1. Models of Baumol and Miller-Orr of managing the cash balance on the current account
Calculation of the optimal cash balance
Cash as a type of current assets is characterized by some features:
routine - cash is used to pay off current financial obligations, so there is always a time gap between incoming and outgoing cash flows. As a result, the company is forced to constantly accumulate free cash on a bank account;
precaution - the activity of the enterprise is not strictly regulated, therefore, cash is necessary to cover unforeseen payments. For these purposes, it is advisable to create an insurance cash reserve;
speculative - funds are needed for speculative reasons, since there is always a small probability that an unexpected opportunity for profitable investment will appear.
However, cash itself is a non-profitable asset, so the main goal of the cash management policy is to maintain it at the minimum required level, sufficient for the effective financial and economic activities of the organization, including:
timely payment of suppliers' invoices, allowing you to take advantage of the discounts they provide on the price of the goods;
maintaining a constant creditworthiness;
payment of unforeseen expenses arising in the course of business activities.
As noted above, if there is a large amount of money on the current account, the organization has the costs of missed opportunities (refusal to participate in any investment project). With a minimum supply of cash, there are costs to replenish this stock, the so-called maintenance costs (sales expenses due to the purchase and sale of securities, or interest and other costs associated with raising a loan to replenish the balance of funds). Therefore, when solving the problem of optimizing the balance of money on the current account, it is advisable to take into account two mutually exclusive circumstances: maintaining current solvency and obtaining additional profit from investing free cash.
There are several basic methods for calculating the optimal cash balance: mathematical models of Baumol-Tobin, Miller-Orr, Stone, etc.
Baumol-Tobin model
The most popular model of liquidity management (cash balance on the current account) is the Baumol-Tobin model, built on the conclusions that W. Baumol and J. Tobin came to independently in the mid-1950s. The model assumes that a commercial organization maintains an acceptable level of liquidity and optimizes its inventory.
According to the model, the enterprise begins to operate with the maximum acceptable (expedient) level of liquidity for it. Further, as the work progresses, the level of liquidity decreases (money is constantly spent over a certain period of time). The company invests all incoming cash in short-term liquid securities. As soon as the level of liquidity reaches a critical level, that is, it becomes equal to a certain predetermined level of security, the company sells part of the purchased short-term securities and thereby replenishes the cash reserve to its original value. Thus, the dynamics of the company's cash balance is a "sawtooth" graph (Fig. 1).
Rice. 1. Schedule of changes in the balance of funds on the current account (Baumol-Tobin model)
When using this model, a number of limitations are taken into account:
1) at a given period of time, the organization's need for funds is constant, it can be predicted;
2) the organization invests all incoming funds from the sale of products in short-term securities. As soon as the cash balance falls to an unacceptably low level, the organization sells part of the securities;
3) the receipts and payments of the organization are considered constant, and therefore planned, which makes it possible to calculate the net cash flow;
4) the level of costs associated with the conversion of securities and other financial instruments into cash, as well as losses from lost profits in the form of interest on the proposed investment of free funds, can be calculated.
According to the model under consideration, to determine the optimal cash balance, you can use the optimal order lot (EOQ) model:
F - fixed costs for the purchase and sale of securities or servicing the loan received;
T - the annual need for funds necessary to maintain current operations;
r - value of alternative income (interest rate of short-term market securities).
Miller-Orr model
The disadvantages of the Baumol-Tobin model noted above are eliminated by the Miller-Orr model, which is an improved EOQ model. Its authors M. Miller and D. Orr use a statistical method when building a model, namely the Bernoulli process - a stochastic process in which the receipt and expenditure of funds over time are independent random events.
When managing the level of liquidity, the financial manager must proceed from the following logic: the cash balance changes chaotically until it reaches the upper limit. As soon as this happens, it is necessary to buy enough liquid instruments in order to return the level of funds to some normal level (point of return). If the stock of funds reaches the lower limit, then in this case it is necessary to sell liquid short-term securities and thus replenish the stock of liquidity to the normal limit (Fig. 2).
