Discursive method of calculating interest. Interest Calculation Methods
The basis of any credit operation, that is, the transfer of money in debt to the borrower from the lender, is the desire to receive income. The absolute amount of income received by the lender for the transfer of money in debt is called interest money or percent. The origin of this name is due to the fact that the amount of the loan payment is usually determined as the corresponding percentage (in the mathematical sense) of the loan amount.
A loan can be paid at the end of the loan term or at the beginning (advance interest income). In the first case, interest is accrued at the end of the term based on the amount of the amount provided, and the amount of the debt, together with the interest, must be returned. This method of calculating interest is called discursive.In the second case, interest income is paid in advance (paid at the beginning of the term), while the debtor is given an amount reduced by its amount, and only the original loan is subject to repayment at the end of the term. Interest income paid in this way is called discount(i.e., a discount on the amount of the loan), and the method of calculating interest - 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 loan interest. When using antisipative percentages, the full name is used.
Types of Interest Rates
Consider first the discursive method, when interest is accrued at the end of the loan term. On the quantitative side, a credit operation is characterized by the following basic relationship:
where R- initial amount (loan amount); I - interest income - the amount of the loan fee; S - amount to be repaid (total cost of the loan).
Amount of loan fee 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 falls is also called interest calculation period(the term "conversion period" is often found). The interest rate refers to the entire period of the loan agreement.
Since the terms of loans vary over a wide range (from several days to tens of years), to compare the conditions of various loans, the interest rate is set in relation to a certain base period. The most common annual base period - in this case, they talk about the annual interest rate. If the conversion period coincides with the base, then the annual interest rate coincides with actual(1.1.2). If the transaction period 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 par.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 repaid 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 \u003d 1 /m; m - the number of interest calculation periods in a year, or the frequency of interest calculation.
The law of accrual at a simple interest rate. Discounting future and current value of money
Interest income under the law of simple interest is calculated on the basis that the nominal interest rate does not depend on the interest calculation period:
Amount S also called the accumulated (accumulated) value of the original amount R.Using formulas 1.1.1, 1.1.6, we obtain:
where s(T) = l + iT- accumulation factor (coefficient), or accumulation factor for the period T.
Knowing the amount invested Rand interest rate i, it is easy to calculate by formula (1.1.7) the value S for an arbitrary term loan agreement. The accumulation factor does not depend on the size of the initial amount and shows how many times the initial capital has grown. It is he who characterizes the profitability of a credit operation, allowing you to determine what the unit amount will turn into at 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 unit amounts, then multiplying the result by the initial value and obtaining the value of the accumulated amount.
When working out various kinds of financial transactions, one often has to solve the inverse problem: it is known how much in the future is needed to obtain some result, the desired value is its current value. In other words, the task is as follows: what amount should be invested today in order to receive a given value after a certain time interval? In this situation, the present value of the monetary amount is a projection of its predetermined future value. This projection of the amount from the future is now called discounting.The name of the term comes from the word "discount" - a discount on the price of a debt obligation in case of advance payment of interest for using a loan. Discounting and building are mutually inverse processes. The simple interest rate discount formula is as follows:
(1.1.8)
where v = 1/(1 + iT) - discount factor for the period T.
In the English language literature, the combination of letters is traditionally used to indicate the accumulated amount Fv (from Future Value of Money - future value of money); to indicate fair value - PV (fromPresent Value of Money - the true value of money).
The terms “accrual” and “discount” are also used in a broader sense, as a means of determining any value at a certain arbitrary point in time, regardless of the particular type of financial transaction involving interest. This calculation is called bringing the cost indicator to a given point in time. The accumulated, or future, value of the sum of money means the projection of the amount currently set for a certain time interval forward, into the future. Discounting is the projection of the amount specified at some point in time in the future, to a certain time interval back, in the present.
Bringing a sum to a certain point in time consists in multiplying it by a cast factor, which is equal to either the build-up multiplier when converting to a future point in time, or a discount factor when casting to a previous (present) moment in time. It is convenient to combine the beginning of the timeline with the point in time when the sum is specified. Then the buildup corresponds to the positive part of the time axis, and the discount to the negative. In this case, the reduction factor r (t) can be written as
(1.1.9)
where s (t) \u003d s (T) is the accumulation factor; v ( ׀ t ׀ ) \u003d v Т - discount factor; T \u003d ׀ t ׀ - the value of the calculation period (the value of the time interval on the numerical axis, taken modulo).
The dependence of this factor on time, i.e. from the value of the interest accrual period T \u003d ׀ 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, interest is calculated separately for each period during which the interest rate is constant, and then at the end of the loan term the interest calculated for individual periods is added up.
