Discursive way of calculating interest. Interest calculation methods
At the heart of any lending operation, that is, transferring money to a borrower from a lender, is the desire to receive income. The absolute value of the income received by the creditor for the transfer of money into debt is called interest money or percent. The origin of this name is due to the fact that the amount of payment for a loan is usually determined as the corresponding percentage (in a mathematical sense) of the loan amount.
The loan fee can be charged both at the end of the loan term and at its beginning (advance interest income). In the first case, interest is charged at the end of the term based on the amount of the amount provided, and the amount of debt is subject to return, along with interest. This method of calculating interest is called decursive. In the second case, interest income is received in advance (paid at the beginning of the term), while the debtor is given the amount reduced by its value, and only the original loan is subject to return at the end of the term. Interest income paid in this way is called discount(i.e., a discount on the loan amount), and the method of calculating interest is antisipative.
In world practice, the decursive method of calculating interest has become more widespread, therefore, the term "decursive" is usually omitted, speaking simply about interest or lending interest. When using antisipative percentages, the full name is used.
Types of interest rates
Let us first consider the decomposition method, when interest is calculated at the end of the loan term. From the quantitative point of view, a credit operation is characterized by the following basic relationship:
Where R- the initial amount (loan amount); I- interest income - the amount of the loan payment; S - the amount to be returned (the full cost of the loan).
Loan payment amount I usually defined as a percentage of the amount of the loan itself - i T. This ratio is called the interest rate, more precisely, the interest rate for the period T:
(1.1.2)
The time period at the end of which interest income is received is also called interest period(the term "conversion period" is often used). The interest rate applies to the entire validity period credit agreement.
Since the terms of loans vary in a wide range (from several days to tens of years), in order to compare the terms of various loans, the interest rate is set in relation to a certain base period. The most common is the annual base period - in this case, they speak of the annual interest rate. If the conversion period coincides with the baseline, then the annual interest rate coincides with actual(1.1.2). If the term of the transaction has a different duration, then the annual interest rate, which serves as the basis for determining the interest rate for the period (actual interest rate), is called nominal. The interest rate for the period is calculated by the formula
Where i- nominal annual interest rate; T- the term of the agreement, after which the loan must be returned together with interest.
If the conversion period fits an integer number of times a year, then the rate for the period is calculated by the formula
Where T = 1 /m; m - the number of periods of interest calculation per year, or the frequency of interest calculation.
The law of increasing at a simple interest rate. Discounting; future and present value of money
Interest income by law simple interest calculated on the assumption that the nominal interest rate does not depend on the period of interest accrual:
Amount S also called the accumulated (accrued) value of the original amount R. Using formulas 1.1.1, 1.1.6, we get:
Where s(T) = l + iT- multiplier (coefficient) of the accumulation, or accumulating multiplier for the period T.
Knowing invested amount R and the interest rate i, it is easy to calculate by the formula (1.1.7) the value S for an arbitrary term of the loan agreement. The growth factor does not depend on the value of the initial amount and shows how many times it has grown initial capital... It is he who characterizes the profitability of a credit operation, allowing you to determine what a single amount will turn into by the end of the term (or after any period of time T). In financial mathematics, it is customary to calculate the results of financial transactions for single amounts, then multiplying the result by the initial value and obtaining the value of the accrued amount.
When working out various kinds of financial transactions, it is often necessary to solve the inverse problem: it is known what amount in the future is needed to obtain a certain result, the desired value is its current value. In other words, the problem is posed as follows: how much should be invested today in order to get a given value after a certain time interval? In this situation, the current value sum of money is a projection of its given future value. Such a projection of the sum from the future to the present is called discounting. The name of the term comes from the word "discount" - price discount promissory note with advance payment of interest for the use of the loan. Discounting and accumulation are mutually opposite processes. The formula for discounting at a simple interest rate is as follows:
(1.1.8)
Where v = 1/(1 + iT) - discount multiplier for the period T.
In the English-language literature, the combination of letters is traditionally used to denote the accrued amount FV (from Future Value of Money - future value of money); to indicate the present value - PV(fromPresent Value of Money is the real value of money).
The terms "accumulation" and "discounting" are also used in a broader sense, as a means of determining any value at some arbitrary point in time, regardless of the specific type of financial transaction involving the accrual of interest. Such a calculation is called bringing the cost indicator to a given point in time. The accrued, or future, value of a monetary amount means the projection of the currently set amount for a certain time interval forward into the future. Discounting is the projection of an amount given at a certain point in time in the future, at a certain time interval back, into the present.
Bringing a sum to a certain point in time consists in multiplying it by a reduction factor, which is equal to either the increment factor when reduced to a future point in time, or a discount factor when reduced to a previous (present) moment in time. It is convenient to combine the beginning of the time scale with the moment in time when the amount is set. Then the accumulation corresponds to the positive part of the time axis, and discounting - the negative. In this case, the reduction factor r (t) can be written as
(1.1.9)
where s (t) = s (T) is the build-up factor; v ( ׀ t ׀ ) = v Т - discount factor; T = ׀ t ׀ - the value of the calculation period (the value of the time interval on the numerical axis, taken modulo).
