254 J. Duan et al. International Review of Economics and Finance 8 1999 253–265
and regulators of deposit-taking institutions. First, it has been argued that maturity mismatching, in conjunction with increased interest rate volatility in the 1980s, was a
major factor in precipitating the Savings and Loan debacle.
1
Second, in response to legislation, the Federal Reserve Board and the Office of Thrift Supervision have
implemented regulations to link banks’ capital requirements to their interest rate risk exposures, a development that makes the problem of managing interest rate risk
increasingly important to both banks and regulators.
2
To manage interest rate risk, banks choose asset and liability portfolios to control changes in the value of a target variable that result from changes in interest rates.
Although a number of target accounts are discussed in the banking literature, a useful and widely employed target is the market value of bank equity.
3
This target gains added importance because it is the exposure of equity value to changes in interest
rates that bank regulators target when adjusting capital standards for interest rate risk. Unfortunately, conventional models of bank interest rate risk management do
not take into account an important characteristic of equity, viz., limited liability. The reason is that conventional models of bank interest rate risk exposure largely rely on
tools adapted from the management of interest rate risk for portfolios of default-free, fixed-income securities.
4
Existing research implies, however, that limited liability may play an important role in interest rate risk management.
In this paper, we compare the conventional and limited liability models and their implications for gap management at banks. As the prototype of the limited liability
case, we adopt the options-based model of bank interest rate risk exposure developed by Duan, Moreau, and Sealey 1995; cited hereafter as DMS, where bank equity is
valued as a call option and interest rates are stochastic.
5
We present numerical compari- sons of the options-based and the conventional models. The results show that the two
approaches can give significantly different values for a bank’s interest rate risk expo- sure, especially during periods with above-average interest rate volatility, andor for
banks with above average credit risk.
The remainder of the paper is organized as follows: Section 2 presents a brief overview of the DMS measure of a bank’s interest rate risk exposure, which is based
on their model of bank equity valuation under limited liability and stochastic interest rates. A comparable version of the conventional model is also developed. Section 3
discusses some comparative properties of the options-based and the conventional models. Specific numerical comparisons of the models are presented in Section 4. The
conclusions and implications are discussed in Section 5.
2. Alternative models of bank interest rate risk
In this section, we present two alternative models of bank interest rate risk exposure. The first model employs an options-based approach where bank equity has limited
liability. The second model is what we refer to as the conventional approach, which ignores limited liability. For both models, we use the interest rate elasticity of bank
equity as the relevant target variable for a bank’s gap management.
J. Duan et al. International Review of Economics and Finance 8 1999 253–265 255
2.1 The options-based model of bank interest rate risk The options-based model employed here is developed primarily in the work of
DMS 1995.
6
In their model, bank equity has limited liability and is valued as a call option where the instantaneous interest rate, r
t
, follows a mean-reverting stochastic process such as that employed by Vasicek 1977 and others.
7
Banks have a planning horizon that extends over the period [0, T], hold assets with market value A, equity
capital with market value E, and deposit liabilities with a t 5 0 value of L and maturity T.
Deposits are assumed to be insured and earn a fixed, risk-free rate of return denoted by R, and equity has limited liability in the event of bankruptcy. At maturity, the
bank’s liability to depositors is X ; Le
RT
. The DMS 1995 valuation equation for bank equity, at time t, is as follows:
E
t
5 A
t
N h
t
2 rXPr
t
,tNh
t
2 d
t
, 1
where h
t
; 1
d
t
ln
3
A
t
P r
t
,trX
4
1 d
t
2 ;
d
2 t
; w
2 A
y
2
1 c
2
t 1 2w
A
y
2
3
t q
1 1
q
2
e
2
q
t
2 1
4
1
3
y
2
t q
2
1 2
q
3
e
2
q
t
2 1 1
1
1 2 e
2
2q
t
2q
3
24
; P
r
t
, t is the price per dollar of bank deposits at time t with t ; T 2 t to maturity;
w
A
is the instantaneous interest rate elasticity of bank assets; c is the instantaneous credit risk of bank assets; y is the instantaneous volatility of the interest rate; q is a
positive constant measuring the magnitude of the mean-reverting force of the interest rate; and r is a slack variable used to account for different closure rules enforced by
regulators.
