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Chicago
orking Paper Series
A Framework for Estimating the Value
and Interest Rate Risk of Retail Bank
Deposits
David E. Hutchison and George G. Pennacchi*
Working Papers Series
Issues in Financial Regulation
Research Department
Federal Reserve B a n k of Chicago
D e c e m b e r 1992 (WP-92-30)
FEDERAL RESERVE B A N K
OF CHICAGO
Comments Welcome
A Framework for Estimating the Value and
Interest Rate Risk of Retail Bank Deposits*
by
David E. Hutchison
Department of Finance
Washington State University
Richland, WA 99352
(509) 375-9263
and
George G. Pennacchi
Department of Finance
University of Illinois
Urbana, IL 61801
(217) 244-0952
November 1992
Abstract
Traditional measures of interest rate risk are based on the assumption that
prices of financial assets and liabilities are set in perfectly competitive markets.
However, recent empirical evidence on the pricing of retail bank deposits suggests that
many banks exercise market power. In this paper we provide a framework for valuing
a bank’s rent attributable to issuing demandable deposits. We also derive the interest
rate sensitivity (duration) relevant to these retail bank deposits. Using monthly survey
data on NOW accounts and MMDAs, we estimate the value of retail deposit rents and
their durations for over 200 commercial banks.
*We are very grateful to Herb Baer and Nancy Andrews of the Federal Reserve Bank of Chicago
for their generous help in carrying out the empirical work of this paper. We are also indebted to Herb
Baer, Bjorn Flesaker, Francis Longstaff, and participants of seminars at the Federal Reserve Board and
the Federal Reserve Bank of Chicago for their valuable comments.
L 1 B R A R t
.
I Introduction
.
Models of interest rate risk that generate the concepts of bond duration and immunization
commonly assume that fixed-income security prices are determined in perfectly competitive markets.
Individuals who trade in these securities are assumed to be price-takers who treat security supply as
infinitely elastic. For money managers trading in instruments such as Treasury securities and highgrade corporate bonds, these competitive market assumptions can be a useful abstraction. Hence, it is
not surprising that models of interest rate risk have been most productively applied to these
competitive trading environments.
However, for a number of financial institutions the competitive paradigm is less defensible.
Mounting empirical evidence supports the hypothesis that many banks exercise market power in some
of their deposit markets.1 This empirical research consistently finds that a bank’s behavior in setting
retail deposit interest rates, such as NOW account and MMDA rates, is correlated with measures of
market concentration. Retail deposit interest rates tend to be lower, and adjust more slowly and less
completely to changes in market interest rates, in more highly concentrated geographic markets. An
implication of this slow adjustment is that spreads between market interest rates and retail deposit rates
widen as market interest rates rise and compress as market interest rates fall. These empirical
characteristics of interest rates are consistent with optimal bank deposit pricing behavior predicted by
some theoretical models.2
When changes in the spread between market and deposit interest rates are not met with equal
and opposite changes in the volume of retail deposits issued, the current profitability of retail deposits
will vary with movements in market interest rates. For many types of deposits, this is likely to be the
'Empirical studies examining the relationship between deposit rates and competitive market interest
rates include Ausubel (1990), Berger and Hannan (1989), Hannan and Berger (1991), Hannan and
Liang (1990), Neumark and Sharpe (1992), and Sharpe and Diebold (1990).
2 Hutchison (1992) for an illustration in the context of an intertemporal asset pricing model.
See
2
case. The same macroeconomic factors that cause market interest rates to rise may cause the demand
for deposits to increase as well. For banks with market power, optimal deposit pricing may then
imply a widening spread between market and deposit rates, raising the current profitability of retail
deposits. Hence, the present value of a bank’s future deposit profits, i.e., its monopoly rent, will
change as well. This implies that demandable retail deposits, despite having zero stated maturities,
expose a bank to interest rate risk through changes in the value of its rent from issuing these deposits.3
A proper measure of the duration (interest rate sensitivity) of these deposits must incorporate the
change in the value of the rent associated with issuing these deposits.
The complexities in estimating the duration of retail bank deposits have been noted by other
researchers.4 Flannery and James (1984) offer an indirect solution that utilizes bank holding company
data. They relate estimates of the interest rate sensitivities of bank stocks to the cross-sectional
variation in the banks’ holdings of different types of deposits. This enables them to infer the interest
rate sensitivities of various classes of bank deposits. However, their method may be limited by its
identifying assumption that durations of particular types of retail deposits are the same across different
banks. If banks have differing degrees of deposit market power, this assumption may be violated. In
the present paper, the direct method used to estimate durations places no restriction on an individual
bank’s market power.
3
Fixed maturity deposits, such as consumer CDs, are subject to two types of interest rate risk: one
due to changes in the term structure that affect equivalent maturity competitively priced bonds, and the
other due to changes in the value of the rent associated with these deposits. In related work, we have
examined the value and duration of these rents. The results are comparable to those presented here.
4 example, Kaufman (1984, p.26) states "A number of types of bank deposit accounts, such as
For
demand deposits, savings, NOWs, SNOWs, and MMDAs, do not have specific maturity dates....What
are the durations of such deposit accounts?...If a bank’s deposit rates lag increases in market rates, all
deposits will not leave the bank immediately. ’Core’ deposits will remain for some time and flow out
only slowly. It may be possible to assign accurate probabilities to the timing of net deposit outflows,
depending on the difference between the market and deposit rates....If interest rates increase, it is then
possible to value these deposits at less than their par value." He later states "The correct duration
awaits additional research."
3
Our paper extends traditional duration theory to situations in which assets trade in imperfectly
competitive markets, focusing on the retail bank deposit market. The market for retail bank deposits is
modeled as having a downward sloping demand curve that is a function of the deposit interest rate, the
competitive market interest rate, and other local market variables. We derive a bank’s optimal deposit
interest rate and profit at each point in time as a function of the competitive market rate and other
exogenous variables. These results are incorporated into a continuous time model of the term structure
of interest rates, and contingent claims theory is used to value the bank’s stochastic future stream of
profits from deposit issue. This value is a function of the current competitive market interest rate, and
by simple differentiation we obtain the sensitivity of the value of the retail deposit rent to changes in
the market interest rate. A measure of retail deposits’ duration then results. Following this, we
employ our valuation technique to estimate the rents and interest rate risk of NOW accounts and
MMDAs for a sample of several hundred U.S. commercial banks.
