Note: this is for teaching older teens, not a technical accounting class.
APR is (total interest and fees) ÷ (principal x time). No problem there. The problem comes from making a down payment on a mortgage. No matter what your down payment is, the APR stays the same because it is a function of the loan amount and doesn’t address that you are not paying interest on the money you paid for the down. ISTM that in this case, the APR would be a better measure of interest paid if it were based on the price of the home and not the mortgage principal you borrowed.
If your formula is correct, this can’t be right. More down payment means lower principal and lower total interest, but I don’t think the relationship is linear, since amortization is nonlinear (could be wrong there).
I thought the APR basically amortized the upfront fees over the life of the loan or something.
An amortized payment is proportional to the principal so if you double the principal [insert math here] you get twice the total interest. It is not proportional to other variables such as the term or nominal interest rate.
I’m having a hard time figuring out what you mean. When you say “down payment”, do you really mean “additional principal payment” after the loan is closed? If you’re really referring to the original down payment at the time of purchase, then I don’t understand you at all.
The APR would be affected by the size of the down payment, to the extent that the fees (in the numerator) don’t change in proportion to the change in principal (in the denominator). But beyond that, it shouldn’t be impacted, because the APR is about the cost of the loan relative to the size of the loan, not relative to the size of your purchase.
Why is that a “problem”? A larger downpayment means you are borrowing less money, so at the same interest rate you are paying less interest. Are you confusing APR with the absolute size of the mortgage payment? The latter will of course be smaller. And in fact a lender will usually offer a lower APR on the smaller amount of money you need to borrow if you can make a larger downpayment, because the loan is less risky (the loan-to-value ratio is lower).
Why do you think this makes sense? The denominator for the interest rate you are paying on a loan is the size of the loan, not the (higher) value of the collateral. The purpose of an APR - and the strict rules for how an APR is calculated and presented by a lender - is to ensure that non-experts can compare like-with-like when shopping around for the best deal from a lender.
Perhaps you want some other measure of whether a house is “affordable” to a potential buyer given their circumstances, but it is not correct to call that an APR. Something like what you suggest would muddy the waters on the effective interest rate that is being charged on the actual amount borrowed, precisely the kind of deceptive practice among lenders that the concept of the APR is designed to prevent.
Yeah, that makes sense. I’m not sure I understand your overall question.
APR is weird, because for some loans, the APR can be less than the interest rate – I think 5-1 ARMS used to be like that, because at the time of the quote, the rate it would reset to was lower than the current rate.
To expand on my earlier comment, in the bad old days lenders would market their offering based on the size of the initial monthly mortgage payment, allowing them to hide massive fees or engage in other deceptive practices about the true economics of the loan. And borrowers with little expertise in finance were frequently suckered by thinking that this was a good metric for which mortgage company was giving them the best deal on the loan. It is not. That’s what the APR is for.
To be clear, the OP’s formula is not the correct math, it is an oversimplification. The APR takes into account exact dates of all cash flows associated with the loan to provide a net all-in effective rate of compounding interest on the loan.
But OP’s formula provides a correct conceptual representation of which variables are in the numerator and the denominator of the overall calculation, appropriate to the stated teaching context.
Since this is simplified for teaching, I’m assuming no fees. In that case, yes total interest paid is proportional to the principal. For example I buy a house for $300000 and I pay $250000 in interest over the term of the mortgage (APR = 2.78%). I make a 20% down payment and so my loan is now $240000. The total interest likewise reduces by 20% (I have checked the math, this statement is true) to $200000, giving an APR of … 2.78%.
Yes I recognize that this means that any down payment will not change the APR but this contradicts the concept I’m teaching students the concept that lower APR on the same loan (but changing a variable) implies saving more money. I certainly understand what is going on here but I fear my students won’t and ISTM that teaching them
higher down payments = lower interest
and
APR as a truer measure of the cost of a loan
that it makes more sense to base the APR on the price of the house and not the loan principal for the purpose of teaching students how to pay off loans.
No, once again - what you need to do is be clear about when you’re talking about an interest rate and when you’re talking about the absolute amount of interest. The rate (APR) is the appropriate metric when comparing loans in the marketplace, that it the explicit purpose of the APR. The absolute size of the morgtage payment is the way to assess whether you can afford to pay off a loan given your personal circumstances.
Riemann I know you’re right, but I’m still not happy with it. I tell my students all the time to trust the math and I just know a down payment saves you money and therefore lowers your APR for reasons (but it doesn’t).
It seems like a pretty elementary mathematical principal that you should be teaching to teens to distinguish between a rate/ratio and an absolute amount.
I’m not sure why it’s complicated or counterintuitive to explain:
a larger downpayment means you are borrowing less money, so at the same interest rate (APR) the absolute amount of interest is less.
There is, as a I mentioned, a subtler second-order effect that a larger downpayment means the loan-to-value ration is smaller, making the loan less risky for the mortgage company, which may mean that they do offer you a lower APR for the smaller loan. But that’s a distinct and less elementary concept.
The other important thing that you should be teaching teens is that ownership of the house is an equity investment. Borrowing rates do not apply to equity investments. Borrowing rates apply to loans. Therefore it would be quite wrong to associate the APR with the value of the house. It should be associated with the size of the loan.
It might be helpful to analyze the economics of the loan and the purchase of the house as two separate transactions. They are linked only in the sense that the house is then hypothecated as collateral, something that backs up your ability to repay the loan. Collateral is always worth more than the value of a loan, sometimes substantially more. If the loan were from a family member, this hypothecation step might not even occur, the loan really could be completely separate from the purchase of the property, the family member might just trust you to pay it back eventually. But of course a commercial lender won’t just “trust” you with a huge loan without some asset as collateral - a backstop if you fail to meet your obligations.
Actually, I think I’ve got it and it is something I teach my students but your answers helped me frame it better in my mind in very specific financial terms. What makes two loans different - thus appropriate to analyze with APR? Changing the rate, term, payments per year (and if I remember my accounting class paying at the beginning of the period vs. the end but clearly my class isn’t THAT pedantic). However if the only difference are the principals is not a different loan, just amount borrowed, thus APR shows exactly what it should.
If you want to show your students that putting a larger down payment will save them money, don’t divide the interest by the principal, just show them the interest payments. The difference in interest payments between 20% down and 25% down will be dramatic, I bet.
Riemann you are not wrong, but I am teaching high school students who have had absolutely no financial education whatsoever either from home or school. I’m teaching the difference between simple and compounded interest, how to amortize a loan, how term life insurance works. AND this is a math class not a finance elective so my curriculum has to be tied into the math standards somehow. If I taught how increasing equity affect my house’s LTV I’d first of all have to relate it to math (OK that’s not too hard) but I’m starting with kids that don’t know how houses work as an investment. This means I have to pick and choose the most bang for my buck in limited time.
I already do. The problem is this.
Lesson 1: Show how interest changes when you increase the term of your car loan. (Goes up)
Lesson 2: Now calculate the APR for each term (Goes up)
Student conclusion: higher APR means more real cost to the longer termed loan
Lesson 3: Show how your total interest when you increase the down payment on a house (Goes down)
Lesson 4: Now calculate the APR for each down (Stays the same)
Student conclusion: ??? Shouldn’t APR go down?
This is very confused. You are using the word “interest” without making clear whether you mean the interest rate, the size of the periodic payments, or the total amount of interest over the lifetime of a loan.
If you increase the term of a loan, the size of the periodic payments of interest+principal goes down, although of course the total amount of interest over the lifetime of the loan increases. The APR may or may not change, depending on whether the lender requires a higher rate of interest for a more risky longer-term loan.