Authentic payments. Annuity loan payments
In this financial blog article, we will study what an annuity payment is. An annuity, or financial annuity, in banking is a schedule for repaying a debt and interest on it, when payments are made in equal amounts at equal time intervals. The annuity scheme is an alternative to the differentiated scheme, when the principal loan amount is paid in equal installments, and interest is charged on the balance.
It is noteworthy that, despite the fact that differentiated payment is a classic, more and more banks are switching to an annuity lending scheme. This is convenient both for the banks themselves, since it provides greater interest benefits, and for clients, since the amount of the payment is easy to remember or fix, and to determine the payment amount there is no need to communicate with bank representatives every month. From the client’s point of view, such a payment is much more convenient than a differentiated payment in case of early repayment of the loan. And the longer the loan amount, the more profitable the annuity scheme.
It should be remembered that annuity payments are used not only in lending. In a broad spectrum, this concept includes several points.
- In particular, an annuity is a term government loan, for which the amount and interest are paid annually;
- An annuity is also a type of contract under which a person, when taking out insurance after a certain period, receives the right to payment of the amounts established by the contract - for example, after retirement.
- Finally, the cost of a chain of regular insurance payments, which are made during the period specified in the contract, is also considered an annuity.
In addition, the annuity payment schedule can be used to accumulate a certain amount at the required point in time: such a scheme is convenient due to the equal contributions.
In general, the formula for calculating the payment amount under an annuity scheme is very simple: it is the amount of the loan amount used to repay and interest accrued for the current period. As for the repayment process itself, according to the time of payments they are divided into pre-numerando annuity (payments at the beginning of the first period) and post-numerando annuity (payments at the end of the period).
Calculation of annuity payments
There are several ways to calculate the monthly payment under an annuity scheme.
- The first of them is the simplest and most convenient for the average person - contacting a bank. Consultants, as a rule, do not refuse calculations to their clients.
- The second method is designed for active Internet users - this is a loan calculator. As a rule, such services are built into most banking websites; this is global practice. It is convenient because the consumer can analyze and compare offers from various banks without leaving home. In addition, there are universal loan calculators and entire websites created on a universal basis. As a rule, they are “tailored” to different lending currencies; exchange rates are provided daily by Sberbank of Russia, so there are practically no failures in their operation.
- Finally, if you don’t want to go to the bank, and for one reason or another you don’t trust online calculators or don’t have access to them, there is a third way - a formula that is easy to use with just paper and pencil on hand. The banking formula that allows you to find out the size of the loan payment is quite complex, but its use is not at all necessary. There is a simpler, “consumer” way to calculate this value. We will talk about it further.
In order to calculate the interest component of the payment under the annuity scheme, you need to take the loan balance for the stated period, multiply it by the annual percentage and divide by 12 (as you might guess, this is the number of months in a year). And in order to determine the part of the payment that goes to repay the debt, you only need to do nothing - subtract the accrued interest from the amount of the monthly payment. In this regard, it is important to remember that due to the fact that the part of the payment that goes to repay the principal debt depends on payments made earlier, the schedule must be calculated sequentially from the very first payment. Sometimes it is also necessary to find out the amount of overpayment on annuity loans - for this it is also not necessary to run to the bank, it is enough to multiply the amount of the monthly payment by the number of periods (months) and subtract the total loan amount from this product.
As a conclusion, let’s weigh the pros and cons of annuity lending.
This scheme is suitable for clients who:
- cannot pay large amounts every month, especially in the early stages. And this very favorably distinguishes the annuity scheme from the differentiated one. The most striking example of such an application is mortgage lending. Mortgages involve “long” large loans, and payments in equal installments are usually convenient for consumers;
- have a permanent, stable income and, as a result, have the opportunity to calculate the family budget;
- have the option of early repayment or take out a small loan.
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At the same time, the main disadvantage of the annuity loan repayment scheme is that it is significantly more expensive. True, this high cost is rather nominal: the fact is that with an annuity, the consumer pays off the debt more slowly and, as a result, pays high interest on the loan. In this regard, there is an interesting point, which is a way out of the situation: if you take a loan from a bank according to an annuity scheme, and repay it according to the classical - differentiated - scheme, that is, pay off the body of the loan ahead of schedule, then the annuity turns into a differentiated loan. Otherwise, there are no significant differences between these two schemes, and by and large the choice of scheme lies with the client. Annuity lending in a nutshell is just an opportunity to pay smaller amounts on the loan body. This is a convenient and modern way that makes it possible to obtain loans even for those consumers who lack the opportunity to work under a differentiated scheme.
Russians actively use loans, but not everyone understands the intricacies of lending. As a result, they may agree to a bad deal. When choosing a program, it is important to pay attention not only to the interest rate, the presence of commissions, but also to what scheme the debt is repaid. Most borrowers prefer annuity loan payments, but this option is more expensive.
