Finance is all about numbers.
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Luckily, you don’t have to be a mathematician to find your way around the figures.
Would you like to know what your net worth means and how to calculate it?
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How about the difference between simple and compound interest – or what the heck the rule of 72 is and what it means for your money?
Here are 12 financial formulas you should know:
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Calculating your cash flow is one of the most simple formulas and most likely one of the first ones you learned in high school.
It’s down to how much you’re bringing in and how much you’re spending – if you’re carrying a negative balance at the end of this formula, you need to re-evaluate your finances.
Income – Expenses = Cash Flow
2. How to Calculate Your Net Worth
Your personal net worth, simply, is what you own minus what you owe.
Again, you’re looking for a positive balance here.
According to the Forbes Billionaires List, Bill Gates currently tops it at the time of this writing with a net worth of $75 billion.
Remember: You might not be starting there, but neither did he.
Assets – Debts = Net worth
3. How to Calculate Simple Interest
Simple interest applies when you need to find out the interest being charged on a loan.
It’s also referred to as principal interest because the interest accrued is based only on the principal amount.
I = p x r x t
I = Interest
p = principal amount
R = rate of interest charged per year (as a decimal number)
T = How long the money is borrowed or invested for (in years)
4. How to Calculate Compound Interest
Compound interest refers to calculating the compounded interest, not just the interest gained on the principal invested or borrowed amount.
A = P (1 + r/n) nt
A = The amount earned after interest
P = The principal amount
r = The annual interest rate (as a decimal)
n = The number of times the interest is compounded (per year)
t = How long the money is borrowed or invested for (in years)
5. How to Calculate Price to Earnings Ratio
Here’s a formula that’s useful if you’re watching stocks or companies – use it to calculate the relationship between a company’s share prices and per-share earnings.
This can tell you if stocks are over- or undervalued.
Price to Earnings Ratio = Price per share / Earnings per share
For example, Microsoft Corporation on the 20th of July 2016 traded at $56, 52 per share, and the EPS was listed at $0.69 according to this piece on News is Money.
6. How to Calculate the Break-even Point
Knowing when you’re breaking even is an essential part of a successful business venture.
Use this formula to calculate when you’ll finally be breaking even – or to find out if you already have:
Break-even point (in dollars) = Fixed expenses / Gross profit margin (in percentage)
7. How to Calculate Net Income?
Here’s a simple way to figure out your business’s net income:
Net income = Revenue – Expenses
8. How to Calculate the Variation of Investment
Calculating the variation of your investment tells you what’s happened to your investment over a period of time: Has it gone up, or has it tanked?
Expressed as a decimal ratio, the result tells you how much more (or less) your purchase is worth.
Purchase price variation = (Current price – purchase price) / purchase price
9. The Rule of 72
The rule of 72 is a useful trick that tells you how many years your investment will need to double in value at a specific annual return rate.
Simply, how long will it take to double what you put into it?
The same formula can also be used to figure out how long it could take you to double your debt – steer carefully!
Years needed to double your investment = 72 / compound interest rate (per year)
10. Your Basic Liquidity Ratio
Basic liquidity ratio tells you how long (in months) a family will be able to cover their expenses with the assets they have.
This is useful for personal finance when the breadwinner is no longer able to bring in money (which can sometimes happen for a varying amount of reasons).
In the case of companies, it’s used to calculate how long a company can survive off their current assets should they need to.
Here’s how:
Basic liquidity ratio = Monetary assets (in dollars) / Monthly expenses (in dollars)
The end-result is expressed as a decimal, and according to Xplained.com, it’s advised to have a basic liquidity ratio of at least three months.
Free Financial Calculators
Still too much for you?
Have a look at these financial calculators to help you out:
Takeaway
Get on the same page with your finances.
Share more financial formula’s in the comments!
by M. Bourne
Interactive exploration
See an applet where you can explore Simpson's Rule and other numerical techniques:
In the last section, Trapezoidal Rule, we used straight lines to model a curve and learned that it was an improvement over using rectangles for finding areas under curves because we had much less 'missing' from each segment.
We seek an even better approximation for the area under a curve.
In Simpson's Rule, we will use parabolas to approximate each part of the curve. This proves to be very efficient since it's generally more accurate than the other numerical methods we've seen. (See more about Parabolas.)
We divide the area into `n` equal segments of width `Delta x`. The approximate area is given by the following.
Simpson's Rule
Area `=int_a^bf(x)dx`
`~~(Deltax)/3(y_0+4y_1+2y_2+4y_3+2y_4+` `{:...+4y_(n-1)+y_n) `
where `Deltax = (b-a)/n`
Note: In Simpson's Rule, n must be EVEN.
See below how we obtain Simpson's Rule by finding the area under each parabola and adding the areas.
Memory aid
We can re-write Simpson's Rule by grouping it as follows:
`int_a^bf(x)dx` `~~(Deltax)/3[y_0+4(y_1+y_3+y_5+...)` `{:+2(y_2+y_4+y_6+...)+y_n]`
This gives us an easy way to remember Simpson's Rule:
`int_a^bf(x)dx` `~~(Deltax)/3['FIRST'+4('sum of ODDs')` `{:+2('sum of EVENs')+'LAST']`
Example using Simpson's Rule
Approximate`int_2^3(dx)/(x+1)` using Simpson's Rule with `n=4`.