In general form at time intervals N, each of which will have its own interest rate, the accrued interest for the whole period will be
where k – sequence number of the time interval; i k, T k – accordingly, the nominal interest rate and the duration of the time interval (in years).
Sometimes in the literature there is a statement that (1.1.10) is the sum of interest accrued in each time period. However, according to the simple interest scheme, interest is accrued and paid only upon expiration of the loan agreement; their accrual and addition to the amount of the principal debt within the loan term is not provided. In this regard, a distinction should be made between the calculation and calculation of interest. Interest calculation - it 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. Accrualsame percent - this is a specific accounting operation, as a result of which the loan fee should be either transferred to the creditor or added to the amount of the principal debt. Therefore, it is incorrect to talk about interest accrual when the interest rate changes within the loan term (since no accounting operations are carried out in this case); we can only talk about calculating interest for a given period.
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Most business operations (acquisition of fixed assets, purchase / sale of securities, leasing, obtaining / repayment of bank loans, analysis of investment projects, etc.) generate cash flows. The implementation of these operations is accompanied by many payments and cash receipts, forming a cash flow distributed over time.
In this regard, in the process of financial management of the enterprise there is a need for special calculations related to the movement of cash flows in different periods of time. The key role in these calculations is played by the time value of money. 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 in the financial market, which is the rate of loan interest or the rate of return on government securities.
The Time Value of Money (TVM) principle has two important consequences:
- the need to take into account the time factor, especially when conducting long-term financial transactions;
- incorrect summation of monetary values \u200b\u200brelating to different periods of time.
Consider the individual elements of the methodological tools of 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, on bonds and bills).
Simple percentage - the amount of income accrued to the principal amount of capital in each interval for which further calculations are not performed.
Compound interest - the amount of income accrued in each interval, which is not paid, but is added to the main amount of capital (deposit) in the subsequent payment period.
Interest rate - the specific indicator, according to which the amount of interest per unit of capital (deposit) is paid within the prescribed time. In practice, the interest rate expresses the ratio of the annual amount of interest income to the volume of principal.
Future value of money (Future Value, FV) - the amount of funds currently invested in which they will turn into after a certain period of time, taking into account the selected interest rate.
The true value of money (Present Value, PV) - the amount of future cash (deposit), given taking into account the specific interest rate to date.
Value addition (compounding - compounding) - the process of recalculating the present value of cash (deposit) in their future value in a specific period of time by adding to the initial amount of accrued interest.
Discount value (discounting) - the process of bringing the future value of cash (deposit) to its present value by excluding from the future amount the corresponding amount of interest (discount). Through this financial transaction, comparability of the present value of future cash flows is achieved.
Accrual period - the total period of time during which the process of increasing or discounting the amount of money (deposit).
Accrual interval - This is the minimum period after which interest is accrued.
Interesting accrual method - a method in which interest is accrued 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 percentage ratio of the amount accrued for a certain interval of income to the amount available at the beginning of this interval.
Antisipative method (preliminary) of interest calculation - This is the way in which interest is accrued at the beginning of each accrual interval. The amount of interest money is determined based on the accumulated amount. The interest rate will be the percentage ratio of the amount of income paid for a certain interval to the accumulated amount received after this interval. The interest rate determined in this way is called the discount rate, or antisipative interest.
Simple Interest Growth
Simple interest is used in short-term financial transactions, the term of which is less than a year or equal to it.
The increase at the annual rate of simple interest is carried out according to the formula:
FV \u003d PV (1 + r × n), (1)
where FV is the future value;
PV - initial cost;
n is the number of periods (years);
r is the interest rate.
Example 1
The client made a contribution to the bank in the amount of 10,000 rubles. at 12% per annum for a period of five years. By the formula (1) we find:
FV \u003d 10,000 (1 + 0.12 × 5) \u003d 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 duration of its operation 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 value of the operation can be determined:
The time of the deposit (loan) can be calculated either taking into account the exact number in months, or assuming that the estimated duration of any month is 30 days.
As a result, specific calculations for calculating 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 the year (exact percentages);
365/360 - the exact number of days of the operation and the financial year (12 months of 30 days);
360/360 - the approximate number of days of the operation (the month is assumed to be 30 days) and the financial year (ordinary interest).
For the same conditions for calculating interest, settlements on these options lead to slightly different financial consequences.
Example 2
The joint-stock company received a loan in 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 is used: February 15 - the 46th day of the year, April 15 - the 105th day of the year. Hence the exact loan term is 59 days. Then, by the formula (3) we find:
Interest rate discounting
There are two ways to discount.