What is this?
Repayment of the loan received is carried out according to a specific schedule. In banking practice, two repayment schemes are used: annuity and differentiated (standard). Just 10 years ago, the second option was most often used. Nowadays, most institutions do not leave the client the right to choose and offer only annuity payments. A differentiated schedule is currently offered by only three financial organizations: Nordea, Rosselkhozbank and Gazprombank.
Translated from Latin, “annuity” means “annual”, that is, it describes a certain option for repaying debt. In this case, the amount of monthly contributions is fixed and does not change during the entire period of the agreement.
Difference between annuity and differentiated payment
Every month the client is required to make a payment that contains interest for the past 30 days and part of the principal debt, that is, the “body of the loan.” In an annuity payment, the ratio of interest and “body” constantly changes depending on the accrued interest. In the first year, the principal debt is repaid with minimal payments. The loan balance is large, and therefore high interest rates are charged. With each month, the proportion of body and percentage changes towards the first indicator.
According to the standard schedule, the monthly payment amount is constantly changing. The loan body is repaid in fixed installments. The amount of the loan received is divided by the number of months for which the agreement was concluded. The result obtained is a fixed part of the mandatory payment. Interest for the previous month is added to this amount.
In the first year, the amount of differentiated payment will be maximum. As the accrued interest decreases, the amount of the required contribution also decreases.
What is more profitable for the borrower?
If we compare the two types of schedules, then from the point of view of the overall overpayment, the standard one is more profitable. According to it, the loan body is repaid in large payments. Since the amount of debt decreases faster, the amount of accrued interest decreases faster.
Both repayment schemes have their pros and cons.
Benefits of an annuity:
- It is easy for the client to remember the fixed fee. He does not need to constantly monitor the schedule. To simplify the debt repayment procedure, you need to use the “autopayment” service. It is connected via a salary card. On the specified date, funds will be transferred to the loan account automatically. The payer does not need to do anything for this.
- This option is suitable for borrowers with limited income. The amount of the annuity payment is significantly less than the differentiated one (in the first months). Therefore, when approving an application, the bank makes fewer demands on its solvency.
- This scheme is the best choice for long-term lending. Due to rising inflation, wages will increase. The client will not have problems repaying the loan.
But annuity also has significant disadvantages. Firstly, there is a large overpayment, especially for consumer loans issued for a long period. Secondly, the annuity delays the payment of the principal debt.
This scheme is absolutely unprofitable for borrowers who are not constrained by income and plan to repay the debt ahead of schedule.
Considering the differentiated scheme, one can note its main advantage - debt repayment is carried out in large payments. The amount of accrued interest decreases faster compared to an annuity.
But the amount of monthly contributions will be quite large at first. Banks have stricter requirements for client income. The likelihood of your application being rejected also increases.
If the borrower receives a high income and wants to repay the loan as quickly as possible, then it will be more profitable for him to choose a differentiated schedule. For clients who have average earnings, are raising minor children, and are paying off other loans, it is better to opt for an annuity.
Formulas for calculating loan payments
To finally decide on the repayment scheme, it is necessary to perform calculations. As a result, the client will have a clear idea of the amount of overpayment for each of the schedules.
To calculate the annuity, a rather complex formula is used:
X = S * (P + (P/(1+P)N-1)
X is the amount of the monthly payment;
P - rate per month (annual rate must be divided by 12);
N – number of months of debt repayment according to the schedule.
To determine the interest part of the payment, you need to multiply the debt balance as of a specific date by the annual rate and divide by 12. The result obtained is the amount of interest for the past month:
Pn = Sn * P / 12
Рn – interest for one month;
Sn – size of the loan balance;
P – interest rate, per annum.
To find out the amount that will be allocated to the “body”, you need to subtract the accrued interest from the total amount of the mandatory contribution:
x is the amount of the mandatory contribution;
рn – interest on a certain date;
s – the amount of funds allocated to repay the loan body.
Procedure for calculating differentiated payment
In this case, determining the amount of the monthly contribution is much easier.
First you need to find out the part that goes to repay the principal debt. To do this, the amount of the issued loan is divided by the number of months of repayment according to the schedule.
NP=OK*(PS/12)
NP - interest for the previous month;
OK – loan balance;
PS – interest rate in per annum.
- loan amount – 500,000 rubles;
- the contract is concluded for 60 months;
- interest rate – 28% per annum.
Substituting these values into the loan calculator, we get the following results:
- The amount of the monthly contribution under the annuity schedule will be 92 rubles, and the total overpayment will be 434,073.97 rubles.
- In the first month, only 3901.25 rubles will be used to repay the principal debt, and 67 rubles will be spent on interest.
- Next month, this ratio will change insignificantly: body - 3992.28 rubles, interest - 11575.64.
