Simple Interest is Too Complicated

The only virtue of simple interest is that it is easier to calculate the amount owing than when we use compound interest. However, in today’s world, computers do our calculations for us and this advantage means very little. Despite its name, I’ll show that simple interest is far more complex than compound interest in important ways.

A Basic Example

Let’s start with an easy example to illustrate the difference between simple and compound interest. You borrow $10,000 from Uncle Jack to be paid back in 10 years. Uncle Jack isn’t a very loving uncle and knowing you have no other options he charges you 10% interest each year.

If Uncle Jack charges simple interest, then your debt rises by $1000 each year for a total of $20,000 after 10 years. The 10% interest is always charged “simply” on the original principal amount. To put this into a formula, if the interest rate is r=0.10 per year, the number of years is t=10, the initial loan amount is M=$10,000, and the future value after t years is F, then we have

F = M(1 + tr).

This formula works even if the time t is not a whole number. If you pay back Uncle Jack after two and a half years, it makes sense to use t=2.5 so that you would owe a total of $12,500.

If Uncle Jack prefers compound interest (compounded yearly), then your debt still rises by $1000 the first year to $11,000, but in the second year the 10% interest is calculated on the $11,000 debt. So, the interest is $1100 for a total debt after two years of $12,100. Each year your debt is multiplied by (1+r)=1.1. After 10 years, you owe $25,937. Expressing this as a formula we get

F = M(1+r)^t.

This formula also works if the time t is not a whole number. Such calculations are difficult to do by hand, but spreadsheets have no trouble with them.

So far, based on the formulas, I seem to be proving that compound interest is more complicated than simple interest. But that’s going to change.

A Simple Interest World

Imagine a world where all banks structure their loans with simple interest rather than compound interest. Suppose that there are two competing banks called Bank-5-5 and Bank-7-25. Bank-5-5 only offers 5% simple interest loans for 5 years, and Bank-7-25 only offers 7% simple interest loans for 25 years. Suppose that these banks will offer these same deals indefinitely (interest rates never change).

You need to borrow $100,000 and not pay it back for 25 years. Should you take the 7% loan for 25 years or a series of 5-year 5% loans? The first impression is that the 5% deal is better because the interest rate is lower. But all is not what it seems.

Let’s start with the 7% loan. Using our simple interest formula with M=$100,000, t=25, and r=0.07, we get a final debt after 25 years of $275,000.

Now let’s look at a series of five 5-year loans to cover the 25 years. After the first 5 years, the simple interest formula says the debt grows to $125,000. Now we have to roll that debt into a new 5-year loan. The new value for M is $125,000, and after 5 more years the debt becomes $156,250. Continuing this way until the end of the 25 years, the total debt comes to $305,176.

So, it turns out that the single 7% loan is actually better than the series of 5% loans. This calculation might not seem too bad, but suppose the loans are structured like a mortgage where we make monthly payments along the way. Which deal is better in this case? I had to break out a spreadsheet for this one. It turns out that the series of 5% loans is better with monthly payments of $556.50 instead of $577.70 for the 7% loan.¹

This interplay between simple interest rate, duration of the loan, and timing of payments makes it difficult to figure out which deal is better. This is analogous to trying to describe the motion of the planets. If you view the Earth as stationary, then the motion of the other planets seems very complex. But if you see the Earth as just another planet moving around the sun, then the paths of all the planets can be described simply as (roughly) ellipses. Just as simple interest is superficially simpler than compound interest, treating the Earth as stationary is only superficially simpler than treating it as though it moves around the sun.

Bank Incentives in a Simple Interest World

Let’s look at things from the bank’s point of view after 20 years into a 7% simple interest 25-year loan. Each dollar of the initial loan that hasn’t been paid off already has grown by 20x7%=140% to $2.40. Over the course of the twenty-first year, each dollar of initial loan still outstanding will grow from $2.40 to $2.47. This is an increase of only 2.9%.

If the bank could persuade you to pay your debt off right now, it could put the money to work in a new loan that would make 7% initially instead of the paltry 2.9%. In a simple interest rate world, banks would always prefer shorter loans if they could lend the money out again at the same simple interest rate.