Mathematical discounting - a method based on solving a problem inverse to determining future value. When making calculations, the interest rate is used here.
Given the previously accepted notation, the discount formula for the rate r will be:
(4)
The income of the bank (FV - PV) is called the discount, and the used rate of reduction r is called discursive interest rate.
Example 3
What price will an investor pay for a coupon-less bond, the nominal value of which is 500 thousand rubles and the maturity term is 270 days, if the required rate of return is 20%?
According to the formula (4) when using ordinary interest:
PV \u003d 500 / (1 + 0.2 × 270/360) \u003d 434.78 thousand rubles;
exact percent:
PV \u003d 500 / (1 + 0.2 × 270/365) \u003d 435.56 thousand rubles.
Bank discounting is used for banking accounting of bills, with interest accrued on the amount payable at the end of the transaction. When making calculations, the discount rate d is used:
(5)
When discounting at the discount rate, the temporary base 360/360 or 360/365 is most often used. The reduction rule 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 in the bank after 270 days at a simple discount rate of 20%. How much will the holder of the bill receive?
We use the formula (5), considering that n is the time difference between the moment of accounting and the maturity of the bill:
PV \u003d 500 (1 - 0.2 × 90/360) \u003d 475 thousand rubles.
Application of the two discount methods considered to the same amount leads to different results, even at r \u003d d. The discount rate gives a faster reduction in the amount than the usual.
Example 5
A promissory note in the amount of 100 thousand rubles. with payment after 90 days is taken into account at the bank immediately after receipt. It is necessary to determine the amount received by the owner of the bill at an interest / discount rate of 15%.
When using the interest rate by the formula (4):
PV \u003d 100 / (1 + 0.15 × 90/360) \u003d 96.39 thousand rubles.
When using the discount rate according to the formula (5):
PV \u003d 100 (1 - 0.15 × 90/360) \u003d 96.25 thousand rubles.
The discount rate d is also used to increase at simple interest (for example, when determining the future amount of the contract):
(6)
Change the conditions of example 5 as follows.
Example 6
How much should a bill be issued in order for the supplier to complete the price of goods (100 thousand rubles) if the discount rate is 15%?
Using the formula (6), we determine the future value (par value) of the bill:
FV \u003d 100 / (1 - 0.15 × 90/360) \u003d 103.896 thousand rubles.
The interest rate r or discount rate d can be determined from the ratios (1) and (5):
(7)
(8)
Example 7
A short-term liability with a maturity of 90 days was purchased at a price of 98.22 units. from face value. It is necessary to determine the profitability of the operation for the investor.
It is (using ordinary interest):
The duration of the operation in days is determined as follows:
Example 8
It is necessary to determine the term of ownership of the obligation at a value of 98.22 units repaid at par if the required rate of return is 7.2%.
Interest rate equivalencer andd
Equivalent interest rates - these are different types of bets, the use 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 terms of a financial transaction and a tool is required for the correct comparison of different interest rates.
The derivation of the equivalence formulas is based on the equality of the corresponding accumulation factors:
1 + n × r \u003d (1 - n × d) - 1. (11)
Given the formula (11) for operations with a duration of less than a year, the equivalence relations will take the form:
- the time base is the same and 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 payment term on the bill is 250 days. In this case, the simple interest rate is measured with a temporary base of 365 days, and a simple discount rate - with a temporary base of 360 days. What will be the profitability measured in the form of a simple interest rate, accounting for a bill of exchange at a simple discount rate of 10%?
Using formula (14) for r for given time bases, we obtain:
r \u003d 365 × 0.1 / (360 - 250 × 0.1) \u003d 0.1089, or 10.89%.
Suppose that the present value of a bill is 100,000 rubles. Then its nominal value according to the formula (3) will be:
Compound interest accrual
Compound interest is applied, as a rule, in financial transactions, the term of which is more than a year. At the same time, the base for calculating interest is both the initial amount of the financial transaction and the amount of interest already accumulated by this time.
The compound interest growth is as follows:
FV n \u003d PV (1 + r) n. (16)
Compound interest accrual implies reinvestment of income or capitalization.
Compound interest may be calculated not once, but several times a year. In this case, the nominal interest rate j is negotiated - the annual rate at which the interest rate applied at each accrual interval is determined.
With m equal accrual intervals and a 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 accumulated 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 of 200 thousand rubles. determine the accrued amount in five years using a complex interest rate of 28% per annum. Solve an example for cases where interest is accrued for six months, quarterly.
