Carlos Corona
CMU
Lin Nan
Purdue
Gaoqing Zhang
Minnesota
SWUFE workshop
June 14th, 2017

Dodd-Frank Act (DFA) requires the Fed to conduct stress tests
annually on large bank holding companies (BHC)

stress tests evaluate a bank’ resilience to adverse scenarios

DFA mandates the disclosure of stress tests
Stress test report, 2014: “The Federal Reserve believes that disclosure of
stress tests results provides valuable information to market participants and
the public, enhances transparency, and promotes market discipline.”

we examine the effect of disclosing stress tests on banks’ risk
decisions in the prospect of a bailout by regulators

by 2009, FED had committed $7.77 trillion for bailouts

it was claimed that the bailouts avoided the negative
externalities associated with a massive bank failure

banks may have anticipated bailouts and have incentives to
coordinate to make concurrent failures more likely

we formally examine the role of disclosing stress tests in
facilitating risk-taking coordination

how does stress test disclosure affect the coordination among
banks in taking risk?

how does stress test disclosure affect banks’ risk, the
likelihood of bailouts and the rate of bank failure?

a coordination role of stress test disclosure
o in a strong banking system, disclosure disciplines risk
o In a vulnerable banking system, disclosure aggravates risk

disclosing stress tests increases average risk level and bailouts,
unless bank failure externalities are sufficiently severe

disclosing stress tests increases expected bank failures
stress test disclosure
 disclosure by financial institutions
 regulatory bailouts

The model

a continuum of risk neutral banks, 𝑖 ∈ 0,1 , each endowed
with a loan with binary outcome: success or failure

macroeconomic state 𝜔 ∈ 𝐺, 𝐵

banks’ type 𝜃𝑖 ∈ 𝐿, 𝐻 .
o 𝜃𝑖 determines bank 𝑖’s outcome at 𝜔 = 𝐵.
o stress tests report {𝜃𝑖 }

each bank 𝑖 takes a loan risk decision: 𝑆𝑖 ∈ [0,1]

a regulator decides whether to bail out banks or not
0
Each bank 𝑖
observes and
reports 𝜃𝑖 ∈ {𝐿, 𝐻}
to regulator.
{𝜃𝑖 }𝑖∈[0,1] disclosed
(disclosure scenario)
1
Each bank 𝑖
takes risk
𝑆𝑖 ∈ [0,1].
2
Macro state
𝜔 ∈ {𝐺, 𝐵}
realized
Loan outcomes
realized
Regulator decides
bailout

loan payoffs depend on:
o macro state 𝜔 ∈ {𝐺, 𝐵}, with
Pr 𝐺 = 𝑞,
o bank type 𝜃𝑖 ∈ {𝐻, 𝐿}, with
Pr 𝐿 = 𝑝 ~ 𝑈[0,1],
1
1 − 𝑆𝑖
o loan success/failure, with Pr 𝑠𝑢𝑐𝑐𝑒𝑠𝑠 =


𝑖𝑓 𝜔 = 𝐺
𝑖𝑓𝜔 = 𝐵
loan payoffs
𝜃𝑖 = 𝐻
𝜃𝑖 = 𝐿
Project Success
𝑆𝑖
𝑆𝑖
Project Failure
𝐾
0
𝐾: minimum capital a bank needs to survive.
Bank fails without bailout

regulator decides whether to bail out all banks or none

not bailing out a bank implies assuming the cost of bank failure
externalities
o social cost of 𝑛 ∈ [0,1] bank failures is 𝐶 𝑛
o 𝐶 0 = 0, 𝐶 1 = ∞, 𝐶 ′ 0 = 0, 𝐶 ′ 𝑛 > 0, 𝐶 ′′ 𝑛 > 𝑜 for 𝑛 ∈ (0,1]

bailing out a bank implies injecting a capital 𝐾 to the bank and
assuming an additional social cost of 𝜆 𝐾
o 𝐾 > 0, capital injected in each bailed out bank
o 𝜆 > 0, social cost per unit of injected capital
𝜃𝑖 = H
𝜃𝑖 = L
𝑝
1−𝑝
Bank 𝑖
𝑆𝐿
𝜔=B
𝜔=G
𝑞
Regulator
Bailout
No Bailout
0
𝜔=B
1−𝑞
𝑆𝑢𝑐𝑐
1 − 𝑆𝐿
𝑆𝐿
𝐾
𝑞
𝜔=G
𝑆𝑢𝑐𝑐
𝐹𝑎𝑖𝑙
0
1−𝑞
𝑆𝐻
1 − 𝑆𝐻
𝑆𝐿
𝑆𝐿
𝑆𝐻
𝑆𝐻
𝐹𝑎𝑖𝑙
𝑆𝐻
𝐾
The equilibrium

if 𝑛 banks fail, the regulator chooses between
o allowing banks to fail cost of 𝐶(𝑛)
o bailing out all failing banks
Cost
cost of 𝑛𝜆𝐾
𝐶(𝑛)
𝑛𝜆𝐾
0