The plan of the paper is as follows. Section II presents a general model and introduces the
contingent claims techniques that are used to value the rent associated with issuing demandable retail
bank deposits. Section III describes additional parametric assumptions used to empirically implement
the model. We derive the value of the monopoly rent as an infinite series of generalized
hypergeometric functions. Section IV confronts the model with data on NOW account and MMDA
interest rates and deposit balances from a sample of over 200 commercial banks. Section V provides
concluding remarks.
II. The General Model
Since previous empirical studies have found a link between bank market power and retail
deposit rate setting, we assume that individual banks face a downward sloping demand curve for retail
deposits. The quantity of retail deposits demanded, D, is given by
4
(1)
D(t) = D(r(t),rd(t),x(t))
where r(t) denotes the time t rate of return on a competitive market, short (instantaneous) maturity
default-free bond, rd is the deposit interest rate set by the bank at time t, and x(t) is a vector of other
(t)
variables affecting demand that are assumed to be independent of r(t). If the bank’s marginal revenue
from investing its deposits equals the market interest rate, r(t), then at each point in time the bank will
set rd as the value that solves
(t)
(2)
“ ax MO - rd(t) - c(t)]D(t)
where c(t) denotes the non-interest cost of issuing a retail deposit and is assumed to be independent of
rd The solution to problem (2) is5
(t).
<
3)
rd(t) = r(t) - c(t) - D/(aD/ard)
Note that if a bank’s retail deposit market is perfectly competitive, 3D/3rd = °° and thus rd(t) would
equal the competitive rate r(t) - c(t). A less than perfectly elastic demand for deposits implies a
deposit interest rate that is below r(t) - c(t).
This equilibrium deposit pricing behavior can be embedded into a model that values a bank’s
monopoly rent from issuing retail deposits. This monopoly rent can be considered a part of a bank’s
^his first order condition assumes that the bank is always at an interior solution. We do not
consider the possibility it may be optimal for a bank to completely cease issuing deposits.
5
"going concern value" or "charter value."6 Once we obtain a valuation formula for a bank’s retail
deposit monopoly rent, we can examine the sensitivity of this rent to changes in the market interest
rate, r(t). The result gives us a measure of the duration of retail bank deposits.
Define f(t) as the instantaneous cashflow (profit) received by the bank from issuing deposits in
amount D(t) and using the proceeds to invest in short (instantaneous) maturity Treasury bills paying
rate of return r(t). Substituting equations (3) into (2), we have
<
4>
f(r(t),x(t)) = [r(t) - rd - c(t)]D(t) = D2/(dD/dr;)
*(t)
*
where rd(r(t),x(t)) is the bank’s profit maximizing retail deposit interest rate.
The value of the bank’s monopoly rent from issuing retail deposits is then the present value of
the stochastic cashflow, f(r(t),x(t)), for all future dates. We can value this stream of cashflows by
applying techniques used to price interest rate dependent claims. For analytical simplicity, we will
make the assumption that the risk associated with changes in the variable x(t) bears no risk premium,
i.e., it can be considered as "idiosyncratic" or "non-systematic" risk.7 Hence, the only risk "priced" by
investors is assumed to derive from changes in market interest rates, r(t).
In order to determine the present value of risky cashflows to be received at future dates T, we
employ a version of Longstaff s (1990) Separation Theorem which is a stochastic interest rate
extension of the Cox and Ross (1976) risk-neutral valuation approach. This theorem states that when
valuing interest-sensitive risky cashflows the operation of "discounting" can be separated from the
6 some instances, the rent associated with issuing retail deposits might be more accurately
In
described as a quasi-rent. There may be fixed costs of entering a retail deposit market, such as
building branch offices, that could offset the acquired rent from issuing deposits at below market
interest rates.
7 can be viewed as reflecting local market conditions.
x(t)
6
operation of risk-adjusted "expectations-taking.” Specifically, let us define P(t,T) as the price at time t
of a default-free discount bond that pays $1 at time T and, therefore, has a time until maturity defined
by x = T - 1 Then the present (time t) value of f(r(T),x(T)), for T > t, can be expressed as
.
(5)
P(t,T)Et[f(rt(T),x(T))]
where rt(T) is an appropriately transformed market interest rate process whose form depends on x.
The description of this transformation is given in Longstaff (1990, p.102).8 Now define F as the
present value of the stream of cashflows, f(r(T),x(T)), for all future dates, T. F is given by
(6)
F * / “p(t,s)E t[f(V t)(s),x(s))]ds
Once F can be evaluated, we can calculate the duration of retail bank deposits as well. Let Dc be the
value of competitively priced, instantaneous maturity bonds equal to the level of deposits currently
issued by the bank.9 Then L = F - Dc is defined as the net value of the bank’s retail deposit liability.
We can now use Cox, Ingersoll, and Ross’s (1979) definition of duration which is a measure of an
asset’s risk from interest rate changes. In this sense, the duration of L is equal to the maturity of a
zero coupon bond that has the same rate of return sensitivity to changes in the market interest rate,
8
The transformed process for r(t) is that which changes the probability distribution for r(T) into
one (an equivalent martingale measure) that allows f(rT
(T),x(T)) to be valued by finding its expected
value. Longstaff (1990) calculates the appropriate transformed process for r(t) when it follows the
square-root process of Cox, Ingersoll, and Ross (1985b). In the Appendix, we calculate the
appropriate transformed process for r(t) when it follows the Omstein-Uhlenbeck process of Vasicek
(1977).
9 is equal to the "book value" of the bank’s retail deposits, D(r,rd,x).
