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Chapter 10 An Introduction to Working Capital and Cash Management A. Introduction to Current Assets and the Treasurer’s Role When appropriate inputs can be estimated, the capital budgeting techniques discussed in the previous chapter work very well for determining whether to purchase specific assets or to invest in specific projects yielding observable cash flows. This is often the case for takeover decisions, project or division start-up decisions and certain types of fixed asset decisions. However, in many instances, it is quite difficult to determine precisely the cash flows or discount rates that can be associated with many types of assets. This is particularly true with most types of current assets. For example, financial managers of firms know that they must maintain cash balances; however, they usually are unable to determine returns and NPV's for these balances. Without discount rates and net present values, how does one determine whether to increase the cash balance or vary accounts receivable policy? Fortunately, there exist a number of additional techniques that enable the firm to determine appropriate levels for the various types of current assets. The cash management decisions are one of the primary functions of the corporate treasurer. The treasurer’s office is also responsible for most of the decisions affecting other current assets. More generally, the functions of the corporate treasury are to obtain financing, manage cash and near cash (marketable securities) accounts and direct relationships with the various financial institutions with whom the firm does business. In the global business or in the firm with foreign suppliers or clients, the treasurer is often responsible for managing currency exchange and risk. In financial institutions and larger non-financial firms, the treasurer will also be responsible for managing interest exposure and currency exposure. Use of derivative securities (securities whose values are derived from the values or rates of other securities) may be required for these functions. In some instances, certain of these functions will be managed by other areas in the corporate structure; in most larger corporations, these functions are managed within the single corporate treasury department. In addition to the three functions listed above, the treasurer will also be responsible for dividend disbursement, credit management, insurance management and pension management for the firm. In some instances, the treasurer will also play a major role in forecasting and investor relations. B. Introduction to Cash Management Cash is the lifeblood of economic activity. Without cash, business cannot operate. Firms maintain cash balances for four primary purposes:1 1. transactions needs 2. precautionary purposes 3. speculative purposes 4. compensating balance purposes Firms require cash balances to pay balances owed to suppliers, wages owed to employees, taxes payable to governments, etc. Cash balances maintained to conduct such transactions reflect 1The first three of these motives were proposed by John Maynard Keynes in his masterpiece The General Theory of Employment, Interest and Money.

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the firm’s transactions demand for cash. These cash balances may be maintained in cash registers, vaults, non-interest bearing checking accounts interest bearing checking accounts, money market accounts and in "float" (checks waiting to be cleared by the banking system or otherwise delayed in processing). Some firms may maintain balances in savings accounts that pay interest at rates below competitive returns available elsewhere. Most firms will maintain the bulk of their cash balances in deposits or accounts at financial institutions. This transactions demand for cash balances accounts for the majority of balances maintained by most firms. Some firms may maintain some cash balances for emergencies or simply to ensure that they never run out. These precautionary balances or cash reserves usually account for a smaller percentage of the total levels maintained by the firm. In addition, many firms maintain speculative balances to enable themselves to quickly take advantage of new investments, shifts in interest rates and fluctuations in exchange rates. For example, when interest rates are low, firms maintain higher speculative cash balances; when interest rates rise, firms commit their speculative balances to the higher yield securities. However, money market accounts allow firms to maintain highly liquid interest-bearing assets that can be quickly converted into longer-term interest-bearing debt certificates. Firms that maintain checking accounts with commercial banks are frequently required to maintain minimum balances with their banks. These minimum balances, referred to as compensating balances, are required to compensate commercial banks for the checking account services they provide the firm or to avoid fees normally imposed for checking and other bank services. Such compensating balances generally do not accrue interest or accrue interest at below-market rates. Thus, the commercial bank obtains the free (or inexpensive) use of the firm's money in exchange for providing “free” checking account services. Most firms will prepare cash budgets, projecting cash inflows and cash outflows over given time periods. In many instances, the firm's cash outflows for a particular day or week will exceed its inflows for that period. The firm will wish to maintain a positive cash balance (surplus) to ensure its ability to operate during these periods. Many firms find the use of various cash management models helpful for determining optimal cash balances, how much additional cash will be required when balances run low and when these balances should be replenished. Perhaps, the more popular of the cash management models are based on the Baumol and Miller-Orr models. As with all financial constructs, these cash management models require several assumptions which are often not applicable in reality. In many cases, the fact that certain assumptions may be violated in reality will barely affect the use or implications of these models. However, if necessary, these models are simple and flexible enough to be easily adapted to more realistic conditions. Equally important is that these models make it easy to see how certain factors impact the firm’s optimal cash balances. Both the Baumol and Miller-Orr models assume that the firm incurs an opportunity cost by maintaining cash balances. That is, when the firm maintains a cash balance, it forgoes the opportunity to invest money in interest or return-bearing assets. Thus, the firm's opportunity cost is the foregone interest or returns on money it could have invested more profitably elsewhere. Larger cash balances will imply larger foregone interest opportunity costs. Secondly, these models

