A New Approach to the Valuation of Banks

8/10/2019 A New Approach to the Valuation of Banks

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1 IntroductionThe principal function of nancial markets is the efficient and intertemporal alloca-tion of capital, but, if let alone, nancial markets tend to imperfections and exhibitfrictions in performing this function. Financial intermediaries in general, and banksin particular, owe their existence to these inefficiencies in the distribution of capitalfrom those with a surplus to those with a need, i.e. reducing transaction costs, andat the same time efficiently selecting among those with capital needs according totheir respective risks, thereby solving problems of asymmetric information. Most of the reasons literature has brought forward for the existence of banks can and havebeen subsumed under these two broad categories. 1 The importance of these services

has been underpinned in several studies showing that the level of development andsophistication of nancial intermediaries, among which banks certainly belong tothe most important, can have a signicant impact on economic growth. 2

When we dene a banks business as accepting (shorter-term) deposits and issu-ing (longer-term) loans in our framework, this is congruent to common denitionsin the literature .3 As such, the bank is one of the oldest institutions of nancialintermediation and still plays a prominent role in the economy for the allocation of capitalthis also in spite of the discussion surrounding the disintermediation of thenancial sector .4 Nonetheless, technological advances have had a profound effect onbanking and have lead to a worldwide consolidation in the sector. 5 In Germany,consolidation is also underway but still lagging behind. Germanys comparably stillvery fragmented banking market is often attributed to the institutionalized segre-gation in three pillars .6 However, the strict segregation of this three-pillared systemis currently up for discussion, possibly opening the gates for further consolidationin the largest European banking market.

Given these considerations, the value of banks is clearly a question of interest,be it for shareholder value-oriented management or in the course of a merger oracquisition. Although rm valuation is one of the core problems of corporate -nance and has attracted extensive coverage in the literature, we argue that a banks

1 See e.g. Santomero (1984) or Bhattacharya and Thakor (1993) for surveys and Freixas andRochet (1997) as a more comprehensive textbook.

2 For a review of this issue, see e.g. Levine (1997).3 See e.g. Freixas and Rochet (1997). In a classication of banks, this refers mainly to com-

mercial banks, but not exclusively. For example, in the regulatory framework of many countries,a commercial bank is clearly distinct from e.g. a savings institutions or a savings and loan as-sociation. However, all three would fall within our denition of a bank as an institution issuingloans nanced by deposits. The facts that e.g. thrifts in the U.S. mainly issue loans in the form of mortgages, or savings institutions in Germany are obliged to serve common welfare, are technicaldetails with little relevance for our purposes.

4 See e.g. Schmidt et al. (1999) for a study countering the common arguments of disintermedi-ation and supporting the importance of banks.

5 See e.g. Berger et al. (1999) and a May 2006 special issue of The Economist titled Thinkingbig: A survey of international banking.

6

See Hackethal (2004) for an overview and Decressin et al. (2003) for a detailed analysis of theGerman banking system.

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business exhibits peculiarities that do deserve a special treatment in the approachto its valuation. Although this problem obviously does not require a separate andnovel pricing theory, certain deviations from standard methods seem not only ap-propriate but also necessary to us; to say it with Damodarans (2002) words: Thebasic principles of valuation apply just as much for nancial service rms as theydo for other rms. There are, however, a few aspects relating to nancial servicerms that can affect how they are valued. 7 We will demonstrate both why andhow standard valuation approaches should be modied to account for bank-specicbusiness risks. Specically, we address interest rate risk as a major characteristicof the banking business. The bank faces this type of risk not only directly in theprocess of maturity transformation but also indirectly in the determination of price

margins and the attraction of business volume, all affecting the value of the bank.The high interest-rate sensitivity of a banks market value has been noted many

times before in the literature; for example, early empirical studies supporting such arelationship are Flannery and James (1984 a ) for U.S. banks and Bessler and Booth(1994) for German banks. Accordingly, there exist several authors who have takenup this nding and suggested special bank valuation approaches. However, in theprocess of reviewing these approaches, we will show that their models do not reachvery far in incorporating bank-specic risks. Hence, to our knowledge, there existsno common framework to value a bank which adequately accounts for these features.

We propose an alternative and new valuation model for banks based on term

structure models of the interest rates. These models allow us to directly account forinterest rate risk but let us avoid the problem of having to explicitly forecast inter-est rate developments, a problem inherent in all other approaches. Originally, termstructure models were developed for the valuation of interest-rate sensitive deriva-tives. In the same spirit, we model banks as a portfolio of interest-rate contingentclaims and value bank equity as a call option on the portfolio value. In other words,we build on Mertons (1974) structural model of the rm and extend its standardasset process of the rm for an application to the dynamics of the banking rm.For expository reasons, we will set up our model rst in discrete time to gain anintuition for the banks business model and will then derive a valuation model in acontinuous-time setting.

The paper proceeds as follows. In Section 2, we review the existent literature onbank valuation and identify shortcomings of the present approaches. In section 3, wereview the banks business model and motivate a distinct approach to its valuation.We feel that this is necessary given the fact that this problem has attracted littleattention in the literature so far. In the fourth section, we sketch a simple version of our model in discrete time and in section 5, we extend this model to the mathematicsof continuous time in order to derive a solution. Section 6 concludes.

7 Damodaran (2002), p. 603.

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2 Survey: Literature on bank valuation

2.1 Why banks are specialLittle has been written on the valuation of banks. This fact allows for two conclu-sions: Either this question is of minor relevance and the valuation of banks deservesno special attention, or, alternatively, bank valuation exhibits particularities andproblems that have not found appropriate attention in the literature so far. Inseizing this problem, we follow the latter view and rely on the few existing paperscontributing to this issue and on those explicitly mentioning the valuation of banksas one of the unresolved issues in nancial research, as e.g. Copeland et al. (2005)do.8

In corporate nance, it is not unusual to specify valuation models for particulartypes of rms. For example, to mention just two of them, Brennan and Schwartz(1985) propose a real-options based valuation approach to natural resource compa-nies, explicitly modeling the options to temporarily close, reopen and shut-down themine, depending on the market price of the resource and Kronimus (2002) develops amodel suitable to the traits of young, fast-growing rms, including little or negativecurrent earnings but fast revenue growth. These models do not aim at introducing anew paradigm in asset pricing theory. Rather, the common feature of these modelsis the attempt to better grasp the underlying characteristics of the businessonwhich equity is the residual claimas when compared to standard approaches. Forthe same reasons, one can argue for a special valuation approach for banks.

The characteristics of the banking business motivating a distinct valuation ap-proach can be subsumed in four categories. First, due to their central role for theeconomy, banking is typically a heavily regulated industry, covering a wide rangeof provisions, such as e.g. market entry, deposit insurance, reserve requirements, orcapital structure. 9 Second, banks operate on both sides of their balance sheets, ac-tively seeking prots not only in lending but also in raising capital, a duality whichhad been practiced for long but had not been fully understood by economists un-til the late 19th century: A Banker is a trader whose business consists in buyingMoney and Debts, by creating other debts. 10 From a nancial accounting point

of few, this implies relatively few xed assets, resulting in a low operating lever-age, and relatively high nancial leverage, resulting in a comparably higher earningsvolatility. 11 Third, and as a consequence of the previous point, banks are exposedto credit default risk, but in contrast to other rms, they also actively seek this kindof risk as part of their business model. Last but not least, the prot and the valueof the bank is strongly dependent on interest rate risk.

Tackling each of these four points is of unequal complexity. Solving the rst8 See Copeland et al. (2005), p. 872.9 See e.g. Carey and Stulz (2006).

10 MacLeod (1875), p. I:275.11

It can be shown that banks optimally operate at high leverage because of deposit insuranceschemes granted to them in almost all industrialized countries, which is essentially a put optiongiven to the bank, see e.g. Buser et al. (1981).

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issue is straightforward, since the regulatory framework is usually known and thesame for all banks. Regulations have to be accounted for, but they merely imposedeterministic structural restrictions on a valuation model. Similarly, the secondpeculiarity does not pose large problems either. Instead of applying the entitymethod, which measures the cash ows to all claimants of the rm and discountsthem at the weighted-average cost of capital, one has to rely solely on the equitymethod, treating interest expense as a cost, more precisely an operating cost in thecase of banks. Adequately measuring the credit default risk of a banks assets is amore challenging issue. Nonetheless, its effect on bank value is obviousa downsiderisk proportional to the credit risk of the loan portfolioand one can readily applymodels of credit portfolio management to account for its effect on the value of the

entire bank. 12Much more interesting are the effects of interest rate risk on bank value, since

they surface at many different corners of the banks business. Most obviously, theslope of the term structure determines the protability of the maturity mismatchbetween a banks assets and liabilities. In addition to this,there exist further andmore subtle effects, though. For example, bank rates adjust sluggishly to changesin market rates, and do so asymmetrically, i.e. the speed of adjustment is differentwhen rates are rising as when they are falling, resulting in a time-varying spreadwhich is typically larger in a high interest-rate environment. These empirical traitshave been documented in several studies, such as e.g. Hannan and Berger (1991),

Ausubel (1991), or Neumark and Sharpe (1992), and current data in the latestmonthly report of the Bundesbank (2006) indicates similar evidence. Besides, thedemand for deposits and loans depends on both, the market rate and the rate a bankcharges or pays, respectively, relative to the market rate. Also, loan demand anddeposit demand typically have a negative correlation. Taken together, the effects of interest rate risk on bank value are signicant but nontrivial.

