Bank Regulation under Fire Sale Externalities

Bank Regulation under Fire Sale Externalities Gazi Ishak Kara 1 S. Mehmet Ozsoy 2 1 Office of Financial Stability Policy and Research, Federal Reserve Board 2 Ozyegin University May 17, 2016 Disclaimer: The analysis and the conclusions set forth are those of the authors and do not indicate concurrence by other members of the research staff or the Board of Governors. Kara and Ozsoy (Fed/OzU) Bank Regulation 1 / 36

Motivation: Background The recent crisis was characterized by severe liquidity problems. The regulation before the crisis was predominantly micro-prudential and focused on capital requirements. Basel III supplements capital regulations with liquidity requirements (such as LCR and NSFR) and focuses on macro-prudential measures. Kara and Ozsoy (Fed/OzU) Bank Regulation 2 / 36

Research Questions This paper investigates the optimal design of capital and liquidity regulations in a model characterized by fire sale externalities. Our research questions are: Can we trust the institutions to properly manage their liquidity, once excessive risk taking has been controlled by the capital requirement? (Jean Tirole, 2011) What are -if any- the advantages and disadvantages of liquidity requirements that supplement the capital regulations? Kara and Ozsoy (Fed/OzU) Bank Regulation 3 / 36

Sketch and the Timing of the Model t=1 t=2 Good times 1-q t=0 Banks choose risky and safe assets Raise funds from consumers Bad times q t=1 Investment is distressed Fire-Sales t=2 Agents A continuum of banks with a unit mass. A continuum of consumers with a unit mass. A continuum of outside investors with a unit mass. A financial regulator (e.g. a central bank). Kara and Ozsoy (Fed/OzU) Bank Regulation 4 / 36

Related Literature Financial Regulation Holmstrom and Tirole (1998), Acharya (2003), Farhi and Tirole (2009), De Nicolo, Gamba and Lucchetta (2012), Goodhart et al (2013), Kashyap, Tsomocos and Vardoulakis (2014), Ahnert (2014), Walther (2014) Asset Fire Sales Williamson (1988), Shleifer and Vishny (1992, 2011), Kiyotaki and Moore (1997), Lorenzoni (2008), Gai et al. (2008), Korinek (2011), Stein (2012) Incomplete Markets Hart (1975), Stiglitz (1982), Geanakoplos and Polemarchakis (1986) Kara and Ozsoy (Fed/OzU) Bank Regulation 5 / 36

Key Results Under fire sale externalities, banks overinvest in the risky asset and underinvest in the liquid asset in the unregulated competitive equilibrium. Kara and Ozsoy (Fed/OzU) Bank Regulation 6 / 36

Key Results Under fire sale externalities, banks overinvest in the risky asset and underinvest in the liquid asset in the unregulated competitive equilibrium. When we regulate capital but not liquidity, banks undermine the regulation by taking more risk through the liquidity channel. Pre-Basel III regulatory framework, with its reliance only on capital requirements, was inefficient and ineffective in addressing systemic instability caused by fire sales. Kara and Ozsoy (Fed/OzU) Bank Regulation 6 / 36

The Model: Basic Setup Three dates: t = 0, 1, 2. Two goods: - A consumption good (liquid/safe asset) - An investment good (illiquid/risky asset) Consumers are endowed with e units of consumption goods at t = 0 but none at t = 1 and t = 2. Banks can convert consumption goods into investment goods one-to-one at t = 0. Banks choose risky asset level, n i, at t = 0, which pays a return of R at t = 2. Kara and Ozsoy (Fed/OzU) Bank Regulation 7 / 36

Technology and Notation I Safe assets: Banks are endowed with a storage technology with unit returns. A bank chooses how much safe assets to hold per unity of risky assets, b i [0, 1]. A bank hoards total safe assets of n i b i at t = 0. The total assets of a bank is n i + n i b i = (1 + b i )n i. Kara and Ozsoy (Fed/OzU) Bank Regulation 8 / 36

Technology and Notation II Assets Risky assets (n i ) Cash (n i b i ) Liabilities Deposits (L i ) Equity (E) Banks are endowed with E units of fixed equity capital. Banks raise L i = (1 + b i )n i E units of consumption goods from depositors. Risk weighted capital ratio of bank is E /n i. Capital regulation limits risky investment n i since the equity is fixed. Kara and Ozsoy (Fed/OzU) Bank Regulation 9 / 36