DC
7
r(t), as does L. Mathematically, L is defined to have a positive duration of T* - t if [dL(t)/dr(t)]/L(t) <
0, and a negative duration of -(T*-t) if [dL(t)/dr(t)]/L(t) > 0, where T* solves the following expression:
(7)
1
dP(t,T*) ,
P(t,T*)
5r(t)
dL ,
L dr(t)
,1
Note that since deposits are assumed to be demandable, in the absence of any monopoly rents they
would have zero duration. This implies dDc
/dr(t) = 0, so that dL(t)/dr(t) = dF(t)/dr(t).
in . The Empirical Method
To evaluate the monopoly rent, F, given by equation (6), we need to make specific
assumptions regarding the form of the bank’s demand function, D(r,rd,x), and the stochastic processes
for r(t) and x(t). This will then imply an equilibrium process for rd(t), the bank’s optimal deposit
interest rate. Our choice of parameterization attempts to satisfy a number of goals. First, we want to
make parametric assumptions that will allow us to obtain a computationally practical way of evaluating
F and dF/dr(t). Second, we want to allow for sufficient generality so that our model can capture the
most important regularities of the time series for bank and market interest rates, as well as bank
deposit balances.1 Third, we wish to estimate the parameters of these time series using standard
0
econometric techniques.
With these goals in mind, we make the following assumptions. Let x(t) be a 2x1 vector equal
to (v(t) 11(0)'. v(t) is a variable that affects the elasticity of deposits with respect to the variable rd
1We specify a particular model that gives rise to equilibrium processes for deposit interest rates
0
and deposit quantities, equations (13) and (17), below. These processes then determine the equilibrium
process for bank profits that is used in our empirical work. However, there may be other models,
perhaps having more realistic behavioral assumptions, that could result in the same equilibrium
processes. Hence, our empirical results may be more general than implied by the paper’s particular
model.
8
and r|(t) is a variable that affects the demand for deposits without changing the elasticity with respect
to rd. This distinction will be clarified below. We can now define a 3x1 vector s(t) = (r(t) x(t)')'
which is assumed to follow a trivariate Omstein-Uhlenbeck process:
(8)
ds(t) = [ a + p s(t)]d t + osdZ8
where a is a 3x1 vector of constants, P is a 3x3 matrix of constants, and (asdZs)(asdZs)’ = Esdt with
Es being a 3x3 matrix with diagonal elements osi2, i=l, 2, and 3. For simplicity, it is assumed that
r(t), v(t), and t|(t) are independent in the sense that P and Zs tire diagonal matrices.
Under the assumption that the inverse deposit interest elasticity, D/(3D/3rd), is a linear function
of r(t) and v(t):
(9)
D/(aD/ard) = -d0 - c(t) + ( l - d ^ r d ) - v(t)
then the bank’s optimal retail deposit rate, given by equation (3), takes the form
(10)
rd = do + d ^ t ) + v(t).
*
Given the parameters d0, d,, and the market interest rate r(t), equation (10) shows that rd(t) reveals the
unobserved elasticity shift variable v(t). Further, equation (10) implies that r(t) and rd(t) also follow a
joint bivariate Omstein-Uhlenbeck process. To see this, note that equation (10) implies that the retail
deposit rate follows the process
9
drd = d,dr(t) + dv(t) = djotj + Pnr]dt
(ID
+ [a2 + P22v]dt + djOgjdZgj + o^dz^
Substituting for v(t) from (10) into (11), we see that (11) can be written in the form
(12)
drd
where a2 = dia, +
+ dj(pjj
- d0 2 and
P2
p22)r(t) + P22rd(t)]dt + o2dzj
a 2 = (d,2 l + a s2
os2
2)^.
Hence, if we define R(t) = (r(t) rd as the vector of the exogenous market rate, r(t), and the
(t))'
endogenous retail deposit rate, rd the process for R(t) can be written as:
(t),
(13)
dR(t) = [A + B R(t)]dt + odZ
where the newly defined elements are a, = a,, bu = pn, b1 = 0, b2 = d1
2
1
(Pn_p2 )> 12 = ?2
2 *2
2>
= ° si>
and a 1 = d,a,2. Equation (13) implies that R(t) has a discrete-time multivariate normal distribution.
2
While we have assumed a form for the inverse retail interest rate elasticity of deposits,
D/(9D/9rd), we have not yet fully specified the form of the demand function for deposits, D(r,rd,x).
Note that equation (9) is a first order differential equation whose solution can be written as
(14)
f ___
a
D (r,rd,x) = K (r,x)e"do' c
( l-d i)r
v
where K(r,x) is any arbitrary function that is independent of the bank’s choice of rd but which may be
dependent on any set of exogenous variables r and/or x. Due to our freedom in choosing K, we can
allow deposit demand to take on essentially any set of values. In other words, the specification of the
demand elasticity given by (9) places no constraint on the equilibrium level of deposits demanded.
10
Therefore, we will choose a form for deposit demand that facilitates estimation. The form of the
function, K, is assumed to be:
4q +
(15)
+v
K(r,x) = [kjr + k^do + d^ + v) + riJe** - V 0**1
where p represents the trend growth rate in deposit demand. This implies that
r <o<i
d“(* +^r+v)
(16)
D(r,rd,x) = [kjr + k ^ + d^ + v) + q]e
This functional form is convenient because it implies that when banks optimally set the retail deposit
rate according to equation (10), the level of bank deposits satisfies the following relation:
(1?)
D(r,rd,x) = [kjr(t) + k,r4
*(t) + utt)]©**
Equation (17) indicates that detrended deposit balances can be described by a linear regression model.
Given the assumed process for r|(t) in equation (8), the error term in this regression model will follow
a discrete-time AR(1) process, allowing for persistence in local market shocks. This model will be
used to forecast deposit demand based on the expectation of future values of r(t) and rd(t) that satisfy
the process given in equation (13).
In order to compute the bank’s expected future profits, we also need to specify a functional
form for the bank’s non-interest expense, c(t). Allowing for required (non-interest-bearing) reserves on
retail deposits, we assume c(t) = q + (l-p)r(t), where (1-p) is the deposits’ required reserve ratio.
11
Given this parameterization, along with equations (13) and (17), we can now find the expectation of
f(r(T),x(T)) given in equation (4):
Et[f(rt(T),x(T))] = E .K p r/IV rJ D -s X k ^ C r) + ^ ( T ) + nCTDe**1]
= ( - ?kjEt[rT
(T)] - ckjEjIr^CT)] - sEt[ri(T)]
(18)
+ pk1Et[rT
(T)2] - k,Et[rd/T )2] + (pk2-k 1)Et[r<
Jt(T)rT
(T)]
+ Et[t1(D]Et[prt(T)-r(It(T)]}e'‘T
where rx is the appropriately transformed process of r(t), and rd(T) = d0 + djrx + v(T). The
(T)
x
(T)
formulas for the moments of these variables are given in the Appendix.
