Are you interested in the interest of Dr. Mathematics and the importance of the inside-of-the-brackets decisions?

You will be in shock to notice that things are always made to injure the common people to best that they can in democracy and capitalism (also in the first world nations!). To the least, they are made to confound us all, and therefore leave us with no chance of arguing.

Basically, there is the standard formula, through which common people calculate interest, for instance in loans and applications, and the mathematical theories that should be behind it.

We are talking about the compound interest, which differs from the simple interest because it is calculated over the current value, rather than over the initial value.

With the simple interest, if we invest, say, AU$ 1,000.00 today, at the rate of, say, 5% per annum, and the money remains applied for 12 months, then we get to withdraw, supposing that we take all out, AU$ 1,050.00 at the end of the 12 month period.  

To calculate the amount of the withdrawal, we could have done 1000 x 0.05 + 1000 = 1000 (1+0.05), that is, A = P (1 + r n). 

Most of us, however, would have reasoned that 10% of 1000 would be going back from the end of the 1000 one spot (10/100 = 1/10), so that 10% would be 100. Since 5% is half of 10%, we get half of 100 or simply 50.




If it were compound interest, in this particular case, we would have gotten the same amount of money at the end of the period, and we would have calculated the amount of money to withdraw through the following formula: A = P (1 + r/n)^nt

(http://qrc.depaul.edu/StudyGuide2009/Notes/Savings%20Accounts/Compound%20Interest.htm, 29/06/2013).

In the just-mentioned formula, we would have made P be equal to AU$ 1,000.00, since that is the money that we have invested, our principal, r = 0.05, since that is the percentage per annum, our interest rate, n = 1, since the frequency of addition of interest per year is one (once a year we have the interest calculated and paid to our account), and t = 1, since the money would have rested in the application for one year. See: A = 1000 (1+0.05/1)^1x1= 1000 (1.05) = 1,050.

Now suppose that we duplicate the period of time in both situations (and that is when things are going to start looking really bad for those making the rules in democracy and capitalism). 

We would then get A = 1000 (1+0.05x2) = 1100, therefore AU$ 1,100.00 with simple interest and A = 1000 (1+0.05/1)^1x2=1000x1.05²=1102.5, therefore AU$ 1,102.50, with compound interest.

So far, no problems, we all think.

Not really.

If we go for the reasoning behind this formula, that of the compound interest, we will actually notice that the formula is not accurate.

Compound interest should be a mathematical calculation made always over the current amount, so that we would do 1000 (1.05) =1,050 to get the first year total and [1000 (1.05)](1.05) =1,102.50 to get the second  year’s total.

Well, you would say, things are still all the same.

Indeed, that is why nobody has ever thought that there could be some mistake with the formula or its application.

However, when you start playing with frequency and duration, things change quite a lot.

Consider, for instance, a rate of 5% per quarter instead of 5% per year, ceteris paribus (other things being equal), and apply the formula. Then calculate it all mathematically and observe the difference in values.

In simple interest, no problems whatsoever: 1000 (1+0.05x4)=1200 (four quarters per year).

The formula we have can only be applied when the interest comes measured by the year, so that we have to first convert 5% per quarter into x% per year to then apply our formula.

Mathematically, we will look for an amount x that satisfies the following equation:

(1.05)⁴=(1+x)¹. 

Our x is (1.05)⁴ -1, therefore 0.21550625.

X would then be 21.55%, that is, 5%, quarterly, of interest is the same as 21.55% per year of interest (compound).

Now, we can use our formula:

1000 (1+0.21550625/4)^4x1≈1233.55.

Using mathematical reasoning, we would get 1000 (1.05)⁴ ≈ 1215.51.

Now our formula does not look that good, is it not?

Indeed.

The higher the frequency in the year, the worse it will look, quite sincerely.

Notice that if we simply multiply 5% by four and plug in the formula, we get: 1000 (1+0.2/4)^4x1 ≈ 1215.51.

Also, if we simplify the ratio 0.21550625/4 through making the division and rounding the result mathematically, we get 1.05 for the sum, therefore we get the same value, 1215.51, in the end.

Notice that Mathematics will tell us to commit the least amount of mistake as possible, therefore to round just the final result, as we have done when we got 1233.55 as an answer.

That means that our financial formula is an approximation to the calculation and only works as in Mathematics under special conditions.

The question that emerges is then why we would not be using the mathematical calculation, which seems to be so clear. 

Notice that we will be losing in our investment if they apply Mathematics, but we will be paying less in our loans, unless they round it all still inside of the brackets, before raising to the exponent, when it is then all the same.

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