Interest and interest calculation

The effect of interest calculation on your money

Interest is nothing more than the fee for using money. However, it matters how we calculate this fee. The way interest is calculated has a strong effect on the final result.

Markings during interest calculation

Before we get into it, let’s look at which abbreviation means what.

FV: Future Value. You will have that much money at the end of the period in question.
PV: Present Value. You have that much capital right now. They are also used to denote C as Capital.
r: growth rate. You can express this in % or even decimal form. If you see it without additional information, it always expresses an annual rate. Even if you are counting 3 months.
n: number of periods. They are even denoted by t as time. By default, here too, it denotes the number of years.

The method of interest is important, so the central bank also prepared a very useful and well-done training booklet on this.

Simple interest

Simple interest can be find on bank deposits and interest payments on bonds most often. You take your money to the bank, deposit it, and then at the end of the period you get the interest earned. You pick up this interest and look for another place for it, only the original capital continues to work.

Calculation of simple interest

FV = PV * ( 1 + r*n)

Here, pay attention to the sequence of operations in parentheses. That is, first the multiplication and then the addition comes.

Example of simple interest calculation

You have 100.000 forints, which you commit for 5 years to a place where you are guaranteed to receive 10% interest. If you use simple interest, you will pick up your earned interest every year and only the share capital will continue to work.

Since there is no increase in capital, it is irrelevant for the simple interest of 10% working for 5 years or 5% working for 10 years in terms of the end result.

FV = 100.000 * (1+0,1*5)

If you complete the operation, you will receive that you received HUF 150.000 by the end of the period using simple interest.

150.000 = 100.000 * (1+0,1*5)

Not bad, but let’s look further.

Compound interest

As long as only your original principal works at simple interest, your earned interest already works at compound interest. That is, what you get interest you don’t pick up, but you add to your capital to get interest on it as well. This will increase your wealth faster.

You can also encounter this method with a bank deposit, when on the balance sheet date of the deposit, the earned interest is not transferred to your current account, but is put back to the deposit in the same way. In addition, you may encounter the principle of compound interest even in the case of securities, when you do not live up to the profits made, but in the same way you buy some kind of security from it in order to win on it.

Calculation of compound interest

FV = PV * (1+r)ⁿ

You can see that we are no longer talking about smooth multiplication, but exponentiation. If we replace the numbers in the previous example, the difference is already obvious.

161.051 = 100.000 * ( 1+0,1)⁵

However, it matters how many times you capitalize the compound interest in a year.

Calculate the change in your own money with the Silver Moon compound interest calculator.

Calculation of compound interest by recapitalization several times a year

When applying this, pay special attention to the fact that the interest rate is given on an annual basis. Thus, if you capitalize your money several times a year and worked with compound interest, reduce the interest rate proportionally by increasing the number of periods.

If I want to write with a formula, I can do it like this:

FV = PV * (1+ r/m)^m*n

Here, m denotes the number of periods within a year. That is, if you deposit again every six months, then 2, if quarterly, then 4, if monthly, then m will be 12, while if, say, weekly, it will be 52.
With the help of the formula, you can even calculate daily interest, as does the bank if you fall into late payment.

Calculation of compound interest for a six-month term

If you deposit your money for semester periods of 5 years, you have two tasks. The first is to double the number of periods, as you will receive an interest credit twice as often. The other thing to do is to halve the interest rate accordingly, as you will only receive half of the annual credit in six months.

162.890 = 100.000 * ( 1+0,1/2)^2*5

Calculation of interest on a monthly deposit

This is done similarly to the half-yearly calculation, with the difference that here 12 payments are made within a year, but the amount of interest earned will be only twelfth as much as the annual rate.

164.530 = 100.000 * ( 1+0,1/12)^12*5

You can see that you can achieve even greater results here without changing the capital you invest or the time horizon.

Significance of the difference between the methods of interest

If you take a close look at the difference between a simple interest rate, a compound interest rate, and a one-year compound interest rate, you can see that you have reached a different end result. Compound interest means to you as if you had put your money to a higher interest rate with the same risk.

This is also true on the credit side, as you also pay the interest there on a monthly basis. The longer the term, the longer you pay interest to the bank. Since the higher result achieved was also true on the investment side, it will also be true on the loan side. Even if the bank writes out that the interest is 10%, if you pay monthly, he will win a little more.

However, it does matter if your money only grows numerically, or even in real terms.

The real interest rate

When you put your money anywhere, your main question is not really how much interest you get on it. Although you ask this and this is also written on the prospectuses, behind it is usually how much better you do if you don’t spend your money now, but invest anywhere.

This is shown by the change in purchase value, i.e. at the end of the commitment period you can buy more or less of your money compared to today’s value. This shows us the real interest.

Real interest rate = ((1+ Nominal interest rate) / (1 + Inflation rate)) – 1

Nominal interest is the interest you receive numerically. This is usually stated in the prospectuses. Now, however, we do not want to see a change in numbers, but a change in purchasing power. That is why we adjust for inflation.
The inflation rate is harder to grasp than the nominal interest rate, and it is a little different for everyone. Depending on what and where you usually buy.

