Foreign researchers in the field of stock management emphasize the importance of models for calculating the optimal stock Money, developed by W. Baumol and J. Tobin.

It is noted that U. Baumol was the first to emphasize the similarity of inventories of tangible assets and cash reserves and considered the possibility of applying the inventory management model to calculate the company's cash balance. The Baumol model, as well as the Miller-Orr model, does not take into account the possibility of attracting borrowed funds.

1. Model of Baumol - Tobin

W. Baumol rightly argues that the company's cash can be regarded as a stock of money, the owner of which is ready to exchange them for labor, raw materials and other types of tangible assets. Cash in hand is essentially no different from the shoemaker's stock of shoes, which he is willing to exchange for retailer's money. Therefore, methods for determining the optimal size of stocks can be applied to calculate the stock of cash that is optimal for the company at the available costs.

W. Baumol's model is described in detail in the November issue of the journal for 1952 1811. The model developed by W. Baumol is based on the assumption that transactions are made continuously and in a situation of complete certainty. Assume that the company is required to pay daily during the period T cash in total R. The company has the opportunity to replenish the cash reserve at the expense of funds raised in debt (by placing a bonded loan) or on stock market by selling securities. In either case, the company bears the cost of servicing the debt or the opportunity cost of selling valuable papers and related to the refusal of the company from income on securities.

Let us consider a situation where a company sells short-term financial investments in income-generating securities, and then sells them to replenish the stock of cash. In this case, let's say e - profitability of financial investments in securities (reflecting the profit for each ruble invested in securities), and b- costs associated with the transaction for the sale of securities. It is interesting to note that U. Baumol calls such costs "broker's fee", emphasizing that such a phrase should not be taken literally 181, p. 5461. Such costs include all costs associated with short-term financial investments, which are conditionally considered constant for the ongoing operation to raise funds (in this case, the sale of securities). Period T divided into equal intervals t. The amount of money raised evenly over the period T to replenish the cash reserve, denote C. Considering this value, U. Baumol uses the term "withdrawal" ( withdrawal), assuming that cash is withdrawn from a financial investment by selling securities.

Thus, the total volume of transactions R predetermined, but the magnitudes? d and b - are constant. The amount of funds C, attracted to replenish the cash reserve, is reduced evenly until the complete depletion of the supply of money, and then the withdrawal of funds is again made. Average cash reserve С avg in the interval t equals

Then the company's opportunity cost of terminating the financial investment over time is T(in terms of inventory management, such costs reflect the cost of storage for a certain time) will be

Number of transactions for the sale of securities during the time T equals /us, and the costs associated with the transaction for the sale of securities are b rubles per transaction. Hence, the total cost of raising funds is equal to

^, r.l = *?? (3.3)

Therefore, the total costs /%, including the costs of holding and raising funds, will be

A company's total cost of changing its cash balance over time T:

(3.4) where E - profitability of financial investments in securities per day;

T - cash reserve planning period, days.

Based on the fact that the company seeks to reduce the cost of attracting and storing a stock of cash, the optimal amount of cash balance C wholesale will correspond to the minimum total cost. Consider the change in the stock of cash over time T when replenishing the stock by the optimal value C opt at time points t v t 2 and d 3 when the cash is completely used up by the time (Fig. 3.1).

We study the expression (3.4). The first term depends on C linearly and increases with an increase in the cash balance, and the second term, on the contrary, decreases with an increase in C (Fig. 3.2).

It can be seen from the graph that there is such an optimal value of the cash balance C opt, at which E takes the minimum value. Indeed, consider /' as a function of C and, equating the derivative of / with respect to C to zero, we obtain

Then, the optimal value of the cash reserve

Rice. 3.1.

• 1, 3, 5, 7 - uniform spending of funds for payments with a total volume R;
• 2, 4, 6 - replenishment of the cash reserve at the expense of funds received from the sale of securities

Rice. 3.2.

The second derivative of Y 7 with respect to C, equal to

is positive, we have a minimum at С = С opt.

Thus, at constant transaction costs and the return on securities, the size of the cash reserve varies in proportion to the square root of the volume of payments that the company undertakes to make over a certain period of time.

J. Tobin, independently of W. Baumol, developed a similar demand for money model, showing that the cash reserves intended for transactions depend on changes in the interest rate 11021. J. Tobin's model proceeds from the premise that the company chooses between bonds and cash . At the same time, J. Tobin notes that bonds and cash are the same assets, with the exception of two differences. First, bonds are not a means of payment. Second, bonds are profitable and cash yields are zero. Unlike W. Baumol, J. Tobin used the portfolio approach to prove his position.

Following the reasoning of J. Tobin, the following options for making transactions for the acquisition of bonds and their subsequent sale are possible. For example, a company does not buy bonds immediately, after receiving cash, but after some time, and sells bonds without waiting for the cash to be completely spent. This approach is not optimal for the company, since postponing the purchase of bonds leads to a shortfall in interest on them. It is more rational for the company to purchase bonds immediately at the time of receipt of funds in the logistics system and sell them later, due to the expenditure of funds. In this case, the company will receive more high percent on bonds. .

W. Baumol used the idea of ​​minimizing the total cost of registration and storage of inventories, considering the opportunity costs of storing funds and the cost of attracting financial resources. The main idea of ​​Baumol's model is that there is an opportunity cost of holding money - the interest income that can be earned on other assets. However, holding cash reserves reduces transaction costs. When the interest rate increases, the company will tend to reduce the amount of funds due to the increase in the opportunity cost of holding money. Based on the calculations, Baumol and Tobin proposed a formula for calculating the demand for

money ( M), which is the average cash balance:

The above formula is called the square root rule 149, p. 762].

Example 3.1

Let's say that the company has the opportunity to purchase securities with a yield of 0.022% per day (8.03% per year). At the same time, the fixed costs of transactions by the company are 1.2 thousand rubles. for every operation. Let us determine the optimal balance of funds evenly spent during the quarter, given that the total amount of all company payments for the quarter is 90,000 thousand rubles. Having carried out calculations according to the formula (3.6), we obtain C opt \u003d 3302.9 thousand rubles. (Fig. 3.3):

1 2-1.2 90 000 V 90 0.00022

3302.9 (thousand rubles).

At the same time, the minimum costs of the company, calculated by formula (3.4), are equal to 65.4 thousand rubles:

TE,C BP-- + - 2 C

• 1,2-90 000 3302,9
• 90 0,00022-3302,9 - ! --+

65.4 (thousand rubles).

