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Asset Encumbrance, Bank Funding and Fragility∗
Toni Ahnert Kartik Anand Prasanna Gai James Chapman†
March 15, 2017
Abstract
We propose a model of asset encumbrance by banks subject to rollover risk
and study its influence on fragility, funding costs, and welfare. A banker’s en-
cumbrance choice trades off expanding profitable investment funded by cheap
secured debt with potentially greater fragility due to unsecured debt runs. We
derive several testable implications about the privately optimal level of encum-
brance. Deposit insurance or wholesale funding guarantees induce excessive
encumbrance and fragility. To eliminate risk shifting, regulation imposes limits
on encumbrance or a tax on encumbrance with a lump-sum rebate. We relate
our policy implications to the current regulatory debate in several jurisdictions.
Keywords: asset encumbrance, wholesale funding, rollover risk, fragility, se-
cured debt, unsecured debt, encumbrance limits, encumbrance surcharges.
JEL classifications: G01, G21, G28.∗We thank Jason Allen, Xavier Freixas, Douglas Gale, Itay Goldstein, Agnese Leonello, Frank
Milne, Jean-Charles Rochet and seminar participants at Alberta School of Business, University ofAmsterdam, Bank of Canada, Carleton, Danmarks Nationalbank, ECB, Frankfurt School of Finance,Humboldt University Berlin, Monetary Authority Singapore, HEC Montreal, McGill, Queen’s, Re-serve Bank of New Zealand, Wellington and AFA 2017, EEA 2016, FDIC Bank Research Conference2015, NFA 2015, the RIDGE 2015 Workshop on Financial Stability for comments. These views areof the authors and do not reflect those of the Bank of Canada or Deutsche Bundesbank.†Ahnert: Bank of Canada, 234 Laurier Ave W, Ottawa, ON K1A 0G9, Canada. Email: tahnert@
bankofcanada.ca. Anand: Deutsche Bundesbank, Wilhelm-Epstein-Strasse 14, 60431 Frankfurt,Germany. Email: [email protected]. Gai: University of Auckland, 12 Grafton Rd,Auckland 1142, New Zealand. Email: [email protected]. Chapman: Bank of Canada.
1
1 Model
Our analytical framework builds on Rochet and Vives (2004). There are three dates,
t = 0, 1, 2, and a single good for consumption and investment. The economy is
populated by a large mass of investors, who are each endowed with a unit at t = 0
and are indifferent between consuming at t = 1 and t = 2. Investors differ in their
risk preferences: a first clientele is risk-neutral, while a second is infinitely risk-averse.
The latter group may be thought of as pension funds or large institutional investors
mandated to hold high-quality and safe assets (IMF, 2012). All investors have access
to a storage technology at t = 0, which yields a gross return, r > 0, at t = 2.
The economy also comprises a unit mass of identical risk-neutral bankers with
access to profitable and high-quality investment opportunities at t = 0. Bankers can
invest their own funds, E ≥ 0, at t = 0 in order to consume at t = 2. But they can
also obtain funding by issuing unsecured and secured debt to investors at t = 0. Each
investment matures at t = 2 with a gross return, R > r. Premature liquidation of the
investment at t = 1 yields a fraction ψ of the return at maturity, where 0 < ψR < r.
Given the segmented investor base, bankers issue unsecured demandable debt
to risk-neutral investors and asset-backed secured debt to risk-averse investors. In the
spirit of Rochet and Vives (2004), an exogenous quantum of unsecured debt, U ≡ 1,
can be withdrawn at t = 1 or rolled over until t = 2. The rollover decision is made by
a group of professional fund managers, indexed by i ∈ [0, 1], who typically prefer to
roll over funds but are penalized by investors if the bank fails. Fund managers face
strategic complementarity in their decisions in the sense that individual manager’s
incentive to roll over increases in the proportion of managers who roll over. A man-
ager’s conservatism, 0 < γ < 1, is crucial to this decision.1 The more conservative1The conservatism ratio, γ ≡ c
b+c ∈ (0, 1), derives from managerial compensation in Rochet
1
the manager, the higher is γ and the less likely that unsecured debt is rolled over.2
The face value of unsecured debt, DU , is independent of the withdrawal date.
A banker attracts secured funding from risk-averse investors by encumbering –
or ring-fencing – a proportion α ∈ [0, 1] of assets into a bankruptcy-remote entity
on its balance sheet.3 We denote by S ≥ 0 the total amount of secured funding
raised, and by DS the face value of secured debt at t = 2. Table 1 illustrates the
balance sheet of the representative bank at t = 0 once funding is raised, investment
I ≡ E + S + U is made, and assets are encumbered.
Assets Liabilities(encumbered assets) αI S(unencumbered assets) (1− α)I U
E
Table 1: Balance sheet at t = 0 after funding, investment, and asset encumbrance.
We suppose that a bank’s balance sheet is subject to shock, A, at t = 2. The
shock may enhance the value of assets, in which case A < 0. But the crystallization of
operational, market, credit or legal risks may require write downs, corresponding to
A > 0. The shock has a continuous probability density function f(A) and cumulative
distribution function F (A), with decreasing reverse hazard rate, ddA
f(A)F (A)
< 0. This
common condition supports the concavity of the expected equity value in the level of
asset encumbrance and therefore equilibrium uniqueness.
and Vives (2004). In bankruptcy, a manager’s relative compensation from rolling over is negative,−c < 0. Otherwise, the relative compensation is positive, b > 0.
2Reviewing debt markets during the financial crisis, Krishnamurthy (2010) argues that investorconservatism was an important determinant of short-term lending behavior. See also Vives (2014).
3Secured funding typically takes the form of covered bonds, which are backed by assets thatremain on the issuing bank’s balance sheet, or RMBS (residential mortgage-backed securities), whichare generally off-balance sheet instruments. To the extent that the issuing bank offers implicitor explicit guarantees, RMBS also affect encumbrance. Term repos (repurchase agreements) alsocontribute to asset encumbrance levels.
2
Bankers and investors are protected by limited liability. The value of bank
equity at t = 2 is E2(A) ≡ max{0, RI−A−UDU −SDS}. Table 2 shows the balance
sheet at t = 2 for a small shock and when all unsecured debt is rolled over. Since
encumbered assets are ring-fenced, the shocks affects only unencumbered assets.4
Assets Liabilities(encumbered assets) RαI SDS
(unencumbered assets) R(1− α)I − A UDU
E2(A)
Table 2: Balance sheet at t = 2 after a small shock and unsecured debt is rolled over.
If a proportion ` ∈ [0, 1] of unsecured debt is not rolled over at t = 1, the banker
liquidates an amount ` UDUψR
to meet withdrawals. A bank is illiquid at t = 1 and
closed early if the liquidation value of unencumbered assets is insufficient to meet
withdrawals:
R(1− α)I − A <`UDU
ψ. (1)
In the event of early closure, encumbered assets are legally separated from the bank’s
other assets. This separation is automatically triggered since the encumbered assets
are bankruptcy-remote, and unsecured debt holders cannot lay claim to them. The
encumbered assets maturity at t = 2 and yield αλR, where λ ∈ [ψ, 1] measures
the efficiency of managing encumbered assets to maturity after early closure.5 In
stark contrast, bankruptcy costs are assumed to be so high that unsecured debt
holders recover zero in bankruptcy. The illiquidity threshold for the shock is AIL(`) ≡
R(1− α)I − ` UDUψ
.
4Our modeling of the shock is also consistent with the notion of ‘replenishment,’ whereby creditand market risks are expunged from the encumbered assets and concentrated onto the unencumberedpart of the bank’s balance sheet. Such mechanisms are present for covered bonds and repos.
5Similar to Bolton and Oehmke (2016), the efficiency loss 1−λ may capture the loss of economiesof scale between the bank and the bankruptcy-remote vehicle stemming, for example, from joint ITand risk management systems.
3
If the bank is liquid at t = 1, then the total value of bank assets is RI− ` UDUψ−A
at t = 2. The bank is insolvent at t = 2 if it is unable to repay its secured debt
holders and the proportion 1− ` of unsecured debt holders:
RI − A− ` UDU
ψ< SDS + (1− `)UDU . (2)
Upon repaying secured debt holders at t = 2, the banker uses any residual encum-
bered assets (due to over-collateralization) to repay remaining unsecured debt. The
insolvency threshold of the shock is thus AIS(`) ≡ RI−SDS−UDU
[1 + `
(1ψ− 1)]
.
At t = 1, fund managers receive a noisy private signal about the shock upon
which they base their rollover decisions. Specifically, they receive the signal
xi ≡ A+ εi, (3)
where εi is idiosyncratic noise drawn from a continuous distribution G with support
[−ε, ε] for ε > 0. The idiosyncratic noise is independent of the shock, and is indepen-
dently and identically distributed across fund managers.
