I had a good chuckle while reading this epic discussion thread on the Bogleheads Investment Forum: Does a home mortgage use Simple or Compound Interest?

It sounds a like factual question, as in "Is Miami located to the north or south of Boston?" The answer shouldn’t be ambiguous or subject to opinion or interpretation. You look at a map and say "south" and everybody would agree. Yet as I’m writing this, there are more than 100 replies, and still growing, by the smartest people offering opposite answers, assisted by graphs, math equations, and numeric examples. Some say it’s simple interest; some say it’s compound interest.

Someone answered by saying it’s a **compound interest loan that doesn’t compound**. If it doesn’t compound, does it make it a simple then? Or is it like the difference between 0 and null?

To answer the question we first need to understand what is a simple interest loan, what is a compound interest loan, and what are the characteristics of each.

### Simple Interest Loan

In a simple interest loan, the interest in a second period is not affected by the interest in the previous period. Suppose we have a 3-year $100,000 simple interest loan at 1% annual interest. The interest for each of the 3 years is $1,000 for a total of **$3,000**.

If the interest rate is 2% a year, the interest over the life of the loan would be **$6,000**, exactly twice as much as in the 1% loan.

If the rate is still 1% a year but the term of the loan is 6 years instead of 3 years, the total interest over the life of the loan also doubles.

Same goes with a 3% loan or a 9-year loan. You just multiply the principal by the rate and the years to get the total interest.

### Compound Interest Loan

In a compound interest loan, the unpaid interest at the end of the first period is added to the principal for the second period, allowing the interest to compound. In a 3-year $100,000 compound interest loan at 1% annual interest rate, the interest for the first year is $1,000, the second year $1,010, the third year $1,020.10, for a total of **$3,030.10**. That’s more than the total interest paid on a comparable simple interest loan.

If interest rate is twice as high at 2%, the total interest over the life of the loan is **$6,120.80**, which is more than twice the total interest on a 1% loan, due to compounding interest.

If the term of the loan is twice as long at 6 years at 1% interest rate, the total interest over the life of the loan is **$6,152**, also more than twice the total interest on a 3-year loan at the same rate, again due to compounding interest.

A higher rate or a longer term in a compound interest loan costs more than just a straight multiple.

### Home Mortgage

In a typical home mortgage, your monthly payment first covers the interest for that month, with the remainder being applied to principal. Interest does not add to the principal for the next month. This led to the answer that it’s a **compound interest loan that doesn’t compound** because you pay the interest for each month in full, leaving nothing to compound in the next month.

If the mortgage is interest-only — yes, there are those mortgages — it behaves exactly like a simple interest loan. If the rate is twice as high, your total interest in each period and over the life of the loan is twice as much. If the term of the loan is twice as long and the rate is the same, your total interest over the life of the loan is also twice as much.

### Principal Payments

Paying down principal by an amortization schedule makes it more tricky. Even though interest still doesn’t carry over from month to month — and if you skip a payment, you are not charged more interest the next month — the loan no longer behaves like a simple interest loan.

Doubling the interest rate more than doubles the total interest over the life of the loan. The total interest of a 30-year mortgage at 8% is **2.3 times** that of a 30-year mortgage at 4%.

Doubling the length of the loan also more than doubles the total interest over the life of the loan. The total interest of a 30-year mortgage at 4% is **2.2 times** that of a 15-year mortgage at the same rate.

Making a principal payment early has a compounding effect. If you pay $1,000 extra in month 13, you not only stop paying interest on that $1,000 but you also cause more of your subsequent regular payments to go toward principal, further reducing the interest you pay.

These characteristics make a typical home mortgage with amortized payments *behave* more like a compound interest loan, but it doesn’t make it one. **The compounding effect comes from varying principal payments, not from compounding interest**.

Between two mortgages, if you keep principal payments the same, they behave like simple interest loans.

If you have a 8% loan and a 4% loan, and you just go by the amortization schedules, you are paying less toward principal each month on the 8% loan, at least in the first half of the loan term, even though your monthly mortgage payment is higher. Those lower principal payments compound, resulting in your paying more than twice as much in total interest on the 8% loan versus the 4% loan.