𝑛
𝑛
regulator bails out failing banks if and only if 𝑛 ≥ 𝑛, where 𝑛
satisfies 𝐶 𝑛 = 𝑛𝜆𝐾.

high-type bank,
Π𝐻 (𝑆𝐻 ) = 𝑞𝑆𝐻 + 1 − 𝑞

1 − 𝑆𝐻 𝑆𝐻 + 𝑆𝐻 𝐾
low-type bank,
Π𝐿 (𝑆𝐿 ) = 𝑞𝑆𝐿 + 1 − 𝑞
where, 𝑛 = 𝑝𝑆𝐿
1 − 𝑆𝐿 𝑆𝐿 + 𝑆𝐿 𝐾 Pr(𝑛 ≥ 𝑛)

regulator chooses risk (𝑆𝐻 , 𝑆𝐿 ) to maximize welfare:
bank payoffs,
1 − 𝑝 Π𝐻 (𝑆𝐻 ) + 𝑝 Π𝐿 (𝑆𝐿 )
minus social costs,
−(1 − q){Pr(𝑛 < 𝑛)𝐶 𝑛 + Pr(𝑛 ≥ 𝑛)𝑛 1 + 𝜆 𝐾},

regulator chooses,
𝑆𝐿𝐹𝐵
1
1
𝐾
𝐹𝐵
<
< 𝑆𝐻 =
+
2(1 − 𝑞)
2(1 − 𝑞) 2
𝑝~ 𝑈[0,1]

regulator does not disclose stress tests
uncertain

high-type bank chooses risk to maximize Π𝐻 𝑆𝐻
𝑆𝐻𝑁

1
𝐾
=
+ = 𝑆𝐻𝐹𝐵
2(1 − 𝑞) 2
low-type bank chooses risk to maximize Π𝐿 𝑆𝐿
𝑁 𝑁
1
𝐾
Pr(
𝑛
≥
𝑛|𝑆
1
𝐻 , 𝑆𝐿 )
𝑁
𝑆𝐿 =
+
>
> 𝑆𝐿𝐹𝐵
2(1 − 𝑞)
2
2(1 − 𝑞)

regulator discloses stress tests
banks know 𝑝 when
choosing risk

high-type bank chooses
𝑆𝐻𝐷
1
𝐾
=
+ = 𝑆𝐻𝐹𝐵
2(1 − 𝑞) 2

low-type bank chooses
𝑆𝐿𝐷
=
𝐷 𝐷
1
𝐾 Pr(𝑛≥𝑛|𝑝,𝑆𝐻
,𝑆𝐿 )
+
2(1−𝑞)
2
Pr 𝑛 ≥ 𝑛 𝑝, 𝑆𝐻𝐷 , 𝑆𝐿𝐷
𝑝𝑆𝐿𝐷
0
If
<𝑛
1
If 𝑝𝑆𝐿𝐷 ≥ 𝑛
=
1
=
2(1 − 𝑞)
1
𝐾
𝐷2
𝑆𝐿 =
+
2(1 − 𝑞) 2
𝑆𝐿𝐷1