Table 3 summarizes the timeline of events in the model.
t = 0 t = 1 t = 2
1. Issuance of secured 1. Balance sheet shock realizes 1. Investment maturesand unsecured debt 2. Private signals about shock 2. Shock materializes
2. Investment 3. Unsecured debt withdrawals 3. Debt repayments3. Asset encumbrance 4. Consumption 4. Consumption
Table 3: Timeline of events.
4
2 Equilibrium
Our focus is on the symmetric, pure-strategy, perfect Bayesian equilibrium of the
model. Without loss of generality, we study threshold strategies for the rollover of
unsecured debt (Morris and Shin, 2003). Thus, fund managers roll over unsecured
debt if, and only if, their private signals indicate a healthy balance sheet, xi ≤ x∗.
Definition 1. The symmetric, pure-strategy, perfect Bayesian equilibrium comprises
a proportion of encumbered assets (α∗), an amount of secured debt for each bank
(S∗), face values of unsecured and secured debt (D∗U , D∗S), and critical thresholds for
the private signal (x∗) and balance sheet shock (A∗) such that:
a. at t = 1, the rollover decision of fund managers x∗ and the run threshold A∗
are optimal, given the level of asset encumbrance and secured debt (α∗, S∗) and
face values of debt (D∗U , D∗S);
b. at t = 0, each banker optimally chooses (α∗, S∗) given the face values of debt
(D∗U , D∗S), the participation of secured debt holders, and the thresholds (x∗, A∗);
c. at t = 0, secured and unsecured debt are priced by binding participation con-
straints, given the bankers’ choices (α∗, S∗) and the thresholds (x∗, A∗).
We proceed to construct the equilibrium in four steps. First, we price secured
debt. Second, we derive the optimal rollover decision of fund managers. Third, we
characterize the optimal asset encumbrance choice of the banker and, in doing so,
obtain the endogenous level of secured debt issuance. And, in a final step, we price
unsecured debt.
5
2.1 Pricing secured debt
A secured debt holder either receives the face value DS or an equal share of the value
of encumbered assets. Since early closure at t = 1 occurs with positive probability
for a large balance sheet shock, competitive pricing of secured debt by infinitely risk-
averse investors implies a binding participation constraint, r = min{DS, λ
RαI∗(α)S
}.
Lemma 1. Asset encumbrance and cheap secured debt. Secured debt is cheap,
D∗S = r, and the maximum issuance of secured debt tolerated by risk-averse investors
is S ≤ S∗(α) = αλzI∗(α), where z ≡ R/r is the relative return and I∗(α) = U+E1−αλz
is total investment. Greater encumbrance increases secured debt issuance and invest-
ment, dS∗
dα= dI∗
dα= λzI∗(α)
1−αλz > 0.
Competitive pressures push the face value of secured debt to the outside option
of investors. Since infinitely risk-averse investors evaluate the secured debt claim at
the worst outcome (the bank fails and is closed early, but the encumbered assets
remain legally separated), the maximum level of secured debt increases in the level of
asset encumbrance. As we make clear below, a banker always chooses this maximum
level of secured debt for a given level of asset encumbrance, since it reduces fragility
and increases the expected equity value.
2.2 Rollover risk of unsecured debt
Asset encumbrance and secured debt issuance fundamentally alter the dynamics of
rollover risk, as shown in Figure 1. Panel A shows the illiquidity and insolvency
thresholds, AIL(`) and AIS(`), that obtain without any asset encumbrance and se-
6
cured debt issuance, that is α = 0 = S. It thus recovers the dynamics in Rochet and
Vives (2004) for the case where liquid cash reserves are set at zero. An illiquid bank
at t = 1 is always insolvent at t = 2. And so, in this case, the insolvency threshold is
the relevant condition for analysis.
Panel B shows the illiquidity and insolvency thresholds in the case of asset en-
cumbrance and secured debt issuance. Over-collateralization means that the thresh-
olds do not coincide at ` = 1. Additional assets worth RαI∗(α) − rS∗(α) = Rα(1 −
λ)I∗(α) become available to service unsecured debt withdrawals at t = 2, which are
not available at t = 1 because of encumbrance. As a result, a bank that is illiquid at
t = 1 can, nevertheless, be solvent at t = 2, that is Rα(1 − λ)I∗(α) ≥ (1 − `)UDU .
A sufficient condition for the illiquidity threshold to be the relevant condition for
analysis is a requirement for an upper bound on the face value of unsecured debt:
DU ≤ DU ≡ (1− λ)RαI∗(α). (4)
We suppose that this condition holds and later verify that it does in equilibrium.
Henceforth, we focus on the limit of vanishing noise about the balance sheet
shock, ε→ 0. Thus, the rollover threshold converges to the run threshold, x∗ → A∗.
Proposition 1. Run threshold. There exists a unique run threshold
A∗ ≡ R(1− α)I∗(α)− γ UDU
ψ. (5)
Fund managers roll over unsecured debt at t = 1 if and only if A ≤ A∗, such that
early closure occurs if and only if A > A∗.
Proof. See Appendix A.
7
Balance Sheet Adjustment (A)
UnsecuredDebtW
ithdrawals(`)
Illiquid andInsolvent
Liquid butInsolvent
Liquid andSolvent
Illiquiditythreshold (t = 1)
Insolvencythreshold (t = 2)
(a) Without asset encumbrance (α = 0 = S)
Balance Sheet Adjustment (A)
UnsecuredDebtW
ithdrawals(`)
Illiquid andInsolvent
Illiquid butSolvent
Liquid andSolvent
Insolvencythreshold (t = 2)
Illiquiditythreshold (t = 1)
Overcollateralization: R,(1! 6)I$(,)
(b) With asset encumbrance
Figure 1: Asset encumbrance and secured debt issuance alters the run dynamics. Thefigure depicts the illiquidity and insolvency thresholds as a function of the proportionof withdrawing investors ` for two cases: without asset encumbrance in panel A andwith asset encumbrance in panel B. Panel A replicates the results of Rochet andVives (2004) without liquid asset holdings and with a balance sheet shock, wherethe relevant condition is the insolvency threshold. As depicted in panel B, over-collateralization shifts the insolvency threshold to the right. Therefore, the upperbound DU ensures that the relevant condition is the illiquidity threshold.
8
Proposition 1 uses global games techniques to pin down the unique incidence of
an unsecured debt run by fund managers. More conservative fund managers decrease
the threshold of the balance sheet shock above which a run occurs, ∂A∗
∂γ< 0. A
higher return on investment increases the value of unencumbered assets as well as the
amount of secured debt raised for a given level of asset encumbrance. Both effects act
to reduce run risk, so ∂A∗
∂R> 0. A higher liquidation value of investment decreases the
extent of strategic complementarity among fund managers and increases the amount
of secured debt issued, given encumbrance. Since both these effects also lower run
risk, it follows that ∂A∗
∂ψ> 0. Similarly, an increase in the efficiency of managing
encumbered assets until maturity in the case of bank closure allows the banker to
raise more secured funding. Hence, the stock of unencumbered assets increases and
run risk decreases, ∂A∗
∂λ> 0. A higher cost of funding decreases the amount of secured
debt raised for a given level of encumbrance. This reduces the value of unencumbered
assets, so ∂A∗
∂r< 0. And a better capital of banks reduces run risk through its effect
on increased investment and unencumbered asset values, implying ∂A∗
∂E> 0.
Lemma 2 links the fragility in unsecured debt markets to secured debt issuance,
especially the efficiency of managing encumbered assets in early closure.
Lemma 2. Asset encumbrance and fragility. If encumbered assets are managed
efficiently until maturity in early closure, λz > 1, greater asset encumbrance reduces
the bank’s run risk. Conversely, if λz < 1, greater asset encumbrance heightens bank
fragility:dA∗
dα
(λz − 1
)≥ 0, (6)
with strict inequality whenever λz 6= 1.
Proof. See Appendix A.
9
Greater asset encumbrance affects the run threshold in two opposing ways.
First, for a given level of investment, greater asset encumbrance reduces the stock
of unencumbered assets, which heightens run risk. Second, greater encumbrance
allows the banker to issue more secured funding, which increases the total size of
investment. As a result, the stock of unencumbered assets increases and run risk is
reduced. Thus, the overall effect depends on the relative size of these two effects.
Whether the second effect dominates depends on the efficiency of managing
assets until maturity in the event of early closure. If encumbered assets are poorly
managed, then for each unit of secured funding raised, the banker must encumber
more than one unit of assets. In this case, the stock effect dominates and greater
asset encumbrance increases run risk. In contrast, if encumbered assets are managed
well, then for each unit of secured funding raised, the banker needs to encumber less
than one unit of assets. As such, greater encumbrance reduces the run risk.
