Economic Profit and Performance Measurement in Banking, Exams of Business

Download Economic Profit and Performance Measurement in Banking and more Exams Business in PDF only on Docsity! Ralph C. Kimball Economist, Federal Reserve Bank of Boston, and Associate Professor of Fi- nance, Olin Graduate School of Busi- ness, Babson College. The author wishes to thank Richard Kopcke, John Jordan, and Eric Rosengren for helpful comments. The opinions expressed are those of the author and not necessarily those of the Federal Reserve Bank of Boston or the Federal Reserve System. Economic Profit and Performance Measurement in Banking Successful bank operation requires managers to weigh complex trade-offs between growth, return, and risk. In recent years banks increasingly have adopted innovative performance metrics such as risk-adjusted return on capital (RAROC) and economic value added (EVASM)1, which assist managers in making such difficult and complex decisions. These innovative measures all share as a basis the concept of economic profit, rather than accounting earnings. By forcing line manag- ers to include the opportunity cost of equity when making investment and operating decisions, banks expect to elicit better decision-making by managers. By implementing performance measurement and incentive systems driven by economic profit and allocated equity capital, senior managers also hope to align managerial behavior more closely with the interests of shareholders. This article analyzes the use of economic profit for measuring the performance of banks. In particular, since economic profit cannot be calculated without some imputation of equity, the article focuses on the allocation of equity capital to products, customers, and businesses. The first section of the article describes the use of economic profit to evaluate performance, to price transactions, and to reward managers. The second section describes in detail one performance measurement and incentive system, known as EVASM, which has been adopted by a considerable number of both banks and other companies. The third and fourth sections discuss the shortcomings of performance metrics founded on economic profit, which may distort banks’ investment and operating decision- making. These metrics assume that it is possible to allocate earnings and equity capital to lines of business, products, and customers in a way that isolates the economic revenues and costs of each activity. However, if lines of business are related, either in the production of output or in their use of capital, then this isolation may not be possible, and these methods of measuring performance may mislead managers. The conclusion argues that banks need to recognize the ambiguities inherent in the calculation of economic profit and be prepared to create and apply multiple specialized performance metrics. I. Economic Profit and Performance Measurement in Banks Economists and accountants differ on the proper definition of profit. To the accountant, profit is the excess of revenues over expenses and taxes and is best measured by earnings. To the economist, earnings fails to include an important expense item, the oppor- tunity cost of the equity capital contributed by the shareholders of the firm. A firm earns economic prof- its only to the extent that its earnings exceed the returns it might earn on other investments. Thus, earnings will always exceed economic profits, and a firm can be profitable in an accounting sense yet unprofitable in an economic sense.1 This conceptual difference has important practical implications. If managers attempt to maximize earn- ings (or growth of earnings) rather than economic profit, they will invest additional units of equity capital so long as the marginal contribution to earn- ings is positive. But if they do so, the marginal contribution of the last unit of equity capital will be zero and less than its opportunity cost, and the aver- age return to equity capital may be greater or less than its opportunity cost depending upon how much eq- uity is used. In contrast, a manager who maximizes economic profits will add units of equity capital only until the marginal contribution of capital is equal to its opportunity cost, and the average return to equity capital will equal or exceed its opportunity cost. As a result, firms that make business decisions without explicitly incorporating the opportunity cost of equity will be inefficient users of equity capital, engaging in investment projects that generate low returns to shareholders.2 In 1995, a year of robust earnings, one study estimated that fewer than half of the 1,000 largest industrial and nonfinancial firms earned sufficient returns to cover their opportunity cost of capital (see Ross 1997). Banks and other companies have begun to ad- dress this issue by incorporating an explicit opportu- nity cost of equity into their decision processes. In particular, a number of banks have incorporated a measure of economic profit in three key areas: strate- gic decision-making, product pricing, and perfor- mance evaluation and incentive compensation. Strategic Decision-Making Businesses with different risk characteristics re- quire different proportions of equity to achieve the same risk exposure. Evaluating businesses only on the level and rate of growth of their reported earnings fails to take into account differences in their use of equity, and the fact that shareholders may