Compound interest - what is it and how can it make you financially free?

The concept of compound interest

Why do investors claim that it is possible to create capital even with small amounts? By putting 5,000 rubles a month into an account, will you really save anything significant?

Firstly, depending on what you mean by essential. Some people want to buy an apartment, others want to buy a bicycle, others create passive income for retirement. Secondly, you can actually create capital with small amounts. Investors don't lie because they are already well acquainted with the magic of compound interest.

In the article I will definitely show how this works with numbers and specific examples. In the meantime, let's remember our childhood. In winter, many of us made a snowman. They took a small lump of snow, rolled it, and it grew into a big lump. The same thing happens with our money, which we do not put in the nightstand, but make it work for us. Compound interest helps with this.

Compound interest is a percentage that is calculated from the original amount, then added to it, then calculated from the new amount, taking into account previously accrued income, and so on until the end of the billing period. In the banking industry, this process is called capitalization.

In the 1st year, a small amount grows with a small income. In the 2nd year, income will be accrued to “Amount + Income for the 1st year”, in the 3rd year – “Amount + Income for the 1st and 2nd years”, etc. I will show you with a simple example. The figures are conditional, given for ease of understanding the process, and have nothing to do with real deposits.

You deposited 10,000 rubles into your account at 10% per annum. A year later, 11,000 rubles were withdrawn. Now let’s say that they didn’t withdraw it, but left it on the account at the same 10% per annum. Only already 11,000 ₽. After a year, the bank charges another 10% on them. And now you see 12,100 ₽ on your account. Looking ahead, I will say that in 10 years it will be 25,937.42 rubles, and in 40 years – 452,592.56 rubles. Notice that you are not doing anything other than withdrawing money.

The numbers increase significantly if you regularly top up your account, but more on this is yet to come and will definitely use examples.

Calculation formulas

Since there is a compound interest rate, it means there is also a simple interest rate. It's unfair if we don't deal with our hero's younger brother.

Simple interest

Simple interest is accrued every billing period (month, quarter, year) only on the initial amount. It does not give any “snowball” effect. The amount increases slowly.

Calculation formula:

SN = SP * (1 + % st * N), where

• SN – amount at the end of period N;
• SP – initial amount of capital;
• % st – interest rate (income);
• N – billing period.

The formula is valid if we are talking about accruing income once a year. For example, we deposited 100,000 rubles into an account at 10% per annum for 10 years. At the end of the term you will receive: 100,000 * (1 + 0.1 * 10) = 200,000 rubles.

In real life, the concept of simple % is used, for example, in economic calculations for bank deposits without taking into account capitalization. The contract must indicate the annual interest rate. Interest is accrued for each day the money is on deposit. And the investor can receive income monthly, quarterly or once a year.

In this case, the formula will take the form:

SN = SP * (1 + % st * D / 365), where

• D – the number of full days the money is on deposit.

For example:

1. We deposited 100,000 rubles into the account at 10% per annum for 91 days. At the end of the term you will receive: 100,000 * (1 + 0.1 * 91 / 365) = 102,493.15 rubles.
2. For 180 days: 100,000 * (1 + 0.1 * 180 / 365) = 104,931.51 RUR.
3. For 2 years (730 days): 100,000 * (1 + 0.1 * 730 / 365) = 120,000 ₽.

Compound interest with income accrued once a year

Using the compound interest method, when income is calculated once a year, the future amount is determined by the formula:

SN = SP * (1 + % st)N

Example. They deposited 100,000 rubles in the bank at 10% per annum for 2 years. The future value of the deposit will be: 100,000 * (1 + 0.1)2 = 121,000 rubles.

Compound interest with income accrued more than once a year

Income can be accrued monthly, quarterly or twice a year. The formula changes:

SN = SN * (1 + % st / K)N*K, where

• K – frequency of income accrual (12, 4 or 2 times a year).

Example. They deposited 100,000 rubles into the bank at 10% per annum for 2 years with monthly interest accrual. The future value of the deposit will be: 100,000 * (1 + 0.1/12)24 = 122,039.1 rubles.