When can we expect encumbered assets to be well managed? In the spirit of
Bolton and Oehmke (2016), who study the resolution of a global bank, the bank would
need to set up redundant systems ex ante. For example, access to critical IT and risk
management infrastructures after early closure may be restricted for the bankruptcy-
remote entity, which adversely impacts how encumbered assets are managed. To
overcome this issue, redundant systems need to be setup for the bankruptcy-remote
vehicle, which is costly. Indeed, as Gorton and Souleles (2007) describe, bankruptcy-
remote special purpose vehicles typically are ‘robot-firms’ (p.550) that exist only on
paper and have no independent employees. Encumbered assets are typically held in
a special purpose vehicle, which motivates the following assumption.
Assumption 1. Managing encumbered following early bank closure is costly:
λz < 1. (7)
10
2.3 Optimal asset encumbrance and secured debt issuance
The representative banker chooses a level of asset encumbrance to maximize the
expected equity value of the bank, taking as given the face value of unsecured debt
DU , and subject to the run threshold, A = A∗(α), and the (maximum) amount of
secured debt that can be raised, S ≤ S∗(α). Since a higher level of secured debt for
a given level of asset encumbrance both increases the expected equity value of the
bank and lowers fragility, we obtain S = S∗(α). The banker’s problem reduces to
maxα
π ≡∫E2(A)dF (A) =
∫ A∗(α) [RI∗(α)− UDU − S∗(α) r − A
]dF (A). (8)
Figure 2 shows the relationship between encumbrance and expected equity value.
It illustrates the existence of a unique interior solution for asset encumbrance, taking
into account the effect of encumbrance on both the amount of secured debt raised
and bank fragility.
α*(��)
����� �����������
�������������������
Figure 2: A unique interior privately optimal level of asset encumbrance: expectedbank equity and asset encumbrance for an example with R = 1.5, r = 1.1, E = 0.5,ψ = 0.6, λ = 0.66, γ = 0.8, DU = 3.3, and the balance sheet shock follows theLogistic distribution with mean -3 and scaling parameter 1.
11
Proposition 2. Asset encumbrance schedule. There is a unique asset encum-
brance schedule, α∗(DU). If fund managers are sufficiently conservative, γ > ψ, then
the schedule decreases in the face value of unsecured debt, dα∗
dDU≤ 0, and an interior
solution for DU < DU < DU is implicitly given by:
F (A∗(α∗))
f(A∗(α∗))=
(1− λz)
λ (z − 1)
[RI∗(α∗)α∗(1− λ) + UDU
(γ
ψ− 1
)]. (9)
Proof. See Appendix B.
The banker balances the marginal benefits and costs of asset encumbrance when
choosing the privately optimal level. The marginal benefit of encumbrance is an
increase in the amount of secured funding obtained. Since secured debt is cheap and
investments are profitable, the equity value of the bank – conditional on it surviving
an unsecured debt run – is increased. But the marginal cost of encumbrance is an
increase in bank fragility and, therefore, a lower probability of surviving an unsecured
debt run. Consequently, a higher face value of unsecured debt exacerbates rollover
risk and lowers the run threshold, inducing the banker to encumber fewer assets.
2.4 Pricing of unsecured debt
The repayment of unsecured debt depends on the size of the balance sheet shock.
Without a run, A ≤ A∗, unsecured debt holders receive the promised payment DU ,
while for larger shocks, A > A∗, bankruptcy occurs and they receive zero. Thus, the
value of an unsecured debt claim is V (DU , α) ≡ DU F (A∗(α,DU)), and competitive
pricing implies that it equals the cost of funding for any given level of encumbrance:
r = D∗U F (A∗(α,D∗U)). (10)
12
Proposition 3. Private optimum. If bank capital is scarce, E < E, and managers
conservative, γ ≥ γ˜, then there exists a unique face value of unsecured debt, D∗U > r.
If funding is costly, r > r˜, asset encumbrance is interior, α∗∗ ≡ α∗(D∗U) ∈ (0, 1).
Proof. See Appendix C.
Figure 3 shows the privately optimal allocation and its construction. The condi-
tion γ ≥ γ˜ ensures that the schedule D∗U(α), which is derived from the market clearing
condition for the face value of unsecured debt, is upward-sloping in the vicinity of
the asset encumbrance schedule. This, in turn, leads to a unique characterization
of the joint equilibrium for the level of asset encumbrance, α∗∗, and the face value
of unsecured debt, D∗∗U . Finally, the condition that the bank’s capital must satisfy
E < E ensures that D∗U < DU(α∗), and that the illiquidity condition is more binding
that the insolvency condition for the bank. Finally, the lower bound on the cost of
funding ensures that the equilibrium face value of unsecured debt is sufficiently high,
such that the equilibrium level of asset encumbrance is an interior solution.
��(α
*)
α*(��)
��* (α)
��**
α **
���� ����� �� ��������� ����
����������������
Figure 3: Privately optimum of asset encumbrance and face value of unsecured debt:an example with R = 1.5, r = 1.1, E = 0.5, ψ = 0.6, λ = 0.66, γ = 0.8, and theshock follows a Logistic distribution with mean -3 and scaling parameter 1.
13
3 Testable implications
Our model yields a rich set of comparative static results that yield testable implica-
tions. Parameter changes affect the unique, interior equilibrium in two ways. First,
for a given face value of unsecured debt, the banker trades off heightened fragility
against the funding of investments with a positive net present value. And second,
the equilibrium face value of unsecured debt changes with underlying parameter val-
ues, influencing the incentives of investors to participate in the debt contract. We
summarize the main results in Proposition 4, before discussing each in turn.
Proposition 4. The banker’s privately optimal level of asset encumbrance, α∗∗, de-
creases in the cost of funding, r, and in the conservatism of fund managers, γ. It
increases in the profitability of investment, R, improvements in the shock distribution,
the liquidation value, ψ, and the efficiency with which encumbered assets are managed,
λ . Greater bank capital, E, has an ambiguous effect on asset encumbrance.
Proof. See Appendix D.
Monetary policy. A lower cost of funding, perhaps brought about by easier mon-
etary conditions, increases secured debt issuance and increases the benefits of asset
encumbrance. Since the required face value of unsecured debt is also lowered, the two
effects combine to increase asset encumbrance. Thus dα∗∗
dr< 0.
Our model suggests that less restrictive monetary conditions lead to a shift to-
ward secured lending. This implication appears consistent with the stylized evidence
in the wake of the global financial crisis. Since 2007/8, central banks in advanced
countries have run expansionary monetary policy and there has been an increased
14
appetite for safe assets among investors (IMF, 2013; Caballero et al., 2016). Juks
(2012) and Bank of England (2012) document a clear, increasing trend in the encum-
brance ratios of Swedish and UK banks following the implementation of extraordinary
monetary policy measures in response to the crisis.
Market stress. The conservatism parameter γ can be broadly interpreted as a
measure of market stress. Krishnamurthy (2010) documents how, during the global
financial crisis, fund managers turned conservative and became less inclined to roll
over unsecured debt. Faced with a deterioration in investor sentiment, the bank
is more fragile for any given level of encumbrance. The banker responds to the
heightened fragility in a precautionary fashion, lowering the extent of encumbrance
and forgoing profitable investment from the issuance of secured debt, in order to
induce rollovers by fund managers. The combined effects of increased fragility and
the greater face value of unsecured debt reduce encumbrance, so dα∗∗
dγ< 0.
Increased market stress thus induces a reduction in the share of secured debt
(as a proportion of total debt). A joint analysis of banks’ borrowing and lending
behavior during both normal and crisis episodes would shed light on this implication.
Profitability. Higher returns on investment or a more favorable distribution of the
balance sheet shock, in the sense of a first-order stochastic dominance shift according
to the reverse hazard rate, reduces fragility and induces the banker to encumber
more. The model implies, therefore, that banks with less risky balance sheets or more
profitable assets increase the share of secured debt on their balance sheet. Consistent
with these implications, DiFilippo et al. (2016) document that banks with higher risks
reduce their share of secured debt.
15
The channel identified in our model is quite distinct from theories based on
asymmetric information (e.g., Stiglitz and Weiss 1981; Freixas and Jorge 2008). An
asymmetric information approach might suggest that banks with low risk have less
incentive to borrow in unsecured markets, since lenders do not know the borrower’s
quality and therefore over-charge low-risk borrowers. Empirical work in the spirit of
DiFilippo et al. (2016) may clarify the relative importance of our proposed mechanism.
Liquidation values. Higher liquidation values decrease the degree of strategic com-
plementarity among fund managers for any given level of asset encumbrance, reducing
illiquidity at t = 1 and, hence, bank fragility. The banker, therefore, encumbers more
assets and increases investment with positive NPV. Lower fragility, in turn, reduces
the face value of unsecured debt required for investors to participate, increasing en-
cumbrance still further. Chen et al. (2010) identify strategic complementarities in
mutual fund investment and show that illiquid funds face greater strategic comple-
mentarities than liquid funds because of the higher costs associated with redemptions.
