Public Liquidity, Runs and Financial Crises

Wenhao Li

August 27, 2018 Public Liquidity Supply around the 2008 Financial

Dramatic increase in public liquidity, including reserves and treasuries.

0.45 Bank Reserves Treasuries Held By Domestic Private Investors 0.4 0.35 0.3 0.25 0.2 0.15

Liquidity/GDP ratio Liquidity/GDP 0.1 0.05 0 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015 How relevant is public liquidity supply to the economy?

Quantify the liquidity channel: by holding more public liquidity, reduce losses during bank runs and fire sales.

I Holmstrom & Tirole 1998: Public liquidity alleviates systemic liquidity shocks.

Results: high relevance to both asset prices and output.

Different from credit channel, collateral channel, etc. Powerful without nominal and monetary frictions. Contributions

The first quantification of the liquidity channel.

Connection to the macroeconomy.

I Advancing QE1 to 2008 ⇒ ↑ 0.9% GDP

I Liquidity is effective only against bank run shocks, not non-financial .

I Liquidity wealth effect, crowding out investment in the long-term

Explanatory power on asset prices.

I Model explains 43% monthly variations in liquidity premium. Benchmark: ∼ 10% for macro models (Del Negro et al. 2017).

I Model explains half of variations in the credit spread. Related Literature

Liquidity premium of U.S. treasury securities.

I Importance: (1) Indicator of financial distress (Longstaff 2004); (2) Informative on “moneyness” of the treasuries (Nagel 2016, Li 2018).

I Supply-premium relationship (Krishnamurthy and Vissing-Jorgensen 2012).

Financial sector balance sheet and macro.

I Bernanke and Gertler 1995; Kiyotaki and Moore 1997; Adrian and Shin 2010; Gertler and Karadi 2011; He and Krishnamurthy 2013; Brunnermeier and Sannikov 2014. Di Tella 2018.

Bank runs.

I Diamond and Dybvig 1983; Drechsler, Savov, and Schnabl 2014; Gertler, Kiyotaki, and Prestipino 2017; Moreira and Savov 2017. Outline

1 Model

2 Mechanism

3 Solution Method and Calibration

4 Asset Pricing Implications

5 Macroeconomic Implications Model Structure

Banks Households Government

Government Government No equity Bonds Bonds Wealth issuance (Bankers) Productive Capital Lump sum Government Wealth Taxation Bonds Productive Capital (lending to Bank Debt Bank Debt firms) Model Structure

Banks Households Government

Government Government Bonds Bonds Wealth (Bankers) Productive Capital Fire-Sale Lump sum Government Wealth Taxation Bonds Productive Capital Bank Run (lending to Bank Debt Bank Debt firms) Model Setup

Exogenous variables or functions in color.

Individual banker: maximize Z ∞ −ρt b E[ e log(ct )] 0 s.t. banker budget constraint.

Individual household: maximize Z ∞ −ρt h E[ e log(ct )] 0 s.t. household budget constraint. Investment and Production Technology

Firms = Productive capital. Modigliani-Miller holds.

K Firms make investment µt (growth of capital) according to the Q-theory.

Evolution of individual capital

dkj,t K K = µt dt − δdt + σ dBt − κ˜j,t dNt kj,t | {z } |{z} | {z } | {z } growth depreciation short-term fluctuations crisis shock

with crisis intensity λ> 0, and i.i.d. idiosyncratic shockκ ˜j,t ∈ {0, 1},

P(˜κj,t = 1) = θ ∈ (0, 1) Banker and Household Returns on Productive Capital

Bankers: the unit output of capital is A¯, with return

¯ K ¯K d(pt kj,t ) (A − φ(µt ))kj,t dRj,t = + dt pt kj,t pt kj,t | {z } | {z } ”capital gain” ”dividend”

Households: K K d(pt kj,t ) (A − φ(µt ))kj,t dRj,t = + dt pt kj,t pt kj,t

¯ ¯K K A > A ⇒ dRj,t > dRj,t ⇒ Bankers have incentives to raise funding from households. Government Liquidity and the Liquidity Premium

Government Bonds.

I Total supply Qt Kt . Balanced budget with lump-sum taxation. g g I All short-term with return dRt = rt dt

Illiquid assets.

I Selling price is only π ∈ (0, 1) fraction of the normal price. illiq illiq I Short-term with return dRt = rt dt.