If you actually keep principal payments the same by making extra principal payments on the 8% loan, your 8% loan will be exactly twice as costly as the 4% loan but not more than twice. That’s the classic trait of a simple interest loan.

### Conclusion

A typical home mortgage is still a **simple interest loan** even though it feels like compound interest. The compounding feel comes from varying principal payments. If you don’t let the principal payments vary, as in an interest-only loan (zero principal payment), or by equalizing the principal payments, the loan interest itself doesn’t compound.

Why does a seemingly simple factual question elicit completely opposite answers? Because it focuses people’s attention on the wrong thing. The interest doesn’t compound. The principal payments do. A $1,000 principal payment saves interest on that $1,000 and causes higher principal payments the next year, and higher the following year, and so on.

This is the same situation as in asking whether the 401k loan interest is double taxed. There are two taxes, unrelated, and you still pay the same two taxes whether you borrow from your 401k or not. Focusing on the wrong thing leads people down the wrong path.

In practice though, you are better off treating the mortgage as compound interest even though it actually isn’t. Lowering the rate has a compounding effect. Shortening the term has a compounding effect. Pre-paying principal also has a compounding effect.

[Photo credit: Flickr user *hannah8ball*]

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Carl says

Uhm… I think your examples are wrong.

On simple interest:

“Suppose we have a 3-year $100,000 simple interest loan at 1% annual interest. The interest for each of the 3 years is $1,000 for a total of $3,000.” I didn’t think you paid interest on principle that had been paid off in previous years. The interest is recalculated after each payment only on the unpaid principle (assuming no late payments). I’m obviously assuming you are making payments that cover more than just the interest.

The same applies with the compound interest example.

You never pay interest on money you have already paid back. Or said another way, Interest only accrues on the outstanding balance.

Harry says

Carl – That example has just one balloon payment at the end of the term. Same for the next example for compound interest. I need to start with the simplest form to show the difference between simple interest and compound interest. Once you start making payments in the middle it gets murky.

Lei says

Understandable article. Thank you! My question is: for paying extra principle each month, does it make difference paying on the 1st day or paying on the last day of the month? In other words, is interest calculated on daily basis?

Harry says

Lei – The required monthly payments are due on the 1st of each month, usually with a grace period to the 15th of the month. It doesn’t matter whether you make the required monthly payments on the 28th of the previous month or the 12th of each month. You are not charged less interest for paying early or more interest for paying late (but still within the grace period). The extra principal payments, however, are calculated daily. The day your extra principal payment hits, you stop paying interest on that amount for the rest of the month and beyond. If you want to be anal, you make the extra principal payment separately (and make sure it’s marked as extra principal payment) and you pay as early as you have the money for it.

Rob says

My mortgage payment is $2000 per week. My payment is due on the 1st of the month. I have a grace period until the 15th. My payment includes principal, interest, and escrow. I pay $1050 every 2 weeks to the mortgage bank through my online banking service of my personal bank; one payment before the 1st of the month and the 2nd payment before the 15th. I noticed the online detail on the mortgage bank’s website each month the first payment shows up as “lockbox” on the day it is received and the next day a new transaction shows up as principal curtailment for $1050. When the 2nd payment arrives it shows as “lockbox”. I call the bank after I see the 2nd payment arrives but definitely before the 14th of each month and have the 1st $1050 reapplied to the monthly payment along with $950 of the amount in “lockbox”. The extra $100 is applied as principal curtailment. The breakdown on my monthly “paper” statement shows (example w/ rounded numbers; each month my escrow distribution stays same, interest paid goes down slightly, principal paid increases slightly) $500 to principal, $1000 to interest, $500 to escrow, and $100 extra principal. My intent is to have the first $1050 payment (paid before the 1st of the month) applied as principal (which it appears to be doing), thus reducing my principal balance for approximately 14 days each month. Then I call on the day before the grace period expires and have it reapplied as the monthly payment, along with a $950 portion of the 2nd $1050 payment. I go through this extra effort each month because it appears to me that I benefit financially by having my principal balance lowered by $1050 for 14 days each month. In your opinion should this strategy financially benefit me more than just paying the $2000 mortgage payment + $100 extra to principal each month before the 15th + an additional $2000 principal only payment each year when I get my taxes back. I was quite surprised when I first noticed on my bank’s online detail of my account that the $1050 payment I make each month before the 1st was automatically applied to principal. This discovery is what prompted me to develop this strategy that seems to maximize my benefit financially with the only down side being a 5 minute call with a customer support person at the bank each month to be sure everything gets applied properly. Thanks for thinking this through with me. I am trying to minimize the overall interest paid and pay the loan off quicker.

