A new approach to the valuation of banks∗

Michael Adams†and Markus Rudolf‡

First Version: November 2006

This version: August 2010

Abstract

We argue that the business model of the bank exhibits such pe-culiarities that it deserves a special treatment in the approach to itsvaluation as well. In particular, the exposure to interest rate risk is amajor characteristic of its business, not only in the process of maturitytransformation but also as a major determinant of price margin andbusiness volume. There exists no common framework to value a bankwhich adequately accounts for these features. We propose a valuationmodel for banks based on Merton’s (1974) structural model of thefirm, which we adapt to the banking firm by the help of term struc-ture models of the interest rates. In this setting, we interpret banksas a particular portfolio of long and short positions in interest-ratesensitive assets. In doing so, we are able to show another peculiarityof bank valuation, i.e. that the exercise price of a call option on thefirm value, representing the bank’s equity, is not the face value of bankliabilities but their economic value.

JEL classification: C22, G12, G13, G21.

Key Words: matched maturity marginal value of funds, MMMVF,bank valuation, term structure, model

∗We are grateful to Stewart C. Myers, Matthias Muck, and participants of the 2005Burgenland doctoral seminar for helpful comments and the WHU USA foundation for itsfinancial support. Any remaining errors are our own.

†Große Gallusstraße 10-14, 60272 Frankfurt, Germany E-mail: michael.adams@db.com.‡WHU - Otto Beisheim Graduate School of Management, Endowed Chair of Fi-

nance, Burgplatz 2, 56179 Vallendar, Germany, Tel.: +49-(0)261-6509-421, e-mail:markus.rudolf@whu.edu.

1

1 Introduction

In corporate finance, it is not unusual to specify valuation models for partic-

ular types of firms. For example, to mention just one of them, Brennan and

Schwartz (1985) propose a real-options based valuation approach to natu-

ral resource companies, explicitly modeling the options to temporarily close,

reopen and shut-down the mine. For similar reasons, one can argue for a

special valuation approach for banks. Indeed, Copeland, Weston and Shastri

(2005) (p. 872) address bank valuation as one of the unresolved issues in

financial research.

The characteristics of the banking business motivating a distinct valuation

approach can be subsumed in four categories. First, banking is a heavily

regulated industry.1 Second, banks operate on both sides of their balance

sheets, actively seeking profits not only in lending but also in raising capital2.

This is the aspect this paper concentrates on. Third, banks are exposed to

credit default risk, but they also actively seek credit risk as part of their

business model. Last but not least, the profit and the value of the bank is

much more dependent on interest rate risk than other industries.

The impact of the interest rate risk on bank value is mainly driven by the

bank’s assets and liabilities mismatch. Though, there exist further effects.

Bank rates adjust asymmetrically to changes in market rates. This results in

a time-varying spread which is typically larger in a high interest-rate envi-

ronment; see e.g. by Hannan and Berger (1991), Ausubel (1991), or Neumark

and Sharpe (1992), and data in the monthly report of the Bundesbank (2006).

Moreover, the demand for deposits and loans does not only depend on the

market rate but also on the rate a bank charges or pays. Also, loan demand

1See e.g. Carey and Stulz (2006).2see MacLeod (1875), p. I:275.

2

and deposit demand typically have a negative correlation. Consequently,

interest rate effects on bank value are significant and nontrivial.

In contrast to existing discounted cash flow based frameworks like Samuel-

son (1945), Flannery and James (1984a), and (1984a, 1984b), we propose a

valuation model for banks based on Merton’s (1974) structural model of the

firm, which we adapt to the banking firm by the help of interest rate term

structure models. In this setting, we interpret banks as a particular portfolio

of long and short positions in interest-rate sensitive assets and thus aim at

integrating the interest rate-sensitivity into the valuation approach.

The paper is structured as follows. The next section outlines the busi-

ness model of a bank. Section 3 derives a structural firm valuation model

and applies it to banks. Section 4 provides a valuation approach for the

deposit business, section 5 for the loan business, section for the ALM busi-

ness. Finally, the valuation model of the entire bank is presented in section

7. Section 8 concludes the paper.

2 The business model of a bank

To describe the bank in terms of our portfolio view, we make several sim-

plifying assumptions. First, we abstract from taxes, reserve requirements,

minimum capital requirements and other regulatory factors that could im-

pact the bank’s profit and portfolio value. Second, we assume that there is

no depreciation, amortization, or other non-cash item, which provides the

useful equality of free cash flow to equity (FCFE) and bank profits. Finally,

concerning the definition of our bank’s profits, we assume that there are only

variable cost for the beginning3. When accounting for all of this, our bank’s

3Neglecting issues of fixed cost is not as absurd as it may seem at first sight and will

greatly simplify our analysis. We show how to introduce fixed cost later without changing

3

profit, πbank, can be stated in its simplest form as difference between the

return on assets and the costs of liabilities,

πbank = rcL × L− rcD ×D, (1)

where L denotes the loan volume and D the deposit volume. rcL and

rcD, indicate the rates the bank actually earns after variable cost of service

and net of servicing fees charged to customers, while individuals receive the

deposit rate rD and pay rL on their loans.