- In the last month, the proportion will be equal: body - 15212.98 rubles, and interest - 354.94.
Results according to the differentiated scheme:
- The amount of the first payment is 20,000 rubles, and the last one is 77 rubles.
- The total overpayment is 355833.33 rubles.
- In the first month, the loan amount will decrease by 8333.33 rubles, and interest will be 11666.67.
- In the second payment, the amount of repayment of the principal debt remains unchanged - 8333.33 rubles, and interest - 11472.22.
- In the last month, the borrower will be charged 194.44 rubles in interest.
![](https://i1.wp.com/temabankov.ru/wp-content/uploads/2018/12/%D0%BF%D0%BE%D0%B4%D1%81%D1%87%D0%B5%D1%82-%D0%B2%D1%8B%D0%BF%D0%BB%D0%B0%D1%82-%D0%BF%D0%BE-%D0%B4%D0%B8%D1%84%D1%84%D0%B5%D1%80%D0%B5%D0%BD%D1%86%D0%B8%D1%80%D0%BE%D0%B2%D0%B0%D0%BD%D0%BD%D0%BE%D0%B9-%D1%81%D1%85%D0%B5%D0%BC%D0%B5-932x622.jpg)
With the same initial data, the amount of overpayment according to the annuity schedule is greater by 78,240.64 rubles. The difference is quite significant. If possible, you should always choose the standard scheme.
Possibility of switching from annuity payment to differentiated
At any time during loan repayment, the client can contact the bank with an application to change the repayment schedule. Of course, if this is not prohibited by the provisions of the loan agreement.
The borrower also needs to document his income. After changing the payment scheme, the amount of the monthly contribution under the standard schedule will be significantly higher than under the annuity. The bank must make sure that the client has a sufficient level of solvency and will be able to pay new payments without problems.
After this, you must sign an addendum to the current loan agreement and a new schedule. Only borrowers with a positive credit history can count on this privilege. If the client has previously been late, the bank will not agree to change the repayment schedule.
Not every bank offers its borrowers to choose a loan repayment scheme. As a rule, this condition is an integral part of a specific loan program and does not depend on the will of the borrower. But if this does happen, a citizen is unlikely to be able to immediately understand all the nuances of calculating and collecting interest payments in order to choose the most favorable conditions for himself. Therefore, this article will discuss the choice of repayment scheme. Annuity and differentiated payments are available to individuals.
Differentiated payment (classic loan repayment scheme). What is this?
The amount of the differential payment changes every month, and downward: the first payment is the largest, and the last is the smallest. This name comes from the Latin differentia - “difference, difference.” This repayment scheme is considered classic.
Why are payments different? When drawing up a payment schedule, the entire amount of debt (the body of the loan) is divided into equal parts, the number of which depends on the number of months of lending. As a result, each month accounts for the same “piece” of the principal debt. If the loan amount does not allow creating equal shares based on the number of months, then the remaining indivisible rubles or kopecks are reflected in the last payment.
For each piece of the principal debt, the interest due is added - a bank fee for the service provided to you, usually they are displayed in the adjacent column of the payment schedule. Interest is calculated on the balance of the loan debt. Since the loan body systematically decreases every month, the amount of interest will also decrease. Consequently, the total payment amount will also decrease.
On the one hand, this scheme is more pleasing, because every month you have to pay less and less. On the other hand, it is not very convenient for forgetful citizens who will find it difficult to keep track of the cost of the next payment - they will have to keep the repayment schedule before their eyes.
In addition, the solvency of a potential borrower is calculated in relation to these very first payments. This means that your earnings must exceed the amount of the first payment at least 2 times. And this is not the whim of a particular bank - the law establishes a rule according to which loan payments cannot exceed half of the monthly salary. Otherwise, the bank may refuse to lend or reduce the loan amount, which does not always please borrowers.
In fact, another form of payment is often used - annuity.
What is an annuity payment?
The word annuity is derived from the Latin annuus - “annual, annual”. Such a repayment scheme implies that throughout the entire loan term you will make exactly the same amount of payments every month. This will be the main difference from a differentiated system.
Interest here is also charged on the balance of the debt, but in the first months of repayment it practically does not decrease. The very first payments are mainly interest plus a tiny part of the loan body. Only after a year or two, or maybe more (depending on the loan term), will you begin to repay your principal debt. It is due to this that the equivalence of the amounts contributed is achieved.
This repayment method is attractive from a stability point of view. There is no need to look at the payment schedule every month and clarify the next installment amount, because it is constant. In addition, the first payment is always lower than the first differentiated one, which plays a significant role in determining solvency. With the annuity system, you can get a much larger amount on credit, and this is especially true for those who want to take out a mortgage. This repayment method also has a disadvantage - the overpayment for it is significantly higher compared to the previous method.
So which method is more profitable for the borrower? Let's analyze this below.