According to the formula (16) for compound interest rates:
Fv \u003d 200 (1 + 0.28) 5 \u003d 687.2 thousand rubles.
According to the formula (17) for the accrual over half year:
Fv \u003d 200 (1 + 0.28 / 2) 10 \u003d 741.4 thousand rubles.
According to the same formula for quarterly accrual:
Fv \u003d 200 (1 + 0.28 / 4) 20 \u003d 773.9 thousand rubles.
If the term of the financial transaction n in years is not an integer, the accumulation multiplier k is determined by the formula:
k \u003d (1 + r) n a (1 + n b × r), (18)
where n \u003d 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 a corresponding non-integral exponent. However, this method is approximate. The larger the values \u200b\u200bincluded in the formula, the greater the error in the calculations.
Example 11
The initial amount of debt is 50,000 thousand rubles. It is necessary to determine the accrued amount after 2.5 years using two methods for calculating compound interest at a rate of 25% per annum.
By the formula (18) we obtain:
FV \u003d 50,000 (1 + 0.25) 2 (1 + 0.5 × 0.25) \u003d 87,890.6 thousand rubles.
For the second method, we use the formula (16) with a non-integral exponent:
FV \u003d 50,000 (1 + 0.25) 2.5 \u003d 87,346.4 thousand rubles.
When using the approximate method, the lost profit could be about 550 thousand rubles.
If compound interest is accrued several times a year and the 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 accrual 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 conditions of financial transactions involving different periods of interest calculation. In this case, the corresponding interest rates lead to their annual equivalent by the formula:
The resulting value is called the effective interest rate (effective percentage rate - EPR), or the comparison rate.
Example 12
On a four-year deposit of 10,000 rubles. compound interest is accrued quarterly at a rate of 2.5%, that is, at the rate of 10% per annum. Will a deposit of 10,000 rubles be invested for the same period at 10% accrued once a year?
Calculate the effective bid for both operations:
- quarterly: EPR \u003d (1 + 0.1 / 4) 4 - 1 \u003d 0.103813;
- annually: EPR \u003d (1 + 0.1 / 1) 1 - 1 \u003d 0.10.
Thus, the conditions for placing the amount of 10,000 rubles. for a four-year deposit at 2.5% accrued on a quarterly basis will be equivalent to an annual rate of 10.3813%. Therefore, the first operation is more profitable for the investor.
If the EPR value is known, the nominal interest rate can be determined as follows:
Compound interest discounting
Consider the use of complex interest rates for mathematical discounting:
If interest will be charged m times a year, then formula (22) will take the form:
Example 13
The Bank accrues interest on the deposit at a compound interest rate of 20% per year. What amount should be put on deposit, 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 annual interest accrual according to the formula (22):
PV \u003d 10 000 / (1 + 0.2) 10 \u003d 1615.1 thousand rubles.
With quarterly interest calculation according to the formula (23):
PV = 10 000 / (1 + 0.2 / 4) 40 \u003d 1420.5 thousand rubles.
Using a complex discount rate
To calculate the discount operation at a complex discount rate, the following formula is used:
PV n \u003d FV n (1 - d) n. (24)
Example 14
The owner of a bill with a nominal value of 500 thousand rubles. and a circulation period of 1.5 years he offered it to the bank immediately for accounting, that is, 1.5 years before maturity. The bank agreed to account for the bill at a complex discount rate of 20% per annum. It is required to determine the discount received by the bank and the amount issued to the holder of the bill.
Using the formula (24), we find:
PV \u003d 500 (1 - 0.2) 1.5 \u003d 357.77 thousand rubles.
The bank discount will be: 500 - 357.77 \u003d 142.23 thousand rubles.
For these conditions, we determine the amount that the holder of the bill would have received if the bank had recorded the bill at a simple discount rate of 20%. For this we use the formula (5):
PV \u003d 500 (1 - 0.2 × 1.5) \u003d 350 thousand rubles.
The bank discount will be 500 - 350 \u003d 150 thousand rubles.
Thus, it is more profitable for a bank to take into account a bill of exchange at a simple discount rate.
If discounting at a complex discount rate is made m times a year, the calculation formula will be as follows:
Example 15
We preserve the conditions of the previous example, but let the calculation of the discount be done quarterly, that is, m \u003d 4.
By the formula (25) we get:
PV \u003d 500 (1 - 0.2 / 4) 6 \u003d 367.55 thousand rubles.
The bank discount will be: 500 - 367.55 \u003d 132.45 thousand rubles.
Bank income at a quarterly discount will be less than at an annual discount, by: 142.23 - 132.45 \u003d 9.78 thousand rubles.
