Cash balance optimization (Baumol model)

One of the main tasks of managing cash resources is optimizing their average balance. We are talking about the total balance in bank accounts and cash). First of all, the question arises: why do cash remain idle and not used in in full, for example, for purchasing valuable papers, generating income in the form of interest. The answer is that cash has absolute liquidity compared to securities.

The financial manager is faced with the task of determining the size of the cash reserve, based on the fact that the price of liquidity does not exceed the marginal interest income on government securities.

Thus, the standard policy regarding absolutely liquid assets in a market economy this is the case. The company must maintain a certain level of free cash flow, which, for insurance purposes, is supplemented by a certain amount of funds invested in liquid securities, that is, in assets that are close to absolutely liquid. When necessary or at certain intervals, securities are converted into cash; When excess cash accumulates, it is either invested on a long-term basis or in short-term securities, or paid out as dividends.

From the perspective of investment theory, cash represents one of the special cases of investing in inventory. Therefore, general requirements apply to them:

A basic reserve of cash is required to carry out current calculations;

Certain funds are required to cover unexpected expenses.

It is advisable to have a certain amount of free cash to ensure possible or projected expansion of their activities.

The difficulty of optimizing the level of the average cash balance of an organization is due to the dialectical contradictory unity of its goals, which consists in the need to simultaneously maintain high business activity and a stable financial position.

The essence of this contradiction is also manifested in the contradictory unity of requirements for the optimal level of cash balance in the short and long term.

In the short term, from a liquidity perspective, it is necessary to maximize cash balances (to maintain solvency); from the standpoint of business activity - minimization (money should change natural form into commodity goods, then they become capital and can make a profit). With this approach, it is clear that in the long term, liquidity and business activity inseparable. Sufficient business activity is the reason for generating financial result, which means an increase in the cash balance, and therefore solvency. Only sufficient solvency makes it possible to finance the continuous production process in a timely manner and in the required amount.

In the theory of financial management, there are two methods for determining the optimal amount of funds: the Baumol model and the Miller-Or model. We will consider the Baumol model.

In the Baumol model, it is assumed that the enterprise begins to operate with the maximum and appropriate level of funds for it, and then constantly spends them over a certain period of time. The company invests all incoming funds from the sale of goods and services in short-term securities.

As soon as the cash reserve is depleted, that is, it becomes equal to zero or reaches a certain specified level of safety, the company sells part of the securities and thereby replenishes the cash reserve to its original value.

According to Baumol's model:

1) minimum balance monetary assets is taken as zero:

2) the optimal (aka maximum) balance is calculated using the formula:

where V is the projected need for funds in the period (year, quarter, month);

с - expenses for converting cash into securities;

r - acceptable and possible interest income for the enterprise on short-term financial investments, for example, in government securities.

For an enterprise, the optimal cash balance is the amount of 220,857 rubles.

Thus, the average cash holding is Q/2,

The total number of transactions to convert securities into cash is equal to:

Total costs of implementing such a management policy in cash will be:

The first term in this formula represents direct expenses, the second is the lost profit from keeping funds in a current account instead of investing them in securities.

ST=13785*104+13*110428.5=1433640+1435570.5=2869210.5rub

The costs of implementing this policy amounted to 2,869,210.5 rubles.

The disadvantage of the model is that it poorly describes the situation of returning funds from short-term financial investments.

There is no single way to determine the optimal cash balance. The trade-off decision depends on your money management strategy. With an aggressive strategy, the priority is business activity, and with a conservative strategy, a sufficient value of indicators financial condition characterizing liquidity, solvency and financial stability.

Baumol's model is simple and quite acceptable for enterprises, cash expenses which are stable and predictable. In reality, this rarely happens - the balance in the current account changes randomly, and significant fluctuations are possible.

One of the most famous money management models is the Baumol model. It was developed in 1952 by William Baumol (W.J. Baumol) based on the inventory management model EOQ (Economic Order Quantity). Basic assumptions of Baumol's model:

1. The enterprise’s sustainable need for funds;

2. Everything cash receipts the company immediately invests in highly liquid securities;

3. The cost of converting investments into cash does not depend on the amount being converted (fixed for one transaction);

4. The company begins operations with maximum reasonable cash balances.

Baumol's model is applicable in cases where an enterprise can predict its cash needs with a sufficient degree of certainty. In this case, as already noted, it is assumed that the enterprise begins to operate with the maximum appropriate level of funds Q+m. Then the enterprise evenly (due to sustainable needs) spends these funds over a certain period of time (see Fig. 8.5).

Rice. 8.5. Changes in enterprise cash balances according to the Baumol model

As soon as cash balances fall to the minimum allowable safety stock m, the company sells part of its short-term investments and restores its cash reserve to its initial level.

In this case, it is assumed (see assumption 2) that the funds received by the enterprise as a result of the sale of products, goods, and services are transferred as received into short-term investments.

Let us introduce the following notation:

V- the projected total need for funds for the period (usually a year);

c- costs of converting short-term investments into cash (transaction costs);

r- average annual return on short-term investments.

The number of conversions of securities into cash during the period will be .

Total enterprise costs TC related to cash management for the period will be:

where the first term represents transaction costs and the second term represents opportunity costs.