A cash reserve of 200 thousand rubles will lead to a total cost of the company in the amount of 542 thousand rubles, and if the company holds a cash reserve of 10,000 thousand rubles, then its total costs will be 110 thousand rubles. The company will be able to minimize its total costs by forming a cash reserve at the level of 3302.9 thousand rubles. (Table 3.2)

Table 3.2

The change in costs in the micrologistics system depending on the cash supply according to the Baumol model with E= 0.022% per day, thousand rubles

• - total costs of the company;
• - the cost of raising funds;
• - the cost of holding funds

Rice. 3.3. The change in the company's costs depending on the cash balance according to the Baumol-Tobin model with E = 0.022% per day, thousand rubles

The value of the cash reserve increases with an increase in the cost of transactions with securities and the volume of payments, and decreases with an increase in the profitability of financial investments. If we substitute into the model the profitability of securities less than that accepted in the calculations and equal to 0.0137% per day (5% per year), and the fixed costs of transactions by the company in the amount of 1.8 thousand rubles. for the operation and the amount of the company's payments - 280,000 thousand rubles. per quarter, we can conclude the following:

Cash reserve in the amount of 200 thousand rubles. will lead to the full costs of the company, equal to 2521 thousand rubles, and in the amount of 12,000 thousand rubles. - to total costs 116 thousand rubles; the minimum cost of the company is achieved in the range between 6,000 thousand and 10,000 thousand rubles. Baumol's model based on the given data makes it possible to calculate the cash reserve that minimizes the company's total costs (111 thousand rubles). Thus, the optimal cash reserve is equal to 9042 thousand rubles.

The model for calculating the optimal cash balance of Baumol - Tobin is deterministic, which limits its application in practice.

2. Model of Miller and Orr

One should agree with Burnell K. Stone 11011 that two completely different logistical approaches to managing cash reserves can be distinguished: a model in conditions of complete certainty, proposed by W. Baumol, and a model for calculating the cash reserve in a situation of uncertainty, developed by American economists Merton Miller (Merton H. Miller) and Daniel Orr (Daniel Opt) and published in the issue of the magazine Quarterly Journal of Economics for August 1966. Based on a later publication by M. Miller and D. Orr, which contains additional evidence for the applicability of the stochastic cash management model, we can generally formulate the similarities and differences between these models. M. Miller and D. Orr, as well as W. Baumol, emphasize that the company's cash reserve depends on the opportunity costs of storing cash and the costs of making securities purchase and sale transactions. However, unlike the Baumol-Tobin model, the stochastic model assumes the probabilistic nature of the behavior of the company's cash flows.

The Miller-Orr stochastic model is based on three main assumptions. In this case, the first assumption repeats the assumptions of the developers of deterministic models.

• 1. Similar to the assumptions considered earlier in the W. Baumol and debt accumulation models, M. Miller and D. Orr theoretically assume that a company uses two types of assets (bank deposits, securities and cash), enters into transactions to transfer one type of asset in another without delay in time and spends at the same time a constant amount that does not depend on the volume of the transaction.
• 2. There is a minimum level of cash that the company strives to maintain. In practice, the company follows the terms of the agreement with the bank, stipulating the obligation of the company not to reduce the amount of funds in the current account below a certain amount.
• 3. In contrast to the Baumol-Tobin model, the stock of funds changes randomly, since the magnitude of cash flows cannot be predicted based on previous values.

Let's take a closer look at the third assumption. The Miller-Orr model assumes that an increase or decrease in the stock of cash by a certain amount (T) for a short period of time (1/G of a working day) can be considered as the appearance of some event when P independent retests according to the Bernoulli scheme (P - number of days). If the probability of increasing the cash reserve by the amount T rubles is R, then the probability of reducing the stock by the same amount T calculated as q = 1 -R. Then the distribution of the company's net cash flow (the difference between inflow and outflow) will have an average r p and dispersion a 2 „ equal

p /7 = ntm(p-q), o 2 n =4ntpqm 2 .

M. Miller and D. Orr proceed to consider the case of equal probabilities of inflow and outflow of funds:

dya = 0, 0^=/7D7 2 /,

In this case

o 2 \u003d ^ \u003d t 2 g. (3.10)

Thus, the cash flows are normally distributed with zero mean and constant variance.

At the same time, the Miller-Orr model overcomes the drawback of the Baumol-Tobin model associated with the assumption of a uniform expenditure of funds during the planning period (Fig. 3.1). Indeed, the most common is uneven cash flow of companies during the period T(Fig. 3.4).

If receipts exceed cash outflows, then the cash reserve C increases, on the contrary, if the cash outflow exceeds the inflow, the value of C decreases. The stock of funds C decreases and increases irregularly, but when it reaches the top point C max at the end of the interval /., the company makes a short-term financial investment, reducing the excess cash. At the end of the interval / 2, when the stock of funds becomes minimal

Rice. 3.4.

1 - implementation of short-term financial investments in securities in the amount M 2 - sale of securities in order to replenish the cash reserve by the amount M

with t1n, the company replenishes its cash balance by selling securities.

In accordance with the Miller-Orr model, the stock of funds changes within the limits established by the upper limit C max and the lower limit C m1n. At the same time, the zero value of the cash reserve is considered as the lower bound in , and in some positive value, which is the result of the calculation of the model. The arguments of M. Miller and D. Orr about the random walk of the value of the stock of funds within the established limits are based on the conclusions of V. Feller on the theory of random walks and the ruin problem.

According to the classical ruin problem, the player wins or loses money with the probabilities R And c respectively. According to the condition of the problem, the initial capital of the player is equal to G and he plays against an opponent with initial capital but-1 . Therefore, the total capital of the two players is equal to but. The game continues until the player's capital either increases to but, or will not decrease to zero, i.e. until one of the two players goes bankrupt. The unknowns in the problem are the probability of ruining the player and the probability distribution for the duration of the game. V. Feller gives an analogy, using the concept of a wandering point leaving the initial position r and making single jumps in a positive or negative direction at regular intervals. If the test is terminated when the point first reaches either the value but, or 0, then we say that the point performs a random walk with absorbing screens at points with values ​​o and 0. A modification of the classical ruin problem is the problem in which the absorbing screen is replaced by a reflecting one. In game terminology, this corresponds to an agreement under which the player who loses the last ruble is returned this ruble to him by the opponent, which makes it possible to continue the game.

It can be concluded that the Miller-Orr model is a problem of wandering the value of the company's net cash flow with two absorbing screens: the upper Cmax and the lower Cm1. If we designate the cusp C opt, then the mathematical expectation M(S) the duration of the stock change C before touching one of the screens (upper or lower) is equal to

M(S)= C opt (C max - C 0PT), (3.11)

if condition (3.9) is satisfied.

The objective function in the model is the expected value of the total costs

bm 2 1 e d (x + 2C)

• (3.12)
• * = C max ~ From

The first term in (3.12) reflects the costs of raising funds, and the second - the opportunity costs of holding cash.

After finding partial derivatives E(P) in C and X and equating them to zero, we get

E HER) _ bm 2 12E th dS ~ C 2 x + 3

• (3.13)
• (3.14)

E? (/ g) ? t 2 G E

----=--~-n--= and

Eh x 2 C 3

( ST 2 1 33

• 4?I
• (3.16)
• (3.17)

h ”"max ~^opt in

However, expressions (3.16) - (3.17) are valid if the minimum cash balance is zero: C t[n = 0. Otherwise (if C 1 > 0), the values ​​C opt and C max should be determined as follows:

FROM =C +

• (b b m 2 ^

G b b m 2 ^

Consequently, expressions (3.16)-(3.17) are a special case (with a zero lower limit of the money supply) of the general case described by (3.18)-(3.19) for C. > 0.