02.05.2020

25 426

10 most profitable bank deposits today

Criteria for choosing a bank for a deposit and an overview of the best offers.

Types of bets

Most often, the rate appears in the loan agreement and financial agreement. When signing such a document, the borrower undertakes an obligation to the lender to pay a specific amount. It is defined as the ratio of interest paid over a fixed period of time to the amount of the loan. Called the rate, it is calculated as a percentage.

Methods for calculating interest vary and depend on the terms of the contract. Rates can be applied to the same initial amount throughout the entire loan period or to the amount with interest accrued in the previous period.

The first calculation option is called a simple interest rate, the second - complex. A simple rate applies to the same initial amount of debt throughout the entire term, i.e. the initial base (money amount) is always the same (without taking into account its sequential repayment). This method of calculation is used in consumer lending.

Compound is applied to the increased loan amount, i.e. to the amount increased by the amount of interest accrued for the previous period. Therefore, the initial base is constantly growing.

In addition to simple and complex, there are several other types of bets. Additionally there are:

• Fixed. Set as a specific number in financial contracts.
• Variable. It changes discretely over time and does not have a specific numerical characteristic.
• Floating. Tied to a certain value that changes over time, it consists of a base and an addition to it (margin). The base is the initial value, the margin is a variable that depends on conditions such as the duration of the transaction, the financial situation of the borrower, etc.

Also in economics there is the concept of nominal, ordinary, exact and real interest. They all have their own characteristics.

Key parameters influencing the calculation result

The amount that a depositor or investor will receive at the end of the settlement period depends on a number of key parameters:

1. Interest rate is the return you receive from investing your funds in a particular instrument, or the fee for using “other people’s” money (for example, a loan). The higher the %, the more you will earn or pay.
2. Settlement period – the period (days, months, years) during which it is planned to receive income or pay for loan funds. The higher it is, the greater the amount accumulated or paid to creditors.
3. Starting capital is the amount that you initially allocated for savings or received on credit.
4. Frequency of additional contributions. Over a short period of time, the effect of additional amounts deposited into the account is insignificant. The snowball begins to grow at a noticeable pace from the 5th–7th year of accumulation or repayment.
5. The frequency of % accrual is daily, monthly, quarterly or annually. The more often, the higher the rate of increase in the amount.

It is not necessary to independently calculate the amounts using the above formulas and play with changing key parameters. There are numerous online calculators on the web, into which all you have to do is plug in the numbers. As a last resort, you can enter the formulas into Excel once and consider different calculation options. In further examples I will use an online calculator.

Mechanism of operation

So far we have looked at how compound interest works in theory. Let's look at what they are in practice, using the example of bank deposits and investments.

Using the example of a bank deposit

When choosing a bank deposit, the depositor must pay attention to several parameters: the reliability of the bank, its participation in the state insurance system, conditions for replenishing and withdrawing money, the minimum amount in the account. But the main one is the interest rate and the conditions for its calculation.

The compound interest mechanism is connected to deposits with interest capitalization. And the rate itself, which will be valid on your account, is called effective. If you do not plan to withdraw accrued income during the entire accumulation period, then it is logical to choose a deposit with capitalization.

Let's compare the income received on the deposit with interest accrued annually, quarterly, monthly and daily. Initial conditions:

• amount – 400,000 ₽;
• % rate – 4% per annum;
• deposit term: 1, 2 and 3 years.

The amount that the investor will receive at the end of the term will be:

In investments

Compound interest works not only in banking, but also in the investment sector. If in banks the process of calculating interest on interest is called capitalization, then in investments it is called reinvestment, i.e. re-investment. But the essence remains the same.

Long-term investors are well aware of the compound interest mechanism and try to use it to the maximum. Let's look at how it works in various investment instruments.

• Bonds

The yield of a bond consists of two sources – growth in quotes and coupons. The latter are paid as a percentage of the face value of the security. As a rule, once every six months.