Samuelson (1945) is among the pioneers to argue for the (positive) dependenceof a banks value on the level of interest rates. However, for several decades, hispurely theoretical reasoning has not been adequately matched with supporting em-pirical results. Contemporary surveys on the interest-rate dependency of banksequity value typically start with Flannery and James (1984 a , 1984b), whose em-pirical studies could conrm Samuelsons earlier suggestions. For a sample of U.S.banks, Flannery and James (1984 a ) nd that bank stock returns and interest ratechanges are highly correlated. They introduce the interest rate as an additionalrisk factor in a two-factor intertemporal capital asset pricing model, where the in-vestment opportunity set varies with the level of interest rates, as rst suggestedin Mertons (1973a ) ICAPM. Furthermore, their results are consistent with whatthey call the maturity mismatch hypothesis, i.e. that cross-sectional variations ininterest rate sensitivity result, at least in part, from differences in the assets and

12 This is not to mean that measuring credit portfolio risk is straightforward. Rather, we implythat there are already abundant suitable models for this task, a review of which would deservea paper in its own right. See e.g. Crouhy et al. (2000) or Uhrig-Homburg (2002) for a survey of credit risk models.

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liabilities structure. In addition, Flannery and James (1984 b) show that the interestrate sensitivity does not depend on the stated maturities but on the effective matu-rities of instruments on the bank balance sheet. Usually, the bulk of bank liabilitiesconsist of core deposits, to which we will refer in the following simply by deposits.This is for ease of exposition and is not to be confused with the general meaningof deposits, which also includes deposits with a pre-dened maturity, for examplecerticates of deposits (CDs) or time deposits .13 These (core) deposits have a veryshort or no maturity but their effective maturity tends to be longer, which can beconcluded from two observations: First, although depositors are allowed to with-draw their balances every day, the half-life or decay rate of balances tends to besignicantly longer; second, banks take advantage of this sticky behavior and adjust

bank rates in response to market rate changes only sluggishly, especially in times of rising interest rates. Consequently, a higher share of core deposits in bank liabilitiesresults in a decrease of both the maturity mismatch and interest rate sensitivity of bank stock returns, ceteris paribus. In the following, many authors rened thesemethods and added new aspects, of which the overwhelming part found supportingevidence consistent with Flannery and James .14 However, in concluding it shouldbe remarked that these approaches draw a fairly general picture, since the inclusionof a second interest rate risk factor in a multi-factor model only accounts for the theinterest-rate related business risk of the banking industry in general.

2.2 Existing approaches to bank valuationInterestingly, one nds mostly German authors who take up the problem of valu-ing banks, with many Ph.D. theses published on this topic, whereas in the Anglo-American literature, we are only aware of textbooks and guides for practitioners,although the previously mentioned Copeland et al. (2005) list it as one of the unre-solved problems of nancial theory.

Among the German authors, early treatments of Zessin (1982) and Adolf et al.(1989a , 1989b) were followed by several theses during the 1990s on the shareholder

13 For a more detailed characterization, it can be said that in the U.S., four types of deposits

are characterized as non-maturing core deposits: demand deposit accounts (DDAs), negotiableorder of withdrawal (NOW) accounts, money market deposit accounts (MMDAs), and passbookaccounts. DDAs are standard accounts held for transaction purposes and represent the greatestpart of money supply in the U.S. NOW accounts are savings accounts that serve for transactionthrough negotiable orders of withdrawals, on behalf of the holder and payable to third parties.MMDAs pay a higher interest rate than traditional deposits but are limited in the number of transactions they allows per month. Passbook accounts have a legally required short noticationperiod before withdrawals.

14 For the U.S. market, these include e.g. Saunders and Yourougou (1990) and Elyasiani andMansur (1998). For Europe and Germany in particular, see for example , Bessler and Booth(1994), Oertmann et al. (2000), or Bessler and Opfer (2003). A particularly interesting study isthat of Kane and Unal (1990), who identify off-balance sheet sources why bank stocks might seeminsensitive to interest rate shocks while in fact they are not. This reconciles contradicting resultsof several previous studies such as Lynge and Zumwalt (1980), Chance and Lane (1980), or Kaneand Unal (1988).

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value of banks, including Kummel (1995), Behm (1994), Hohmann (1998), andKirsten (2000), and by one thesis of Sonntag (2001), which deals exclusively withthe particularities of bank valuation. Textbooks and practitioners guides includee.g. Miller (1995), Johnson (1996), Copeland et al. (2000), Damodaran (2002), andKoch and MacDonald (2005). Irrespective of their origin, the reasons all these bringforward for a special bank valuation approach largely fall within the four categoriesmentioned in the preceding section, among which, in accordance with our remarks inthe previous section, most stress the interest rate risk as one of the most importantvariables affecting the value of bank equity.

Although Zessin (1982) emphasizes the interest rate risk as a major determinantof bank equity value, his propositions to account for it are rather crude. First, he as-

sumes that the level of interest rates is mainly driven by macroeconomic factors andproposes expert surveys and macroeconomic outlooks as basis for an interest rateforecast. Then, assuming that there is little room for individual pricing behaviorin the oligopolistic banking market, he suggests a regression analysis to determinethe relationship between bank rates and market interest rates. Finally, this methodshould be repeated in a scenario-like analysis .15 A serious drawback of such an ap-proach is the fact that the predictive power of expert surveys is often disappointing;of the many studies coming to this result, see e.g. Brooks and Gray (2004). Fur-ther, there is little room for cross-sectional and intertemporal differentiation amongbanks. Last but not least, Zessin mentions the interrelated problem of forecasting

business volume on the asset and liability side on one hand and respective interestmargins on the other, but offers little to overcome it .16 In the remainder, he discussesfurther aspects in which banks distinguish themselves from other rms, though theoverall valuation procedure is of little difference to rm valuation methodologies.

Adolf et al. (1989a , 1989b) also suggest a combination of forecasts of the interestrate and business volume, and scenario techniques to grasp the inuence of interestrate risk on bank earnings, and gradually rene the previous approach in that theylook at strategic business units instead of the bank in its entirety. They introducea maximum payout assumption, according to which the bank retains cash owsonly up to the regulatory capital required for its business in the next period andpays out all remaining cash ows. Additionally, they suggest the application of aninstrument from internal bank controlling, the transfer pricing based on matched-maturity marginal value of funds (MMMVF), in the valuation process. 17 In so doing,they are able to separately value the asset business, the liability business and thetreasury. Such a procedure offers several advantages, most notably the considerationof different risk factors for the asset and the liability business, and the possibility toisolate the interest rate risk from maturity mismatches in the treasury.

15 Zessin (1982), p. 121127. Later, he includes a description of several instruments to measureinterest rate risk, but does so only in a separate analysis, i.e. not incorporating these methods inthe valuation process.

16 See Zessin (1982), p. 126.17

Their adaption of the MMMVF is the Marktzinsmethode as outlined e.g. in Schierenbeck(1997). We will return to this model in more detail in the following section.

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This idea is further outlined and worked on in Behm (1994) and Kummel (1995),of which the latter elaborates on alternative forecasting methods for business vol-umes, interest rates and, depending thereon, bank rates .18 He focuses clearly on thedevelopment of interest rates as a key driver of bank value, adding further complex-ity with his point that the bank rate is not a passive function of the market ratebut also dependent on management decisions and business strategies .19 To accountfor the effects of interest rate changes, he proposes another tool of bank controlling,the balance sheet of interest rate elasticities. 20 Without delving into a detailed de-scription here, one can characterize this tool as offering the advantage to include theresponsiveness of bank rates to changes in market rates, which is not accounted forin purely market-based measures of interest rate risk such as e.g. duration. Although

this offers a potential enhancement in accuracy and elasticities seem to be relativelystable, 21 it merely shifts the forecasting problem since it implicitly assumes thatother market parameters, for example intensity of competition or customer behav-ior, remain constant over time as well. Ultimately, K ummels (1995) approach doesnot offer alternatives to forecasting interest rates, and for this he relies on the samemethodologies as previous authors, implying the same serious drawbacks. Sonntag(2001) also builds his model upon the transfer pricing of MMMVF; his innovationis the valuation of bank cash ows based on the method of certainty equivalents. 22