The company's control actions on the value of the cash reserve for the general case can be formulated as follows (Fig. 3.5):

1) if the value of the money supply C increases to the upper limit C max » then the company should invest the excess cash in short-term financial investments at the end of the period in the amount C -C(rub.);

Rice. 3.5.

• 1 - implementation of short-term financial investments in the amount of C max - C 0PT; 2 - sale of securities in order to replenish the cash reserve by the amount C opt - C t; P
• 2) if the value of the stock C decreases to the lower limit C min , then the company should replenish the cash reserve by selling securities at the end of the period t2 in volume With opt - Cmin(rub.).

Example 3.2

Suppose that the dispersion of the planned daily cash flow is 70 thousand rubles, the minimum balance of funds under the terms of the agreement with the bank is 200 thousand rubles, and the annual rate of return on securities and fixed costs for transactions with securities are the same as in the previous example. Let us determine the optimal cash balance and the upper limit of the cash reserve.

According to formulas (3.18) - (3.19), we get C opt \u003d 265.9 thousand rubles, and C max \u003d 397 ’ 7 THOUSAND - RU 6 "

from = from +

"" OPT "" "PPP 1

f b bm 2 t^

3-1,2-70 4 0,00022

265.9 (thousand rubles),

C = FROM +3

"“"tah ^tt 1 ^

G bt 2 ^

3-1,2-70 4 0,00022

397.7 (thousand rubles).

If we substitute in the model under consideration a lower value of the return on securities - 5% per year, and take fixed costs for transactions by the company in the amount of 1.8 thousand rubles. per operation, the variance of the planned daily cash flow is 8100 thousand rubles. and the minimum balance of funds under the terms of the agreement with the bank is 45,000 thousand rubles, then the control effects of the micrologistics system on the value of the cash reserve should be formulated as follows:

• 1) if the cash reserve reaches the maximum value C max 46,292 thousand rubles. the company should purchase securities in the amount of 861 thousand rubles, which is the difference between the maximum value of the stock (46,292 thousand rubles) and the return point of the value of the cash reserve C opt (45,431 thousand rubles), i.e. take action 1 at the end of the period
• 2) if the company's cash reserve reaches the minimum value C m1p, equal to 45,000 thousand rubles, then the company should, on the contrary, sell securities, seeking to increase the stock of money from the value (45,000 thousand rubles) to the point of return of the value cash reserve by 431 thousand rubles, i.e. perform action 2 at the end of period G 2 .

Thus, M. Miller and D. Orr, taking into account the company's desire to reduce total costs, including the costs of attracting and opportunity costs of holding funds, proposed an approach to managing cash reserves that is completely opposite to the deterministic approach of W. Baumol. Limitation practical application Miller-Orr model is associated with the theoretical assumptions of the model, for example, with the complete unpredictability of cash flows. Such an assumption means that the company does not have the ability to plan cash inflows and outflows with a sufficient degree of certainty, which is not always true. Companies know the exact timing of the payment of dividends, wages, payments to creditors, tax payments. In addition, the model does not take into account seasonal fluctuations in demand for the company's products and services. Therefore, considering the behavior of a company's net cash flow as a random walk of a certain point between absorbing screens should be recognized as not completely reliable, but to some extent close to reality.

An extension of the Miller-Orr model to predict a company's net cash flow was proposed by Burnell C.

Stone (Bernell K. Stone) . In contrast to the considered stochastic model for calculating the optimal cash balance, B. Stone's model assumes the possibility of a company's cash flow forecasting with a sufficient degree of certainty.

3. Improved Miller-Orr model

for a transitional economy

The transformed Miller-Orr model for cash reserve planning in a transitional economy was proposed by E.Yu. Krizhevskaya 1391. In conditions of high inflation and the absence of state guarantees for investments in investment funds Krizhevskaya is recommended to invest free cash on currency market. The alternative costs of holding cash are the company's losses from cash depreciation, therefore, in the model under consideration, instead of the profitability of short-term financial investments E a inflation rate used E i.

In the model under consideration, the fixed costs of the company for the conclusion of transactions b are replaced by the costs of converting ruble cash into currency values? . expressed as a percentage of the amount

^ -^kon (Snah Servants) ^^konSzht -

In contrast to the Miller-Orr model, the term for holding funds in financial instruments is limited to seven business days, i.e. conversion costs increase three times compared to formula (3.20) and are equal to

b = 6E con C opt. (3.21)

Then, in accordance with the model of cash management in the conditions of their depreciation, the Miller-Orr model considered earlier, we will formulate as follows:

FROM =3 FROM

^max -^opt'

where E - the costs of converting funds in rubles into currency values; o - standard deviation of the cash flow from the average value, calculated by the formula (3.10), from which it follows

o \u003d l / / l 2 /.

A company that has a stable net cash flow in the planned period is recommended to place free cash on deposit in a bank, and in the process of calculating C opt use the following formula:

where E- the profitability of investing funds in a bank on a foreign currency deposit, and the costs of converting ruble cash into currency values? ko|1 are calculated by formula (3.20).

When applying this model, it should be remembered that the opportunity cost of holding cash is estimated at the rate of the highest return on the financial investment that the company refuses. In the Miller-Orr model, such opportunity costs are calculated based on the return on short-term financial investments E. Therefore, it may not be sufficiently justified to add an interest rate to a foreign currency deposit. E to the rate of inflation E and in the denominator of the fraction of the expression under the square root sign in (3.24).

Note that the model under consideration has the following drawback. In the process of transforming the Miller-Orr formula, the company's fixed and volume-independent costs of making deals b are replaced by conversion costs expressed as a percentage of the transaction amount. However, the full cost formula underlying the reasoning of M. Miller and D. Orr is the sum of the costs of raising funds and the opportunity costs of storing cash. At the same time, the cost of raising cash is equal to the product of the company's fixed costs for concluding transactions b on the number of transactions. Therefore, it is not possible to derive the transformed formula (3.22), if we substitute into expression (3.12) instead of the fixed costs of making transactions b variable costs for converting ruble cash into currency values? con (expressed as a percentage of the transaction amount). Therefore, the replacement of fixed costs by interest must be justified.

It can be concluded that the improved Miller-Orr model for a transitional economy is a special case of the approach formulated by M. Miller and D. Orr for practical application in conditions of high inflation and FROM . = 0.

This model assumes that the organization starts operating with a maximum level of cash that is constantly spent over a certain period of time. As soon as the stock of funds reaches a certain limit, the organization replenishes them.

This model is used in case of stability of receipts and expenditures of funds, taking into account the fact that the storage of all monetary assets is carried out in the form of short-term financial investments and the change in the balance of funds occurs from the maximum amount to zero.

Calculation of the maximum and average balance is carried out according to the formula:

R o

Software up to- the planned volume of cash turnover;

P d

If there is a very large amount of money in the account, the organization has the cost of unused opportunities or lost profits. These costs are also called forced costs. If the stock of cash is too small, the organization incurs costs to replenish this stock, which are also called maintenance costs or maintenance costs of a cash replenishment operation.