When discounting with interest for periods less than a year, the term “effective compound discount rate” may be used. The effective compound discount rate equivalent to the complex discount rate for a given value of m is determined by the formula:
d eff \u003d 1 - (1 - d / m) m. (26)
Example 16
Debt obligation with a nominal value of 500 thousand rubles. due in five years. The compound discount rate is 20% per annum. Interest accrual quarterly. It is required to determine the true value of the liability and the effective discount rate.
Using formulas (25) and (26), we obtain:
PV = 500 (1 - 0.2 / 4) 20 \u003d 179.243 thousand rubles.
d eff = 1 - (1 - 0.2 / 4) 4 \u003d 0.18549, or 18.549%.
Substituting the value of 18.549% in the formula (24), we obtain:
PV \u003d 500 (1 - 0.18549) 5 \u003d 179.247 thousand rubles.
The discrepancy between the values \u200b\u200bof the present amount calculated by these formulas are within the accuracy of the calculation.
Determination of the interest rate and duration of the operation
With the known values \u200b\u200bof 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 the formula (27) we find:
r \u003d (14,641/10,000) 1/4 - 1 \u003d 0.1, or 10%.
The duration of the operation is determined by the logarithm:
Example 18
The amount of 10,000 rubles placed with the bank at 10% per annum amounted to 14,641 rubles. It is necessary to determine the duration of the operation.
By the formula (28) we find:
n \u003d log (14,641/10,000) / log (1 + 0,1) \u003d 4 years.
Conclusion
The given calculation formulas describe the mechanism of the influence of the time factor on the result of financial transactions. Their use will help to avoid mistakes and losses in the face of a decrease in the purchasing power of money.
E. G. Moiseeva,
Cand. econ. Sciences, Arzamas Polytechnic Institute
Concept estimates of the value of money over time plays a fundamental role in the practice of financial computing. It determines the need to take into account the time factor in the process of carrying out any long-term financial transactions by evaluating and comparing the value of money at the beginning of financing with the value of money when they are returned in the form of future profit.
In the process of comparing the value of money when investing and returning, 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 currently invested in which they will turn into 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 current period. The determination of the true value of money is associated with the process of discounting this value.
There are two ways to determine and calculate interest:
1. Interesting accrual method. Interest is accrued at the end of each accrual interval. Their value is determined based on the amount of capital provided. A destructive interest rate (loan interest) is the percentage expressed as the ratio of the amount accrued for a certain interval of income 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 accrual. Interest is accrued at the beginning of each accrual interval. The amount of interest money is determined based on the accumulated amount. Antisipative rate (discount rate) is the percentage of the amount of income paid for a certain interval expressed in percentage terms to the value of the accumulated amount received after this interval (S). In countries of developed market economies, the antisipative method of interest calculation 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, the most important element of entrepreneurial economic activity and enterprise management is planning, including financial.
Financial planning is the planning of all revenues and areas of expenditure of funds of the enterprise to ensure its development. Financial planning is carried out through the preparation of financial plans of different contents and purposes depending on the tasks and objects of planning. Financial planning is an important element of the corporate planning process. Each manager, regardless of his functional interests, should be familiar with the mechanics and the meaning of the implementation and control of financial plans, at least as much as concerns his activities. The main objectives of financial planning:
Providing a normal reproduction process with the necessary sources of financing. Moreover, target sources of financing, their formation and use are of great importance;
Compliance with the interests of shareholders and other investors. A business plan containing a similar justification of an investment project is for investors the main document that stimulates investment;
The guarantee of the fulfillment of obligations of the enterprise 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 with the specified parameters;
Identification of reserves and mobilization of resources in order to efficiently use profits and other income, including non-operating;
The 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 is sent to the reserve fund. If expenses exceed incomes, the amount of lack of financial resources is compensated by issuing securities, obtaining loans, receiving charity contributions, etc.
Planning methods are specific methods and techniques for calculating indicators. When planning financial indicators, the following methods can be used: normative, settlement and analytical, balance sheet, method of optimization of planned decisions, economic and mathematical modeling.
The essence of the normative method of planning financial indicators is that based on pre-established norms and technical and economic standards, the needs of an economic entity for financial resources and their sources are calculated. Such standards are tax rates, rates of tariff contributions and fees, depreciation rates, requirements 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 its change indices 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 an analysis of their dynamics and relationships. This method is based on expert judgment.
The essence of the balance method of planning financial indicators lies in the fact that by constructing balances achieved by linking the available financial resources and the actual need for them. The balance method is used primarily when planning the distribution of profits and other financial resources, planning the need for the receipt of funds in financial funds - accumulation fund, 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 from them.