To determine the amount of replenishment of cash balances Q opt., with which TC minimally differentiate the function TC(Q) By Q:

Equating expression (8.2) to zero, we find the value Q, corresponding to the minimum of the function TS:

A graphical illustration of cost minimization using the Baumol model is presented in Figure 8.6.

Rice. 8.6. Minimizing costs according to the Baumol model

Graphs in Fig. 8.6 are built under the following conditions: V= 2000 thousand rubles, c= 0.1 thousand rubles, r= 5%, m= 50 thousand rubles.

Calculation using formula (8.8.3) showed that Q opt≈ 89.44 thousand rubles. The same result can be obtained graphically with an acceptable degree of accuracy.

Miller-Orr model

In 1966, Merton Miller and Daniel Orr (M.H.Miller, D.Orr) developed a money management model that is much closer to reality than Baumol's model. It helps answer the question: how should a company manage its cash reserves if it is impossible to predict the daily outflow or inflow of cash? Miller and Orr used the Bernoulli process to build the model - a stochastic process in which the receipt and expenditure of money from period to period are independent random events.

The basic premise of the Miller-Orr model is that the distribution of daily cash flow balances is approximately normal. Actual value the balance on any day may correspond to the expected value, be higher or lower than it. Thus, the cash flow balance varies randomly from day to day; no tendency for its change is envisaged.

The model is implemented in several stages [ Kovalev]:

1. The minimum amount of funds is established ( L), which it is advisable to constantly have in the current account (determined by experts based on the average need of the enterprise to pay bills, possible demands of the bank, creditors, etc.).

2. Based on statistical data, the variation in the daily receipt of funds to the current account (σ 2) is determined.

3. Opportunity costs are determined r- expenses for storing funds in a current account (usually they are taken in the amount of the daily income rate on short-term securities traded on the market) and expenses c on the mutual transformation of cash and securities (this value is assumed to be constant per transaction).

4. Calculate the range of variation in the balance of funds on the current account R according to the formula

5. Calculate the upper limit of funds in the current account H, above which it is necessary to convert part of the funds into short-term securities:

H=L+R(8.5)

6. Determine the return point ( Z) - the amount of funds balance on the current account, to which it is necessary to return if the actual balance of funds on the current account goes beyond the boundaries of the interval ( L, H):

An example of a graph depicting the dynamics of funds using the Miller-Orr model is presented in Fig. 8.7.

Rice. 8.6. Dynamics of enterprise cash balances using the Miller-Orr model [Kovalev, p. 547].

At a moment in time t 1 there is a purchase of securities in the amount of ( H – Z), and at the moment t 2 securities are sold with net proceeds ( Z – L).

When using the Miller-Orr model, you should pay attention to the following points [ Brigham, Gapenski, p.312-313].

1. The target account balance is not the average between the upper and lower limits, since its value more often approaches its lower limit than its upper limit. If you set the target balance to average between limits, it will minimize transaction costs, but if it is set below the average level, the result will be a reduction in the magnitude of opportunity costs. Based on this, Miller and Orr recommend setting the target balance in the amount if L= 0; this minimizes overall costs.

2. The size of the target cash balance and, therefore, the limits of fluctuation, increase with growth c and σ 2 ; increase c makes it more expensive to reach the upper limit, and a larger σ 2 leads to both being reached more often.

3. The target balance decreases as it increases r; since if the bank interest rate increases, then the value of opportunity costs increases and the company tends to invest funds rather than keep them in an account.

4. The floor does not have to be zero, but can be positive if the firm has to maintain a compensating balance or management prefers to maintain a safety stock of cash.

5. Experience in using the described model has shown its advantages over purely intuitive money management; however, if the company has several alternative options for investing temporarily free funds, and not the only one in the form of purchasing, for example, government securities, then the model ceases to work.

6. The model can be supplemented with the assumption of seasonal fluctuations in revenue. In this case, cash flows will not follow a normal distribution, but will take into account the likelihood of an increase or decrease in the balance of funds, depending on whether the company is experiencing a period of decline or recovery. Under these assumptions, the target cash balance will not always be between the upper and lower limits.

Stone's model

Stone's model, unlike the Miller-Orr model, places more emphasis on managing the target residue rather than defining it; at the same time, they are similar in many ways [ Brigham, Gapenski, p. 313-314]. The upper and lower limits of the account balance are subject to change depending on information about cash flows expected in the next few days. The concept of Stone's model is presented in Fig. 8.7. Just like in the Miller-Orr model, Z represents the target account balance that the firm strives for, and H And L- the upper and lower limits of its fluctuations, respectively. In addition to those indicated, Stone’s model has external and internal control limits: N And L- external, and ( H – X) And ( L + x) - internal. Unlike the Miller-Orr model, where immediate action is taken when control limits are reached, this does not always happen in the Stone model.

Rice. 8.7. Dynamics of cash balances using the Stone model [Brigham, Gapensky, p. 313].

Let's assume that the account balance has reached the outer upper limit (point A in Fig. 8.7.) at the moment of time t. Instead of automatically converting the value ( H– Z) from cash into securities, the financial manager makes a forecast for the next few days (in our case, five). If the expected balance at the time ( t+ 5 ) will remain above the internal limit ( H– x), for example its size is determined at the point IN, then the sum ( H– Z) will be converted into securities. Further dynamics of the cash balance in this case will correspond to the thick line starting at the time t.If the forecast shows that at the moment ( t+ 5 ) value cash balance will correspond to the point WITH, then the firm will not buy securities. Similar reasoning is true for the lower limit.