We count the benefits
So what is more profitable - annuity or differentiated payment? It all depends on what exactly you are used to calling benefit.
Annuity is beneficial, as we have already said, from the point of view of memorability. With a differentiated payment, the amount is unstable and changes every month. But this, of course, is unimportant.
If we consider the benefit regarding the amount of the loan received, then preference should be given to the annuity repayment scheme. The credit load is distributed evenly, and the borrower will be able to count on a higher loan amount, which is sometimes important!
Differentiated contributions, on the contrary, are characterized by a high credit load in the first months (or even years) of repayment, and only then the reduction in payment will become noticeable. Take the same mortgage - it is unlikely that you will pay the very first installments on it if you choose a differentiated repayment scheme.
The benefit may also depend on the period during which you plan to actually repay the loan. In our country, early repayment is not uncommon. But it will not be profitable if you chose annuity payments during the period of receiving the loan. It turns out that you have already paid the bank a huge amount of interest, but the principal debt has remained practically unchanged. Early repayment in this case will lead to the loss of money precisely on the interest that you paid in advance - in fact, you will repay the loan amount ahead of schedule and gain little. Therefore, with this scheme, it is advisable to repay the loan for the entire planned period.
With differentiated payments, the story is different - the loan body is gradually repaid in equal shares, and early repayment of at least part of the debt reduces the amount of accrued interest and, accordingly, all subsequent payments.
Table 1. Payment of a loan of 1 million rubles in annuity payments
Table 2. Payment of a loan of 1 million rubles in differential payments
Credit term | Bid | Differentiated payment | Overpayment | |
---|---|---|---|---|
First | Last | |||
5 years | 15% | 29167 | 16875 | 381250 |
10 years | 15% | 20833 | 8437 | 756250 |
15 years | 15% | 18056 | 5625 | 1131250 |
20 years | 15% | 16667 | 4219 | 1506250 |
30 years | 15% | 15278 | 2813 | 2256250 |
If we take banal mathematical calculations, then with the same amount, term and lending rate, the overpayment under the annuity system will be higher than under the differentiated one. And sometimes the difference in the overpaid amounts is very, very significant - pay attention to the conditional examples of different repayment schemes for the same amount of 1 million rubles with the same rates (to simplify) and different lending periods.
If you know for sure that you will repay the loan ahead of schedule and are able to pay the first highest installments, then it is better to give preference to differentiated payments.
As you can see, the benefits vary, but there are just a ton of nuances. Therefore, when determining the repayment scheme you need, ask bank employees to make a preliminary printout of payments for the requested loan. Then you will be able to assess your real capabilities and make the only right choice, if the bank can offer it to you.
Annuity is a term that has several different meanings. In the broadest interpretation, it can be represented as a certain instrument that serves to carry out financial activities.
Multiple annuity values
For example, the first meaning that the concept of annuity has is one of the types of government loans, and urgent ones. Such loans can be placed with the condition that interest payments will occur annually, and a certain part of the loan will be repaid.
At the same time, an annuity is cash payments that are equal to each other and are paid to repay loan obligations and interest on it. Such payments are made after a certain period of time.
Annuity concept
Let's look at the concept of annuity in more detail.
Annuity, or, as it is also called, financial annuity, is a generalized term that describes the schedule according to which the repayment of any financial instrument occurs, and the concept of annuity implies the payment of not only some part of the principal debt, but also the payment of remuneration - interest on its use. The main feature of an annuity is that payments in this case are equal to each other and are made at absolutely equal time intervals. The annuity schedule is quite complicated. It differs significantly from a schedule that reflects the payment of the due amount in full and at the end of the term during which the instrument was valid, and from a schedule that reflects the periodic payment of interest only and the process of repaying the amount towards the principal debt at the end of the instrument. There is a special annuity formula. We present it below.
Thus, it can be established that an annuity-type payment in its structure consists of two parts: a part reflecting the principal debt, and a part reflecting the remuneration for the use of loan funds.
Annuity Examples
In the most general sense, an annuity can be understood not only as an instrument of a financial nature, but also as the actual amount of payment, which has a certain frequency, and the type of schedule that reflects the repayment process.
- An annuity is a government term loan of a certain type, for which an annual payment of some part of the principal debt and interest for the use of the loan itself occur.
- Equal cash payments, the payment of which is expected at equal time intervals. Moreover, such payments include the amount used to repay part of the principal debt and the amount used to pay interest.
- The concept of annuity is also used in insurance, in particular in life insurance. In this case, we mean a contract that an individual enters into with an insurance company. Such an agreement gives an individual the right to receive regular payments when an earlier agreed time occurs. For example, after retirement.
- An annuity schedule can also be used to accumulate a specified amount of money by a certain point. In this case, it is assumed that equal deposits are made into a deposit account, against which remuneration is accrued.