Dependence of this factor on time, i.e. from the value of the interest accrual period T = ׀ t ׀ defined by formula (1.1.9) is shown in Fig. 1.1.1 for a rate of 30% per annum.
Variable interest rate
Often during the term of the loan agreement, the interest rate changes. In this case, the interest is calculated separately for each period during which the interest rate is constant, and then at the end of the loan period, the interest calculated for the individual periods is summed up.
IN general view at time intervals N, on each of which its own interest rate will be applied, the accrued amount of interest for the entire period will be
where k – the ordinal number of the time interval; i k, Tk – respectively, the nominal interest rate and the duration of the time interval (in years).
Sometimes in the literature there is an assertion that (1.1.10) is the amount of interest accrued in each time period. However, according to the simple interest scheme, the accrual and payment of interest is assumed only after the expiration of the loan agreement; their accrual and addition to the principal amount within the term of the loan is not provided. In this regard, a distinction should be made between the calculation and the calculation of interest. Interest calculation - this is a mathematical operation to determine the amount of interest-bearing money for any time period, as well as for the entire term of the loan agreement. Accrual the same percent - this is specific accounting transaction, as a result of which the payment for the loan must either be transferred to the lender or added to the principal amount. Therefore, it is incorrect to talk about the accrual of interest when the interest rate changes within the loan term (since no accounting operations are carried out in this case); we can only talk about the calculation of interest for a particular period.
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Most business transactions(purchase of fixed assets, purchase / sale valuable papers, leasing, receipt / repayment bank loans, analysis of investment projects, etc.) generate cash flows. The implementation of these operations is accompanied by many payments and receipts Money, forming a cash flow distributed over time.
In this regard, in the process of managing the finances of an enterprise, it becomes necessary to carry out special calculations related to the movement cash flows at different times. Estimating the value of money over time plays a key role in these calculations. The concept of such an assessment is based on the fact that the value of money changes over time, taking into account the rate of profit prevailing on financial market, which is the lending rate or the rate of return on government securities.
The Time Value of Money (TVM) principle has two important implications:
- the need to take into account the time factor, especially when conducting long-term financial transactions;
- incorrect summation of monetary values related to different periods of time.
Consider the individual elements of the methodological toolkit for the value of money.
Percent- the amount of income from the provision of capital in debt or payment for the use of loan capital in all its forms (deposit and credit interest, for bonds and bills).
Simple interest- the amount of income accrued to the principal amount of capital in each interval, for which no further calculations are made.
Compound interest- the amount of income accrued in each interval, which is not paid, but added to the principal amount of capital (contribution) in the subsequent payment period.
Interest rate- specific indicator, according to which in deadlines pay the amount of interest per unit of capital (contribution). In practice, the interest rate expresses the ratio annual amount interest income to the amount of the principal debt.
Future value of money(Future Value, FV) - the amount of money invested at the moment, into which they will turn after a certain period of time, taking into account the selected interest rate.
The real value of money(Present Value, PV) - the amount of future funds (deposit), given taking into account a specific interest rate to date.
Increase in value(compounding) - the process of recalculating the present value of monetary funds (contribution) into their future value in a specific period of time by adding the accrued interest to the initial amount.
Cost discounting(discounting) - the process of bringing the future value of funds (deposit) to their present value by excluding the corresponding amount of interest (discount) from the future amount. Through such a financial transaction, comparability of the present value of the forthcoming cash flows is achieved.
Accrual period- the total period of time during which the process of increasing or discounting the amount of money (deposit) is carried out.
Accrual interval- this is the minimum period after which the interest is accrued.
Decursive way of calculating interest- the way in which interest is calculated at the end of each accrual interval. Their value is determined based on the amount of capital provided. Accordingly, the decursive interest rate is the ratio, expressed as a percentage, of the amount of income accrued over a certain interval to the amount available at the beginning of this interval.
Antisipative method (preliminary) interest calculation Is a method in which interest is calculated at the beginning of each accrual interval. The amount of interest money is determined based on the accrued amount. The interest rate will be the ratio, expressed as a percentage, of the amount of income paid over a certain interval to the amount of the accrued amount received after that interval. The interest rate determined in this way is called the discount rate, or anti-sipative interest.
Increase by simple interest
Simple interest is used in short-term financial transactions, the period of which is less than a year or equal to it.
The accrual at the annual rate of simple interest is carried out according to the formula:
FV = PV (1 + r × n), (1)
where FV is the future value;
PV is the initial cost;
n is the number of periods (years);
r - interest rate.
Example 1
The client made a deposit to the bank in the amount of 10,000 rubles. at 12% per annum for a period of five years. By formula (1) we find:
FV = 10,000 (1 + 0.12 × 5) = 16,000 rubles.
The amount of accrued interest will be 6,000 rubles. (16,000 - 10,000).
If the duration of a short-term operation is expressed in days, then the period of its execution is adjusted as follows:
where t is the number of days of the operation;
B - time base (number of calendar days in a year).
Then the future cost of the operation can be determined:
The time of the deposit (loan) can be calculated either taking into account the exact date in months, or assuming that the estimated duration of any month is 30 days.