In order to separately identify the sources of cash flows on both sides of

the balance sheet, we apply an instrument from internal bank controlling, the

transfer pricing based matched maturity marginal value of funds (MMMVF),

in the valuation process4. This requires to break up the profits of entire bank

into the parts of its constituent business units, i.e. a loan business (LB), a

deposit business (DB), and a treasury department which is in charge of the

asset-liability management of the bank (ALM). The LB is in charge of issuing

loans and measuring and managing credit risk. The DB issues transactions

and savings deposits, providing its customers with an access to the payment

system and the bank with an internal funding of the LB. The ALM’s main

task in our simple model of the bank is to manage the interest rate risk or,

more precisely, term structure risk stemming from the differing maturities

of the assets and liabilities or the positive duration gap between loans and

deposits. Applying the MMMVF transfer pricing framework, the profit of

our major results here.4The MMMVF framework allows to achieve a separation of the bank’s profit into

two parts: One depending on general capital market conditions as captured in the term

structure of interest rates and the other one depending on the bank’s market power in

loan and deposit markets.

4

the bank can be restated as

πbank = (rcL − rτ=2)× L+ (rτ=1 − rcD)×D + (rτ=2 × L− rτ=1 ×D)

= πLB + πDB + πALM ,(2)

where rτ=2 signifies the default-risk-free capital market rate for a longer

maturity and rτ=1 represents a comparable rate for a shorter maturity.5 In

this representation, the bank’s profit is the sum of profits of LB, DB and

ALM, πLB, πDB, and πALM , respectively. An additional task of the ALM

would be the equalization of any imbalances arising temporarily between the

volume of assets and liabilities by use of external financing, taking long or

short positions in the interbank market. Again for reasons of simplification,

we start developing our model with the additional assumption of L = D,

which simplifies equation (2) to6

πbank = (rcL − rτ=2)×D + (rτ=1 − rcD)×D + (rτ=2 − rτ=1)×D. (3)

From this restatement of profits, the economic value of the bank’s equity,

VE , can be obtained as the sum of discounted future cash flows (DCF),

originating from its three business units, which is

VE = NPV (πbank) = NPVLB +NPVALM +NPVDB, (4)

5These can be thought of as average effective maturities of loans and deposits, where

deposits typically have a shorter maturity than loans.6We assume this for expository reasons and not to restrict the generality of our model.

For L = D, one might e.g. implicitly assume that the equity of the bank equals its non-loan

assets such as cash and property. Additional asset balances would have to be included in

the valuation of the entire bank, of course.

5

where

NPVLB =

T∑

t=1

πLB,t

(1 + rLB)t=

T∑

t=1

(rcL,t − rτ=2,t)×Dt

(1 + rLB)t, (5a)

NPVALM =T∑

t=1

πALM,t

(1 + rALM)t=

T∑

t=1

(rτ=2,t − rτ=1,t)×Dt

(1 + rALM)t, (5b)

NPVDB =

T∑

t=1

πDB,t

(1 + rDB)t=

T∑

t=1

(rτ=1,t − rcD,t)×Dt

(1 + rDB)t, (5c)

and where rLB, rALM , and rDB signify the required rates of return of the

three respective business units. T is the valuation time horizon; typically,

T = ∞.

In equations (5), the value attributable to the deposit business is a func-

tion of the outstanding balances and the relative cost savings originating from

the issuance of deposits, that is, the deposit spread. In the same fashion, the

value of the loan business depends on loan volume, which we set equal to

the deposit volume, and the average realized loan spread. The contribution

of the ALM also is a function of average business volume and additionally

of the spread between two market rates of differing maturities, depending on

the slope of yield curve. The net present value for each business unit can

then be calculated by discounting the annual cash flow profits at a discount

rate that provides an appropriate return on equity.

Consequently, the definition of the bank’s profit in the MMMVF frame-

work and the resulting net present value of the bank in equations (4) and (5)

allow us to identify five types of parameters that affect it and which we will

have to account for in a valuation model:

1. The deposit volume, D, representing the business volume of the bank,

2. the loan rate, rL, respectively the loan rate net of cost, rcL,

3. the deposit rate, rD, respectively the deposit rate net of cost, rcD

6

4. the required return on equity of all three business units, rLB, rDB, and

rALM

5. the market rate rτ=1 for a short maturity and a long maturity (rτ=2).

Among these, the last applies to all banks in general and represents their

duration-dependent interest rate risk exposure. Opposed to this, the former

four are bank-specific, thereby determining the idiosyncratic component of

each bank’s firm value.

To assume a constant interest rate like in equation (5), would disregard

the factor that is at the heart of a bank’s business. Not only the discounting

factor but also all three sources of a bank’s cash flow (LB, ALM and DB)

are highly interest rate sensitive. Therefore, we propose a bank valuation

model which accounts for interest rate sensitivity in that it views the bank

as a portfolio of interest-rate contingent claims. In the following, we lay out

the ground for such a model, which is based on Merton’s (1974) structural

model of the firm.

