## Tuesday, January 8, 2013

### 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))]. #### 17 comments: 1. Careful there Mr. Numbers, you are confusing we simple folk with Math Degrees! You'll have to show me how to get math formulas up on your site, I never succeed when I try. 2. @Big Cajun Man: I've seen you do math. If you didn't follow this, either I explained it poorly or you're not trying :-) I did the equations with italics, subscripting and superscripting -- nothing fancy. The only exception is at the end with the sum symbol (HTML &#8721). 3. Hmmm, your examples of rolling over short duration simple interest rate loans are really just compounding, aren't they? Interesting you didn't mention continuous compounding, I guess that might get even more mathy for a general blog. But if we standardized on rates quoted with continuous compounding there could be no messing with compounding intervals and all rates would be directly comparable. By the way, there are some nice tools out there for rendering LaTeX math into graphics files, HTML, etc., like http://www.codecogs.com/latex/eqneditor.php. You could nicely render your last sum with P=\frac{M}{\sum_{n=1}^{12t}\frac{1}{1+\frac{nr}{12}}} I can't embed HTML in a comment to show you the result, but here is an example URL generated by this application: http://latex.codecogs.com/gif.latex?P=\frac{M}{\sum_{n=1}^{12t}\frac{1}{1&plus;\frac{nr}{12}}} 4. For a mortgage, isn't the compounding applicable only if you allow the interest to compound? For someone who pays interest plus principal every month it doesn't matter if the interest is simple or compounding because there would be no interest balance carried over. If I take our mortgage rate of 2.99% and put it through your formula the monthly rate x 12 works out to 2.97%. If that is what gets applied in our payments it looks like a small decrease (not counting the time value of the earlier payments compared to the later ones). If that's the typical result I don't see this practice being banned any time soon. 5. @Greg: Yes, when we roll-over simple interest loans, we are effectively compounding at some interval. It is the playing around with compouding intervals that makes it hard to compare interest rates. If we stop talking about simple interest, we don't have to talk about compounding intervls any more. You're right that if we advertised continuous compounding rates, everything would be comparable. However, this would fail on the test of clarity for the average person. A$100 debt with 10% per year interest would grow to more than $110 after a year, which is counter to what the average person would expect. I think it's better to always advertise compounded rates. Thanks for the pointer on turning LaTeX into html. I use LaTeX all the time. For the most part I try to avoid using too much math in my posts, but I may try using LaTeX the next time it comes up. @Simply Rich Life: No, compounding is still applicable even if you pay the interest each month. A better way to think about this is to imagine your debt puffing up slowly over time. Each payment takes away a portion of the debt that corresponds to a smaller payment against the original debt size. The portion that you don't pay off then continues to puff up. The compounding issue gets to the heart of how fast the debt is puffing up. The true puffing up rate is determined by the compounded interest rate rather than the simple interest rate. It's true that the compounding effect is small on small interest rates. However, it grows quickly as the interest rate rises. Department store credit cards used to charge 2.4% per month, which is often called 28.8% per year. However, the true compounded interest rate is 32.9%. I certainly don't expect a ban any time soon. Banks wouldn't want it for 2 reasons. The first is that it would prevent the added profitability they get from the current system. The second is the cost of changing software and advertising practices. 1. If you carry a balance of$10,000 on a department store card, it incurs 2.4% interest every month, and at the end of the month you pay exactly $240 to keep the balance level, do you not pay$2,880 over the full year?

That's an unlikely example for a credit card but it would seem that all mortgages except the ones in negative amortization would be handled similarly.

If on the other hand lenders are managing to charge you interest on something you have yet to borrow or have already paid off they are pulling quite the trick :)

2. @Simply Rich Life: The $240 payments are made at different times. It does not make sense to simply add them because$240 right now is worth more than $240 a year from now. From the point of view of the end of the year, compounding the payments at 2.4% per month, the future value of the payments is$3292.28.

For more sensible interest rates we get similar results; the compounded interest rate is the one you are really paying.

I decided to take a look at how this simple/compound interest issue affects a mortgage at 2.99% as you said yours is. Assuming that is uses semi-annual compounding, the true compounded rate is 3.10235%. Assuming a $250,000 principal and 25-year amortization, the monthly payments are$1181.83. However, if the true compounded rate were 2.99%, then the payments would be $2.82 per month less. If we take the present value of the actual payments at a true compounded rate of 2.99%, the result is$250,598. So, the difference is about $600. This isn't much over 25 years, but it isn't nothing. If we repeat this exercise with a mortgage rate of 10%, the difference in present value goes up to$4551. So, the higher the interest rate is, the bigger the difference this simple/compound interest issue makes.

3. Right, the time value of money will affect the result. As long as your opportunity cost for capital doesn't run at 30% and the interest payments aren't excessively large the effect is minimal but it is there.

At least in this case the lender can truthfully claim that the amount of interest we pay over a year is 2.99% of the balance (I checked the payments and it's actually 2.97%, matching the results of your monthly payment formula). It's a little better than loans where the interest is compounding and the amount paid is higher than the stated interest rate. Even those would have an extra cost since they probably require monthly payments too.