Bank resolution planning. A better management of encumbered assets in the
event of early closure implies that, for each unit of secured funding raised, the banker
needs to encumber fewer assets. Hence, more unencumbered assets are available to
meet the withdrawals from unsecured investors, which also decreases the face value
of unsecured debt. Taking both effects into account, encumbrance levels increase,dα∗∗
dλ> 0. That is, our model implies that banks with better resolution planning,
everything else equal, have higher shares of secured funding.
In its resolution plan, or living-will, a bank must make contingency arrangements
to ensure that access to shared-services between entities is not disrupted in the event
of its failure. These living-wills are typically highly confidential documents reviewed
16
by the relevant banking authorities, who may publicly comment on any deficiencies
they find.6 Such public information can be used by investors to price debt contracts.
Capital buffers. Our model suggest that the effect of an increase in bank capital
on asset encumbrance is ambiguous. There are two opposing effects. On the one hand,
more capital enables the bank to withstand larger balance sheet shocks and so lowers
fragility. While this “loss absorption” effect induces greater encumbrance, the bank
risks losing more of its own funds in bankruptcy. The result of such “greater skin in
the game” is to lower encumbrance. Figure 4 illustrates how these two opposing effects
induce a non-monotonic relationship between bank capital and asset encumbrance.
For a uniformly distributed shock, the loss absorption effect dominates (Lemma 3).
���� �������
����������������
Figure 4: Non-monotonic relationship between bank capital and encumbrance: anexample with R = 1.5, r = 1.2, ψ = 0.6, λ = 0.66, γ = 0.8, and the shock follows theLogistic distribution with mean -3 and scaling parameter 1.
Lemma 3. If the balance sheet shock is uniformly distributed over [−AL, AH ], then
the privately optimal level of asset encumbrance increases in bank capital.
Proof. See Appendix E.6For example, during the 2015 annual review of The Bank of New York Mellon Corporation’s
(BNYM) living-will, the Board of Governors of the Federal Reserve System and the Federal DepositInsurance Corporation communicated that BNYM had failed to identify critical shared-services thatwould be essential for surviving entities to have continual access.
17
4 Policy implications
We turn to normative implications of asset encumbrance. Our welfare criterion is
constrained efficiency defined by the allocation of a planner who takes the information
structure as given (that is, the incomplete information of fund managers about the
balance sheet adjustment). Since the pricing of debt is competitive and externalities
across banks are absent, the allocation in Proposition 3 is constrained efficient.
However, unsecured debt holders enjoy the benefits of explicit (or perhaps im-
plicit) public guarantee schemes in many jurisdictions. Such schemes, which usually
apply to retail deposits, often extend to unsecured wholesale depositors during times
of crisis. The analysis of these guarantee schemes has not yet incorporated its effects
on asset encumbrance on bank balance sheets. One concern is that while guarantees
are designed to reduce bank fragility ex post, they may induce excessive risk taking
ex ante. By externalizing the costs to the guarantor for servicing the guarantees when
the bank fails, banks have greater incentives to choose a (constrained) inefficiently
high level of asset encumbrance. Our model provides a natural framework to examine
these issues and explore different prudential regulations to correct this externality.
Let a fraction 0 < m < 1 of unsecured debt be guaranteed and the guarantor
be deep-pocketed (e.g., the government). Withdrawing guaranteed debt at t = 1 is a
weakly dominated for unsecured debt holders. Let the face value of guaranteed unse-
cured debt be DG. If the fraction ` of unsecured non-guaranteed debt is withdrawn
at t = 1, then the bank is illiquid whenever
R(1− α)I − A ≤ `(1−m)UDU
ψ, (11)
where the guarantee reduces the illiquidity of the banker at t = 1 for given funding
18
choices. Similarly, the bank is insolvent at t = 2 whenever
RI − A− `(1−m)UDU
ψ≤ SDS + (1− `)(1−m)UDU +mUDG, (12)
where the guarantee reduces the insolvency of the banker at t = 2, since guaranteed
unsecured debt is not subject to rollover risk at t = 1 and cheaper in any equilibrium.
The equilibrium in secured debt and guaranteed unsecured debt is as follows.
Backed by the guarantor, guaranteed debt is safe, D∗G = r, in any equilibrium. The
balance sheet adjustment has full support, so the bank is closed early at t = 1 with
positive probability for any given guarantee coverage. Moreover, since the guarantee
has no bearing on how encumbered assets are managed following early closure, Lemma
1 continues to hold, implying D∗S = r and S∗ = S∗(α) and I∗ = I∗(α).7
Figure 5 shows how introducing the guarantee affects run dynamics. The illiq-
uidity and insolvency thresholds, respectively, depend on the volume of withdrawals:
AIL(`) = R(1− α)I∗(α)− `(1−m)UDU
ψ, (14)
AIS(`) = R(1− αλ)I∗(α)− UDU(1−m)
(1 + `
[1
ψ− 1
]). (15)
Guarantees reduces the the responsiveness of both thresholds to changes in the with-
drawal volume, with a stronger effect on the illiquidity constraint. Moreover, the
insolvency threshold shifts outwards, while the illiquidity threshold pivots outwards.7The upper bound on the face value of unsecured debt is more relaxed with guarantees. Thus,
the conditions previously imposed continue to suffice for the illiquidity threshold to be more bindingthan the insolvency threshold, irrespective of guarantee coverage. To see this, the bound is
DU ≤ DU (m) ≡ 1
1−m
[Rα(1− λ)I∗(α)
U−mr
]. (13)
A higher coverage of the guarantee relaxes the condition in equation (13), since ∂DU (m)/∂m =(DU (m)− r)/(1−m) ≥ 0 because DU ≥ r in any equilibrium.
19
Balance Sheet Adjustment (A)
UnsecuredDebtW
ithdrawals(�)
IlliquidityThresholds
InsolvencyThresholds
m=0m>0
Figure 5: The impact of guarantee coverage on the run dynamics. The figure showsthe illiquidity and insolvency thresholds as a function of the withdrawal volume `before (solid lines) and after (dashed lines) the introduction of the guarantee.
Paralleling the previous arguments, and maintaining vanishing private noise, a
funds manager withdraws unsecured debt without guarantee at t = 1 whenever
A > A∗m ≡ R(1− α)I∗(α)− γ(1−m)UDU
ψ. (16)
Conversely, such debt is rolled over if A ≤ A∗m. The direct effect of the guarantee is to
increase the run threshold and therefore to decreases the incidences of early closure,∂A∗
m
∂m= γUDU
ψ> 0, which is the intended purpose of introducing a guarantee. However,
the introduction of a guarantee also affects the incentives of the banker to encumber
assets and, as a result, the pricing of unsecured debt. We proceed with these ex-ante
effects of the guarantee. The banker’s problem in the presence of the guarantee is:
α∗m ≡ maxα
πm(α) =
∫ A∗m(α) [
RI∗(α)−(1−m)UDU−S∗(α) r−mUr−A]dF (A). (17)
20
Proposition 5. Privately optimal encumbrance with guarantees. If the cov-
erage of the guarantee satisfies m ≤ m, then there exists a unique equilibrium for
the level of asset encumbrance and face value of unsecured debt. The introduction of
guarantees increases the level of asset encumbrance, α∗∗m > α∗∗.
Proof. See Appendix F.
The introduction of a guarantee has two intuitive effects, shown in Figure 6.
First, the stock of unsecured debt that may be withdrawn at t = 1 is reduced, which
lowers the incidence of runs. As a result, the banker has more incentives to encumber
assets, which shifts the encumbrance schedule α∗m(DU) outward. Second, unsecured
debt without a guarantee is repaid with greater probability, since the run threshold
increases. This, in turn, reduces the face value of unsecured debt and shifts the
participation constraint of investors D∗U(α) inward. In sum, the introduction of the
guarantee unambiguously increases the encumbrance level, α∗∗m , but has an ambiguous
effect on the equilibrium face value of unsecured debt, D∗∗m ≡ D∗U(α∗∗m ).
α�* (��)
α�* (��)
��* (α� � = �)
��* (α� � > �)
α�**
α�**
���� ����� �� ��������� ����
����������������
Figure 6: The privately optimal allocation before and after the introduction of aguarantee scheme. The parameter values are as in Figure 3 and coverage is m = 0.04.The introduction of the guarantee unambiguously increases asset encumbrance.
21
The banker does not account for the cost of servicing the guarantee. These
costs, however, are incurred by the guarantor: the amount is mUr upon failure of the
bank, which occurs with probability 1−F (A∗m). Thus, a planner’s asset encumbrance
schedule for a given face value of unsecured debt, α∗P (DU), accounts for the expected
cost of providing the guarantee:
α∗P (DU) ≡ maxα
πm(α)−[1− F (A∗m(α))
]mUr. (18)
While the asset encumbrance schedules of the banker and planner differ, the participa-
tion constraint of unsecured debt holders is the same and given by D∗U(α). Therefore,
any difference in allocations is due to differences in the respective asset encumbrance
schedules. The face value of unsecured debt chosen by the planner, denoted by D∗∗P ,
solves the fixed point problem D∗∗P ≡ D∗U(α∗P (D∗U)). Figure 7 shows the asset en-
cumbrance schedules and the pricing of unsecured non-guaranteed debt as well as the
privately and socially optimal levels of asset encumbrance and face values of debt.