stress-test disclosure coordinates risk-taking decisions:
o low-risk equilibrium: if 𝑝 <
o high-risk equilibrium: if 𝑝 >
o for
𝑛
1
𝐾
+
2(1−𝑞) 2
<𝑝<
𝑛
1
2(1−𝑞)
𝑛
1
2(1−𝑞)
𝑛
, 𝑆𝐿𝐷1 =
1
and
2(1−𝑞)
𝐷2
,
𝑆
𝐾
𝐿
+
1
2(1−𝑞)
1
2(1−𝑞) 2
=
, both equilibria coexist
no bailout
𝐾
2
+ and bailout
1
𝑆𝐻𝑁 = 𝑆𝐻𝐷 = 𝑆𝐻𝐹𝐵
𝑆𝐿𝐷2
Risk
𝑆𝐿𝑁
𝑆𝐿𝐷1
𝑆𝐿𝐹𝐵
0
0
𝑛
1
𝐾
+2
2(1 − 𝑞)
p
𝑛
1
2(1 − 𝑞)
1
𝑆𝑖 ∗ 𝑆−𝑖 =
𝐾 𝐼𝑝𝑆−𝑖>𝑛
1
+
2(1 − 𝑞)
2
1
𝑆𝑖
45°
∗
𝑆𝑖 ∗ (𝑆−𝑖 )
0
0
𝑆𝐿𝐷1
𝑛
𝑝
1
𝑆−𝑖
𝑆𝑖 ∗ 𝑆−𝑖 =
𝐾 𝐼𝑝𝑆−𝑖>𝑛
1
+
2(1 − 𝑞)
2
1
𝑆𝑖
45°
∗
𝑆𝑖 ∗ (𝑆−𝑖 )
0
0
𝑛
𝑝
𝑆𝐿𝐷2
1
𝑆−𝑖
𝑆𝑖 ∗ 𝑆−𝑖 =
𝐾 𝐼𝑝𝑆−𝑖>𝑛
1
+
2(1 − 𝑞)
2
1
𝑆𝑖
45°
∗
𝑆𝑖 ∗ (𝑆−𝑖 )
0
0
𝑆𝐿𝐷1
𝑛
𝑝
𝑆𝐿𝐷2
1
𝑆−𝑖


upon stress test disclosure, each bank 𝑖 sees 𝑥𝑖 = 𝑝 + 𝜀𝑖
o 𝜀𝑖 ∈ [−𝜂, 𝜂] is independent across banks
in the limit as η
0, there exists a unique equilibrium such that,
o if 𝑝 < 𝑝 then
𝑆𝐿𝐷1
o if 𝑝 ≥ 𝑝 then
𝑆𝐿𝐷2
=
1
2(1−𝑞)
=
1
2(1−𝑞)
and no bailout.
𝐾
2
+ and bailout.
1
𝑆𝐻𝑁 = 𝑆𝐻𝐷 = 𝑆𝐻𝐹𝐵
𝑆𝐿𝐷2
Risk
𝑆𝐿𝑁
𝑆𝐿𝐷1
𝑆𝐿𝐹𝐵
0
0
𝑝=
𝑛
1
𝐾
+4
2(1 − 𝑞)
1
p
Equilibrium analysis

the ex ante average risk,
o 𝐸 𝑆𝐿𝑁 = 𝑆𝐿𝑁
o 𝐸 𝑆𝐿𝐷 = 𝑝 𝑆𝐿𝐷1 + (1 − 𝑝)𝑆𝐿𝐷2
o 𝐸 𝑆𝐿𝐷 < 𝑆𝐿𝑁 if and only if 𝑛 < 𝑛𝑇

disclosure disciplines risk-taking when the externalities of
bank failure is sufficiently severe

comparing bail-out likelihood is equivalent to comparing risk
𝑆𝐿
1
𝐾 Pr(𝑛 ≥ 𝑛)
=
+
2(1 − 𝑞)
2

the ex ante expected bank failure is always larger with stress
test disclosure:
𝐸 𝑝𝑆𝐿𝐷 > 𝐸 𝑝𝑆𝐿𝑁
o𝐸
𝑝𝑆𝐿𝑁
o𝐸
𝑝𝑆𝐿𝐷
=𝐸 𝑝
𝑆𝐿𝑁
=𝐸 𝑝 𝐸
1
4
=
𝑆𝐿𝐷
𝑆𝐿𝑁
2
+ 𝐶𝑜𝑣
𝑝, 𝑆𝐿𝐷
o 𝐶𝑜𝑣 𝑝, 𝑆𝐿𝐷 = 𝑝 1 − 𝑝 𝐾 > 0
=
𝐸 𝑆𝐿𝐷
2
+ 𝐶𝑜𝑣 𝑝, 𝑆𝐿𝐷

a novel role of stress-test disclosure in facilitating banks’
coordination in risk-taking decisions

disclosing stress tests increases average risk level and bailouts,
unless bank failure externalities are sufficiently severe

disclosing stress tests increases expected bank failures
Appendix

each large BHC is evaluated under 3 scenarios: baseline,
adverse, and severely adverse.

results are disclosed for each BHC and in aggregate.

three Scenarios: Baseline, Adverse and Severely Adverse

each including trajectories for 28 variables, including trends
in: GDP growth, Unemployment, Inflation, Stock Prices, Real
Estate Prices, Interest rates.

examples of variable trends for 2016