The minimum value of the cash balance on the current account is taken at the level of the insurance stock, and the maximum - at the level of its triple size. However, when deciding on the range (the difference between the upper and lower limits of the cash balance), it is recommended to take into account the following: if the daily volatility of cash flows is large or the fixed costs associated with buying and selling securities are high, then the company 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.
When using this model, one should take into account the assumption that the costs of buying and selling securities are fixed and equal to each other.
Rice. 2. Graph of changes in the balance of funds on the current account (Miller-Orr model)
The following formula is used to determine the cusp point:
where Z is the target cash balance;
δ2 - dispersion of the daily cash flow balance;
r is the relative value of opportunity costs (per day);
L - the lower limit of the cash balance.
The upper limit of the cash balance is determined by the formula:
The average cash balance is found by the formula:
C \u003d (4Z - L) / 3
Miller-Orr model. The model developed by M. Miller and D. Orr is a compromise between simplicity and everyday reality. It helps answer the question of how a company should manage its cash supply if it is not possible to accurately predict cash inflows or outflows on a daily basis. 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. Their basic premise is that the distribution of daily cash flow balances is approximately normal. The actual value of the balance on any day may correspond to the expected value, be higher or lower than it. Thus, the cash flow balance varies by day randomly; no trend is foreseen.
The logic of actions of the financial manager to manage the balance of funds on the current account is as follows. The account balance fluctuates randomly until it reaches the upper limit. As soon as this happens, the company begins to buy highly liquid securities in order to return the stock of cash to a certain level (point of return). If the cash reserve reaches the bottom limit, then the company sells the previously accumulated securities, replenishing the cash reserve to a normal level.
When deciding on the range of variation (the difference between the upper and lower limits), it is recommended to follow the rule: if the daily volatility of cash flows is large or the fixed costs associated with buying and selling securities are high, then the company 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.
The implementation of the model is carried out in several stages.
Stage 1 . Set the minimum amount of cash (FROMmin) , which it is advisable to always have on the current account. It is determined by an expert, based on the average need of the company to pay bills, the possible requirements of the bank, creditors, etc.
Stage 2 . According to statistical data, the variation of the daily receipt of funds to the current account is determined (VAR).
Stage 3 . Determine the cost of keeping funds in the current account (Zs) (usually they are taken as a sum of daily income rates on short-term securities circulating in the market) and expenses for the mutual transformation of cash and securities (Z). It is assumed that the value Z constant; an analogue of this type of expenses, which takes place in domestic practice, are, for example, commissions paid at currency exchange offices.
Stage 4 . Calculate the range of variation of the cash balance on the current account (R) according to the formula:
Stage 5 . Calculate the upper limit of cash on the current account ( FROMmax), above which it is necessary to convert part of the funds into short-term securities:
Cmax= Cmin+R.
Stage 6. Define a cusp (FROMr ) - the value 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 interval:
Cr = (Cmin+ 1 / 3 Cmax).
The following data necessary to optimize the company's cash balance was taken as initial data:
minimum cash reserve (FROMmin) - 10,000 thousand tenge;
costs of converting securities (Z)- 25 thousand tenge;
· interest rate: r= 11.6% per year;
· standard deviation per day - 2,000 thousand tenge.
Using the Miller-Orr model, it is necessary to determine the policy for managing funds on the company's current account.
Solution
1. Calculation Zs . :
Zs = r / 365 = 11.6 / 365 = 0.03% per day.
2. Calculation of daily cash flow variation (VAR) (thousand tenge):
VaR = (2000) 2 = 4 000 000.
3. Calculation of the range of variation (R) (thousand tenge):
4. Calculation of the upper limit of cash and the point of return (thousand tenge):
FROMmax = 10 000 + 18 900 = 28 900.
FROMr = 10 000 + 1 / 3 X 18 900 = 16 300.
Thus, the balance of funds on the company's current account should vary in the range of 10,000,000 - 28,900,000 tenge); when going beyond the interval, it is necessary to restore funds on the company's current account in the amount of 16,300,000 tenge.