have dif- ferent required rates of return reflecting the risk of the equity invested. Thus, when allocating scarce re- sources or when deciding to enter or exit a new line of business, managers compare a return on equity (ROE) for the business unit relative to an appropriate hurdle cost of equity. Business units earning an ROE in excess of a risk-adjusted opportunity cost of that equity are candidates to receive additional resources, while those earning less than this opportunity cost of equity are candidates for corrective action. In recent years, such calculations have been extended from lines of business to products, distribution channels, and even customers. Pricing As noted above, different products, customers, or transactions will absorb different amounts of equity capital, with larger and more risky transactions requir- ing more equity than smaller, less risky ones. To ensure that a transaction is profitable, managers must assign the appropriate amount of capital and a re- quired contribution to equity must be calculated and incorporated in the price applied to the transaction. This use of allocated capital to ensure adequate pric- ing was first implemented by Banker’s Trust in its RAROC system, which subsequently has been adopted by many other commercial banks. In the RAROC system, the required rate on a loan comprises a cost of funds, a charge for non-interest expenses, a premium for credit risk, and a capital charge. The great contribution of the RAROC system was to include explicit charges for both the credit risk premium and the use of capital. By doing so, it ensures that banks price individual loans to cover credit risks and generate an adequate return for shareholders. An example of the use of the RAROC system to price loans is shown in Table 1. The capital charge is determined as the product of the proportion of equity capital assigned to support the loan and the required pre-tax hurdle rate on equity. As shown in Table 1, a loan rate of 11.25 percent will permit the bank to earn a 15 percent return on the equity required to back the loan. If the bank can obtain a rate greater than 11.25 percent, then it will earn an economic profit, while a 1 EVASM is a registered servicemark of Stern Stewart & Co. 2 While conventional capital budgeting models such as net present value or internal rate of return explicitly include a cost of equity capital, many decisions taken outside the capital budgeting process, such as product pricing or entry or exit from a particular line of business, may not. July/August 1998 New England Economic Review36 mance this period but depress it in succeeding peri- ods.6 For example, in banking, current-period operat- ing earnings can be enhanced by cutting service levels and relaxing credit standards, but such actions will depress future operating earnings as disgruntled cus- tomers switch to competitors and credit losses in- crease. Indeed, incentive systems resemble tax sys- tems in that one of the biggest challenges in designing such systems is to identify and close loopholes that facilitate gaming. As with any other incentive system that focuses solely on current-period performance, an EVASM- based system can be manipulated to maximize current incentive compensation at the cost of future reported performance. In many firms, this time horizon prob- lem is addressed by creating an incentive compensa- tion account for each manager into which both posi- tive and negative annual payments are made. Managers are permitted to withdraw only a maximum percentage of the balance in the incentive account in any one year.7 By creating a rolling five-year time horizon for the effective vesting of the incentive com- pensation, the manager’s incentives to manipulate short-term performance at the cost of long-term per- formance is limited, since any increase in this period’s incentive compensation would be offset by negative incentive compensation in the succeeding periods.8 III. Related Operations and EVASM To be effective in reducing agency costs and facilitating the devolution of decision-making, any performance measurement and incentive system must apply not just to senior management, but also at the divisional, product, and customer levels. Only by application at the business unit level can a perfor- mance measurement system be expected to affect the behavior of managers at these levels. However, appli- cation of any measurement and incentive system based on economic profit, whether EVASM or another, to subunits of a bank is based on a key assumption: that it is possible to isolate the earnings contribution of each business unit of the bank and the proportion of the bank’s equity capital it uses. In effect, calculation of economic profit at the business unit level views the firm as being the aggregation of individual units, and Lines of business, divisions, products, or other subunits are related operationally when the level of activity in one unit affects the earnings of another. Relatedness can affect revenues as well as expenses. the earnings and equity capital of the firm as being the sum of the individual earnings and equity capital used by the subunits. But this “the whole is the sum of the parts” assumption may not be valid if either the earnings of one unit are affected by the actions of another or the economic risks faced by different units are imperfectly correlated. This section discusses the effects of related operations upon