# What is an "Effective interest" ?

By | 13/06/2011

Many people do not know what is an effective interest, and how to calculate it.

Actually, it is a very simple concept, and it is very easy to calculate.

First, you need to understand that interests are expressed in annual terms. When the bank says that it will loan you a sum of money at an interest rate of 4%, this is the annual interest rate.

Now, interest is a logarithmic function. Hey, no need to close the window!!! I did not say a dirty word!

In a laymen language – you cannot just add or subtract interest rates, because interest rates are expressions of a product, not a sum. For instance, if the bank lends you 5,000,000$for one year, at an interest rate of 4%, the interest will be: $5,000,000\\cdot4\%=5,000,000\cdot0.04=200,000\$ In order to calculate the interest we multiplied the principal by the factor 0.04 which represents $4\%=\frac{4}{100}$. In all the following examples I will use a loan for demonstrations, but you can do the same calculations on deposits as well, in this case, you earn the interest rather than paying it. Now the bank tells you that the interest rate is 4% per annum, same as before, but now it states that the interest is calculated on a quarterly basis. This means, that for every quarter you will pay 1%. Remember, you cannot add interest rates ! this means, that paying 1% per quarter is not the same is paying 4% per annum. The reason is that after the first quarter, you will owe the bank 5 million dollars plus interest of 1%, or 50 thousand dollars. For the next quarter, you will also pay 1%, but on an initial principal of 5 million dollars and 50 thousand dollars. So after the second quarter you will owe the bank: $5,050,000\\cdot101\%=5,050,000\cdot1.01=5,100,500\$ There's another 500$ there ! You'd expect that after half a year you will owe the bank half the interest, 2%, or 100,000$, over the 5 million principal, but because you cannot sum interest rates, you would owe 100,500$ and not 100,000$. This residual element, in our case 500$, is called the logarithmic error, it is the "error" added when adding interest rates using simple addition.

In the year-end, you will need to pay the bank: $5,000,000\\cdot101\%\cdot101\%\cdot101\%\cdot101\%=5,000,000\cdot1.01^4=5,203,020.05\$

The additional 3020.05$we incurred over the basic calculation of 4% per annum on 5 million dollars are the logarithmic error. This logarithmic error will increase when: • Dividing the year to smaller periods of time. • Interest rate is higher Pay attention that the increase in the logarithmic error is also not linear. If the logarithmic error on a 5M$ loan at 4% was 3020.05\$, on 8% it will not be twice as much.

Now for the punch:

The effective interest is the interest, expressed in annual terms, that will take this logarithmic error into account.

To calculate this interest we divide the interest the bank say by the periods of time, and then we multiply the percentages.

When the bank tells you that you will pay 4% annually calculated on per quarter basis, it actually tells you that you will pay: $I_{effective}+1=101\%\cdot101\%\cdot101\%\cdot101\%=1.01^4=1.04060401= 104.060401\%$

or 4.060401% per annum, effectively.

Now what will happen if the bank will calculate the interest on per month basis: $I_{effective}+1=(100+\frac{4}{12})\%\cdot(100+\frac{4}{12})\%\cdot(100+\frac{4}{12})\%\cdot(100+\frac{4}{12})\%+...=(1+\frac{4}{12})^{12}=1.00333334^{12}=1.0407415429=104.07415429\%$

or 4.074% effective per annum, higher than on per quarterly basis.

Lets take annual interest of 12% calculated monthly. Try to guess how much will the effective interest be: $I_{effective}+1=(100+\frac{12}{12})^{12}\%=1.01^{12}=1.1268250301=112.68250301\%$

12.68%! the bank "forgot" to mention 0.68% !

So the basic formula for calculating effective interest, or the real interest you will pay for a loan: $I_{effective}=(1+\frac{I_{per-annum}}{n})^n$

where "n" is the number of periods in a year (for quarters, n=4, for months, n=12, for days, n=365).

That's it. You can stop here. You know all there is about effective interest. You got what you want… now before you leave:

The more the bank will divide the year, the higher the interest rate will be, but not to infinity.

For the math geeks (like me), lets see what happens if the bank decides to divide the year to seconds, or even milliseconds, or even nanoseconds, or…. ok you got the idea. $\lim_{n\to\infty}I_{effective}=\lim_{n\to\infty}(1+\frac{I_{per-annum}}{n})^n=\lim_{n\to\infty}((1+\frac{I_{per-annum}}{n})^\frac{n}{I_{per-annum}})^{I_{per-annum}}=e^{I_{per-annum}}$

The limit of the effective interest is $e$, the natural logarithm (2.71828…), by the power of the annual interest stated by the bank.

See? Wasn't that hard!