Not surprisingly, the practitioners guides offer little methodological innovation.Their focus is more applied in nature and their suggestions seem to be easier to

implement. According to them, the two most prominent problems in bank valuationare to determine the quality of loans and to understand the role of interest rate riskand maturity mismatches. As to the former, they discuss the problems for outsidersto appropriately value the loan portfolio, which is further severed by managerialfreedom in the share set aside for loan loss provisions; for the latter, they also proposea model based on internal transfer prices. For example, Copeland et al. (2000)compare a net income model for non-nancial rms with a spread model specicallyfor banks, where the latter is largely equivalent to the transfer pricing model basedon MMMVF mentioned above. In so doing, they summarize the advantages of thespread model in explicitly requiring a forecast of four parameter: (1) the spreadsbetween respective bank rates and market rates, and their elasticity to changes inmarket rates, (2) the dynamics of inows and outows of funds over time and atrespective rates, (3) the substitution between bank products as interest rates change,and (4) the portion of prots purely due to maturity mismatch and how sustainableit is over time.23 Nonetheless, they admit at the same time that [i]t is not easy to

18 See Kummel (1995), p. 59 and pp. 6698.19 See Kummel (1995), p. 71 and p. 97.20 The Zinselastizit atsbilanz as proposed e.g. in Rolfes (1999), part 3.21 See e.g. Rolfes and Schwanitz (1992).22 He further claims that one of the main results of his treatise is to have shown the irrelevance

of banks in the business of maturity transformation if one remains in a neoclassic framework, aclaim that we nd hard to approve. See e.g. the rich literature on the existence of banks as quotedin the beginning.

23 See Copeland et al. (2000), p. 434.

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build all of these variables into a forecast and do not offer alternatives to interestrate forecasts, either.

To summarize these works, it can be said rst, that there exist reasons whybank valuation represents a problem that is distinct in certain respects from rmvaluation; second, that these reasons can be grouped in four categories; third, thatamong these four problems, interest rate risk is the most important one; and fourth,that the methods proposed to account for this kind of risk are unsatisfactory andleave room for improvement.

3 Stylized business model of the bankA banks valuation begins with understanding its business model, which reects itsoperating activities, sources of revenues and costs structure. Once having estab-lished a suitable business model in this section, we will value the banks equity as acall option on the banks asset value in the next, following the insights of Mertons(1974) structural model of the rm. The Merton model provides a risk-neutral valu-ation framework within which precise solutions for the valuation of various nancialclaims on the rm (or asset) value can be derived. This precision comes at a priceand indeed merely shifts the ambiguity of rm valuation to the specication of thebusiness model that generates the rm value. Hence, in this section we characterizethe banks business model and in so doing lay out the foundation for a model of thebanking rm value.

A bank is an institution whose current operations consist of granting loans andreceiving deposits from the public. 24 As such, it seeks to make a prot on both sidesof the balance sheet, a fact that is sometimes referred to as duality of the bankingbusiness. For now, and without loss of generality, we abstract from taxes, reserverequirements and other sources of income; further, we will also neglect default riskof bank customers for the beginning. Then, a banks prot, , can be stated inits most simple form as difference between the return on assets and the costs of liabilities, net of general and administrative costs, C ,

= r A A r L L C = N II C. (1)

Here, the difference rA A r L L is also known as net interest income, NII .Obviously, the factors determining the current NII are the realized spread and theexisting business volume, which can expressed in terms of average interest-earningassets as we abstract from reserve requirements. 25 Future NII is driven by theexpected spread, the expected growth in interest-earning assets, and additionallythe development of the term structure of interest rates, as bank assets and liabilitiestypically differ in maturity and composition of xed and oating rate products.

24 Freixas and Rochet (1997), p. 1.25 Accounting for reserve requirements would result in a deduction from liabilities and hence

a difference between interest-earning assets and interest-bearing liabilities. Introducing such adeterministic feature later does not pose a problem but doing so now would further complicate ourexposition at this point.

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A theoretical model for the exercise of market power by the banking rm in animperfectly competitive Cournot-type of market has rst been established by Klein(1971) and Monti (1972), who formalize the idea of a positive relationship betweenthe relative NII and the market power of a bank, ceteris paribus. 26 The assump-tion that banks operate in an imperfectly competitive market is intuitive. In theliability business, banks offer signicant economies of scale in storing valuables andin providing an efficient access to the payment system. In the asset business, banksoffer economies of scale in credit risk assessment, loan procurement and creditormonitoring. 27 Naturally, the larger their customers are, the lower will be the advan-tages banks can offer in terms of economies of scale and accordingly, the lower theirmarket power will be. For example, while it would not make sense to access capital

markets for a single consumer loan, corporate customers will often nd it cheaperto issue bonds instead of nancing their projects by bank loans.

Another important measure is the ratio of the NII to total income. Followingthese thoughts, this ratio can also take on two interpretations. First, it is clearlya measure for the interest-rate dependency of a banks incomethe more income abank can generate from service fees and provisions, the less it will depend on interest-rate sensitive income. Second, this ratio can also be seen as a proxy for the customergroups a bank services and the market power it can exercise with them. While thelatest monthly report of Bundesbank (2006), gives an overview of this measure foraggregate groups of German banks, 28 let us pick out just two of them, Hamburger

Sparkasse (Haspa) and Deutsche Bank (DeuBa), to make our point more obvious.Haspa is the largest German savings institution in Germany and mainly serves retailcustomers and regional SMEs; in retail banking, it has a local market share above70%.29 On the other hand, DeuBa is the largest German bank and operates as afully-integrated universal bank, offering commercial banking, investment banking,and asset management on a global scale and under one roof .30 While DeuBas com-mercial banking unit also serves retail clients, its corporate clients will typically belarger than those of Haspa. Not surprisingly, Haspas net interest income accountedfor 62.7% of its gross prot in 2005, whereas this gure is only 23.4% in the caseof DeuBa.31 Certainly, the difference in this ratio reects the underlying differencesin banking services offered. However, it is also conceivable that DeuBa is just notable to extract an adequate NII from its corporate customers and therefore has to

26 Relative NII is to mean the net interest margin of a bank, which one obtains by dividingthe net interest income by the amount of average earning assets. For a comprehensive summaryof the Monti-Klein model, see e.g. Freixas and Rochet (1997), section 3.2.

27 These are also the economic justications brought forward for the existence of banks. See e.g.Allen and Santomero (1998) for an overview.

28 See Bundesbank (2006), p. 23.29 See company information at www.haspa.de . Haspa is the largest savings institution in terms

of assets.30 See company information at www.db.com . DeuBa is the largest German bank both in terms of

assets and market value.31

Net interest income is presented in % of total income net of interest expense in the denominator.This denominator can be likened in some sense to the gross prot of a non-nancial rm. Figuresfor calculation are taken from the annual reports for 2005.

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i l

V BANK = NPV (r A, =2 A r L, =1 L C)

Fristentransformationsbeitrag

r A, =2 = f (r A, =2, t-1 , r, X r A )

r L, =1 = f (r L, =1 , t-1 , r, X r L )

Value of bank equity asan interest rate

contingent claim

A = f (At-1

, [r A,

=2 r], X

A )

L = f (Lt-1 , [r r L, =1 ] , X L )

Figure 1: The stylized valuation model of the bank

turn to other services as well.How do these insights bring us any further? They do so in two respects. First,

the market power is a central feature of our bank model. Therefore, it is importantto grasp its importance for the bank value, which is also underscored by the sheernumber of studies analyzing this question .32 Second, since our model is primarilyconcerned with the interest rate risk in the business of commercial banks, this seg-mentation of banks offers helpful background information on the models relevance.Further, in such a segmentation, it should be stressed that our model can be appliedto the valuation of universal banks as well, this for valuing the commercial bankingunit in a sum-of-the-parts approach. 33

The basic idea of a N II -based approach is depicted in gure 1. There, the banksequity value V BANK results as the net present value of the NII net of costs, as longas we neglect items like depreciation, amortization, and changes in working capitalso that the net income can be used as a proxy for cash ows. Besides, the gure listsalready some of the variables that affect the constituents of the NII ; for example,the asset volume A is assumed to be a function of its past value A t 1, the spread thebank charges above comparable rates, rA r , and further yet unidentied factorssummarized in the vector X A . Although this framework is suitable and has beenproposed in several works on bank valuation, it also has serious drawbacks. A majorweakness of valuing a bank based on the overall NII as shown in gure 1 is that itignores differences in the maturity composition of assets and liabilities and the factthat banks are typically exposed to a positive duration gap, i.e. that the average

32 There exist numerous papers analyzing almost as many factors impacting the N I I relative toasset volume, see e.g. Ho and Saunders (1981), McShane and Sharpe (1985), Kolari et al. (1988),Allen (1988), Zarruk (1989), Moore, Porter and Small (1990), Ausubel (1991), Hannan (1991),Hannan and Berger (1991), Neumark and Sharpe (1992), Calem and Mester (1995), Scholnick(1996), Heffernan (1997), Angbazo (1997), Wong (1997), Sharpe (1997), Berger and Hannan (1998),Demirg uc-Kunt and Huizinga (1999), Kahn et al. (1999), Saunders and Schumacher (2000), Bergeret al. (2000), Lim (2001), English (2002), Stanhouse and Stock (2004), Hanweck and Ryu (2004),Maudos and de Guevara (2004), Brewer III and Jackson III (2006).