Interpretation of real interest rate

The real interest formula takes into account actual growth and the actual impact of inflation. That is, we are not looking at the interest rate and the ratio of the inflation rate, but what happened to the purchase value at that time. Therefore, it is necessary to add 1 to the fraction. In the end, however, we are interested in the real interest rate, so we subtract 1, which indicates 100% (i.e. invested capital). Since a subtraction has taken place, its value can be negative, positive, or point 0.

Value of 0: the purchasing power does not change, ie today you can buy exactly the same amount of it as in 1 year.
Negative value: the purchase value of your money decreases, i.e. either choose a more profitable investment or spend the money now.
Positive value: the purchase value of your money increases, i.e. you have invested in a good place in this respect.

Calculation of real interest rates

Suppose inflation is 4%. As both inflation and interest are expressed on an annual basis, the real interest rate will normally be expressed on an annual basis.
If we carry over the numbers of the example just above, the formula will look like this:

5,77% = ((1+0,1 / 1+0,04 )) -1

Here, with this example, the value of your money has increased.

The value of real interest rate on bank deposits

This 4% inflation is broadly in line with the level reported by the CSO. If we look at a traditional bank deposit, with a peak interest rate of around 0.5%, which is common today, this calculation is transformed as follows:

-3,37% = ((1+0,005) / (1+0,04)) -1

You can see that your money loses its value when tied up in the bank.

The interest factor

The interest rate factor shows how many times your money has increased over a given period of time.
For simple interest and compound interest, you can count on pre-specified interest rates, which are usually given to you by Unified Deposit Index Rate(You can find it as EBKM in Hungarian banks). Life, on the other hand, can make a difference and this information is not necessarily available to you. In such cases, it may be necessary if you can determine the rate of growth yourself.

Calculation of the interest factor at a simple interest rate

Then you don’t have too much to do, you divide the amount you reach by the end of your invested capital, and then you annualize it. That is, you divide by the number of years.

((FV / PV) -1) / n = ((150,000 / 100,000) -1) / 5 = 10%

Here, -1 also subtracts the original principal so that you only have the interest rate left for you.

At a compound interest rate, it is no longer that simple because we have powered the rate of growth there.

Calculation of the interest factor at compound interest rate

So if we want to get an exact amount, we need to do the power backwards, that is, we need to draw a radical. Since the power has been the exponent here for a number of years, do the root subtraction accordingly. That is, if your money worked for 5 years, you raised it to the 5th power, and now you draw a 5th root from it.

⁵√(161.051 / 100.000 ) -1

If you do the operation, you can see that 10% comes out here as well.

Continuous interest

You may also encounter continuous interest rates as logarithmic returns, although you will not necessarily encounter them in everyday life. There are so many reasons for this, there are more convenient solutions for you.

The development of continuous interest rates

Since the 1960s, interest rates ceiling have been introduced in several places on both the deposit and credit side. That is, you could not achieve more than a certain increase in the banks on your investment, nor could the bank ask you for more.
However, this regulation said nothing about the frequency of interest payments. In the case of compound interest, you have already seen that the more often the interest payment is made, it is as if a higher annual interest rate has been offered. Thus, at first half-yearly and then monthly interest payments were introduced everywhere.
Once this spread, one eye-catching company introduced a steady interest rate. That is, it is as if payments come continuously and evenly throughout the year.
This is shown by the logarithmic yield.

Calculation of continuous interest

Since the continuous interest rate is practically the compound interest rate spinned slightly up, so its formula will be based on this as well.

FV = PV * (1*r/m)^m*n

Since we set the interest rate on an annual basis, and we still want to say the rate for 1 year, the value of n will be 1, that is, we can omit it from the inscription.

FV = PV * (1*r/m)^m

The m, on the other hand, goes to infinity because we assume interest payments at arbitrarily small intervals (i.e., at all times).
You could say that this requires a lot of computational tasks and other work, after all, here m goes to infinity. This, in turn, has already been greatly simplified by smart people with the calculation of the limit value.
Before we get into the math in calculating the limit, let’s accept the mathematicians calculation. The limit is 2,718r.

And 2,718 is a mathematical constant, which is the basis of the natural-based logarithm. This called Euler ‘s number, so denote by e.

So with a constant interest rate, your annual interest rate is er.

If m goes to infinity, i.e., there is an interest payment at arbitrarily small intervals, then its formula looks like this:

FV = PV * e^r*n

An example of continuous interest

If you commit the HUF 100.000 of the first example to continuous interest for 1 or 5 years, it will look like this. With a 1-year commitment, your money is as follows:

FV = 100.000* e^0,1 = 111.626

For a 5-year commitment:

FV = 100.000* e^0,1*5 = 164.863

You can see that here, too, there is a minimal discrepancy in the end result.

Summary

You use compound interest most often for bank deposits and loans, after all, the bank determines everything based on this. If you want to use it for your own benefit, choose to capitalize as often as possible when investing. The more times your money is capitalized, the greater your available benefits will be. A theoretical top of this is the constant interest rate at a given interest rate. You won’t encounter this very much in real life, as it is impossible to make an interest payment in practice. It is also used by financial professionals for option pricing and comparisons.

If you would like to make the right decision or just be in the picture contact us to arrange an appointment for this.