What Is Financial Mathematics And What Is It For?
Financial mathematics is an area of practical application of mathematics , which consists of calculations aimed at better organization and greater control of money .
More than a science, it is a very useful tool in everyday life, both to take care of personal accounts and those belonging to a company.
Using different formulas, which we will talk about later in this article, it is possible to have a comprehensive view of finances, use money well , increase its value and avoid losses.
It is also from the instruments of financial mathematics that dreams are realized.
To understand better, just remember the importance of organization and planning when taking out a loan or obtaining financing, whether for the purchase of a vehicle or property.
Unless you have the entire amount to make the cash payment, you will have to make calculations to understand the impact of this financial product and its installments on your personal budget.
This requires basic knowledge of percentage, interest and formulas that allow you to understand exactly the size of the account.
Always remembering that, in this type of operation, the final cost is different from the contracted one, precisely due to the incidence of interest.
Another good example is investing , when the numbers play in your favor.
You can plan for your retirement by leaving money in savings. But it is important that this decision is made after comparing profitability with other options.
Thus, it identifies the gains that it will obtain in a certain period.
And you can only do that from financial math instruments.
Are you going to say that, now, she doesn’t seem even more interesting for your day to day ?
But its importance goes beyond and appears prominently in the corporate world, as we will see below.
How Important Is Financial Mathematics In The Corporate World?
By observing the examples brought in the previous topic, regarding the application of financial mathematics in a personal scope, you can already have an idea of its importance for companies .
The truth is that the entrepreneur does not need to master mathematics, but is committed to understanding and knowing how to use some of its formulas for routine tasks.
The best example, without a doubt, is cash flow .
This is the tool that records the company’s cash inflows and outflows. That is, your income and expenses.
It is from this that the manager identifies how the financial health of the business is going, on what he has been spending more than he should and, thus, where the savings opportunities are.
There, we already have a sample that there is no way to grow, or even survive as a company, without strict control of finances.
And it gets even worse when taking out loans without knowing the reality of cash.
Or, who knows, designing a new product or opening a branch, without designing how the business will perform in the coming months and years.
It all depends on the tool we are talking about in this article: financial mathematics.
You can be a great administrator, pay bills on time, collect customers and receive on time, negotiate advantageous conditions with suppliers and have high levels of productivity and efficiency in the company.
All this is valid to achieve the objectives proposed for her.
On the other hand, everything can go down the drain in a single unplanned move, which disregards your financial capacity in the medium and long term.
What financial math does is help you understand how money behaves.
And for a business to grow sustainably and achieve longevity, there is nothing more important.
What Is PMT In Financial Mathematics?
PMT are payments of the same amount , that is, recorded by the cash flow (personal or business) on a recurring basis.
They can appear in different formulas used precisely to have a closer understanding of the financial reality and make projections based on it.
It is important to note that the PMT does not only refer to payments made, but also received.
However, in both, the main characteristic is repetition, especially monthly, but also annually. Therefore, they are also treated as part value .
Examples of payments of the same amount are:
- Fixed loan or financing installment
- Fixed purchase installment with a supplier
- Fixed installment of receipt from a customer.
We can still cite an example .
Think of a sale worth R$ 2,000.00, which you paid in installments on the bank slip in four fixed installments, always due on the 25th.
Thus, for the next four months, R$ 500.00 should always enter your cashier on the 25th.
This value is a PMT.
What if the charge had interest? In that case, the PMT would be just one of the elements to consider in the calculation .
Later, when we talk about the main mathematical formulas, we will present examples in which the PMT appears.
What Is PV In Financial Mathematics?
PV, in English, means present value. That is, in financial mathematics, the acronym is known as present value .
But what does that mean?
There is no mystery: it is the value you have at the moment and from which you start in a mathematical operation.
There are formulas aimed at finding present value, but we’ll focus on an example of its most common application.
Let’s assume you make an investment of R$10,000 in Treasury Direct, a fixed income modality.