Liquidity premium is illiq g `t = rt − rt Bank Runs: Asset Destruction

휅෤ = 1 휃 푗,푡 Capital destroyed

Banks: Shock 푑푁푡 hits

휿෥풋,풕: private info 휅෤푗,푡 = 0 1 − 휃 Capital not destroyed Bank Runs: Funding Withdrawal

휅෤ = 1 휃 푗,푡 Capital destroyed

Banks: Shock 푑푁푡 hits

휅෤푗,푡 = 0 1 − 휃 Capital not destroyed

Sticky deposits: No run. 1 − 훽

Not knowing which Active deposits: banks are solid, they 훽 have incentives to run. Bank Runs: Consequences

Bankruptcy and 휅෤ = 1 휃 푗,푡 Liquidation, Capital destroyed “run deposits” first

Banks: Shock 푑푁푡 hits

휅෤푗,푡 = 0 Fire Sales 1 − 휃 Capital not destroyed

Sticky deposits: No run. Funding 1 − 훽 withdrawal

Not knowing which Active deposits: banks are solid, they 훽 휿෥풋,풕 revealed have incentives to run. Endogenous Credit Limit and Dominant Strategies

Proposition 1 (Endogenous Credit Limit) Banks cannot raise debt during bank runs.

Intuition: during crisis shock dNt at time t, the expected loss of lending to a bank is O(1), but the benefit is O(dt).

Proposition 2 (Bank Run Game) If banks take leverage to hold productive capital, running on banks is a

weakly dominant strategy, for active deposits during crisis shocks dNt . Bank Runs and Fire Sales

Price of capital Crisis Shock 푝푡− Fundamental decline 푝 휅푡− = 푝푡− − 푝푡

푝푡 Market pressure 0 훼 푝푡

0 (1 − 훼 )푝푡 Time Banker Budget Dynamics

Individual banker j’s wealth dynamics

b dwj,t K R g g illiq illiq f d d b = xt µt + xt rt dt + xt rt dt + xt rt dt − xt dRt wj,t− | {z } | {z } | {z } | {z } | {z } capital return govt bond return illiquid asset return interbank lending return interest payment

b 0 ct K p α − b dt − (1κ˜j =1(1 − ε) + 1κ˜j =0(xt−κt− + 0 ∆xt− )) dNt . wt 1 − α | {z } | {z } consumption loss during a crisis

Banker’s effective funding withdrawal

d g illiq + ∆xt− = ( βxt− − (xt− + πxt− ) ) | {z } | {z } total funding withdrawal total liquidity insurance

g β < 1: higher xt− ⇒ smaller ∆xt−. Production

Then total output is  Yt = ψt A¯ + (1 − ψt )A Kt , with A¯ > A | {z } per unit productivity of capital

where ψt is the fraction of bank holding of productive capital,

K wt xt ψt = K K wt xt + (1 − wt )yt

Liquidity supply affects

1) banker wealth wt drops in crises (↓ fire sales); K 2) bank lending xt (↓ risks). The Aggregate State Variable

b b h Aggregate state: banker wealth share wt = Wt /(Wt + Wt ), and aggregate capital Kt .

Aggregate growth of banker wealth share

dwt w b h b h = µt dt + (1 − wt )(σt − σt ) dBt − (1 − wt−)h(κt− − κt−) dNt wt− | {z } | {z } exposure to productivity shock exposure to bank run shock

Aggregate growth of productive capital follows

dKt K K = µt dt − δdt + σ dBt − θdNt Kt | {z } |{z} | {z } | {z } growth depreciation short-term fluctuations capital destruction loss Equilibrium Definition

Markov equilibrium with state variables wt and Kt , such that

I Consumption and portfolio choices are optimal.

I Market clearings: productive capital, government bonds, and the illiquid assets (zero supply)

I Aggregate wealth

∞ Z h b h −rs (s−t) Wt + Wt = pt Kt + Q(wt )Kt − Et [ e τs ds] |{z} t real wealth | {z } liquidity wealth

I Government budget balance

g Et [d(Q(wt )Kt ) + τt dt − Q(wt )Kt rt dt] = 0

I Resource constraint: Production = consumption + investment costs.

¯ b h K (ψt A + (1 − ψt )A) · Kt = Ct + Ct + φ(µt )Kt Outline

1 Model

2 Mechanism

3 Solution Method and Calibration

4 Asset Pricing Implications

5 Macroeconomic Implications Liquidity Premium ∼ Price of Liquidity Service

Liquidity Premium ℓ = 푟푖푙푙푖푞 − 푟푔

Liquidity Supply

Equilibrium Liquidity Demand

0 Bank Holding of Public Liquidity

Figure: Determination of the liquidity premium at certain state (w, K) Liquidity Premium: Impact of Supply

Liquidity Premium ℓ = 푟푖푙푙푖푞 − 푟푔

Larger public liquidity supply

0 Bank Holding of Public Liquidity

Figure: Determination of the liquidity premium at certain state (w, K) Liquidity Premium: Impact of Demand

Liquidity Premium ℓ = 푟푖푙푙푖푞 − 푟푔

Higher Leverage, more bank debt

0 Bank Holding of Public Liquidity

Figure: Determination of the liquidity premium at certain state (w, K) Equilibrium Determination of Crisis Severity