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assume that the firm incurs transactions costs when it liquidates assets in order to obtain cash. (The Miller-Orr model assumes that the firm also incurs transactions costs when it converts surplus cash to return-bearing securities.) That is, when the firm sells assets (such as marketable securities) to replenish its cash balances, it incurs a transactions cost (such as a fixed brokerage fee). The more often the firm must transact to replenish its cash balances, the higher will be its total transactions costs. Hence, these models are structured to find the cash balance that minimizes the total costs associated with maintaining and obtaining cash. C. The Baumol Cash Management Model The Baumol Cash Management Model is deterministic in that it assumes that the firm's demand for cash over the relevant time period is known with certainty. For example, Polk Company management may know with certainty that the firm will require $100,000 over the next year. If the company’s cash balance declines by equal amounts at all times (for example, the firm spends the same amount each day), simply starting the year with a cash balance of $100,000 will mean that the firm, over the course of the year, will maintain an average cash balance of $50,000. Thus, the firm begins the year by ordering an initial cash balance of $100,000 and permits it to continue to drop until the balance reaches zero. On the other hand, if the firm were to begin the year with a cash balance of $10,000 then obtain cash nine more times over the next year (for a total of ten orders), its average cash balance will be only $5000. The average cash balance declines as the firm orders cash more frequently. Smaller average cash balances allow more capital to be invested in return- and interest-bearing assets. Decreasing the average cash balance will increase the number of transactions that the firm executes for cash and the firm’s overall transactions costs. At the same time, the firm’s average cash balance will decline as will the interest or returns that it foregoes (See Figure 1). Where (c) is the amount of cash obtained each time the firm replenishes its balance and (x) is the total cash demand over the relevant time period, the average cash balance (avg) can be determined as follows:

(1) avg = 2c .

The firm's foregone interest or return cost (ic) is simply the average cash balance times the interest or return rate the firm could have earned had it otherwise invested in marketable securities:

(2) ic = ic⋅

2 = avg i⋅ ,

where (i) is the interest rate foregone by the firm when maintaining cash balances. Cash balances require the firm to forego interest or returns on marketable securities or other assets. In our example, if the annual interest rate foregone by the Polk Company were 10%, its total foregone interest cost over the year would be $500.

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0 C* C

Mi ni mumcashbal ance

cost

Tot al cash

bal ance cost ( $) For egone

i nt er est cost ( i c)

Tr ansact i on

cost ( t r c)

Cost s

Figure 1: Costs associated with obtaining and maintaining cash balances

Figure 2: Cash balances over time: Baumol Model

In Figure 2, the firm begins the time period with a cash balance of (c). It then spends this money in equal amounts each day until the balance reaches zero. At this point, the firm cannot operate unless it obtains additional cash. To obtain this cash, the firm engages its broker to sell marketable

Cash Balance

C*

2*C

0

Time

5

securities; the receipts of the sale are used to replenish cash balances and the process continues. Each time the broker is engaged to sell marketable securities enabling the firm to replenish its cash balances, a transactions cost is incurred. In the Baumol Model, any revenues received by the firm are immediately converted into return-bearing assets; thus, the firm must liquidate a portion of its return-bearing assets to obtain cash. When the firm sells marketable securities to raise cash, it will incur a fixed brokerage fee (B) for each transaction. The total number of transactions executed by the firm per time period is simply the total cash demand over that period divided by the cash order quantity:

(3) tr = cX

Therefore, the total transactions costs incurred by the firm over the relevant time period is determined:

(4) trc = BtrBcX

⋅=⋅ .