In this case, your PV is just 10,000. It is from this that you start to find out what the yield will be 12 months from now, for example, in a formula that considers the monthly correction promoted by the incidence of interest.
The present value can also be in debt incurred, such as a loan contracted to purchase machinery and equipment for the company.
What doesn’t change is that the VP will always be the moment value, the starting point of an equation.
Basic Concepts Of Financial Mathematics
Whether composing formulas or in the day-to-day management of finances, some concepts are at the base of the application of financial mathematics.
Therefore, understanding what they mean is one of the first steps to demystify any operation in this area.
Check out the details below and you will realize that the fundamentals of this discipline are quite simple.
Starting with one of the most popular words in the investment world: capital.
Capital (C)
Capital is the present value , which corresponds to an investor’s initial amount or the initial cost of a product or service, in cash and without fees.
Imagine, for example, that a small bakery wants to buy a new mixer that initially costs R$ 220.00 – this value is the capital.
Generally, the product becomes more expensive if the purchase is made in installments, as interest is added.
Let’s say the mixer has been divided into 10 installments, at R$ 23.00 in installments.
At the end of the purchase, the bakery will have paid R$ 230.00, that is, R$ 10.00 more than the capital.
Interest (J)
Interest corresponds to the amount remunerated by the capital , that is, it is a kind of payment for the use of capital.
One of the most intuitive ways to understand interest is to think about the loan of capital, which is remunerated through interest.
When we borrow any amount from a bank, financial institution or even an acquaintance, a fee is charged for the period in which we keep that amount.
The corresponding amount is interest.
Even when we pay in installments on a credit card and there is interest – as in the example above -, we are, in practice, taking a loan from the card administrator, which adds a value to what we initially took.
Interest can work in our favor when we choose good financial investments, which yield values on our capital.
Interest Rate (I)
The interest rate is the percentage that determines the additional amount of capital invested or borrowed initially.
This percentage is always related to a previously defined period, which can be daily, monthly, yearly, etc.
To simplify the use of the interest rate in financial math formulas, it is common to convert periods into months.
Amount (M)
We call an amount the total amount paid through an operation, including principal and interest.
Thus, considering the example given above, the amount would be calculated as follows:
- C = 220
- J = 10
- M = C + J = 220 + 10 = BRL 230.00.
Addition
When a product or service becomes more expensive, we call it added value .
Unlike interest, the accrual does not remunerate a capital investment, but adds an amount for the purchase of a product/service.
Returning to the mixer example, there would be an increase if the initial price (R$ 220.00) was changed, for example, to R$ 225.00, even for a cash purchase.
The addition can represent a simple search for an increase in profit or an adjustment arising from the increase in raw material costs, taxes, labor, among other factors.
Discount
Represents the amount or percentage taken from the initial price of a product or service.
If, instead of becoming more expensive, the mixer cost R$ 215.00, there would be a discount of R$ 5.00.
We could also calculate the percentage that this discount represents.
It is worth remembering that the capital corresponds to 100% of the value of the mixer, that is, we want to solve the sentence:
- X% of 220 = 5
We then have:
- X/100 * 220 = 5
- 220X/100 = 5
- 220X = 500
- X = 500/220
- X = 2.27.
Profit
Profit is the amount earned from a trade transaction , excluding the initial cost or purchase price of an item.
Let’s say the bakery purchased the mixer for R$220.00, but decided to sell the item at a time when its model was out of stock.
Thus, he passed on the mixer for R$ 240.00.
The profit was 240 – 220, that is, the sale generated RS 20.00 for the bakery.
Main Financial Math Formulas
Now that some of the key concepts are clear, as well as the definition and importance of financial mathematics, let ‘s get into the practical realm .
Remembering that, as much as you don’t like numbers, calculations and formulas, mathematics plays a fundamental role in your pocket.
As we’ve already seen coming this far, the health of your budget depends on it.
And if you own a company or hold a management position in it, knowledge is mandatory.
Otherwise, it may be one more to have to close the doors early.
So, know now the main formulas of financial mathematics.