33 The sum-of-the-parts approach is a common methodology for valuing a multi-product rm.See e.g. Copeland et al. (2000), ch. 14.

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duration of assets is longer than that of liabilities. Yet this composition accounts inlarge parts for the changes in the N II in reaction to changes in the term structure of interest rates. Also, a banks market power in issuing loans might be much differentfrom that in issuing deposits, a difference which remains undetected in this setting.

Considering these problems, we follow a common practice in bank managementand attribute the prots of entire bank to its constituent business units based onMMMVF-transfer prices, i.e. the opportunity costor, more appropriately, the op-portunity rateof an investment in bonds with a comparable risk prole. 34 Al-though these transfer prices were originally developed as a tool of bank controlling,they can serve useful purposes in bank valuation and have been applied to bankvaluation before, as mentioned in the previous section reviewing the literature.

Furthermore and again simplifying, we assume that a bank consists of threebusiness units, an asset business unit (ABU), a liability business unit (LBU), and atreasury department which is in charge of the asset-liability management of the bank(ALM). The ABU is in charge of issuing loans and, principally, of measuring andmanaging credit risk, although we abstract from any issues of credit risk in our modelso far. The LBU issues transactions and savings deposits, providing its customerswith an access to the payment system and the bank with an internal funding of theABU. The ALM mainly has two task in our simple model of the bank. First, it hasto equalize any imbalances between assets and liabilities with external nancing,taking long or short positions in the interbank market. 35 Second, it is in charge

of hedging interest rate risk, most notably term structure risk stemming from theduration gap between the two sides of the balance sheet. Given this internal bankstructure and further assuming that there exists market rates of various maturities at which the bank can lend and borrow in the interbank market, r , the prot inequation 1 can then be restated as

= ( r A r =2 ) A + ( r =1 r L ) L + ( r =2 A r =1 L) C. (2)

Here, r =2 signies the market rate for a longer maturity and r =1 represents acomparable rate for a shorter maturity. Implicitly, this means of course that the re-spective bank rates have similar maturities. The market rates then serve as internal

transfer prices of funds within the bank. In this way, the prot of the bank can beinterpreted as the sum of the prots of the three business units dened above, netof cost.36 This conceptual view is depicted in gure 2.

Thus, the prots of both ABU and LBU in this single-period model are concep-tually driven by the same two factors as the NII of the entire bank: The spread

34 See e.g. Rolfes (1999) or Dermine (2005).35 Although we allow the volume of assets and liabilities to differ within a period, with the

difference offset in the interbank market, we rule out the case of permanent or even increasingdeviations, i.e. that the two sides of the banks balance sheet have different long-term growthrates.

36 It is desirable to spread costs on the level of business units already. While this represents a

sound theoretical approach, we are aware of the fact that there might be more than one viable wayto split up costs between the units in an empirical implementation of the model. In the derivationof our bank value, we will largely ignore issues with costs.

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BANK = r A, =2 A r L, =1 L C

Assets and liabilitiesAssets

Asset Business Unit(ABU)

Asset-LiabilityManagement (ALM)

Liability Business Unit(LBU)

Liabilities

ABU = (r A, =2 r =2 ) A ALM = r =2 A r =1 L LBU = (r =1 r L, =1 )L

r A, =2 r L, =1r =2 r =1

BANK

Figure 2: The stylized business units of a bank

they can earn from their funds compared to the respective MMMVFs and the assetor liability volume of their business. In a multi-period model, we will need a thirdvariable accounting for growth in business volume. Then, the bank derives its valuefrom three sources: Value of expected market power in the asset business and theliability business, and expected growth of the balance sheet.

4 Risk-neutral valuation of the bank

Existing bank valuation approaches value these units based on their discountedcash ows and all run into the problem of having to forecast the development of theinterest rate if they wish to consider its effects on cash ows. Relying on the sameMMMVF-framework, we propose a different approach based on a structural modelof the rm. Whereas Black and Scholes (1973) and Merton (1973 b) remarked alreadythat their option pricing theory can be applied to a variety of nancial contracts,notably to the most basic claims in the form of debt and equity, it remained toMerton (1974) to develop a structural model of the rm in which he valued corporatedebt as a contingent claim on the rms asset value, V , which he assumed to followthe standard Brownian motion

dV V

= dt + dW, (3)

where is a drift factor and dW is a standard Wiener process. According to thisapproach, corporate debt can be valued as a portfolio of riskless debt and a shortposition in a put option on the rms asset, representing the default risk of the rm.The strike price of the call option representing rm equity and of the put optionimplied in rm debt are both the face value of outstanding debt. Since the put-call-parity holds in this case, too, a structural model of the rm always implies a uniedapproach to the valuation of a rms equity and debt.

Derivative pricing rests on the well-known principles of no-arbitrage and risk-neutral pricing, according to which a hedging or replicating portfolio ought to havethe same price tag attached as the prices of its constituent parts. The internal

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hedging activities of the ALM and the hedging costs it charges to the other twounits can be used to value the portfolios required to hedge the activities of the ABUand LBU. Then, these two units can be valued along the lines of the risk-neutralvaluation approach. The activities of the remaining ALM are the simplest to value,akin to an long-for-short yield curve swap. Thus, we suggest to value the entirebank as a portfolio of interest-rate contingent claims with the value of bank equityresulting as a call option on such a portfolio. In this fashion, and over and abovethe advantages which we have already mentioned, our approach circumvents anothercommon and nontrivial problem, i.e. the estimation of cost of equity for the threebusiness units on a stand-alone basis.

For this, and as conceptual link between our and a DCF-approach, it will be

useful to measure the economic value of assets and liabilities, as opposed to theirbook values. In the proposed framework, the existence of a positive spread has theeffect that the economic value of assets and liabilities does not equal its respectivebook value anymore. The relationship between the economic value of assets, V A ,and liabilities, V L , to their respective book or face value, A and L , can be expressedas:

V L = L NP V (LBU ), (4a)V A = A + NP V (ABU ), (4b)

i.e. the economic value of a banks liabilities (assets) is its face value reduced (in-creased) by the net present value of the additional prots it is able to generate. Theimpact of equations 4 is intuitive: If a bank is able to charge a larger spread on itsassets, their value will increase; similarly, if a bank is able to issue liabilities at alower spread, this will also increase the banks value. One can clarify this even moretalking the viewpoint of the hedging model based on MMMVFs. Then, since theexcess-spread on assets is inversely related to the bonds price, the ABUs businesscan be likened to buying a bond below par from the banks creditors and selling itat par to the ALM, and inversely, the LBUs business can be seen as buying bondsfrom ALM at par and selling them above par to depositors. The difference of bothplus the result from maturity transformation is the residual claim of the banksshareholders, V Bank , which they will only exercise if it is positive, of course. Thisimplies the boundary condition

V Bank = max [NP V (ABU ) + NP V (ALM ) + NP V (LBU ); 0] (5a)= max [V A + V ALM V L ; 0]. (5b)

This is an interesting result and deserves further comment. Obviously, the ex-ercise price of the option is V L , i.e. the economic value of liabilities and not theirface value, L, as in the case of Mertons (1974) model for non-nancial rms. Thishighlights the difference between banks and non-banks and offers an intuitive in-

terpretation: If the assets of a non-nancial rm will not suffice to cover the rmsnotional liabilities, L, the shareholders will choose not to exercise their option andrather will default. In contrast, if a banks assets will not cover the face value of

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its liabilities, L, shareholders might still be willing to exercise the call option inthe boundary case. They will avoid a liquidation of the bank as long as the valuederived from the liability business is large enough to cover the gap to the face valueof liabilities:

V Bank = max [V A + V ALM + NP V (LBU ) L ; 0]. (6)

Having established this basic boundary condition for the banking rm, we needto establish a stochastic process describing the evolution of the rm value as theunderlying. For this, we have to identify variables affecting the value of the aboveportfolios representing the business units and the specication of processes describ-

ing the stochastic behavior of these variables. In our approach, we assume that ourmodel bank offers just two products: A non-maturing deposit liability in its LBUand a a non-maturing overdraft facility in its asset business. Then, valuing a banksbusiness units requires explicit modeling of three factors for each unit:

1. ABU and LBU:

(a) Volume of assets and liabilities and their growth rates

(b) Average effective maturities and durations of assets and liabilities

(c) Changes in bank rates, both for bank assets and liabilities, in response

to changes in comparable market rates2. ALM:

(a) Volume of assets and liabilities and their growth rates

(b) Effective duration gap resulting from the difference between average ef-fective durations of assets and liabilities

(c) Dynamics of the term structure of market rates

As to a banks balance sheet size and growth rate, it is possible to deduce es-timates from past data; however, regarding the growth rate, this requires as wellexplicit assumptions on overall market growth rates and growth rates of competi-tors. The average effective maturities and durations also are at the core of bankvalue and, accordingly, its estimation is a key aspect of our valuation approach.While hedging against (parallel) shifts in the yield curve requires knowledge of theduration, this measure is based on an estimation of the effective maturity. 37

Banks further distinguish themselves from each other in the spreads they chargeon their assets above respective market rates and they pay on their liabilities belowcomparable rates. We call this the market power of banks. The sources of a banksmarket power can be manifold, one can think of it as e.g. the monopoly power of aregional bank in a geographically fragmented market or quality of service resulting

in above-average customer loyalty. Although we will incorporate this feature in37 See e.g. Mays (1997).