��* (α� � > �)
α�* (��)
α�* (��)
��** ��
**
α�**
α�**
���� ����� �� ��������� ����
����������������
Figure 7: Constrained inefficiency after the introduction of the guarantee scheme: thesocially optimal asset encumbrance schedule lies below the privately optimal schedulefor any given face value of debt, α∗P (DU) ≤ α∗m(DU). Since the competitive pricing ofunsecured and non-guaranteed debt is the same for the planner and the banker, theprivately optimal levels of asset encumbrance and face value of debt are excessive.
22
Proposition 6. Constrained inefficiency. The decentralized equilibrium with guar-
antees is constrained inefficient. The privately optimal asset encumbrance and the face
value of unsecured non-guaranteed debt are excessive, α∗∗m > α∗∗P and D∗∗m > D∗∗P .
Proof. See Appendix G.
We next study the efficacy of several policy tools aimed at curbing the excessive
encumbrance of assets to restore constrained efficiency in the presence of guarantees.
In particular, we consider (i) a limit on asset encumbrance, α ≤ α, as an additional
constraint to thee banker’s problem; (ii) a lump-sum tax or transfer T ; and (iii) and
a linear tax (or subsidy) on asset encumbrance with rate τ > 0, where the revenue
raised (or required) may be transferred back (taxed) lump sum. Let α∗R(DU) denote
the banker’s optimal asset encumbrance schedule subject to regulation.
Proposition 7. Optimal regulation. In the presence of the guarantee, m > 0, a
planner achieves the constrained efficient allocation, (α∗∗P , D∗∗P ), by imposing:
a. a limit on asset encumbrance at α∗∗P ;
b. a transfer T = Umr to the banker at t = 2;
c. a linear tax on asset encumbrance imposed at t = 2, combined with a lump-sum
rebate of the generated revenue, T = ατ . The optimal rate is
τ ∗(DU) =(1− λz)RI∗(α∗P )Umrf(A∗m(α∗P ))
(1− λzα∗P )F (A∗m(α∗P )), (19)
which depends on the the face value of unsecured non-guaranteed debt.
d. a linear tax on asset encumbrance at t = 2 that is not contingent on the face
value of debt, combined with a lump-sum rebate of the generated revenue. That
is, there exists a unique rate τ ∗ > 0 such that α∗∗R (τ ∗) = α∗∗P and D∗∗R (τ ∗) = D∗∗P .
23
Proof. See Appendix H.
Figure 8 shows how a limit on asset encumbrance achieves constrained effi-
ciency. When the constraint α ≤ α∗∗P is added to the banker’s problem, the banker’s
constrained encumbrance schedule for any given face value of debt is
α∗R(DU) ≡
α∗m(DU) α∗m(DU) < α∗∗P
if
α∗∗P α∗m(DU) ≥ α∗∗P .
(20)
As a result, the constrained banker chooses the efficient level of asset encum-
brance and, therefore, unsecured non-guaranteed debt is also priced at the efficient
level. In sum, when the regulator controls the level of asset encumbrance directly at
t = 0, a limit on encumbrance achieves constrained efficiency. We proceed with tools
that do not require the regulator to observe the level of encumbrance at t = 0.
α�* (��)
α�* (��)
��* (α� � > �)
��**
α�**
���� ����� �� ��������� ����
����������������
Figure 8: A limit on asset encumbrance achieves constrained efficiency.
24
Figure 9 shows how a transfer to the banker at t = 2 achieves constrained effi-
ciency. When the banker receives the market value of non-guaranteed unsecured debt,
Umr, it internalizes the effect of its failure on the guarantor, that is the social cost of
bank failure. A benefit of this transfer is that the regulator does not need to observe
the banker’s level of asset encumbrance at all. However, the size of this transfer may
be quite large and its implementation may be subject to political constraints.
��* (α� � > �)
α�* (��)
α�* (��)
��** ��
**
α�**
α�**
���� ����� �� ��������� ����
����������������
Figure 9: A transfer of the market value of guaranteed unsecured debt achievesconstrained efficiency by aligning the socially and privately optimal asset encumbranceschedules. That is, it induces the banker to internalize the social cost of its failure.
We turn to revenue-neutral regulation. Specifically, we consider a linear tax
τ on asset encumbrance imposed at t = 2, combined with a lump-sum rebate of
T = τα. A higher tax rate reduces the privately optimal encumbrance of assets, since
the banker’s equity value contingent upon no early closure decreases in encumbrance.
When the tax rate can be contingent on the face value of unsecured non-guaranteed
debt, the optimal rate τ ∗(DU) ensures that the privately and socially optimal asset
encumbrance schedules are aligned for any face value of unsecured debt. This situation
parallels that of a transfer in Figure 9. As a result, constrained efficiency is attained.
25
When the tax rate cannot be contingent on the face value of debt, the privately
and socially optimal encumbrance schedules cannot generically be aligned, as shown
in Figure 10. Since a higher tax rate reduces asset encumbrance, there exists a unique
rate at which the constrained efficient allocation is achieved. Basically, the privately
optimal asset encumbrance schedule has to be shifted in by the marginal tax on
encumbrance by enough to cross through the constrained efficient allocation.
α�* (��)
α�* (��)
��* (α� � > �)
��**
α�**
���� ����� �� ��������� ����
����������������
Figure 10: A linear tax on asset encumbrance combined with a lump-sum rebateachieves constrained efficiency.
We have imposed the linear tax on asset encumbrance and its lump-sum rebate
at t = 2. This may be viewed as requiring the least information by the regulator,
since asset encumbrance has to be observed at t = 2 only. Our timing assumption
is without loss of generality, however. If a tax and lump-sum rebate scheme were
imposed at t = 1, the banker would internalize the effect of asset encumbrance on
heightened illiquidity and thus a greater incidence of early closure, which induces a
banker to encumber fewer assets. The credibility of a regulatory measure perceived
to heighten fragility at t = 1 precisely when a bank may be illiquid can be debated.
26
If a tax and lump-sum rebate scheme were imposed at t = 0, the available re-
sources for investment would be affected. The banker would internalize the effect of
great encumbrance on reduced profitable investment, which also exacerbates illiquid-
ity at t = 1. Both channels induce the banker to encumber fewer assets. The full
rebate of the tax revenue generated ensures that the actual investment size at t = 0,
actual illiquidity at t = 1, or actual equity value at t = 2 is unchanged, while aligning
the incentives of the banker with that of the planner.
Relation to current regulation Several jurisdictions have introduced measures
to restrict asset encumbrance. These measures comprise limits on encumbrance (Aus-
tralia and New Zealand), ceilings on the amount of secured funding by banks (Canada
and US), and including encumbrance levels in the calculation of surcharges on the
premia paid to the deposit insurance fund (Canada).
These caps or ceilings on encumbrance are comparable to the limits on encum-
brance discussed here. In practice, these limits are applied as a percentage of the
bank balance sheet, ranging from 4% to 8%. Finally, the ongoing Canadian policy
debate about surcharges on asset encumbrance is similar to a tax on asset encum-
brance. The model suggests that a constrained efficient outcome is obtained only
when the surcharge is accompanied by a lump-sum transfer from the planner to the
banker. The absence of such a transfer, however, would lead to an equilibrium with
heightened fragility for the bank.
27
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29
A Proof of Proposition 1
This proof is in three steps. First, we show that the dominance regions at the rollover
stage based on the illiquidity threshold are well defined for any choice of bankers and
face values of debt. If the balance sheet shock were common knowledge, the rollover
behavior of fund managers would be characterized by multiple equilibria, as shown
in Figure 11. If no unsecured debt is rolled over, ` = 1, the bank is liquid at t = 1
whenever the shock below a lower dominance bound A ≡ R(1 − α)I − UDUψ
. For
A < A, it is a dominant strategy for fund managers to roll over. If ` = 0, the bank is
illiquid whenever the shock is above an upper dominance bound A ≡ R(1− α)I. For
A > A, it is a dominant strategy for managers not to roll over.
- A
A A
Liquid Liquid / Illiquid Illiquid
Roll over Multiple equilibria Withdraw
Figure 11: Tripartite classification of the balance sheet shock
Second, in any rollover stage, it suffices to establish the optimality of threshold
strategies for sufficiently precise private information. Morris and Shin (2003) and
Frankel et al. (2003) show that only threshold strategies survive the iterated deletion
of strictly dominated strategies. Specifically, we consider the case of vanishing private
noise, ε → 0. Therefore, each fund manager i uses a threshold strategy, whereby
unsecured debt is rolled over if and only if the private signal about the balance sheet
shock is below some signal threshold, xi < x∗.