Simple Interest
Simple interest is a correction calculated over an initial amount , expressed as a percentage.
This is an addition that, as the name implies, is quite simple to perform.
This could be an extra charge or receipt for not being fully disbursed at the time
Starting from a present value, an interest rate is applied that also takes into account the period of the operation.
Valid for term sales and investments (cash in) and for purchases in installments and loans (money out).
The simple interest formula is very lean and considers four variables:
- Capital (C) : the present value, which refers to the total amount of the transaction
- Interest (J) : increase on capital
- Time (t) : the duration of the operation (usually expressed in months)
- Rate (i) : percentage that determines the amount of interest levied on the transaction.
Thus, we arrive at the following formula: J = C * i * t .
Simple interest example
You made a purchase worth R$1,000.00. That’s the capital.
The interest rate applied is 2% per month. For the calculation, the percentage is converted to a decimal number, that is, 2/100 = 0.02.
The operation was scheduled for five months. That’s the time.
Therefore, the formula to be applied is as follows:
- J = 1000 x 0.02 x 5 = 100 .
In other words, the final cost of the operation with the addition of simple interest will be R$ 100, which means that its purchase in installments will represent an expense of R$ 1,100.00.
Compound Interest
Compound interest represents interest on interest, that is, it is applied to the amount of each period.
The best way to understand is to compare it with simple interest.
Looking at the previous example, you can see that there is a clear forecast about the increase even before the operation is carried out, with interest levied on the total value of the operation.
In the case of compound interest, this changes a little bit.
What happens is that each month, a correction is applied to the current capital.
This makes the profitability of an investment more attractive, but, on the other hand, it can raise a debt if this is the method of correction used.
In the case of compound interest, a new element is added to the formula:
- M : corresponds to the final amount.
The others remain: capital (C), interest rate (i) and time (t).
The formula is now as follows:
- M = C x (1 + i) t
Example of compound interest
For this example, we will consider a financial investment in the amount of R$ 2,000.00 for one year, with compound interest of 2% per month.
So we have the following:
- M = ?
- C = 2,000
- i = 2% = 2/100 = 0.02 (decimal)
- t = 1 year = 12 months.
So, let’s apply the formula:
- M = 2,000 x (1 + 0.02)¹²
Now, let’s calculate:
- M = 2,000 x 1.02¹²
- M = 2000 x 1.268242
- M = 2,536.48.
See, then, that the application of compound interest for the period of 12 months resulted in a yield of R$ 536.48.
Percentage
A percentage, also called a percentage, is a centesimal ratio .
That is, a unit of measurement that presents the proportion or relationship between two values from a fraction that has 100 as the common denominator.
Within financial mathematics, it can be very useful to identify, for example, how much of your budget is committed to a certain expense or what is the main source of revenue in percentage terms.
It would be interesting to discover, for example, that two customers represent 56% of your revenue, wouldn’t you say? But how much is this 56%?
The percentage can be found from different calculations.
One of the simplest is to multiply the percentage you want to find out by the present value.
To follow the same example, let’s assume that your monthly billing is R$ 14 thousand.
Therefore, 56% will be equivalent to the following:
- 56 x 10,000 = 784,000 / 100 = BRL 7,840.00
Interesting and easy, isn’t it?
Use the percentage on a day-to-day basis to calculate discounts and profits .
An example: your competitor has been offering a 10% cash discount and you are considering offering 12%. How much would that represent in practice?
Considering a sale worth R$ 890, we have the following:
- 12 x 890 = 10,680 / 100 = 106.80
- 890 – 106.8 = 783.20
Therefore, you will offer BRL 106.80 off and set the sale price at BRL 783.20.
Ratio And Proportion
Here, we have two other important concepts in the universe of financial mathematics.
The ratio is used in the comparison of two quantities (A and B).
Its calculation consists of dividing one by the other.
A very practical example of everyday life is the speed with which we travel from home to work.
If you print an average speed of 40 km/h, know that this value is the ratio of two quantities: distance (A) and time (B).