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calibrating our model on past data and hence accept the market power of a bankas given, we should bear in mind that in applying our model to the long run, wehave to additionally assume strategic factors that could erode or increase a banksmarket power.

An additional relevant variable for the valuation of the ALM is the duration gapresulting as the difference of effective durations of the ABU and LBU. Since theALM absorbs all the interest rate risk stemming from maturity transformation, it isalso here that the dynamics of the term structure are relevant. The term structuremodel serves as a stochastic measure for interest rate risk and as such supersedesthe problem of inaccurate forecasts of future market rates. Indirectly, the dynamicsof the term structure will impact the value of all three business units, since we will

model the asset and the liability rate of the bank as a function of market rates.The fact that we deal with two different maturities at the same time recommends

the use of a two-factor interest rate model, which is better suited to grasp thedynamics of the yield curve. In such a two-factor model, the rst factor is theshort rate, just as in single-factor models, and the second factor is often chosen tobe either a stochastic mean reversion speed or stochastic volatility. In comparison,single-factor models with constant volatility and reversion speed often exhibit poorts on empirical data in many yield curve environments .38 One suitable two-factormodel is the one proposed by Hull and White (1994), which offers computationaladvantages when compared to other models and allows an efficient implementation

in discrete time, as Muck and Rudolf (2005) could show.Of course, such dynamics are the same for all market participants and as suchdo not represent a variable in which one bank can distinguish itself from others.Finally, it is obvious that these variables have many interrelations which we have toaccount for as well. For example, the effective maturity of a product will determinethe appropriate market rate for transfer pricing, and changes in bank rates will havean inuence on business volume.

In the following, we begin sketching the valuation of the LBU, will then showhow the valuation of the ABU mirrors the approach of the LBU and nally will turnto the ALM.

4.1 Valuation of the liability business unit (LBU)We mentioned already that typically, the largest portion of bank liabilities consistof core deposits, i.e. deposits that have either no or a very short maturity, thusrepresenting the centerpiece of a banks liability side prots. Although depositsfactually have no maturity, i.e. mature daily, and should therefore be very interestrate sensitive, they often are not and behave more like longer-maturing liabilities.This is attributed to the fact that the deposit rate is administered rate, i.e. set bythe bank in an imperfectly competitive market. If one assumed that the statedmaturity dened the effective maturity of deposits, this meant that deposits would

38 See Rebonato (1998) for a discussion of theoretical and practical issues of various single-factorand two-factor models.

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be withdrawn as soon as they matured, leaving little room for the premium earnedby the bank. Empirical data contradicts this, as banks typically enjoy high retentionrates for their deposits. For example, estimates for U.S. thrift institutions in studiesperformed by the Office of Thrift Supervision (1994, 2001) indicate that yearlyretention rates for certain deposit types are above 75%.

Hence, the assumption of a longer effective maturity is in accordance with ob-served behavior of depositors, lower interest rate sensitivity of deposit rates andvolumes, and subsequently higher premium earned by banks on their liabilities. Inthis manner, the estimate for effective deposit maturities affects the value of theLBU in two ways. First, it determines the time horizon during which the bank isable to extract a premium on a given deposit base without attracting new deposits.

Besides deposit volume growth, this is an important measure for the business vol-ume of the LBU. Second, in our transfer pricing model of the bank, the appropriatemarket rate r =1 should be one with a duration comparable to the duration of de-posits. Although duration and maturity are obviously different measures and mostbanks confuse the length of time a dollar stays in the demand deposit (the maturity)with the sensitivity of balance values to changes in interest rates (the duration), 39the close relation between the two is obvious as well: An increase in the depositseffective maturity due to a change in market power entails an increase in the effec-tive duration of deposits. In turn, a larger duration of a banks deposits raises theduration of the transfer price r =1 at the same amount, which increases the prot

of the LBU as long as the term structure is upward sloping.In nancial research, the analysis of deposits as interest-rate sensitive claims israther young. Following the development of the option pricing theory, a plethoraof quantitative models for the valuation of interest-rate sensitive claims sprang up,whereas the valuation of deposits seemed to be the sleeping beauty and it tookmore than 20 years to kiss her awake. During the 1990s, several authors started todevelop original approaches to their valuation, accounting for both interest rate riskand the problem of indetermined maturity. To our knowledge, these include Office of Thrift Supervision (1994, 2001), Selvaggio (1996), Hutchison and Pennacchi (1996),Jarrow and van Deventer (1998), and OBrien (2000). 40 Shortly reviewing these willcertainly help us for our own approach and we will do so along the four key issues of deposit valuation: (i) The valuation approach chosen, (ii) the way in which interestrate risk is accounted for, (iii) the method chosen to model deposit volume and itsgrowth, and (iv) the method chosen to model deposit rates. An overview of thiscomparison is shown in gure 3.

39 Copeland et al. (2000), p. 439.40 Extensions to their approaches, most notably to the one of Jarrow and van Deventer (1998),

were developed by Goosse et al. (1999), de Jong and Wielhouwer (2003), Sheehan (2004) (nicht inbib), Laurent (2004), and Kalkbrener and Willing (2004).

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Valuation approach Interest rate

risk

Deposit volume

(function of)

Deposit balance growth

(function of)

Deposit rate

(function of)(1) OTS (1994,

2001)DCF /OAS-IRR Deterministic 1 Past balance, market

rate, deposit rate,retention rate

None Past deposit rates,interest rate, asymm.parameter

(2) HP (1996) Contingent claims(general equilibrium)

Vasicek (1977) Market rate, depositrate, exogenousdemand, trend gowth

Historical trend growthrate

Market rate, depositdemand elasticity

(3) Selvaggio(1996)

DCF / OAS-IRR CIR (1985) Past balance, marketrate, nominal income,seasonal dummy

Nominal income Market rate, pastdeposit rate,volatility parameter

(4) JvD (1998) Contingent claims(no-arbitrage)

HJM (1992) Past balance, retentionrate, growth rate,market rate

Growth rate net ofretention rate, marketrate

market rate, mean-reversion to averagedeposit rates

(5) O'Brien(2000)

Contingent claims(no-arbitrage)

CIR (1985) Past balance, marketrate, deposit rate,(regional) income

Historical trend growthrate

market rate, mean-reversion to cond.expected deposit

Model

Figure 3: Overview of deposit valuation modelsRemarks: 1 : Expected spot rates are inferred from the current forward rate curve; OAS-IRR:option-adjusted spread internal rate of return; CIR: Cox, Ingersoll, and Ross; HJM: Heath,

Jarrow, and Morton.

4.1.1 Valuation approach

The Office of Thrift Supervision (OTS) (1994, 2001) developed its Net Portfolio

Value Model (NPVM) in 1994 and added an update to some of its equations in2001.41 The OTS framework applies a discounted cash ows approach to the valua-tion of deposits. It estimates the cash ows generated by several deposit types on amonthly basis, net of costs. Instead of determining an appropriate discount rate forobtaining the net present value of deposits, the OTS uses price data from depositsales or purchases as an input parameter. With this given as an estimate of the netpresent value, it discounts the cash ows with LIBOR plus a spread that equalizesthe discounted cash ows to the value estimate. Hence, the NPVM can be likenedto an internal rate of return analysis, where the value of deposits is expressed as anoption-adjusted spread (OAS) above LIBOR. 42

Selvaggio (1996) applies a valuation methodology similar to that of the NPVM, inthat he estimates the OAS above the zero coupon rate that equalizes the discountedfuture cash ows of deposits to their empirically observed values.

Hutchison and Pennacchi (1996) (HP) develop a contingent claims valuationmodel for deposits and are the only ones building their model on a particular equi-librium model of the economy which allows for the existence of banks. Specically,

41 Compared to the original approach, the basic structure and approach remained the same inthe version of 2001, though, and the NPVM of both years will be presented as one model here. Asthe OTS is a regulatory and supervisory agency for thrift institutions in the U.S., the features andassumptions of this model, unlike those of the remaining models, have direct implications for theoperation of U.S. thrifts.

42

LIBOR was introduced in the 2001 framework. The NPVM of 1994 used rates paid on sec-ondary certicates of deposits.

