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Working Paper 9110
ON FLEXIBILITY, CAPITAL STRUCTURE, AND INVESTMENT DECISIONS FOR THE INSURED BANK by Peter Ritchken, James Thomson, Ray DeGennaro, and Anlong Li
Peter Ritchken is an associate professor at the Weatherhead School of Management, Case Western Reserve University. James Thomson is an assistant vice president and economist at the Federal Reserve Bank of Cleveland. Ray DeGennaro is an assistant professor in the Department of Finance at the University of Tennessee. Anlong Li is a graduate student at the Weatherhead School of Management. The authors thank Andrew Chen, Myron Kwast, and Lucille Mayne for helpful comments and suggestions. Working papers of the Federal Reserve Bank of Cleveland are preliminary materials circulated to stimulate discussion and critical comment. The views stated herein are those of the authors and not necessarily those of the Federal Reserve Bank of Cleveland or of the Board of Governors of the Federal Reserve System. July 1991
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Most models of deposit insurance assume that the volatility of a bank's assets is exogenously provided. Although this framework allows the impact of volatility on bankruptcy costs and deposit insurance subsidies to be explored, it is static and does not incorporate the fact that equityholders can respond to market events by adjusting previous investment and leverage decisions. This paper presents a dynamic model of a bank that allows for such behavior. The flexibility of being able to respond dynamically to market information has value to equityholders. The impact and value of this flexibility option are explored under a regime in which flatrate deposit insurance is provided.
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I. Introduction
Almost all models of deposit insurance take the underlying source of risk, namely, the volatility of the bank's assets, to be exogenously provided.
Within this framework, the relative merits of the firm
increasing its volatility and leverage can be easily explored.
The
disadvantage of this approach is that it is static and does not recognize the fact that equityholders can respond to market events by dynamically adjusting previous investment and leverage decisions.
Such
dynamic behavior can lead to changing levels of portfolio risk over time, with commensurate effects on the value of deposit insurance. This 2 is the classic moral hazard problem. The objective of this paper is to establish a model that identifies how equityholders select a capital structure and investment policy under a flatrate deposit insurance regime. The model we consider is dynamic and
explicitly
incorporates
the
flexibility
option
that
allows 3
shareholders to adapt their asset portfolio decisions to market events.
We investigate how this flexibility option affects portfolio decisions and risktaking. Our findings show that with no opportunities to revise portfolio decisions, optimal bank financing and investment policies are bangbang;
that is, shareholders will either fully protect the charter
value or fully exploit the insurance subsidy granted by the insurer.
A
special case of our oneperiod model reduces to the model developed by
The 1iterature on deposit insurance using an option pricing framework was pioneered by Merton 119771. For a review of the literature, see Flood [ 19901. The moral hazard problem has been well discussed by Kane [19851. Fixedrate deposit insurance gives bank owners strong incentives to increase risk. Kane illustrates that the incentive scheme can become so socially perverse that projects with a negative net present value may be optimally selected. The term "flexibility option" is derived from the asset option pricing literature and has been discussed by Breman and Schwartz [19851, McDonald and Siege1 [1985, 19861, Kester [19841, and Triantis and Hodder [19901.
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Marcus t19841. However, unlike his model, ours allows equityholders to select risks dynamically and therefore allows moral hazard incorporated.
to be
With a finite number of portfolio rebalance points
remaining before an audit, bangbang policies may no longer be optimal and interior solutions may exist.
Finally, we investigate how the
flexibility option granted to equityholders affects the value of deposit insurance.
We show that ignoring the flexibility option leads to
understating the value of deposit insurance.
In particular, as the
number of portfolio revisions allowed prior to an audit date increases. a bank's ability to exploit the insureddeposit base increases.
This
can only be to the detriment of the flatrate deposit insuree. This paper is organized as follows.
Section I1 develops a
oneperiod model of a banking firm in which the equityholders optimally select their capital structure and their investment policy over the time remaining before an audit.