Thus, the main feature of Stone's model is that the company's current actions are determined by its forecast for the near future. Consequently, reaching the cap will not trigger an immediate transfer of cash into securities if relatively high cash burns are expected in the coming days; thereby minimizing the number of conversion operations and, consequently, reducing costs.

Unlike the Miller-Orr model, the Stone model does not specify methods for determining target cash balances and control limits, but they can be determined using the Miller-Orr model, and x and the period for which the forecast is made - with the help of practical experience.

A significant advantage of this model is that its parameters are not fixed values. This model can take into account seasonal fluctuations, since the manager, making a forecast, evaluates the characteristics of production in individual periods.

The disadvantage of Stone's model is the emergence of subjectivity. If the manager makes a mistake with the forecast, the company will incur storage costs. excess amount funds (in the case of an upper limit) or by short period of time will lose liquidity (in the case of a lower limit). However, correct short-term forecasting of the size of the cash balance can reduce transaction costs.

Simulation modeling

Simulation modeling is the most accurate of the models considered, but at the same time the most labor-intensive. The modeling technique is described by Brigham and Gapenski ([ Brigham, Gapenski, p. 314-316].

Modeling begins with drawing up a preliminary cash flow budget. After this, an assumption about the probabilistic nature of the indicators is introduced into the forecasting methodology.

It is expected to calculate the volume of monthly sales ( S) random variable with normal distribution. Let us denote the coefficient of variation in the volume of monthly sales as CV, and its standard deviation is as s S. We will also assume that over time the relative variability of sales volume is constant.

Then the standard deviation of sales volume for i The th month will be equal to:

Where S i- volume of sales i month.

The receipt of revenue from sales is associated with the actual, and not with the expected volume of sales, that is, the payment receipt scheme is based on information about actual sales that took place in the past.

The essence of the Monte Carlo method is based on studying the operation of a model of a system when it receives random input data that has specified characteristics (type of distribution, dispersion, etc.) and restrictions. In our case, it is necessary to model (at a given level of significance) the value of the enterprise's possible cash shortage by month and plan the corresponding values ​​as the target balance. The key indicator here is the significance level set by the manager - the probability with which the results obtained (target remainder) are statistically significant. The recommended level is around 90%.

Brigham and Gapensky emphasize that it is possible to introduce the assumption that monthly sales volumes are dependent on each other; that is, for example, if the actual implementations in i-month will be below their expected level, this should serve as a signal of a decrease in sales revenue in the following months. IN in this case cash flow uncertainty will increase and, therefore, to ensure the desired level of security, it is necessary to set the target cash balance at a relatively higher high level [Brigham, Gapenski, p. 316].

The main advantage of simulation modeling is the relatively high accuracy of the results obtained.

However, it should be noted that the use of this method for financial forecasting in practice it is almost impossible without the use of a computer. In addition, to obtain reliable results, it is advisable to have information on the company's cash flows for at least two previous years to obtain a representative sample of the initial data.

Accounts receivable management.

Accounts receivable, or accounts receivable, are one of the most important and significant elements current assets enterprises. Modern trade practice increasingly relies on the buyer receiving a deferred payment for the delivered products, which results in the formation of significant accounts receivable for the seller (supplier).

Level accounts receivable enterprise is determined:

· Type of products sold

· Degree of market saturation with this type of product

· Payment system adopted at a particular enterprise

General economic factors

Accounts receivable management is a classic example of a trade-off between risk and profitability: the optimal level of accounts receivable is determined based on the trade-off between increasing sales volume and, consequently, profit as a result of decreasing credit requirements to customers, and in parallel with rising costs of financing an increasing level of receivables and an increase in probable losses on bad debts. At the same time, the basic laws of financial management are clearly followed: the expected profitability changes in inverse proportion to the liquidity of the asset (in this case, accounts receivable) and in the same direction as the risk. At the same time, attempts popular in the domestic literature to classify debts for shipped products as an object of accounts receivable management, which significantly exceed in their urgency the industry average indicator of the receivables circulation period, or even a period of 12 months, are obviously untenable: such “receivables” are already cannot be considered as an integral part of current assets.

An important element of accounts receivable management is the ranking of accounts receivable according to the timing of their occurrence (compiling a so-called “aging register” of accounts receivable), as well as monitoring its turnover (turnover of funds in settlements). The latter is carried out on the basis of a number of turnover indicators, which are discussed in the corresponding section of the course.