Taking into account these types of costs, an optimization model is built that determines the frequency of replenishment and the optimal size of the cash balance, at which the total costs will be minimal.

Miller-Orr model

In the Miller-Orr model, receipts and expenditures of funds are stochastic, i.e. independent random events. The main feature of this model is the presence of a certain insurance stock of funds, at the level of which the minimum size cash balance. The maximum amount of cash balance is set at the level of three times the size of the insurance stock.

The cash balance changes until it reaches the upper limit. In this case, the excess cash is withdrawn and invested, for example, in short-term financial instruments. If the cash balance reaches the lower limit, then cash is replenished by selling a part of short-term instruments.

The range of fluctuations in the cash balance between the minimum and maximum levels is calculated by the formula:

KO- the range of fluctuations in the balance of cash;

R o– expenses for servicing one cash replenishment operation;

d2- standard deviation of the daily volume of money turnover;

P d- the level of loss of alternative income during the storage of funds (average% rate for short-term financial investments).

The calculation of the maximum and average balance is carried out according to the formulas:

Despite the clear mathematical apparatus for calculating the optimal amounts of cash balances, both of the above models (the Baumol Model and the Miller-Orr Model) are still rarely used in domestic financial management practice, in particular, for the following reasons:

· a chronic shortage of current assets does not allow organizations to form the balance of funds in the required amount, taking into account their reserve;

· the slowdown in the payment turnover causes significant (sometimes unpredictable) fluctuations in the amount of cash receipts, which, accordingly, is reflected in the amount of the balance of monetary assets;

· a limited list of circulating short-term stock instruments and their low liquidity make it difficult to use indicators related to short-term financial investments in calculations.

3. Differentiation of the average balance of monetary assets in the context of national and foreign currencies. Such differentiation is carried out only by organizations that lead outwardly economic activity. The purpose of such differentiation is to isolate their currency part from the general optimized need for monetary assets in order to ensure the formation of the currency funds necessary for the organization. The basis for the implementation of such differentiation is the planned volume of spending funds in the context of internal and external economic operations in the course of operating activities. In the calculations, formulas are used to determine the need for operating and insurance balances of monetary assets with their differentiation by type of currency.

4. The choice of effective forms of regulation of the average balance of monetary assets. Such regulation is carried out in order to ensure the constant solvency of the organization, as well as to reduce the estimated maximum and average need for cash balances.

The main method of regulating the average balance of cash assets is to adjust the flow upcoming payments(postponement of certain payments by prior agreement with counterparties). This adjustment is carried out in the following steps.

At the first stage on the basis of the plan (budget) for the receipt and expenditure of funds in the coming quarter, the range of fluctuations in the balance of the organization's monetary assets in the context of individual decades is studied. This range of fluctuations is determined in relation to the minimum and average indicators of the balances of monetary assets in the forthcoming period.

At the second stage ten-day periods for spending funds (in relation to their receipts) are regulated, which allows minimizing the balance of cash assets within each month and for the quarter as a whole. Optimality criterion This stage of regulation of the flow of forthcoming payments is the minimum level of the root-mean-square (standard) deviation of the ten-day values ​​of the balance of the organization's monetary assets from their average size.

At the third stage the values ​​of the balances of monetary assets obtained as a result of regulation of the flow of payments are optimized taking into account the envisaged size of the insurance balance of these assets. First, the maximum and minimum balances of monetary assets are determined, taking into account the new range of their fluctuations and the size of their insurance stock, and then their average balance (half the sum of the minimum and maximum balances of monetary assets).

The amount of monetary assets released during the ten-day adjustment of the flow of payments is reinvested in short-term financial instruments or in other types of assets.

There are other forms of operational regulation of the average balance of monetary assets, providing both an increase and a decrease in its size. These forms are considered as part of the management cash flows organizations.

5. Ensuring profitable use of the temporarily free balance of monetary assets. At this stage of the formation of the monetary asset management policy, a system of measures is developed to minimize the level of losses of alternative income in the process of their storage and anti-inflationary protection. The main activities include:

coordination with the bank settlement service organization, conditions for the current storage of the balance of monetary assets with the payment of deposit interest on the average amount of this balance (for example, by opening a checking account in a bank);

· the use of short-term monetary investment instruments (first of all, deposits in banks) for temporary storage of insurance and investment balances of monetary assets;

· the use of highly profitable stock instruments for investing the reserve and the free balance of monetary assets (government short-term bonds; short-term bank certificates of deposit, etc.), but subject to sufficient liquidity of these instruments in the financial market.

6. Construction of effective systems of control over the organization's monetary assets. The object of such control is the aggregate level of the balance of monetary assets that ensure the current solvency of the organization, as well as the level of efficiency of the formed portfolio of short-term financial investments - cash equivalents of the organization.

Monetary assets play a decisive role in the process of ensuring solvency for two types of financial obligations of the organization - urgent (with a maturity of up to one month) and short-term(with a deadline of up to three months); Current responsibility with a maturity of up to one year are provided mainly by other types of current assets. Control over the aggregate level of the balance of monetary assets while ensuring the solvency of the organization should be based on the following criteria:

· urgent liabilities ≤ balance of monetary assets

· current liabilities ≤ balance of cash assets + net realizable value of current accounts receivable

Control over the level of efficiency of the formed portfolio of short-term financial investments - cash equivalents of the organization should be based on the following criteria:

· the level of profitability of the portfolio as a whole and its individual instruments ≥ the average market level of profitability of short-term investments with an appropriate level of risk

· rate of return of each investment instrument > inflation rate

[Kovalev, 1999]. The essence of these models is to give recommendations on the range of variation of the balance of funds, going beyond which involves either the conversion of funds into liquid securities, or the reverse procedure.

NB average cost reserves can be calculated using the Baumol model

The most popular theory of demand for money, which considers it from the point of view of optimizing money reserves, is based on the conclusions reached independently by William Baumol and James Tobin in the mid-1950s. Today this theory is commonly known as the Baumol-Tobin model. They pointed out that individuals maintain stocks of money in the same way that firms maintain stocks of goods. At any given moment, a household is holding a portion of its wealth in the form of money for future purchases.

At the same time, it is possible to obtain an algebraic expression for the demand for money in the Baumol-Tobin model. This equation is interesting because it allows you to represent the demand for money as a function of three key parameters of income, interest rate and fixed costs.

There are theories of the demand for money that emphasize such a function of money as a medium of exchange. These theories are called transactional demand theories for money. In them, money plays the role of a subordinate asset, accumulated only for the purpose of making purchases. Thus, the Baumol-Tobin model analyzes the benefits and costs of holding cash. The benefit is that there is no need to visit the bank for each purchase (transaction). The total costs are determined by the shortfall in interest on possible savings accounts (d), and the client's time to visit the bank based on his earnings (F). If Y is the amount of annual spending on purchases planned by the individual, then at the beginning of the year this amount will be equal to Y, at the end of the year - 0, and its average annual value - Y / 2. If an individual visits the bank not once a year, but N times, then the average annual value of the amount of cash in his hands will be Y / (2xN). The interest not received by him will be (rxU) / (2x.N), and the costs of visiting the bank will be equal to FxN. The greater the number of visits to the bank (N), the higher the costs associated with this, but the smaller the amount of lost interest.