In addition to finding a value for Et[f(rx
(T),x(T)], we need an expression for the value of a
default-free discount bond maturing at date T, P(t,T), in order to utilize equation (6) which gives the
value of the bank’s monopoly rent, F. Given the assumed process for the instantaneous maturity
interest rate r(t) in equation (13), Vasicek (1977) shows that the equilibrium value of P(t,T) is
2
(19)
P(t,T) = exp
(l -eb l )
|T
(r(t)
bl
i
~ r j - xr. + —±- (l - e b‘,T)'
4b u
where r„ is the constant yield on a long-term (infinite maturity) discount bond." Inserting the
expressions for P(t,T) in (19) and Et[f(rt(T),x(T)] in (18) into equation (6), the value of the bank’s
monopoly rent is determined. The Appendix shows that the integral in (6) has a solution that can be
"Vasicek (1977) shows that if the market price of interest rate risk is a constant, equal to q, then
the equilibrium value of r„ = -a^b-^ - q ^ /b ^ - a l/b^. The Vasicek model of the termstructure is consistent with a Cox, Ingersoll, and Ross (1985a) general equilibrium model of asset
pricing. This is demonstrated by Campbell (1986) for the case of the univariate Vasicek model and by
Pennacchi (1991) for the case of a bivariate extension of the Vasicek model.
12
expressed in terms of a series of generalized hypergeometric functions. dF/dr(t) can be calculated in a
similar manner.
Note that equation (19) implies
(20)
1
dP(t,T) _ 1 - ebl|t
P(t,T) dr(T)
bu
so that the duration of retail bank deposits is ±x* where x* solves
(21)
l - e ”" '' .
bn
1
dP .
F -D ° dr(t)
and where it should be emphasized that dF/dr = dF/dr + (dF/dr^drJdr = dF/dr + d,(dF/9rd), i.e., we
account for the bank’s optimal adjustment of its deposit interest rate, drjdr = d,, when measuring the
sensitivity of the rent to changes in market interest rates. Solving for x* we obtain:
(22)
+ bnl
JLfi
F - D c <fc(t) )
IV. Empirical Results
IV.A Data and Parameter Estimation
A series for the instantaneous maturity, competitive interest rate, r(t), was constructed using
end of month Treasury bill prices collected from the Center for Research in Security Prices (CRSP)
bond file. The average of bid and ask prices of 30, 90, 180, and 345 day Treasury bills, over the
period 1968 through 1990, were used to find maximum likelihood estimates of the parameters of the
13
Vasicek (1977) term structure model. A single state variable Kalman filter was used to compute the
likelihood function following the methodology described in Pennacchi (1991).1 Table 1 reports the
2
results of this estimation. Note that -a,/bu = .0813 is the estimate of the unconditional (long run)
mean of r(t). -bn = .09800 measures the speed of mean reversion of r(t) to its unconditional mean,
o, = .02432 is an estimate of the annualized instantaneous standard deviation of changes in r(t).
These parameter estimates then provided the basis for constructing a "smoothed" time series of the
instantaneous interest rate, r(t), that was used in the deposit interest rate and deposit demand
regressions described below.1
3
Along with the parameter estimates in Table 1, we required an estimate of the yield on an
infinite maturity zero coupon bond, r„, in order to evaluate the term structure of bond prices, P(t,T),
used to calculate a bank’s monopoly rent. Data on the average of bid and ask prices of 119 zero
coupon Treasury stripped bonds, whose maturities ranged from 51 days to just under 30 years, were
obtained from the Wall Street Journal for December 31 (the last trading day) of 1990. The sum of
squared deviations between these prices and the theoretical Vasicek (1977) model prices, given by
equation (19), were used to estimate the parameter r„, fixing the other parameter estimates to those
given in Table 1. We obtained an estimate of r„ = .08809, with a standard error of .00055.
Retail deposit interest rate and quantity (account balance) data were obtained from the
"Monthly Survey of Selected Deposits" collected by the Board of Governors of the Federal Reserve
System and analyzed at the Federal Reserve Bank of Chicago. The MMDA series begins in October
I2
The basis of the estimation was a state space model with the state transition equation being the
stochastic process for r(t) and the measurement equations being the prices of 30, 90, 180, and 345 day
Treasury bills which were assumed to measure their corresponding theoretical Vasicek (1977) model
prices with error.
1A smoothed estimate of the unobserved instantaneous maturity interest rate, r(t), is an optimal
3
estimate based on the full sample of Treasury bill prices. See Harvey (1981) for a discussion of
smoothing.
14
1983, and extends through December 1990, providing a maximum of 87 monthly observations per
bank. Prior to January 1986, both NOW account and Super NOW account data were collected, but the
Super NOW account series was discontinued in January 1986, when legislative changes left no
distinction between NOW and Super NOW accounts. For the sake of consistency, we use only the
NOW account series beginning in January 1986 and extending through December 1990, providing a
maximum of 60 monthly observations per bank. For both the MMDA and NOW account series, our
analysis was limited to a subset of the 416 banks that reported continuously over the sample period.
Prior to July 1989, balances in both MMDA and NOW accounts were aggregated, whether or
not banks offered multiple account categories within each account type. Since this date, data has been
collected for several categories of NOW and MMDAs, allowing for different tiers of minimum
balances and associated interest rates. Again to maintain consistency, we restricted our sample to
those NOW or MMDA accounts at banks that reported data for only one NOW or MMDA category in
the post July 1989 period. This left us with 244 NOW account series and 216 MMDA series.1
4
Using these NOW and MMDA deposit interest rate series along with the aforementioned series
for the market interest rate, r(t), we estimated the continuous-time parameters of equation (13) via an
ordinary least squares regression on its discrete-time analogue.1
5
Deposit demand, equation (17), was estimated in two parts. First we used ordinary least
squares to estimate
1We did not attempt to adjust the times series of deposit quantities to take into account the effect
4
of bank mergers or spin-offs. For this reason, and others mentioned below, our empirical results
should be viewed as illustrative rather than the most precise possible.
1Pennacchi (1991) gives the discrete-time process that corresponds to the continuous-time process
5
of an equation such as (13). The discrete-time process takes the form of a vector AR(1) process with
normally distributed error terms. The Appendix also gives the moments of the discrete-time process in
terms of the continuous time parameters.