As a result, specific calculations for the accrual of interest can be carried out in three ways:
365/365 - the exact number of days of the operation and the actual number of days in a year (exact percentages);
365/360 - the exact number of days of the operation and fiscal year(12 months x 30 days);
360/360 - the approximate number of days of the operation (the month is taken equal to 30 days) and the financial year (ordinary interest).
For the same conditions of interest accrual, the settlement of these options leads to slightly different financial consequences.
Example 2
The joint-stock company received a loan from the bank in the amount of 200 thousand rubles. at 15% per annum for a period from February 15 to April 15. Determine the amount to be returned to the bank.
First, you need to determine the number of days the loan will be used: February 15 is the 46th day of the year, April 15 is the 105th day of the year. Hence the exact term of the loan is 59 days. Then, by formula (3), we find:
Discounting at simple interest
There are two ways to discount.
Mathematical discounting - a method based on solving the problem inverse to determining the future value. The interest rate is used here for settlements.
Taking into account the previously adopted designations, the discount formula at the rate r will look like:
(4)
The bank's income (FV - PV) is called the discount, and the used reduction rate r is called the decursive interest rate.
Example 3
What price will an investor pay for a zero-coupon bond with a par value of RUB 500,000 and a maturity of 270 days if the required rate of return is 20%?
According to the formula (4) when using ordinary interest:
PV = 500 / (1 + 0.2 × 270/360) = 434.78 thousand rubles;
exact percentages:
PV = 500 / (1 + 0.2 × 270/365) = 435.56 thousand rubles.
Bank discounting is applied in bank accounting for promissory notes, with interest accruing on the amount payable at the end of the transaction period. When making calculations, the discount rate d is used:
(5)
When discounting at a discount rate, the most commonly used time base is 360/360 or 360/365. The reduction rate d used in this case is called the antisipative interest rate.
Example 4
A promissory note in the amount of 500 thousand rubles. with a maturity of one year is recorded with the bank after 270 days at a simple discount rate of 20%. How much will the owner of the bill receive?
We use formula (5), taking into account that n is the difference in time between the moment of accounting and the maturity of the bill:
PV = 500 (1 - 0.2 × 90/360) = 475 thousand rubles.
Applying the two discounting methods discussed above to the same amount leads to different results, even when r = d. The discount rate gives a faster reduction in the amount than the usual one.
Example 5
A promissory note in the amount of 100 thousand rubles. with payment after 90 days is recorded in the bank immediately upon receipt. It is necessary to determine the amount received by the holder of the bill at an interest / discount rate of 15%.
When using the interest rate according to the formula (4):
PV = 100 / (1 + 0.15 × 90/360) = 96.39 thousand rubles.
When using the discount rate according to the formula (5):
PV = 100 (1 - 0.15 × 90/360) = 96.25 thousand rubles.
Discount rate d is also applied to accrue at simple interest (for example, when determining the future amount of the contract):
(6)
Let's change the conditions of Example 5 as follows.
Example 6
For what amount should a bill of exchange be issued so that the supplier, having carried out the accounting operation, received the value of the goods (100 thousand rubles) in full, if the discount rate is 15%?
Using the formula (6), we determine the future value (par) of the bill:
FV = 100 / (1 - 0.15 × 90/360) = 103.896 thousand rubles.
The value of the interest rate r or discount rate d can be determined from the relations (1) and (5):
(7)
(8)
Example 7
The short-term liability with a maturity of 90 days was purchased at a price of 98.22 units. from par. It is necessary to determine the profitability of the operation for the investor.
It is (using ordinary percentages):
The term of the operation in days is determined as follows:
Example 8
It is necessary to determine the period of ownership of an obligation worth 98.22 units, repayable at par, if the required rate of return is 7.2%.
Equivalence of interest ratesrandd
Equivalent interest rates- these are the rates different kind, the application of which under the same initial conditions gives the same financial results.
Equivalent interest rates need to be known in cases where there is a possibility of choosing the conditions of a financial transaction and a tool is required to correctly compare different interest rates.
The derivation of the equivalence formulas is based on the equality of the corresponding growth factors:
1 + n × r = (1 - n × d) - 1. (11)
Taking into account formula (11), for operations with a duration of less than a year, the equivalence ratios take the form:
- the time base is the same and is equal to B (360 or 365 days):
- the time base of the rate r is 365 days, and d is 360 days:
Example 9
The due date for the bill is 250 days. At the same time, the simple interest rate is measured with a time base of 365 days, and a simple discount rate - with a time base of 360 days. What would be the return, measured in terms of the simple interest rate, of accounting for the promissory note at a simple discount rate of 10%?
Using formula (14) for r for given time bases, we get:
r = 365 × 0.1 / (360 - 250 × 0.1) = 0.1089, or 10.89%.
Let us assume that the present value of the bill is 100,000 rubles. Then its nominal value according to the formula (3) will be:
Increasing by compound interest
Compound interest is used, as a rule, in financial transactions, the period of which is more than a year. In this case, the basis for calculating interest is both the initial amount of the financial transaction and the amount of interest already accumulated by this time.
The increase in compound interest is as follows:
FV n = PV (1 + r) n. (sixteen)
Compound interest growth implies reinvestment of the income received or capitalization.