Doug says

I pay extra mortgage principal payments each month. At my bank, the principal payments are ‘effective’ as of the first of that same month, regardless of which day I pay it – at least that’s what my bank statement tells me. I see that as an extra bonus.

Junsheng says

I liked some other stuff you wrote but this is not correct. Let’s say I have a mortgage of 100k with APR of 12%. If the mortgage were indeed simple interest and I pay a first payment of $1010 after one month, that should have returned to the lender $1000 principal + $10 interest on that $1000 principal for the month. For the next month, my principal used for interest calculation would have been reduced to 99k (sure I still owe another $990 of interest from that $99k for the 1st month).

But that is not reality. The reality is the $1010 payment will be considered $1000 “interest” and $10 “principal”. Note that it is only convenient to think a part of the payment being interest and the rest being applied towards principal in this case. The fact of the matter is, after one month before I make any payment, I simply *owe $101k* to the lender. There is no distinction of the $1000 interest from the $100k principal, because they would equally earn interest going forward, until I pay some of it off. This is compound interest.

Harry Sit says

I don’t follow how you are able to pay down $1,000 principal with a $1,010 payment when the interest due in the first month is $1,000. The lender will always collect the interest first. Any excess is then applied to principal.

Suppose you have an interest-only mortgage of $100k at 12% APR as in your example. Think about what if you don’t make the first payment. Putting aside the issue of late fees, how much do you need to pay in the second month to catch up to current? The answer is $2,000. The interest you didn’t pay in the first month will not increase the interest in the second month. No interest-on-interest. That makes it a simple interest loan. It’s different than a credit card loan.

Junsheng says

You overlooked the time value of money. By requesting interest on full principal amount be paid each month, it is already compound interest. To make a point, suppose the lender made one hundred such $100k loans at 12% APR. After one month, the lender can make a new loan using the 100 x $1000 interest payments he will have received. That is interest on interest.

In my book, real simple interest is like the counterpart of CD. For the sake of argument, think of my $100k loan as one hundred bills with 12% APR each at $1000 face value that matures in 30years with no prepayment penalty. In my prior example, after one month, I choose to pay off one such bill, which costs me $1000(1+12%/12) = $1010, and reduces the principal amount by $1000.

Harry Sit says

What the lender does with the money received is not your business. When it comes to determining whether a loan is a simple interest loan or a compound interest loan, it only matters whether *you* are paying interest-on-interest on this loan. If yes, it’s a compound interest loan; if not, it’s a simple interest loan.

Requiring payments received be applied to interest first before reducing principal doesn’t change whether a loan is simple interest or compound interest. Going back to the simple example in this article, $100,000 simple interest loan at 1% annual interest for 3 years, if I add the requirement that any money received before the end is applied to interest first, the loan is still a simple interest loan. 3 years, $3,000 in interest. If you do a partial payment in the 26th month, the interest for the second year is the same as the interest for the first year. It doesn’t compound.

CDs by the way do compound. Many CDs pay interest monthly. The interest you receive in the second month is higher than the interest you receive in the first month. If the CD only pays interest annually, the interest you receive in the second year is higher than the interest you receive in the first year.

Junsheng says

It’s not my business, true.. But that was not the point. The point was, if the lender keeps lending out the interest payment he received, he earns compound interest. Guess who’s paying the bills? The borrowers as a whole. Equivalently, forget about the lender lending the interest out, but think of the *loss* of time value of the (interest) money the borrowers paid.

If you take the formula for mortgage payment (P) for a given loan amount (L) and interest rate (I), it has compound interest written all over it: L = P * sum_j (1+I)^-j. Here j goes from 1 to the total number of payment cycles (typically months).

Harry Sit says

Then by your definition as long as the borrower is required to pay interest before the final loan due date, there is no simple interest loan, because the lender will be able to lend out the money received and the borrower will lose the time value on the money paid. That’s not the definition of a simple interest loan as commonly understood.