3 Structural, risk-neutral firm valuation ap-

proach to the bank

The book value of deposits equals the face value of this liability, i.e. 1 ¤

deposited is recorded at 1 ¤ in the bank’s books. As long as the deposit

rate deviates from its MMMVF or opportunity rate, though, the economic

value of this deposit will also deviate from its book value. The ability of the

bank to raise liabilities at rates below comparable market rates creates an

intangible asset (market power). The value NPVDB of this intangible asset

is positive and equals the net present value of profits derived from the ability

7

BANK

VL

VD

assets liabilities

NPVLB

NPVDB

E

BANK

L

D

assets liabilities

E

NPVLB

NPVDB

(a) book value of the bank (b) economic value of the bank

VE

Figure 1: Book value and economic value of bank assets and liabilities

Remarks: E stands for book equity and VE for its economic value.

to issue at below market rates as defined in equation (5). This provides the

link between the book value, D, and the economic value, VD, of deposits:

VD = D −NPVDB. (6)

The relation between economic and the book value is depicted in Figure

1. The same argumentation applies vice versa for bank assets as well. In

the absence of credit risk (or after accounting for the expected loss), the

economic value of the loan portfolio, VL, results as the sum of its book value,

L, and the net present value of the profits NPVLB:

VL = L+NPVLB. (7)

Finally, the economic value of the ALM, VALM , equals its net present

value

VALM = NPVALM , (8)

because it does not have a net book value. By assumption, the book or

face value of the ALM’s long position is offset exactly by its short position.

8

Changes in the slope of the term structure affect only the ALM’s economic

value, which is reflected in NPVALM .7

According to Black and Scholes (1973) and given a planning horizon T ,

the value of debt, VB,T , and equity, VE,T , at the end of the planning horizon

can be described by option-like payout profile,

VB,T = B∗ −max[D − VA,T ; 0], (9a)

VE,T = max[VA,T −D; 0], (9b)

where VA,T is the economic value of the bank’s assets at the planning horizon

T and B∗ represents a risk-free bond with a face value that equals the face

value of the bank’s liabilities, which in our case consist exclusively of deposits

D. The economic asset value of the bank, VA, can be defined as

VA = A+NPV (πbank), (10)

where NPV (πbank) is the present value of the bank’s future profits as

defined in equation (4) and A is the asset book value. For the non-banking

firm, the economic asset value or market value of assets equals the firm value.

In the case of the bank, however, the firm value exceeds the market value of

the bank’s assets because the bank also generates value on the liability side.

Indicating the bank’s firm value as its asset value, VA, allows to simplify a

comparison between the structural model of the firm and the structural model

of the banking firm. Moreover, with this notation we want to highlight the

consequences of this important difference between the bank and the non-

bank, which will become apparent shortly. From equations (10) and (4)

7This value is positively correlated with the slope of the yield curve implying that banks

are usually more profitable in an upward sloping yield curve environment.

9

follows

VA = A +NPVLB +NPVDB +NPVALM , (11)

which we can state alternatively in terms of the units’ economic values

as previously defined in equations (6), (7), (8),

VA = VL +NPVDB + VALM = L+NPVLB +NPVDB + VALM , (12)

where, by assumption, A = L = D. This result can be seen as an

expression for the bank’s economic asset value in the MMMVF framework.

In order to obtain the boundary conditions for the banking firm, we combine

this term in (12) with the payout profiles for the non-banking firm in (9).

This lets us obtain the equivalent expressions for the banking firm, when

equation (6) is substituted in,

V bankB,T = B∗ −max[VD,T − VL,T − VALM,T ; 0], (13a)

V bankE,T = max[VL,T + VALM,T − VD,t; 0]. (13b)

In this equation, we can see an interesting particularity of the banking

firm: For non-banking firms, the exercise price of the options is the face or

book value of debt. However, when applying the structural model to the

banking firm, the exercise price of the options in the structural model is

not the book value of liabilities, D, but their economic value, VD. This is a

consequence of the duality of the bank’s business model, which also seeks to

extract a profit from issuing its own liabilities. An alternative representation

highlights the difference, which we obtain by substituting back equation (6)

into (13),

V bankB,T = B∗ −max[DT −NPVDB − VL,T − VALM,T ; 0], (14a)

V bankE,T = max[VL,T + VALM,T +NPVDB,T −DT ; 0]. (14b)

10

In other words, should the bank’s asset value, VA, fall below the book

value of its debt, D, shareholders will not choose to default but will rather

be willing to uphold the bank’s business operations as long as the net present

value extracted from the deposit base, NPVDB, is large enough to cover a

potential decrease of the rest of the banks assets, VL and VALM , below the

book value of debt (deposits), D. For creditors of the bank, the decrease

of the put option’s exercise price from D to VD also increases the value of

their debt for they hold a short position in that option. The additional value

gained in the liability business increases the distance to default of the bank,

thereby decreasing its probability of default and increasing the value of its

debt, V bankB,T . The next step is to model the underlying economic value of a

bank’s assets and liabilities.

4 Valuation approach to the deposit business

The analysis of deposits as interest rate sensitive claims includes Hutchison

and Pennacchi (1996), Selvaggio (1996), Jarrow and van Deventer (1998),

O’Brien (2000), Kalkbrener and Willing (2004), and Dewachter, Lyrio and

Maes (2006). Although we will draw on elements and insights from all of

those, the model of Jarrow and van Deventer (1998) seems to be the most

useful for our problems.8 Hutchison and Pennacchi (1996)’s equilibrium ap-

proach would require the design of an analogous equilibrium in the loan

market before we could extend the model to the entire bank; besides, equi-

librium models often exhibit a poorer fit in empirical implementations when

compared with arbitrage-free models because the former impose more struc-

8A valuation based on this model yields at the same time an estimate for the deposit

balance’s effective duration, which is needed for selecting the appropriate MMMVF trans-

fer rate.