In this case, the firm invests either all or
none of its wealth in risky assets. preferable.
No
interior solutions are
Moreover, under certain assumptions, we show that the
equityholders' interests are best served by supplying the minimum amount of capital. Section I11 extends the analysis to the twoperiod case and shows that interior solutions may be optimal.
Section IV considers the
case in which multiple portfoliorevision periods remain prior to the audit. Numerical illustrations are provided to highlight the fact that the
value
of
deposit
portfoliorevision
insurance
increases
opportunities.
Section
with
V
the
number
discusses
of
policy
implications and concludes the paper.
11. A OnePeriod Model of a Banking Firm Consider an insured bank with one period remaining until an audit by the insuring agency. At the initial time, t=O, the deposit base is 1a and the capital supplied by the shareholders is a.
Deposits are fully
insured by the agency, which levies a fixedrate premium per dollar deposited. Let P(t) be the value of this deposit insurance net of the premium.
P(t) can be viewed as governmentcontributed capital.
Since
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the deposits are insured, their value at the end of the period is (la)e'*T,
where r* is the rate of return on the deposits.
For
simplicity, we assume that deposit inflows and outflows are equal over this period. Depositors, unlike the bank, may be faced with high transaction costs and may be unable to hold the riskless asset directly.
Moreover,
bank deposits may have unique characteristics, such as convenience yields, that make assets.
them
lessthanperfect
substitutes for riskless
In either case, barriers to entry, such as the need for a
government license or charter, allow banks to raise deposits at rates below the riskfree rate, r.
This positive spread produces an
intangible asset, or charter value, in the form of future monopoly rents.
If the charter obtains its value solely from monopolistic rents
attributable to the interestrate spread, and if this spread remains constant or grows over time, then the charter value equals the deposit base, D(O) = 1a.
In general, however, due to deregulation or increased
competition from other financial intermediaries, monopolistic rents are likely to erode over time.
Usually, the rents are taken to be some
function of the deposit base at time t.
For example, Marcus [I9841
assumes that the charter value is a fraction of the deposit base. Let C(0) represent the present value of this charter. If the bank fails the audit, it loses its charter. option on the charter.
Thus, at time 0, the bank holds a call
Let G(O) be the value of this claim.
In what
follows, we assume that the liability g r o s at the riskfree rate; that is, r* = r, with the capitalized value of the deposit spread reflected in the charter value. We assume that the bank invests 1q in riskless discount bonds and q in risky securities.
Assuming no dividends, the risky portfolio
follows a diffusion process of the form
where p and
(I.
are the instantaneous mean and volatility, respectively,
and dz is the Wiener increment.
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The bank's balance sheet at time 0 can be summarized as follows: Assets Tangible Assets Riskless Asset Risky Asset
Liabilities and Net Worth 1q q
Intangible Assets Government Subsidy P(0) Charter Value G(O)
I
Deposits D(O)=lu Shareholdercontributed Capital u Governmentcontributed Capital P(0) Charter Value G(0)
I
Total = 1 + P(O) + G(O)
Shareholder Equity Total = 1  u + E(0)
E(0)
Clearly, E(0) = u + P(0) + GIO). The initial value of the bank's tangible assets is V(0) = 1. Given q, the value of these assets follows the process
Conditional on the capital structure decision, a, and the investment decision, q, the value of the tangible assets of the firm at time T is
where x is a normal random variable with mean p

s2/2 and variance s2 .
At the audit date, T, the deposit base is D(T) = (1alerT. If the liquidation value of the marketable assets, V(T), is less than the deposit base, then the bank is declared insolvent and the shareholders receive nothing.
If, however, the bank is declared solvent, the
equityholders receive a claim worth V(T)
 D(T)
+ G(T).