the calculation of economic profit, while Section IV discusses issues associated with the allocation of equity capital to business units. Lines of business, divisions, products, or other subunits are related operationally when the level of activity in one unit affects the earnings of another. An extreme example of related operations is the produc- tion of joint products, where a process results in the production of two separate products in fixed relative proportions. A classic example of joint production used in many textbooks is the slaughter of a steer, resulting in both beef and leather. Neither product can be produced without the other, and the volume of each is more or less fixed with respect to the volume of the other. But operations can be related in many circumstances other than strict joint production. In many situations business units share common expense bases, products, distribution channels, or customers. 6 When, presumably, the manager would no longer be with the firm. 7 For example, suppose a manager earns an incentive compen- sation bonus in 1997 of $10,000. The $10,000 is deposited as deferred compensation into an incentive compensation account. The man- ager is allowed to withdraw only 20 percent of the account in any one year, so that a maximum of $2,000 can be withdrawn in 1997 with a balance of $8,000 carried over to the next year. If in 1998 the manager earns another $5,000 in incentive compensation, the bal- ance in the account will be $13,000, and the manager may withdraw a maximum of 20 percent, or $2,600. On the other hand, if the manager’s unit does poorly and the manager earns an incentive compensation payment of a negative $5,000, then the balance in the account declines to $3,000 and the manager may withdraw only $600. 8 Such systems work only if the manager is willing to accept negative incentive compensation in poor years, and if any remain- ing balance in the incentive compensation account is forfeited if the employee leaves the firm. The latter condition also acts as “golden handcuffs” to reduce turnover of key managers. July/August 1998 New England Economic Review 39 For example, bank products such as credit cards and home equity loans may share the same revolving loan system used to process account pay- ments and statements. Simi- larly, advertising that stresses a bank’s willingness to lend may affect more than one loan product. The existence of shared expense bases means that these costs must be divided among the subunits that share them, if the earnings of each are to be calculated. If the expenses of the shared cost center vary directly with volumes, they can be allo- cated to the subunits in pro- portion to their usage. But in most cases, the expenses of the cost center are relatively fixed or vary less than pro- portionately with volumes. Cost allocations then become arbitrary and can lose their economic usefulness. Con- sider the example of a bank operating a revolving loan system used by two loan products: credit cards and home equity loans. An economist would argue that each product or transac- tion should be charged its marginal cost. But in the case of the revolving loan system, most of the costs are fixed in the form of system development and mainte- nance, and the cost of executing an additional trans- action or adding a product is almost zero. Thus, the manager for either product could argue that it should not be allocated any of the costs of the system since the marginal cost of adding the product or transaction to the system, once it exists, is zero. In reality, the costs of such systems are usually allocated on the basis of usage, so that each product is effectively charged an average cost per transaction times the number of transactions executed. Even if one is willing to overlook the distortions introduced to decision-making by the use of average costs rather than marginal costs, one is still left with the result that changes in the volume of activity of one product will affect the costs of the other. For example, in the case of the revolving loan system, should the credit card product increase its volume while the home equity product did not, then the fixed costs of the revolving loan system would be spread out over a larger number of transactions, and the average cost per transaction would fall. If both products are charged the equivalent of average transaction cost times the number of transactions, then the allocated expenses of the home equity product will fall, solely because it shares an information system that has economies of scale.9 Relatedness can affect revenues as well as ex- penses. For example, to the extent that advertising is positively correlated with sales, all the business units 9 Management accountants try to mitigate this relationship between volume variances and allocated expenses by allocating expenses on the basis of budgeted volumes rather than actual ones. That is, the expenses of the revolving loan system would be allocated on the volumes the credit card and home equity products expect to occur rather than actual ones. Thus, unexpected increases or decreases in volumes do not affect the amount allocated. How- ever, this approach is a short-term remedy at best. While an unexpected increase in the volumes of the credit card business will not affect the costs allocated to the home equity product this period, next period the credit card business will revise its planned volumes upward to match the actuals, and at that point the proportion of systems costs allocated to the home equity product will decline. 