Baumol model. According to W. Baumol, the balance of funds on the account is in many respects similar to the balance of inventory, therefore, the model of the optimal batch of the order can be used to optimize it. The optimal amount of funds in the account is determined using other variables C - the amount of cash in liquid securities or as a result of a loan C / 2 - the average balance of funds in the account C - the optimal amount of cash that can be received from the sale of liquid securities or loan C /2 - the optimal average balance on the account F - transaction costs for the purchase and sale of securities or servicing the loan received for one operation T - total

So, in accordance with the Baumol model, the balances of DA for the coming period are determined in the following amounts

The most widely used for these purposes is the Baumol Model, which was the first to transform the previously considered EOQ Model for cash balance planning. The starting points of the Baumol Model are the constancy of the cash flow, the storage of all reserves of monetary assets in the form of short-term financial investments and the change in the balance of monetary assets from their maximum to a minimum equal to zero (Fig. 5.17.)

Figure 5.17. formation and spending of the balance of funds in accordance with the Baumol Model.

Taking into account the losses of the two types considered, an optimization Baumol Model is constructed, which allows determining the optimal frequency of replenishment and the optimal size of the cash balance, at which the total losses will be minimal (Fig. 5.18.)

The mathematical algorithm for calculating the maximum and average optimal cash balances in accordance with the Baumol Model has the following form

An example is to be determined on the basis of the Baumol Model, the average and maximum amount of cash balances based on the following data, the planned annual volume of the company's cash turnover is 225 thousand conventional units. den. eg. the cost of servicing one operation of replenishment of funds is 100 conventional units. den. eg. the average annual interest rate on short-term financial investments is 20%.

In accordance with the Baumol model,

Baumol's model is simple and quite acceptable for enterprises whose cash costs are stable and predictable. In reality, this rarely happens, the balance of funds on the current account changes randomly, and significant fluctuations are possible.

What is the fundamental difference between the Baumol model and the Miller-Orr model?

The Baumol model is an algorithm that allows you to optimize the size of the average balance of the company's cash assets, taking into account the volume of its solvent turnover, the average interest rate on short-term financial investments and the average amount of costs on short-term investment operations.

Baumol model. Suppose that the organization has a certain amount of cash, which is constantly spent on paying supplier bills, etc. In order to pay bills on time, a commercial organization must have a certain level of liquidity. As a price for maintaining the required level of liquidity, the possible income from investing the average cash balance in government securities is taken. The basis for this decision is the assumption that government securities are risk-free (that is, their degree of risk can be neglected). Cash received from the sale of products (goods, works, services), a commercial organization invests in government securities. At the moment when the funds run out, there is a replenishment of the stock of funds to the initial value.

When a household takes the entire required amount with the help of one large-scale withdrawal M = P x Q, they are provided with their own needs, but interest is lost. In the Baumol-Tobin model, we can obtain an algebraic expression for the demand for money MD = M/2. The peculiarity of the equation is that it allows us to represent the demand for money (in terms of one visit to the bank) as a function consisting of three key parameters of fixed costs Pb, income Q, interest rate r

The Baumol model assumes that when an excess of money appears in the account in excess of the calculated amount of the optimal stock, it uses it to buy short-term securities in order to generate income, and when the stock of money decreases, it sells some of these securities, increasing the stock of money to the optimal level.

The Baumol model is suitable for stable predictable cash expenditures and receipts, it does not take into account seasonal or random fluctuations, i.e., it simplifies the real situation. Later, other models were developed that take into account the daily variability of cash flows (for example, the Miller-Orr model, 1966). However, all formalized models have certain limitations, therefore, in the practice of cash management, they are used as auxiliary to establish the optimal amount of cash.

Let us turn to the analysis of the properties of the transaction demand function for money obtained from the Baumol-Tobin model. First, as follows from formula (4), the demand for money depends negatively on the interest rate. This is because an increase in the interest rate leads to an increase in forgone interest payments and thus encourages the individual to go to the bank more often and hold less cash.

In addition to the two traditional factors discussed above that affect the demand for money, we can single out one more parameter that, according to the Baumol-Tobin model, affects

Thus, the velocity of circulation of money depends positively on

Enterprise Finance

refinement of the Baumol-Tobia model for cash management

A.G. MNATSAKANYAN, Head of the Department of Finance and Credit, Doctor economic sciences, Professor

IN AND. RESHETSKY, Candidate of Physical and Mathematical Sciences, Associate Professor, Department of Financial Management, Baltic Institute of Economics and Finance,

Kaliningrad

Optimal cash management decisions are made based on several models. The choice of one or another model depends on the specifics of the practical problem of financial management being solved. Among them, the Baumol-Tobin model occupies a special position and belongs to the classical results of financial management, since it has an important theoretical significance.

The Baumol-Tobin model is discussed in many books on economics and finance (sometimes called the "Baumol model"). At the same time, the authors focus on explanations on the practical use of the main results and usually neglect, to one degree or another, the detailed conclusion and calculation of the main results (this is typical not only for this model), which often leads to unconscious replication of erroneous results. However, the logic of the derivation of the main result (formula) is extremely important both methodologically and for the correct use of the model, since it must always specify the conditions for the applicability of this model, its essence and be given detailed description internal picture of the financial process. The logic for obtaining the main result, i.e. the formula, is a description of the technology for managing the corresponding financial process. There are no perfect control technologies, and any of them can become even more perfect. The word "model" is not a very good term, as it emphasizes unreality, far-fetchedness. More

it is correct to speak here about technology, but not about model. In this article, we will adhere to the generally accepted terminology, as it is convenient for drawing parallels and comparing our results with the results following from the work of Baumol-Tobin.

The Baumol-Tobin model is most important not from a practical, but rather from a theoretical point of view, as it underlies the development of many other economic and financial concepts and financial technologies. In particular, this concerns the technology for determining the demand curve for cash balances, as well as building stochastic cash management models. For objectivity, we note that the Baumol-Tobin model was based on Wilson's ideas on inventory management.

Therefore, we will once again, but in more detail, describe the principle of operation of this model (technology) and analyze its vulnerabilities in order to obtain more correct and accurate results, which are given below. We only note that this model has one important shortcoming (delusion), which is of a fundamental and general nature, concerning the time horizon of financial planning. In the general case, this horizon cannot be short, which is proved in our article. In this case, the order of consideration will be as follows.

1. At the very beginning, it will be revealed that in the Baumol-Tobin model the opportunity cost of the costs associated with the under-received

finance and credit

interest income on a bank deposit (or any other asset). In reality, these costs are much higher than previously thought.

2. It will be shown that this model is approximate (time linearization), so the results can (but not desirable) be applied only at sufficiently low interest rates (note that in Russia these rates still remain relatively high) and a small number of bank visits N in order to withdraw money from a deposit account. Note that the Baumol-Tobin model, which is obviously approximate, does not imply a quantitative criterion for this approximation. Therefore, the conditions for its applicability remain unclear.