A very popular tool for monitoring accounts receivable is to compare the average repayment period with the average repayment period of debt on supplier accounts ( accounts payable). Despite all the conventions of such a comparison (due, in particular, to the different nature of obligations and in some cases different volumes), it can show whether the enterprise is a net creditor financing investments in working capital its customers, or, conversely, a net borrower using the funds of its counterparties. It should be noted here, however, that the popular arguments among many domestic theorists about the management of receivables based on an analysis of the operating and financial cycles of an enterprise2, in practice, face significant limitations. The operating cycle of an enterprise is, as is known, equal, on the one hand, to the sum of the duration of the production process3 and the average repayment period (circulation period) of receivables, and on the other hand, to the sum of the duration of the financial cycle and the average repayment period (circulation period) of debt on supplier accounts (accounts payable ). If we approach the problem of managing receivables “mechanically”, then the task of minimizing the duration of the financial cycle4 (namely, for this period the enterprise’s funds are diverted from circulation and the enterprise has to use financing through own funds or attract a loan) can be solved in two ways5. On the one hand, it is possible to tighten the conditions for the sale of products on credit, which should reduce the period of circulation of receivables, but at the same time reduce the volume of sales (profit). On the other hand, you can “delay” the payment of supplier bills. Within certain limits, this may “work”, but if this technique is abused, the supplier will be objectively forced to reconsider the terms of delivery or simply include the cost of financing its increased receivables in the delivery price. The result is increased costs and decreased profits. The art of management here consists precisely in avoiding, if possible, both dangers.

From a practical point of view, the most important tool for managing receivables of an enterprise is its credit policy, represented by two interrelated activities: providing deferred payments and debt collection.

The credit policy of an enterprise involves making decisions on five main issues [ Levy, Sarnat]:

1. Determination of the period for which payment is expected to be deferred;

2. Determination of lending instruments, i.e. legal form processing a commercial loan;

3. Formation credit standards- a set of criteria and procedures for determining “good” and “bad” in terms of providing a deferment on customer payments;

4. Collection policy - certain procedures for monitoring receivables and procedures for action in cases of delays in payments must be established;

5. Incentives that can be offered to customers to speed up payment of bills (usually discounts).

In conditions developed countries the seller will rely on knowledge credit history client for study financial statements client, etc. In domestic conditions, the main sources of information about the creditworthiness of clients are

· Own experience companies

· Information from confidential sources - for example, a bank where a potential client is served.

· Information from supplier companies that have already worked with this client.

For large contracts, special investigations by the security service may be carried out.

An analysis of the current situation in Russia shows that spontaneously, based on the interaction of market factors, domestic enterprises are developing their own credit policy, already quite comparable with that which has developed in countries with developed market economy. The result is the establishment of a certain balance between sales on prepayment terms, with payment upon delivery and with deferred payment - a balance, the violation of which in one direction leads to a drop in sales volume, in the other direction to an unjustified increase in the risk of non-receipt of payment.

Inventory Management

To provide effective management cash flows V foreign practice The most widely used models are the Baumol model and the Miller–Orr model.

The first was developed by W. Baumöl in 1952, the second by M. Miller and D. Orr in 1966. The direct application of these models in domestic practice is still difficult due to insufficient development of the securities market, therefore we will give only a brief theoretical description of these models.

Baumol's model

It is assumed that the enterprise begins to operate with the maximum and appropriate level of cash for it and then constantly spends it over a certain period of time. The company invests all incoming funds from the sale of goods and services in short-term securities. As soon as the cash reserve is depleted, i.e. becomes equal to zero or reaches a certain specified level of security, the company sells part of the securities and thereby replenishes the stock of funds to the original value. Thus, the dynamics of the balance of funds in the current account is a “sawtooth” graph.

Rice. 6.3.

Thus, in accordance with the Baumol model, cash balances for the coming period are determined in the following amounts:

• a) the minimum cash balance is assumed to be zero;
• b) the optimal (aka maximum) balance is calculated using the formula

where DAmax is the maximum cash balance in the planning period; Рк – the average amount of expenses for servicing one operation with short-term financial investments; Oda – total cash expenditure in the coming period; SPKfv – interest rate on short-term financial investments in the period under review;

c) the average cash balance in accordance with this model is planned as half of its maximum balance (DAmax: 2).

Miller–Orr model is a more complex calculation option optimal size cash balances. The model is based on a certain unevenness in the receipt and expenditure of funds, and, accordingly, their balance, and also provides for the presence of an insurance reserve.

The minimum limit for the formation of the cash balance is taken at the level of the insurance balance, and the maximum limit is three times the amount of the insurance balance.

The logic of the financial manager’s actions to manage the balance of funds on the current account is presented in Fig. 6.4 and is as follows - the account balance changes chaotically until it reaches the upper limit. Once this happens, the company begins to purchase a sufficient amount of securities in order to return the cash reserve to some normal level (the point of return). If the cash reserve reaches the lower limit, then the company sells its securities and thus replenishes the cash reserve to the normal limit.

Rice. 6.4.

When deciding on the magnitude of variation (the difference between the upper and lower limits), it is recommended to adhere to the following policy: if the daily variability of cash flows is large or fixed costs associated with the purchase and sale of securities are high, then the enterprise should increase the range of variation, and vice versa. It is also recommended to reduce the range of variation if there is an opportunity to generate income due to the high interest rate on securities.

In accordance with the Miller-Orr model, cash balances for the upcoming period are determined in the following amounts in several stages.