Compound interest can be calculated not once, but several times a year. In this case, the nominal interest rate is negotiated j - annual rate, at which the value of the interest rate applied at each accrual interval is determined.
With m equal accrual intervals and the nominal interest rate j, this value is considered equal to j / m. Then, if the term of the financial transaction is n years, the expression for determining the accrued amount (16) will take the form:
With an increase in the number of accrual periods m the future value of FV mn also increases.
Example 10
The initial investment amount is 200 thousand rubles. determine the accrued amount in five years using a compound interest rate of 28% per annum. Solve an example for cases where interest is calculated on half-yearly basis, quarterly.
According to formula (16) for compound interest rates:
FV= 200 (1 + 0.28) 5 = 687.2 thousand rubles.
According to the formula (17) for accrual by half-year:
FV= 200 (1 + 0.28 / 2) 10 = 741.4 thousand rubles.
Using the same formula for the quarterly charge:
FV= 200 (1 + 0.28 / 4) 20 = 773.9 thousand rubles.
If the term of a financial transaction n in years is not an integer, the accumulation factor k is determined by the formula:
k = (1 + r) n a (1 + n b × r), (18)
where n = n a + n b;
n a is an integer number of years;
n b is the remaining fractional part of the year.
In practice, in this case, formula (16) is often used with the corresponding non-integer exponent. However, this method is approximate. The larger the values of the quantities included in the formula, the greater the error in the calculations will be.
Example 11
The initial amount of the debt is 50,000 thousand rubles. It is necessary to determine the accrued amount in 2.5 years using two methods of calculating compound interest at a rate of 25% per annum.
By formula (18) we get:
FV = 50,000 (1 + 0.25) 2 (1 + 0.5 × 0.25) = 87,890.6 thousand rubles.
For the second method, we use formula (16) with a non-integer exponent:
FV = 50,000 (1 + 0.25) 2.5 = 87,346.4 thousand rubles.
Using the approximate method, the lost profit could be about 550 thousand rubles.
If compound interest is calculated several times a year and total number of accrual intervals is not an integer (mn is an integer number of accrual intervals, l is a part of the accrual interval), then expression (17) takes the form:
(19)
For an integer number of calculation periods, the compound interest formula (16) is used, and for the remainder, the simple interest formula (1).
In practice, it is often necessary to compare the terms of financial transactions that provide for different periods of interest accrual. In this case, the corresponding interest rates lead to their annual equivalent using the formula:
The resulting value is called the effective percentage rate (EPR), or comparison rate.
Example 12
For a four-year deposit of 10,000 rubles. compound interest is charged on a quarterly basis at a rate of 2.5%, that is, at the rate of 10% per annum. Will a deposit of 10,000 rubles, invested for the same period at 10%, accrued once a year, be an equivalent investment?
Let's calculate effective rate for both operations:
- quarterly: EPR = (1 + 0.1 / 4) 4 - 1 = 0.103813;
- annually: EPR = (1 + 0.1 / 1) 1 - 1 = 0.10.
Thus, the conditions for placing the amount of 10,000 rubles. on a deposit for a period of four years at 2.5%, accrued quarterly, will be equivalent to an annual rate of 10.3813%. Consequently, the first operation is more profitable for the investor.
If the EPR is known, the nominal interest rate can be determined as follows:
Compound discounting
Consider the use of complex interest rates in mathematical discounting:
If interest will be calculated m times a year, then formula (22) will take the form:
Example 13
The Bank charges interest on the deposited amount at a compound interest rate of 20% per year. What amount should be deposited, provided that the depositor expects to receive 10,000 thousand rubles. after 10 years? It is required to consider two options for calculating interest - annual and quarterly.
With the annual calculation of interest according to the formula (22):
PV = 10,000 / (1 + 0.2) 10 = 1615.1 thousand rubles.
With quarterly interest accrual according to the formula (23):
PV = 10,000 / (1 + 0.2 / 4) 40 = 1,420.5 thousand rubles.
Using a complex discount rate
To calculate the discount operation at a complex discount rate, the following formula is used:
PV n = FV n (1 - d) n. (24)
Example 14
The owner of the bill with a par value of 500 thousand rubles. and with a circulation period of 1.5 years offered it to the bank immediately for accounting, that is, 1.5 years before maturity. The bank agreed to post the promissory note at a compound discount rate of 20% per annum. It is required to determine the discount received by the bank and the amount issued to the owner of the bill.
Using formula (24), we find:
PV = 500 (1 - 0.2) 1.5 = 357.77 thousand rubles.
The bank's discount will be: 500 - 357.77 = 142.23 thousand rubles.
For these conditions, we will determine the amount that the owner of the promissory note would receive if the bank had accounted for the promissory note at a simple discount rate of 20%. For this we use the formula (5):
PV = 500 (1 - 0.2 × 1.5) = 350 thousand rubles.
The bank's discount will be 500 - 350 = 150 thousand rubles.
Thus, it is more profitable for the bank to account for the bill at a simple discount rate.
If discounting at a complex discount rate is performed m times a year, the calculation formula will be as follows:
Example 15
Let's keep the conditions of the previous example, but let the calculation of discounting be done on a quarterly basis, that is, m = 4.