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 indicators and the factors that determine them. This relationship is expressed through an economic-mathematical model. The economic-mathematical model is an exact mathematical description of the economic process, i.e. a description of the 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 prospective (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 degree. Four main types of managerial activity can be distinguished within the framework of the strategic planning process: resource allocation; adaptation to the external environment; internal coordination; organizational strategic foresight. The system of current planning of financial activities of the company is based on the developed financial strategy and financial policy for certain aspects of financial activity. Linking each type of investment with a source of funding. To do this, usually use the estimates of the formation and expenditure 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 own resources.
Current financial plans of an entrepreneurial firm are developed on the basis of data that characterize: the financial strategy of the company; financial analysis results for the previous period; planned volumes of production and sales of products, as well as other economic indicators of the operating activities of the company; a system of norms and standards developed by the company for the costs of individual resources; current tax system; current system of depreciation rates; average rates of credit and deposit interest in the financial market, etc. Operational financial planning consists in the preparation and use of the plan and report on cash flows. A payment calendar is compiled on the basis of a real information base of the company's cash flows. In addition, the company must draw up a cash plan - a cash turnover plan that reflects the receipt and payment of cash through the cash desk.
Accrual of simple rates is used, as a rule, for short-term lending.
We introduce the following notation:
S - accrued amount, p .;
P - the initial amount of debt, p .;
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 \u003d P + P i \u003d P (1+ i);
at the end of the second year:
S2 \u003d S1 + P i \u003d P (1+ i) + P i \u003d P (1+ 2 i); at the end of the third year:
S3 \u003d S2 + Pi \u003d P (1+ 2 i) + P i \u003d P (1 + 3 i) and so on. At the end of the term n: S1 \u003d P (1+ n i).
This is a build-up formula 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 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 is the accumulation rate at a simple interest rate:
KH \u003d (1+ n i).
Consequently,
Si \u003d P Kn.
Task 5.1
The bank issued a loan of 5 million rubles. for six months at a simple interest rate of 12% per annum. Determine the repayable amount.
DECISION:
S \u003d 5 million (1 + 0.5 ¦ 0.12) \u003d 5,300,000 p.
If the period for which the money is borrowed is specified in days, the accumulated amount will be equal to S \u003d P (1 + d / K i),
where d is the duration of the term in days;
K is the number of days in a year.
The value of K is called the time base.
The time base can be taken equal to the actual duration 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 taken to be 30 days. In both cases, the date the loan was issued and the date of 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 amount 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 preferred options from the point of view of the bank and the borrower.
DECISION:
The exact number of days of the loan is from 12.03. until 12/25:
20+30+31+30+31+31+30+31+30+25=289.
Estimated number of loan days:
20+8-30+25=285;
a) Exact interest and exact number of days of the loan:
S \u003d 200,000 (1 + 289/366 ¦ 0.07) \u003d 211 016 p.;
b) ordinary interest and the exact number of days of the loan:
S \u003d 200,000 (1 + 289/360 ¦ 0.07) \u003d 211,200;
c) ordinary interest and the approximate number of loan days:
S \u003d 200,000 (1 + 285/360 ¦ 0.07) \u003d 211 044;
d) the exact interest and the approximate number of days of the loan:
S \u003d 200,000 (1 + 285/366 ¦ 0.07) \u003d 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) - the exact interest with the approximate number of days of the loan.
Therefore, from the point of view of the bank as a creditor, option b) is preferable, and from the point of view of the borrower, option d) is preferred.
It must be borne in mind that in any case, ordinary interest is more beneficial to the lender, and accurate to the borrower (at any rates - 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 \u003d P (1 + I nt it),
t \u003d 1
where N is the number of interest calculation intervals;
nt is the duration of the tth accrual interval;
it is the interest rate on the tth accrual interval.
Task 5.3
The Bank accepts deposits at a simple interest rate, which is 10% in the first year, and then increases by 2 percentage points every six months. Determine the size of the contribution of 50 thousand rubles. with interest after 3 years.
Decision:
S \u003d 50,000 (1 + 0.1 + 0.5 0.12 + 0.5 0.14 + 0.5 0.16 + 0.5 0.18) \u003d 70,000 p.
Using the formula for the accrued amount, you can determine the loan term under other specified conditions.
Loan term in years:
S - P N \u003d.
P i
Determine the loan term in years, for which the debt is 200 thousand rubles. will increase to 250 thousand p. when using a simple interest rate - 16% per annum.
DECISION:
(250,000 - 200,000) / (200,000 0.16) \u003d 1.56 (years).