Types of Annuities
Annuities can be classified into two types, depending on the time when the first payment occurs:
- If the payment is made at the end of the first period, then such an annuity is called postnumerando.
- If the payment is made at the very beginning of the first period, then such an annuity is called prenumerando.
Still, most often an annuity is a certain way of returning loan funds. Therefore, in this article we will focus on this meaning of this concept.
Today, only a small part of Russian banks prefers to use a different loan repayment scheme. Using the annuity method allows the bank to receive a guaranteed profit. This is due to the fact that the annuity schedule is structured in such a way that the bank first returns interest for the use of credit funds, and only then the credit body, that is, the amount of the principal debt, is paid.
Annuity formula
The formula by which annuity is calculated is quite complex. Her recording has various representations.
One of them: PI = (S * pr/12) / (1 - 1 / (1 + pr/12) N), in this formula:
- Pl - directly represents the annuity payment itself.
- S - total amount of credit funds.
- Pr is the interest rate or annuity coefficient used on the loan.
- N is the total number of periods during which repayment will be made (months are most often used).
Its functions
It is worth noting that throughout the entire period the size of the payment does not change, but its structure differs significantly from the structure of another, the same payment. The payment made in the first month of repayment mainly consists of interest, and payments made towards the end of the payment period mainly consist of the amount used to repay the loan. This is how cash flow management works.
In order to determine what structure a certain payment has, it makes sense to use this particular formula. It clearly reflects the percentage that is included in it. To make this calculation, you need to take the principal balance and multiply it by 1/12 of the annual loan rate.
An example that clearly reflects the method of calculating an annuity
The formula that we gave above will be much clearer if you apply it in practice by analyzing the corresponding example.
Suppose a bank client applies for a loan. The loan amount is one hundred thousand rubles, the loan period is 12 months, the interest rate on the loan in this case is 24 percent per annum. In accordance with the formula, you can calculate what the current value of the annuity will be:
(100000 * 0,24/12)/(1 - 1)/(1 + 0,24/12) 12 = 2000/0,2115 = 9457.
Thus, the client will have to transfer exactly this amount, in the amount of 9,457 rubles, to the bank every month in order to repay the loan.
100000 * 0,24/12 = 2000.
It turns out that as part of the first payment of 9,457 rubles, only 2,000 rubles will go towards paying interest on the loan. Accordingly, an amount of 7457 will be used to repay the principal debt.
After the first payment is made, the amount of the total debt will decrease and amount to 92,543 rubles:
100000 - 7457 = 92543.
From this amount you can calculate the interest part for the next, second, loan payment:
92543 * 0,24/12 = 1851.
This means that the second payment includes interest in the amount of 1851 rubles and the principal debt of 5606 rubles.
It is in this way that the calculation is made for each payment for the entire loan term.
Automatic payment calculation method
Undoubtedly, making such calculations is quite labor-intensive. The formula for calculating annuity can be useful only in order to understand the principles of its calculation. As for practice, it makes no sense to calculate payments using a calculator. Modern technologies make it possible to easily automate the calculation process, which makes cash flow management easier.
When a client applies for a loan from a bank, an employee of the credit institution will make a printout specifically for him, reflecting all the data on the annuity schedule. It will reflect all the necessary data: the payment amount, the dates when payments should be made, as well as the payment structure reflecting the amount of interest and the amount of principal for each payment.
In addition, you can find a special calculator on the Internet. It will be enough to enter in the appropriate fields such data as the total loan amount, its term, and rate. After which the calculator will instantly make the appropriate calculation of the annuity and display all the information of interest: the amount of payment that will have to be made every month and an approximate schedule for repaying the loan.
An office program such as Excel can also make a similar calculation. This program provides a function called PMT - it will help calculate the size of the annuity. But, unfortunately, with this calculation method it is impossible to obtain an approximate repayment schedule.
Pros of an annuity
The annuity method is not always beneficial for the client, although it is convenient. When using an annuity, there will be no confusion with the size of the payment and the timing of its payment, because an annuity always has a fixed amount of payments that must be made monthly. This method will avoid the need to contact the bank every month in order for its employees to calculate the next payment.
This method is convenient if the borrower has a low income.
An alternative scheme, called differential, involves a monthly recalculation of the payment amount. This has to be done because with such a scheme, the amount of the principal debt decreases every month, and accordingly, you have to pay less interest for using a smaller amount. That is, each subsequent payment will be less than the previous one. However, the first payments under such a scheme are very high, and not every borrower can afford this.
Disadvantages of Annuity
During the first half of the term for which the loan is issued, the payment structure contains mainly interest. This is why the annuity scheme is very profitable for banks. It is best to repay the loan ahead of schedule in the first half of the term, since then it makes no practical sense, because most of the interest has already been paid. Repaying the loan early in the second half of the term will not bring benefits to the borrower, since the funds contributed to repay interest on the loan will not be returned.