Third, we characterize this threshold equilibrium. For a given realization A ∈
[A,A ], the proportion of fund managers who do not roll over unsecured debt is:
`(A, x∗
)= Prob
(xi > x∗
∣∣A) = Prob (εi > x∗ − A) = 1−G(x∗ − A
). (21)
30
A critical mass condition states that illiquidity occurs when the shock equals A∗,
where the proportion of managers not rolling over is evaluated at A∗:
R(1− α)I − A∗ = `(A∗, x∗
)UDU
ψ. (22)
The posterior distribution of the shock conditional on the private signal is derived
using Bayes’ rule. An indifference condition states that the manager who receives
the threshold signal xi = x∗ is indifferent between rolling and not rolling over:
γ = Pr (A < A∗|xi = x∗) . (23)
Using the definition of the private signal xj = A+ εj, the conditional probability is
1− γ = Pr (A ≥ A∗|xi = x∗) = Pr (A ≥ A∗|xi = x∗ = A+ εj) , (24)
= Pr (x∗ − εj ≥ A∗) = Pr (εj ≤ x∗ − A∗) = G(x∗ − A∗
). (25)
The indifference condition implies that x∗−A∗ = G−1(1−γ
). Inserting this expression
into `(A∗, x∗
), the proportion of managers who do not roll over when the shock equals
the threshold level A∗ is perceived by the indifferent manager to be:
`(A∗, xi = x∗
)= 1−G
(x∗ − A∗
)= 1−G
(G−1
(1− γ
))= γ. (26)
The run threshold A∗ stated in Proposition 1 follows. The run threshold varies with
the face value of unsecured debt and asset encumbrance according to:
dA∗
dDU
= −γUψ
< 0, (27)
dA∗
dα= R (λz − 1)
I∗(α)
1− αλz. (28)
31
B Proof of Propositions 2
The banker’s problem is given in (8). The total derivative dπdα, which takes indirect
effects via A∗(α) and S∗(α) into account, yields dπdα
= RI∗(α)1−αλz f(A∗)G(α), where
G(α) =F (A∗)
f(A∗)λ(z − 1)− (1− λz)
[RI∗(α)α(1− λ) + UDU
(γ
ψ− 1
)]. (29)
If an interior solution 0 < α∗ < 1 exists, it is given by G(α∗) = 0. This solution is a
local maximum, since
dG
dα=dF (A∗)f(A∗)
dA∗︸ ︷︷ ︸+
dA∗
dα︸︷︷︸−
λ(z − 1)− (1− λz)R(1− λ)I∗(α)
1− αλz< 0, (30)
where the first term is positive since f(A) satisfies has decreasing reverse hazard rate.
Using the implicit function theorem, we obtain dα∗
dDU< 0 for the interior solution since
dG
dDU
=dF (A∗)f(A∗)
dA∗dA∗
dDU
λ(z − 1)− (1− λz)U
(γ
ψ− 1
)< 0 (31)
where we used γ > ψ. Hence, the weak inequality dα∗
dDU≤ 0 follows for the entire
schedule of asset encumbrance.
We next study whether the solution toG(α∗) = 0 is interior. An interior solution
requires two conditions, G(α = 0) > 0 and G(α = 1) < 0, which we consider in turn.
First, evaluating the implicit function when there is no encumbrance, α = 0, yields
F (A∗(0))
f(A∗(0))λ(z − 1)− (1− λz)
(γ
ψ− 1
)UDU , (32)
which is strictly decreasing in DU . To ensure α∗ > 0 requires the face value of
unsecured debt to be bounded from above, DU < DU , where DU is uniquely and
32
implicitly defined by
F (R(U + E)− γψUDU)
f(R(U + E)− γψUDU)
λ(z − 1)− (1− λz)
(γ
ψ− 1
)UDU = 0. (33)
Second, evaluating the implicit function when all assets are encumbered, α = 1,
yields
F (A∗(1))
f(A∗(1))λ(z − 1)− (1− λz)
[R(1− λ)(U + E)
1− λz+
(γ
ψ− 1
)UDU
], (34)
which is also decreasing in DU . To ensure that α∗ < 1, we need that the expression
in Equation (34) is strictly negative. This, in turn, requires that the face value
of unsecured debt is bounded from below, DU > DU , where DU is uniquely and
implicitly defined by
F (− γψUDU)
f(− γψUDU)
λ(z − 1)− (1− λz)
[R(1− λ)(U + E)
1− λz+
(γ
ψ− 1
)UDU
]= 0 . (35)
It readily follows that DU < DU .
C Proof of Proposition 3
The proof is in five steps. First, we ensure that the equilibrium level of asset encum-
brance is interior, α∗∗ ∈ (0, 1). Since α = 0 implies DU = 0, there is no equilibrium
consistent with our supposition DU ≤ DU , so it must be the case that α∗∗ > 0 (we
verify the supposition below). If α = 1, then the run threshold and value of an
unsecured debt claim are A∗(1) = −γDUψ
and V (1, DU) = DUF (A∗(1)), respectively.
The value of the unsecured debt claim attains a maximum at DU = Dmax, which is
33
uniquely and implicitly defined byF(− γψDmax)f(− γψDmax)
− γψDmax = 0. Since V (α,DU) decreases
in α, if we impose that the risk-free rate must satisfy r > r˜ ≡ V (1, Dmax), then any
solution for the equilibrium level of asset encumbrance is interior, if it exists.
Second, we show that the equilibrium face value of unsecured debt satisfied
D∗U > r. To this end, note that while ∂V∂DU
has an ambiguous sign, in general, the
derivative evaluated along the asset encumbrance schedule is
∂V
∂DU
∣∣∣∣α∗(DU )
= f(A∗)
[1− λzλ(z − 1)
R(1− λ)α∗I∗(α∗) + β0UDU
], (36)
which is non-negative whenever β0 ≡ 1−λzλ(z−1)
(γψ− 1)− γ
ψ≥ 0 that is equivalent to
γ ≥ γ˜ ≡ 1−λz1−λz−λ(z−1)
ψ. Having established conditions under which V increases in DU ,
at least in the vicinity of the asset encumbrance schedule, it readily follows that DU =
r always violates the participation constraint, V (DU = r) = rF (A∗(DU = r)) < r.
Consequently, we must have that D∗U > r.
Third, we establish that, at least locally, the intersection between the asset
encumbrance schedule and market clearing condition for unsecured debt holders yields
an unique joint equilibrium. From Proposition 2 we know that α∗(DU) decreases in
DU . Also, from the second step of this proof, we provided sufficient conditions that
ensure, at least in the vicinity of the the asset encumbrance schedule, the market-
implied face value of unsecured debt, D∗U(α), increases in α. It immediately follows
that there is at most only one intersection between these two curves, thus establishing
uniqueness of the joint equilibrium.
Fourth, we need to demonstrate that the equilibrium specified above exists. To
this end, define T (DU) = r/F (A∗(α∗(DU), DU)) as a mapping from the set U of face
values of unsecured debt into itself. If U is a closed an compact set, then by Brouwer’s
34
fixed-point theorem, there exists at least one fixed-point for the mapping. The lower
bound on DU is, by definition, r. For the upper bound, note that if the banker could
pledge all assets to unsecured investors, then DU ≤ RI∗(α) − A. If we truncate the
shock distribution at some arbitrary −AL < 0, we have a well defined upper bound
on DU .
As the fifth and final step we verify our prior supposition that DU ≤ DU ≡
αRI∗(α)(1 − λ). Denoting the run threshold, evaluated at DU = DU , by A∗(α) ≡
A∗(DU(α)), we can express this by
A∗(α) = RI∗(α)
[1− α
(1 +
γ
ψ(1− λ)
)]. (37)
This run threshold decreases in the level of asset encumbrance:
dA∗
dα= − RI∗(α)
1− αλz
[1− λz +
γ
ψ(1− λ)
]< 0. (38)
Next, suppose we define α∗ as the equilibrium level of asset encumbrance eval-
uated at DU = DU(α∗). This is implicitly defined by
F (A∗(α∗))
f(A∗(α∗))=
1− λzλ(z − 1)
γ(1− λ)
ψRα∗I∗(α∗) . (39)
Since the left-hand side decreases in α, while the right-hand side increases, if follows
that α∗ is uniquely defined.
We can now translate the condition DU ≤ DU into a condition for the pricing
of the unsecured debt claim:
r ≤ V (α∗, DU(α∗)). (40)
35
A stricter sufficient condition is to demand that r ≤ V (1, DU(1)). If we plug in the
condition for α∗ into this condition, we obtain
r ≤ ψλ(z − 1)
γ(1− λz)
F(−R(1+E)
1−λzγψ
(1− λ))2
f(−R(1+E)
1−λzγψ
(1− λ)) , (41)
where right-hand side decreases in the bank’s initial capital, E. This suggests that
by imposing an upper bound, E ≤ E, on the bank’s capital will ensure that, in
equilibrium, D∗U ≤ DU(α∗). The upper bound on capital is implicitly defined as
ψλ(z − 1)
γ(1− λz)
F(−R(1+E)
1−λzγψ
(1− λ))2
f(−R(1+E)
1−λzγψ
(1− λ)) − r = 0. (42)
D Proof of Proposition 4
This proofs is in two steps. First, we show the effect of a parameter on the schedule
of asset encumbrance α∗(DU). Second, we show that this direct effect is reinforced
by the indirect effect via the equilibrium cost of unsecured debt D∗U .