The effect of compound interest can be observed on coupon payments, but only in one case - if you do not spend the profit received on current consumption, but re-invest it in investments, i.e. reinvest. It is clear that the income from one bond can buy little. But if there are several dozen or hundreds of securities, then the amount is sufficient to purchase several more bonds.

Complete information about current strategies that have already brought millions of passive income to investors

For example, the owner of one OFZ-26212-PD will receive 35.15 rubles twice a year. In a year he will earn 70.3 rubles. You cannot buy a new OFZ with this money. If there is not one bond, but, for example, 50 pieces, then the income for the year will be 3,515 rubles. You can buy 3 more OFZs for RUB 1,085.81/piece. (quote as of October 27, 2020).

If you do not hold a bond until maturity, but are trying to make money on rising quotes, then in this case it is better to reinvest the profit received from resale to enable the compound interest mechanism.

• Stock

Exactly the same effect as described in the previous example can be achieved by reinvesting income from shares in the purchase of new shares. To do this, dividends received do not need to be withdrawn from the account, but reinvested.

Not all issuers pay dividends. Some investors buy growth stocks into their investment portfolios, i.e. securities that may increase in price in the future. Bought cheaper, sold more expensive is one of the investment strategies. Compound interest will be earned if the profit received from resale increases the capital in the investment, and not the number of items in the wardrobe.

The snowball mechanism works similarly with other investment instruments. The effect can be enhanced if you invest in an individual investment account, then each income tax refund (maximum 52,000 rubles per year) must be returned to the brokerage account again and securities purchased.

Percentage calculations in the life of our school

We encounter percentages in mathematics, chemistry, physics, and geography lessons. Teachers, head teachers, social educators, caretakers, librarians, accountants, and medical workers use percentage calculations in their work.

In the work of a teacher, percentages are very often encountered, not only in lessons, but also when calculating the quality of knowledge, the quality of student performance, and when analyzing the results of activities.

Teacher's task : Results of the performance of students in our class in mathematics for the 1st half of the year: “5” - 1, “4” - 9, “3” - 8, “2” - 0. Find the quality of knowledge and the quality of performance of students in our class. Solution: 10:18·100% = 55.6% - quality of knowledge Task of the head teacher: There are 537 students in our school, of which 259 students are at stage I, 244 students are at stage II, 34 students are at stage III. The percentage of primary school students to the total number of students is 48.2%, the percentage of secondary school students to the total number of students is 45.4%, and the percentage of high school students is 6.4%. Let's conduct a comparative analysis of academic performance over the last two years.

We can conclude that there is a positive trend in the quality of academic performance, since the increase was 3.2%. The number of students with one C grade has increased. But the level of learning decreased by 0.1%. Analysis of the State Examination results in mathematics in our school for two years.

Based on these data, the following conclusions can be drawn: Positive dynamics are observed. The training indicator has increased by 11.6%, quality by 0.5%. The primary score increased by 37.5%. Our school hosts many clubs and sports sections. Attendance at clubs and sections by students in grades 5–11 in the 2016-2017 school year. The year is presented in the form of a table and diagram (Appendix 3)

After analyzing this data, I came to the conclusion that most students attend electives, studios and teams.

The librarian determines the percentage of textbooks provided to students, the percentage of library attendance by students and teachers, and uses percentage calculations in reports. Librarian's task: Our library has a total of 12,921 books, of which 8,185 are textbooks, fiction - 4,395 books, methodological literature - 341 books. What percentage of textbooks make up the number of all books? Solution: 8185+4395+341=12921 book 8185:12921 · 100% = 63.35%

Accountant's task: Calculating the wages of a 1st category teacher with a workload of 27 hours. The salary of a category 1 teacher is 6,578 rubles. The payment for the watch will be: 6578:18*27=9867 rubles. Regional coefficient: 9867*0.3=2960.1 rubles. Northern coefficient: 9867*0.3=2960.1 rubles. Total: 15787.2 rubles. Income tax: 15787.2*0.13=2052.33 rubles.