If you don’t agree that a $100k loan at 1% interest for 3 years with $1,000 payable at the end of each year plus a final payment of $100k at the end is a simple interest loan, we don’t really have a common ground at the root level. The whole discussion about whether something is a simple interest loan or not becomes moot when there is no agreement on the definition of a simple interest loan.

If you create your strict definition for the color ‘green’ and you say the color of the traffic light isn’t green because it’s off a shade, you are right. When others say the light is green, they are also correct.

David says

Can my lender add unpaid interest to my loan balance on an automobile installment loan.

Olabode says

Harry, I appreciate your write-up. Its educative and interesting based on the scenarios stated therein. I think the high interest rate ab initio in the life cycle of a mortgage loan is as a result of the borrower paying smaller principal which increases gradually month on month. And the interest payments also reduces month on month.

@Junsheng, Please clearly state your argument and the mortgage payment formula. I will be glad to have you do that.

Thanks guys

Jose says

Hi, if I have a mortgage on which I make advance payments on the principal every month. Does it matter if make that payment at the beginning of the month or if in turn I make the payment at the end of the month, before the next payment is due?

Thanks.

Harry Sit says

I don’t think it matters. However, holding on to the extra principal payment for extra 20 days doesn’t really earn you much interest anyway. If you keep $1,000 for 20 days at 1%, you will get only $0.55 extra before tax.

Steven Charles Scott says

Good stuff, Harry. When I read some of the comments to your article it reminds me of the simple fact that “intelligent” people read more (and most times conclude before reading) than actually just looking and confirming what is actually happening in practice i.e. empirical evidence is big here). Your CD rebuttal is spot on, anyone can see that easily in their own CD sitting in the bank over two months. And naturally that is also obvious (after good blogs such as these) when looking at your mortgage statement over time.

WR says

jun sheng just shut the fuck up alr lol… clearly doesnt even understand what simple interest is. going by his argument, there is no such thing as a simple interest loan in any way since the lender can do what ever he wants with the earned interest LOL..

michael ryan says

OK theres something missing here which is why youre all still confused although the author is correct. its the temporal relation between the interest rate and the payment schedule. In his example of an interest only loan the interest is “compounding” at the one year time frame in one example and is not in the other. Its pretty clear but both loans principle is only due at loans end (3 years) and thats a different time frame from either interest schedule.

Put aside the question of what time of month you make your payments for a moment youre getting ahead of yourselves. lets look at the two loans again more closely and imagine them this time as not balloon loans but a typical mortgage that will be paid off by the end of the loans term in this case 3 years whats most important to understand is that you are not paying interest on money you no longer owe principal you have already paid back. The math is a bit tricky but to allow you to have the same monthly bi monthly annual etc payment they figure that out. say in the non compounded 3 year loan you wanted annual payments they would be less than the authors example because at the end of the first year you would owe the interest on the entire 100k but at the end of the second year you would only owe interest on something close to 66,660, and at the end of the third year on about 33,333. in short as you repay the loans principal you are paying interest on less money. Its a differential equation where the parts are all moving simultaneously but a simple concept and it applies to both compounded and simple interest loans. Imagine again this loan as compound interest the important thing to note first is compounded over what period of time, a real simple interest loan is in a sense a compounded interest loan compounded over the time period (term) of the loan but the authors example is a more standard understanding- the interest is calculated at a rate per year, but you do not pay interest on the interest, I havnt read the fine print but Im pretty sure you actually would pat interest on the interest of any normal mortgage if you actually paid less than the principle and the amount of interest that was calculated to keep you ahead of it. This is what the authors trying to make you see, the bank has already figured out how much interest you must pay and how much principle and if you pay a bit late more of you payment will go toward interest and if you missed a payment not only will you owe interest on that amount of principal longer you now missed paying that interest on last months balance and its essentially become principal instead of changing your payment amount every time you are a day late or early or even miss they will just adjust the length of the loan on paper which is why when you refi or wish to pay off they must give you a calculation they call the payoff.