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they found their approach on Hutchisons (1995) Cournot-type equilibrium modelwith imperfect competition. In this way, the spread of the deposit rate below themarket rate is internalized as the optimal spread, i.e. the prot-maximizing spreadbased on the trade-off between price and volume as captured by the price elasticityof deposit demandor deposit supply, to be precise. The value of the deposit busi-ness is then obtained as the discounted value of future oligopoly rentsor, moreappropriately, oligopsony rentswhere the rents are modeled as interest-rate sensi-tive claims. HP are able to derive a closed-form solution for their deposit valuationmodel.

Jarrow and van Deventer (1998) (JvD) develop a contingent claims valuationframework for interest rate sensitive claims with exercise of market power in a no-

arbitrage setting and apply it to the valuation of demand deposits. 43 They showthat the valuation of these instruments can basically be understood as being akinto the valuation of a particular swap of which the principal is a function of pastmarket rates. Applying the term structure model of Heath et al. (1992), they showhow to obtain a basic closed-form solution for the pricing of deposits. Their workconcentrates on the theoretical derivation of the model; an empirical implementationfollowed with Janosi et al. (1999) for the deposit data of one (anonymous) bank.

OBrien (2000) values deposits based on an arbitrage-free interest rate contingentclaims framework which is similarly to the one of JvD and applies it to a data sampleof NOW accounts and MMDAs of 74 U.S. commercial banks over 8 years.44 Like the

NPVM of the OTS and unlike the models of JvD and HP, however, OBrien allows foran asymmetric adjustment of deposit rates in accordance with empirically observedbehavior. Specically, in an environment of rising rates, the spread between marketrates and deposit rates tends to increase whereas in times of falling rates, it tendsto decrease. In the empirical implementation of his model, he nds the asymmetricfeature in deposit rate adjustments to be statistically highly signicant. However,due to the asymmetry he introduced, OBrien is not able to derive a closed formsolution and so has to rely on a Monte Carlo simulation of 1000 different paths.

4.1.2 Interest rate risk

There is no interest rate risk inherent the NPVM. Given the forward rates as ob-served in the market, the OTS derives one static scenario of future spot rates onwhich all calculations are based. Selvaggio (1996) uses Monte Carlo simulation tech-niques and generates 300 interest rate paths to account for interest rate risk. Hebases his simulation on the term structure model of Cox et al. (1985). While someparameters are taken from a previous estimation of this model by Chan et al. (1992),he calibrates the model on the LIBOR term structure of Eurodollar. HP model themarket rate using an Ornstein-Uhlenbeck process as specied by Vasicek (1977).

43 Although they derive their model for deposits, they state already that it can be applied in theinverse case to credit card loans as well.

44

NOW accounts are negotiable order of withdrawal accounts and MMDA stands for moneymarket deposit accounts. See also a precedent footnote.

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Since they are able to derive a closed-form solution to the valuation problem, nosimulations are needed. Consistent with JvDs arbitrage-free valuation approach,they model the term structure of interest rates according to the no-arbitrage ap-proach of Heath et al. (1992). However, they show that, in principle, any otherterm structure model could be used as long as it species a SDE of the spot rate.OBrien applies the underlying SDE of the market rate from Cox et al. (1985), i.e.an one-factor mean-reverting interest rate process which rules out negative interestrates.

4.1.3 Deposit volume and growth

A common way to model deposits with no or short maturity is to assume thatthey are withdrawn as soon as they mature and that the bank then issues newdeposits. The empirically observed stickiness of core deposit is accounted for by anautoregressive (AR) process of future deposit volume. For example, in a rst-orderautoregressive process, or AR(1) process, the deposit volume next period is largelydetermined by the volume in this period, L t ,

L t +1 = + L t + X + t , (7)

where is a parameter vector and X is a vector with other variables determiningdeposit volume, such as e.g. the level of interest rates, the spread between market

rates and deposit rates, or nominal income. Most deposit valuation approaches applysuch an autoregressive model and distinguish themselves in the relevant factors theyidentify and include, where the selection of relevant factors can have a signicantinuence on the dynamics of the calibrated process. A cautionary remark should berepeated: While such AR-processes are appropriate short-term forecasting, longer-term forecasts are more uncertain since these models take a market structure asgiven which might be due to change. 45

In combining the elements of both stated and effective maturity, one can ap-propriately quantify the cash ow streams of deposit in a convenient fashion. Eachperiod, t, the bank has cash outows of consisting of previous periods deposit vol-

ume plus interest claims thereon and cash inows of consisting of new deposits. Thispattern repeats itself up to the forecasting or valuation horizon T is reached andis exemplied in table 1. The length of t can be chosen in accordance with stateddeposit maturity and data availability.

t = 0 t = 1 t = 2 . . . t = T 1 t = T + L0 + L1 + L2 . . . + L (T 1)

L0 L1 . . . L (T 2) L (T 1) L0 r L, 0 L1 r L, 1 . . . L (T 2) r L, (T 2) L T 1 r L, (T 1)

+ L0 r =1 ,0) + L1 r =1 ,1) . . . + L (T 2) r =1 ,(T 2) + L (T 1) r =1 ,(T 1)

Table 1: Cash ow streams of the LBU45 The same remark applies to the models of the deposit rate.

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Similarly to deposit rates, the NPVM denes the level of deposit balances as afunction of the spread between current deposit rates and market rates, the sensitivityof deposit rate to changes in the market rate, the institutions retention rate, and anindustry-average of the retention rate. Additionally, the NPVM assumes that thedifference between the last two factors converges to zero over time. Given its dataon retention rates, the OTS assumes a maturity of 30 years for deposits. There isno growth assumption in the OTS framework, i.e. only the presently held depositbalances are valued, with levels declining as indicated by the retention rate, orrespectively, its counterpart, the decay rate.

Selvaggio species a partial adjustment model of deposits balance levels withthe market rate, the nominal income, and a seasonal dummy as determining factors,

yielding an in-sample t of R2 = 98% for the four-year period between February1991 and February 1995. Just as the OTS, he assumes a maturity of 30 years forhis sample of demand deposits. Insofar as the functional form of deposit balancelevels in this model depends on the nominal income, among other factors, there is animplicit growth assumption made, although it might be criticized for being rathervague.

HP specify the equilibrium deposit balance level as a function of the marketrate, the (optimal) deposit rate, an exogenous demand variable, and a trend growthrate. HP include this trend growth variable in the calibration of their model onempirical deposit data in order to captures the long-run equilibrium growth rate of

the aggregate deposit volume.JvD model deposit balances as a function of the market rate, the change in themarket rate from the preceding period, the previous deposit balance, and a timetrend which is supposed to take up other (macroeconomic) variables not explicitlyaccounted for. Janosi et al. (1999) implement JvDs model in specifying depositbalances additionally as a function of average past balance levels, a retention rateof old balances, a growth rate of new balances, and its sensitivity to changes inthe market rate. Growth in deposit balances is accounted for in the model asdescribed in the functional specication of deposit balances already. A retentionrate measures the decay in old deposit balances and an exponential growth ratemeasures the increase in balances through new deposits. The growth rate is modeledto be sensitive to changes in the market rate.

OBrien (2000) species the level of deposit balances as depending on the currentdeposit rate, the current market rate, recent deposit balance levels, and a measureof nominal income which is interpolated of national data and regional data for thecounty where the bank is situated. Concerning the maturity of deposits, the authorgroups maturities and measures the contribution to the overall deposit premiumalong these maturity ranges. He nds that over 40% of premiums are contributedby maturities beyond 10 years, thus supporting the assumption of very long effectivematurities of core deposits. OBrien includes expected growth in deposits througha trend growth parameter.

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4.1.4 Deposit rate behavior

In the NPVM, deposit rates are a function of three factors, i.e. past deposit rates,the market rate, and a third factor accounting for whether the current deposit rateis above or below its long-term average on such deposits. With the last factor,the NPVM accounts for the asymmetry empirically observed in the adjustment of deposit rates. In this framework, deposit rates are a function of the market rate andin that way also deterministic.

In Selvaggios (1996) methodology, the deposit rate results as the sum of themarket rate and the OAS, which he calls the demand deposit premium. It isa function of past demand deposit premiums, the market rate as simulated in hisMonte Carlo model, and an additional parameter that adjusts the OAS dependenton the level of LIBOR volatility.

HP are able to derive optimal deposit rates as an endogenous function of theinterest rate and the elasticity of demand for deposits. Both variables are simulatedas mean-reverting Ornstein-Uhlenbeck processes and calibrated on historical data.

As for deposit balances already, JvD specify deposit rates as function of the mar-ket rate, the change in the market rate from the preceding period, and the previousdeposit rate. In the implementation of their model by Janosi et al. (1999), the de-posit rate is calibrated on the data as a similar function, while adding additionallya mean-reverting function of its long-run average, thus adding the long-run averageand the speed of adjustment as additional variables.

