7,998.40. APR = 7998.40 ÷ (50000 x 6) = 2.6661%
Show me your calculations that show the two APRs are the same
Also, I never implied Reimann agreed with me. I’ve said like two or three times here that he corrected my misconception (or rather misrationalization) about home mortgages, down payments and APRs.