Direct effects Regarding the direct effect via α∗(DU), we take DU as given and
use the implicit function theorem, whereby dα∗
dy= −
dGdydGdα
for y ∈ {γ, r, R, ψ,E, F (·)}:
36
dG
dγ=
dF (A∗)f(A∗)
dA∗dA∗
dγλ(z − 1)− (1− λz)U
DU
ψ< 0, (43)
dG
dr=
dF (A∗)f(A∗)
dA∗dA∗
drλ(z − 1)− F (A∗)
f(A∗)λz
r(44)
− λz
r
[R(1− λ)α(1− α)I∗(α)
1− αλz+DU
(γ
ψ− 1
)]< 0
dG
dR=
dF (A∗)f(A∗)
dA∗dA∗
dRλ(z − 1) +
F (A∗)
f(A∗)
λ
r(45)
+λ
rDU
(γ
ψ− 1
)+ (1− λ)αI∗(α)
{λz − (1− λz)
1− αλz
}.
The expression in Equation (45) has an ambiguous sign in general. Therefore,
we evaluate this derivative at α∗ by substituting F (A∗)/f(A∗) from the first-order
condition in Equation (9). Grouping terms, we obtain that dG/dR > 0 as long as
(1− λz)
[z
1− z− 1
1− α∗λz
]+ λz > 0 . (46)
The condition in Equation (46) is toughest to satisfy for α∗ = 1. Evaluating it at this
bound, we get the sufficient condition (1 − λz)[
zz−1− 1]
+ λz > 0, which requires
z > z − 1, which is always true.
For the derivative with respect to ψ, we obtain
dG
dψ=dF (A∗)f(A∗)
dA∗dA∗
dψλ(z − 1) + (1− λz)UDU
γ
ψ2> 0. (47)
37
The derivative with respect to λ yields
dG
dλ=
dF (A∗)f(A∗)
dA∗dA∗
dλλ(z − 1) +
F (A∗)
f(A∗)(z − 1) + zRI∗(α)α(1− λ)
+ zUDU
(γ
ψ− 1
)+
(1− λz)(1− αz)
1− αλzRαI∗(α) > 0 . (48)
For the derivative with respect to E, we obtain
dG
dE=
dF (A∗)f(A∗)
dA∗dA∗
dEλ(z − 1)− 1− λz
1− αλzR(1− λ)α Q 0. (49)
Finally, suppose that the balance sheet shock distribution F stochastically dominates
the distribution F according to the reverse hazard rate. This implies that
f
F≥ f
F, (50)
or, equivalently, F/f ≥ F /f . Let G(α∗) = 0 denote the implicit function defining
the privately optimal level of asset encumbrance, α∗, under the balance sheet shock
distribution F . Thus, G(α) ≤ G(α) for all levels of encumbrance. Since dG/dα < 0
and dG/dα < 0, the privately optimal levels of encumbrance satisfy α∗ ≥ α∗.
Indirect effects The indirect effects arise from the equilibrium face value of un-
secured debt. For any given level of asset encumbrance, it is given by the implicit
function J(α,D∗U) = 0 where
J ≡ −r +DUF (A∗(α,DU)) (51)
38
Using the implicit function again, and noting that ∂J∂DU
∣∣∣α∗> 0, we obtain the desired
reinforcing effects since:
∂J
∂R= DUf(A∗)
dA∗
dR> 0 (52)
∂J
∂ψ= DUf(A∗)
dA∗
dψ> 0 (53)
∂J
∂λ= DUf(A∗)
dA∗
dλ> 0 (54)
∂J
∂γ= DUf(A∗)
dA∗
dγ< 0 (55)
∂J
∂E= DUf(A∗)
dA∗
dE> 0 (56)
∂J
∂r= −1 +DUf(A∗)
dA∗
dr< 0. (57)
Finally, note that an improvement in the distribution of the balance sheet shock also
increases J . Taking the direct and indirect effects together, we arrive at the total
effects reported in Proposition 4.
E Proof of Lemma 3
If the balance sheet shock is uniformly distributed in the interval [−AL, AH ], then
the equilibrium level of asset encumbrance, for a given DU is
RI∗(α∗)
[1− α∗
{1 +
(1− λ)(1− λz)
λ(z − 1)
}]= UDU
{γ
ψ+
(γ
ψ− 1
)(1− λz)
λ (z − 1)
}− AL ,
(58)
which uniquely defines α∗. It readily follows that dα∗/dE > 0 by the implicit function
theorem, since the left-hand side of Equation (58) decreases in the level of asset
encumbrance and increases in bank capital.
39
F Proof of Proposition 5
To derive the private optimum with guarantees, we first determine the conditions
under which the banker’s asset encumbrance schedule, α∗m(DU), has a unique interior
solution. Second, we show that there exists a unique crossing between the banker’s
asset encumbrance schedule and the unsecured debt holder’s participation constraint
that yields the overall equilibrium. Finally, we derive how the optimal level of asset
encumbrance depends on the coverage of the guarantee.
Banker’s asset encumbrance schedule. Taking the derivative of the banker’s
objective function with respect α, the optimal level of encumbrance, for a given face
value of unsecured debt, is implicitly defined by Gm(α∗m(DU)) = 0, where
Gm(α) ≡ F (A∗m)
f(A∗m)λ(z−1)−(1−λz)
[Rα(1− λ)I∗(α) + (1−m)UDU
(γ
ψ− 1
)−mUr
].
(59)
The solution is a local maximum since
dGm
dα=dF (A∗
m)f(A∗
m)
dA∗m︸ ︷︷ ︸+
dA∗mdα︸ ︷︷ ︸−
λ(z − 1)− (1− λz)R(1− λ)I∗(α)
1− αλz< 0 . (60)
For an interior solution, α∗m(DU) ∈ (0, 1), we require: (i) Gm(α = 0) > 0 and
(ii) Gm(α = 1) < 0. We consider each in turn. First,
Gm(α = 0) =F (A∗m(0))
f(A∗m(0))λ(z − 1)− (1− λz)
[(1−m)UDU
(γ
ψ− 1
)−mUr
], (61)
which decreases in DU . Therefore, as long as DU < DU(m), then α∗(DU) > 0. The
upper bound is implicitly defined by
40
F(R(U + E)− γ
ψ(1−m)UDU
)f(R(U + E)− γ
ψ(1−m)UDU
) λ(z−1)−(1−λz)
[(1−m)UDU
(γ
ψ− 1
)−mUr
]= 0 ,
(62)
where DU(m) > DU(0), for all m > 0. Second, to ensure that the banker never wants
to encumber all assets, we require that Gm(α = 1) < 0, where
Gm(α = 1) =F (A∗m(1))
f(A∗m(1))λ(z − 1)− (1− λz)
[R(1− λ)(U + E)
1− λz
+ (1−m)UDU
(γ
ψ− 1
)−mUr
], (63)
which also decreases in DU . As such, as long as DU > DU(m), then the banker never
wants to encumber all assets. The lower bound is implicitly defined by
F(− γψ
(1−m)UDU
)f(− γψ
(1−m)UDU
) λ(z − 1) − (1− λz)
[R(1− λ)(U + E)
1− λz
+ (1−m)UDU
(γ
ψ− 1
)−mUr
]= 0 , (64)
where DU(m) > DU(0) for all m > 0. It also follows that DU(m) < DU(m) for all m.
This establishes the uniqueness of the banker’s asset encumbrance schedule.
Participation constraint of unsecured and non-guaranteed debt holders.
The value of the unsecured debt contract without a guarantee is V (α,DU ,m) =
DUF (A∗m). Since the debt is competitively priced, the face value of unsecured debt
in equilibrium satisfies
r = D∗UF (A∗m(α,D∗U)). (65)
41
Following the previous reasoning laid out in Appendix C, we need to show the fol-
lowing: (i) The intersection between the banker’s asset encumbrance schedule and
the unsecured investors’ participation constraint, if it exist, must admit a solution
for the level of asset encumbrance satisfies 0 < α∗∗ < 1; (ii) the equilibrium face
value of unsecured debt is strictly greater than the risk-free rate, r; (iii) the existence
and (iv) uniqueness of the intersection between the asset encumbrance schedule and
participation constraint, and finally (v) that the face value of unsecured debt is low
enough to ensure that the illiquidity condition is more binding than the insolvency
condition for the bank. We consider each in turn.
Step 1. If α∗∗ = 0, then this implies that DU(m) = 0, for all m. This cannot
be a solution since it violates are supposition that DU ≤ DU(m), where DU ≥ r.
