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.
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.
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.
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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