From the formula for the accrued amount, you can determine the rate of simple interest, as well as the initial amount of debt.
Decide for yourself
Task 5.5
When issuing a loan of 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%.
Task 5.6
A 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 return amount should be 500 thousand rubles. with a temporary base of 360 days.
ANSWER: 480 000R.
The operation of finding the initial amount of debt at a known repayable is called discounting. In a broad sense, the term “discounting” means determining the value P of a 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 converting a cost 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 time factor in cost calculations. The discount factor is always less than one.
Discount rate formula at a simple interest rate:
P \u003d S / (1 + ni), where 1 / (1 + ni) is the discount coefficient.
Related topic The decursive method of calculating simple interest:
- 1. The concept and methodological tools for assessing the value of money over time.
- 2.3. Determination of current and future cash flows
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Determination of the unsatisfactory structure of the enterprise balance sheet by 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 May 25, 94, No. 498, the degree of insolvency of enterprises should be assessed according to three criteria characterizing an unsatisfactory balance sheet structure:
1. current ratio;
2. the ratio of own funds;
3. the recovery or loss of solvency ratio.
The basis for recognizing the balance sheet structure of the enterprise as unsatisfactory, and the enterprise as insolvent, is the fulfillment of one of the following conditions:
The current 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 following decisions are made by territorial insolvency and bankruptcy agencies of enterprises: On the recognition of the balance sheet structure as unsatisfactory, therefore, the company is insolvent. On the existence of a real opportunity for the debtor enterprise to restore its solvency. The existence of a real possibility of losing the solvency of the enterprise if it is in the near future unable to fulfill its obligations to creditors. These decisions are made regardless of whether the company has external signs of insolvency established by law.
Current ratio characterizes the general security of the enterprise with working capital for conducting business activities and the ability of the enterprise to timely repay urgent obligations \u003d tech assets / tech liabilities.
Equity Ratio characterizes the availability of the company's own funds necessary to ensure its financial stability \u003d (current liabilities-tech assets) / total value of tech assets.
Declaring an enterprise insolvent does not always mean declaring it 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 ensure measures to prevent the insolvency of the enterprise, as well as stimulate the enterprise to independently overcome the crisis. If at least one of the above two ratios does not meet the normative values, the solvency recovery ratio for the coming period of 6 months is calculated. 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 for the coming period of 3 months is calculated.
Solvency recovery ratio defined 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.
K1F - the actual value of the current ratio at the end of the reporting period.
K2F - the actual value of the current ratio at the beginning of the reporting period.
T - reporting period in months
2 - current ratio ratio
(for 6 months)\u003e 1, then the company has a real opportunity to restore its solvency in a fairly short period.
If the solvency recovery ratio< 1, то у предприятия нет реальной возможности восстановить свою платежеспособность на данный момент и за достаточно короткий срок.
The solvency loss ratio is determined by:
If the coefficient of loss of solvency (3 months)\u003e 1, this indicates the presence of a real opportunity for the company to lose solvency.
If there are grounds for declaring the balance sheet structure unsatisfactory, but if a real opportunity to restore solvency is identified, the territorial bankruptcy agency decides to postpone the decision to declare the balance sheet structure unsatisfactory and the company insolvent for up to 6 months.
If there are no such grounds, then one of two decisions is made:
If the solvency recovery ratio is\u003e 1, then no decision is made to declare the balance sheet structure unsatisfactory, and the company insolvent.
If the solvency recovery ratio< 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 территориальный орган по банкротству, но только в том случае, если доля государственных предприятий в общей собственности более 25%.
A number of enterprises may be insolvent in connection with the debt of the state to this enterprise. In this case, an analysis is made of the dependence of the solvency of the enterprise at the moment and the debt of the state to the enterprise.
Interest - income from the provision of capital in debt in various forms (loans, credits, etc.), or from industrial investments or financial. character.
Interest rate - this is a value characterizing the rate of interest calculation.
Currently, there are two ways to determine and calculate interest:
The discursive way. Interest is accrued at the end of each accrual interval. Their value is determined based on the amount of capital provided. Accordingly, the decursive interest rate (percentage) is the percentage ratio of the amount accrued for a certain interval of income to the amount available at the beginning of this interval.
Antisipative (preliminary) method.A preliminary percentage is accrued at the beginning of each accrual interval. The amount of interest money is determined based on the accumulated amount. The interest rate will be the percentage ratio of the amount of income paid for a certain interval to the amount of 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, is measured in monetary units (e.g., rubles) and is denoted by I. If we designate the future amount S and the present (or initial) P, then I \u003d S - P. The interest rate i is a relative value, measured in decimal fractions or%, and determined by dividing interest by the initial amount:
In addition to interest there discount rate d (another name is the discount rate), the value of which is determined by the formula:
where D is the sum of the discount.