It is obtained from the division between them. That is: you covered 10 kilometers in 0.25 hours (15 minutes).
The proportion corresponds to the equality or equivalence of ratios .
Following the previous example, we can say that the average speed of 40 km/h (which represents the ratio) is the same as someone who travels 20 kilometers in 0.5 hours (30 minutes).
How about an exercise? Below, you can see two ratios that are equivalent and you need to find the value of X in the proportion:
- 2 / 7 = 12 / X
For the calculation, we can apply the rule of three (we will talk more about it later), being:
- 12 x X = 7 x 12
- 2X = 84
- X = 84 / 2
- 42.
Simple And Compound Rules Of Three
The rule of three, which we used to calculate the proportion in the previous exercise, is more present in your life than you might think.
You can use this easy and practical formula to solve a series of equations in everyday life, including percentages.
In its classical concept, it applies to problems involving two or more directly or inversely proportional quantities.
Considering its simple format, you need three elements to discover a fourth , unidentified.
Shall we go to an example?
Assuming you sell 50 items of a particular product every month and that represents R$ 2,500.00 of revenue in the period, how much will your billing be if you sell 60 items?
Look at the equation below:
- 50 = 2,500
60 = X
In the simple rule of three, you must cross-reference the information. Ie:
- 60 x 2,500 = 150,000
- 000 / 50 = 3,000
That’s it: when you sell 60 items, your billing will be R$ 3,000.00. Simple, isn’t it?
The compound rule of three model adds two more elements , totaling six in the same equation (only one unknown)
Following the example, you now have the number of items and the amount to receive in two different situations.
But how about also including the expense you have today to buy the items to resell and check how much you will have to pay after the move?
Observe the equation below, which opens with the current expense (R$ 1,000.00):
- 000 = 50 = 2,500
X = 60 = 3,000.
To calculate, first solve for the known part by multiplying 50 by 2500 and 60 by 3000.
In this case, we will have the following:
- 000 = 125,000
X = 180,000
Now the rule of three becomes simple:
- 000 x 180,000 = 180,000,000
- 000,000 / 125,000 = 1,440
In other words, your expenses with suppliers will rise from R$ 1,000.00 to R$ 1,440.00.
Want more proof of how financial mathematics is a powerful management tool ?
By selling ten more items, you will have an increase of BRL 500.00 in income and BRL 440.00 in expenses. That is, a profit of BRL 60.00 – BRL 6.00 per item.
With this information, you can now decide if it is really worth promoting this change, considering the existence of other expenses, such as the payment of employees.
Fractions
Finally, let’s quickly talk about fractions, which are nothing more than numbers expressed by the ratio of two others.
It ‘s a form of division , just like your pizza that comes in eight slices. In this case, each slice is equivalent to ⅛ of the total.
Fractions can also be presented in graphs, which helps in their visualization and understanding.
There are different types of fractions : equivalent, apparent, mixed, proper and improper.
We will not go further in their differentiation, but it is worth presenting a practical example to see how they can be useful in everyday life.
Assuming your digital marketing budget is BRL 25,000 for 2019 and that 100% of the amount needs to be divided equally into 12 different actions, including content marketing, sponsored links, social media and email marketing campaigns.
In practice, it means that each action will account for 1/12 (one twelfth) of the total marketing budget.
From the results, you will be able to identify if the division proved to be the most correct and, if not, adjust your planning for the following period.
Financial Math Exercises
Once you know the concepts and applications of this field, it’s time to put what you’ve learned into practice.
As an exact science, financial mathematics gets simpler and clearer as it is trained .
With that in mind, we’ve put together 10 exercises to reinforce knowledge, including some possibilities for resolution.
We recommend that you start by trying to solve the problems yourself and then check the answers and reasoning used to arrive at them.
Come on?
Exercise 1 – Rule Of Three And Reason
(Source: Financial Mathematics Workbook of the e-Tec Network , distributed by the Ministry of Education)
In a basketball tournament, a certain team played four matches in the first round and won three. What is the percentage of wins obtained by this selection in this phase?
