Elements of financial mathematics. Initial amount of money (present, modern, current, reduced) - the amount of capital available at the initial point in time (or the amount of capital invested in the transaction in question) Method of calculating pr
Antisipative method
Anticipatory interest rate (discount rate or anticipatory interest) is the ratio of the amount of income accrued for a certain interval to the accrued amount received at the end of this period. With the anticipatory method, the accumulated amount received at the end of the period is considered the amount of the received credit (loan), which the borrower is obliged to repay. He receives an amount less than the lender's interest income. Thus, interest income (discount) is accrued immediately, i.e. remains with the lender. This operation is called discounting at a discount rate, commercial (banking) accounting.
Discount- income received at the discount rate, as the difference between the amount of the repaid loan and the amount issued: D = F - R.
Simple discount rates
If you enter the notation:
d, % - annual discount rate;
d- relative value of the annual discount rate;
D- the amount of interest money (discount) paid for the period (year);
D- the total amount of interest money (discount) for the entire accrual period;
R - the amount of money issued;
F- amount returned (loan amount);
k n - growth factor;
P - number of accrual periods (years);
d- duration of the accrual period in days;
TO - length of year in days K = 365 (366), then the anticipatory interest rate can be expressed as
Then at
Then (6.20)
Example. The loan is issued for 2 years at a simple discount rate of 10%. The amount received by the borrower P = 4 5,000 rub. Determine the amount returned and the discount amount.
Discount: rub.
Hence the inverse problem.
Example. The loan is issued for 2 years at a simple discount rate of 10%. Calculate the amount received by the borrower and the amount of the discount if you need to return 50,000 rubles.
Discount: rub.
If the accrual period less than a year, That
From here, 
Example. The loan is issued for 182 days of an ordinary year at a simple discount rate of 10%. The amount received by the borrower R = 45,000 rub. Determine the amount returned.
Complex discount rates
If the loan is repaid after several accrual periods, then income can be calculated using the method of complex discount rates.
If you enter the notation:
d c , % - annual discount rate;
d c - relative value of the annual discount interest rate;
f - the nominal discount rate of compound interest used in the interval calculation of the discount, then when calculating the accrued amount but at the end of the first period, the accrued amount
At the end of the second period 
Through P years, the accumulated amount will be . (6.23)
Then the increase coefficient is . (6.24)
Example. The loan is issued for 3 years at a compound discount rate of 10%. The amount received by the borrower P = 43,000 rub. Determine the amount returned and the discount amount.
P is not an integer, then the increase coefficient can be represented as follows:
(6.25)
Where p = p c + d/K - the total number of accrual periods (legs), consisting of integer and non-integer accrual periods; p c D- number of days of non-integer (incomplete) accrual period; K = 365 (366) - number of days in a year; d c - relative value of the annual discount interest rate.
Example. The loan is issued for 3 years 25 days at a complex discount rate of 10%. The amount received by the borrower P = 45,000 rub. Determine the refundable amount and the discount amount.
Discount amount D = F - P = 62,151 - 45,000 = 17,151 rubles.
If the discount rate during periods nv ..., n N different d 1 d 2 , ..., d N , then the formula of the accrued amount takes the form
Example. The loan is issued at a complex discount rate of 10.9.5.9%. The amount received by the borrower, P = 45,000 rubles. Determine the amount returned.
When interest is calculated at intervals during the period m times the formula of the accrued amount
Example. The amount received by the borrower is 10,000 rubles. issued for 3 years, interest is accrued at the end of each quarter at a nominal rate of 8% per annum. Determine the amount to be refunded.
If the number of compounding periods N is not an integer, then the increase coefficient can be represented as
(6.28)
Where p c - the number of whole (full) periods (years) of accrual; T - number of accrual intervals in the period; R - the number of whole (full) accrual intervals, but less than the total number of intervals in the period, i.e. R<т; d - the number of days of accrual, but less than the number of days in the accrual interval.
Example. The loan is issued for 3 years 208 days (183 + 25 days) at a compound discount rate of 10%. Payment by half-year (T = 2). The amount received by the borrower R = 45,000 rub. Determine the amount returned and the discount amount.
In addition, you can define other parameters:
(6.30)
Inverse problem:
Example. The loan is issued for 3 years at a compound discount rate of 10%. The amount to be returned is F= 45,000. Determine the amount received by the borrower.
