Interest Rates and Compounding

Professor Matthew Spiegel
March 20, 2005

Interest rates appear everywhere.  You see them quoted in the newspaper, on line, and posted within nearly every financial institution you deal with.  One thing most of these “interest rates” have in common is that they are not truly interest rates.  This in turn has lead to mass confusion regarding the “magic of compounding” and the difference between monthly and annual rates.  Well, in point of fact there is no “magic” and compounding is very easy once you know what an interest rate truly is.  This note discusses what an interest rate is, how compounding works, and what many of those numbers you see quoted really mean.  At the end the reward will be a better understanding of how mortgage, and credit card interest rates are calculated.

Interest Rates

An interest rate represents the rate at which an investment grows.  For example, imagine that you deposit $100 into a bank account that pays 5% interest per year.  Then at the end of one year you will have $105 (which is $100×1.05).  This is how a number that is truly an interest rate works.  Unfortunately, to go any further examples with numbers have to yield to some simple algebra.  If you find algebra intimidating consider this: a little algebra will make you a much more sophisticated financial consumer and that can only help make you a richer at day’s end!

Throughout this note an interest rate is represented by the letter r.  In the example given above, r equals .05.  The money (cash) you have in period t is represented by C_t.  Period zero always stands for “today.”   Thus, in the example C₀ equals 100 and C_1 year equals 105.  With these symbols in hand a number r is an annual interest rate if (and only if) it satisfies the relationship:

C₀(1+r) = C_1 year.

While this seems pretty simple it is easy to get fooled especially when more than one time period is involved.

Not So Magical Compounding

If you learned about “compounding” in high school or college then you probably recall some confusing formula and the teacher saying something like you earn more if the bank “compounds” your investment more frequently.  If so, then your education provides yet further evidence that those who do not understand finance should be banned from teaching it.  In fact, compounding is very simple if you know arcane facts like how many months there are in a year.  Did you say 12?  Well that is right!

Once again an example will help point the way.  Here is a very difficult question: will you have more, less, or the same amount of money if you invest for one year or for twelve months?  Did you say the same?  Well right again!  (You are doing really well on this test.)  Investing your money for twelve months is exactly the same as investing for one year.  How we measure time (months or years) cannot possibly matter.  All that counts is the total length of time.  Unfortunately, this necessitates adding a superscript to the interest rate symbol (r^{time period}) that will allow us to specify the period of time it covers.  Going back to the definition of an interest rate if the monthly rate equals  then at the end of one month you must have

To make this concrete suppose the monthly interest rate equals 1% (in your dreams!).  Then if you start with $100 at the end of one month you will have $100×1.01 = $101. 

Now consider what happens if after one month you reinvest your money (now at $101) for a second month.  In that case you will have

at the end of two months.  In our example, the $101 investment becomes $101×1.01 = $102.01 in the second month.  If you use the first equation to replace the C_1 month term in the second equation you then get:

and this can be written more compactly as:

What if after two months you then invest your funds for yet another month?  In that case you will have

Notice how this works.  For every additional month you invest you simply multiply your initial purchase by .  So if you invested your money for twelve months you would have:

Going back to the example, this means the initial $100 investment turns into $100×1.01¹²  = $113.81 after twelve months.  Now suppose you invested your money at the yearly rate of r^yearly for one year instead.  Since r^yearly is an interest rate then by the definition of an interest rate it must satisfy

But, earlier on you pointed out that investing for twelve months is exactly the same as investing for one year.  That must mean that C_12 months = C_1 year.  If that is true then the above two equations imply that

or after canceling out the C₀ terms,

This is the “magic of compounding.”  There is nothing more or less to it.  You can read the above equation as saying, “If you invest at the monthly rate for twelve months you will have just as much money as if you invest for one year at the yearly rate.”  If you can read a calendar you can handle compounding. 

Returning to our example, the above equation says 1.01¹² = 1+r^yearly or after a little bit of algebra r^yearly = .1381.  In percentage terms this implies r^yearly = 13.81%.  Notice that the annual interest rate is greater than 12 times the monthly interest rate.  If there is any “magic” to compounding this is it.  But it is not really very magical.  This is just a mathematical relationship that ensures that what matters is the length of time you invest and not whether you are counting in months or years.

Here are some other examples. 

• If you invest your money for seven days at the daily rate you must have just as much money as if you invest for one week at the weekly rate,

• If you invest your money for 365 days at the daily rate you must have just as much money as if you invest for one year at the yearly rate,

• If you invest your money for 24 months at the monthly rate you must have just as much money as if you invest for two years at the yearly rate,

So far all of our examples have involved going from short periods to long periods (e.g. weeks to years).  But, you can go the other way as well.  A day is one seventh of a week.  Thus, it must be that

Similarly, a month is one twelfth of a year which means

By now you are (hopefully) a compounding expert!  Able to go from short time periods to long ones and back in a single equation!