Cost of funding and operating a bank Banks initial equity is sufficiently large to avoid default in equilibrium. As a result, deposits are safe, and the net interest rate on deposits is zero. The operational cost of a bank is Φ((1 + b i )n i ), where Φ ( ) > 0 and Φ ( ) > 0. Φ( ) is convex, that is, Φ ( ) > 0 and Φ ( ) > 0. Van den Heuvel (2008) and Acharya (2003, 2009). The total cost of a bank is D((1 + b i )n i ) = Φ((1 + b i )n i ) + (1 + b i )n i. Kara and Ozsoy (Fed/OzU) Bank Regulation 10 / 36

Liquidity Shock at t = 1 Good state (with probability 1 q): - No liquidity shock. - Bank s assets yield Rn i + n i b i units of consumption goods at t = 2. Kara and Ozsoy (Fed/OzU) Bank Regulation 11 / 36

Liquidity Shock at t = 1 Good state (with probability 1 q): - No liquidity shock. - Bank s assets yield Rn i + n i b i units of consumption goods at t = 2. Bad state (with probability q): - Investment distressed, has to be restructured to remain productive. - Restructuring costs are c units per risky asset. - Banks can use safe assets n i b i to carry out the restructuring. - A limited-commitment prevents banks from raising external finance. - Banks sell risky assets to investors to raise liquidity (fire sales). The net expected return on the risky asset is positive: R > 1 + qc. Kara and Ozsoy (Fed/OzU) Bank Regulation 11 / 36

Outside Investors Problem Outside investors are endowed with large liquid resources at t = 0 and 1. They can purchase assets from banks and employ them in a technology F. F is concave (F > 0 and F < 0), and satisfies F (0) R. They choose how much investment goods y to buy from banks at t = 1 max y 0 F (y) Py First-order condition: F (y) = P. Outside investors demand function y = Q d (P) F (P) 1 is downward sloping! Assume outside investors demand is elastic to rule out multiple equilibria: ɛ P,y = y P P y = F (y) yf (y) > 1 Kara and Ozsoy (Fed/OzU) Bank Regulation 12 / 36

Crisis and Fire-Sales A bank decides what fraction of investment to sell (1 γ i ) max 0 γ i 1 π i = Rγ i n i + P(1 γ i )n i + b i n i cn i subject to the budget constraint P(1 γ i )n i + b i n i cn i 0. Kara and Ozsoy (Fed/OzU) Bank Regulation 13 / 36

Crisis and Fire-Sales A bank decides what fraction of investment to sell (1 γ i ) max 0 γ i 1 π i = Rγ i n i + P(1 γ i )n i + b i n i cn i subject to the budget constraint P(1 γ i )n i + b i n i cn i 0. In equilibrium c P R. Hence, the BC binds, and we obtain and the total supply of assets is 1 γ i = c b i P (1 γ)n = c b n = Downward Sloping Supply P Kara and Ozsoy (Fed/OzU) Bank Regulation 13 / 36

Asset Market Equilibrium at t=1 P R Supply P* Demand c Total fire-sales n Q Supply of assets, (1 γ)n = c b P n, is a function of n and b. Kara and Ozsoy (Fed/OzU) Bank Regulation 14 / 36

Asset Market Equilibrium: Comparative Statics P R Supply Supply c Demand n n Q Lemma: A higher initial risky investment (n) or a lower liquidity ratio (b) increases the severity (lower asset prices) and the cost (more asset fire-sales) of financial crises. Kara and Ozsoy (Fed/OzU) Bank Regulation 15 / 36

Understanding externalities higher initial risky investment (n) Kara and Ozsoy (Fed/OzU) Bank Regulation 16 / 36

Understanding externalities higher initial risky investment (n) but P n < 0 so P Kara and Ozsoy (Fed/OzU) Bank Regulation 16 / 36

Understanding externalities higher initial risky investment (n) but P n < 0 so P P = more fire sales, as P(1 γ i )n i = cn i b i n i Banks create negative externality on each other Similarly, as P b > 0, lower a liquidity ratio (b) implies more fire sale At a lower price, banks have to sell more asset Kara and Ozsoy (Fed/OzU) Bank Regulation 16 / 36

What we do next We will compare and contrast: Competitive Equilibrium: No regulation (n, b). Constrained Planner s Problem: (n, b ). Kara and Ozsoy (Fed/OzU) Bank Regulation 17 / 36