Price Decline Jump 휅푝

Individual Aggregate consequence choice of fire-sale

Equilibrium Productivity

0 Bank Holding of Productive Capital

Figure: Determination of price decline κp in a bank run at state (w, K) Liquidity Supply and Severity of the Crisis

Price Decline Jump 휅푝

Larger Liquidity Supply

Productivity

0 Bank Holding of Productive Capital

Figure: Influence of liquidity supply at state (w, K) Liquidity Externality

Price Decline Jump 휅푝

Aggregate consequence of fire-sale

Considering externality

Productivity

0 Bank Holding of Productive Capital

Figure: Liquidity Externality Outline

1 Model

2 Mechanism

3 Solution Method and Calibration

4 Asset Pricing Implications

5 Macroeconomic Implications Solution Method

Main challenge: Jump depending on the unknown function p(w). I have designed an efficient functional iteration algorithm. Generality: more state variables; other macro finance/asset pricing/dynamic corporate finance models with endogenous jumps.

Price 푝(푤)

푤푡−

푝 휅푡−

푤푡 푤 휅푡−

Banker Wealth Share 푤 Calibration: Connections between Model and Data

Banks Households

Wealth Govt Bonds (Bankers) Govt Bonds Productive Capital Less Liquid Govt Assets MODEL Sticky Deposits Wealth Sticky Bank Debt Productive Capital

Active Runnable Active Deposits Bank Debt Calibration: Connections between Model and Data

Bank Holding Companies, Households, MMF, Pensions, Depository Institutions, Broker Dealers etc. Mutual Funds, etc.

Treasuries Equity Treasuries, Bank Reserves Corporate Bonds etc. Illiquid Govt Liabilities Insured DATA Deposits, $14t Wealth Long-term in 2007 Insured Bank debts Deposits, Long-term (syndicated debts , commercial loan, etc.) ABCP, Repo, ABCP, Repo, MBS, etc. and other $4.6 t and other wholesales in 2007 wholesales funding funding Calibration

Parameters Choice Quantity to be matched Target Model

Production and Investment: A¯ Banker productivity 0.15 Investment to capital ratio 11% 11%

A Household productivity 0.125 Difference of return (A¯ − A)/pt 0.024 0.025 to prime loan rate - MMF rate

Bank Run and Fire Sales: λ Bank-run arrival rate 2.5% Frequency of financial crises ?0.025 0.025 β Fraction of bank run 25% Bank runnable funds/assets ?25% 25% ? α0 Fire sale market pressure. 21% Price pressure during fire sales. 21% 21%

θ Crisis shock size 10−5 Negligible value < 0.01 < 0.01 ε Bankruptcy leftover 10−3 Negligible value < 0.01 < 0.01

For Asset Prices: π illiquid asset resellability 11% Match average liquidity premium 22 bps 22 bps

Other macro parameters: Depreciation rate δ, discount rate ρ, growth volatility σK , investment adjustment cost χ are set to the standard values in the macro literature. Outline

1 Model

2 Mechanism

3 Solution Method and Calibration

4 Asset Pricing Implications

5 Macroeconomic Implications Data and Measurements

Public liquidity supply (1920-2016)

I Treasuries held by domestic private investors + Bank reserves

Financial sector leverage (1970-2016).

I Leverage data from He, Kelly, Manela (2017)

Liquidity premium (1920-2016)

I 1991-2016: Principal component of maturity matched Refcor Bonds - Treasury spreads (1-5 years), and Repo 3M collateralized - Treasury 3M

I 1920-1990: Banker Acceptance 3M - Treasury 3M. (Nagel 2016)

t t Method: At each month t, set Q (·) = Qdata, solve model and find w with t lvg(w) = lvgdata. Liquidity Premium – Time Series Predictions and Data

43% explained = 10% + 29% + 4% |{z} |{z} |{z} liquidity supply financial sector leverage interaction

1.6 Data Model Prediction 1.4 Spike in liquidity demand:

)

% 1.2 crisis and fragile banks Spike in liquidity

( supply

m

u

i 1

m

e r 0.8

P

y

t

i

d 0.6

i

u

q

i

L 0.4

0.2

0 1970 1975 1980 1985 1990 1995 2000 2005 2010 2015 2020 Impact of Public Liquidity Supply on Liquidity Premium

Strong interaction effects: liquidity supply is more valuable in distress periods.