By formula (25) we get:
PV = 500 (1 - 0.2 / 4) 6 = 367.55 thousand rubles.
The bank's discount will be: 500 - 367.55 = 132.45 thousand rubles.
The bank's income with quarterly discounting will be less than with annual discounting, by: 142.23 - 132.45 = 9.78 thousand rubles.
When discounting with interest for periods of less than one year, the term “effective compounding rate” may be used. The effective compound discount rate, equivalent to the compound discount rate for a given value of m, is determined by the formula:
d eff = 1 - (1 - d / m) m. (26)
Example 16
Debt with a par value of RUB 500 thousand. must be repaid in five years. The compound discount rate is 20% per annum. Interest accrual on a quarterly basis. It is required to determine the present value of the liability and the effective discount rate.
Using formulas (25) and (26), we get:
PV = 500 (1 - 0.2 / 4) 20 = 179.243 thousand rubles.
d eff = 1 - (1 - 0.2 / 4) 4 = 0.18549, or 18.549%.
Substituting the value 18.549% into formula (24), we get:
PV = 500 (1 - 0.18549) 5 = 179.247 thousand rubles.
The discrepancy between the values of this amount, calculated using these formulas, are within the accuracy of the calculation.
Determination of the interest rate and the term of the transaction
With the known values of FV, PV and n, the interest rate can be determined by the formula:
Example 17
The amount of 10,000 rubles placed in the bank for four years amounted to 14,641 rubles. It is necessary to determine the profitability of the operation.
By formula (27) we find:
r = (14,641 / 10,000) 1/4 - 1 = 0.1, or 10%.
The duration of the operation is determined by taking the logarithm:
Example 18
The amount of 10,000 rubles placed in the bank at 10% per annum amounted to 14,641 rubles. It is necessary to determine the date of the operation.
By formula (28) we find:
n = log (14,641 / 10,000) / log (1 + 0.1) = 4 years.
Output
The above calculation formulas describe the mechanism of the influence of the time factor on the result of financial transactions. Their use will allow avoiding errors and losses in conditions of decrease purchasing power money.
E. G. Moiseeva,
Cand. econom. Sciences, Arzamas Polytechnic Institute
Concept estimating the value of money over time 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 profit.
In the process of comparing the cost of funds when investing and returning them, it is customary to use two basic concepts - the future value of money and their present value.
Future value of money (S) - the amount of funds invested at the moment, 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.
Present value of money (P) - the amount of future cash receipts, given taking into account a certain interest rate (the so-called "discount rate") to 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 way of calculating interest... Interest is calculated at the end of each accrual interval. Their value is determined based on the amount of capital provided. Decursive interest rate (loan interest) is the ratio of the amount of income accrued over a certain interval to the amount available at the beginning of this interval (P), expressed as a percentage. In world practice, the deccursive method of calculating interest has become the most widespread.
2. Antisipative method(preliminary) interest accrual. Interest is calculated at the beginning of each accrual interval. The amount of interest money is determined based on the accrued amount. The anti-sipation rate (discount rate) is the percentage ratio of the amount of income paid over a certain interval to the amount of the accrued amount received after that interval (S). In developed countries market economy the antisipative method of calculating interest was used, as a rule, during periods of high inflation.
66. Financial planning at the enterprise. To manage is to foresee, i.e. predict, plan. Therefore, an essential element of entrepreneurial economic activity and enterprise management is planning, including financial.
Financial planning is the planning of all income and directions of spending the enterprise's funds to ensure its development. Financial planning is carried out by drawing up financial plans of different content and purpose, depending on the tasks and planning objects. Financial planning is an essential element of the corporate planning process. Every manager, regardless of his functional interests, should be familiar with the mechanics and rationale of executing and controlling financial plans, at least as far as his activities are concerned. The main tasks of financial planning:
Providing the normal reproduction process with the necessary funding sources. At the same time, target sources of financing, their formation and use are of great importance;
Respect for the interests of shareholders and other investors. A business plan with a similar rationale investment project, is for investors the main document that stimulates capital investment;
Guarantee of the fulfillment of the company's obligations to the budget and extrabudgetary funds, banks and other lenders. The capital structure optimal for a given enterprise brings the maximum profit and maximizes payments to the budget with the given parameters;
Identification of reserves and mobilization of resources for the effective use of profits and other income, including non-operating income;
Ruble control over financial condition, solvency and creditworthiness of the enterprise.
The purpose of financial planning is to link revenues to the required expenditures. If income exceeds expenses, the excess amount is sent to reserve fund... If expenses exceed income, the amount of the lack of funds 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, method of optimizing planning decisions, economic and mathematical modeling.
The essence of the regulatory method for 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, norms depreciation charges, standards of the need for working capital, etc.
The essence of the calculation and analytical method for 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 the indicators can be established indirectly, based on the analysis of their dynamics and relationships. This method is based on expert judgment.
The essence of the balance sheet method of planning financial indicators lies in the fact that by constructing balance sheets, a linkage of available financial resources and the actual need for them. The balance method is used primarily in planning the distribution of profits and other financial resources, planning the need for receipts of funds in financial funds - an accumulation fund, a consumption fund, etc.