0 10 20 30 40 50 60 70 80 July/August 1998 New England Economic Review40 may benefit from a corporate advertising campaign. Similarly, to the extent that customers prefer to pur- chase on a relationship basis and cluster their product purchases with one supplier, the acquisition of a new customer by one unit may enhance the revenues of other units. This is illustrated in Figure 1, which compares the propensity of the retail customers of a large money center bank to purchase a second product within 18 months of purchasing different initial prod- ucts. As the figure shows, 65 percent of customers who opened a transactions account as their first product purchased a second product within 18 months, com- pared to only 5 percent of customers who took out a mortgage as their first product. Thus, it would appear that the initial sale of a transactions account has positive externalities for other retail products and that this positive impact is much larger than those con- nected with the initial sale of other products. In such cases the existence of shared expenses or revenues can make it impossible to isolate costs or revenues in an economically meaningful way. In the extreme case of joint production, the sale of the two products generates two streams of marginal revenue, but there is only a single shared marginal cost. As a result, economists have long recognized that it is impossible to meaningfully calculate the profitability of either of the joint products, and that the condition for profitability maximization is the production of the joint products until the sum of their marginal reve- nues just equals the joint marginal cost. In the case of relatedness arising through sharing, something simi- lar occurs. Ideally, the revenues and expenses allo- cated to a business unit would consist not only of those that can be directly traced to a change in the volumes of that unit, but also of the incremental revenues and expenses of other subunits that result from the change in volumes of the first unit. Banks often attempt to accomplish this by implementing transfer pricing systems. Thus, a retail branch that serves corporate customers of the middle market group may receive an internal credit to cover the asso- ciated incremental expenses. But relatedness often ap- pears in subtle and intangible ways, so that it is unlikely that transfer pricing can capture all of the effects. Relatedness would not be an issue in performance measurement if the degree of relatedness was small or if positive and negative effects for each unit canceled each other out. While no empirical data exist that would permit us to measure the effects of related operations on reported revenues or expenses, intu- itively one can expect relatedness to exist and to increase in importance, the smaller the subunit being considered. Moreover, there is reason to believe that the effects of relatedness are not unbiased, but instead act to cause some subunits to systematically underes- timate their contribution to earnings and others to overestimate it. Indeed, some form of related operations is a necessary condition for different lines of business to exist or different products to be produced in the same firm. If no relatedness is present between lines of business or products, then each business or product could operate independently with no loss in value, and there is no economic rationale for joining them in the same firm. Increasingly, this argument is being accepted by managers, as demonstrated by the in- creased number of spin-offs and sales of “nonstrate- gic” businesses or products. It is only when benefits exist from joint operation that lines of business or products should be combined in one firm, so that these benefits can be captured.10 In effect, multi- divisional or multi-product firms exist because relat- edness causes the value of the whole to be more than the sum of the value of the parts (see Zimmerman 1997). Where relatedness exists, any performance metric that is calculated only on the allocated revenues and expenses of a single business unit, such as EVASM, will be an inaccurate measure of that unit’s contribution. The contributions of business units that generate neg- ative expense or positive revenue effects for other units will be underreported, while the contributions of subunits that enjoy either lower expenses or higher revenues as a result of the activities of others will be exaggerated. Managers attempting to maximize unit EVASM will underinvest in units that generate positive externalities and overinvest in units that receive them. The failure to incorporate relatedness into the calcu- lation of EVASM leads to a “management myopia” where each manager is trying to maximize business unit EVASM but not bankwide EVASM. As discussed in the box, “Relatedness and Incentive Systems,” incen- tive systems can be constructed to encourage managers to take into account the effects their decisions will have on other business units, but such incentive systems are complex and usually are only partially effective. 