As already noted, Western experts have developed other approaches to managing the target balance of funds, in particular, the Stone model, which is a development of the Miller-Orr model, has gained some popularity.
Baumol-Tobin model. The most popular model of liquidity management (cash balance on the current account) is the Baumol-Tobin model, built on the conclusions that W. Baumol and J. Tobin came to independently in the mid-1950s.
Using the Baumol-Tobin model, one can determine the optimal amount of a company's cash that it should keep under certainty. The Baumol-Tobin model relies heavily on the assumption that a possible alternative to holding money is the use of marketable securities and/or interest-bearing deposits.
According to the model, the company starts operating with the maximum acceptable (expedient) level of liquidity. Further, as the work progresses, the level of liquidity decreases (money is constantly spent over a certain period of time). The company invests all incoming cash in short-term liquid securities. As soon as the level of liquidity reaches a critical level, that is, it becomes equal to a certain predetermined level of security, the company sells part of the purchased short-term securities and thereby replenishes the cash reserve to its original value. Thus, the dynamics of the company's cash balance is like a "sawtooth" graph.
The Baumol-Tobin model is used when there is a high level of certainty that a company may need cash.
Suppose you want to determine how much cash the company should have. At the same time, the total costs should be minimized, which consist of conversion costs and costs that arise due to the fact that the company refuses part of the income from marketable securities, since it keeps funds in cash.
When building the model, it is assumed that for some time (for example, a month) the company has a stable need and demand for cash. At the same time, cash is received by selling marketable securities. When cash runs out, the company sells marketable securities to raise cash.
The total costs can be represented as:
Total costs =B x (T / C) + r x (C / 2),
where B X (T/C) are the total transaction costs for the period, while AT– total costs associated with the sale of securities (transaction costs); T/C- the number of transactions for the sale of marketable securities (equal to the ratio of the total demand for cash in the period ( T) to the cash balance ( FROM);
r X (S/2)- the amount of income that the company refuses, keeping its funds in cash, while r– interest rate on marketable securities; ( C / 2) is the average cash balance.
On the one hand, the more cash, the higher the income that the company refuses, simply by keeping its funds in cash or on current accounts. On the other hand, the higher the cash balance, the fewer transfers into marketable securities are needed and the lower the conversion costs.
In accordance with the Baumol-Tobin model, the company's costs for the sale of securities in the case of keeping part of the funds in highly liquid securities are compared with the lost profit that the company will have if it refuses to hold funds in securities, and therefore will not have interest and dividends on them. The model allows you to calculate the amount of money that would minimize both transaction costs and lost profits. The calculation is carried out according to the formula:
C = √2 x B x T / r.
The disadvantage of the Baumol-Tobin model is the assumption of predictability and stability of the cash flow. In addition, it does not take into account the cyclical and seasonal nature of most cash flows.
Let us determine the optimal balance of funds according to the Baumol-Tobin model, if the planned volume of the company's cash turnover is 50 million tenge, the cost of servicing one cash replenishment operation is 400 tenge, the level of losses of alternative income when storing funds is 10%.
Using the formula, we calculate the upper limit of the company's cash balance (thousand tenge):
C=√2 X 0,4 X 50 000 / 0,1 = 632,46.
Thus, the average cash balance will be 316.23 thousand tenge (632.46 / 2).
Let's assume that the company's cash expenses during the year will be 1,500 million tenge. The interest rate on government securities is 8%, and the costs associated with each of their sale is 25,000 tenge.
Calculate the upper limit of the company's cash balance (million tenge):
C=√2 X 1 500 X 0,025 / 0,08 = 30,62.
The average amount of funds on the current account is 15.31 million tenge (30.62 / 2).
The total number of transactions for the transformation of securities into cash for the year will be (million tenge):
1 500 / 30,62 = 49.
Thus, the company's policy on managing cash and cash equivalents is as follows: as soon as the funds on the current account run out, the company sells part of its liquid securities in the amount of approximately 30 million tenge. This operation is performed approximately once a week. The maximum amount of funds on the current account will be 30.62 million tenge, the average - 15.31 million tenge.