3. In conclusion, for the first time, exact results will be obtained in the form of a transcendental equation, which allows making optimal decisions for any interest rates and any number of visits to the bank N to withdraw money from a deposit account (i.e., in the most general case). It will be shown that the Baumol-Tobin model is a special case of these general results, and this can serve as an additional proof of their validity, i.e. our results are reduced to the results of the Baumol-Tobin model as the interest rate tends to zero.

As usual, here we will understand money as the most liquid type of assets, usually referred to in macroeconomics as M1, which includes cash and money on settlement, current and other demand accounts. This money brings either very little income, or no income at all. There are other monetary aggregates M2, MZ, etc., which are less liquid, but with the same degree of risk, can bring significant income over time: term savings deposits, government bonds, certificates of deposit. Despite the great variety of different types of assets that can generate income over time, the population still keeps part of its funds or assets in the form of cash, or rather in the form of M1. This means that the population shows a non-zero demand for cash. Economists were faced with the task of quantifying this demand. The usefulness of money is generally determined, as is well known, by three functions: a medium of circulation, a measure of value, and a means of preserving income. It is obvious that cash, as a means of circulation, surpasses all other monetary aggregates, since it is absolutely liquid. But cash

as a means of preserving income is worse than other forms of money. Demand theories for money based on its role as a medium of exchange are called transactional demand theories for money. Cash is needed to make purchases or, in general, to make transactions. Among the various transactional theories of demand for money, the Baumol-Tobin model is still the most widely known and popular, although it appeared more than half a century ago - in 1952. In addition to determining the money demand curve, this model allows you to optimally manage the cash balances of companies (cash balances ), as well as citizens. The desire for optimality should set the parameters of the demand curve. Companies should appropriately forecast their cash balances at the optimum level. Based on knowledge of the company's future cash needs, the manager must decide how much cash balances to have. Excess cash can be invested in high-quality short-term securities, pay dividends, create additional reserves, etc. A lack of cash forces the company to borrow, sell securities, as it is necessary to pay bills and be prepared for various unexpected situations. All these measures relate to such an important element of company management as cash management, the task of which is to determine the optimal value of the cash balance. Cash balance is the amount of cash that changes over time. household(family) or company. The same problems have to be solved not only by companies, but also by the government, administrations of regions, cities, etc.

The main advantage of cash is its convenience, since it eliminates the need to go to the bank for every purchase and incur some costs associated mainly with the loss of time. Cash could be placed in a bank, invested in bonds or even stocks and have a corresponding additional income. Therefore, we can say that cash brings losses in the form of unearned interest (the alternative cost of money is always present as the other side of the coin). That is, you always have to pay for the convenience of cash, but not overpay. The task of each person (manager) is to reduce the total costs to a minimum. Suppose that a person knows (planned on the basis of pre-

future experience) that during the next period T0 = 1 (for example, five years, a year, a month, etc.) he will need S0 rubles in cash. Note that S0 here has the financial meaning of cash flow, since this amount refers to a conventional unit of time T0 (for example, to a year). It is natural to assume that he will spend this amount S0 evenly, for example, daily at So/365 rubles.

There are several options for cash management. You can withdraw the entire amount of S0 at the beginning of the year and then spend it evenly throughout the year. The average annual amount, in the sense of the arithmetic average, that a person will have during the year will be + 0) = S0 / 2. As usual, we take one year as the unit of time. This is done for illustrative purposes only. In fact, our approach provides for the possibility of choosing any arbitrary unit of time.

The second cash management option is to visit the bank twice during the year. At the beginning of the year, the first half of the amount equal to S0/2 is withdrawn, which is evenly spent during the first half of the year, decreasing to zero. At this time, the other half, located in the bank, brings interest income. Consequently, during the first half of the year, on average, there will be an amount of cash on hand equal to ^¿ / 2 + 0) / 2 = S0 / 4 (this is the arithmetic mean, which is legitimate here due to the hypothesis of even spending of cash, which leads to to the amount of cash available in the form of an arithmetic progression). At the end of the first semester, the second amount S0/2 is immediately withdrawn from the bank account for expenses in the next second semester. Consequently, during the second half of the year, on average, there will be an amount of cash equal to ^¿/2+0) /2=S0/4, which is the same as in the first half of the year. If during each half year the average amount of cash on hand was equal to So / 4, then the average annual amount of cash will be S0 / 4, which is obvious.

Similarly, you can consider a three-time, four-time visit to the bank. In general, when visiting Iraz Bank during the year, the amount S0 / N will be withdrawn each time. This amount will be spent during the period 1/I = T, changing during this time from the value of S0/N to zero.

Consequently, in the general case, the average annual amount of cash will be ^¿/N + 0) /2 = S0/2N (this is the average amount of a decreasing arithmetic progression). From this formula, it can be seen that the more I, the less the average annual amount "per

hands”, which means less loss from unearned interest. This is the rather non-obvious logic underlying the Baumol-Tobin model. Therefore, below we will analyze and correctly determine these losses more carefully and present more convincing justifications for this logic.

Alternative value of cash. Now it is necessary to determine the losses from holding cash on hand. Usually, in the economic literature, without proof, on an intuitive level, it is assumed that these losses are proportional to the product of the bank rate R0 by the average annual amount of cash S0/2N. However, this statement is erroneous, which leads to an underestimation of losses relative to their true value (the authors of this model followed the logic of the Wilson model related to inventory management). Losses from cash storage, or rather their correct calculation, may have independent economic significance that is not related to this context. In particular, the underestimation of these losses can be misleading to managers who will not pay attention to such "little things" and will ignore cash management. In addition, experimental verification of the cash demand curve did not confirm the theoretical result, as shown in . Therefore, a corresponding exact calculation of these losses is proposed below.

Let R0 be the annual bank rate, or the rate of return of an alternative investment of money. The Baumol-Tobin model "by default" assumes that this interest rate R0 is set relative to a conditionally unit period Т0, i.e. R0 = R0(T0), where Т0 = 1. This circumstance should also be taken into account when using this model, otherwise gross miscalculations are possible. For example, if the planned period T0 = 6 months, then the rate ^ should be determined relative to the period of 6 months, which in the Baumol-Tobin model is assumed to be equal to one. This is a clear drawback of this approach, since certain difficulties arise, which often lead to errors. All these difficulties could be easily bypassed if the equality T0 = 1 was not required. However, for now we will stick to the traditional approach. These problems are covered in more detail in the works. Let us assume that this rate is small enough, only in this case it is possible to apply simple interest, which is done by default in the Baumol-Tobin model. Let's explain this below.

finance and credit

At the beginning of the year, during the first visit to the bank, the amount S0 / N will be withdrawn from the account, the interest income on which during the year would be L ^ / N if this amount were in the bank, i.e. it represents the loss from the first withdrawal of the amount S0 /N. Therefore, the cost of the first withdrawal of the amount from the bank account will be:

where multiplication by one is left for clarity, since it should be borne in mind that this is the time T0 = 1.