Today it is not enough to calculate simple or complex interest; not a single bank uses them in their pure form. It is more profitable for banks to use not only different types of interest calculations, but also different calculation concepts, which in turn strongly depend on the terms of the contracts. Let's consider the main method (concept) of calculating interest rates, this is the method of decursive calculation of interest.
Today, this is the most common method of calculating interest, which is used in world practice. The basis of this concept is “from present to future”, where at the end of a specified time interval interest is accrued or accrued interest is paid on the base deposit. For decursive interest calculation, both a simple interest calculation and an accrual rate are used, in other words, a complex interest calculation is used. Below is a graphical display of the income on the deposit depending on the chosen method of interest calculation and its term.
In the case of low interest rates, the decursive method is more beneficial to the borrower than to the lender. And this method is best used for short-term financial transactions. Moreover, it is advisable to invest for a period of no more than a year, with interest payments at the end of each time interval. Ideally, the decursive method is used when it coincides with the interest calculation interval. However, this does not mean that decursive interest cannot be used in any other cases. It all depends on the agreement of the parties involved in the financial transaction.
Stay up to date with all the important events of United Traders - subscribe to our
Concept estimates of the time value of money plays a fundamental role in the practice of financial computing. It predetermines the need to take into account the time factor in the process of carrying out any long-term financial transactions by assessing and comparing the cost of money at the beginning of financing with the cost of money when it is returned in the form of future profits.
In the process of comparing the value of money when investing and returning it, it is customary to use two basic concepts - the future value of money and its present value.
The future value of money (S) is the amount of funds currently invested into which they will turn after a certain period of time, taking into account a certain interest rate. Determining the future value of money is associated with the process of increasing this value.
The present value of money (P) is the sum of future cash receipts, given taking into account a certain interest rate (the so-called “discount rate”) for the present period. Determining the present value of money is associated with the process of discounting this value.
There are two ways to determine and calculate interest:
1. Decursive method of calculating interest. Interest is calculated at the end of each accrual interval. Their value is determined based on the amount of capital provided. The decursive interest rate (loan interest) is the ratio, expressed as a percentage, of the amount of income accrued for a certain interval to the amount available at the beginning of this interval (P). In world practice, the decursive method of calculating interest is most widespread.
2. Antisipative method(preliminary) interest calculation. Interest is calculated at the beginning of each accrual interval. The amount of interest money is determined based on the accrued amount. The anticipatory rate (discount rate) is the ratio, expressed as a percentage, of the amount of income paid for a certain interval to the amount of the accrued amount received after this interval (S). In countries with developed market economies, the anticipatory method of calculating interest was used, as a rule, during periods of high inflation.
66. Financial planning in an enterprise. To manage means to foresee, i.e. predict, plan. Therefore, the most important element of entrepreneurial economic activity and enterprise management is planning, including financial planning.
Financial planning is the planning of all income and areas of spending of an enterprise's funds to ensure its development. Financial planning is carried out through the preparation of financial plans of different contents and purposes, depending on the objectives and objects of planning. Financial planning is an important element of the corporate planning process. Every manager, regardless of his functional interests, must be familiar with the mechanics and meaning of the implementation and control of financial plans, at least as far as his activities are concerned. Main tasks of financial planning:
Providing the normal reproductive process with the necessary sources of financing. At the same time, targeted sources of financing, their formation and use are of great importance;
Respect for the interests of shareholders and other investors. A business plan containing such a justification for an investment project is the main document for investors that stimulates capital investment;
Guarantee of fulfillment of the enterprise’s obligations to the budget and extra-budgetary funds, banks and other creditors. The optimal capital structure for a given enterprise brings maximum profit and maximizes payments to the budget under given parameters;
Identification of reserves and mobilization of resources in order to effectively use profits and other income, including non-operating ones;
Ruble control over the financial condition, solvency and creditworthiness of the enterprise.
The purpose of financial planning is to link income with necessary expenses. If income exceeds expenses, the excess amount is sent to the reserve fund. When expenses exceed income, the amount of the lack of financial resources is replenished by issuing securities, obtaining loans, receiving charitable contributions, etc.