Cash is vital to the operation of any business and is an integral part of its working capital. At the same time, the following features are characteristic of cash:
- loss of purchasing power due to inflation;
- ability to generate income.
Due to the features listed above, there is an objective need to justify the optimal balance of funds, which will not be excessive and at the same time will be sufficient to maintain solvency. allows you to calculate its value, subject to certain provisions.
Initial provisions of the Baumol model
- cash flows are not subject to fluctuations, that is, it is initially assumed that cash is spent evenly;
- spending of funds is carried out to zero balance;
- there is some uncertainty in the flow of funds;
- the possibility of using a credit line or overdraft is not expected;
- the opportunity cost of maintaining a cash balance does not change;
- surplus funds are invested in liquid securities;
- when buying and selling liquid securities into cash, certain transaction costs arise.
Calculation of the optimal cash balance
The value of the optimal cash balance, according to the Baumol model, depends on two factors: the cost of one cash replenishment transaction and the opportunity cost of maintaining it. In this case, the total cost function can be represented as follows:
where C- cash balance;
F– transaction costs of replenishing the balance of funds;
T- annual need for cash;
k– the opportunity cost of maintaining the cash balance (the interest rate on liquid securities).
From the resulting equation, we can express the optimal cash balance ( English Optimal Cash Balance, OCB):
Graphically, these dependencies can be expressed as follows:
Example. The company's need for cash is 75,000 USD. per week, transaction costs for the purchase and sale of securities are 800 USD, and the interest rate on liquid securities is 9% per annum.
The company's annual cash requirement is $3,900,000. (75000*52). In this case, the optimal cash balance in accordance with the Baumol model will be 263,312.24 c.u.
Interpretation of the Baumol model
Provided that the initial provisions of the Baumol model are met, the resulting optimal cash balance is sufficient to maintain the solvency of the business. When the condition of uniform spending of funds is met, there is no need to maintain the insurance balance, so their minimum balance will be equal to 0.
Since the expenditure of funds to zero balance is carried out over a certain period of time, all receipts received should be invested in liquid securities. When the cash balance reaches zero balance, it is necessary to replenish it to the optimal one by converting liquid securities.
Foreign researchers in the field of inventory management emphasize the importance of models for calculating the optimal cash reserve developed by W. Baumol and J. Tobin.
It is noted that W. 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 the stock market by selling securities. In either case, the company bears the cost of servicing the debt or the opportunity cost that arises from the sale of the securities and that is associated with the company's forfeiting income from the securities.
Let's consider a situation of realization by the company of short-term financial investments in profitable securities, and then their subsequent sale for replenishment of a stock of cash resources. 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 money demand 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 a higher interest 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 value of all payments by the company 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 money 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 t1n. At the same time, the zero value of the cash reserve is considered as the lower limit 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 a-1 . Therefore, the total capital of the two players is equal to a. The game continues until the player's capital either increases to a, 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 a, 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 ~ C
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 "
With = With +
"" 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 into 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, on the contrary, should sell securities, trying 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 cost 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. The limitation of the practical application of the Miller-Orr model is related to the theoretical assumptions of the model, for example, 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, it is recommended to Krizhevskaya to invest free cash in the foreign exchange 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 С opt use the following formula:
where E- the profitability of investing money 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.
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 a decrease in the period) of replenishment of funds. Let's designate 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).
W.Baumol drew attention to the fact that the dynamics of the target cash balance (CA) is similar to the dynamics of inventory and proposed a model for optimizing the target balance of CA, based on the Wilson model.
Assuming that:
1. The need of the enterprise for DS within a certain period (day, week, month) is known and constant;
2. Cash receipts for the same period are also known and constant, then the changes in the target VA balance will look like this (see Fig. 7):
src="/files/uch_group42/uch_pgroup67/uch_uch6621/image/761.gif">
1 week 2 weeks 3 weeks Time
Rice. 7. Dynamics of the DS balance on the current account
At the end of the first week, you will either have to sell the existing securities (for the amount of the weekly need for DC), or take out a loan for the same amount. And that's what you have to do every week.