The second visit to the bank will occur after a period of time T = 1/N, and the amount S0/N will be withdrawn again. The entire unit period (one year, for example) is divided into N equal intervals. During one period T, this amount brings interest income, but in the remaining ^ - 1) periods, each of which is equal to T \u003d 1/N, no interest income will be received, which will amount to losses equal to:

^i -^.^=^ ^(1 -1),

where the factor (1-1/^ describes the time similar to the unit in the previous expression, i.e. the time during which this amount could have been on a bank deposit, but was not. After the time 2T, the third visit to the bank should occur, and again withdrawn the amount S0/N. Losses from lost interest income in this case will be:

S0 i - 90 i.- \u003d 90 i0 (1 - -).

N 0 N ^ N N 0 N Further consideration can be carried out by analogy. In the general case, after j periods there will be 0 + 1) -th visit to the bank and the amount S0 / N is withdrawn from the account, where y = 1, 2, ... W The cost of lost interest income in this general case will be equal to:

90I0 - 90 I \u003d ^ R0 (1 -C.

N N N N N In particular, when y = N, this general formula can be used to determine the losses from the last N^0 on the cash withdrawal account, which will be: i 50 i N -1 = 50 i (1 N -1) i

This result is pretty obvious. Indeed, the amount S0/N will be withdrawn from the account at the beginning of the last ^th period and will not generate income only during the time 1/K. and obtained on the right side of the last equality. During the first (N-1) periods, this amount will still earn interest

income. The cost of this cash will be the most minimal compared to all the others. The maximum loss will be given by the very first withdrawal of cash from the account.

Let us now find the total loss of interest income from interest not received, designated as C (N), for the entire planning period (one year). To do this, let's sum up all the losses for each individual cash withdrawal that were obtained above:

) = N 1 + ~N Ro(1 -N +

+^ Ro(i - -2) + ...+^ Ro(i - N^) =

1 + (1 - -) + (1 - -) + (1 - -) +... + (1 - N-1)

Above, obvious algebraic transformations were performed in order to extract the sum of the terms of an arithmetic progression. Each subsequent term of the progression (they are in parentheses) is obtained from the previous one by subtracting the value 1 / K. We describe in detail all these stages of calculations, since it was here that the first mistake was made more than half a century ago and then repeated many times in books and articles. Using the formula for the sum of members of an arithmetic progression, we find the alternative cost of cash:

С1(N) = -°- R0 1 N 0 2

N = R0(1 + N) = 2N 0

= -~ R + - S0 R0. 2N^200

Our result (1) differs from similar expressions in that a new term has appeared to the right of the last equal sign. Previously, only the first term £0L0/2W was present in these costs. It is strange that for such a long time no attention was paid to this error. In addition to the computational proofs of the correctness of expression (1), which were presented above in full detail, we can also consider the financial meaning of this expression and its predecessor. As usual, in such cases, one should resort to some extreme cases of verification, where detailed calculations are not required. For example, in the case of a single visit to the bank, it follows from formula (1) that with N=1, the opportunity costs will be

C1 (1) \u003d I + - S0 I \u003d ^ I.

The dependence of costs on the number of visits to the bank

The validity of this result leaves no doubt. This is equal to the interest income for the year on the amount of the deposit Sg, the profitability of which is equal to B.a If we use the previous result, we will get only half of the actual costs.

The second extreme case is an infinitely large number of visits to the bank N, at which the minimum cost (1) is achieved. If all losses were reduced only to this type of costs, then the minimum of these losses would be achieved with the maximum possible number N of visits to the bank during a conditionally single period (year). Theoretically, this value can be equal to infinity (i.e., arbitrarily large), then the costs will be due only to the second term SgRg/2 of equality (1). That is, even with an infinitely large value of A, this type of cost will not be reduced to zero, but will be equal to 0.5^^. So far, this is the main difference between our results and the results of the Baumol-Tobin theory, from which it directly follows that in this case these costs will be reduced to zero. The erroneousness of such conclusions seems obvious, given that the problem is reduced to a continuous annuity. For a sufficiently large value of N, we can assume that the withdrawal of amounts occurs continuously. The amount of Sg in the account will continuously decrease to zero during the year, which will be the reason for the loss of interest income.

This blunder is quite obvious from simple qualitative considerations, if it is correct to proceed to the continuous calculation of interest income, and, as can be seen from this expression, for N > 1, the contribution of the second term to these losses is always higher than the first term in formula (1). That is, the losses from lost interest income are actually much higher than previously thought. These differences are visually represented by the C(W) plot (dashed line).

This graph does not asymptotically tend to the abscissa axis (zero value), as it was supposed earlier, but approaches the horizontal straight line C1(da) = SgRg/2 (dash-dotted line). Note that sometimes in the economic literature, the dependence of costs on the value of the cash balance, and not on N, is built, which does not change the essence of the problem.

Having Full description costs in the form of formula (1), we obtain additional opportunities in making optimal decisions on managing the company's cash balances. Withdrawing money from an account makes sense if it can be reinvested at a higher rate of return (or utility for individual), which is assumed by default in the Baumol-Tobin model. Knowing the costs (1), they can be compared with the income that can be received from reinvestment. That is, we get the possibility of optimal management not only of cash, but also of any other assets. Withdrawing money from the account will make sense if the net present value is at least zero. Further details can be omitted, since the costs (1) are estimated here approximately, as shown below. Further more accurate results will be obtained. The underestimated level of costs in the Baumol-Tobin model may lead some managers to ignore them and not apply methods of optimal cash management. In addition, this error is also of a logical nature, distorting some of the qualitative ideas of investment analysis.

Some refinements of the model. Let us show that when obtaining the result (1), simple (approximate) interest was actually used, so formula (1) inaccurately estimates the imputed costs due to the lost interest income. In addition, we will take one more step towards a more adequate solution of this problem.

If N is the number of annual visits to the bank, then the period of time T (measured in years) between each visit to the bank will be equal to

T = - (year). N

Note that N is a flow quantity, and its dimension must correspond to the number

bank visits per unit of time (for example, one year). The amount £ regularly withdrawn from the account is equal to:

For m periods, each of which is equal to T, interest income must be accrued on the amount of £, equal to:

S(1 + R0)mT -S and mTR0S = m

where the approximate equality is obtained up to linear terms of the expansion into a series (simple percent) . The expression to the left of the equals sign is exact. With regard to our problem, t is the number of periods during which the amount £ = S0 / N was not in the account, and therefore it is lost interest income. For the first amount withdrawn m = N for the second m = N- 1), for the third m = N- 2), etc. These values ​​should be alternately substituted into expression (A), which will give the corresponding imputed costs that were obtained when deriving formula (1).

In addition to loss of interest income, there is another component of the total costs C2(I), directly related to the process of withdrawing funds from the account that brings interest income. As shown above, the cost C1 decreases with an increase in the number of visits to the bank N. However, with an increase in N, the cost C2(I) associated with visiting the bank increases.