What we do next We will compare and contrast: Competitive Equilibrium: No regulation (n, b). Constrained Planner s Problem: (n, b ). How can we implement constrained planner s allocations? Complete Regulation: Both capital ratio (E /n i ) and liquidity ratio (b i ) are regulated, as in Basel III. Partial Regulation: Only capital ratio (E /n i ) is regulated, i.e. pre-basel III regulation. Kara and Ozsoy (Fed/OzU) Bank Regulation 17 / 36

Full insurance is not optimal Proposition It is optimal for both banks in the unregulated competitive equilibrium and the constrained social planner to take fire sale risk; that is, to set b i < c. The amount (c) and frequency (q) of the aggregate liquidity shock are exogenous in the model, but whether and to what extent a fire sale takes place are endogenously determined. In principle, it is possible to insure banks perfectly against the liquidity shock by setting b i = c. However, liquidity has an opportunity cost in terms of forgone investment in the risky asset, which has a higher expected return. Kara and Ozsoy (Fed/OzU) Bank Regulation 18 / 36

Competitive Equilibrium Banks problem at t = 0: max Π i (n i, b i ) = (1 q)(r + b i )n i + qrγ i n i D(n i (1 + b i )) n i,b i where γ i = 1 c b i P as obtained from banks problem at t = 1. Kara and Ozsoy (Fed/OzU) Bank Regulation 19 / 36

Competitive Equilibrium Banks problem at t = 0: max Π i (n i, b i ) = (1 q)(r + b i )n i + qrγ i n i D(n i (1 + b i )) n i,b i where γ i = 1 c b i P as obtained from banks problem at t = 1. First-order conditions with respect to n i and b i are respectively: (1 q)(r + b i ) + qrγ i = D (n i (1 + b i ))(1 + b i ) (1 q)n i + qr 1 P n i = D (n i (1 + b i ))n i Kara and Ozsoy (Fed/OzU) Bank Regulation 19 / 36

Constrained planner s problem Constrained planner s problem at t = 0: max n,b where γ = 1 c b P W (n, b) = (1 q)(r + b)n + qrγn D(n(1 + b)) + e as obtained from banks problem at t = 1. Kara and Ozsoy (Fed/OzU) Bank Regulation 20 / 36

Constrained planner s problem Constrained planner s problem at t = 0: max n,b where γ = 1 c b P W (n, b) = (1 q)(r + b)n + qrγn D(n(1 + b)) + e as obtained from banks problem at t = 1. First-order conditions with respect to n, b are respectively: { (1 q)(r + b) + qr γ + n c b (1 q)n + qr P 2 { 1 P + c b P 2 P } n P b = D (n(1 + b))(1 + b) } n = D (n(1 + b))n Kara and Ozsoy (Fed/OzU) Bank Regulation 20 / 36

Functional-form Assumptions Outside investors technology: F (y) = R ln(1 + y). For this return function we obtain the (inverse) demand function as P = F (y) = R 1 + y and hence y = F 1 (P) = R P P Q d (P) Kara and Ozsoy (Fed/OzU) Bank Regulation 21 / 36

Implementing Constrained Planner s Allocations Proposition n > n b < b Banks overinvest in the risky asset and underinvest in the liquid asset in the unregulated competitive equilibrium. The inefficiency is created by the fire sale externality. The constrained efficient allocations can be implemented using both A minimum risk weighted capital ratio requirement: E /n i E /n n i n A minimum liquidity ratio requirement: b i b Kara and Ozsoy (Fed/OzU) Bank Regulation 22 / 36

Partial Regulation: Regulating only capital Regulator moves first and sets n. Given n i = n, banks choose the liquidity ratio (b i ) to maximize their expected profits. FOC of banks problem wrt b i yields: (1 q) + qr 1 P = D (n(1 + b i )) = b i = D 1 (1 q+q R P ) n 1 The regulator maximizes: max n W (n) = (1 q)(r + b(n))n + qrγn D((1 + b(n))n) + e Using γ = 1 c b(n) P the first-order condition with respect to n is: ( (1 q)[r + b(n) + nb c b dp (n)] + qr[γ + n P 2 dn + 1 ) P b (n) ] = D (n(1 + b(n)))[1 + b(n) + nb (n)] Kara and Ozsoy (Fed/OzU) Bank Regulation 23 / 36

Partial Regulation Proposition Banks decrease their liquidity ratio as the regulator tightens the limit on risky investment, i.e. b i (n) > 0. Stricter limits on risky investment lower liquidity ratios. Banks are restricted to take risk on the investment side, they switch to the liquidity channel. Kara and Ozsoy (Fed/OzU) Bank Regulation 24 / 36