15
(23)
ln[D(t)]
= k + (t + e
i
(t)
Under the assumed processes for r(t), rd and r|(t), ln[k,r(t) + k2
(t),
rd(t) + t|(t)] is stationary. Hence the
procedure in (23) will produce unconditionally unbiased estimates of the deposit demand trend, p.
Using these estimates of p, we then detrended the deposit demand series in order to estimate the lq
parameters in equation (17). Based on the assumed process for r|(t), equivalent to a discrete-time
AR(1) process, we used the Prais-Winsten procedure (also called the two-step full transform method)
to estimate the parameters and autoregressive coefficient of1
6
( )
24
D(t)e-^ = kir( ) + k ^ t ) + T ( )
t
]t
IV. B. Estimates of Rents and Durations from Retail Deposit Issue
The parameters required to evaluate a bank’s retail deposit rent and duration are listed in the
first column of Table 2. Except for the market interest rate parameters discussed in the previous
section and the non-interest cost parameters, q and p, these parameters were estimated for NOW
accounts of 244 banks and MMDAs of 216 banks. The average values of these NOW account and
MMDA estimates are reported in the second and third columns of Table 2. A few of the parameter
estimates are worth comment. Note that the deposit interest rate’s instantaneous standard deviation,
G2 is
,
larger for MMDAs than NOWs, .007418 versus .004124. Also as expected, a measure of the
1Essentially, this procedure is generalized least squares employing the residuals from an OLS
6
estimation of the detrended demand equation in order to estimate the unknown parameter of the AR(1)
process describing r|(t). The only distinction between this method and that of Cochrane and Orcutt is
the use of a full transformation matrix rather than the elimination of the first observation. According
to Judge et al. (1985), this procedure has been found to have good statistical properties relative to
other techniques.
16
responsiveness of the deposit interest rate to the market rate, d,, is greater for MMDAs than NOWs,
.82920 versus .40126. The values of r(t), rd(t), and r|(t) reflect values for the last trading day of
1990.1
7
We did not attempt to estimate the non-interest cost parameter, q, for each individual bank.
This would involve measuring a bank’s variable costs attributable to checking services, deposit
services, and account maintenance, net of service and handling charges, for its NOW or MMDA
deposits. This data was not available to us. Instead, we utilized summary data from the Federal
Reserve Board’s Functional Cost Analysis (FCA) which performs this cost allocation for NOW and
MMDA accounts of approximately 150 banks. However, there are at least two reasons why this data
is less than ideal and why our use of this data will contribute to the imprecision of our resulting
estimates Of deposit rents and durations. First, the FCA cost estimates are averages across banks. As
we show below, calculations of deposit rents and durations are sensitive to the level of non-interest
costs, and individual banks are likely to differ in their cost efficiency. Second, this data does not
separate non-interest costs into its variable and fixed components, though only variable costs are
relevant to our calculation of deposit rents and durations.1 We attempted to compensate for this
8
second deficiency by (somewhat arbitrarily) assuming that one-half of the reported costs are variable in
nature. For NOW accounts, we used 50% of the average non-interest cost given in the 1986-1991
9
FCA reports, resulting in q = .0066972 (67 basis points).1 Due to incomplete reporting on MMDA
nDue to confidentiality requirements on deposit quantity data, we are unable to report the actual
level of banks’ deposits, and hence values for ri(t), corresponding to the last trading day of 1990.
Hence, we assume that each bank’s r|(t) equals its long-run mean value of -a3 3
/p3 .
1For example, the FCA data includes expenses related to office overhead (office furniture and
8
equipment, occupancy expense), data services, and publicity and advertizing. These costs are likely to
have a significant fixed component. As mentioned earlier, fixed costs are relevant for determining
whether rents associated with retail deposits are true monopoly rents or quasi-rents. However, for the
purpose of calculating a retail deposit’s duration, only variable costs should be considered.
l9Each year’s FCA expense estimates are broken down into three bank deposit size-based
categories: banks with deposits up to $50 million; $50-$200 million; and greater than $200 million.
We averaged the estimates across these three categories and across the years 1986-91.
17
accounts prior to 1991, we used only the 1991 FCA report to obtain an MMDA estimate of < =
;
.0045166 (45 basis points).20
The average parameter estimates in Table 2 were used to construct the graphs in Figures 1 to
4. Figures 1 and 2 graph rents per deposit, the duration of the rents (alone), and the duration of
deposits (including the value of rents) as a function of the growth of deposit demand, p. Because it is
difficult to obtain a precise measure of this trend, p, given the length of our sample period, it is
worthwhile to see how NOW and MMDA rents and durations vary for a range of different growth
rates. As intuition would tell us, rents per deposit are increasing functions of the growth in deposit
demand. Quite simply, this reflects the higher level of future profits that a bank with a growing
deposit market expects to receive. Indeed, these rents become unbounded as the growth rate
approaches the infinite maturity yield, r„ = .08809. Examining the durations of rents in Figures 1 and
2, we see that the duration of NOW rents is smaller than the duration of MMDA rents for the same
rate of deposit growth. This is what one would expect given the prior empirical evidence that NOW
rates are less responsive to market interest rates. In addition, the duration of NOW rents is much more
sensitive to changes in deposit growth than is the duration of MMDA rents. Indeed, as can be seen
from Figure 1, for small deposit growth rates the NOW rent’s interest rate sensitivity can be so large
that duration becomes undefined.2
1
What is perhaps more interesting is that both NOW and MMDA durations are increasing
functions of deposit demand growth, being negative for low growth rates, then positive for higher
20Prior to 1991, MMDA service and handling charges were not reported.
2 This is due to the fact that a bond in the Vasicek (1977) model (as well as other general
1
equilibrium bond pricing models) has a sensitivity to interest rates, (l/P)dP/8r = (l-exp(bnx)/bn, that
is a bounded function of bond maturity. In other words, the most this sensitivity can be is l/bu as
T—
Hence, duration is undefined for interest sensitivities of greater magnitude.
18
growth rates.2 The economic intuition for this is as follows. Because MMDA rates, and especially
2
NOW account rates, adjust less that one-for-one with market interest rates, a rise in the market interest
rate increases a bank’s interest margin and, hence, its profitability.2 This first effect would tend to
3
increase the value of the bank’s rent from deposit issue, thus giving the bank a negative duration.