In OBriens (2000) framework, deposit rates are a mean-reverting function of past deposit rates and a conditional deposit equilibrium rate 46 , an asymmetric adjustment feature as mentioned above, and market rates as obtained by the SDE.

4.1.5 Valuing the LBU

Deposit valuation models are a sensible starting point to model the LBUs valueas part of the banking rms value. However, for our purposes only some of theseapproaches offer useful insights and modeling tools, and some adjustments have tobe made. For example, our initial requirement of accounting for interest rate risk

through term structure models is satised by four of the ve approaches, namelyHutchison and Pennacchi (1996), Jarrow and van Deventer (1998), OBrien (2000),and Selvaggio (1996), who all value deposits as interest rate contingent claims andaccount thereby directly for the effects of interest rate changes on deposit values. Incontrast, the model of the Office of Thrift Supervision (1994, 2001) does not accountfor interest rate risk at all.

Hutchison and Pennacchi (1996) nest their model within a general equilibriumframework of deposit banking, which allows additional economic interpretation butfurther complicates the model setting; likewise, it would take an extended equilib-rium model to account for an additional equilibrium in the market of bank loans.

46

As the author shows, when including asymmetric adjustment of deposit rates, the unconditionalexpected deposit rate and long-run expected rate deviate from the unconditional mean equilibriumrate.

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Since this required extension to the general equilibrium model raises an additionalissue for a complete and integrated framework of bank valuation, HPs approachappears to be the more cumbersome than direct way to reach our ends. Selvaggio(1996), Jarrow and van Deventer (1998) and OBrien (2000) seem to be the mostpromising approaches for our purposes and we follow their general valuation ap-proach. In contrast to them, as we will see later, valuing bank assets and liabilitiesat the same time requires term structure models which allow for richer dynamics of the yield curve.

Based on the previous models, the deposit valuation formula can be derived asa hedging portfolio of the cash ow pattern in table 1 in a risk-neutral valuationprocedure, which yields 47

V L, 0 = E P

0

T 1

t =0

L t (r =1 ,t r L,t )g(t + 1)

, (8)

where E P

0 is the martingale expected value given the risk-neutral probability mea-sure P and g(t) represents the money market account with g(0) = 1. The depositliability value based on the result in equation 8 can be likened to the value of anexotic interest rate swap, lasting for T periods, receiving oating at r t and payingoating at r L , and with an alternating principal of L t . An alternative representationof this result is

V L = L0 + E P

0

T 2

t =0

L ( t +1) L tg(t + 1)

T 1

t =0

L t r L,tg(t + 1)

L (T 1)g(T )

. (9)

for which JvD off an intuitive interpretation. According this restatement of equa-tion 8, the value of the deposit liability is the sum of initial liability volume, plusthe present value of any changes in volume over time, minus the present value of total costs, and minus the present value of deposit volume at maturity or valuationhorizon. Hence, the deposit value is that of a series of T 1 single-period, risk-freebonds paying below risk-free interest rates. Consequently, all of these bonds willhave a price below par and shorting them can derive positive value, since the pro-ceeds can be invested in the risk-free asset. This arbitrage transaction representsthe liability business of banks. In the risk-neutral valuation framework, the LBUsvalue is equivalent to the absolute value of costs of the hedging strategy that offsetsthe LBUs expected cash ow streams. For the derivation of a valuation formula of the LBU, the functional representations of the deposit rate and the deposit volumecan be substituted into either equation 8 or 9.

4.2 Valuation of the asset business unit (ABU)In our model, the only credit product a bank offers is a non-maturing overdraftfacility. We have chosen this representative product for several reasons, one of the

47 See e.g. OBrien et al. (1994).

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being the similarity of its modeling to the modeling of deposits, and another onebeing that the rates a bank charges on its customers overdrafts are well in excess of the assumed credit risk, hence supporting our earlier argument of a banks marketpower. We chose this product for ease of modeling and exposition although, inprinciple, our approach can be applied to all kinds of credit products when allowingfor their respective special characteristics and features.

While there exists no liquid market for bank overdrafts, in the U.S. there is amarket in asset-backed securities based on credit card loans, which in many respectsare very similar to overdraft facilities. For example, Ausubel (1991) shows in a studyon credit card loans that not only credit card interest rates are high and sticky butalso that credit card loans change hands in interbank transactions at considerable

premiums, often surpassing 20%. This indicates that banks earn disproportionatelyhigh interest rates on credit card loans on a risk-adjusted basis, i.e. after accountingfor default risk. Indeed, in their deposit valuation model, Jarrow and van Deventer(1998) mention already the applicability of their approach to credit card loans, whichwe argue that this observation can in principle be extended to all kinds of bank loans.

When neglecting credit default risk, obtaining a valuation formula for the loanbusiness is a straightforward matter; table 2 and equation 10 are mirror images of the ones for the LBU.

t = 0 t = 1 t = 2 . . . t = T 1 t = T A0 A1 A2 . . . A(T 1)

+ A0 + A1 . . . + A(T 2) + A(T 1)+ A0 r A, 0 + A1 r A, 1 . . . + A(T 2) r A, (T 2) + L A 1 r A, (T 1)

A0 r =2 ,0 A1 r =2 ,1 . . . A(T 2) r =2 ,(T 2) A(T 1) r =2 ,(T 1)

Table 2: Cash ow streams of the ABU

For simplicity of modeling, we have chosen the same time intervals as for theliabilities. Nonetheless, this a sensible assumption and only affects the frequency of steps in the autoregressive process, whereas the resulting effective maturity of assetsshould remain unaffected. Thus, the relevant rate would still typically be one of

longer effective maturity, = 2. Assuming a loan business with one type of loanmaturing t and a exogenously given loan volume of the bank, At , (compare alsotable 2) the economic value of the banks assets is analogously given by

V A = A0 + E P

0

T 2

t =0

At A( t +1)g(t + 1)

+T 1

t =0

A t r A,t C ABU,tg(t + 1)

+ A(T 1)

g(T ).

(10)We argued above that we abstract from credit risk in our valuation model. This

assumption need not be detrimental to obtaining a realistic bank value. For example,we may assume that the rate a bank charges on a loan can, at least theoretically,

be divided into two constituent parts, one representing the market price of creditrisk and the other attributable to the banks market power in the loan market. Analternative method would be to assume that a constant fraction of the assets is

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lost, which according to Ausubel (1991) seems to be in accordance with empiricalobservations. More elaborate ways to include credit default risk of bank customerswould be desirable but would also require to introduce more state variable. Fornot complicating matters further, we leave this aside for now. Without introducingmore state variables at this time, we could include a conditional expected loss, EL (t),driven by our state variable, the interest rate. The return on the assets would then just be net of this expected loss. When adding this to 10, we obtain gives

V A = A0 + E P

0

T 2

t =0

A t A( t +1) EL ( t )g(t + 1)

+T 1

t =0

A t r A,t C ABU,tg(t + 1)

+ A(T 1)

g(T ).

(11)

4.3 Valuation of the asset and liability management unit(ALM)

The ALMs residual prot is represented by

ALM =T 1

t =0

(r =2 A r =1 L), (12)

This reects the idea that the ALMs primary goal is not prot generation butmanaging the residual imbalance between asset and liability volume and duration.As such, this unit isolates and internalizes the interest rate risk stemming fromchanges in the slope and the curvature of the term structure, essentially representinga complex yield curve swap with time-varying notional principal, similarly to LBUand the ABU but, in contrast to them, based on market rates and not bank rates. Forthis, we require either perfect elasticity in the interbank market or that the banksfunding requirements are small relative to the capacity of the interbank market, i.e.in any case the ALM can take net positive and negative position without affectingthe market rate.

In order to avoid strenuous positions that might expose the bank to liquidity risk,

we further assume that the bank does not voluntarily grow the business volume onone side of its balance sheet wile leaving the other behind. Rather, asset and liabilityvolume will balance in the long run with short-run deviations due to stochasticallyarriving transactions. This can be achieved by introducing a dened maximumdifference for the difference between assets and liabilities, ,

= A L , (13)

which may be thought of as maximum imbalance allowed by the banks risk commit-tee. Should it be reached, this condition will limit the further growth of the larger of assets or liabilities. Attaching a precise value to this limit will depend on the bank

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at hand for valuation; for now, we conne ourselves to a general formalization,

At +1 = A + A A t + A X A + A,t if t +1

if t +1 > (14a)

L t +1 = L + L L t + L X L + L,t if t +1

if t +1 > (14b)

The ALM is slightly easier to value than either LBU or ABU. Like the otherunits, it can be modeled as a complex swap The ALMs value can then be obtainedby

V ALM = E P

0

T 1

t =0

(r =2 ,t At r =1 ,t L t )

g(t + 1). (15)

The ALM concentrates the benecial effect of combining the activities of lendingand borrowing, which have been argued to be manifold; for example, Diamond andDybvig (1983) and Kashyap et al. (2002) propose that the combination offers anatural hedge against liquidity risk, see also Strahan (2005) for a recent survey of empirical evidence. In our model, this idea is intuitively exemplied by the swap.