Then DSav = , where DS is the weekly (monthly, etc.) need;
DSav - the average balance of money in the current account.
A large balance of DC reduces the cost of selling securities or servicing a loan (the so-called transaction costs), but on the other hand, it also reduces the possible income from securities (because the money is not moving).
The value of these possible incomes can be conditionally taken in the amount of income brought by liquid securities. But at the same time, the availability of securities (credit) will require additional (transactional) costs.
Then the total amount of costs (ZDSob) for maintaining the target balance of CA will be the sum of:
Variable cost (loss of profit) (ZDSper);
Fixed value of transaction costs (ZDSpos);
ZDSob \u003d ZDSper + ZDSpos;
ZDSper = * r,
where DS / 2 - the average balance of money in the current account;
r is the yield on securities.
ZDSpos \u003d F * k,
where F is the amount of transaction costs for one cycle of replenishment of funds on the current account;
k is the number of DS replenishment cycles per year.
But we know that the annual need for DS is equal to:
PDS \u003d k * DS;
Hence: k = ; Substitute the equivalent of "k" in the formula for ZDSpos: ZDSpos = * F;
Or in general terms: ZDSob = * r + * F;
Since we need to minimize the remainder of the DS, we differentiate the value of ZDSob with respect to the DS and equate to zero:
R / 2 - PDS * F / DS2 = 0,
where X = DS; Y = ZDSob;
Hence: DSmin = ; This is the Baumol formula.
Example: Let F = $150; PDS = 100 thousand dollars * 52 weeks = 5200 thousand dollars; r - 15% per annum, or 0.15; Then: DSmin = = $101980
Average current account balance DSav = $50,990, or approximately $51,000.
The disadvantages of the Baumol model are:
1. Assumption of stability and predictability of cash flows;
2. Failure to take into account the cyclicality and seasonality of fluctuations in the need for DS.
If these conditions are required to be taken into account, then other methods for calculating the optimal value of the target balance of DS must be applied.
Review questions
1. What is Net Working Capital (NFL) and how is it calculated?
2. What do DFTs show?
3. What determines the DFT?
4. What are the types of working capital management policies?
5. What is the main issue in the process of managing accounts payable?
6. How are receivables managed?
7. How are the minimum costs for maintaining the necessary inventories determined?
8. What is the basis of enterprise cash management?
1. The enterprise has the following annual financial balance:
ASSETS LIABILITIES
Fixed assets 3500 Equity 2000
Stocks of raw materials 400 Reserves 1000
Production in progress 200 owes. 2000
Stocks goth. products 600 Short term debt 1000
Accounts receivable 1800 Accounts payable 1200
Short term financial investment 200
Other tech. assets 300
Cash 200
Total assets 7200 Total liabilities 7200
b) determine current financial needs;
c) determine the cash surplus/deficit and the amount of new credit required;
2. The need for cash from the enterprise - 1000 thousand rubles. per month. It is expected that products shipped to consumers will be paid evenly. The annual interest rate is 20%. The cost of each loan operation or withdrawal of money from the account is 100 rubles.
Required:
a) determine the optimal amount of the cash balance of funds;
3. The enterprise has the following performance characteristics:
Annual sales on credit - 5 million rubles.
The period of repayment of receivables - 3 months.
Profit rate - 20%
The company is considering an offer for discounts of 4 / 10, gross 30. It is expected that the repayment period will be reduced to two months.
You want to determine whether it is worth implementing such a discount policy?
4. The enterprise uses 400 units. material per month. The cost of each order is 200 thousand rubles. The cost of storing each unit of material is 10 thousand rubles.
Define:
a) What is the value of the optimal order?
b) how many orders should be made per month?
c) how often do you need to place orders for the supply of material?
5. Sales on credit from the enterprise amount to 500 thousand rubles. The payment period is 90 days. The cost price is 50% of the sales price.
It is required to determine the average investment in receivables.