This is, in fact, the case for banks with low deposit demand growth rates. However, there is a second
effect working in the opposite direction. An increase in the market interest rate reduces the present
discounted value of the bank’s future profits (cashflows) from retail deposit issue. For banks with
high deposit demand growth rates, i.e., banks whose future profits contribute significantly to the value
of its monopoly rent, this second effect dominates, causing the value of its rent to fall when market
interest rates increase, i.e., giving the bank a positive duration.
Figures 1 and 2 also show the duration of NOW and MMDA deposits, i.e., the duration of L =
F - Dc which is given by equation (22) above. Since the competitive component of these demandable
deposits, Dc, has a zero duration, we see that the signs of deposit durations are opposite to the signs of
rent durations and have a magnitude that depends on the size of the rents per deposit. For the
parameter estimates used in Figures 1 and 2, these deposit durations are small in magnitude, except for
large growth rates of deposits when the value of rents per deposit become large.
Figures 3 and 4 consider the value of NOW and MMDA rents as a function of a bank’s non
interest cost per dollar of deposit, q.24 As expected, rents decline as q increases from 0 to .02 (0 to
200 basis points). Durations of deposits increase, which imply that durations of rents are decreasing,
2 Note that by referring to a deposit rent having a negative duration of x years, we mean that its
2
interest rate sensitivity is of equal magnitude, but of opposite sign, to that of an x year maturity zero
coupon bond.
“Cashflows from deposit issue will be expected to increase at all future dates, but more so in the
near-term than farther in the future. This is because of the mean-reverting behavior of market interest
rates.
2 A zero growth in deposit demand, p=0, is assumed in Figures 3 and 4.
4
19
as this cost increases. The intuition for this is similar to that of a declining growth rate. An increase
in market interest rates increases the bank’s lending-borrowing spread due to the slow adjustment of
NOW and MMDA rates (effect 1), which tends in increase current bank profitability. But, the greater
market rate also reduces the present discounted value of future profits (effect 2). When < is relatively
;
small, making future profits significant, the second effect dominates, giving the bank’s rent a greater
(more positive) duration and, hence, the bank’s deposit a smaller (more negative) duration. When
c,
is
large and the value of future profits small, an increase in the market interest rate increases the value of
the bank’s rent, giving rents a smaller (more negative) duration and deposits a larger (more positive)
duration.
We now turn to results using individual bank parameter estimates rather than averages of
individual bank estimates. Figures 5 to 12 report distributions (histograms) for individual bank NOW
and MMDA rents and deposit durations. The first four histograms, Figures 5 to 8, were constructed
using the actual estimated growth in deposit demand, p, for each individual bank. However, we had to
exclude those NOW and MMDA observations for which the estimated growth rate, p, exceeded r„ =
.08809 since the estimated value of these rents would be unbounded. Based on this truncated
subsample of 98 NOW accounts and 188 MMDAs, we observe that the median NOW account rent per
deposit (median = 3.49%) is slightly smaller than the median rent for MMDAs (median = 4.97%).
However, there is significant variation in these NOW and MMDA rents across individual banks.
Comparing durations, we are not surprised to find NOW account deposit durations are generally much
higher (median = 11.12 years) relative to those of MMDAs (median = .29 years). Similar to the rent
estimates, durations also vary significantly across individual banks.
In Figures 9 to 12, we calculate the distribution of rents and durations for the individual bank
NOW accounts and MMDAs, but under the assumption that the growth in deposit demand is zero for
each bank’s accounts. Doing this allows us to use the full sample of 244 NOW accounts and 216
20
MMDAs. Assigning all banks the same growth rate has the effect of reducing the variation in the
estimates of the individual bank’s deposit rents and durations. NOW accounts have a median rent of
6.55% and a significantly smaller median duration of 6.69 years. MMDAs have a slightly higher
median rent of 7.88% and a median duration of .37 years.
While it is difficult to calibrate the precision of our estimates, they certainly appear to be in a
range that most would consider reasonable. It is interesting to note that our median NOW and
MMDA rent estimates are comparable to estimates obtained by Berkovec and Liang (1991) from data
on failed bank merger premiums. Based on prices paid to the FDIC for banks that failed over the
period January 1987 to October 1990, Berkovec and Liang (1991) present a hedonic approach to
estimating deposit premiums that takes into consideration the FDIC’s failure resolution method. For
bank failures resolved using a purchase and assumption transaction, they estimate a NOW account
premium of 4.9% (versus our actual growth and zero growth medians of 3.49% and 6.55%,
respectively). For MMDAs, they find a premium of 4.0% (versus our actual growth and zero growth
medians of 4.97% and 7.88%, respectively).
V. Conclusion
This paper analyzes the interest rate risk of retail bank deposits by recognizing the existence of
market power associated with these bank liabilities. Accounting for retail deposit rents is essential to
correctly evaluating a deposit’s duration. By valuing this rent using contingent claims techniques and
deriving its interest rate sensitivity, we have provided a framework for evaluating interest rate risk.
Given additional parametric assumptions, we presented a practical method of estimating retail deposit
rents and their deposit durations. Our results have a number of potential applications. Examples
include: calculating the "going concern” value of a distressed bank’s retail deposits so that regulators
can be make a better decision regarding the cost of keeping a bank open or closing it via a purchase
and assumption or liquidation; calculating a merger premium associated with one bank purchasing
21
another bank’s retail deposit market; measuring the value and duration of a bank’s "core" deposits, as
part of an exercise to estimate the duration of a bank’s overall net worth. This last application should
be of interest to bank managers seeking to limit interest rate risk as well as regulators attempting to set
interest rate risk-based capital standards.
The valuation technique presented here could potentially be extended to other fixed-income
assets or liabilities traded in imperfectly competitive markets. One class of callable assets might be
revolving lines of credit to consumers or small or medium-sized firms. In principle, incorporating
default-risk in the valuation of the rents from these assets is not an insurmountable task. However,
there are likely to be additional empirical complications in estimating rents from these assets since the
(promised) interest rates charged on these loans will not be equivalent to their actual rates of return.
Tackling this problem is left to future research.