Let E(T) be the
shareholders' equity at time T. Then, we have
E(T) =
{
:(TI
 D(T)
+
G(T)
if V(T) > D(T) otherwise
(1
Using standard option pricing methods, shareholder equity at time 0 is
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given by
E(0)
2
E(a,q;O) = a + G(a,q;O) + P(a,q;O)
where ifqra ifq
Shareholders will raise capital provided the marginal benefit of each incremental dollar raised is positive.
Since we assume all financial
assets are fairly priced, the tangibleasset portfolio has zero net present value, and the shareholders' objective is reduced to maximizing Z(a.q), where Z(a,q) = E(a,q;O)
a
Equation (4)clearly illustrates the tradeoff faced by the shareholders. Specifically, in selecting the optimal capital and investment decisions, the shareholders trade off the value of the call option on the charter (which is maximized by reducing default risk) and the value of the put option (which is maximized by increasing default risk). G(a,q;O) and P(a,q;O), we obtain
Let
Substituting for
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= Max {Z(a,q))
z(a*,q*)
o=aa 0lqS1 Given that
the
insurer charges a flatrate insurance premium
independent of the portfolio composition, the equityholders' objective is to select the investment and capital parameters, q and a, such that Z(a,q) is maximized. The Investment Decision
To investigate the optimal controls, first fix a and note that
az
[N(dl
 N(d2) I  aC(0)2
for q r a
(qa)
0
(7)
otherwise
az
If a were negative, then  > 0 and hence q* = 1.
as
Insolvent banks are
driven to extreme risk. This strategy is optimal because shareholders receive nothing unless the audit is passed. Indeed, for this case the firm may even select projects with a negative net present value to an allequity firm, provided their volatilities are sufficiently large. For a > 0, the sign of
is indeterminate. aq derivative of equation (7)for q r a, we obtain
By taking the second
Then, the function Z(a,q) is convex in q over the interval [a,lI. Figure 1 illustrates possible functions for any given a.
,
Given that the function is flat in q over the interval [O,al, the
*
optimal investment in risky assets, q , is either in that interval or at unity, depending on the value of a. Specifically, ~ ( a , ~ =* ) Max {Z(a,O), Z(a, 1))
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where
*
and a is that value of a chosen such that B
We conclude that for any capital structure decision, the optimal investment decision is either q = 0 or q = I . Firms ~ with capital lower than a* will shift thefr portfolio out of the riskfree asset into the B
risky investment. Firms with capital greater than a* will protect their B
charter value by increasing their riskfree holding and decreasing their investment in the risky portfolio. As an example, assume the charter value is some fraction, f, of the deposit base. Then C(0) = f(1a) Figure 2 traces out the breakeven point for given values of f and cr. Note that as cr increases, banks take on riskier positions.
Therefore,
for higher levels of asset risk, the range of capital structures and charter values over which the bank will risk its charter is larger. The graph highlights the fact that investment decisions depend critically on financing decisions in our model.
4~ctually,the optimal investment decision, q, is either anywhere in the interval [O,al or 1. Since equityholders are indifferent between investments in the range [O,al,we restrict attention to 0. It is worth noting that if the risky investment is a positive net present value project, then the optimal investment, q*, will be either at a or at unity, depending on which offers the greater value.
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The Financing/Capital Decision
We now turn to the financing decision. From the above analysis, we have Z(u*,q*) = Max (Max {Z(u,O)), Max {Z(u,l))) o=u=1 0*=1 with
Assume the charter value is some fraction f of the deposit base. Then
For small charter value f, i.e., when 1f
2
Q N(E)/N(), the 2 2
Z(u,O)
curve is uniformly higher than Z(u,l). The optimal capital structure Q should be u = 0 with q = 0. On the other hand, when 1f S N(;)/N(%), the curves E(a,l) and E(u,O) have a unique intersection point for 0 1.
u
Before the intersection, Z(u,1) is convex, decreasing, and above
Z(u,O).