4. @Simply Rich Life: I've never heard of mortgage advertising using compounded rates. I would have expected that your rate uses semi-annual compounding, and that your compounded rate is (1+0.0299/2)^2-1 = 3.01235%. If we then take this rate and compute the monthly rate, we get 0.24763%. Using "simple" interest and multiplying this by 12 gives 2.9715%. We could have done this directly as 12*((1+0.0299/2)^(1/6)-1) = 2.9715%. If this is how your calculations went, then you are in fact paying a yearly rate of about 3.01%. If this isn't the case, then I'm interested in hearing where you got your mortgage so that I can check out their advertising.

5. As far as I know it uses semi-annual compounding (the results matching your formula seem to confirm that). The compounded rate would be 3.01% then so it's slightly misleading.

When it comes to mortgages I don't expect that has much of an effect on people which is likely why there isn't much objection to the practice. Leaving aside the small size of the dollar amount in our situation, rates can still be compared directly to find the lowest one since the payment structures of mortgages don't vary much. It's also easy to bypass the whole question by getting projections of the end result such as the remaining amortization if you make an extra payment, which has a much bigger impact on most peoples' financial decisions.

With higher interest rates and more variation it could be a bigger issue. In that case maybe the lenders that use calculations closer to true compound interest would lobby for more accurate representation form the others.

6. @Simply Rich Life: There are some variable-rate mortgages in Canada that use monthly compounding. So, there can be some confusion when comparing rates, particularly when interest rates are higher. I agree that there is unlikely to be any widespread objection to advertising lower than the true interest rate, mainly because so few people understand the issue. You're right that it is possible to bypass the issue by getting projections, but most people have already made their decision before they look at a projection table.

7. Yeah, you're mixing two different types of interest. The 7% example is simple interest, the 5% example you're calling simple interest is actually compound interest.

If I recall my exams, it's 5% annual interest, compounded every 5 years, and a variation of the compound interest formula you gave will calculate it.

8. In terms of the mortgage rates, I believe that simply rich is correct - the advertised rate is the 'effective' rate taking into account the semi-annual compounding. i.e. they would advertise the 3.01 not the 2.99 rate.

6. @Life Insurance Canada: The point of the article is to explain that compound interest is always involved. So, yes, I'm mixing two kinds of interest. I'm illustrating the confusion that ensues when trying to compare different compounding intervals.

You're wrong about advertised mortgage rates. The actual compounded rate is never advertised (or at least I've never seen examples of it being advertised). What is usually advertised is double the half-year rate, which is slightly less than the compounded full-year rate.

1. @Life Insurance Canada: Upon reflection, I think what you call the effective rate may be different from what I mean by the compound rate. For Simply Rich's example, there were 3 "rates": 2.97%, 2.99%, and 3.01%. In all cases the monthly rate is 0.2746%. If we multiply this by 12, we get the yearly rate of 2.97% as though this were a car loan that is compouned monthly. If we compound it for 6 months and double it, we get 2.99%, which is the semi-annual rate that gets advertised for mortgages. This may be what you called the "effective rate". If we compound out the monthly rate for 12 months we get 3.01%. This is the most logical way to think about a yearly interest rate, but this compoinded rate is not what is advertised.

2. The effective rate of interest is a specific term with a specific calculation. It's the amount of money earned over the course of a year. If you have $100 mortgage and don't make any payments and owe 7% at the end of the year, the effective rate of interest is 7% no matter how it's compounded. I could be wrong on the advertised mortgage rates, but I'm surprised. I assumed there were regulations surrounding this,and that everyone automatically advertised effective rates of return, basically removing the semi-annual compounding and ending up with the intuitive definition of interest for a year. In fact years ago when I built a mortgage calculator for my website (which matches to the penny the calcs from the banks) I'm pretty sure that's what I did. If you're matching bank calcs from a spreadsheet, you will end up with differences. That's because in addition to the semi-annual compounding, Canadian mortgage calcs have a rounding in there somewhere - without the rounding your numbers will likely be off. I'd have to go back and check to confirm, but I'd rather not :). I learned interest theory from basic principles so I have a bad habit of starting with a sum of a series rather than a formula when doing these calcs. I suppose a call to a bank would tell you if they're advertising the effective rates of interest, but I figure the problem is going to be finding someone at a bank that knows what the heck you'd be talking about with a questions like that. 7. @Life Insurance Canada: Thanks for the clear explanation of what you mean. What you call the "effective rate" is the same as what I call the "compound rate" and others call CAGR and other names. It's clear to me that you understand the issues (although we may be using different terms). I think you'll be surprised to see how rates are advertised after you investigate. I agree completely that if a$100 debt grows to \$107 after a year with no payments, then the yearly interest rate should be 7%, no matter how it is compounded.

However, the advertised mortgage rate for this example would be 6.88%. This is because banks take the half year rate (3.44% per half year compounds to 7% per year), and they double it to get the rate they advertise. Most people I explain this to are very surprised to learn that the advertised rate is less than the interest that builds up over a year without any payments.

I have also built formulas that exactly match (to the penny) all the mortgage payments I've ever had. I haven't ever noticed the rounding errors you're talking about. My calculations always exactly matched the bank calculations once I understood how the bank calculations work.

I also derive amortization formulas from basic principles (summing a series) rather than memorizing them. I actually went through the painful process of talking to bankers about how interest is calculated many years ago. This was back in the days when a branch manager actually had some authority. I was put in contact with a bank employee who actually knew how the math worked. Since then I have just confirmed that my math matches actual payments.