10 In the strategic planning literature the effect of relatedness is captured in the concepts of core competencies and horizontal strategies. A core competency is a skill or activity that cuts across lines of businesses or products and is the basis for the competitive advantage of the firm. A horizontal strategy is one that is built around a core competency. See Prahalad and Hamel (1990) and Porter (1985), Chapters 9–11. July/August 1998 New England Economic Review 41 IV. Allocating Equity Capital If some variation of economic profit is to be calculated at the business unit level, then the bank’s equity capital, as well as its earnings, must be disag- gregated and divided among the business units. This allocation of equity capital is critical, since without it the opportunity cost of equity cannot be calculated. In any firm, equity has two different functions: as a source of funding to purchase equipment, premises, and inventory, and as a cushion to protect debt holders against loss in the event of operating losses. Because banks hold relatively few of their assets in the form of real assets, equity’s function as a cushion for economic risk is especially important in banking. The proportion of equity needed to support a line of business, product, or customer within the bank will depend upon the riskiness of the activity, with riskier activities requiring additional capital.11 Thus, the amount of equity capital allocated to a particular business unit will depend both on the scale of opera- tions (for example, the amount of assets held) and the riskiness, so that a small but risky subunit could require as much equity as a large but low-risk one. Stand-Alone Allocation Methods One approach to allocating equity bases the allo- cations on the capital structure of independent “pure play” peers. To do so, a bank would construct for each line of business a group of publicly traded peers and allocate capital according to the average capital ratio of the peer group.12 For example, the mortgage bank- ing business would be assigned equity as though it were an average, independent, publicly traded mort- gage banker. While this approach has the advantage of being based on objective market data, actual imple- mentation quickly reveals several drawbacks. The number of independent publicly traded peers may be small or in some cases nonexistent, and these peers may differ in important respects from the business being analyzed. And even if a sufficient number of publicly traded peers exist, their capital ratios may vary significantly, so that management must choose among a possible range of capital allocations rather than a closely clustered point estimate. This approach is illustrated for a fictional Consol- idated Amalgamated Bank in Table 2. The Consoli- dated Amalgamated Bank is constructed from data for three separate publicly traded monoline lenders: a mortgage banker, a credit card bank, and a subprime consumer lender.13 In Table 2, the capital allocated under the peer group approach is assumed to be the same as the units’ actual equity capital in their true identity as publicly traded independent firms. As shown there, while the bank as a whole has an equity-to-asset ratio of about 15 percent, the equity- to-asset ratios of the individual businesses vary from about 10 percent for the credit card business to 33 percent for the subprime lending business. While the peer group method of allocation clearly differentiates among the lines of business in terms of the amount of capital allocated, it does not necessarily result in equal probabilities of insolvency across dif- ferent lines of business.14 For example, if consumer finance companies have on average a higher probabil- ity of insolvency than do mortgage banks, then allo- cation of equity capital based on the average of their respective capital structures will result in a higher 11 Riskiness is usually measured as the volatility of returns, for example, the standard deviation of the return on assets. 12 Allocations to products and customers would usually reflect the line of business to which they belong. 13 This approach was necessary because no bank publishes line-of-business results on a quarterly basis over a sufficient time period to permit calculation of expected returns and their covari- ance. 14 Insolvency for a line of business should be interpreted as the probability that the losses of the line of business will exceed the equity capital allocated to it. Table 2 Peer Group Approach to Allocating Equity Capital for Consolidated Amalgamated Bank Line of Business (1) Assets ($millions) (2) Equity ($millions) (3) Equity/Assets (Percent) (4) Return on Assets (Percent) (5) sROA (6) Z-Ratio Credit Cards 20,261 2,018 9.96 4.94 1.08 13.80 Mortgage Banking 11,314 1,949 17.23 4.96 2.78 7.98 Subprime Lending 5,072 1,666 32.77 14.67 7.96 5.96 Total 36,647 5,633 15.37 5.99 1.29 16.56 Source: Compustat and author’s calculations. July/August 1998 New England Economic Review44 probability of insolvency for the bank’s consumer lending business than for its mortgage origination business. One index of the probability of insolvency is the Z-ratio,15 defined as: Z 5 (ROA* 1 K)/sROA (1) where ROA* 5 the pretax expected return on assets, usually defined as the historical mean ROA, K 5 the ratio of equity capital to assets, and sROA 5 the standard deviation of ROA. Thus, the Z-ratio is a function of the normal profit margin of the bank, the variation in that profit margin, and the equity capital available to absorb that varia- tion. In effect, the Z-ratio measures the number of standard deviations by which ROA would have to decline before the book equity capital of the bank would be exhausted. The relationship