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.

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.