• 1. The minimum amount of funds (He) is established, which it is advisable to constantly have in the current account.
• 2. Based on statistical data, the variation in the daily receipt of funds to the current account is determined ( V).
• 3. Expenses (Px) for storing funds in a current account and expenses (Pt) for the mutual transformation of cash and securities are determined.
• 4. The range of variation in the balance of funds on the current account is calculated ( S) according to the formula:

5. Determine the upper limit of funds in the current account (), above which it is necessary to convert part of the funds into short-term securities:

6. Find the point of return (TV) - the amount of the balance of funds on the current account, to which it is necessary to return if the actual balance of funds on the current account goes beyond the boundaries of the interval ():

On first At this stage, ten-day terms for spending funds are regulated (in connection with their receipts), which allows minimizing the balance of monetary assets within each month (quarter).

On second stage, the size of the average balance of monetary assets is optimized taking into account the envisaged reserve stock of these assets. In this case, the maximum balance of monetary assets is first determined, taking into account the unevenness of payments and the reserve stock, and then their average balance (half the sum of the minimum and maximum balances of monetary assets).

The amount of monetary assets released in the process of adjusting the flow of payments is reinvested in short-term financial investments or in other types of assets.

Ensuring the acceleration of the turnover of monetary assets determines the need to search for reserves for such acceleration in the enterprise. The main of these reserves include:

• a) acceleration of cash collection, which reduces the balance of monetary assets in the cash register;
• b) reduction of cash payments (cash cash settlements increase the balance of funds in the cash register and reduce the period of use of own funds for the period of processing supplier payment documents);
• c) reducing the volume of settlements with suppliers using letters of credit and checks, since they divert monetary assets from circulation for a long period due to the need to pre-reserve them in special bank accounts.

Ensuring the effective use of temporarily free cash balance can be achieved through the following measures:

• a) agreeing with the bank on the conditions for the current storage of the balance of funds with the payment of deposit interest;
• b) the use of high-yield short-term stock instruments to place a reserve of monetary assets, but subject to their sufficient liquidity in the stock market.

Minimization of losses of used funds from inflation is carried out separately for funds in national and foreign currencies.

Anti-inflation protection of monetary assets is provided if the rate of return on the temporarily free balance used is not lower than the inflation rate.

Enterprise finance

clarification 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 of the 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 model or another depends on the specifics of the practical financial management problem being solved. Among them, the Baumol-Tobin model occupies a special position and belongs to the classical results of financial management, since it has 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 place the main emphasis 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 the unconscious replication of erroneous results. However, the logic of derivation of the main result (formula) is extremely important both in methodological terms and for the correct use of the model, since it must always stipulate the conditions of applicability of this model, its essence and give 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 controlling the corresponding financial process. No control technology is perfect, and any of them can become even more perfect. The word “model” is not a very good term, as it emphasizes unreality and artificiality. More

It’s correct to talk about technology here, not about the model. In this article we will adhere to generally accepted terminology, as this is convenient for drawing parallels and comparisons of our results with the results arising 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, since 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 the construction of stochastic cash management models. For objectivity, we note that the Baumol-Tobin model was based on Wilson’s ideas on management material reserves.

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. Let us only note that this model has one important flaw (misconception), which is fundamental and general in nature, relating to the time horizon of financial planning. IN general case this horizon cannot be short, which is proven in our article. In this case, the order of consideration will be as follows.

1. At the very beginning, it will be revealed that the Baumol-Tobin model incorrectly determines the opportunity cost of costs associated with lost

finance and credit

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

2. It will be shown that this model is approximate in nature (linearization in time), so the results can (but are undesirable) be applied only at sufficiently low interest rates (note that in Russia these rates still remain relatively high) and a small number of visits to bank N for the purpose of withdrawing 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 one to make optimal decisions for any interest rates and any number of visits to bank N to withdraw money from the 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 additional evidence of their validity, i.e., our results reduce to the results of the Baumol-Tobin model as the interest rate tends to zero.

As usual, here we mean money as the most liquid type of asset, usually designated in macroeconomics as M1, which includes cash and money in settlement, current and other demand accounts. This money brings either very low income, or they don’t bring it 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: urgent savings deposits, government bonds, certificates of deposit. Despite the great diversity various types assets capable of generating income over time, the population still stores part of its funds or assets in the form of cash, or more precisely in the form of M1. This means that the population has a non-zero demand for cash. Economists were faced with the task of determining the quantitative characteristics of this demand. The usefulness of money in general is determined, as is known, by three functions: a medium of exchange, a measure of value, and a means of preserving income. It is obvious that cash, as a medium of exchange, is superior to all other monetary aggregates, since it is absolutely liquid. But cash

as a means of preserving income, it is inferior to other forms of money. Theories of the demand for money based on its role as a medium of exchange are called theories of transactional demand for money. Cash is needed to make purchases or, in general, to carry out transactions. Among the various transaction theories of demand for money, the Baumol-Tobin model still remains the most widely known and popular, although it appeared more than half a century ago - in 1952. In addition to determining the demand curve for money, this model allows you to optimally manage companies' cash (cash balances ), as well as citizens. The desire for optimality should be set by the parameters of the demand curve. Companies must forecast their cash balances accordingly at an optimal level. Based on knowledge of the company's future cash needs, the manager must decide how much cash balance to hold. Excess cash can be invested in high-quality short-term securities, pay dividends, create additional reserves, etc. Lack of cash forces the company to turn to loans, 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 comes down to determining the optimal cash balance. Cash balance is the amount of cash available to a household (family) or company that changes over time. The same problems have to be solved not only by companies, but also by the government, regional and city administrations, etc.