We haven't seen how to integrate this using algebraic processes yet, but we can use Simpson's Rule to get a good approximation for the value.
Answer
Here is the situation.
`Δx = (b − a)/n = (3 − 2)/4 = 0.25`
`y_1= f(a + Δx) = f(2.25)` `= 1/(2.25+1) = 0.3076923`
`y_2= f(a + 2Δx) = f(2.5)` `= 1/(2.5+1) = 0.2857142`
`y_3= f(a + 3Δx) = f(2.75)` `= 1/(2.75+1) = 0.2666667`
`y_4= f(b) = f(3)` `= 1/(3+1) = 0.25`
So
Area ` = int_a^bf(x) text[d]x`
`approx 0.25/3 (0.333333+4(0.3076923)` `+2(0.2857142)+4(0.2666667)` `{:+0.25)`
`=0.2876831`
Notes
1. The actual answer to this problem is 0.287682 (to 6 decimal places) so our Simpson's Rule approximation has an error of only 0.00036%.
2. In this example, the curve is very nearly parabolic, so the 2 parabolas shown above practically merge with the curve `y=1/(x+1)`.
Don't miss...
There is an interactive applet where you can explore Simpson's Rule, here:
Background and proof for Simpson's Rule
We aim to find the area under the following general curve.
We divide it into 4 equal segments. (It must be an even number of segments for Simpson's Rule to work.)
We next construct parabolas which very nearly match the curve in each of the 4 segments. If we are given 3 points, we can pass a unique parabola through those points.
NOTE: We don't actually need to construct these parabolas when applying Simpson's Rule. This section is just to give you some background on why and how it works.
Let's start with the first 2 segments on the left. We take the end points, and the middle point as shown:
We can take measurements (using an overlaid grid) and observe these three points to be:
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`(x_0,y_0) = (−1.57, 1)`
`(x_1,y_1) = (−0.39, 1.62)`
`(x_2,y_2) = (0.79, 2.71)`
Using these 3 points, we use the general form of a parabola, `y=ax^2+bx+c`, and substitute the known `x`- and `y`-values, as follows.
`1=a(-1.57)^2+b(-1.57)+c`
`1.62=a(-0.39)^2+b(-0.39)+c`
`2.71=a(0.79)^2+b(0.79)+c`
This gives us a set of 3 simultaneous equations in 3 unknowns, which we can solve using these algebraic methods. Doing so gives us:
`a=0.17021`, `b=0.85820`, `c=1.92808`.
So the parabola passing through those 3 points is
`y=0.170x^2+0.858x+1.93`
Note: Of course, we are using full calculator accuracy throughout, but final results are rounded.
Here is what that parabola looks like:
We can see the parabola passes through the 3 points, and it is close to our original curve, and so it's a good approximation for the curve in that portion of the graph. As usual, the more divisions we take, the more accurate it will be.
We do the same process for the final 2 segments, and get a parabola that passes through the 3 points shown, and which looks like this:
There are noticeable gaps between the oriignal curve and our parabolas. We only have to halve the segment size to get a much better fit, as we can see in this next image. The parabola is almost identical to the curve.
See an applet that explores this concept here:
Proof of Simpson's Rule
We consider the area under the general parabola `y=ax^2+bc+c`.
For easier algebra, we start at the point `(0,y_1)`, and consider the area under the parabola between `x=-h` and `x=h`, as shown. (Note that `Delta x = h`.)
We have:
`int_(-h)^h (ax^2+bx+c) dx `
`= [(ax^3)/3 + (bx^2)/2 + cx]_(-h)^h `
`= ((ah^3)/3 + (bh^2)/2 + ch)-` `(-(ah^3)/3 + (bh^2)/2 - ch) `
`= (2ah^3)/3 + 2ch`
`=h/3(2ah^2 + 6c)` (getting it into a convenient form)
Our parabola passes through `(-h,y_0)`, `(0,y_1)`, and `(h,y_2)`. Substituting these `x`- and `y`-values into the general equation of our parabola, we get:
` y_0 = ah^2 - bh + c`
`y_1 = c`
`y_2 = ah^2 + bh +c`
Solving these gives us
`c = y_1` (from the second line)
and
`2ah^2 = y_0 -2y_1 + y_2` (by adding the first and 3rd line)
Substituting these into `A = h/3(2ah^2 + 6c)` from above, we have:
`A = h/3(2ah^2 + 6c)`
`= h/3(y_0 -2y_1 + y2 + 6y_1)`
`= h/3(y_0 + 4y_1 + y_2)`
The parabola passing through the next set of 3 points will have an area of:
`A = h/3(y_2 + 4y_3 + y_4)`
Adding the 2 areas, we get:
`A= h/3(y_0 + 4y_1 + 2y_2 + 4y_3 + y_4)`
Say we have 6 subintervals. We just find the areas under the 3 resulting parabolas, and add them to obtain:
`A = h/3[y_0 + 4y_1 + 2y_2 + 4y_3 +` ` 2y_4 +` ` 4y_5 +` `{: y_6]`
We could keep going by creating more and more segments, and adding the areas as we go along. and we would obtain Simpson's Rule:
`int_a^bf(x)dx` `~~(Deltax)/3(y_0+4y_1+2y_2+4y_3+` `2y_4...+` `4y_(n-1)+` `{:y_n) `
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