If the cash order quantity for the Polk Company were $10,000, the total number of transactions for cash executed by the firm would be 10. If the brokerage fee per transaction were $50, the total transactions costs incurred by the Polk Company would be $500. The total costs associated with cash balances for a firm is simply the sum of its foregone interest and brokerage costs:

(5) $ = trcicicBcX

+=⋅+⋅2

The firm's primary objective with respect to cash management policy should be to minimize these costs yet remain able to operate effectively. If Polk Company management realized that it requires $100,000 in cash over a given year, its objective will be to choose a cash order quantity enabling it to minimize the total costs associated with obtaining and maintaining cash balances. Therefore, it will minimize ($) with respect to (c). This optimum (c) level will enable the firm to determine its optimum average cash balance (simply c*/2) and to determine the optimum number of security sales for cash over the year (X/c). The optimal cash order quantity is given by Equation 6:

(6)i

BXc 2* =

Since c* is that order quantity that enables the firm to minimize the total costs associated with its obtaining and maintaining cash balances, c* is the optimum cash order quantity. The Polk Company will be able to determine its optimum cash order quantity using Equation (6), given its known cash demand of $100,000, the ten percent interest rate on marketable securities, and the $50 brokerage cost per transaction:

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.000,10$000,000,100$1.

000,100$50$2* ==⋅⋅

=c

The company will maintain an average cash balance of $5,000 (from Equation [1] ) and will engage its broker ten times over the course of the year (from Equation [3]) to liquidate marketable securities. Since the company will order cash ten times over the year, it will, on average, order cash every 36.5 days. Polk's total foregone interest cost will be $500; its total transactions costs over the year will be $500. The total cost to the Polk Company of obtaining and maintaining cash balances will be $1000. No level of (c) will result in lower total cash balances costs. Notice that the foregone interest cost is equal to the total transactions costs in the Polk Company example. This equality will hold only when the firm chooses optimum cash order quantities and allows its cash balances to diminish to zero before replenishing them. In the following section, the firm will not permit its cash balance to decline below some minimum

Derivation Box 1 Deriving the Baumol Cash Management Model

The derivation of Equation (6) results from the minimization of ($) in Equation (5) with respect to (c). To minimize ($), we need only to find the derivative of ($) with respect to c, set it equal to zero and then solve for (c). First, for the sake of simplicity, Equation (5) will be re-written:

(7) $ = BXc-1 + 2ic .

Now we find the derivative of ($) with respect to (c) and set it equal to zero:

(8) dcd$ = -BXc-2 +

2i = 0 .

We now solve algebraicly for (c):

(9) BXc-2 = 2i ,

(10) c-2 = BXi

2 ,

(11) c2 = 2BX , i

(6) *2 ci

BXc ==

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acceptable level; therefore, the optimum cash order quantity will not result in the firm's transactions costs equaling its foregone interest costs. The Baumol Model With Demand Uncertainty In the previous section, we noted that the Baumol Model requires that the firm know with certainty its total cash demand. We further noted that the firm must spend its cash inventory evenly over the time period. These assumptions may be violated in reality. In this case, the firm may wish to establish some minimum acceptable cash balance, or precautionary balance to ensure that it never runs out of cash and be unable to operate. This precautionary motive for maintaining cash balances may be accompanied by speculative motives or by the need to maintain compensating cash balances. Several methods exist to enable the firm to establish this minimum acceptable level. One method requires that the firm first determine its expected cash demand and order levels and then determine its maximum demand or usage level. For example, the Polk Company determined its expected annual cash demand level to be $100,000. Its cash order quantity was $10,000, thus its expected cash demand for every 36.5 day period was $10,000. However, if the company could require as much as $12,000 over this 36.5 day period to cover higher than expected expenses, it may wish to establish a $2000 minimum acceptable cash balance. Where (min) is the minimum acceptable cash balance, the company's average cash balance will rise to: (12) AVG = min + c + min = c + 2min . 2 2 The maximum cash balance is now (c) plus (min) and the average cash balance is simply the sum of the maximum and minimum balances divided by two. The total costs associated with obtaining and maintaining cash balances are determined as follows:

Figure 3: Cash balances over time, Baumol Model with minimum acceptable balances

Cash Balance

C*+Min

22* MinC +

Min

Time

8

(13) $$ = icBcX

⋅+

+⋅2min2 .

Even with the non-zero minimum acceptable cash balance, we find that our optimal cash order quantity is the same as before when we allowed cash balances to decline to zero:2

(6) i

BXc 2* = .