Therefore, the optimal capital structure is again u = 0 with q
= 1, and the optimal financing decision is for equityholders to provide
the minimal amount of capital; that is, z(u*,~*) = Max Z(u,q) = Max {Z(0,1), Z(0,O)) a,q
111. Extension to the TwoPeriod Case
We have seen that with no opportunities to revise portfolios, the optimal portfolio decision is always bangbang.
If a portfolio
revision opportunity exists prior to the audit date, then the optimal solution may not be bangbang. This is illustrated below.
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Let the current values of the bank's deposits and assets be 99 and 100, respectively, and let f equal 6 percent.
For simplicity, assume
that the riskfree rate and the deposit rate of return are both zero. Furthermore, assume that the riskyasset returns are either 20 percent or 20 percent in the next two periods. in each period is 0.5.
The probability of an up move
Finally. assume that the bank can revise its
portfolio at the beginning of each period and that the audit is at the end of the second period. TABLE 1: Comparison of BangBang Strategies with an Interior Strategy
STRATEGY IN PERIOD 1
OPTIMAL STRATEGY IN PERIOD 2
EQUITY VALUE Eo
0
1 in upstate 1 in downstate
13.47
1
0 in upstate 0 in downstate
13.47
7/8
0 in upstate 1 in downstate
13.705
Table 1 shows the equity values associated with a few decisions in period 1, followed by optimal decisions in period 2. From our previous analysis, the optimal policy for period 2 is bangbang.
It is apparent
that given an initial strategy q0 = 0 (or qo = 11, the ability to switch decisions in the next period is valuable. equity for the strategies q
0
Note that the values of the
= 1 and qo= 0, followed by optimal
decisions in the next period, happen to be the same (13.47).
However,
the strategy qz = 7/8, followed by optimal decisions in the next period, leads to a higher equity value of 13.705. We now extend our model to two periods, where the time to an audit is t and where portfoliorevision opportunities exist at times t 2
0
and
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Let V,O, El(), D J O , and C ( 1 be the portfolio J value, shareholder equity, deposit level, and present value of the
tl, respectively.
j = 0,1,2. Finally, let q and q be the fraction
charter at times t
1
0
j'
of funds invested in the risky portfolio at times to and ti. When the risky portfolio follows a geometric Wiener process, then the value of the equity with one period to go, El(V1), is given by
VIN(dll) El(V1) =
v1 

D~ +
(Dl  GlIN(d12)
c1
for Vl s Vl for
v1
> V* 1
where
and V* satisfies the condition 1
The value of Vl, of course, depends on the initial decision qo; that is,
where
t
= tito. Given an initial capital structure, a, and a portfolio
decision, qo, the initial equity value, ~
~ la),( is qgiven ~ by
where go is the expectation operator taken over the riskneutralized process, dS/S = rdt + cdz. The optimal q, qo, is ~~(~i= l aMax ) (Eo(qo)} osq I 1 0
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Numerical methods are used to solve this optimization problem.
Assuming
capital structure decisions are made only at the initial period, the initial joint capital structure and investment problem is given by Z(q*,u*) = Max 0<4=1
{
Max {~~(q~lu)} 1 osqsi
For the more general nperiod problem, numerical procedures based on backward dynamic programming can be used to obtain the optimal value of the equity and the optimal control policy q(.).
IV. Numerical Results In this section, we illustrate how the asset flexibility option affects the behavior of the banking firm under flatrate deposit insurance. Consider a bank with deposits equal to (1u) and a charter Assume the riskless rate, r, is 10 percent. value equal to f(1a). Figure 3 depicts the net present value of the bank as a function of u for the cases where zero, two, and four revision opportunities are allowed before the audit date. The curved segment of the function corresponds to the range of u values where the bank optimally places the charter at risk.
Conversely, the linear segment of the function
corresponds to the range of u values where the bank's optimal portfolio decision is to set q < u to ensure that the charter value is captured. Figure
3
illustrates how
the
number
of
opportunities affects the net present value (NPV).
portfoliorevision Over the range of u
where the NPV function curves, the charter is placed at risk. increases, two events occur.