Therefore α∗∗ = 0 cannot be an equilibrium. Next, suppose that α∗∗ = 1. In this
case, the value of the debt holder’s claim is V (1, DU ,m), which is concave in DU .
The claim attains its maximum value for DU = Dmax, which is implicitly defined
byF(− γψ (1−m)UDmax)f(− γψ (1−m)UDmax)
− γψ
(1 −m)UDmax = 0. Since the investor’s claim decreases in
the level of asset encumbrance, it follows that if the free rate must satisfy r > r˜m ≡V (1, Dmax,m), then any solution for the equilibrium level of asset encumbrance is
interior, if it exists.
Step 2. We wish to establish that D∗U > r. To this end, note that while ∂V∂DU
has
an ambiguous sign, in general, the derivative evaluated along the asset encumbrance
schedule is
∂V
∂DU
∣∣∣∣α∗(DU )
= f(A∗)
[1− λzλ(z − 1)
R(1− λ)α∗I∗(α∗) + β0UDU
], (66)
which is non-negative whenever β0 ≡ (1−m)[
1−λzλ(z−1)
(γψ− 1)− γ
ψ
]−m r
DU≥ 0. This
42
is equivalent to requiring that γ ≥ γ˜ ≡ 1−λz1−λz−λ(z−1)
ψ and m ≤ m ≡1−λzλ(z−1)(
γψ−1)− γψ
( γψ−1)[ 1−λzλ(z−1)
−1]. It
now readily follows that DU = r always violates the participation constraint, V (DU =
r) = rF (A∗(DU = r)) < r. Consequently, we must have that D∗U > r.
Steps 3-4. We establish the existence and uniqueness of the equilibrium, which
is given by the intersection of the asset encumbrance schedule and participation con-
straint. These steps are identical to the those in Appendix C and are thus omitted.
Step 5. We verify our prior supposition that DU ≤ DU(m). Denoting the run
threshold, evaluated at DU = DU(m), by A∗m(α) ≡ A∗m(DU(α)), this is given by
A∗m(α) = RI∗(α)
[1− α
(1 +
γ
ψ(1− λ)
)]+γ
ψmUr , (67)
which decreases in the level of asset encumbrance.
Next, suppose α∗m is the equilibrium level of asset encumbrance evaluated at
DU = DU(m, α∗m). This is implicitly defined by
F (A∗m(α∗m))
f(A∗m(α∗m))=
1− λzλ(z − 1)
γ
ψ[Rα∗m(1− λ)I∗(α∗m)−mUr] . (68)
Since the left-hand side decreases in α, while the right-hand side increases, if follows
that α∗ is uniquely defined. We can now translate the condition DU ≤ DU(m) into a
condition for the pricing of the unsecured debt claim:
r ≤ V (α∗m, DU(m, α∗m),m). (69)
A stricter sufficient condition is to demand that r ≤ V (1, DU(m, 1),m). If we plug in
the condition for α∗ into this condition, we obtain
43
r ≤ ψλ(z − 1)
(1−m)γU(1− λz)
F(γψ
[mUr − (1− λ)R(U+E)
1−λz
])2
f(γψ
[mUr − (1− λ)R(U+E)
1−λz
]) , (70)
where right-hand side decreases in the bank’s initial capital, E. This suggests that
by imposing an upper bound, E ≤ Em, on the bank’s capital will ensure that, in
equilibrium, D∗U ≤ DU(m,α∗m). The upper bound on capital is implicitly defined as
ψλ(z − 1)
(1−m)γU(1− λz)
F(γψ
[mUr − (1− λ)R(U+Em)
1−λz
])2
f(γψ
[mUr − (1− λ)R(U+Em)
1−λz
]) − r = 0. (71)
Comparative statics. Finally, we derive how the optimal level of asset encum-
brance depends on the guarantee coverage. As before, we decompose these into
direct effects on the banker’s asset encumbrance schedule, and indirect effects from
the equilibrium face value of unsecured debt.
Direct effects. The derivative of the implicit function, Gm(α) with respect to
the guarantee coverage is
dGm
dm=dF (A∗
m)f(A∗
m)
dA∗m
dA∗mdm
λ(z − 1) + (1− λz)U
[DU
(γ
ψ− 1
)+ r
]> 0 . (72)
Since we previously established that dGm/dα < 0, it follows from the implicit function
theorem that dα∗m/dm > 0 for any given DU . Thus, the banker’s asset encumbrance
schedule shifts outwards following an increase in the guarantee coverage.
Indirect effects. These arise from the equilibrium face value of unsecured debt,
which is given by the implicit function Jm(α,D∗U(m)) = 0 where
Jm ≡ −r +DUF (A∗m(α,DU)) (73)
44
Using the implicit function again, and noting that dJmdDU
∣∣∣α∗> 0, we obtain
dJmdm
= DUf(A∗m)dA∗mdm
> 0 . (74)
Therefore, the face value of unsecured debt is reduced as the coverage of the guarantee
raises. This, in turn, reinforces the banker’s incentives to encumber more assets.
G Proof of Proposition 6
Taking the derivative of the planner’s objective function with respect to the level of
asset encumbrance, we obtain the first-order condition
dπmdα
+ f(A∗m)dA∗mdα
mUr = 0 . (75)
The planner’s asset encumbrance schedule, α∗P (DU), is given by GP (α∗P ) = 0, where
GP (α) ≡ F (A∗m)
f(A∗m)λ(z − 1)− (1− λz)
[Rα(1− λ)I∗(α) + (1−m)UDU
(γ
ψ− 1
)],
(76)
which is decreasing in α. As before, α∗P (DU) > 0 as long as DU < DUP (m), which is
implicitly defined by
F(R(U + E)− γ
ψ(1−m)UDUP
)f(R(U + E)− γ
ψ(1−m)UDUP
) λ(z−1)− (1−λz)
[(1−m)UDUP
(γ
ψ− 1
)]= 0 .
(77)
45
It follows that DUP (m) < DU(m) for all m. Next, α∗P (DU) < 1 as long as DU > DUP ,
which is given by the solution to
F(− γψ
(1−m)UDUP
)f(− γψ
(1−m)UDUP
) λ(z − 1) − (1− λz)
[R(1− λ)(U + E)
1− λz
+ (1−m)UDUP
(γ
ψ− 1
)]= 0 , (78)
where DUP (m) < DU(m) for all m.
Comparing equation (76) to the implicit function that provides the banker’s
encumbrance schedule in equation (59), we have that GP (α∗m) < 0 for any given DU .
Hence, for any given m and DU ∈ [DU(m), DUP (m)], α∗m(DU) > α∗P (DU) follows.
For the rest of the normative analysis, we assume that the asset encumbrance
schedule yields interior solutions that are local maximums. Bounds similar to those
above, and those derived in the previous section, can be derived analogously.
H Proof of Proposition 7
With a limit on asset encumbrance, the banker’s constrained problem is given by
α∗m ≡ maxα∈[0,α∗∗
P ]πm(α) =
∫ A∗m(α) [
RI∗(α)− (1−m)UDU −S∗(α) r−mUr−A]dF (A).
(79)
Since the banker’s problem is concave around α∗m, its unconstrained optimum, its
marginal profit at α = α∗∗P is strictly positive. It follows that the constrained optimum
is the encumbrance schedule stated in equation (20).
For the tax and transfer schemes, suppose that the regulator imposes a linear
46
tax τ > 0 on asset encumbrance at t = 2 combined with a lump-sum transfer T .
Then, the privately optimal asset encumbrance schedule subject to this regulation is
given by the implicit function GR(α), where α∗R(DU) solves GR(α∗R(DU)) ≡ 0:
GR(α) ≡ F (A∗m)
f(A∗m)λ(z − 1)− F (A∗m)
f(A∗m)
1− αλzRI∗(α)
τ + · · · (80)
−(1− λz)
[Rα(1− λ)I∗(α)− τα + T + (1−m)UDU
(γ
ψ− 1
)−mUr
].
Consider the pure transfer scheme, τ = 0. Comparing GR(α)|τ=0 with GP (α) in
equation (76), the two asset encumbrance schedules are the same whenever T = Umr.
Next, consider a revenue-neutral scheme, T = τα for all α. We evaluate α∗R(DU)
at T = τα and solve for the optimal tax. Equalizing α∗R = α∗P – that the equality of
the socially optimal asset encumbrance schedule and the privately optimal schedule
subject to regulation – we obtain τ ∗ as stated in equation (19). This tax rate depends
on the face value of unsecured and non-guaranteed debt.
Finally, consider a revenue-neutral scheme with linear tax on encumbrance that
is independent of DU . Using the implicit function theorem, we obtain that dα∗R
dτ< 0
since dGRdα
< 0 by optimality and
dGR
dτ
∣∣∣∣T=τα
= −1− αλzRI∗(α)
F (A∗m)
f(A∗m)< 0. (81)
47