This means the teacher will receive 13,734.86 rubles. Monthly contributions to the funds will be: to the pension fund - 22%; social insurance fund - 2.9%; social fund. accident insurance - 0.2%; federal health insurance fund - 5.1%. Total: 30.2% or 15787.2*0.302=4794.91 rubles. The amount of the salary of this teacher for the school, as for a legal entity, will be: 15787.2 + 4794.91 = 20582.11 rubles

During my research work, I came across percentages where I least expected to see them. In literature lessons we will study the work of F.M. Dostoevsky "Crime and Punishment". This novel was written by the writer in 1866 and here we also encounter percentages.

The old woman Alena Ivanovna gave the main character, student Rodion Raskolnikov, money as a mortgage and took interest for it. I will give a short excerpt from this work: “Here, father: if there is a hryvnia per month per ruble, then for one and a half rubles you will be charged fifteen kopecks per month in advance, sir. Yes, for the previous two rubles, you still owe twenty kopecks in advance on the same account. And in total, therefore, thirty-five.” This once again proves that the scope of interest calculations is not limited.

Conclusion

After conducting research work, we came to the conclusion that modern man is very closely connected with percentages. Interest is used in various spheres of human life: in financial and economic (banks), social (population distribution), political (voting), utilities (increasing and decreasing the cost of electricity and rent), in commodity industries (sales, discounts), in scientific ( chemistry, physics - efficiency value).

Studies have shown the importance of percentage in the life of a modern person. Percentages are one of the complex topics in mathematics; each version of the Unified State Exam test tasks in mathematics contains a percentage problem. Therefore, you need to know and be able to use this topic as best as possible. Every modern student should be able to competently and economically carry out basic percentage calculations.

Knowledge about percentages is necessary for every person, since we constantly encounter percentages in everyday life. People today cannot live without knowledge of percentages! Thus, in the course of carrying out this work, I was able to prove that interest is not an abstract concept, but a constant companion of our lives.

Bibliography

1. Vilenkin N.Ya., Zhokhov V.I. Mathematics: textbook for grades 5-6. – M.: Mnemosyne, 2005.-280c.
2. Depman I.Ya., Vilenkin N.Ya. Behind the pages of a mathematics textbook. - M.: Education, 1999.-287c
3. Dostoevsky F.M. Crime and Punishment: A Novel. At 6 o'clock with an epilogue - M.: Enlightenment, 1982
4. Kramor V.S. We repeat and systematize the school course of algebra and the beginnings of analysis. – M.: Education, 1990.-416c.
5. Kuznetsova L.V., Bunimovich E.A., Pigarev B.P., Suvorova S.B. A collection of tasks for conducting a written exam for a basic school course. – M.: Bustard, 2001.-192c.
6. Savin A.P. Encyclopedic dictionary of a young mathematician. – M.: Pedagogy, 1985.-352c.
7. Friedman L.M. Studying mathematics: a book for students in grades 5-6.-M.: Education, 1995.-255c.
8. Yashchenko I.V., Shestakov S.A., Trepalin A.S., Semenov A.V. GIA 2014. Mathematics 9th grade. State final certification. Typical test tasks. – M.: Publishing House “Exam”, 2014.-78c.
9. Internet

If you liked the page, share it on social networks:

How to create passive income for retirement even with 1,000 rubles in your pocket

The retirement age has been increased, the funded pension has been frozen, pension reform is regularly carried out and conditions are changed. All these chaotic movements only indicate that the leadership does not have a clear action plan and vision of how pensions should be calculated in our country.

What conclusion should the ordinary citizen draw from all this? There is only one way - to save for retirement on your own. And compound interest will help with this. Using specific calculations, we’ll see how to create passive income even with 1,000 rubles a month. But first, a wonderful tale from Bodo Schaefer’s book “Mani, or the ABC of Money.”