The essence of the method for optimizing planning decisions is to develop several options for planned calculations in order to choose 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 relationship between financial performance and the factors that determine them. This relationship is expressed through an economic and mathematical model. The economic and mathematical model is a precise mathematical description economic process, i.e. description of the factors characterizing the structure and patterns of changes in this economic phenomenon using mathematical symbols and techniques (equations, inequalities, tables, graphs, etc.). Financial planning can be classified into perspective (strategic), current (annual) and operational. The strategic planning process is a tool to assist in making management decisions... Its task is to provide innovations and changes in the organization to a sufficient extent. There are four main types of management activities in the strategic planning process: resource allocation; adaptation to external environment; internal coordination; organizational strategic foresight. Routine planning system financial activities the firm is based on the developed financial strategy and financial policy for certain aspects of financial activities. Each type of investment is linked to the source of funding. For this, they usually use estimates of education and spending of funds of funds. These documents are necessary to monitor the progress of financing the most important events, to select the optimal sources of replenishment of funds and the structure of investing their own resources.
The current financial plans of an entrepreneurial firm are developed on the basis of data that characterize: financial strategy firms; results financial analysis for the previous period; planned volumes of production and sales of products, as well as other economic indicators the operating activities of the firm; a system of norms and standards for the costs of individual resources developed at the firm; the current taxation system; the current system of rates of depreciation deductions; average rates of credit and deposit interest in the financial market, etc. Operational financial planning consists in drawing up and using a plan and a cash flow statement. The payment calendar is compiled on the basis of the real information base of the company's cash flows. In addition, the company must draw up a cash plan - a plan for the turnover of cash, reflecting the receipt and payment of cash through the cashier.
The accrual of simple rates is applied, as a rule, for short-term lending.
INTRODUCING DESIGNATIONS:
S - accrued amount, p .;
P is the initial amount of debt, p .;
i - annual interest rate (in unit fractions);
n - loan term in years.
At the end of the first year, the accrued debt amount 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 the term n: S1 = P (1+ n i).
This is a formula for building up at a simple interest rate. It must be borne in mind that the interest rate and the term must correspond to each other, i.e. if the annual rate is taken, then the term should be expressed in years (if quarterly, then the term - in quarters, etc.).
The expression in parentheses represents the increment factor at a simple interest rate:
KN = (1+ n i).
Hence,
Si = P Kn.
Task 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 amount to be repaid.
DECISION:
S = 5 million (1 + 0.5 ¦ 0.12) = 5,300,000 rubles.
If the period for which the money is borrowed is set 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 K value is called the time base.
The time base can be taken equal to the actual length of the year - 365 or 366 (then the percentages are called exact) or approximate equal to 360 days (then these are ordinary percentages).
The value of the number of days for which money is borrowed can also be determined accurately or approximately. In the latter case, the duration of any whole month is assumed to be 30 days. In both cases, the date of issue of money in debt and the date of their return is considered one day.
Task 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 repaid amount with different options for the time base with the exact and approximate number of days of the loan and draw a conclusion about the preferred options from the point of view of the bank and the borrower.
DECISION:
The exact number of days of the loan from 12.03. on 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 the loan:
S = 200,000 (1 + 289/366 ¦ 0.07) = 211,016 p.;
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 the loan:
S = 200,000 (1 + 285/366 ¦ 0.07) = 210,863.
Thus, the largest accrued amount will be in option b) - ordinary interest with the exact number of loan days, and the smallest - in option d) - exact interest with an approximate number of loan days.
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).
It should be borne in mind that the lender is in any case more profitable ordinary interest, and the borrower - accurate (at any rates - simple or complex). In the first case, the accrued amount is always higher, and in the second case, it is less.
If interest rates at different intervals of accrual during the term of the debt are different, the accrued amount is determined by the formula
N
S = P (1 + I nt it),
t = 1
where N is the number of interest calculation intervals;
nt is the duration of the t-th accrual interval;
it is the interest rate on the t-th calculation interval.
Task 5.3
The bank accepts deposits at a simple interest rate, which in the first year is 10%, and then every six months it increases by 2 percentage points. Determine the size of the contribution at 50 thousand rubles. with interest in 3 years.
Decision:
S = 50,000 (1 + 0.1 + 0.5 0.12 + 0.5 0.14 + 0.5 0.16 + 0.5 0.18) = 70,000 rubles.
Using the formula for the accrued amount, you can determine the term of the loan under other specified conditions.
Loan term in years:
S - P N =.
P i
Determine the term of the loan 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.
DECISION:
(250,000 - 200,000) / (200,000 0.16) = 1.56 (years).
From the formula for the accrued amount, you can determine the simple interest rate as well as the original amount of the debt.
Decide on your own
Task 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%.
Target 5.6
The loan, issued at a simple rate of 15% per annum, must be repaid in 100 days. Determine the amount received by the borrower and the amount of interest received by the bank, if the amount returned should be 500 thousand rubles. with a time base of 360 days.
ANSWER: 480,000 rubles.