8
Note that d
t
is the pricing-relevant risk for bank assets.
9
Let w
O E
denote the instantaneous interest rate elasticity of bank equity derived from the equity pricing model in Eq.1. Then, the bank’s interest rate risk can be expressed
as follows: w
O E
t
5 X
t
[w
A
2 w
L
t] 1 w
L
t 2
where V
t
; [Nh
t
]A
t
E
t
, and w
L
t is the instantaneous interest rate elasticity of bank liabilities with maturity t.
10
Note that w
A
and w
L
t are elasticities and are negative in sign.
The interpretation of Eq. 2 is based on the intuition behind option pricing with stochastic interest rates.
11
The pricing-relevant risk for an option contract with stochas- tic interest rates is the risk exhibited by the asset value process in a nume´raire economy,
where the price of a zero-coupon, default-free instrument, with t to maturity, is used as the nume´raire. The underlying asset and the nume´raire instrument have interest
rate elasticities of w
A
and w
L
t, respectively. Thus, the interest rate risk exposure at time t of the normalized asset is [w
A
2 w
L
t], referred to here as the interest rate elasticity gap
, which is the negative of the bank’s duration gap. Since V
t
is the option
256 J. Duan et al. International Review of Economics and Finance 8 1999 253–265
elasticity, the first term on the RHS of Eq. 2 is the interest rate risk exposure of equity in the nume´raire economy. Adding the interest rate elasticity of the nume´raire
bond yields the interest rate risk exposure of equity in the original economy.
2.2 The conventional measure of interest rate risk In this section, we derive an expression for the conventional measure of the interest
rate elasticity of bank equity. As discussed above, the focus of our investigation is the impact of limited liability on a bank’s interest rate risk exposure, a property that
the conventional model does not take into account. There are two possible approaches to the derivation of the conventional model, both of which lead to expressions of
similar form.
12
To maintain continuity with the options-based model of DMS 1995 shown above, and to make the subsequent comparisons as straightforward as possible,
our derivation here is based on the assumptions used by DMS 1995, adapted to the case of unlimited liability.
In Eq. 1, limited liability is reflected through the option pricing terms Nh
t
and N
h
t
2 d
t
, both of which are equal to one in the absence of limited liability. Thus, the equity valuation equation for the conventional model, can be written as follows:
E
t
5 A
t
2 rXP r
t
,t, 3
where all terms are defined earlier. Applying Ito’s lemma to Eq. 3 yields dE
t
5 Q
t
dt 1 w
A
A
t
dr
t
2 rX ]
P r
t
,t ]
r
t
dr
t
1 cA
t
dW
t
, 4
where W
t
is a Weiner process independent of the interest rate, and Q
t
is used to denote all terms associated with dt, which are all locally deterministic and are not
related to changes in the interest rate. Let w
C E
t
denote the instantaneous interest rate elasticity of equity as measured by the conventional approach. Then, Eq. 4 can be rearranged to yield:
w
C E
t
5 w
A
A E
t
2 rP r
t
,t X
E
t
w
L
t, 5
where w
L
t ;
3
] P
r
t
,tPr
t
,t ]
r
t
4
. Finally, substituting for rXPr
t
, t from Eq. 3 and simplifying yields
w
C E
t
5 A
t
E
t
[w
A
2 w
L
t] 1 w
L
t. 6
Eq. 6 can be interpreted as a continuous time version of the conventional model of bank interest rate risk exposure when interest rates follow a mean reverting stochastic
process.
13
Comparing equations Eqs. 2 and 6, it is evident that the conventional model is identical to the options-based model when Nh
t
5 1, that is, unlimited liability is assumed.
14
J. Duan et al. International Review of Economics and Finance 8 1999 253–265 257
3. Comparative properties of the options-based and conventional models
