Low Interest Rate Policy and Financial Stability David Andolfatto Fernando Martin Aleksander Berentsen The views expressed here are our own and should not be attributed to the Federal Reserve Bank of St. Louis or the Federal Reserve System.
Motivation Does low-interest-rate monetary policy (LIMP) promote financial stability or instability? There are opposing views. One view is that LIMP induces reach-for-yield into higher risk assets. Promotes instability, esp. if financed with short-term debt (Stein, 2013). Another view is that LIMP encourages banks to hold more cash reserves. Promotes stability, since banks more likely to become narrow banks (Cochrane, 2014).
What we do The question we address requires a model of money and banking. Money is modeled as an exchange medium (Lagos-Wright, 2005). Dynamic general equilibrium framework (easy to make DSGE). Steady-state analysis so R policy equivalent to Π policy (Fisher effect). Bank is modeled as risk-sharing arrangement (Diamond-Dybvig, 1983). Demandable debt is a solution to private information problem, but opens the door to bank-runs (multiple equilibria).
Model Discrete time, infinite horizon, subjective discount factor β. Each period divided into 3 subperiods: morning, afternoon, evening. Two groups of agents: investors and workers. Investors face i.i.d. shock that determines type within period. impatient/patient want consumption in afternoon/evening. π denotes fraction of impatient, shock realized in afternoon.
Model Investors produce output in morning (consumer and capital goods) and consume in afternoon/evening (depending on type). Capital goods k yield afternoon output ξk or evening output Rk (assume 0 ξ < 1 < R). Workers want to consume in morning, able to work in afternoon. Afternoon output can be costlessly stored to the evening. Since ξ < 1, afternoon output supplied more effi ciently by workers relative to scrapping capital produced by investors.
Model Investor and worker preferences, resp. E 0 t=0 β t [ y 0,t + πu(c 1,t ) + (1 π)u(c 2,t ) ] E 0 t=0 β t [ c 0,t y 1,t ] Resource constraints (assuming no scrapping), c 0,t + k t = y 0,t πc 1,t y 1,t (1 π)c 2,t Rk t
Effi cient allocation Investors produce consumer and capital goods in morning. Consumer goods used to reward workers in the morning. Capital goods used to finance investor consumption in the evening. Workers produce output in the afternoon to service impatient investors. Effi ciency implies risk-sharing among investors (workers are risk-neutral).
Steps in our analysis Step 1: Competitive monetary eqm with ex post securities market. Investors accumulate money and capital in the morning, then trade money for capital in the afternoon after realizing their types. Markets are incomplete along two dimensions. No ex ante insurance market & ex post securities market closed with probability η (market freeze). Industry standard is η = 1 (illiquid capital). Liquidating capital when sec. mkt. closed means scrapping (firesale).
Steps in our analysis Step 2: Construct a Diamond-Dybvig bank assuming no bank runs. Demonstrate two dimensions along which welfare improved. (i) improved risk-sharing and (ii) bypass sec. mkt. (avoid firesales). Demonstrate that (i) above entails fractional reserve banking. Bank funds assets (cash + capital) with liabilities redeemable on demand for cash. Optimal contract anticipates that only fraction π will withdraw early, but fractional reserves implies bank run eqm.
Steps in our analysis Step 3: Construct Diamond-Dybvig bank assuming runs occur w.p. θ. We think of societies as potentially differing in θ (social fragility). Bank offers investors deposit contracts that maximizes ex ante welfare for a given parameter configuration (θ, µ, η). We identify regions in the parameter space where narrow banking is preferred. We investigate how these regions depend on interest (inflation) rate policy.
Steps in our analysis Step 4: In an extension, we plan to specify more general belief functions. i.e., let θ be a function of economic variables. Jeremy Stein specification θ is decreasing function of yield differential between money and capital. John Cochrane specification θ is decreasing fuction of currency-deposit ratio.
Workers Workers in our model behave the same in all versions of the model. Workers have linear preferences, their decisions essentially pin down eqm pricing relationships. Let p m t, pa t denote price of output in morning and afternoon, resp. The following no-arbitrage-condition must hold: 1/p a t = β/pm t+1, or (p a t /p m t ) = ( p m t+1 /pm t ) /β = (µ/β) = R n R n is the nominal interest rate and R n = µ/β is the Fisher equation.
Monetary equilibrium with securities market New money injected as lump-sum transfers to workers each morning. Let p k t denote price of capital in security market (if it is open) and let ρ p k t /pa t. Let ( c 1, c 2 ) denote allocation when the securities market is open (ĉ 1, ĉ 2 ) when closed. Let k I denote sales of capital by impatient depositor, let k P denote purchases of capital by patient depositor (assuming market is open).
