A lot of math actually originates in dealing with money and games. What I call money math here is nothing beyond what most people learn in school and may have forgotten or just aren’t comfortable doing anymore. But if, like me, you’re aiming to make large amounts of money, you will sooner or later need to learn and understand the basic techniques described here - especially in order to avoid mistakes and being ripped off by those who do know them.
Probably you already know all of those techniques. But knowing is not enough, you need to feel at home with them and be willing to check up on the numbers you encounter in banking and finance. Whether you’re comparing loans or bonds or just checking if your bank is calculating the interest of your loan correctly, some math will definitely be needed. Even though most people understand percentage and interest, many are prone to make mistakes and not notice or not notice other people’s mistakes.
Banks profit on customers who don’t notice mistakes!
Banks often reckon with this and deliberately include small errors (that benefit them) in their calculations for their customers. This is most common with interest payments for credits or loans because people often don’t check or can’t because they don’t know the math (which is what the banks are counting on). Sure it’s dishonest and outright despicable but it makes sense. You see, when you do catch them, all you can do is complain and they’ll correct it. Out of the thousands or millions of little mistakes they make, only a small number will be found out and need to be corrected and the rest is free money for them. Basically it’s the same as stealing very small amounts so that most people don’t notice. Even if they’re caught, nobody can prove that their “mistakes” were deliberate. I dare not think how many loans have been artificially made more expensive this way over the years. Well the only way to protect yourself from this is to look over their shoulder and double check everything that involves numbers - particularly when working out loan interest and things like that.
If you’re fine with money math and don’t need any brushing up then the following problem should be really easy and obvious to you:
If you have $ 100 and lose 10 % in a trade, then you make another trade and gain 10%, how much do you have?
Pick an answer:
If you chose $ 100 for your answer, then you need to brush up on your math skills because the correct answer is in fact $ 99. Read on to find out why it’s not $ 100. Don’t worry, it’s an easy mistake to make even if you’re good at percentages.
Here I will discuss some basic things you need, where they’re needed and what the common pitfalls are. I won’t show you the actual methods and operations yet because you almost certainly know them already so I put them in the appendices for those who want to do some hands-on number crunching. Oh, and I’m assuming you know how to add, subtract, multiply and divide - also with fractions.
Percentage is absolutely vital anything to finances. So much in finance is worked out in percentage: stock prices, interest rates for loans, bonds, credits and inflation. You absolutely cannot get by without a thorough understanding of percentage. Why is percentage so often used? Because it’s far easier to understand changes in prices by percentage than it is to compare them directly.
If you have $ 1000 invested in shares from a company and the share price goes up by $ 1 then that tells you nothing without also knowing the share price. If the share price was $1 before then that means it has doubled and you have $ 2000 but if the share price was $ 100 then you only have $ 1010.
But giving the price change in percentage tells you all you need to know because if it went up by 5% then so did your capital - no matter what the share price was.
Supposing you need a loan. So essentially you’re renting money and you pay a fee for it. Obviously, the more you rent, the higher the fee. But if you don’t use percentage (say 10 %), then the person giving you the loan would have to quote a different fee for every different sum a customer asks for and on top of this, the fee for the loan would change every year. With percentage it’s just one number and that way you can compare different loans far more easily. If one loan costs 8 % and another loan costs 8.5 % then you can instantly tell which loan to go for.
The question I gave you above shows how easy it is to make mistakes with percentage calculations. Because percentage isn’t just a number but a relationship between several numbers, people often make the mistake of treating percentage just as any other number. So when confronted with the situation of losing 10 % and then gaining 10 %, we might forget that, after we lost 10 % we have a new number and 10 % of that new number is different to 10 % of the first number we had. So using our problem above here is what happens exactly:
Now what if we reversed the order - meaning we first gained 10 % and then lost 10 %?
Lets take a look:
So in both cases we end up with less than what we started with. This goes to show that percentage tends to have much more impact when it’s working against you than when it’s working with you. Especially stock market traders have to keep this in mind because it means that it usually takes more winning trades to recover from a loss than it took to make the loss in the first place.
Interest is essentially a kind of rent for money. If you borrow money then you pay interest (such as with a loan) and if you lend money you get paid interest (for example a savings account). The interest rate is given as a percentage of the amount borrowed and gets paid out per year (though this can vary too).
So why don’t we just call it a fee instead of giving it a special term of it’s own?
Well interest has the strange feature that it accumulates. We call this compound interest. It’s basically down to another one of those pitfalls with percentage calculations I mentioned above. You see, because interest is a percentage of the capital owed, when it gets paid out, the capital changes. Let’s look at an example to clarify this:
Supposing you have a savings account with an interest rate of 2 % and you leave $ 100 on it for 2 years. Here is what happens:
This is the effect of compound interest working. Because after the first year you also collect interest on the interest from the first year. So the interest payment is greater every year and the growth of your capital isn’t in a straight line but in a curve which means each year it grows faster than the previous year.
The same works the other way when you take out a loan and pay interest. However, usually people pay off the interest on their loans and credits every year (or month) so that it doesn’t accumulate. Better always check the terms of your loan or you might be surprised.
Because of this nonlinear nature of interest, people often don’t realize the impact it can have in the long run. It’s also a favorite place for banks to make their “mistakes”. So it’s very important to be able to work it out yourself and check up on what they’re doing. See appendix B for how to do this.