5 Complete model in a continuous-time specica-tion

What remains is to tie all ends together and to derive a value for the entire bankconsisting of the three business units. Although the discrete representation in theprevious section is more intuitive for outlining the model, solving the valuationequations is easier in a continuous-time specication.

5.1 Valuation of the LBUFor our exposition of the valuation procedure in continuous time, we can followupto a certain pointJvDs general derivation of a closed-form solution for the valu-

ation of non-maturing deposits. They employ very general autoregressive processesfor the deposit volume growth and the deposit rate. For volume, they assume aprocess of the form

d log L t = L0 + L1 t +

L2 r t dt +

L3 dr t , (16)

and for the deposit rate, they set

dr L,t = L0 + L1 r t dt +

L2 dr t , (17)

where r t represents what we earlier have written as r =1 ,t , i.e. the index is dropped

for ease of notation. Hence, both variables are depending on the market rate, r tand a trend growth rate which is represented by t . These processes are very generaland do not include all the identied variables from our survey of deposit valuation

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models. At this point, however, the focus is on the derivation of a closed-formsolution and introducing additional factors would complicate this endeavaour. Thesolution to these SDEs are

L t = L0exp L0 t + L1

t2

2 + L2

t

0r s ds + L3 (r t r 0) (18)

andr L,t = r 0 + L0 t +

L1

t

0r s ds + L2 (r t r 0) (19)

The extension of the discrete time valuation result for deposits in equation 8 is

V L = E P

0 T

0

L t (r t r L )g(t)

dt , (20)

and, when substituting the solutions in 19 and 18 into the deposit valuation formula20, we obtain

V L = E P

0 T

0

L0exp L0 t + L1t 2

2 + L2

t0 r s ds +

L3 (r t r 0)

g(t)

(r t r 0 L0 t L1

t

0 r s ds L2 (r t r 0)

g(t) dt (21)

As already mentioned in the previous section, one can interpret this solution asa complex interest rate swap with stochastically changing principal, dependent onboth the the average,

t0 r s ds , and level, r t , of past market rates.

Next, we need to specify the dynamics of the interest rate. A common represen-tation is a stochastic mean-reverting differential equation of the short rate r , whichcan be stated as

dr t = a (r t r t ) dt + d W (t), (22)

and, applying the the term structure model of Heath et al. (1992), the solution tothis SDE is given by

r t = f (0, t ) + 2 (e at 1)2

2a 2 +

t

0e a ( t s )d W (s). (23)

Applying this to the deposit value in equation 21 and some tedious manipulations

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yield the following solution:48

V L = L0exp ( L3 r 0)(1 L2 )

T

0exp [ L0 t +

L1 t2

2 ]M (t, L2 1,

L3 )[(t) +

22(t)

L3 + 12(t)(

L2 1)]dt +

+ L0exp [ L3 r 0] T

0exp [ L0 t +

L1 t2

2 ]( r L, 0 L0 t +

L2 r0)M (t,

L2 1,

L3 )dt

L0exp [ L3 r 0] L1

T

0exp [ L0 t +

L1 t2

2 ]M (t, L2 1,

L3 )[2(t) +

21 (t)(

L2 1) + 12(t)

L3 ]dt, (24)

where1

t

0f (0, s )ds +

t0 [

2(1 exp [ a (t s)])2]2

, (25)

21 t

0

2(1 exp [ a(t s)])2

2ds, (26)

2 f (0, t ) + 12 , (27)

22 t

02exp [ 2a(t s)]ds, (28)

12 2(1 exp [ at ])2

2a2 , (29)

M (t, 1, 2) exp 1(t) 1 + 2(t) 2 + 21(t) 21 + 212(t) 1 2 + 22(t) 22

2.(30)

It can be shown that this result allows a closed-form solution which has a ratherbulky format, though. Jarrow and van Deventer (1998) provide a solution to theseintegrals in their appendix. Now, we have a closed-form solution for the depositbusiness. Whether one resorts to this solution or adds complexity with additionalfactors and increases empirical t on the cost of having to rely on numerical methodsis another crucial question which we will not discuss at this point, though. Althoughappealing, this solution is of little help for our problem of valuing the entire bank

and we included it for exemplary reasons only. Implementing the model with asingle-factor term structure model, as JvD do with a one-factor version of HJM,has the already mentioned drawbacks and does not offer yield curve dynamics richenough to allow for the different relevant maturities within the bank. However,resorting to a two-factor model, which is necessary for the valuation of the entirebank, comes at the price of loosing the closed-form solution. Therefore, within thebank, we have to obtain the LBUs value in numerical procedures.

5.2 Valuation of the ABUThe valuation of the ABU follows straightforwardly, mirroring the solution for theLBUs value. Similarly, we can dene general autoregressive processes for the evolu-

48 See Jarrow and van Deventer (1998), pp. 263267 and appendix A.3 for a detailed derivation.

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tion of the loan rate, r A and the asset volume, A , which suffices for the derivation of a expository solution but would await further renement for an empirical implemen-tation covering all relevant factors. Analogous to the previous processes, we denethe asset volume to follow

d log At = A0 + A1 t +

A2 r t dt +

A3 dr t , (31)

and the loan rate to behave according to

dr A,t = A0 + A1 r t dt +

A2 dr t . (32)

To properly value the ABU at the same time as the LBU, we have to rely onthe dynamics of a two-factor model which restricts us from solving for an algebraicsolution; rather, we have to rely in Monte Carlo simulations.

5.3 Valuation of the ALMThe value of the ALM in continuous time is given by

V ALM = E P

0 T

0

(r =2 ,t At r =1 ,t L t )g(t)

dt . (33)

We were not able to derive a closed-form solution to this complex swap and have torely on numerical procedures for its valuation.

5.4 Valuation of the entire bankFor the valuation of such an interest-rate sensitive call option on the portfolio of the three business units, we have to rely on numerical methods. Given the factthat the drift and the volatilities of each new state variable do not contain eachother, we have to resort to a trinomial lattice to match the required moments, andensure recombination. Hull and White (1994) show how to construct such a tree ina trinomial framework. We are currently working on a trinomial tree representation.

6 ConclusionWe argue that the problem of valuing a bank as a rm which is particularly exposedto interest rate risk has not been adequately solved in the literature so far. Wepropose a bank equity valuation model based on the contingent claims theory andderive the banking rm value as constituting of the value of three stylized businessunits, the asset business, liability business, and the asset-liability management. Thevalue of each of these units can be derived in a risk-neutral valuation framework asequivalent to the costs of a hedging strategy that offsets the risk exposure but stillallows for arbitrage prots. There exist several models for the valuation of bankingproducts and we have exemplied our model using the deposit valuation model of

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Jarrow and van Deventer (1998). Although our approach is workable with othermodels as well, the appeal of this particular deposit model is the availability of aclosed-form solution for the economic value of bank assets and liabilities.

Regarding our approach, some caveats and criticism are in order. The appeal of the just proposed valuation model is (a) that it includes interest rate risk but avoidsinterest rate forecasts, and (b) that it provides an elegant and comprehensive risk-neutral framework which circumvents many of the problems of standard valuationapproaches. A necessary assumption for this to work is the tradability of the hedgingportfolio. We argue that such assets are continuously tradable, think e.g. of themarkets for certicates of deposits or collateralized debt obligations. If one ndsthis assumption problematic, a possible alternative comes in the form of almost

good deal boundaries. Cochrane and Saa-Requejo (2001) propose these as a methodto value contingent claims for which a perfectly correlated asset is unavailable.

The paper at hand merely sketches our approach and as such can be likened tothe pizza Margarita of bank valuation: Its got the basics ingredients but a littlemore toppings would be nice. We can think of several extensions to this basic model.For example, this framework should also be suitable to derive a default risk modelof the bank, which we are currently attacking in a follow-on to this paper. Thinkingin the same direction, a less crude balance sheet of the bank might enhance theinsights of the model. One could allow for several different banking products onboth sides of the balance sheet, also including reserve requirements. Besides, a

more rened method for the measurement of credit risk of bank customers, i.e. theexpected loss in the asset business, would certainly improve the model. Finally,a critical question is the inclusion of deposit insurance. It can be shown that thevalue of deposit insurance is akin to a put option and e.g. Ronn and Verma (1986)propose a valuation model for this. Although the benets of this insurance accrueto the deposit holders and not to the shareholders of the bank, the latter might stillbe able to extract value from it, e.g. by assuming a more aggressive leverage on itsbalance sheet. Possible consequence of deposit insurance pose an interesting sideissue.

An empirical implementation of this model would be very interesting as well.For this, one would have to identify the relevant factors for the stochastic processesof the bank rate and business volume. So far, we are aware of only one study whichapplied a deposit valuation model t

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A New Approach to the Valuation of Banks