Planning methods are specific methods and techniques for calculating indicators. When planning financial indicators, the following methods can be used: normative, calculation and analytical, balance sheet, method of optimizing planning decisions, economic and mathematical modeling.
The essence of the normative method of planning financial indicators is that, on the basis of pre-established norms and technical and economic standards, the need of an economic entity for financial resources and their sources is calculated. Such standards are tax rates, rates of tariff contributions and fees, depreciation rates, standards for the need for working capital, etc.
The essence of the calculation and analytical method of planning financial indicators is that, based on the analysis of the achieved value of the financial indicator taken as the base, and the indices of its change in the planning period, the planned value of this indicator is calculated. This planning method is widely used in cases where there are no technical and economic standards, and the relationship between indicators can be established indirectly, based on an analysis of their dynamics and connections. This method is based on expert assessment
The essence of the balance sheet method of planning financial indicators is that by constructing balance sheets, a link is achieved between the available financial resources and the actual need for them. The balance sheet method is used primarily when planning the distribution of profits and other financial resources, planning the need for funds to flow into financial funds - an accumulation fund, a consumption fund, etc.
The essence of the method for optimizing planning decisions is to develop several options for planning calculations in order to select the most optimal one.
The essence of economic and mathematical modeling in planning financial indicators is that it allows you to find a quantitative expression of the relationships between financial indicators and the factors that determine them. This connection is expressed through an economic-mathematical model. An economic-mathematical model is an accurate mathematical description of the economic process, i.e. description of factors characterizing the structure and patterns of change in a given economic phenomenon using mathematical symbols and techniques (equations, inequalities, tables, graphs, etc.). Financial planning can be classified into long-term (strategic), current (annual) and operational. The strategic planning process is a tool that helps in making management decisions. Its task is to ensure innovation and change in the organization to a sufficient extent. There are four main types of management activities within the strategic planning process: resource allocation; adaptation to the external environment; internal coordination; organizational strategic foresight. The system of current planning of the company's financial activities is based on the developed financial strategy and financial policy for individual aspects of financial activities. Each type of investment is linked to a source of financing. To do this, they usually use estimates of the formation and expenditure of funds. These documents are necessary to monitor the progress of financing the most important activities, to select optimal sources of replenishment of funds and the structure of investment of own resources.
The current financial plans of an entrepreneurial company are developed on the basis of data that characterizes: the financial strategy of the company; results of financial analysis for the previous period; planned volumes of production and sales of products, as well as other economic indicators of the company’s operating activities; a system of norms and standards for the costs of individual resources developed by the company; the current tax system; the current system of depreciation rates; average lending and deposit interest rates on the financial market, etc. Operational financial planning involves creating and using a cash flow plan and statement. The payment calendar is compiled on the basis of the real information base of the enterprise's cash flows. In addition, the enterprise must draw up a cash plan - a cash turnover plan that reflects the receipt and payment of cash through the cash register.
Basic concepts and definitions of financial mathematics:
Interest– income from the provision of capital in debt in various forms (loans, credits, etc.), or from investments of an industrial or financial nature.
The initial amount of money (present, modern, current, reduced) is the amount of capital available at the initial point in time (or the amount of capital invested in the operation in question).
Interest rate– a value characterizing the intensity of interest accrual.
Extension (compounding)– an increase in the original amount of money by adding accrued interest.
Accrued (future) amount of money– the original amount of money plus accrued interest.
Discounting– determination of the current financial equivalent of a future monetary amount (bringing a future monetary amount to the present time).
Increment factor– a value showing how many times the initial capital has grown.
Accrual period– the period of time during which interest is calculated. It can be expressed in days or years, and can be either an integer or a non-integer.
Accrual interval– the minimum period of time after which interest is calculated. An accrual period can consist of one or more equal accrual intervals.
Time base for calculating interest T - the number of days in a year used to calculate interest. Depending on the method of determining the duration of a financial transaction, either exact or ordinary interest is calculated.
The following options are possible:
There are several ways to calculate interest and, accordingly, several types of interest rates. Depending on the accrual method used, financial results can vary quite significantly. In this case, the difference will be greater, the greater the invested capital, the applied interest rate and the duration of the accrual period.