22
Appendix
This appendix shows how the value of a bank’s monopoly rent from retail deposit issue, F,
can be computing using equation (6) and the expression Et[f(rt(T),x(T))] and P(t,T) given in
equations (18) and (19), respectively.
Following the line of argument made by Longstaff (1990 p.102), the appropriate
transformation of the market interest rate process, r(t), is
2
<
frx(t) = [at + oxq + -^-(1 - e b,,T) + bn r ]dt + a xdzx
bu
(A.l)
= [alT + bxir j d t + oxdzx
Since, by equation (10), the equilibrium deposit rate is a linear function of r(t), repeating the steps
taken in equations (11) and (12) in the text, we find that the transformed deposit rate, rdt, follows the
process
(A.2)
dTfoCt)
[dxalT + tt2 ^0^22 + ^1 (P 11
P22)rt + P22rdt^^ + ° 2^2
= [32T + b21rt + b22rdT + o2dZj
]dt
Hence, the transformed processes, rt(t) and rd (t), are equivalent to their untransformed processes but
T
with constants added to their unconditional means. One can now calculate Et[f(rt(T),x(T))] given in
equation (18) which depends on some of the first and second moments of rt(T), rdt(T), and t|(T).
These moments are given below. See Pennacchi (1991, p. 63-64) for discussion of the formulas.
23
Et[rT
(T)] = r(t)eb“T + - ^ ( e bnT- l )
D1
1
(A.3)
2
= r(t)eb“T + - -+
-— (eb“T-1 ) - - ^ ( e b“T- l)2
2
-'ll
E^CT)] = rd(t)eb“T +
hi
- l) + r(t)
l2
/2
(eb“T - e ^ )
birb2
2
(A.4)
(eb||T - l) _ (e^ T - l)
b,1
2
2
bU -b2
2
^1
ai b2
x 1
(A.5)
Et[r,(T)2] - E,[rt(T)f -
(e26"' - l)
ZD1
1
Et[rd,(T)2 - Et[rdt(T)f
]
2b,22
’
(e2
^
^
2a1b2 ^(bu +^
2 1
(A.6)
- l
)
bn + b^2
2
bu -b2
2
.2 2
bjiOi
_ j
j
2b,'2
2
(e2b“x - l) _ 2(e<bu 4^ )T - l)
2bn
(>1 - b2 j
*1
2
bn + b22
(e*** - l)
2b-2
'2
L > i ** a t _ i
Qi
t)
\
W
D W T )]
- Et[rt (T)]Et[r^(T)] = a 12^ _ - ----- — ^
bn + bj2
(A.7)
2
b 21g l
b ll
" b 22
(e2b“x - l)
2bi
i
(e®1 +ba)T - l)
1
b ll
+ b 22
24
Et[n(T)] = Ti(t)ep + -^-(ep )
*T
*M
P33
(A. 8)
= [D(t)e->“ - kir(t) - k*rd
(t)]ep + -^-(ep l)
»t
*TP33
Inserting these expressions into equation (18), we find that the expected profit at date T takes the
form:
(A-9)
Et[f(rx(D ,x (D ] = V o + f!eT,T + •• +
where the f,’s are independent of x and functions solely of r(t), rd(t), D(t), and the (constant)
parameters of the model. The y ’s are linear functions of the constants bn, b22, and p3 . Given the
3
functional forms of Et[f(rt(T),x(T))] and P(t,T), the bank’s monopoly rent requires computing integrals
of the form:
(A. 10)
1 s c f “e(Y '*-rjTeC
‘+
l(1'eblIl)(r(t)'0 eC(1 ,U dx
2 'e1 ,)2
0
where c0, cl5 and c2 are constants, with c { = l/bn and c2 = o 2/(4b3 ). Assuming that bn < 0 (which is
a requirement for r(t) to follow a stationary process), and making the change in variables y = i - e
(A. 10) can be written as:
(A. 11)
I s
-bn
r 1(l-y )« -1eC
l7(r(t)-I-)eC
jy2dy
Jo
,
25
where e = (Y +p-rJ/b,,. Note that if we expand the term e Ciy <r<c> ~I- ) around the point r(t) = r„, we
=
obtain the following series:
V
Cyrt-j _
i(()t
1 + C!y(r(t)-rJ
(A. 12)
+ ^ c 2y 2(r(t)-rj2 + ■^c1y3(r(t)-rJ
3
+ ..
.
Hence, the above integral can be written as:
(A. 13)
I=
-bu j=o
4fd(t) -rJ f \ 1 - y r ' y ^ d y
i
Jo
The integral in the right hand side of (A. 13) is now in the form of a Riemann-Liouville (fractional)
integral of order e. See Erdelyi (1954, Chapter XIH). It can be evaluated in terms of a generalized
hypergeometric function as:
(A. 14)
/Ja-yrVe^dy
where 2f2(<o1( < 2; w3, c 4;
o
d
z
)
= £
k=0
(to)k =
1 \G ) j
r(e)r(j+l) p (j+1 j+2. e+j+1 e+j+2 '
j
IXe+j+l)
2 * 2 ’ 2 ’ 2 ’
^2
\
ZlL and where the notation
(o 3) k (<o4) k k!
= t (to+l) . (<o+k-l) , (<o)0 = 1.
o
.
.
Inserting (A.14) into (A.13), the value of I can be written as:
2
J
26
(A.15)
i =
rt
e) i
C° £ cif f ) rV r( r( +1) F fj+l J+2. e+j+1 e+j+2 \
[j!m
J
IXe+j+l) * 1 2 ’ 2 ’ 2 * 2 , *JJ
C
~^ll j o
=
This expression simplifies to
(A. 16)
r(t)-r„Yi
1
F fj+1 j+2.e+j+l e+j+2 )
j=o C bu Y e(e+l) - (e+j) 2 ^ 2 ’ 2 ’ 2 * 2 ’ 2J
I = _f!L £
The duration of retail bank deposits requires computation of the derivative 3F/3r, which, in turn,
requires the derivative 3l/3r. Using (A. 16) we have:
(A.17)
2
*
=
^ 2
-b,2 U
[ r(t)-r» ,
j+1
/j+2 j+3. e+j+2 e+j+3
)
1
( bu
e(e+l) .(e+j+l) 2 H 2 ’ 2 ’ 2 * 2 ’ 2J
27
References
Ausubel, L., 1990, "Rigidity and Asymmetric Response of Bank Interest Rates," mimeo, Northwestern
University, December.