To solve the problem, we can use the simple rule of three .
We then have:
- X% of 4 = 3
- (X/100) * 4 = 3
- 4X/100 = 3
- 4x = 300
- x = 75.
We could also use the concept of reason :
- 3/4 = 0.75
Answer : in the first phase, the winning percentage was 75%.
Exercise 2 – Percentage
(Source: Financial Mathematics Workbook of the e-Tec Network , distributed by the Ministry of Education)
A binder has 25 numbered chips, 52% of which are labeled with an even number. How many chips are labeled with an even number? How many chips have the odd-numbered tag?
Solution :
- Even Tags = 52% of 25 chips = 52% * 25
- 52 * 25 / 100 = 13
- The rest (100% – 52% = 48%) are odd-numbered chips.
We could still calculate the value of 50% and add 2% (1% + 1%), as follows:
- 50% of 25 = 12.5
- 1% of 25 = 0.25.
We then have:
- 12.5 + (0.25 + 0.25)
- 12.5 + 0.5 = 13.
Answer : In this binder, there are 13 even-numbered cards and 12 odd-numbered cards.
Exercise 3 – Percentage And Fees
(Source: Financial Mathematics Workbook of the e-Tec Network , distributed by MEC)
Of the 35 candidates who applied for a contest, 28 were approved. So what was the pass rate?
Solution :
The ratio representing successful candidates would be 28/35.
To get the percentage rate, let’s divide the numerator by the denominator:
- 28: 35 = 0.8
- 0.8 = 80/100 = 80%
Answer : In this contest, 80% of the registered candidates were approved.
Exercise 4 – Compound Interest
(Source: Financial Mathematics Workbook of the e-Tec Network , distributed by MEC)
Capital of R$ 1,400,000.00 was applied to compound interest, at 4% per month, for 3 months. Determine the amount produced in this period.
Separating the data provided in the problem statement:
- C = 1,400,000.00
- i = 4% am (per month)
- t = 3 months
- M = ?
Formula: M = C x (1 + i) t
- M = 1,400,000 x (1 + 0.04)3
- M = 1,400,000 x (1.04)3
- M = 1,400,000 x 1.124864
- M = 1,574,809,600
Answer : The amount is BRL 1,574,809,600
Exercise 4 – Simple And Compound Interest
(Source: Financial Mathematics Workbook of the e-Tec Network , distributed by MEC)
Considering that a person lends to another the amount of R$ 2,000.00, at simple interest, for a period of 3 months, at the rate of 3% per month. How much interest must be paid?
As stated, we have:
- Capital invested (C): BRL 2,000.00
- Application Time (t): 3 months
- Rate (i): 3% or 0.03 per month (am).
Doing the calculation, we will have:
- J = c. i. t
- J = 2,000 x 3 x 0.03
- J = BRL 180.00.
Answer : At the end of the loan, at the end of the three months, the person will pay R$ 180.00 in interest.
Considering the same situation, but with the charge of compound interest, we have:
- Capital Invested (C) = BRL 2,000.00
- Application Time (t) = 3 months.
Conversion to decimal: application rate (i) = 0.03 (3% per month)
Doing the calculations:
- M = 2,000.(1 + 0.03)³
- M = 2,000 . (1.03)³
- M = BRL 2,185.45.
Answer : At the end of the loan, the person will pay R$ 185.45 in interest.
Exercise 5 – Percentage
(Source: Vunesp/ Mundo Educação )
A lawyer hired by Marcos is able to receive 80% of a case valued at R$200,000 and charges 15% of the amount received, as fees. The amount, in reais, that Marcos will receive, deducting the lawyer‘s share, will be:
- a) 24,000
- b) 30,000
- c) 136,000
- d) 160,000
- e) 184,000.
Solution :
1st step: find the amount received in the case:
- 80% of 200,000.00
- 0.8 * 200,000.00 = 160,000.00.