The following diagram gives a general idea of the different methods of calculating interest:
Interest calculation methods |
||||||||||||||||||||||
Decursive |
Antisipative |
|||||||||||||||||||||
|
Simple p/s |
Complex p/s |
Simple p/s |
Complex p/s |
|||||||||||||||||||
|
Accrualn times a year |
Continuous Interest |
|||||||||||||||||||||
The most common is decursive method of calculating interest. With this method the interest I accrued at the end of each accrual interval. Their value is determined based on the amount of capital provided P. Decursive interest rate (loan interest) i represents the ratio, expressed as a percentage, of the income accrued for a given interval (percentage) to the amount available at the beginning of this interval. The interest rate characterizes the intensity of interest accrual.
This incremental operation corresponds to the following mathematical expression:
S = P + I = P + iP = P (1 + i)
The inverse of this operation is the operation discounting, i.e. determining the current value P equivalent to the future amount S:
P = S / (1 + i)
From the point of view of the concept of time value of money, for a given interest rate, the amount P And S are equivalent, we can also say that the sum P is current financial equivalent future amount S.
At antiseptic(preliminary) method, interest is calculated at the beginning of each accrual interval. The amount of interest money is determined based on the amount of future money. Anticipatory interest rate (discount rate) d there will be a percentage ratio of the amount of accrued income to the future amount of money.
In this case, the formula for determining the amount of the accrued amount is as follows:
S = P + I = P / (1 - d)
Accordingly, for the discounting operation, called in this case bank accounting:
P = S (1 - d)
In practice, anticipatory interest rates are usually used when discounting bills of exchange. The interest income received in this case is called a discount - a discount on the future amount.
With both calculation methods, interest rates may be simple, if they apply to the same initial monetary amount throughout the accrual period, and complex, if after each interval they are applied to the amount of the initial capital and interest accrued for the previous intervals.
Formulas for determining the future amount of money for various options for calculating interest for a period n years:
S = P (1 + ni) - for the occasion simple decursive interest
S = P (1 + i) n - for the occasion compound decursive interest
S = P / (1 - nd) - for the occasion simple anticipatory interest
S = P / (1 - d) n - for the occasion compound anticipatory interest
If the accrual period is expressed in days, the simple interest formulas will take the form:
S = P (1 + t/T i)
S = P / (1 – t/T d),
where t is the duration of the accrual period.
Multipliers showing how many times the future amount of money is greater than the amount of initial capital are called accumulation factors. The inverse of the accumulation factors are discount factors, which make it possible to determine the current financial equivalent of a future monetary amount.
In some cases, when analyzing the performance of various financial transactions, it may be useful to determine equivalent interest rates. Equivalent interest rates– these are interest rates of different types, the application of which under the same initial conditions gives the same financial results. In this case, the same initial conditions mean the same amount of initial capital and equal periods for accrual of income. Based on this, one can draw up equivalence equation and derive the ratio for the rates in question.
For example, for simple lending and discount rates such ratios will look like this:
d = i / (1 + ni); i = d / (1 - nd).
The lending rate equivalent to the discount rate reflects the profitability of the corresponding accounting transaction and is useful when comparing the profitability and efficiency of various financial instruments.
Accounting for inflation in financial calculations
Inflation is characterized by a decrease in the purchasing power of the national currency and a general increase in prices. The inflation process affects different participants in a financial transaction differently. Thus, if a lender or investor may lose part of the planned income due to depreciation of funds, then the borrower has the opportunity to repay the debt with money of reduced purchasing power.
In order to avoid errors and losses, inflationary effects must be taken into account when planning financial transactions.
Let us denote by S a the amount whose purchasing power, taking into account inflation, is equal to the purchasing power of the amount S in the absence of inflation. Inflation rate a is the relationship between the inflationary change of a certain value for a certain period and its initial value, expressed as a percentage (a relative indicator is used in calculations):
a= (Sa- S) / S 100%
From here: Sa = S (1 +a)
This means that at an inflation rate of a, prices rise over the period by (1 + a) times. The multiplier (1 + a) is called the inflation index I a.
If the period under consideration consists of several intervals, at each of which the inflation rate is a value, prices as a whole will increase by a factor of (1 + a) n. The overall result is expressed by the following ratio:
Sa= S (1 + a) n
This leads to the first important conclusion regarding the inflation process:
Inflationary growth is similar to the increase in initial capital according to the rule of compound interest. Only in this case we do not receive income, but lose it.