Berger, A., and T. Hannan, 1989, "The Price-Concentration Relationship in Banking," Review of
Economics and Statistics 71, 291-99.
Berkovec, J., and N. Liang, 1991, "Deposit Premiums of Failed Banks: Implications for the Values of
Deposits and Bank Franchises," Proceedings of a Conference on Bank Structure and
Competition, Federal Reserve Bank of Chicago.
Campbell, J., 1986, "A Defense of Traditional Hypotheses about the Term Structure of Interest Rates,"
Journal of Finance 41, 183-193.
Cox, J., J. Ingersoll, and S. Ross, 1979, "Duration and the Measurement of Basis Risk," Journal of
Business 52, 51-61.
Cox, J., J. Ingersoll, and S. Ross, 1985a, "An Intertemporal General Equilibrium Model of Asset
Prices," Econometrica 53, 363-384.
Cox, J., J. Ingersoll, and S. Ross, 1985b, "A Theory of the Term Structure of Interest Rates,”
Econometrica 53, 385-407.
Cox, J., and S. Ross, 1976, "The Valuation Options for Alternative Stochastic Processes," Journal of
Financial Economics 3. 145-166.
Diebold, F., and S. Sharpe, 1990, "Post-Deregulation Bank-Deposit-Rate Pricing: The Multivariate
Dynamics," Journal of Business and Economic Statistics 8, 281-91.
Erdelyi, A., editor, 1954, Tables of Integral Transforms. Volume II, Bateman Manuscript Project,
California Institute of Technology, McGraw-Hill, New York.
Flannery, M., and C. James, 1984, "Market Evidence on the Effective Maturity of Bank Assets and
Liabilities, Journal of Money, Credit and Banking 16, 435-45.
28
Hannan, T., and A. Berger, 1991, "The Rigidity of Prices: Evidence from the Banking Industry,"
American Economic Review 81, 938-945.
Hannan, T., and N. Liang, 1990, "Inferring Market Power from Time-Series Data: The Case of the
Banking Firm," Finance and Economics Discussion Series #147, Board of Governors of the
Federal Reserve System, November.
Harvey, A.C., 1981, Time Series Models, Wiley, New York.
Hutchison, D., 1992, "Retail Bank Deposit Pricing: An Intertemporal Asset Pricing Approach,"
unpublished manuscript.
Judge, G., W. Griffiths, R. Hill, and H. Lutkepolhl, 1985, The Theory and Practice of Econometrics.
2nd. Edition, John Wiley and Sons, New York.
Kaufman, G., 1984, "Measuring and Managing Interest Rate Risk: A Primer," Economic Perspectives
(Federal Reserve Bank of Chicago) 8, 16-29.
Longstaff, F., 1990, "The Valuation of Options on Yields,” Journal of Financial Economics 26,
97-121.
Neumark, D., and S. Sharpe, 1992, "Market Structure and the Nature of Price Rigidity: Evidence from
the Market for Consumer Deposits," Quarterly Journal of Economics, forthcoming.
Pennacchi, G., 1991, "Identifying the Dynamics of Real Interest Rates and Inflation: Evidence Using
Survey Data," Review of Financial Studies 4, 53-86.
Vasicek, O., 1977, "An Equilibrium Characterization of the Term Structure,” Journal of Financial
Economics 5, 177-188.
29
Table 1
Maximum Likelihood Estimates of the Vasicek Model Parameters
(Standard Errors in Parentheses)
- ,
aA
.08131
(.05329)
-,
bi
.09800
a
,
.02432
(.01902)
(.00110)
Data: End of month 30, 90, 180, and 345 day Treasury bill prices over period 1968 through 1990, 267
observations.
30
Table 2
Averages of Parameter Estimates
(Used in Figures 1 to 4)
Parameter
NOWs
rt
()
.06240
.06240
To
o
.08809
.08809
al
.007968
.007968
b„
-.09800
-.09800
MMDAs
.02432
.02432
r ()
dt
.04926
.05648
b2
2
-3.3207
-2.0022
o2
.004124
.007418
c,2
-2.776E-6
-1.4308E-5
o -d0p2
c2
2
.15695
T()
|t
140112.39
414101.81
a3
683845.52
848320.81
P 33
-7.2414
-1.9952
d,
.40126
K
735439.04
-9682.92
k2
-1566220.62
-1833831.38
P
0
0
<
;
.0066972
.0045166
p
.8
8
1.0
*
.00511
.82920
’The estimated average annual growth rate of deposits was .1266 for NOWs and -.0257 for MMDAs.
However, a value of p = 0 was used in Figures 3 and 4.
31
Figure 1
V alue
a n d
D u ra tio n
,
r.\l
r\
'
r\ %
V* lu e /D e p o s its [%], D u ra tio n
a
i
NOW
Figure 2
r\ \
I
_ .
VI lu e /D e p o sits [%], D u ra tio n
a
i
MMDA V alue a n d D u r a t io n
32
Figure 3
NOW Value a n d
D u r a t io n
N o n - In te re s t C o s t / D e p o s it
Figure 4
MMDA Value a n d D u r a t io n
N o n - In te re s t C o s t / D e p o s it
33
Figure 5
R e n ts:
A ctu al
G ro w th
Num ber o f O b s e rv a tio n s
NOW
Figure 6
Num ber o f O b s e rv a tio n s
NOW D u r a tio n s : A c tu a l G r o w th
34
Figure 7
N um ber o f O b s e rv a tio n s
M MDA R e n ts : A c tu a l G r o w th
Figure 8
N um ber o f O b s e rv a tio n s
MMDA D u r a tio n s : A c tu a l G r o w th
D u r a t i o n [Years]
35
Figure 9
R e n ts:
Z e ro
G ro w th
Num ber o f O b s e rv a tio n s
NOW
Figure 10
N um ber o f O b s e rv a tio n s
NGW D u ra tio n s : Z e r o
G r o w th
35
Figure 11
G r o w th
Num ber o f O b s e rv a tio n s
MMDA R e n ts : Z e r o
Figure 12
N um ber o f O b s e rv a tio n s
M M DA D u r a tio n s : Z e r o G r o w th
D u r a t i o n [Years]
@FedFRASER