In fact, with a laser-focus on profitability, banks would think in terms of compound interest. Consider a loan at 10% simple interest for 10 years. Each dollar of the initial loan that remains outstanding would become a debt of $1.10 after year 1, $1.20 after year 2, and so on until it is $2.00 after year 10. Viewed in compound interest terms, the interest rate for each year is as follows:

Year 1: 0.10/1.00 = 10%
Year 2: 0.10/1.10 = 9.1%
Year 3: 0.10/1.20 = 8.3%
...
Year 10: 0.10/1.90 = 5.3%

To judge the profitability of their loans, banks would need to think in compound interest terms like this. Once we see the declining compound rates, it becomes quite obvious that banks in a simple interest world would prefer shorter loans to longer loans.

Compound Interest Makes Comparison Easier

In the real world where banks offer loans with compound interest, consumers can make a simple comparison between loans by looking at the interest rates charged. Other factors such as prepayment terms can make a difference as well, but as long as these other factors are not materially different between two loans, the loan with the lower compound interest rate is usually better. The various factors that go into comparing simple interest loans don’t matter much with compound interest loans.

One thing that can be tricky is determining the actual compound interest rate you’re being charged. It’s important to work this out before comparing loans. That’s the subject of the next section.

Advertised Loan Rates Versus Actual Compound Interest Rates

If your credit card interest rate is 20%, you’d think this means that each $1000 of the balance not paid off through a year would grow to $1200, but this isn’t true. It actually grows to about $1219.39. So, the compound interest rate is actually almost 22%.

When the interest rate is lower, the effect is smaller. For example, an 8% car loan with monthly payments actually has a yearly compound interest rate of 8.3%.

This happens because of a little game that is played with simple interest. You’d think that if simple interest is involved, the interest charges would be lower than they are with compound interest. But this isn’t the case the way that banks play it.

Banks introduce a compounding interval that changes the effective compound interest rate. For most personal loans in Canada the compounding interval is monthly. This means that they take the advertised yearly interest rate, pretend it is simple interest and divide it by 12 to get the monthly interest rate, and then compound it anyway. Neat trick, eh? So, 12% per year compounded monthly is really 1% per month, and this compounds out to 12.7% per year.

With most mortgages in Canada, the compounding interval is 6 months. This is called semi-annual compounding. A 4% mortgage of this type really means 2% interest every half-year. This compounds out to 1.02x1.02 – 1 = 4.04%. This isn’t too bad, but back in the early 1980s when mortgage rates reached 20%, the compounded rate was 21%.

To get to the monthly rate for a mortgage, banks take the semi-annual rate and find the monthly rate that compounds out to this semi-annual rate. Starting with the advertised rate r, the semi-annual rate is r/2, and the monthly rate is

(1 + r/2)^1/6 – 1.

So, it’s not as though the banks don’t know how to work with compound rates; they’re just allowed to stick with this peculiar approach that makes them more money than just using compound interest rates.

To make a meaningful comparison between loans with different compounding intervals and different interest rates, you need to work out the actual yearly compound rates.

A Simpler World with Only Compound Interest Rates

If everyone only advertised compound interest rates, the world would actually be a simpler place. We wouldn’t have to worry about compounding intervals at all. There would be no such thing as dividing an annual rate by 2 to get a semi-annual rate or dividing by 12 to get a monthly rate. Starting with the annual rate r, the only way to compute the monthly rate would be as follows:

(1 + r)^1/12 – 1.

And compounding the monthly rate 12 times would always take you back to the yearly rate r rather than some higher interest rate.

This would eliminate one way for banks and other lenders to advertise lower rates than the actual compound rates they charge. But there are many other tricks that some lenders use, such as adding service fees that don’t count as interest or charging large penalties for paying off a loan early.

Conclusion

Simple interest isn’t very simple. If we actually used simple interest, comparing loans would be a complex endeavour involving calculations with the interest rates, loan duration, and timing of repayments. Even the small use of simple interest in computing monthly interest rates introduces the needless complexity of compounding intervals. Compound interest is far simpler than simple interest.


¹ The formula for the monthly mortgage payment P on a t-year mortgage at yearly simple interest rate r and mortgage principal M is  P = M/[∑_n=1,...,12t (1/(1+nr/12))].