Now, let’s calculate the 15% of the case that will be received by the lawyer:
- 15% of 160,000
- 0.15 * 160,000 = 24,000.
The amount that will remain for Marcos will be the difference between the value of the case and the amount paid to the lawyer:
- 000 – 24,000 = 136,000.
Answer: Alternative C.
Exercise 6 – Simple Rule Of Three
(Source: Financial Mathematics Workbook of the e-Tec Network , distributed by MEC)
Example 1: With a solar absorption area of 1.2 m2, a motorboat powered by solar energy can produce 400 watts per hour of energy. If this area is increased to 1.5 m2, what energy will be produced?
Solution :
To apply the rule of three, let’s relate the known and unknown values:
- 1.2 is for 400
- 1.5 is for X.
We then have:
- 1.2/1.5 = 400/X
- 1.2X = 1.5 * 400
- X = (1.5 * 400)/1.2)
- X = 500.
Answer : The energy produced will be 500 watts per hour.
Exercise 7 – Discount
(Source: Financial Mathematics Workbook of the e-Tec Network , distributed by MEC)
Consider a security whose face value is R$ 10,000.00. Calculate the trade discount to be granted for a bond redemption 3 months before the maturity date, at a discount rate of 5% am
Solution :
- N (nominal value) = 10,000
- i = 5% or 0.05 am
- t = 3.
Discount formula: Dc = N. i. t
- V = 10000. (1 – 0.05.3) = 8,500
- DC = 10000 – 8,500 = 1,500
- Discounted amount = BRL 8,500.00.
Answer : The discount will be R$ 1,500.00.
Exercise 8 – Simple Interest
(Source: Financial Mathematics Workbook of the e-Tec Network , distributed by MEC)
What is the value of a capital that, applied at the simple interest rate of 2% per month, earned R$ 240.00 in interest after one year?
Solution :
As the monthly rate is 2% = 0.02, we must consider, for the time of 1 year, 12 months, since time and rate must be in the temporal reference (in this case in months). So:
Formula: J = C.i.t
- 240 = C. 0.02. 12
- 240 = C. 0.24
- C = 240/0.24
- C = 1000
Answer : The capital invested initially was R$ 1,000.00.
Exercise 9 – Simple Interest
(Source: Financial Mathematics Workbook of the e-Tec Network , distributed by MEC)
Determine the amount corresponding to an application of BRL 450,000.00 for 225 days with a simple interest rate of 5.6% per month.
Data :
- C = 450,000
- i = 5.6% per month
- t = 225 days
- M = ?
Solution 1 :
Note that the rate is expressed in months, while the time is in days.
Therefore, it is necessary to convert one of them so that the calculations are assertive.
Let’s start by transforming time into months, dividing 225 by 30 (since each month has 30 days).
Formula: M = C.(1 +i.t)
- M = 450,000.(1 + 0.056. 225/30)
- M = 450,000.(1 +. 12.6/30)
- M = 450,000.(1 + 0.42)
- M = 450,000.(1.42)
- M = 639,000
- M = 639,000.
Solution 2 :
We can also convert the rate into days, since 1 day is equal to 1/30th month. The rate would be 5.6%/30.
We have:
- M = 450,000.(1 + 0.056/30 * 225)
- M = 450,000.(1 + 0.42)
- M = 450,000.(1.42)
- M = 639,000.
Answer : The amount will be R$ 639,000.00.
Exercise 10 – Simple Interest
(Source: Financial Mathematics Workbook of the e-Tec Network , distributed by MEC)
A person invested BRL 10,000.00 at compound interest of 1.8% pa After how long will it have a total of BRL 11,534.00?
Solution :
- C = 10,000
- i = 1.8% am = 0.018
- M = 11,534.
Formula: M = C.(1 + i)t
- 534 = 10,000.(1 + 0.018)t
- 1.018t = 11534/10,000
- 1.018t = 1.1534
- t = 8.
Answer : After 8 months of application, there will be an amount of R$ 11,534.00.