11

ture on the data.9 The model of O’Brien (2000) is similar to that of Jarrow

and van Deventer (1998) but additionally incorporates the asymmetric ad-

justment feature of the deposit rate, this comes at the price of sacrificing

a closed-form solution to the valuation problem, though. In the trade-off

between allowing for an additional trait of empirical data and obtaining a

closed-form solution, we chose the latter.

4.1 The steps to take in the model

Following these approaches, we define a pattern of deposit cash flows and

then apply a valuation model to them. As can be seen from table 1, the

NPV of the bank’s deposit business actually has two sources; one is the

spread earned in each period, Dt · (rcD,t − rt), and the other is the growth of

the deposit base from period to period D(t+1) −Dt. We will account for this

growth effect later in the derivation of our valuation formula.

t = 0 t = 1 . . . t = T − 1 t = T

New deposits +D0 +D1 . . . +D(T−1)

Deposits withdrawn −D0 . . . −D(T−2) −D(T−1)

Interest paid −D0rcD,0 . . . −D(T−2)r

cD,(T−2) −DT−1r

cD,(T−1)

Interest earned +D0r0 . . . +D(T−2)rT−2 +D(T−1)r(T−1)

Table 1: Cash flow streams of the deposit business

Hence based on this, the valuation process of the deposit business is:

9Duffee (2002) for example finds that martingales possess a higher predictive power

than affine models and Dai and Singleton (2002) have similar concerns related to the

matching of the observed term structure. “In other words, the difference between the

yields predicted by an affine model at the estimated parameter values and the actual yield

data can be substantial.”(Piazzesi (2003), p. 48).

12

1. Choose a term structure model to account for the dynamics of the short

rate, rt. Typical choices include single-factor strictly affine models,

such as Cox, Ingersoll and Ross (1985), multi-factor essentially affine

models, as proposed by Dai and Singleton (2000), and no arbitrage

models, e.g. Heath, Jarrow and Morton (1992) or Miltersen, Sandmann

and Sondermann (1997).

2. Specify a process for the deposit rate, rD,t, and the deposit base, Dt.

3. In the deposit valuation equation, which we will derive within this

section in equations (16) and (17), substitute Dt, rD,t, and rt with the

solutions to the (stochastic) processes of the deposit rate, the deposit

base, and the short rate. If possible, solve for an analytical solution.

4. Calibrate the process specifications under the empirical probability

measure on a suitable data sample.

5. Should a closed-form solution be available, plugging in the estimates of

the parameters yields the deposit value; otherwise, use the estimates

for a Monte Carlo simulation to obtain a distribution of the deposit

value.

4.2 A multi-factor model for deposit valuation

Jarrow and van Deventer (1998) introduce a market segmentation hypothesis

to set up their deposit valuation model: There are two types of agents in

the market, banks and individuals. Both have specific trading limitations.

On one side, banks can issue a limited and exogenously given volume of

deposits, i.e. they are free to issue less but cannot issue deposits beyond this

limit. On the other side, individuals can hold deposits but cannot issue them.

13

Finally, both banks and individuals can buy and sell risk-free bonds without

limitation and do behave rationally in line with the usual assumptions.10

Subsequently, these assumptions allow Jarrow and van Deventer (1998) to

value bank deposits using an arbitrage argument while at the same time

preserving a positive spread for the bank. In the case that deposits pay

less than the market rate, i.e. rD < r, individuals cannot arbitrage this

discrepancy away since they cannot issue bank deposits. As long as rcD < r,

banks can issue deposits and invest the proceeds in the risk-free security,

thereby earning a positive spread, r− rcD > 0. As stated in the assumptions,

however, this does not represent an unlimited arbitrage opportunity since

banks are restricted in this type of transaction by the exogenously given

maximum amount of deposit volume in the market.

As a result, the absence of exploitable arbitrage opportunities in this

segmented deposit market implies that

rD,t ≤ rt for all t, (15a)

rcD,t ⋚ rt for all t, and (15b)

rcD,t < rt for some t. (15c)

Based on the market segmentation, the deposit valuation formula can

be derived as a hedging portfolio which offsets the cash flow pattern of the

deposit business as shown in table 1 in a risk-neutral valuation procedure,

i.e. as expectation, EP∗

0 , under the risk-neutral probability measure, P∗. This

yields

VD = D0 + EP∗

0

[

T−2∑

t=0

(

D(t+1) −Dt

g(t+ 1)

)

−

T−1∑

t=0

(

Dt × rcD,t

g(t+ 1)

)

−D(T−1)

g(T )

]

, (16)

10For a definition of rational behavior in financial theory, see e.g. Ingersoll (1987). The

issue of credit risk when shorting the risk-free bond is discussed later.

14

where g(t) represents the money market account with g(0) = 1. Besides

the market rate, the cash flows of the deposit business depend on the deposit

base and the deposit rate, for both of which we suggested the specification

of a vector autoregressive (VAR) process. Hence, any exogenous explanatory

variable we include in these processes may add an additional state variable

to our “deposit derivative”.11

Jarrow and van Deventer (1998) offer an intuitive interpretation for the

value of the deposit business VD in equation (16): The value of the deposit

liability is the sum of the book value of the initial deposit base, D0, plus

the present value of any changes in the deposit base over time, minus the

present value of total costs, and minus the present value of deposit volume

at maturity or the valuation horizon. In other words, the DB’s economic

value is that of a series of T − 1 single-period, risk-free bonds paying below

risk-free interest rates. Consequently, all of these bonds will have a price

below par and shorting them can derive positive value, since the proceeds

can be invested in the risk-free asset.