between the Z-ratio and the probability of insolvency is an inverse one, with higher Z-ratios indicating a lower probabil- ity of insolvency.16 The last four columns of Table 2 calculate the Z-ratio for each line of business and for the bank as a whole. As shown there, the Z-ratios differ significantly across the lines of business, with the credit card business having a substantially lower probability of exhausting its assigned equity than do the mortgage banking and subprime lending busi- nesses. An alternative approach allocates equity capital based on each business’s cash flow so as to create an equal probability of insolvency. Equation (1) above can be rewritten to express the capital-to-asset ratio required to achieve a given target Z-ratio, as follows: K* 5 Z*sROA 2 ROA* (2) where K* is the required capital-to-asset ratio to achieve a target Z-ratio equal to Z*. In this approach each line of business will be allocated capital until its Z-ratio equals Z*. Application of this approach to Consolidated Amalgamated is illustrated in Table 3, which assumes that each line of business is allocated capital to achieve a Z-ratio of 13.8, the initial Z-ratio of the credit card business. This approach results in substantially higher equity-to-asset ratios for the mort- gage banking and subprime lending businesses. In- deed, the equity capital-to-asset ratio of the subprime lending business increases from about 33 percent under the peer-group method to about 95 percent under the equal probability of insolvency approach. Similarly, if the required equity of the bank as a whole is the sum of the required equity for each of the lines of business, then the bank will require almost 89 percent more equity under the equal probability of insolvency approach than under the peer group ap- proach. 15 This measure was developed by Hannan and Hanweck (1988). Although Hannan and Hanweck called the risk index “g,” in subsequent work it has generally been called “Z.” 16 If the assumption is made that the potential ROAs of the business are normally distributed, then the one-period probability of insolvency can be calculated as a function of the Z-ratio: p 5 1/[2Z2] However, empirical studies indicate that ROAs are not normally distributed, but instead are “fat-tailed,” so that the actual probabil- ity of insolvency may be greater than that calculated using the assumption of normality. Moreover, this one-period probability may understate the true probability of insolvency because it mea- sures the risk of a single-period loss being so large it wipes out equity. In reality, insolvency often occurs after a sequence of smaller losses occurring over several periods, indicating that serial correla- tion between negative shocks may exist. Table 3 Capital Allocations for Consolidated Amalgamated Bank with Equal Probability of Insolvency Line of Business (1) ROA (Percent) (2) sROA (3) Z*-Ratio (4) Equity/Assets (Percent) (5) Equity ($millions) Credit Cards 4.94 1.08 13.80 9.96 2,018 Mortgage Banking 4.96 2.78 13.80 33.40 3,779 Subprime Lending 14.67 7.96 13.80 95.18 4,827 Total Bank 5.99 1.29 27.12 28.99 10,624 Required equity capital for bank to achieve Z* 5 13.80: K 5 (13.80)(1.29) 2 5.99 5 11.81% Equity capital 5 (11.81%)(36,647) 5 $4,329 million. Source: Columns 1, 2, and 3: Table 2 and author’s calculations. Column 4: (Column 3 3 Column 2) 2 Column 1. July/August 1998 New England Economic Review 45 Allowing for Diversification A comparison of the Z-ratios for the bank as a whole with the Z-ratios for the individual lines of business, as shown in Tables 2 or 3, reveals a draw- back to both of these stand-alone methods of allocat- ing capital. The Z-ratio for the bank as a whole is considerably greater than the Z-ratio for any of the three lines of business, indicating that the probability of insolvency for the bank is less than that of any of the lines of business. This occurs because the correlation in the ROAs of the individual businesses is less than perfect. To the extent such correlations are less than perfect, they will tend to dampen the fluctuations in returns for the bank as a whole, so that the risk of the bank will be less than the weighted sum of the risks of the individual businesses. In effect, the business units act as partial natural hedges for each other, reducing the need for equity capital. Thus, a bank with a diversified portfolio requires less equity capital to achieve any given probability of insolvency than do the business units on an aggregated stand-alone basis. This is shown at the bottom of Table 3, where the amount of equity capital needed for the bank as a whole to achieve a Z-ratio of 13.8 is calculated to be only $4.3 billion, less than half of the $10.6 billion calculated as the sum of the stand-alone allocations to the individual businesses. Thus, in those situations where the ROAs of the individual businesses are imperfectly correlated, a discrepancy will result between the sum of the indi- vidual equity allocations to the different lines of busi- ness and the equity capital required when the effects of diversification are incorporated. This discrepancy creates obstacles to the evaluation of businesses and their managers. Ultimately, the larger the capital allo- cation, the more difficult it is for a line of business to earn an economic profit. If capital allocations to indi- vidual businesses exceed the actual capital of the bank, then managers may