60 Public Liquidity/GDP decreased uniformly by 10%

50

40

30

20

10 Liquidity Premium Difference(bps) Premium Liquidity

0 1970 1975 1980 1985 1990 1995 2000 2005 2010 2015 2020 Outline

1 Model

2 Mechanism

3 Solution Method and Calibration

4 Asset Pricing Implications

5 Macroeconomic Implications Matching 2008 : Shocks

Real GDP (Detrended at 1.4%)

푑퐵 푡 -6.3 -0.7 -0.7 -1.0 -0.4 -1.1 -0.7 0.3 2 (%)

푑푁푡 0 1 0 0 0 0 0 0

0

%

n

i

e

g

n

2

a

−

h

C Change Change in %

Contribution

4 − of 푑푁푡 = 3.9%

6

−

2006 2008 2010 2012 2014 Matching 2008 Financial Crisis: Model v.s. Data

Real GDP (% Diff from 2007) Financial Equity (% Diff from 2007)

1 Data Data Model Model 0 0 −1 −20 −2 −3 Change in % Change in % −40 −4 −5 −60 −6 2008 2010 2012 2014 2008 2010 2012 2014

Credit Spread (Diff from 2007) Liquidity Premium (Diff from 2007)

Data Data

Model 0.8 Model 1.0 0.6 0.5 0.4 Change in % Change in % 0.2 0.0 0.0

2008 2010 2012 2014 2008 2010 2012 2014 What if all shocks come from dBt?

Real GDP (Detrended at 1.4%) Financial Equity (Detrended at 1.4%)

1 Data Data Model Model 0 0 −1 −20 −2 −3 Change in % Change in % −40 −4 −5 −60 −6 2008 2010 2012 2014 2008 2010 2012 2014

Credit Spread (Diff w.r.t. 2007) Liquidity Premium (Diff w.r.t. 2007)

Data Data

Model 0.8 Model 1.0 0.6 0.5 0.4 Change in % Change in % 0.2 0.0 0.0

2008 2010 2012 2014 2008 2010 2012 2014 Liquidity Supply and Two Counterfactual Experiments

Total Liquidity/GDP 0.3 Treasuries Held By Domestic Private Investors 0.30 Counterfactual Experiment 1 Bank Reserves Counterfactual Experiment 2 0.25

0.2 0.20 0.15 Liquidity/GDP 0.1 Public Liquidity/GDP Public 0.10 0.05

0.0

2008 2010 2012 2014 0.00 2008 2010 2012 2014 Counterfactual 1: 10% Increase of Liquidity/GDP at 2008

GDP Financial Equity 6

1.5 Counterfactual − Baseline Counterfactual − Baseline 5 4 1.0 3 2 0.5 1 Counterfactual Difference (%) Difference Counterfactual (%) Difference Counterfactual 0 0.0

2008 2010 2012 2014 2008 2010 2012 2014

Credit Spread Liquidity Premium

0.02 Counterfactual − Baseline Counterfactual − Baseline 0.00 −0.02 −0.04 −0.06 Counterfactual Difference (%) Difference Counterfactual (%) Difference Counterfactual −0.08 −0.10 2008 2010 2012 2014 2008 2010 2012 2014 Counterfactual 2: Increase of Liquidity/GDP in 2012,2013

GDP Financial Equity

0.3 Counterfactual − Baseline 1.0 Counterfactual − Baseline 0.8 0.2 0.6 0.4 0.1 0.2 0.0 Counterfactual Difference (%) Difference Counterfactual (%) Difference Counterfactual −0.2 −0.1 2008 2010 2012 2014 2008 2010 2012 2014

Credit Spread Liquidity Premium

Counterfactual − Baseline Counterfactual − Baseline 0.00 0.00 −0.05 −0.05 −0.10 −0.10 Counterfactual Difference (%) Difference Counterfactual (%) Difference Counterfactual −0.15 −0.15 2008 2010 2012 2014 2008 2010 2012 2014 The Liquidity Wealth Effect

Total wealth = productive capital + value of bonds - taxation ∞ Z h −rt t = productive capital + E[ e (··· ) · `t · Qt Kt dt] 0 | {z } liquidity wealth

0.01 liquidity wealth/GDP (%) 10 growth rate of K (% diff) 0 average discounted output (diff)

8 -0.01

-0.02 6

-0.03 4 -0.04

2 -0.05

0 -0.06 0.5 1 1.5 2 2.5 3 3.5 4 public liquidity/GDP Policy Implications from the Liquidity Insurance Channel

Results Policy Implications

A. Injecting liquidity earlier → 50 times benefits. Prompt

B. Liquidity → Vulnerability to real shocks. Only for financial shocks.

C. Liquidity wealth effect → ↓ investment Short-lived Summary: How relevant is public liquidity?

Answer: highly relevant, through the liquidity channel.

Approach: build and quantify a new microfounded macro model.

Results: connection to both asset prices and the macroeconomy.

I Explain 43% variations of monthly liquidity premium, and 47% credit spread.

I Advancing QE1 to 2008 ⇒ ↑ 0.9% GDP, persistent for years.

New implications of liquidity policy: prompt, short-lived, and only against financial crises.