In analogy to the profit of the deposit business in equation (5c), an equiv-

alent but simpler valuation formula is given by

VD = EP∗

0

[

T−1∑

t=0

Dt

(

rτ=1,t − rcD,t

)

g(t+ 1)

]

, (17)

which in continuous time can be represented by (rt: Shortrate)

VD = EP∗

0

[∫ T

0

Dt(rt − rcD,t)

g(t)· dt

]

. (18)

The economic value of the deposit liability in equation (17) or equa-

tion (18) can be linked to the value of an interest rate swap, lasting for T

11We rely on single-factor (affine) term structure models in their specification of deposit

dynamics. A three-factor model based on the term structure model of Heath, Jarrow and

Morton (1992) has been suggested by Kalkbrener and Willing (2004).

15

periods, receiving floating at rt and paying fixed at rcD, and with an alter-

nating principal of Dt.

When specifying a multi-factor model of the term structure of interest

rates, we have the basic choice between exogenous and endogenous models.

Which one to choose is driven by the research focus, that is, economic in-

tuition vs. close fit on the observed term structure. For example, we know

of only two existing papers proposing multi-factor frameworks for the valu-

ation of deposits, where one Dewachter, Lyrio and Maes (2006) relies on an

endogenous multi-factor affine term structure model (ATSM), and the other

(Kalkbrener and Willing 2004) implements its deposit valuation model based

on an exogenous HJM type of model and motivates this with “better cali-

bration results with the non-parametric models”. For our problem at hand,

though, we feel that endogenous models represent a more suitable choice.

Our primary concern is not to exactly price an interest rate derivative based

on the presently given shape of the term structure, but rather the valuation

of an deposit-type of interest rate derivative which depends on the long-run

dynamics of the term structure. With this rationale, we follow Dewachter,

Lyrio and Maes (2006).

Choosing a term structure model Among endogenous models, a com-

mon choice of multi-factor models is the class of essentially affine models,

which can include up to N state factors in their general specification. Set-

ting N = 3 is the usual choice in the literature to closely track the empirically

observed yield curve dynamics.12 Now, we add a fourth risk factor to the spec-

ification of the ATSM which is supposed to capture the time-varying deposit

12Recall the study of Litterman and Scheinkman (1991) and also see Knez, Litterman

and Scheinkmann (1994).

16

spread, rt − rD,t.13 This implies that the state factor process XD

t∈ R

N+1

under the empirical probability measure can be stated as

dXD

t= K

D(XD −XD

t)dt+ΣD

√

SD(XD

t) dWD

t, (19)

where XD ∈ RN+1, ΣD,KD ∈ R

(N+1 )×(N+1 ), dWD

tis a (N+1)-dimensional

independent standard Brownian motion under P, and SD(XD

t) ∈ R

(N+1 )×(N+1 )

is a diagonal matrix of the form

SD(XD

t) =

SD1 0

0. . . 0

0 0 SDN+1

, (20)

where diagonal elements are again given by

SDi = νD

i + υD

i

⊤

XD

tfor i = 1, . . . , N + 1, (21)

with νDi ∈ R and υD

i∈ R

N+1.14 We modify the matrix of mean reversion

speed, KD , to have off-diagonal elements different from zero in row N + 1,

KD =

κD

(1 ,1 ) 0 . . . 0

0. . . 0 0

0 0 κD

(N ,N ) 0

κ(N+1,1) . . . κ(N+1,N) κD

(N+1 ,N+1 )

. (22)

If equation (22) had only diagonal elements, κ(i,i), this would mean that the

mean-reversion speeds of state variables were independent. In the present

13The idea to include the deposit spread as an additional state variable in the term

structure model was first proposed by Dewachter, Lyrio and Maes (2006).14For the following derivations, we remark that in contrast to our heteroskedastic spec-

ification, Dewachter, Lyrio and Maes (2006) assume constant volatility; thus, some of our

results will deviate from theirs, such as e.g. the Riccati equations of the bond pricing

formula (36). For a specification similar to ours, see e.g. Duffee (2002) or Duarte (2003).

17

case, however, where κ(N+1,i) 6= 0, this intuitively means that we relax the

assumption of independence between the state variable N + 1 of the deposit

spread and the rest of the factors driving the term structure of the interest

rate, i.e. the deposit rate may be a function of the market rate.