The main advantage of cash is its convenience, since there is no need to go to the bank for every purchase and incur some costs, mainly associated with wasted time. The cash could be placed in a bank, invested in bonds or even stocks and have the corresponding additional income. Therefore, we can say that cash brings losses in the form of lost interest (the opportunity cost of money is always present as back side medals). That is, you always have to pay for the convenience of cash, but not overpay. The task of each person (manager) is to reduce total costs to a minimum. Let us assume that the person knows (planned on the basis of previous

current experience) that during the next period T0 = 1 (for example, five years, a year, a month, etc.) he will need S0 cash rubles. Note that S0 here has the financial meaning of cash flow, since this amount refers to a conventional time unit T0 (for example, 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 managing cash. 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 an 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 for illustrative purposes only. In fact, our approach provides for the possibility of choosing any conventional unit of time.

The second option for cash management 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, reducing to zero. At this time, the second half, located in the bank, generates interest income. Consequently, during the first half of the year, on average, there will be an amount of cash on hand equal to to the amounts of available cash in the form of an arithmetic progression). After the first half of the year, a second amount of S0/2 is immediately withdrawn from the bank account for expenses in the next second half of the year. Consequently, during the second half of the year, on average there will be an amount of cash on hand equal to ^¿/2+0) /2=S0/4, 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 average annual amount cash will be S0/4, which is obvious.

Similarly, you can consider visiting the bank three or four times. In general, when visiting Iraz Bank during the year, the amount of 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 S0/N to zero.

Therefore, in the general case, the average annual amount of cash will be ^¿/Н + 0) /2 = S0/2N (this is the average amount of a decreasing arithmetic progression). From this formula it is clear that the greater I, the less the average annual amount “per

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

Opportunity cost of cash. Now you need to determine the losses from keeping cash on hand. Usually in the economic literature, without evidence, it is intuitively believed that these losses are proportional to the product of the bank rate R0 and the average annual cash amount 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 associated with inventory management). Losses from storing cash, or rather their correct calculation, can have an independent economic importance, not relevant to this context. In particular, underestimating these losses can mislead managers who will not pay attention to such “little things” and will ignore cash management. In addition, an experimental test of the demand curve for cash did not confirm the theoretical result, as shown in the work. Therefore, the corresponding exact calculation of these losses is proposed below.

Let R0 be the annual bank rate, or the rate of return on an alternative investment. The Baumol-Tobin model "by default" assumes that this interest rate R0 is given relative to a conditionally unit period T0, i.e. R0 = R0(T0), where T0 = 1. This circumstance should also be kept in mind when using this model, otherwise gross miscalculations are possible. For example, if the planning period T0 = 6 months, then the rate ^ must 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 disadvantage of this approach, since certain difficulties arise, which often lead to errors. All these difficulties could be easily avoided if we did not require the equality T0 = 1. However, for now we will adhere to the traditional approach. These problems are covered in more detail in the works. Assuming that this rate is small enough, only in this case can simple interest be applied, which is what is done by default in the Baumol-Tobin model. Let's explain this below.

finance and credit

At the beginning of the year, when you first visit the bank, the amount S0/N will be withdrawn from the account, the interest income on which during the year would be L^/D 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 from a 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 = 1/N, interest income will not be received, which will amount to losses equal to:

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

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

S0 i - 90 i.- = 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 the 0+1) th visit to the bank will occur and the amount S0/N will be withdrawn from the account, where y = 1, 2,...F The cost of lost interest income in this general case will be equal to:

90R0 - 90 I = ^R0 (1 -C.

N N N N N In particular, for y = N from this general formula you can determine the losses from the last N^0 in the cash withdrawal account, which will be: I 50 I N -1 = 50 I (1 N -1) I

This result is quite 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. Therefore, the product of this amount S0/N by the time 1/N and by the rate L^ will give losses, which and obtained on the right side of the last equality. During the first (N-1) periods this amount will still earn interest

income. The costs of this cash will be the lowest compared to all others. The maximum loss will result from the very first withdrawal of cash from the account.

Let's find it now total losses interest income from unearned interest, designated as C (N), for the entire plan 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 isolate 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 this is where 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 the terms 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^2 00

Our result (1) differs from similar expressions in that a new term appears to the right of the last equal sign. Previously, only the first term £0A0/2J was present in these costs. The strange thing is that for such a long time no attention was paid to this error. In addition to the computational proof of the correctness of expression (1), which was presented above in full detail, we can also consider the financial meaning of this expression and its predecessor. As usual, in such cases it is necessary to resort to some extreme cases of verification where detailed calculations are not required. For example, in the case of a one-time visit to the bank, from formula (1) it follows that with N=1, the alternative costs will be

C1 (1) = I + - S0 I = ^ I.

Dependence of costs on the number of bank visits

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

Second extreme case is an infinitely large number of visits to bank N, at which the minimum cost is achieved (1). If all losses were reduced only to this type of cost, then the minimum of these losses would be achieved with the maximum possible number N of bank visits during a conventionally single period (year). Theoretically, this value can be equal to infinity (i.e., arbitrarily large), then the costs will be determined only by 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^^. 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 fallacy of such conclusions seems obvious, given that the problem boils down to a continuous annuity. If the value of N is large enough, we can assume that the withdrawal of amounts occurs continuously. The amount of Sg in the account will continuously decrease to zero throughout the year, which will cause loss of interest income.