Once upon a time there lived a peasant. Every morning he went to the chicken coop to take for breakfast the egg that his chicken had laid. But one day he found in the nest not an ordinary egg, but a golden one. At first he couldn't believe it. Perhaps someone decided to play a cruel joke on him. But the jeweler to whom he brought the egg to show it confirmed that it was made of pure gold. The peasant sold the egg at a profit and organized a big celebration.

The next morning he went to the chicken coop earlier than usual. There was a golden egg in the nest again. This went on for several days. But the peasant was greedy and wanted to get rich quickly. He was angry with his hen because the “stupid bird” could not explain to him how she managed to lay golden eggs. It seemed to him that then he could lay golden eggs himself. Then he would have two eggs every day. And one day the peasant got so angry that he ran into the chicken coop and killed his chicken. There was no one to lay the golden eggs.

The moral of the story is: you shouldn't kill the goose that lays the golden eggs. But to get golden eggs, you must first get a chicken. This is what you should do as soon as possible. Time is the friend of the investor and the enemy of those who put off creating personal capital for later.

Example 1. It is necessary to calculate how much money you need to save in order to live on passive income after a certain number of years. Let’s say we want to receive 50,000 rubles every month in retirement. Let's take into account inflation of 4%.

Let's take the rate of return equal to 10%. Its size depends on the composition of the investment portfolio. If you decide to save in bonds, then you need to pledge a smaller percentage. If you create a balanced portfolio of different instruments (for example, ETFs, stocks and bonds of individual issuers, gold), then 10% is a very conservative estimate. In practice it turns out much more.

Calculation without taking into account inflation: 50,000 * 12 months / 0.1 = 6,000,000 rubles. To take inflation into account, we will use an online calculator. You already need to save 10,000,000 rubles.

Example 2. There is an initial capital of 50,000 rubles with a monthly investment of an equal amount: 1,000 rubles, 5,000 rubles and 10,000 rubles. Profitability – 10%, we will accept annual accrual of %. How much will we save in 10, 20, 30 and 40 years?

What conclusions can we draw from these calculations:

1. We can save up for passive income of 50,000 rubles per month by saving 5,000 rubles for 30 years. If we invest 10,000 rubles each, then after about 23 years we can retire.
2. With a monthly 1,000 ₽ you need to be content with a smaller amount of passive income. For example, to receive 35,000 rubles monthly, you need to save 7,000,000 rubles. The table shows that we will achieve this only in 40 years. But for a monthly increase in pension of 20,000 rubles, you will need to save 4,000,000 rubles in 35 years.

Play around with your numbers in any financial compounding calculator. Some will have a larger initial or monthly amount, others will consider a shorter or longer term, etc.

15.10.2020

3 794

Where to invest little money: 10 best ways to create capital for a novice investor

Where to invest small savings (from 1000 rubles).

Solving concentration and percentage problems

To solve problems from this section, we introduce the basic concepts: Let two different substances A and B with masses mA and mB be given. The mass of a mixture composed of these substances is equal to M = mA + mB. The mass concentration of substance A in the mixture (the proportion of pure substance in the mixture) CA = mA/m = mA/mA + mV. Mass concentrations are related by the equality: CA+ CB =1 The percentage of substance A in a given mixture is calculated by the formula: PA = CA 100%

Problem 1. There is 50g of solution containing 8% salt. You need to get a 5% solution. What is the mass of fresh water that must be added to the original solution?

Solution: Let it be necessary to add x kg of fresh water. We take salt as a pure substance. We will formulate the solution in a table.

Let's make an equation: 0.08 50 = (50 + x) 0.05

50 + x = 80

Problem 2. The solution contains 15% salt. If you add 150g of salt, the solution will contain 45% salt. Find the mass of salt in the original solution.

Solution: Let the mass of the solution be x g. Let us formulate the solution in a table.

Let's create and solve the equation: 0.15x + 150 = (x + 150) · 0.45 0.3x = 82.5 x = 275

Let's find the mass of the pure substance in the original solution: 275 · 0.15 = 41.25 g