But heres the thing they are not trying to make interest on interest in fact they strongly resent people who attempt to skip payments and let the interest accumulate.They instead tried to fugure out an “amortization” schedule that accounted for you paying on time a certain amount that would end up with you only paying interest each month on the principal balance they estimate it will be and enough principal to be paid off by the term of the loan obviously these are moving in opposite directions you payment amount is calculated assuming you pay exactly that amount exactly on time for the entire term in which case you will not be paying interest on interest. if youre still confused lets make it a one year loan of 100k at 10% if it were a ballon loan at the end you owe 100k plus 10k interest=110k now if divide by 12 its 9166.66 but thats wrong because after every monthly payment you are borrowing less money so they have figured it out to be 8791.59 if paid on time you will be paid off in a year and have only paid 10% on whatever you still owed every month which keeps changing

Mick says

Harry, what do you think about the concept of using a HELOC to pay off your mortgage? The concept that was shown to me is to get a 1st position HELOC and pay off your mortgage and also use the HELOC like a savings or checking account. Put all of your money you make into the HELOC each month and then pay your bills and life expenses out of it. Looks like this will pay your mortgage off in 5-10 years and save thousands in interest.

Thanks!

michael ryan says

Thats retarded who telling you this a heloc salesman or some income taxes are unconstitutional nutjob?

mortgage rates are much cheaper than helocs and amortized over longer periods. Now whether you want to have a mortgage debt is a two sided sword. On one hand the rates are actually cheaper than real inflation so in essence youre getting paid to borrow so why pay off a loan when you could use that cash to make a better return than your mortgage interest rate. If you actually can. Most use mortgage to leverage say real estate is appreciating at 5% annually and you have 100 k earning 3% in the bank you could buy a 100 k house and make an extra 25 on your cash or you could get a 400 k morgage @ 3% and buy a 500 k house and make 10% on your 100k. now if you can make say 7% in stocks why pay off a 3% mortgage to save 3% when you could make 7%? The other side is how safe do you want to play it if the stock market seems to risky for you and you are not interested in other investments you may want to pay off your mortgage. but keep in mind a house is really really illiquid and you are putting all your eggs in the home basket if the real estate market tanks you may not lose your home but it might not be worth anything. this sounds confusing but if you put that other money in investments that did better and held their value you could if real estate crashes buy a much better home for pennies on the dollar, you could walk away from your mortgage or renegotiate it with the bank if you had little equity in it.

Ok regarding HELOC they ar basically second mortgages or lines of credit, in some way they sem great no interest unless you use it only interest on whats used. thing is these lenders will reduce that line of credit id realestate prices drop so you only have the line of credit in good times unless you spend it at which point you are paying interest.

But their rates are lower than unsecured credit and often tax deductable.here a good way i think to use them, or at least how i have. So i bought when there was blood in the street in 09 and have a house with a low interest rate and a couple million in home equity its worth five times what i paid for it. great but im worried so much of my paper wealth is in one illiquid property. I dot really want to pay off a 3% mortgage because its free money with inflation more than that and my rents cover the mortgage. but i would like to diversify some of that equity, a cash out refinance would raise my rates and cost quite a bit in fees, but i could get a heloc and use it to take someof the equity out of this property and invest it in something else that would pay more than the heloc interest. I could buy some bare land in the retirement area i have my vacation/ retirement home and build a spec property. A rental in that area would bring in more than the heloc interest especially when rental tax advantages calculated, i could expand my hobby farming business, or start another business if i were sure enough of it to essentially be betting part of my house on, I could even buy gold, or something it might not pay off right away but i would have diversified the home equity. But realize my house is valued at 5x times my mortgage and has rents covering the mortgage I would be only using a bit of that equity to invest in something else. Mot peopleuse helocs to fix up there house which makes sense if they will actually see a return in the matket value which many renovations do not. some just are resigned tothat house and want to live more comfortably rather than selling and moving to a better suited house.

Mick says

Thanks for your reply Michael.

The concept is a 1st position heloc, not a 2nd position. My understanding is in a heloc you only pay interest daily on the current balance. So if you put all of your monthly income directly into the heloc that makes your heloc (ie mortgage) payment and it pays down the balance (so paying less in interest) until you take money out to pay any bills. Supposedly this concept knocks off so much interest as compared to an amortized loan (heavy on interest on the front end) that you can pay it off in 5-10 years.