This gross error is quite obvious from simple qualitative considerations, if we correctly 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, losses from lost interest income are actually much higher than previously thought. These differences are visually represented by the graph C(SH (dashed line).

This graph does not asymptotically tend to the abscissa axis (zero value), as was previously supposed, but approaches the horizontal straight line С1(da) = SgRg/2 (dash-dotted line). Note that sometimes in the economic literature the dependence of costs on the size of the cash balance is constructed, and not on N, which does not change the essence of the problem.

Having a complete description of costs in the form of formula (1), we obtain additional features in making optimal decisions on managing the company's cash balances. Withdrawing money from an account makes sense if it can be reinvested with a higher return (or utility for individual), which is what 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 opportunity to optimally manage not only cash, but also any other assets. Withdrawing money from the account will make sense if the net present value is not less than zero. Further details can be omitted since costs (1) are approximated here, as shown below. More accurate results will be obtained later. The underestimated level of costs in the Baumol-Tobin model may lead to the fact that some managers may ignore them and not apply optimal cash management methods. In addition, this error is also of a logical nature, distorting some qualitative representations of investment analysis.

Some clarifications of the model. Let us show that when obtaining result (1), simple (approximate) interest was actually used, therefore formula (1) does not accurately estimate the opportunity costs due to lost interest income. In addition, we will take one more step towards a more adequate solution to this problem.

If N is the number of annual visits to the bank, then the time period 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 quantity

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

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

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

where the approximate equality is obtained up to linear terms of the series expansion ( simple interest) . The expression furthest to the left of the equal sign is exact. In relation to our problem, m is the number of periods during which the amount £ = S0/N was not in the account, and therefore this is lost interest income. For the first withdrawn amount t = N for the second t = N- 1), for the third t = 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 the loss of interest income, there is another component of total costs C2(I), associated directly with the process of withdrawing funds from an account generating interest income. As shown above, costs C1 decrease as the number of visits to bank N increases. However, as N increases, costs C2(I) associated with visiting the bank increase.

Following tradition, we will give the simplest interpretation of the appearance of costs C2(^ associated with a visit to the bank. Let us 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). They are mainly determined by the loss of time for a trip to the bank and back, waiting in line and processing the withdrawal of money from a savings account, commissions, payment of contracts, etc. For example, if you earn 40 rubles / hour and the total loss of time is 5 hours per visit, the opportunity cost of lost time will be is equal to: 5 hours 40 rubles / hour = 200 rubles. To this amount of losses you should add the direct costs of traveling to the bank and back. In addition, the more often money is withdrawn from the account, the lower the interest rate on time deposits, which should also be included in the costs. The amount of these costs must be calculated by the manager in each specific case separately, which is not the purpose of the article. Annual costs for

a visit 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).

When determining this type of cost, we followed classical approach, talking about withdrawing money from a bank account. However, receiving cash may in practice occur different ways, as already mentioned 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 bankruptcy) the company's profitable risky assets. As a rule, the higher the return on risky assets, the greater P. But in all these cases, the costs of “cashing out” must be determined by formula (2), otherwise a different management technology may be required.

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 it (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 total costs (3), which depend on N. The task of each manager is to calculate the number of visits to bank N at 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 become minimal 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 cash balance that a company should borrow or receive as a result of the sale of securities, then P is the transaction costs of a transaction 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

From this formula it is clear 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 beyond doubt. In the economic literature, the Baumol-Tobin model is also used as a model of 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. In this case, equation (7) is taken as the demand equation. The total costs when equality (5) is satisfied are minimum value, equal to:

TS (Ne) = 2 R £o + \^2РЯо £о,

where expression (5) was substituted into (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 drawbacks that are obvious today, which in no way detract from the merits of this theory, which presents obvious prospects for development and clarification. For example, firstly, discounting of future costs can be fully taken into account. Secondly, the majority of the Russian population receives wages in cash. Other types of income also come in cash. In such cases

we should consider the inverse problem compared to the one discussed above. A person, having received an income, must decide how much money he will keep in cash and how much he will put in a bank savings account that generates interest income. This approach is usually used to describe the first half of a person's life before retirement, when he strives to earn more than he spends during the same time. Above, the Baumol-Tobin model essentially considered a person who is retired and owns money in a savings account.

At the same time, this model has a much broader applied nature. In particular, it concerns the management of a portfolio of securities held in brokerage company or a jar. Securities may have different levels of 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 “cash-out of real estate investments.” The only problem is that the assets being sold are divisible. This is difficult to do in relation to real estate directly, but in principle it is possible.

Literature

1. Braley R. Principles of corporate finance / R. Braley, S. Myers; lane from English M.: Olimp-Business, 1997. 1087 p.

2. Brigham Y. Financial management / Y. Brigham, L. Gapenski. SPb.: Economic school, 1997. T. 2. 668 p.

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

4. Vorst I. Economics of the company / I. Vorst, P. Revent-low. M.: Higher School, 1994. 272 ​​p.

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

6. Mankiw G. N. Macroeconomics / G. N. Mankiw. M.: MSU, 1994. 735 p.

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

8. Reshetsky V. I. Economic analysis and calculation of investment projects / V. I. Reshetsky. Kaliningrad: Yantarny Skaz, 2001. 477 p.