Adding a state variable in such a fashion is an elegant way to install a

functional dependence of the deposit rate on the market rate. An unpleasant

and serious side effect of this procedure is the loss of a unique martingale

probability measure.15 The addition of a state variable to an otherwise com-

plete market leads to market incompleteness. According to the Fundamental

Theorem of Finance, if a market is incomplete, the pricing kernel will be inde-

terminate, and according to the Martingale Representation Theorem, an in-

determinate pricing kernel is equivalent to an indeterminate martingale prob-

ability measure, or infinitely many martingale probability measures.16 This

also implies that arbitrage cannot be precluded. To overcome this problem,

Kalkbrener and Willing (2004) apply a “variance-minimizing” martingale

probability measure, as presented e.g. in Schweizer (1995) or Delbaen and

Schachermayer (1996).17 According to the latter, the variance-minimizing

martingale measure is defined as the measure P∗ which comes closest to the

given martingale measure P∗, where the fit is determined by the time-varying

market price of risk, λ(Xt) ∈ RN+1. For this, recall from the definition of

the Radon-Nikodym derivative, ξt, that the density η of P∗ with respect to

P is given by

η = exp

[∫ t

0

λD(XD

s)dWD

s−

1

2

∫ t

0

∣

∣λD(XD

s)∣

∣

2ds

]

, (23)

15This loss is not a weakness of this approach only. Kalkbrener and Willing (2004) face

the same problem.16Ross (2004) (chapter 1) provides an intuitive derivation of these relationships.17An alternative valuation approach would be good deal bounds. See Cochrane and

Saa-Requejo (2001) for this.

18

where λD(Xt) is the time-varying market price of risk and has the form

λD(XD

t) =

[

λD1 (x

D1,t) . . . λD

N(xDN,t) 0

]⊤

, (24)

which according to Duffee (2002) can be defined as

λD(XD

t) = S

D

tθ + S

D

t

(−1)ΘXt, (25)

where

θ⊤ =[

θ1 . . . θN 0]

, (26)

and Θ ∈ R(N+1)×(N+1) is a matrix in which the elements specify the depen-

dence of the market price of risk, λD, on the vector of state variables with

the exception of row and column N + 1, which contain zeros. In the den-

sity of the Girsanov transformation in (23), P can be seen as the empirical

equivalent of an indeterminate martingale measure. More meaningful is the

variance-minimizing martingale measure, P∗, that can be obtained from its

density with respect to the observable empirical measure, P,

η = exp

[∫ t

0

λD(XD

s)dWD

s−

1

2

∫ t

0

∣

∣λD(XD

s)∣

∣

2ds

]

. (27)

Finally, obtaining dWD

tis straightforwardly possible by

dWD

t= dWD∗

t− λD(Xt)dt. (28)

For the calibration of the model, this basic relationship allows us to mod-

ify the drift and the stochastic term in (19) such that we obtain an empir-

ical state factor process from its risk-neutral equivalent under the variance-

minimizing martingale measure, P∗,

dXD∗

t= K

D∗( ˜XD∗ −XD

t)dt+ΣD

√

SD(XD

t) dWD∗

t, with (29)

KD∗ =K

D +Θ, (30a)

˜XD∗ =(KD +Θ)−1(KDXD − Sθ). (30b)

19

Specifying processes of the deposit rate and deposit base After

we have spent much effort on deriving the term structure model with the

additional state factor “deposit spread”, we can now conveniently formulate

the dynamics of the deposit rate as linear function of the state variables

vector Xt,

rD,t = ϕD0 +ϕD

1

⊤

XD

t+ uD

t , (31)

where ϕD0 is a constant, ϕD

1

⊤∈ R

N+1 is a vector of coefficients with

ϕD

1

⊤

=[

1 . . . 1 ϕ1,(N+1)

]

, (32)

and uDt is the error term.18

Concerning the deposit base, Dewachter, Lyrio and Maes (2006) do not

account for volume growth and concentrate on the decay rate only. They

justify this by limitations of data availability. Their simple assumption is

that deposits increase at the deposit rate, i.e. depositors do not collect the

interest earned, and decrease at the estimated decay rate, rw,

dD(t) = (rD(t)− rw)D(t)dt. (33)

This specification is too simplistic for our purposes. In contrast to this,

while Kalkbrener and Willing (2004) did not spend much time on specifying

an innovative deposit rate process, they propose a more sophisticated evo-

lution of the deposit base, which seems suitable for our problem. In their

model, the deposit base D(t) is the sum of two terms, a deterministic trend

growth function, G(t), and a mean-reverting Ornstein-Uhlenbeck process of

the detrended deposit base, D(t),

D(t) = G(t) + D(t) with (34)

18See Dewachter, Lyrio and Maes (2006) for a slightly different approach to obtain a

deposit rate process. Kalkbrener and Willing (2004) define the deposit rate simply as a

linear function of the short rate r.

20

G(t) =αD0 + αD

1 t, (35a)

dD(t) =µDD(t)dt+ σD dwt, (35b)

with constant σD and the Wiener process dwt under the empirical probability

measure P in an incomplete state space.19 For simplicity, we assume that the

Wiener process of the deposit base is uncorrelated with those of the state

variable processes.20

Deriving a deposit pricing formula For the derivation of a value of the

deposit business, VD, recall from (36) that in N -factor ATSMs, the value of

a risk-free zero bond at time t with time to maturity τ can be obtained by

B∗(t, τ) = exp[

AD(τ)− BD(τ)⊤XD

t

]

. (36)

whereAD(τ) and BD(τ) are ODEs given by the following Riccati equations21:

∂A(τ)

∂τ= −X∗

⊤K

∗⊤B(τ)+

N∑

i=

[Σ⊤B(τ)]i νi − δ (37a)

∂B(τ)

∂τ= −K

∗⊤B(τ)−

N∑

i=

[Σ⊤B(τ)]iυi − δx. (37b)

These ordinary differential equations (ODEs) can be solved by integration

under the initial boundary conditionsA(0) = 0 and B(0) =0N×1, which result

19Kalkbrener and Willing (2004) note that, theoretically, the deposit base in this model

can become negative. To their defense, they propose to floor D(t) at 0 and mention that

this theoretical possibility did not play any role in their empirical study.20Kalkbrener and Willing (2004) allow for correlation but, in turn, limit the number of

state factor to N = 2.21A derivation of this solution to the Riccati problem for the multivariate case can be

found in Cochrane (2001), pp. 374–377.