Annuity indicators
If the annuity is considered from the point of view of the lender, and not the borrower, then it is necessary to evaluate payments to be able to analyze the proceeds.
Few people can use assessments of this kind in everyday life. However, when analyzing and comparing current costs and cash receipts that will occur in the future, they are necessary.
There are two main indicators by which an annuity is assessed. This is the current and future cost.
The future value of an annuity is the sum of absolutely all the elements that make up the annuity. This also includes interest that accrues at the end of the term. The elements, or, as they are also called, members of the annuity, are precisely those equal payments.
This indicator can be used if you need to calculate the amount of the deposit (replenished) that can be accumulated by a certain time if you make a regular deposit of funds at a certain interest rate.
The modern (current) value is a set of annuity elements that are reduced at the time when its implementation began. This indicator is used to assess the feasibility of investing in a certain deposit, which should generate constant and regular income. That is, this assessment allows you to calculate whether future earnings will be higher than the price of the asset itself.
By the way, this assessment can also be used to evaluate what will be more profitable - to make a purchase on credit or pay for it immediately.
In the modern world, where banking products are part of every person's life, understanding the essence of financial mathematics and the ability to do simple financial calculations becomes a necessary skill. But many textbooks and articles on this topic are written in a complex language of financial terms and mathematical formulas. Of course, we cannot do without terms and formulas. However, the essence of calculations can be explained in simple language that anyone can understand. This article is a continuation of the article on discounting cash flows. It will talk about annuity (annuity cash flows). Perpetual annuity, annuity formula - calculation of current and future value using simple examples, explanations for people, not for bankers - you will learn about this by reading this article.
What is an annuity?
Hearing the word annuity, many will think of something super complex and inaccessible to understanding. In fact, everything is simple, only the word is foreign.
An annuity is series identical payments via the same periods of time. This term is a literal "translation" of the English word annuity, which means “fixed sum paid every year”. People who speak English will also remember the word “annual,” which translated means “annual.” Both of these words come from the Latin word annuus– annually. Thus, the word annuity itself contains an indication of the annual frequency of payments.
On a time line (or time scale), annuity cash flows can be depicted, for example, like this (Fig. 1): Currently, an annuity refers not only to a series of identical annual payments, but also to any sequence of payments of the same amount, regardless of their frequency. These can be annual, quarterly, monthly payments. The main thing remains: annuity is some identical payments (cash flows) through the same periods of time. For example, salary. If your salary is constant throughout the year, then the monthly cash flow in the form of salary is an annuity with a monthly payment period. Another example: if you buy something in installments, then your monthly payments to the bank will also be an annuity.
Prenumerando and postnumerando
A few more terms. Annuities can be pre-numerando or post-numerando. These beautiful and mysterious terms just mean the moment of payment: prenumerando means payments at the beginning of each time period, postnumerando- at the end of it. These terms, which apparently came to us from Latin, are used in textbooks or official papers. I will speak in Russian: cash flows with payment at the end of the year or at the beginning of the year.
This article discusses examples of calculating simple annuities in which the payment period and the interest period are equal to each other. That is, if interest is accrued, for example, for a year, then payments will be annual. Or interest is calculated monthly and payments are also made monthly. There are annuities in which these periods are not the same (payment periods and interest periods), but these are more complex calculations. I won't touch on them. Anyone who wants to understand this topic in detail should consult textbooks on financial mathematics.
Discounting and accretion
First, let's remember what discounting and accretion are. This is discussed in more detail in the previous article. It dealt with discounting and increasing a single cash flow, that is, one sum of money. Discounting means calculating the present value of a future cash flow. That is, if you need to save a certain amount by some date in the future, then by using discounting you can calculate how much you need to put in the bank today.
Accumulation is the movement from today to tomorrow: calculating the future value of the money you have today. If you deposit money into a bank account, knowing the bank rate will allow you to calculate how much money you will have in your account at any point in the future.
Compounding and discounting are of course not applicable if you keep your money at home. All these calculations are only valid if you can invest your money: put it in a bank account or buy debt securities.
Discounting and compounding apply not only to one cash flow, but also to a sequence of cash flows, and the cash amounts can be of any size. A special case of such multiple cash flows are annuities.
Annuity formula
Annuity cash flows can also be discounted and increased, that is, their current and future values can be determined.
For example, this is necessary when we need to choose between two options offered to us for receiving money. Without knowing the basic principles of financial mathematics, you can make a mistake and choose an option that is obviously unfavorable for yourself. This is what more knowledgeable participants in the financial market, namely banks, use.
Annuity calculation - discounting
EXAMPLE 1. Let's take an abstract example. Let's say you need to choose which is better:
- (A) receive $100,000 today, or
- (B) 5 times $25,000 at the end of each of the next 5 years.