believe this “ghost capital” unfairly biases downward the reported return on equity of each business. The excess allocated capital can also create strategic issues, since the reported EVASMs of the business units will not sum to the EVASM of the bank. Theoretically it would be possible for each line of business to fail to earn its required opportunity cost of stand-alone equity, while the bank as a whole surpassed its required opportunity cost of equity based on actual equity capital, which includes the effects of diversification. In extreme cases, a bank might choose to exit a business based on an insuffi- cient return to equity earned on allocated capital, when the return on equity on actual capital might be quite satisfactory. Proportional Scaling This problem can be addressed in two ways. The simplest is to scale back the allocations to the individ- ual businesses so that the sum of the allocations equals the actual (diversified) capital of the bank. Thus, if the sum of the individual allocations is 200 percent of the actual capital of the bank, each allocation is reduced by one-half to make the sum of the individual alloca- tions equal to actual capital. This approach is illus- trated for Consolidated Amalgamated in Table 4, assuming that each line of business has the same probability of insolvency (from Table 3) and that the bank as a whole has a target Z-ratio of 13.8. In effect, this approach spreads the reduction in equity capital due to diversification across the lines of business in proportion to their initial stand-alone capital allocations. Table 4 Capital Allocation for Consolidated Amalgamated Bank with Equal Probability of Insolvency and Diversification Effects Line of Business (1) Stand-Alone Equity ($millions) (2) Diversification Effect (3) Equity Allocation with Diversification Effect ($millions) Credit Card 2,018 .4074 822 Mortgage Banking 3,779 .4074 1,540 Subprime Lending 4,827 .4074 1,967 Total Bank 10,624 .4074 4,329 Source: Column 1: Table 3. Column 2: $4,329 (from Table 3) 4 $10,624 (from Table 3). Column 3: Column 1 3 Column 2. July/August 1998 New England Economic Review46 business with a negative correlation with existing businesses can actually reduce the required capital, resulting in negative marginal capital. This is shown in Table 7 for the mortgage banking business. Because the correlation in returns between the mortgage bank- ing business and the subprime lending business is negative (20.53), adding the mortgage banking busi- ness to an existing combination of the credit card and subprime lending businesses actually dampens the variation in the aggregate and therefore reduces the required capital. Moreover, marginal capital is not constant but will vary as the size of the business in question varies relative to the size of the other busi- nesses in the bank. As discussed in the box, “Internal Betas and Marginal Capital,” the marginal capital associated with a given increment in the size of a business increases as the business unit becomes a larger proportion of the bank. Capital Allocations and EVASM Table 8 summarizes the results of Tables 2, 3, 4, 6, and 7 and shows the equity capital allocated to each of Consolidated Amalgamated’s three businesses using each of the capital allocation methodologies discussed above. Depending on the methodology selected, the allocated equity capital, and thus the reported EVASM, of a business unit can vary dramatically. Clearly the capital allocation methodology se- lected will affect not only the reported EVASM of each Table 7 Calculation of Marginal Equity Capital for Consolidated Amalgamated Bank Business Unit (1) Required Equity Capital for Bank with All Three Business Units ($millions) (2) Required Capital Ratio for Bank without Business Unit (Percent) (3) Bank Assets without Business Unit ($millions) (4) Required Equity Capital for Bank without Business Unit ($millions) (5) Marginal Equity Capital ($millions) (6) Marginal Capital Ratio (Percent) Credit Card 4,329 21.74 16,386 3,562 767 3.78 Mortgage Banking 4,329 19.78 25,333 5,012 (683) (6.04) Subprime Lending 4,329 12.55 31,575 3,961 368 7.25 Total Allocated Capital 452 Unallocated Capital 3,877 Total Bank Capital 4,329 Source: Column 1: Table 3. Column 4: Column 2 3 Column 3. Column 2: Author’s calculations, using method from Table 3. Column 5: Column 1—Column 4. Column 3: Compustat. Column 6: Column 5 4 Column 1, Table 2. Table 8 Equity Capital Allocations for Consolidated Amalgamated Bank, by Allocation Methodology Business Unit (1) Stand Alone: Peer Group ($millions) (2) Stand Alone: Equal Probability of Insolvency ($millions) (3) Scaled Diversification ($millions) (4) Internal Betas ($millions) (5) Marginal Capital ($millions) Credit Card 2,018 2,018 822 1,526 767 Mortgage Banking 1,989 3,779 1,540 1,217 (683) Subprime Lending 1,666 4,827 1,967 1,586 368 Unallocated Capital 3,877 Bank Total 5,633 10,624 4,329 4,329 4,329 Source: Column 1: Table 2 Column 4: Table 6 Column 2: Table 3 Column 5: Table 7 Column 3: Table 4 July/August 1998 New England Economic Review 49 Internal Betas and Marginal Capital Internal Betas The risk of a bank (s2 Bank) with n different business units is given by the formula: s2 Bank 5 SSwiwjcovi, j (B-1) where wi is the proportion of assets used by the i-th business unit, and covi,j is the covariance of returns between the i-th and j-th business unit. This rela- tionship is depicted in Table B-1 as the sum