The operation of finding the original amount of debt for a known repayable is called discounting. In a broad sense, the term "discounting" means the determination of the value P of the value at a certain point in time, provided that in the future it will be equal to a given value of S. Such calculations are also called bringing the value indicator to a given point in time, and the value of P, determined by discounting,
called the modern, or reduced, value of the value. Discounting allows you to take into account the factor of time in cost calculations. The discount factor is always less than one.
Simple interest rate discounting formula:
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 in time.
- 2.3. Determination of current and future value of cash flows
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Determination of the unsatisfactory structure of the company's balance sheet according to the criteria of current liquidity, provision with own funds, restoration or loss of solvency
According to the decree of the Government of the Russian Federation of 05.25.94, No. 498, the degree of insolvency of enterprises should be assessed according to three criteria characterizing the unsatisfactory structure of the balance sheet:
1. current liquidity ratio;
2. the ratio of the provision of own funds;
3. coefficient of recovery or loss of solvency.
The basis for recognizing the structure of the company's balance sheet as unsatisfactory, and the company as insolvent, is the fulfillment of one of the following conditions:
The current liquidity ratio at the end of the reporting period has a value of less than 2;
The equity ratio at the end of the reporting period is less than 0.1. Based on these coefficients, the territorial insolvency and bankruptcy agencies of enterprises make the following decisions: On recognizing the structure of the balance sheet as unsatisfactory, therefore, the enterprise is insolvent. On the existence of a real opportunity for the debtor enterprise to restore its solvency. About the existence of a real possibility of loss of solvency of the enterprise, if it will not be able to fulfill its obligations to creditors in the near future These decisions are made regardless of whether the company has external signs of insolvency established by law.
Current liquidity ratio characterizes the general provision of the enterprise with working capital for conducting economic activities and the ability of the enterprise to timely repay urgent liabilities = tech assets / tech liabilities.
Equity ratio characterizes the presence own funds at the enterprise, necessary to ensure its financial stability = (current liabilities-current assets) / total value of tech assets.
Recognition of an enterprise as insolvent does not always mean recognition of it as insolvent, does not entail the onset of civil liability of the owner. This is only recorded in the territorial bankruptcy agency as financial instability.
The normative value of the criteria is established in such a way as to provide measures to prevent the insolvency of an enterprise, as well as to stimulate this enterprise to independently overcome the crisis. If at least one of the above two coefficients does not correspond normative values, the coefficient of recovery of solvency is calculated for the forthcoming period of 6 months. If the current liquidity ratio is greater than or equal to 2, the security ratio is greater than or equal to 0.1, then the solvency loss ratio is calculated for the upcoming period of 3 months.
Solvency recovery rate is determined as the sum of the actual value of the current liquidity of the reporting period and the change in this ratio between the end and the beginning of the period in terms of 6 months.
К1Ф - the actual value of the current liquidity ratio at the end of the reporting period.
К2Ф - the actual value of the current liquidity ratio at the beginning of the reporting period.
T - reporting period in months
2 - the standard of the current liquidity ratio
(for 6 months)> 1, then the company has a real opportunity to restore its solvency in a fairly short period.
If the coefficient of recovery of solvency< 1, то у предприятия нет реальной возможности восстановить свою платежеспособность на данный момент и за достаточно короткий срок.
The loss of solvency ratio is determined:
If the coefficient of loss of solvency (for 3 months)> 1, this indicates that there is a real opportunity for the enterprise to lose solvency.
If there are grounds for recognizing the structure of the balance sheet as unsatisfactory, but if a real opportunity to restore solvency is identified, the territorial bankruptcy agency decides to postpone the decision to recognize the structure of the balance sheet as unsatisfactory, and the enterprise as insolvent for up to 6 months.
If there are no such grounds, then one of two decisions is made:
If the coefficient of recovery of solvency is> 1, then no decision is made to recognize the structure of the balance sheet as unsatisfactory, and the company is insolvent.
If the coefficient of recovery of solvency< 1, тогда решение о признании структуры баланса неудовлетворительной, а предприятие – неплатежеспособным так же не может быть принятым. Однако в виду реальной угрозы утраты платежеспособности оно ставится на учет в territorial body for bankruptcy, but only if the share state enterprises in common ownership over 25%.
A number of enterprises may become insolvent due to the state's debt to this enterprise. In this case, an analysis is made of the dependence of the company's solvency at the moment and the state's debt to the enterprise.
Interest- income from the provision of capital in debt in various forms (loans, credits, etc.), or from investments in production or financial. character.
Interest rate Is a value that characterizes the intensity of interest accrual.
Currently, there are two ways to determine and calculate interest:
Decursive way. Interest is calculated at the end of each accrual interval. Their value is determined based on the amount of the provided capital. Accordingly, the decursive interest rate (interest) is the ratio of the amount of income accrued over a certain interval to the amount available at the beginning of this interval, expressed as a percentage.
Antisipative (preliminary) method. The provisional interest is calculated at the beginning of each accrual interval. The amount of interest money is determined based on the accrued amount. The interest rate will be the ratio, expressed as a percentage, of the amount of income paid over a certain interval to the amount of the accrued amount received after this interval.