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

10. Cheng F. Corporate finance: theory, methods and practice / F. Cheng, J. Li, I. Finnerty. M.: INFRA-M, 2000. P. 685.

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

Baumol model:

Unlike the classical entrepreneurial model, in W. Baumol’s model it is not profit that is maximized, but sales volume. In oligopolistic markets, which in the 20th century. Most, the company strives to maintain its market share, therefore, in an oligopoly, maximizing sales volume becomes the target function of the company.

The Baumol model is an algorithm that allows you to optimize the size of the average balance of an enterprise's monetary assets, taking into account the volume of its payment turnover. In accordance with the model proposed by William Baumol, the balances of the enterprise's monetary assets for the coming period are determined in the following amounts:

a) the minimum balance of monetary assets is equal to zero;

b) the optimal (also, in the interpretation of V. Baumol, the maximum) balance of monetary assets is calculated using the formula:

· where YES is the optimal balance of the enterprise’s monetary assets in the planning period;

· Rk - the average amount of expenses for servicing one transaction of short-term financial investments (fixed amount of expenses for one transaction);

· Oda - the total volume of payment turnover (expenses of means of payment) of the enterprise in the planned period;

· SPk - interest rate on short-term financial investments in the period under review (expressed as a decimal fraction).

c) the average balance of monetary assets in accordance with this model is planned as half of their optimal (maximum) balance.

In Baumol's model, the firm's goal is to maximize total revenue from product sales, which leads to a decrease in profit compared to its maximum level. Obviously, in this case, the sales volume will exceed the sales volume under conditions of profit maximization, which is beneficial, first of all, to the company's managers, since their remuneration is tied primarily to sales volumes. However, the owners of the company may also be interested in maximizing sales revenue; the reasons for this may be that a reduction in sales volumes in the case of profit maximization can lead to:

· reduction of the company's market share, which can be extremely undesirable, especially in conditions of growing demand;

· a decrease in the market power of a firm due to an increase in the market share of other firms;

· reduction or loss of product distribution channels;

· reducing the attractiveness of the company for investors.

From Kovnir slides + extras:

Output at profit maximization will be less than output at revenue maximization. Let's compare the results that a company gets when maximizing total revenue and profit. The marginal revenue of a profit-maximizing firm (MR) is equal to marginal cost (MR = MC > 0). The marginal revenue of a firm that maximizes total revenue is zero (MR = 0). Since the marginal revenue function is decreasing (dMR/dq< 0), и в первом случае предельная выручка больше, чем во втором, то q1 < q2, где q1 - выпуск при максимизации прибыли, q2 - выпуск при максимизации совокупной выручки. Объем производства при максимизации совокупной выручки всегда будет больше, чем при максимизации прибыли.

Williamson model:

O. Williamson's model was based on an analysis of the monopoly position of corporations, which the latter achieve through the process of concentration and centralization. Extracting monopoly profits allows one to deviate from the goal of profit maximization, and justifies the irreducibility of the company’s goal to one indicator. Work on a model of discretionary behavior of a management firm brings O. Williamson to the problems of organizational evolution of a large corporation. During the research process, the question arises: how can the organizational evolution of a large corporation affect the formation of the firm's target function? Answering this question, O. Williamson proposes the idea of ​​“organizational innovation” - major changes in the principles of organizational structure of corporations that have matured historically and become inevitable at a certain stage.

Williamson's model is based on taking into account the interests of managers, manifested in their discretionary (discretionary - acting at their own discretion) behavior in relation to various items of the company's expenses (see figure).

Williamson model

Williamson in his model identifies the following main goals of managers:

a. Salary plus other monetary benefits;

b. The number of employees subordinate to this manager and their qualifications;

c. Control over the company's investment expenses;

d. Privileges - company cars, luxurious offices that exceed in costs those necessary for the company's work. (A form of organizational or managerial slack).

All of these goals increase with firm size. The model focuses on the immediate goals of managers.

Formally, the objective function of managers in the Williamson model includes the following variables:

· S – excess staff costs, defined as the difference between maximum profit (Pmax) and real profit (PA).

· M – “managerial slack”, defined as the difference between real profit (PA) and reported profit (PR) (managers can either hide part of the profit or overstate reported profit compared to the real one).

· I – discretionary investment expenses, defined as the difference between the declared profit (PR) and the amount tax payments(T) and the minimum acceptable level of profit for shareholders (Pmin).

The pursuit of these goals is limited by the need to maintain an acceptable level of reported profit (PR). In this case, the task is written as follows:

Thus, in addition to the output volume (Q), which affects the level of real profit, managers can choose the value:

1) excess staffing costs (S);

2) the amount of expenses for elements of managerial slack (M).

The amount of discretionary investment expenses (I) is determined uniquely, since the minimum profit and tax level are given.

The model is solved by substituting the values ​​of S, M, I into the utility function, followed by differentiating and setting the derivatives with respect to Q, S, and M to zero. This shows that such a firm will have higher staffing costs and greater managerial slack than a firm maximizing profit. The differences with a profit-maximizing firm also lie in the firm’s different reactions to changes in external parameters (changes in demand, tax rates, etc.).