of the terms of a matrix of the business unit variances and covariances,a with each row representing a different business unit. Then the risk contribution of busi- ness 1 can be expressed as the sum of the terms in row 1, weighted by the assets of the business: Risk contribution of business 1 5 w1Swjcov1, j 5 w1cov1, Bank . (B-2) To measure the proportion of total risk contributed by business 1, we divide equation (B-2) by the overall risk of the bank: a Notice that the covariance of a variable with itself equals the variance of the variable. Proportional risk contribution of business 1 5 w1cov1, Bank/s2 Bank 5 w1b1 . (B-3) But this is the internal beta of business 1. Because the proportion of risk accounted for by all the business units in the bank must equal the risk of the bank, then Swibi 5 1 . (B-4) While the internal beta approach divides up the risk of the bank and does so in a way that incorporates the correlation in returns between the business unit and the bank, using the internal beta to allocate capital involves two very restrictive assumptions. First, because the risk of the bank is the weighted sum of the risk contribution of the business units, it already incorporates the risk contribution of busi- ness 1. That is, the risk contribution of each busi- ness is calculated on an ex post basis, assuming that the business is already and will remain a part of the bank. If a new business unit is added (deleted) then the variance/covariance matrix used to calculate the risk of the bank will have to add (delete) both a row and a column and the weights of the original Table B-1 Risk Contribution By Business Unit: The Internal Beta Approach Business Unit 1 2 3 N 1 w1 2s1 2 w1w2cov1,2 w 1 w3cov1,3 — w1wncov1,n Risk Contribution 5 w1Swj cov1,j 5 w1cov1,Bank 2 w2w1cov1,2 w2 2s2 2 w2w3cov2,3 — w2wncov2,n Risk Contribution 5 w2Swjcov2,j 5 w2cov2,Bank 3 w3w1cov1,3 w3w2cov2,3 w3 2s3 2 — w3wncov3,n Risk Contribution 5 w3Swj cov3,j 5 w3cov3,Bank — — — — — N wnw1cov1,n wnw2cov2,n wnw3covn,3 — wn 2sn 2 Risk Contribution 5 wnSwjcovn,j 5 wncovn,Bank Total Contribution 5 SS wiwj covi,j 5 s2 Bank July/August 1998 New England Economic Review50 entries will change so that each row of the matrix, as well as the overall risk of the bank, will change. Second, the calculated risk contributions for each business unit are only valid for the asset weightings used. Any disproportional change in the relative importance of a business unit will change the weights on all of the entries in the variance/ covariance matrix and thus result in a change not only in the internal betas of that business unit, but also in the internal betas of all of the other business units. Thus, capital allocations calculated using the internal beta approach are valid only for a specific mix of business units and cannot be used for other configurations of business units or asset weight- ings. Moreover, the capital allocation and reported EVASM of each business unit will be affected by the activity of the other business units in the bank. Marginal Capital Because a disproportionate change in the activ- ity of one business unit affects the risk weighting of all of the business units, the incremental change in the total risk of the bank is not just the increment in the risk contribution of the particular business unit initiating the change, but also includes the effects on the risk contributions of all of the other business units in the variance/covariance matrix. Except in special circumstances this marginal risk contribu- tion will not be equal to the risk contribution computed using internal betas. This is shown in Figure B-1 for a bank consisting of two business units. Business unit 1 is relatively low-risk and low-return, while business unit 2 is relatively high- risk, high-return. Figure B-1 shows the equity cap- ital-to-asset ratio required to achieve a constant Z-ratio for different asset weightings of units 1 and 2. At point A, 100 percent of the bank’s assets are comprised of unit 1 and the bank’s required capital- to-asset ratio is simply the stand-alone required capital ratio for unit 1. At point B, 100 percent of the bank’s assets are invested in unit 2, and the bank’s required capital-to asset ratio is simply the stand- alone required capital ratio for unit 2. The curve AB represents the equity capital-to-asset ratios for all the weightings of unit 1 and 2 to achieve the same probability of insolvency and is thus an iso-insol- vency curve. It is convex because the returns of the businesses are assumed to be imperfectly positively correlated. As shown in Figure B-1, each point on the iso-insolvency curve shows a different capital-to- asset ratio corresponding to a different mix of business units. If the bank increases the size of unit 2 relative to unit 1 it will move to the right along the curve and its required capital-to-asset ratio will increase. The rate at which the required capital-to- asset ratio increases is equivalent to the marginal capital ratio and can be shown as the slope of a tangent to the iso-insolvency curve. At point C, the required capital-to-asset ratio is OC, but the mar- ginal capital is equal to the slope of the tangent at C, which is greater than OC. Thus, the marginal cap- ital ratio will not equal the capital ratio for the bank as a whole, nor will it be a weighted average of the stand-alone risk of each of the business units. July/August 1998 New England Economic Review 51