As n
First, the range of u values over which
the charter is placed at risk expands. Second, for any given u in this range, the NPV increases. The difference between the NPV curves with n
> 0 and n = 0 represents the value of the flexibility option. The increase in the NPV of equity, due to the flexibility option, is obtained partly at the expense of the deposit insurer. Indeed, the fair
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value of deposit insurance increases with the number of portfolio revisions.
As a result, empirical estimates that ignore the value of
flexibility understate the true value of deposit insurance.
V. Conclusion Optimal
equityholder
decisions
involve
tradeoffs
between
riskminimizing strategies, which reduce the likelihood of losing the charter, and riskmaximizing strategies, which exploit the insurance on the deposit base.
Without the ability to respond dynamically to market
information, optimal financing and investment policies are bangbang; that is, the bank will select extreme positions. Given any flatrate insurance scheme, incentives will exist for firms to revise their portfolios dynamically in response to market information.
These dynamic revisions are aimed at exploiting the
insureddeposit base more fully, while mitigating the likelihood of bankruptcy.
The additional value captured by equityholders responding
dynamically to jointly maximize the charter value and deposit insurance subsidy, beyond the static value, is captured in the value of the asset flexibility option. In the
presence
of
the
asset
flexibility option, portfolio
decisions may not be bangbang and interior solutions may be optimal. The likelihood of an interior solution may increase as the number of portfoliorevision opportunities expands.
Moreover, the value of the
insureddeposit base, provided at a flat rate, increases with the number of portfoliorevision opportunities. Our results suggest that the value of the deposit insurance may be significantly underestimated by
static models because such models
completely ignore the flexibility option.
The findings also suggest
that bank regulators should factor the flexibility option into any riskadjusted capital guidelines, and also into closure policies.
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References
Brennan. M. and E. Schwartz (1985). "Evaluating Natural Resource Investments," Journal of Business, 58 (April), 135157. Flood, Mark J. (1990). "On the Use of Option Pricing Models to Analyze Deposit Insurance," Federal Reserve Bank of St. Louis, Review, 72 (Januaryflebruary), 1935. Kane, Edward J. (19851, The Gathering Crisis in Federal Deposit Insurance. Cambridge, Mass.: MIT Press. Kester, W. C. (1984), "Today's Options for Tomorrow's Growth," Harvard Business Review, 62 (April), 153160. Marcus, Alan J. (19841, "Deregulation and Bank Financial Policy," Journal of bank in^ and Finance, 8 (December), 557565. McDonald, R. and R. Siegel (19851, "Investment and the Valuation of Firms When There is an Option to Shut Down," International Economic Review, 26, 261265. McDonald, R. and R. Siegel (1986). "The Value of Waiting to Invest," Quarterly Journal of Economics, 101 (November), 331349. Merton, Robert C. (1977). "An Analytic Derivation of the Cost of Deposit Insurance and Loan Guarantees: An Application of Modern Option Pricing Theory," Journal of Banking and Finance, 1 (June), 311. Triantis, Alexander J. and James E. Hodder (19901, "Valuing Flexibility as a Complex Option," Journal of Finance, 45 (June), 549565.
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Equity
Equity
Figure 1. The value of equity as a function of the riskyasset portfolio weight, q. There are three possible equity functions. The first panel shows the case where the optimal q equals one. The second and third panels show the cases where the investor is indifferent between values of q in the interval [ O , a ] .
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Figure 2. The breakeven value of aas a function of the charter value, f, and asset volatility, a . For a given a, the values of a for which the bank is indifferent between setting q 0 and q = 1 is a decreasing function of f. The range of (a,f) combinations over which it becomes optimal to risk the charter increases with a .

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Case Parameter = 20% f=5% 0
Figure 3. The impact of flexibility on the net present value (NPV) of equity. The NPV of equity is a decreasing function of initial shareholdercontributed capital, Q. It is an increasing function of the number of revision opportunities for values of Q where deposit insurance has value.