21

from (36) since we know that the initial zerobond price in time 0 is B∗(t, 0) =

1 for a standardized unit of a risk-free zero coupon-bond. Solutions to these

ODEs are finite given some technical regularity conditions on K∗ and Σ.

The Riccati equations can be solved by integration under the initial

boundary conditions AD(0) = 0 and BD(0) = 0N×1. Analytical solutions

for these are not available in our setting. Instead, the numerical proce-

dure will be more cumbersome in the present case, as we allowed here for

a correlation among the state variables in the ODEs. For the calibration

of this model on historical deposit data, a discretization of the time steps

0 = t0, t1, . . . , ts−1, ts = T with step size ∆t = ti − ti−1 is necessary.

Implementing a model describing the deposit behavior of a bank requires

to calibrate the following three processes:

dXD

t=K

D(XD −XD

t)dt+ΣD

√

SD(XD

t) dWD

t, (38a)

rD,t =ϕD0 +ϕD

1

⊤

XD

t+ uD

t , (38b)

D(t) =αD0 + αD

1 t+ µDD(t)dt + σDdwt. (38c)

5 Valuation of the loan business

For ease of modeling and exposition, we assume that the only credit product

a bank offers is a non-maturing overdraft facility, although, in principle, our

approach can be applied to all kinds of credit products when allowing for their

respective special characteristics and features. Regarding a formal valuation

setup for the loan business, a comparable market segmentation argument can

be made for the loan market.22 When abstracting from credit default risk,

obtaining a valuation formula for the loan business is based on table 2 and

22Jarrow and van Deventer (1998) propose this in their paper already; see p. 255.

22

equation (39) which are mirror images of the ones for the DB.23

t = 0 t = 1 . . . t = T − 1 t = T

New loans −L0 −L1 . . . −L(T−1)

Loans paid off +L0 . . . +L(T−2) +L(T−1)

Interest received +L0rcL,0 . . . +L(T−2)r

cL,(T−2) +LT−1r

cL,(T−1)

Interest paid −L0rτ=2,0 . . . −L(T−2)rτ=2,(T−2) −L(T−1)rτ=2,(T−1)

Table 2: Cash flow streams of the loan business

For simplicity of modeling, we have chosen the same time intervals as for

deposits. This is a sensible and unproblematic assumption for it only affects

the frequency of estimation points in the models estimating the loan base

and the loan rate. The resulting decay rate and effective maturity of assets

should remain unaffected of this assumption. Thus, the relevant rate would

still typically be one of longer effective maturity, τ = 2. Bank customers

pay rL on their loans and banks receive this rate less cost, rcL. Assuming a

loan business with only such loans, a planning horizon T , and a exogenously

given loan volume of the bank, Lt, the economic value of the banks assets is

analogously given by

VL = EP∗

0

[∫ T

0

Lt(rcL,t − rτ=2,t)

g(t)dt

]

. (39)

5.1 Valuation procedure

The valuation procedure of the LB follows the same five steps list in the

beginning of section 4, where one should resort to the same choice of term

structure model for reasons of consistency. Moreover, the term structure

23When abstracting from credit risk we assume that credit risk can efficiently be hedged

in the market so that its value impact can be neglected.

23

model should have multiple factors in order to avoid perfect correlations of

yields across maturities. Ideally, the dynamics of the loan rate are integrated

in the estimation of the essentially affine term structure dynamics as done

with the deposit rate already in the multi-factor model specification of the de-

posit business. We propose to add another state variable for the loan spread,

rcL,t − rτ=2,t, to the multi-factor framework presented above in section 4.2.

The implementation of our model with both spreads, the deposit spread and

the loan spread, increases the dimension of our original state vector Xt by

2, from N = 3 to N + 2 = 5, which we can denote in the following with the

superscript LD, XLDt ∈ R

N+2. The process of the state variables under the

empirical probability measure becomes then

dXLD

t= K

LD(XLD −XLD

t)dt+ΣLD

√

SLD(XLD

t) dW LD

t, (40)

where XLD ∈ RN+2, ΣLD,KLD ∈ R

(N+2 )×(N+2 ), dW LD

tis a (N + 2)-

dimensional independent standard Brownian motion under P, i.e. we do not

assume a correlation of the processes other than given by the joint dependence

on the short rate. SLD(XLD

t) ∈ R

(N+2 )×(N+2 ) is again a diagonal matrix

of a form which follows in analogy from equation (20) and equation (21).

We stop here with further derivations to avoid a repetition of trivial results.