Competitive Equilibrium vs Partial Regulation Proposition Denote the outcomes under partial regulation by n, b. Then we have n > n b > b Partial regulation decreases risky investment from inefficiently high levels in the unregulated competitive equilibrium. Banks are less even liquid under partial regulation: They undermine the purpose of regulation. An unintended consequence of capital regulation: Making the system safer allows banks to take more risky on the liquidity side. Kara and Ozsoy (Fed/OzU) Bank Regulation 25 / 36

Comparing Risky Holdings (n) Proposition n > n > n 8 Risky Holdings 7 6 n 5 4 3 0.2 0.4 0.6 0.8 c : size of liquidity shock Kara and Ozsoy (Fed/OzU) Bank Regulation 26 / 36

Comparing Liquidity Hoarding (b) Proposition b > b > b 0.8 0.6 b 0.4 0.2 0.0 0.2 0.4 0.6 0.8 c : size of liquidity shock Kara and Ozsoy (Fed/OzU) Bank Regulation 27 / 36

Fire-sale price of risky asset Proposition P > P > P 1.0 Prices under Fire Sale 0.8 P 0.6 0.4 0.2 0.2 0.4 0.6 0.8 c: size of liquidity shock Kara and Ozsoy (Fed/OzU) Bank Regulation 28 / 36

Severity of the crisis: fraction of risky assets sold Proposition 1 γ > 1 γ > 1 γ 0.7 Fire Sale: fraction of risky assets sold 0.6 0.5 0.4 0.3 0.2 0.1 0.2 0.4 0.6 0.8 c : size of liquidity shock Kara and Ozsoy (Fed/OzU) Bank Regulation 29 / 36

Severity of the crisis: total amount of risky assets sold Proposition 4 (c) (1 γ)n > (1 γ )n > (1 γ )n 3.0 Fire Sale: amount of risky assets sold 2.5 2.0 1.5 1.0 0.5 0.2 0.4 0.6 0.8 c : size of liquidity shock Kara and Ozsoy (Fed/OzU) Bank Regulation 30 / 36

Balance Sheet Size Proposition (1 + b)n = (1 + b )n > (1 + b )n 9 Balance Sheet 8 7 6 5 4 3 2 0.2 0.4 0.6 0.8 c : size of liquidity shock Kara and Ozsoy (Fed/OzU) Bank Regulation 31 / 36

Partial vs Complete Regulation Looking at n > n, one may think that entering the interim period with n rather than n should be safer. However, fire sales are larger under partial regulation: Ratio: 1 γ > 1 γ Level: (1 γ )n > (1 γ )n Level of risky investment is not a good predictor of the stability of the banking system or any individual bank under a potential distress scenario. The important thing is not the level of risky investment; it is how the risky investment is backed by liquid assets. Kara and Ozsoy (Fed/OzU) Bank Regulation 32 / 36

Advantages of Regulating Liquidity More funds for high return projects: n > n More liquidity: b > b Less fire-sales: Ratio: 1 γ > 1 γ Level: (1 γ )n > (1 γ )n Higher fire sale prices: P > P Kara and Ozsoy (Fed/OzU) Bank Regulation 33 / 36

Why not just regulate liquidity? Fire sales are triggered by a liquidity shock in the bad state. Banks are solvent otherwise Can the constrained optimum be implemented using liquidity regulation alone? The answer is negative: n i (b ) > n. Again, when one channel is restricted banks switch to another channel to take heir privately optimal fire sale risk. Kara and Ozsoy (Fed/OzU) Bank Regulation 34 / 36

Interaction with unregulated institutions In practice, banks are not the only ones investing into these risky assets which have fire sale risk Their choices affect the fire sales as well as they are affected from fire sale externalities As regulation tightens risky investment level of banks, unregulated institutions increase their risky investment. Unregulated institutions free ride on the financial stability provided by the regulation Kara and Ozsoy (Fed/OzU) Bank Regulation 35 / 36

Conclusion Under fire sale externalities, banks overinvest in the risky asset and underinvest in the liquid asset in the unregulated competitive equilibrium. Kara and Ozsoy (Fed/OzU) Bank Regulation 36 / 36

Conclusion Under fire sale externalities, banks overinvest in the risky asset and underinvest in the liquid asset in the unregulated competitive equilibrium. When we regulate capital but not liquidity, banks undermine the regulation by taking more risk through the liquidity channel. Pre-Basel III regulatory framework, with its reliance only on capital requirements, was inefficient and ineffective in addressing systemic instability caused by fire sales. Kara and Ozsoy (Fed/OzU) Bank Regulation 36 / 36