Quotes versus Interest Rates

If the compounding equations developed above do not resemble the ones you learned in high school or college that is because the formula you were given has nothing to do with compounding.  Instead it has something to do with certain quotes (often misleadingly called interest rates) that you sometimes see at a bank or other financial institution.  To understand how the “compounding” formula you may have learned came about you have to begin by understanding how one particular bank quote works.  Imagine you are offered a mortgage at an annual rate of “6%.” What does that mean?  If you borrow $100,000 then at the end of one month the bank will say that you owe

So here is the question, is the quoted 6% number an annual interest rate?  No.  If you were to invest $100,000 for twelve months at a rate of .06/12 per month then you would have

which comes to an annual interest rate of 6.167%, which is somewhat greater than the 6% you were quoted.  That seems reasonable.  You take the “annual rate” and divide it by twelve to get the monthly rate.  Well, it is true that .06/12 or .5% is the monthly rate since it fits our definition.  At the end of one month you owe .5% more than you borrowed.  But that means the annual rate is 6.167% no matter what the bank sign may say.  Otherwise investing for twelve months would leave you with more money than investing for one year and that is simply not possible.

At this point you may still believe the bank’s rule given above makes sense.  If so consider another rule you see on occasion.  There are some bank lines of credit which state that if the annual rate is 6% your balance at the end of one day equals   Notice that the “annual rate” is divided by 360 to get the daily rate and not 365.  Now I am not all that good at counting, but the last time I checked years had either 365 or 366 days in them.  I just do not recall ever seeing a year with only 360 days.  Maybe it is just me but I am pretty sure there are none.  So where did the 360 come from?  It is just a rule.  I guess you should be grateful the bank did not divide by only 6!  (That would produce a daily rate of 1% or an annual rate of 3678%!  Wow!)  There is in fact no particularly good reason for the bank’s formula.  It is just a rule and if you do not read the contract you will not know what the rule is.  That is the bottom line.  If you cannot be bothered to read contracts so that you know what rules govern your daily balances and if you cannot be bothered to understand how to go from daily or monthly rates to yearly rates, then you are not going to know what interest rate you are really paying or getting.

At this point I am sure some readers may wish to point out that financial contracts often contain a box with the phrase “annual percentage rate” or APR.  Typically, this box does contain the annual interest rate.  In the case of the bank which divides by 360 the listed APR would be 6.27% which is the annual interest rate.  However, while the APR does occasionally equal the annual interest rate it often does not.  Mortgage contracts, for example, contain an APR that has nothing to do with the annual interest rate.  Instead it is a number that takes the annual interest rate and tries to adjust it for any “points” you may pay.  Another note will go into some of the details, but the APR on a mortgage contract is nearly useless.  If you want to know what the annual interest rate really is just take the monthly rate (which you get by dividing the quoted annual number by 12) and then use our formula for going from monthly to annual rates to convert it to something you can easily understand.

Credit Card Quotes

Credit cards deserve their own section because their contracts make it clear just how important it is to know the difference between a quote and an interest rate.  They are also another example of a contract in which the APR is not an interest rate.  Most credit card agreements state that if you pay off your balance in full each month you will not be charged any interest.  However, if you do not pay off your balance in full then you will owe interest at a rate of X% per month on the entire balance.  The last time I checked, the one exception was the Discover Card.  At the time, their contract required you to pay off your balance in full for two months in a row in order to avoid interest charges.  Unless they have changed their policy be forewarned. 

If your credit card charges 1% a month on unpaid balances it will then list an APR of 12.68% which they calculate by taking 1.01¹² and then subtracting one.  Fair enough, that looks like the right way to convert a monthly rate to an annual one.  But, it ignores the way the contract works which a simple example will make clear.  Suppose that on January 1 you charge $100.  On February 1, you pay off the balance in full ($100) and charge $200.  On March 1 the bank will tell you that your balance is $200.  Since you paid off your January balance in full no interest is due.  Now suppose that on February 1 instead of paying off the entire balance you send in $75.  This means that starting February 1 you have an outstanding balance from the previous month and the bank will begin charging interest.  Since you charged $200 your total balance during the month of February comes to $200 from the new charges plus the $25 unpaid balance for a total of $225.  Thus, on March 1 the bank now tells you that you owe $225×1.01 = $227.25.  As a result, the $25 you borrowed in February cost you $2.25 in interest (the difference between $227.25 and $225).  Going back to the definition of an interest rate the monthly rate must solve $25(1+r^monthly) = $27.25 which implies r^monthly = 9%!  This produces an annual interest rate of (hold on to your chair) 181%!  What happened to the APR of 12.68%?  Had you paid off the January balance in full the $200 you charged would not have incurred any interest.  However, because you did not the $25 you borrowed cost you interest on a balance of $225.  In effect, this multiplied your interest rate by 9 (since $225 is nine times $25) which is how you wound up paying 9% per month rather than the 1% you probably thought you would pay.

As the above example illustrates credit card agreements could honestly state, “the bank guarantees that it will never charge you less than the APR on your unpaid balances.”  Basically, unless you pay off your balance in full each month, the more you pay off the higher the interest rate you are being charged on the unpaid portion.  Do yourself a really big favour and pay those balances in full and avoid all interest charges!  If you cannot do that at the moment then pay off your credit cards as quickly as possible and then only charge as much as you can afford to pay off each month.  Later notes will discuss other ways to borrow funds at terms that are typically a lot more favourable than those associated with credit cards.

©Matthew Spiegel