By help of the variance-minimizing martingale measure, we can once again

obtain a pricing equations although the market is incomplete. The price of

a risk-free zero coupon bond is given by the similar but extended formula

B∗(t, τ) = exp[

ALD(τ)− BLD(τ)⊤XLD

t

]

, (41)

for which a closed-form solution cannot be derived. Hence, we suggest Monte

Carlo simulations of the stochastic processes describing the evolution of the

loan volume and the loan rate with parameters calibrated on historical loan

data. In this context, the far inferior amount of empirical studies on loans

24

as when compared to deposits is suspecting—data availability is certainly an

issue. Ausubel (1991) resorts to credit card loan data of interbank transac-

tions of loan portfolios and also discusses the issue of credit risk. Another

but related point is the higher opacity of the loan business. The appropriate

MMMVF transfer rate can be easily obtained as soon as an estimate of the

effective maturity of loans is available. Then, the empirically given τ can be

used as argument in the following definition of the yield y(t, τ) at time t of

a riskless zero coupon bond maturing τ in ATSMs.

y∗(t, τ) = −lnB∗(t, τ)

τ=

−A(τ) +B(τ)⊤Xt

τ. (42)

When leaving these questions aside, we have to estimate two equations.

Of course, the estimation of the SDE in (40), including N +2 state variables,

serves as basis for the estimation of both rD,t and rL,t and therefore replaces

the estimation of (38a). When comparing the studies on deposit valuation

with Ausubel (1991) and Chatterjea, Jarrow, Neal and Yildirim (2003) on

credit card loan valuation, one finds that the loan rate exhibit traits similar

to those of the deposit rate, respectively. Consequently, a natural choice for

this is a process specification in analogy to the deposit rate:

rL,t = ϕL0 +ϕL

1

⊤

XLD

t+ uL

t (43)

Finally, the assumption L = D alleviates us from estimating a third process.

6 Asset/Liability Management Valuation

The profit of the ALM is defined as

πALM = Dt × (rτ=2 − rτ=1) . (44)

25

As such, this unit isolates and internalizes the interest rate risk stemming

from changes in the slope and the curvature of the term structure, represent-

ing a particular type of yield curve swap with time-varying notional principal.

This makes it similar to the DB and the LB but, in contrast to them, the

spread of the ALM is based on market rates and not bank rates. In turn,

that difference slightly facilitates the valuation of the ALM, for which it can

be shown that its value in continuous time is given by the expectation under

the risk-neutral probability measure, EP∗

0 ,

VALM = EP∗

0

[∫ T

0

(rτ=2,t − rτ=1,t)×Dt

g(t)dt

]

. (45)

Such structures are traded in the market as spreads of constant maturity

swaps, or “yield curve steepeners”: As opposed to a plain vanilla interest rate

swap, here both legs are floating but of differing maturities. Both market

rates can be obtained from the short rate process according to the result in

equation (42) as yields y(t, τ) at time t of a riskless zero coupon bond matur-

ing τ from now. The stochastic process of Dt is proposed in equation (38).

Once again, the distribution of VALM has to be determined by simulation.

7 Valuation of the entire bank

Substituting (6) and (7) into equation (11) and adding precise time subscripts

and taking into consideration that A = L yields the equivalent expression

VA,t = VL,t +Dt − VD,t + VALM,t. (46)

Unfortunately, there is no closed-form solution available for any of the

economic values in (46). Simulating the three components24 of the bank

24Loan, Deposit, ALM business.

26

value is a tedious task, whereas valuing options on the portfolio of all three

is even more complicated, even before taking into account that the strike

prices of both options are time-varying as can be observed in equations (47).

In the specification of VL, VD, and VALM , we assumed independence of the

generating processes. Hence, these three values are correlated insofar only

as each of them is also a function of the short rate. From the boundary

conditions for bank debt, V bankB,T , and bank equity, V bank

E,T , as derived in (13),

the value of our bank’s debt and equity can formally be defined as

V bankB,0 = EP∗

0 [(B∗ −max[VD,T − VL,T − VALM,T ; 0])]× B∗(0, T ), (47a)

V bankE,0 = EP∗

0 [max[VL,T + VALM,T − VD,T ; 0]]×B∗(0, T ), (47b)

where an evaluation of the expectation involves the valuation of interest rate

options with time-varying strike prices, which can be achieved numerically.25

The value components in equations (46) and (47) have to be calculated ac-

cording to the preceding paragraphs:

• The multi-factor economic value of the deposit business value VD,t is

based on equation (36).

• The economic value of the loan business value VL,t is implied by the

numerical solution to equation (41).

• The economic value of the asset and liability management business

value VALM,t is implied by the numerical solution to equation (45).

After having characterized the valuation model in a very stylized way in

section 3, the sections 4 to 6 have provided the formulas needed to fill into

(47).

25While Rebonato (1998) gives a general overview of interest rate contingent claims val-

uation under different term structure models, the valuation of option with time-dependent

strike prices is put forward e.g. in Xia and Zhou (1997).

27

8 Conclusions

We argue that the problem of valuing a bank as a firm which is particularly

exposed to interest rate risk has not been adequately solved in the literature

so far. We propose a bank equity valuation model based on the contingent

claims theory and derive the banking firm value as constituting of the value

of three stylized business units, the asset business, liability business, and the

asset-liability management. The value of each of these units can be derived

in a risk-neutral valuation framework as equivalent to the costs of a hedging

strategy that offsets the risk exposure but still allows for arbitrage profits.

There exist several models for the valuation of banking products and we have

exemplified our model drawing on the deposit valuation model of Jarrow and

van Deventer (1998) and extending it to the entire bank.

28

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32