There is a favorite phrase among fortune builders like us:
Let interest work for you and not against you
What this means is that you should not be borrowing money or having any dept because it costs you a lot of interest. Instead you should be collecting interest because it’s a sure way to increase your capital.
There are many different ways you could round numbers but in finance the widely accepted standard is to round up when the digit is 5 or above and round down when it’s below 5. This means rounding $ 2.333 would yield $ 2.33 and rounding $16.278 would yield $ 16.28.
Usually, in most fields we also say how much we round a number but with finance it’s nearly always to the nearest cent unless stated otherwise.
There is one important thing to note that many people aren’t aware of: always only round your final result - never do your calculations with rounded numbers. Because rounding is an inaccuracy and if you keep using that inaccuracy in calculations with each partial result, that inaccuracy can easily get a lot bigger than it would if you’d only rounded the final result:
We’re adding three numbers together and multiplying the result with another number:
3.249 + 6.148 + 5.338 = 14.735
Now multiplying that with 2.149 gives us 31.665515
Rounding brings 31.67
Now we will perform the same calculation but we will round the numbers to 2 decimal places first:
3.25 + 6.15 + 5.34 = 14.74
Now multiplying with the rounded number 2.15 gives us 31.691
Rounding brings 31.69
One can easily see that rounding your working numbers yields far more inaccurate results than only rounding your final result. The more numbers you use and the more complex the calculation, the more inaccurate the result will be.
If you lose 50 % of your capital, you need to gain 100 % to get all your money back.
At a growth of 10 % per year, your capital will double in 7 years, 3 months and a few days
In June 2011, the inflation of the US dollar was 3.6%. At that rate your money will lose half its value every 20 years! This means that, if you retire in 20 years, you will need to have twice as much coming in every month in order to maintain the same living standard. So make sure you keep inflation in mind when looking at retirement plans.
Most savings accounts give you an interest rate that is BELOW the inflation so keeping your money in such an account means it’s losing value every year. As of now (2011) any savings account in US dollars with less than 3.6% interest is a sure loss.
Calculating percentage is really just a certain combination of multiplication and division.
You typically encounter three kinds of questions in percentage calculation:
We’ll start with the first kind. Here are a few examples followed by a general rule:
How much is 10% from 300?
You start by dividing 300 by 100 and then multiplying the result by 10:
So we divide 300 by 100 and get 3.
Then we multiply 3 by 10 which gives us 30
How much is 3% from 66?
Dividing 66 by 100 gives us 0.66 and multiplying this by 3 gives us 1.98
How much is x % from a number y ?
Divide y by 100 and multiply the result by x.
Actually you don’t even have to do it in that order but it’s better to stick to one method you’re comfortable with.
Now the second kind of percentage problem:
How much percent is 8 from 32?
Divide 8 by 32 and multiply the result by 100.
So 8 divided by 32 is 0.25 and multiplying that by 100 gives us 25.
There fore 8 is 25 % of 32.
How much percent is 5 from 200?
5 divided by 200 is 0.025 and multiplying that by 100 gives us 2.5 %
How much percent is q from r ?
Divide q by r and multiply the result by 100
The third kind of percentage calculation
12 is 5 % of what number?
Divide 12 by 5 and multiply the result by 100.
12 divided by 5 is 2.4 and multiplying that by 100 gives us 240.
So 12 is 5 % of 240.
6 is 20 % of what number?
Divide 6 by 20 and you get 0.3. Multiply that by 100 and we get 30
s is t % of what number?
Divide s by t and multiply the result by 100.
Calculating compound interest is probably the most complex calculation you’ll encounter in finance maths. Luckily, there are online interest calculators in abundance that do that for you. Still it’s better to understand the process yourself because it gives you a clearer picture of what’s actually happening to your money.
Our starting capital is $ 100 and we have a savings account with 4 % interest paid out annually. Let’s work out what happens to our capital after each year:
After the 1st year we get 4 % interest on $ 100. 4 % from 100 is 100 divided by 100 and multiplied by 4 which gives us $ 4. And this we add to our capital which gives us $ 104.
In short we write it like this (the * stands for multiplication instead of using x):
Capital after the 1st year is $ 100 + (100/100)*4 = $ 104.
After the 2nd year we get 4 % interest on $ 104, so we have $ 104 + (104/100)*4 =
After the 3rd year we get 4 % interest on $ 108.16, so we have $ 108.16 + (108.16/100)*4 = $ 112.4864 ( remember not to round until we’re done with all calculations)
As you can see, for each year, we take the capital from the previous year and add the new interest payment to it which we work out using the formula
(capital from previous year / 100 ) * 4.
Now you understand how it works, but with this method, we can only work out what we get from one year to the next. If we want to be able to calculate how much we have after 10 years, or 20 years or n years, then this would be a long and tedious procedure inviting mistakes.
Fortunately there is a formula with which you can work out your capital after n years directly, where n stands for any number of years you choose. Here it is:
Then our formula for working out compound interest is:
Of course, even in finances, things can become more complicated, for example if you have interest payments every quarter rather than every year. Also, ideally we should prove that this formula works for any number of years but since this is about finances as needed by private people, what we’ve covered here should go a long way.