The interest rate shows the degree of intensity of the change in the value of money over time. The absolute value of this change is called percent, measured in monetary units(for example, rubles) and denoted I. If we denote the future amount S, and the modern (or original) P, then I = S - P. The interest rate i is a relative value, measured in decimal fractions or%, and is determined by dividing interest by the original amount:
In addition to interest, there is discount rate d (another name is the discount rate), the value of which is determined by the formula:
where D is the amount of the discount.
Comparing formulas (1) and (2), it can be noted that the sum of interest I and the value of the discount D are determined in the same way - as the difference between the future and present values. However, the meaning of these terms is not the same. if in the first case it comes on the increase in the present value, then in the second, the decrease in the future value, the “discount” from its value, is determined. The main area of application of the discount rate is discounting, a process that is inverse to the calculation of interest. Using the rates discussed above, both simple and compound interest can be calculated. When calculating simple interest, the increase in the initial amount occurs in arithmetic progression, and when calculating compound interest - in geometric progression. The accrual of simple decursive and antisipative interest is made according to various formulas:
Decursive percentages: (3)
antisipative percentages:, (4)
where n is the length of the loan, measured in years.
However, the duration of the loan n does not have to be a year or an integer number of years. Simple interest is most often used for short-term transactions. In this case, the problem arises of determining the length of the loan and the length of the year in days. If we denote the length of the year in days with the letter K (this indicator is called time base), and the number of days of using the loan is t, then the designation of the number of full years n used in formulas (3) and (4) can be expressed as t / K. Substituting this expression in (3) and (4), we get:
for decursive percentages: (6)
for antisipative interest:, (7)
The most common combinations of the time base and loan duration (the numbers in parentheses indicate the value of t and K, respectively):
Exact percentages with exact number of days (365/365).
Ordinary (commercial) interest with exact loan duration (365/360).
Ordinary (commercial) interest with approximate loan duration (360/360).
The inverse problem in relation to the calculation of interest is the calculation of the present value of future cash receipts (payments) or discounting. In the course of discounting at the known future value S and the specified values of the interest (discount) rate and duration of the operation, the initial ( modern, reduced or the current) cost P. Depending on which rate - simple interest rate or simple accounting rate - is used for discounting, two types of it are distinguished: mathematical discounting and bank accounting.
The banking accounting method got its name from the financial transaction of the same name, during which commercial Bank redeems from the owner (takes into account) a promissory note or bill of exchange at a price below par before the expiration of the maturity date indicated on this document. The difference between the par and the redemption price forms the bank's profit from this operation and is called the discount (D). To determine the size of the redemption price (and, therefore, the amount of the discount), discounting is applied using the bank accounting method. This uses a simple discount rate d. The redemption price (modern value) of a bill is determined by the formula:
where t is the period remaining to maturity of the promissory note, in days. The second factor of this expression (1 - (t / k) * d) is called the bank accounting discount factor for simple interest.
Mathematical discounting uses a simple interest rate i. Calculations are performed according to the formula:
The expression 1 / (1 + (t / k) * i) is called the discount factor of mathematical discounting by simple interest.
The main area of application of simple interest and discount rates is short-term financial operations which duration is less than 1 year.
Calculations with simple rates do not take into account the possibility of reinvesting the accrued interest, because the accumulation and discounting are made relatively the same initial amount of P or S. In contrast to them compound interest rates take into account the possibility of reinvesting interest, since in this case the increase is made according to the formula not arithmetic, but a geometric progression, the first member of which is the initial amount P, and the denominator is (1 + i). The accrued cost (the last term of the progression) is found by the formula:
(10), where (1 + i) n is the multiplier of the increment of decomposition compound interest.
By itself, the compound interest rate i is no different from the simple one and is calculated using the same formula (1). The compound discount rate is determined by the formula (2). As in the case of simple interest, it is possible to apply a complex discount rate for the calculation of interest (anti-hypothetical method):
, (11) where 1 / (1 - d) ^ n is the multiplier of compound antisipative interest.
An important feature of compound interest is the dependence of the final result on the number of accruals during the year.
In financial calculations, the nominal compound interest rate is usually denoted by the letter j. The formula for increasing compound interest when calculating them m times a year looks like:
When calculating anti-sipative compound interest, the nominal discount rate is denoted by the letter f, and the accrual formula takes the form:
The expression 1 / (1 - f / m) ^ mn is the accrual factor at the nominal discount rate.
Compound discounting can also be performed in two ways - mathematical discounting and bank accounting. The latter is less profitable for the lender than accounting at a simple discount rate, therefore it is used extremely rarely. In the case of a one-time interest calculation, its formula is:
where (1 –d) n - discount multiplier of banking accounting at a complex discount rate.
for m> 1 we obtain
, (16) where f is the nominal compound discount rate,
(1 - f / m) mn - discount multiplier of bank accounting at a complex nominal discount rate.
Mathematical discounting at a compound interest rate i is much more widespread. For m = 1 we get
, (17) where 1 / (1 + i) n is the discount factor of mathematical discounting at a compound interest rate.
With repeated accrual of interest during the year, the mathematical discounting formula takes the form:
, (18) where j is the nominal compound interest rate,
1 / (1 + j / m) mn - discount factor of mathematical discounting at a complex nominal interest rate.