Maggie Ru says

I think the best standard to apply to decide if an interest rate is a simple interest rate or a compound interest rate is that to see if the interest generated by the principle during a specific time is counted for future calculation of the interest. In other words, if the interest is not generating new interest, then it’s a simple interest rate. If the interest is generating new interest, then it’s a compound interest rate. For example, you have a mortgage of 10K, the first year mortgage total payment let’s say is $5000, and $4000 is paid as interests with only $1000 deducted from the original principle, which gives you 9.9K principle for the beginning of the second year. When bank calculates the interests for the second year, they use 9.9K which in certain way includes the interest $4000 you paid in the first year. So the $4000 interests you paid in the first year actually generate new interests for the second year. In this way, I would say home mortgage interest rate is a compound interest rate.

Harry Sit says

You lost me toward the end. How is it 9.9k when you started with $10k and you paid off $1,000? Wouldn’t it be $9k? Or if you meant you started with $100k borrowed and it becomes $99k after you paid off $1,000 in the first year, in what way does the 99K include the $4000 interest you paid in the first year? $100k – $1k = $99k. The interest number isn’t part of it, whether it was $4,000 or $8,000.

Maggie Ru says

Thank you for the reply and sorry for the typo. What I meant is to start with $10K principle and $100 principle paid off with $400 paid for the interests in the first year leaves 9.9K principle for the second year. What I am trying to say is a simple interest rate actually represents a way of calculating interests. When a rate is a simple interest rate, we can use a formula of I=PRT to calculate the total amount of interests paid for a loan, where P is the principal, R is the annual interest rate in decimal form, and T is the loan period expressed in years. When it comes to the home mortgage, within each year, the R can be considered a simple interest rate when calculating interests for that year and the formula I=PRT applies. But to calculate the total I paid for the mortgage, the formula I=PRT doesn’t apply and it’s much more complicated. From the perspective of viewing simple interest rate as a way of calculating interests, the interest rate for a home mortgage is not a simple interest rate. In addition, I am not denying your point of views, but just provide a different perspective to understand the simple interest rate.

Maggie Ru says

In addition, I think from the banks (lenders) point of view, the mortgage rate is a return rate of banks’ investment. Banks use the rate more like a discount rate (which is a compounding rate) to get present values of future cash flows which help decide how profitable of their loan (investment). In my opinion, a simple interest rate is a very ideal situation in the real lending/borrowing world.

Marc says

I understand that with mortgages we’re never reporting accrued interest since they are already paid… so that technically fits the definition of simple interest. However, why are we saying that in the US, mortgages are compounded 12 times, and in Canada 2 times? Something must still be compounded?

Joseph says

Some of the information here is correct, and some is incorrect or not stated properly. Take it form a mathematician.

First of all, “Does a home mortgage use Simple or Compound Interest” is indeed a factual question, the same as “Is Miami located to the north or south of Boston?” The answer is a simple yes or no. The determination… if you miss a mortgage payment, does the interest get added to the balance that the next month’s interest is calculated on? If the answer is yes, that’s interest on interest, which is compounding. If the answer is no, it is not compounding.

Just because a loan or CD can compound doesn’t mean it will. For a loan that compounds, if you make a payment at the exact moment that interest is being charged and will be added to the balance, and your payment is greater than this interest, you pay it off before it has a chance to be added to the balance and have interest calculated on interest. When this happens, mathematically, there’s no difference between compound interest and simple interest.

Nikko says

This is incorrect. Mortgage amortization has a premise that your interest payment goes lower as the Principal Balance goes down. In your example the Interest went higher every year. That’s false.

It’s compounded interest, because interest is calculated at the frequency of the payment, the more the frequency the more the interest.

Simple interest means the interest is calculated as a whole then spread apart based on the frequency. Big difference, because for simple interest no matter the frequency the interest adds up to the same amount.

For compounded interest the more the frequency the higher the interest. That is why making principal payments reduce interest because at every frequency you calculate the interest based on the principal balance. Lowering it lowers your interest for that frequency.

Naga Rajan says

Very good article Harry.

As far as the type of loan, I would say “*Simple Interest with Varying Principal*”, as I don’t see any compounding at all. Please let me know if I am missing something here. Thanks.