However, since the average cash balance is higher when the firm establishes a non-zero minimum acceptable cash balance, its foregone interest and total cash balances costs will be higher. For example, if the Polk Company were to establish a minimum acceptable cash balance of $2000, its optimum order quantity would still be $10,000; however, its average cash balance would increase to $7000. Therefore, its foregone interest would be $700 and its total cash balance cost would increase to $1200. E. The Miller-Orr Cash Management Model Use of the Baumol Cash Management Model requires that the firm be able to forecast cash demand with some degree of accuracy, that the firm obtain cash only by selling inventories, and that this cash be spent in equal increments every day. The Miller-Orr Model does not require the firm to make a forecast of cash demand; it only requires that the firm be able to associate a variance with uncertainty regarding cash demand or balances. The model further assumes that the firm is able to obtain cash from revenues, but this source of cash may not be sufficient to cover the firm's needs for cash. Therefore, the firm must liquidate securities to obtain cash when revenues are insufficient to cover the firm's cash needs. Because the firm has only limited control over the magnitude and timing of its revenues, it may find its cash balances rising to unacceptably high levels. When the firm's cash balances are too high, it forgoes too much interest and must purchase securities to dispose of the surplus cash. Consider the cash balance activity portrayed in Figure 4. The firm begins the relevant time period with a cash balance of (rtp). From this point, cash balances will decline when the firm incurs costs and will increase when the firm obtains positive cash flows from revenues. When the cash balance reaches the (min) level, they are too low and the firm will sell sufficient securities to raise its cash balance back to (rtp). From this point, cash balances will continue to vary. When the cash balance reaches the (max) level, the firm's balances are too high, and surplus cash must be disposed of by buying marketable securities. Use of the Miller-Orr Model requires that the firm establish minimum and maximum acceptable cash balances and appropriate levels of securities for the firm to sell or purchase when these levels are reached. Formulas will be offered for the 2We derived the Baumol Cash Management Model in Derivation Box 1. Allowing for minimum acceptable cash balances, we find the optimum cash order quantity by setting the derivative of $$ with respect to c equal to zero. Our derivative is exactly the same as in Derivation Box 7.1, so we know that c is exactly the same as when cash balances were permitted to decline to zero.

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determination of these levels, but since their derivations require the use of stochastic calculus, they will not be derived here. The firm should first establish some minimum acceptable cash balance. Determination of this minimum acceptable level should account for the cost to the firm of entirely depleting its cash balances and the length of time elapsing between the point when the firm orders the liquidation of securities and the point when the firm actually receives the receipts from the liquidation. The cost of depleting the cash balance to zero may be a function of interest imposed by a bank for a short-term loan or credit line. Generally, past managerial experience is the best starting point for setting this minimum cash balance.

Figure 4: Cash Balances Over Time: The Miller-Orr Model The maximum acceptable cash balance is determined as follows: (14) max = min + 3z , where (min) is the minimum acceptable cash balance and (z) is determined as follows:

(15) 32

43

iB

z cb

⋅⋅⋅

=σ

where (σ 2

cb) is the variance of projected daily cash balances. The cash level (rtp) is the firm's "return-to-point." This is the cash balance to which the firm returns when either the minimum or maximum acceptable balances are reached. The (rtp) is determined as follows: (16) rtp = min + z .

Cash Balance

Time

ZMinBalance

3+

Min+Z Min

Min

rtp

Max

10

Therefore, when the minimum acceptable level is reached, the firm will sell (z) dollars worth of marketable securities enabling the firm to raise its cash balance to (rtp). Since the difference between the maximum acceptable cash balance and the "return-to- point" is 2z, the firm will buy 2z dollars worth of securities to return to (rtp) when the maximum acceptable level is reached. For example, consider the Taylor Company that is able to obtain a one half of one percent monthly interest rate (or return) on its marketable securities and incurs a $360 brokerage fee whenever it engages its broker to execute a transaction. The company has established a $15,000 minimum acceptable cash balance and has determined that the $22,360 monthly standard deviation associated with previous cash balances appropriately reflects the standard deviation of future cash balances. Since the standard deviation of future cash balances is projected to be $22,360, the projected variance of cash balances is five hundred million dollars. To use the Miller-Orr model, Taylor management first determines (z) to be $30,000 from Equation (15):

3005.4

000,000,500$360$3⋅

⋅⋅=z 000,30$000,000,000,000,27$

02.000,000,540$ 33 ===

= Thus, the firm's maximum acceptable cash balance from Equation (14) is $105,000: max = $15,000 + ( ⋅3 $30,000) = $105,000 . When this maximum acceptable balance is attained, the firm will buy $60,000 (that is, 2z) worth of securities to return to the rtp level of $45,000 determined from Equation (16): rtp = $15,000 + $30,000 = $45,000 . When the minimum acceptable balance is reached, the firm sells $30,000 in marketable securities to return back to the "return-to-point." Although (z) was not derived or proven here to result in optimum maximum and "return-to-point" cash balance levels, its components (B), (i) and ( 2