Comparing formulas (1) and (2), it can be noted that the sum of percent I and the amount of discount D are determined in the same way - as the difference between future and present values. However, the meaning given to these terms is not the same. if in the first case we are talking about an increase in current value, then in the second it is determined the decrease in future value, “discount” from its value. The main area of \u200b\u200bapplication of the discount rate is discounting, the process inverse to the calculation of interest. With the help of the above rates, both simple and compound interest can be calculated. When calculating simple interest, the accumulation of the initial amount occurs in arithmetic progression, and when calculating compound interest - in geometric. The simple decursive and antisipative interest is calculated according to various formulas:
decursive interest: (3)
antisipative percentages:, (4)
where n is the duration of the loan, measured in years.
However, the duration of the loan n need not be equal to a year or an integer number of years. Simple interest is most often used in short-term operations. In this case, the problem arises of determining the duration of the loan and the duration of the year in days. If we denote the duration of the year in days by the letter K (this indicator is called time base), and the number of days of using the loan 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 obtain:
for decursive interest: (6)
for antisipative percentages:, (7)
The most common combinations of the time base and the duration of the loan (the numbers in brackets indicate the values \u200b\u200bof t and K, respectively):
Exact interest with the exact number of days (365/365).
Ordinary (commercial) interest with the exact duration of the loan (365/360).
Ordinary (commercial) interest with an approximate duration of the loan (360/360).
The inverse problem with respect 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 \u200b\u200bof the interest (discount) rate and duration of the transaction, the initial ( modern, refurbished or current) the cost of P. Depending on what kind of rate - a simple interest rate or a simple discount - is used for discounting, there are two types: mathematical discounting and bank account.
The method of banking accounting got its name from the financial transaction of the same name, during which a commercial bank buys from the owner (takes into account) a promissory note or a bill of exchange at a price lower than the nominal value before the expiration date indicated on this document. The difference between the face value and the redemption price forms the bank’s profit from this transaction and is called the discount (D). To determine the size of the redemption price (and, consequently, the amount of the discount), discounting is applied by the method of bank accounting. In this case, a simple discount rate is used d. The redemption price (current value) of a bill is determined by the formula:
where t is the period remaining until maturity of the bill, in days. The second factor of this expression (1 - (t / k) * d) is called the discount factor of bank accounting for simple interest.
For mathematical discounting, the simple interest rate i is used. Calculations are performed according to the formula:
The expression 1 / (1 + (t / k) * i) is called the discount factor of mathematical discounting by simple percent.
The main area of \u200b\u200bapplication of simple interest and discount rates are short-term financial transactions, the duration of which is less than 1 year.
Calculations with simple rates do not take into account the possibility of reinvesting accrued interest, because accruals and discounts are made with respect to a constant initial amount 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 carried out according to the formula not of arithmetic but of geometric progression, the first member of which is the initial sum P, and the denominator is (1 + i). The accrued value (the last member of the progression) is found by the formula:
(10), where (1 + i) n is the multiplier for building up recursive compound interest.
The compound interest rate i itself 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 use a complex discount rate for calculating interest (antisipative method):
, (11) where 1 / (1 - d) ^ n is the multiplier of the increase in compound antisipative percentages.
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 compounding formula for compound interest when calculating them m times a year is:
When calculating antisipative compound interest, the nominal discount rate is denoted by the letter f, and the accumulation formula takes the form:
The expression 1 / (1 - f / m) ^ mn is the accumulation multiplier at the nominal discount rate.
Compound interest discounting can also be done in two ways - mathematical discounting and bank accounting. The latter is less beneficial for the lender than accounting at a simple discount rate, so it is used extremely rarely. In the case of a single interest calculation, its formula has the form:
where (1 –d) n is the discount factor of banking accounting at a complex discount rate.
for m\u003e 1 we get
, (16) where f is the nominal compound discount rate,
(1 - f / m) mn - a discount factor of banking accounting at a complex nominal discount rate.
Mathematical discounting at a compound interest rate i is much more widespread. For m \u003d 1 we get
, (17) where 1 / (1 + i) n is the discount factor of mathematical discounting at a complex interest rate.
With repeated interest accrual during the year, the mathematical discount formula takes the form:
, (18) where j is the nominal compound interest rate,
1 / (1 + j / m) mn is the discount factor of mathematical discounting at a complex nominal interest rate.