Another useful consideration is calculating the rate of return that could offset inflationary losses and provide capital gains.
Let a be the annual inflation rate,
i – desired profitability of a financial transaction (cleared of the influence of inflation)
i a - rate of return compensating for inflation.
Then for the increased amount S, which under inflation conditions will turn into the amount S a, we can write the following expression:
S a = P (1 + i) (1 + a)
The same result can be obtained in another way:
S a = P (1 + i a)
Equating the right-hand sides of the written equalities, we obtain an expression for calculating i a:
ia = i + a + ia
This is the well-known formula of I. Fisher, in which the quantity (a + i a) is "inflation premium" - a necessary addition to compensate for the impact of inflation.
Now we can formulate the second important conclusion:
To calculate the interest rate that compensates for inflation, to to the required rate of return it is necessary to add not only the value of the level inflation, but also the productia.
In real practice, a modification of this formula often turns out to be useful, allowing one to find the real profitability of an operation in conditions of inflationary price increases:
i = (ia - a) / (1 + a)
Most transactions related to capital investment imply in the future not a lump sum receipt of an increased amount, but a whole cash flow of income over a certain period. The main parameters of interest to the investor or lender in this case are the current (present) value of the cash flow, its future (increased) value, as well as the profitability of the financial transaction.
We will use the following notation:
P – the amount of invested capital,
CF k – value of the kth element of cash flow,
i – discount rate (usually a compound interest rate),
A – present value (cost) of cash flow,
S – future value of cash flow,
n – number of cash flow elements.
Present value cash flow is the sum of all its elements reduced (discounted) to the present time:
A = CF 1 / (1 + i) + CF 2 / (1 + i)? + … + CF n / (1 + i) n
Likewise, future value cash flow is the sum of its accrued elements at the time of the last payment:
S = CF 1 (1 + i) n-1 + CF 2 (1 + i) n- ? + … + CF n
Profitability of a financial transaction This is called a decursive interest rate, when discounted at which the present value of the cash flow of income coincides with the amount of invested capital: P = A. To find such a rate, in the general case, you have to solve an equation of the nth degree.
The values of the accumulation and discounting factors in the case of using complex decursive rates can be found in the special tables given in the appendix.
To determine the profitability of a short-term financial transaction (less than one year), a simple interest rate is usually used; for a long-term transaction, a complex one is used.
The calculation of simple rates is usually used for short-term lending.
LET'S INVENT THE NOTATION:
S - accumulated amount, rub.;
P - initial amount of debt, rub.;
i - annual interest rate (in fractions of a unit);
n is the loan term in years.
At the end of the first year, the accumulated amount of debt will be
S1 = P + P i = P (1+ i);
at the end of the second year:
S2 = S1 + P i = P (1+ i) + P i = P (1+ 2 i); at the end of the third year:
S3 = S2 + Pi = P (1+ 2 i) + P i = P (1+3 i) and so on. At the end of term n: S1 = P (1+ n i).
This is the formula for compounding at a simple interest rate. It must be borne in mind that the interest rate and term must correspond to each other, i.e. if an annual rate is taken, then the term must be expressed in years (if quarterly, then the term must be expressed in quarters, etc.).
The expression in parentheses represents the compounding factor at the simple interest rate:
KN = (1+ n i).
Hence,
Si = P Kn.
Problem 5.1
The bank issued a loan in the amount of 5 million rubles. for six months at a simple interest rate of 12% per annum. Determine the repayable amount.
SOLUTION:
S = 5 million (1 + 0.5 ¦ 0.12) = 5,300,000 rub.
If the period for which money is borrowed is specified in days, the accumulated amount will be equal to S = P (1 + d/K i),
where d is the duration of the period in days;
K is the number of days in a year.
The value K is called the time base.
The time base can be taken equal to the actual length of the year - 365 or 366 (then the interest is called exact) or approximate, equal to 360 days (then it is ordinary interest).
The value of the number of days for which money is borrowed can also be determined exactly or approximately. In the latter case, the length of any whole month is taken to be 30 days. In both cases, the date of issuing the money as a loan and the date of its return is counted as one day.