cbσ ) can be intuitively related to these cash balance levels. For example, as transactions costs (B) rise, (z) will rise; consequently, the maximum acceptable and (rtp) cash balance levels will rise. Thus, when transactions costs rise, the firm will wish to make fewer transactions to dispose of or to obtain additional cash. When the maximum acceptable cash balance level rises, the new higher maximum acceptable level will be reached less often than the previous lower level, resulting in the firm needing to dispose of surplus cash less often. When the (rtp) balance rises, the firm will have larger balances at the beginning of the relevant time period and after executing marketable securities transactions. Because these balances are larger, the minimum acceptable cash balance will be reached less often and the firm will execute fewer transactions to obtain cash. The effects of increased interest rates (or returns on marketable securities) will result in decreased levels of (z), (max) and (rtp). Therefore, the firm, on average, will maintain smaller cash balances. The firm sets a lower maximum acceptable balance to better enable it to take advantage

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of the increased interest rates and sets a lower (rtp) to ensure that its average cash balances are smaller. Increased variance or uncertainty regarding future cash balance levels will increase (z), thereby widening the spread between the minimum and maximum acceptable cash balance levels. First, consider the certainty case. Here, the variance of future cash balances is zero, resulting in a (z) level of zero, which will cause the maximum acceptable cash balance to always equal the minimum acceptable balance. In this case, the firm will never have to execute marketable security transactions to dispose of or to obtain cash; cash balances are constant. As the variance of cash balances increase, the firm must execute more marketable securities transactions and will increase the (max) and (rtp) levels to balance out transactions costs with foregone interest costs.

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QUESTIONS AND PROBLEMS 1. What costs or disadvantages might be associated with firms maintaining cash balances that are too small? What costs or disadvantages might be associated with firms maintaining cash balances that are too large? 2. The Capone Company has determined that its operating circumstances are quite suitable for use of the Baumol Cash Management Model. The company consistently earns a five percent annual rate of return on its marketable securities and requires a total of $200,000 in cash each year to maintain its production. Transactions costs are $50 each time Capone liquidates marketable securities. Determine the following for the Capone Company: a. its optimum cash order quantity b. its optimum average cash balance c. the optimum number of securities liquidations for cash per year d. the optimum number of days between orders for cash e. its total annual transactions cost incurred by using the optimum cash order quantity f. its annual foregone returns cost g. the minimum total cost associated with obtaining and maintaining cash balances 3. What would be the total costs associated with cash balances in Problem 2 if Capone orders $10,000 each time it liquidates marketable securities? What will be the total costs if Capone liquidates $25,000 in securities each time it runs out of cash? 4. What will be the optimum cash order quantity if Capone wished to establish a minimum acceptable cash balance of $3,000? What will be its new total costs associated with cash balances? 5. The Nelson Company has determined that its operating circumstances require the use of the Miller-Orr Cash Management Model to manage its cash balances. The standard deviation of the company's daily cash balances has been shown to be $2,000, and management feels this figure also reflects future balance variability. Nelson earns an average daily return of .05% on its marketable securities and incurs an average brokerage fee of fifty dollars each time it engages a securities transaction. Management has determined that it cannot permit the company's cash balance to fall below $5,000. Determine the following for the Nelson Company: a. its "z" value b. its "return-to-point" c. its optimum maximum cash balance d. the optimum dollar value of marketable securities to be sold when the minimum cash balance is reached e. the dollar value of marketable securities to be purchased when the maximum cash balance is attained f. new solutions for parts (a) through (e) if the standard deviation of daily cash balances were to rise to $5,000

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Solutions 1. Too small: can't transact easily, higher order costs for cash, risky Too high: high foregone interest costs, risky

2. a. 000,2005.

000,200502* =××

=c

b. 20,000/2 = 10,000 c. x = 200,000 = 10 c 20,000 d. 365 = 36.5 10 e. x · B = 10 · 50 = 500 c f. 20,000 · .05 = C* · i = 500 2 2 g. 500 + 500 = 1000 3. a. 1250 b. 1025 4. a. same; 20,000 b. X · B + C* + 2 min · i = 500 + 650 = 1150

C* 2

5. a. 32.694,60005.4

000,000,450$33 =

⋅⋅⋅

=z

b. r.t.p. = min + z = 11,694.32 c. max = min + 3z = 25,082.97 d. z = r.t.p. - min. = 6,694.32 e. max - r.t.p. = 2z = 13,388.65 f. a:12,331.06 b: 17,331.06 c: 41,993.18 d:12,331.06 e: 24,662.12