The total is 5 * 25,000 = 125,000, which seems to be better than $100,000. But is it? After all, money also has a “time” value. The bank rate at the moment in a given country, let’s say, is 10%.
Option (B) is a simple annuity option. But not everyone knows that this is exactly what it is called. To compare these two options with each other (which is more profitable?), you need to bring them to the same point in time, since the value of money at different points in time is different. In this case, it is necessary to discount the annuity cash flow (B), i.e. calculate its today's value. If the discounted value of the annuity is greater than $100,000, then the second option is better at a given interest rate.
In the previous article we learned how to discount a single amount. The same calculations can be done this time, but you will have to repeat them 5 times.
On this time scale, in addition to the payment in the amount of 25,000, discount factors corresponding to each period are plotted. given in the previous article about discounting.
If you discount (that is, bring to the current moment) each amount separately, you will get a table like this:
- 25,000*0,9091 = 22,727
- 25,000*0,8264 = 20,661
- 25,000*0,7513 = 18,783
- 25,000*0,6830 = 17,075
- 25,000*0,6209 = 15,523
- Total: 94,770
Here the payment amount is multiplied by the discount factor corresponding to each year. In total, five payments of 25,000 at the end of each year after discounting are worth 94,770, slightly less than 100,000 today. Therefore, 100,000 today at a rate of 10% will be more profitable than the proposed 5 year annuity at 25,000.
This example is important not only to once again demonstrate the time value of money. From the table it becomes clear how the calculation can be simplified discounted value of the annuity. Instead of discounting each amount separately, you can add up all the discount factors and multiply only once:
25.000*(0.9091+0.8264+0.7513+0.6830+0.6209) which is the same as 25.000* 3,7908 =94,770
From this example it is easy to derive the mathematical formula for calculating the discounted value of an annuity.
First, let's remember what the discounting formula looks like:
PV = FV*1/(1+R) n
The discount factor is 1/(1+R)n- this is 0.9091, 0.8264, etc. in our example.
Annuity formula(to calculate the discounted value of annuity cash flows)
PV = FV*
The expression in square brackets can be represented mathematically, but this is unlikely to be necessary for most people. This is called the annuity factor, or annuity discount factor, the exact name is not so important. In the example above, this coefficient is equal to 3,7908 .
It is much more useful to be able to use tables of such coefficients to calculate the present (discounted) value of an annuity cash flow. Such tables allow you to quickly solve simple annuity discounting problems. An example of such a discounting table is given below:
If anyone needs the exact annuity formula, more precisely the formula for the annuity discount factor, then here it is:
Annuity discount factor: 1/R — 1/(R*(1+R) n)
Discounted value of annuity: PV= payment multiplied by coefficient
Annuity calculation - increment
In the example above, we considered the discounted value of cash flow. That is, they brought the value of cash flow to the current point in time. You can also solve the inverse problem - find out future value of the annuity(annuity cash flow).
EXAMPLE 2. In our first example, we can calculate the future value of both options. If we transfer from the field of pure mathematics to the plane of life, then we need to choose which is better:
- (A) deposit $100,000 in the bank today at 10% interest, or
- (B) at the end of each year make contributions in the amount of 25,000.
For the first option, you can use it (it is in the previous article).
For option (A), the future value is calculated simply: $100,000 in 5 years will be equal to 100,000 * 1.6105 = $161,050
For option (B) the situation is somewhat more complicated. We want to know how much we will have in our account in 5 years if we save 25,000 at the end every year. That is, we will make the last payment and immediately calculate how much we have saved. To avoid mistakes, it is better to sign the increment coefficients corresponding to each year on the time scale. The first payment will be made at the end of the first year, which means that after 5 years it will only accrue interest for 4 years. Accordingly, on the second payment we will receive interest for 3 years, on the third - for two years, on the fourth - for one year, and finally, having deposited the money for the fifth time, interest on the last payment will still arise (that is, it will need to be multiplied by 1.10 to the zero power!)
25,000*(1,1) 4 +25,000*(1,1) 3 + 25,000*(1,10) 2 + 25,000*(1,10) 1 + 25,000 (1,10) 0 which is equal to
25,000*1,4641 + 25,000*1,3310 +25,000*1,2100 +25,000*1,1000 + 25,000*1 = 25,000*6,1051 = 152,628
The future value of the annuity (option B) is equal to $152,628, which is significantly less than $161,050 (option A). This means that it is more profitable to deposit $100,000 into a bank account today than to deposit $25,000 at the end each of the next 5 years. This conclusion is valid for a bank rate of 10% per annum.
To calculate the future value of annuity cash flows, there are also tables of coefficients. In this case, this table can be used to calculate annuities with payments at the end of the time interval (i.e. post-numerando).