Problem 5.2
The bank issued a loan in the amount of 200 thousand rubles. from 12.03 to 25.12 (leap year) at a rate of 7% per annum. Determine the size of the repayable amount with various options for the time base with the exact and approximate number of days of the loan and draw a conclusion about the preferable options from the point of view of the bank and the borrower.
SOLUTION:
Exact number of days of loan from 12.03. until 25.12:
20+30+31+30+31+31+30+31+30+25=289.
Approximate number of loan days:
20+8-30+25=285;
a) Exact interest and exact number of days of loan:
S =200,000 (1+289/366 ¦ 0.07) = 211,016 rubles;
b) ordinary interest and the exact number of days of the loan:
S =200,000 (1+289/360 ¦ 0.07) =211,200;
c) ordinary interest and the approximate number of days of the loan:
S= 200,000 (1+285/360 ¦ 0.07) =211,044;
d) exact interest and approximate number of days of loan:
S= 200,000 (1+285/366 ¦ 0.07) =210,863.
Thus, the largest accumulated amount will be in option b) - ordinary interest with the exact number of days of the loan, and the smallest - in option d) - exact interest with an approximate number of days of loan.
Therefore, from the point of view of the bank as a lender, option b) is preferable, and from the point of view of the borrower, option d) is preferable.
It must be borne in mind that, in any case, ordinary interest is more profitable for the lender, and exact interest is more profitable for the borrower (at any rate - simple or complex). In the first case, the accumulated amount is always greater, and in the second case, less.
If interest rates at different accrual intervals during the debt term are different, the accrued amount is determined by the formula
N
S = P (1 + Intit),
t=1
where N is the number of interest calculation intervals;
nt - duration of the t-th accrual interval;
it is the interest rate at the t-th accrual interval.
Problem 5.3
The bank accepts deposits at a simple interest rate, which in the first year is 10%, and then increases by 2 percentage points every six months. Determine the amount of the deposit in 50 thousand rubles. with interest after 3 years.
Solution:
S = 50,000 (1 + 0.1 + 0.5 0.12 + 0.5 0.14 + 0.5 0.16 + 0.5 0.18) = 70,000 rub.
Using the formula for the accrued amount, you can determine the loan term under other specified conditions.
Loan term in years:
S - P N = .
P i
Determine the loan term in years for which the debt is 200 thousand rubles. will increase to 250 thousand rubles. when using a simple interest rate - 16% per annum.
SOLUTION:
(250,000 - 200,000) / (200,000 0.16) = 1.56 (years).
From the formula for the accumulated amount, you can determine the simple interest rate, as well as the original amount of debt.
Decide for yourself
Problem 5.5
When issuing a loan 600 thousand rubles. it is agreed that the borrower will return 800 thousand rubles in two years. Determine the interest rate used by the bank.
ANSWER: 17%.
Problem 5.6
The loan, issued at a simple rate of 15% per annum, must be repaid after 100 days. Determine the amount received by the borrower and the amount of interest money received by the bank if the amount to be returned should be 500 thousand rubles. with a time base of 360 days.
ANSWER: 480,000 RUR.
The operation of finding the original amount of debt against a known repayment amount is called discounting. In a broad sense, the term “discounting” means determining the value P of a cost value at a certain point in time, provided that in the future it will be equal to a given value S. Such calculations are also called bringing a cost indicator to a given point in time, and the value P determined by discounting is
called the modern, or reduced, value of the value. Discounting allows you to take into account the time factor in cost calculations. The discount factor is always less than one.
Discount formula at a simple interest rate:
P = S / (1 + ni), where 1 / (1 + ni) is the discount factor.
More on the topic Decursive method of calculating simple interest:
- 1. Concept and methodological tools for assessing the value of money over time.
- 2.3. Determination of current and future cash flows
- Copyright - Advocacy - Administrative law - Administrative process - Antimonopoly and competition law - Arbitration (economic) process - Audit - Banking system - Banking law - Business - Accounting - Property law - State law and administration - Civil law and process - Monetary law circulation, finance and credit - Money - Diplomatic and consular law - Contract law - Housing law - Land law - Electoral law - Investment law - Information law - Enforcement proceedings - History of state and law - History of political and legal doctrines - Competition law - Constitutional law - Corporate law - Forensic science - Criminology -