Following tradition, we will give the simplest interpretation of the appearance of costs C2(^) associated with visiting the bank. Denote by P the costs of one visit to the bank. Costs P do not depend on the amount withdrawn from the bank account (this is a fundamental condition). Basically, they are determined by the loss of time for a trip to the bank and back, waiting in line and processing the withdrawal of money from the savings account, commissions, paying contracts, etc. For example, if you earn 40 rubles per hour and the total loss of time is 5 hours per visit, the opportunity cost of the lost time will be is equal to: 5h 40 rubles / h = 200 rubles To this amount of losses should be added the direct costs of a trip to the bank and back.In addition, the more often the money is withdrawn from the account, the lower the interest rate on term deposits which should also be included in the costs. The amount of these costs should be calculated by the manager in each specific case separately, which is not the purpose of the article. per year costs for

visits to the bank, which are indicated by C, will be:

C2 (N) = P N. (2)

Obviously, if all losses were reduced only to this type, then their minimum would be achieved with a single visit to the bank at the beginning of the planning period (year).

In determining this type of cost, we followed the classic approach of withdrawing money from a bank account. However, the receipt of cash can in practice occur in different ways, as already discussed above. In general, the application of this technique can require a lot of creative effort and is not limited to bank deposits. It could also be taking out a loan or selling (or selling off in the event of bankruptcy) the company's profitable risky assets. As a rule, the higher this return on risky assets, the greater R. But in all these cases, the costs of "cashing out" should be determined by formula (2), otherwise a different management technology may be required.

The total amount of all costs for the planning period (year) is equal to:

TC(N) = C + C2 = 2 R S + 2 R0 So N-1 + PN. (3)

In this equation, only N depends on the will and desires of the manager (endogenous variable), all other variables do not depend on him (exogenous variables), so they should be considered constant, and the manager can change the variable N as he considers beneficial. The natural desire of the manager is to reduce the total costs (3), which depend on N. The task of each manager is to calculate the number of visits to the bank N, in which these total costs become minimal:

The first order condition for the minimum has

where expression (3) was substituted for TS. Note that there is no contribution to the derivative of total costs from the term A^^, since it does not depend on N. Therefore, the solution obtained by Baumol and Tobin turned out to be correct. Solving equation (4), we find the optimal number of visits to the bank during one year:

at which the total losses will be as low as possible. With this already specific value of N, the optimal amount of cash withdrawn each time from a bank account should be equal to

This formula can also be used to determine the optimal amount of the cash balance that the company should borrow or receive as a result of the sale of securities, then P is the transaction costs of dealing with securities or obtaining loans.

If there are 365 days in a year, then this amount will be withdrawn from the account every 365/^ days. Accordingly, the average annual amount of cash on hand will be

This formula shows that the higher the interest rate, the lower the average annual amount of cash in the hands of the population and firms. The validity of this statement is not in doubt. In the economic literature, the Baumol-Tobin model is also used as a model for the demand for money. Note that it was the demand for cash that initially interested the authors of this model, and not the problem of optimal cash management. Equation (7) is taken as the demand equation. The total costs when equality (5) is fulfilled have a minimum value equal to:

TC (Ne) = 2 R e o +

where expression (5) was substituted in (3) instead of N. It is easy to verify that this is indeed the minimum value by taking the second derivative, which is obviously greater than zero: d2TC/dN2 > 0. Thus, not only the necessary condition for minimum, but sufficient.

The considered model has some shortcomings that are obvious today, which in no way detract from the merits of this theory, which represents obvious prospects for development and refinement. For example, firstly, you can fully take into account the discounting of future costs. Secondly, most of the population of the Russian Federation receives wages in cash. Other types of income also come in cash. In such cases

should consider the inverse problem compared to the one that was considered above. A person, having received income, must decide how much money he will leave in cash, and how much he will put in a bank savings account that brings interest income. This approach is usually applied to describe the first half of a person's life until his retirement, when he strives to earn more than he spends in the same time. Above, in the Baumol-Tobin model, in fact, a person who is retired and owns money in a savings account was considered.

At the same time, this model has a much broader applied character. In particular, it concerns the management of a portfolio of securities held in brokerage company or a bank. The securities may different level liquidity independent of profitability.

With the same success, the Baumol-Tobin model can be used when selling not only securities, but also real estate, which can be called "cashing in real estate investments." The only problem is that the assets being sold are divisible. It is difficult to do this in relation to real estate directly, but in principle it is possible.

Literature

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3. Van Horn J.K. Fundamentals of financial management / J.K. Van Horn. M.: Finance and statistics, 1996. 799 p.

4. Worst I. Economics of the firm / I. Worst, P. Revent-low. M.: high school, 1994. 272 ​​p.

5. Kovalev VV Introduction to financial management / VV Kovalev. M.: Finance and statistics, 1999. 768 p.

6. Mankyu G. N. Macroeconomics / G. N. Mankyu. M.: MGU, 1994. 735 p.

7. Reshetsky V. I. Financial mathematics. Analysis and calculation investment projects/ V. I. Reshetsky. Kaliningrad: BIEF, 1998. 395 p.

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9. Trenev N. N. Financial management / N. N. Trenev. M.: Finance and statistics, 1999. 495 p.

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11. Shim D.K. Financial management / D.K. Shim, D.G. Siegel. Moscow: Filin, 1996. 365 p.

Cash flow management methods.

Baumol's model is simple and sufficiently acceptable for an enterprise whose cash costs are stable and predictable. In reality, this rarely happens; the balance of funds on the current account changes randomly, and significant fluctuations are possible.

The initial provisions of the Baumol Model are the constancy of the cash flow, the storage of all reserves of monetary assets in the form of short-term financial investments and the change in the balance of monetary assets from their maximum to a minimum equal to zero.

Based on the presented graph, it can be seen that if the replenishment of cash balances through the sale of part of short-term financial investments or short-term bank loans was carried out twice as often, then the size of the maximum and average cash balances at the enterprise would be half as much. However, each transaction for the sale of short-term assets or obtaining a loan is associated with certain expenses for the enterprise, the amount of which increases with an increase in the frequency (or reduction in the period) of replenishment of funds. Let's denote this type of expenses with the index "P o" (expenses for servicing one operation of replenishing cash expenses).

Rice. 2.1.1 Formation and spending of the balance of funds in accordance with the Baumol Model.

To save the total cost of servicing replenishment operations, you should increase the period (or reduce the frequency) of this replenishment. In this case, the size of the maximum and average cash balances will increase accordingly. However, these types of cash balances do not bring income to the enterprise; moreover, the growth of these balances means the loss of alternative income for the enterprise in the form of short-term financial investments. The amount of these losses is equal to the amount of cash balances multiplied by the average interest rate on short-term financial investments (expressed as a decimal fraction). Let us designate the size of these losses by the index "P D" (loss of income when storing cash).

The mathematical algorithm for calculating the maximum and average optimal cash balances in accordance with the Baumol model is as follows (2.1.5 and 2.1.6, respectively):

; (2.1.5)

where YES max - the optimal size of the maximum balance of the company's cash assets;

The optimal size of the average balance of the company's cash assets;

Р О - expenses for servicing one operation of replenishment of funds;

P D - the level of loss of alternative income during the storage of funds (average interest rate on short-term financial investments), expressed as a decimal fraction;

PO DO - the planned volume of cash turnover (the amount of money spent).