For math lovers annuity formula to calculate its future value looks like this:
Annuity growth rate: FV = payment multiplied by coefficient,
where the coefficient is: [(1+R)n – 1]/R
It was an annuity with payments at the end of each year ( postnumerando).
EXAMPLE 3. We can consider another example. How much will we accumulate in a bank account if we deposit 25,000 per beginning every year, not at the end? This will be the so-called prenumerando annuity, let's call it option B. This cash flow can be depicted on a time scale in this way:
As can be seen from the figure, payments of 25,000 are made at the beginning of each annual period. For example, you decide to deposit 25,000 into your bank account every year on January 1st. The first payment will give us 5 years of interest, the second will give us 4 years of interest, the third will give us 3 years of interest, the fourth will give us 2 years of interest and finally the payment made at the beginning of the fifth year will give us one year of interest. I took it from the corresponding table, which can be opened via the link.
25,000*1,6105+25,000*1,4641 +25,000*1,3310 + 25,000*1,2100 + 25,000*1,1000 = 25,000* (1,6105+1,4641+1,3310+1,2100+1,1000) = 25,000*6,7156 = 167,890
Thus, if you start depositing 25,000 each year at the beginning of the annual period and do this for 5 years, then after 5 years the amount in the account will be equal to $167,890 . This option B is more profitable than options A and B, which were discussed earlier.
- Option A - $100,000 deposited today will only accumulate 161,050 in the bank account in 5 years
- Option B - $25,000 deposited at the end of each of the next 5 years will only accumulate $152,628 after 5 years
As can be seen from the last two examples, the moment when payments are made is of great importance: at the beginning or at the end of the period. Therefore, if you need to calculate the discounted or future value of any cash flows, it is advisable to draw, on which you note the amounts and coefficients corresponding to each period.
How can these calculations be useful in life?
In the examples above, abstract examples of annuities were discussed. But we also encounter annuity cash flows in real life. For example, it will be interesting to calculate how much you can accumulate in a savings account if you save part of your salary every month. In a similar way, it will be possible to calculate, say, the discounted value of all payments on a car loan. Payments to the bank when buying a car (and not just a car) on credit constitute an annuity. Its discounted (reduced to today) value will be the cost of the purchased car. You can find out exactly how much you overpay when buying a car on credit compared to buying a car and paying the full amount up front. It will also be possible to compare loan offers from different banks. The only problem with such calculations is choosing the correct monthly discount rate.
Perpetual annuity
A perpetual annuity is an annuity whose payments continue indefinitely. In other words, it is a series of identical payments that continues forever. This option is possible if, for example, you have a deposit in a bank, you withdraw only annual interest, and the principal amount of the deposit remains untouched. Then, if the interest rate on the deposit does not change, you will have the so-called.
In the Victorian era, all English aristocrats lived on interest from their capital. The more capital was in the bank, the more money could be spent on life without having to work. Capital was inherited, and theoretically (if there were no bank failures, wars and inflation) this could continue forever.
The future value of a perpetual annuity is meaningless since payments continue indefinitely. However, the present value of a perpetual annuity is a finite amount that can be calculated using the formula:
PV = payment/R,
where R is the bank rate %, PV is the current value
For example, if you want to withdraw interest from your account in the amount of 500,000 rubles per year, and the annual bank rate is 8%, then this means that the deposit amount in the bank account should be equal to:
500,000/0.08 = 6,250,000 rubles (PV).
In this case (unless the bank’s license is taken away or the bank itself goes bankrupt), you can withdraw such interest continuously for an unlimited period of time. The only thing that can disrupt this idyllic picture is inflation, due to which money depreciates. Therefore, over time, the interest withdrawn will bring less and less material benefits.
A philosophical digression for those who have read this far.
For the rent to be eternal, it is necessary to preserve the capital from which we receive this rent. This law applies not only to the financial world. Humanity lives off natural rent - it uses the planet’s resources, which, unfortunately, are exhaustible. If you take too much from nature, natural rent will dry up. The depletion of the earth's resources is happening before our eyes.
In traditional fishing, fish were caught little by little, but this could go on forever. Industrial cities require fish of a certain variety and quality, which is caught by an industrial fishing fleet. Large ships are only after profit and have no respect for the ocean. Currently, 80% of Europe's fishing grounds are depleted. According to scientists, industrial fishing will disappear by 2050. Fishing “rent” will exhaust itself. How many other resources will humanity have left in 35-50 years?
“The world is large enough to satisfy the needs of every person, but too small to satisfy human greed.” Mahatma Gandhi
Planet Earth is ours the only one house. Do we think about it?
You can calculate your potential income on a deposit yourself, without relying on income calculators that are posted on the websites of banking institutions. This article shows, using specific examples, how to calculate income on a deposit with interest capitalization (quarterly, monthly, daily, continuous) and how to calculate the effective rate on deposits with capitalization.
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