Monetary equilibrium with securities market Investor maximizes chooses consumption and x, k, k I, k P to maximize x k +(1 η) [πu( c 1 ) + (1 π)u( c 2 )]+η [πu (ĉ 1 ) + (1 π)u(ĉ 2 )] subject to (β/µ)x + ρk I + ξ ( k k I) c 1 k k I 0 (β/µ)x ρk P + R(k + k P ) c 2 (β/µ)x ρk P 0
Monetary equilibrium with securities market Lemma: In a stationary monetary equilibrium, ξ ρ R for any η [0, 1). Proposition: A stationary monetary equilibrium implements the first best allocation if and only if η = 0, µ = β and u (c 1 )c 1 = u (c 2 )c 2. Result above shows that monetary equilibrium implements the first best allocation in only a razor-edge case. Sec. mkt. always open, Friedman rule, and (e.g.) log preferences. In general, welfare gains will be possible under FRB system.
Fractional reserve bank Bank offers depositors a security that overcomes absence of liquid secondary market for capital. This new security a demand deposit liability grants depositors the option to convert deposits for cash; permits depositors to avoid firesale events that inflicted investors in monetary economy. But even if secondary market is always open (η = 0) a bank can still improve welfare by offering superior risk-sharing. Banks help complete the missing ex ante insurance market.
Fractional reserve bank assuming no bank run Banks offer depositors a contract that maximizes max { x k + πu(c 1 ) + (1 π)u(c 2 )} (β/µ) x πc 1 Rk + [(β/µ) x πc 1 ] (1 π)c 2 c 2 c 1 0 Proposition: In a FRB equilibrium absent bank runs, the Friedman rule µ = β implements the first-best allocation and the currency-to-deposit ratio x/(x + k) increases/decreases with inflation as u (c)c/u (c) 1.
Financial fragility Well-known that demandable debt structures can be fragile (suffer from multiplicity of equilibria). Unique implementation possible with suffi ciently rich contract space. We assume that banks cannot credibly commit to suspend withdrawals when cash reserves are exhausted. Then a bank run equilibrium is possible, with bank run defined as: patient investors misrepresenting themselves as impatient en masse.
Social fragility We index societies by a parameter θ [0, 1]. Formally, θ is probability of sunspot that triggers a bank-run. We think of θ as indexing the propensity of a society to act on rumors and coordinate on bad outcomes. Low θ societies are inherently stable. High θ societies inherently unstable. Distinction between inherent instability and manufactured instability.
Fractional reserve banking when bank runs are expected Bank offers depostors contract that maximizes { x k + (1 θ) [πu(c1 ) + (1 π)u(c 2 )] + (1 η) θ[πu( c 1 ) + (1 π)u( c 2 )] + ηθ[u(πĉ 1 ) + (1 π)u(ĉ 2 )] } (β/µ) x πc 1 Rk + [(β/µ) x πc 1 ] (1 π)c 2 (β/µ)x + ρk B + ξ(k k B ) c 1 (β/µ)x + ρk B + ξ(k k B ) + (R ρ)k P c 2 (β/µ)x + ξk ĉ 1 (β/µ)x + Rk ĉ 2 k k B 0 (β/µ)x + ρk B + ξ(k k B ) ρk P 0 c 2 c 1 0
Fractional reserve banking when bank runs are expected Proposition: In a FRB system with θ (0, 1], then banks are generally able to meet all their obligations in the event of a run if the securities market is open. Formally, c 1 = c 1 and c 2 = c 2. When secondary market is available, orderly disposal of capital is possible, so bank run does not necessarily result in firesales. Proposition: A FRB system with θ (0, 1] implements the first-best allocation if and only if η = 0, µ = β and u (c 1 )c 1 = u (c 2 )c 2. Franchise value of bank W B (θ, η, µ) is decreasing in (θ, η, µ).
Fractional reserve banking when bank runs are expected Benefit of FRB is higher in high inflation (high interest rate) economy. High µ implies low real return on cash (FRB economizes on cash). Benefits of fractional reserve banking are greatly diminished in socially unstable societies (high θ economies). Especially in economies with very illiquid capital markets (high η economies). FRB may be dominated by a narrow banking structure in low µ, high θ, η economies.
Narrow bank Narrow bank: all demandable debt is backed 100% with cash reserves. Unlike FRB, narrow bank can meet short-term obligations in every state of the world. Consequently, runs are impossible (at cost of excess reserves, poorer risk-sharing). W N (µ) max { x k + πu(c 1 ) + (1 π)u(c 2 )} (β/µ) x c 1 Rk + [(β/µ) x πc 1 ] (1 π)c 2 W N (µ) is independent of θ, η and decreasing in inflation µ.
The choice of banking regime Proposition. There exists ˆθ(η, µ) satisfying W B (ˆθ, η, µ) = W N (µ) such that W B (ˆθ, η, µ) > W N (µ) for θ < ˆθ and W B (ˆθ, η, µ) < W N (µ) for θ > ˆθ. Moreover, ˆθ(η, µ) is strictly increasing in µ. Inherently fragile societies prefer more stable banking regimes. Inherently stable societies prefer more unstable banking regimes. Moreover, high inflation economies encourage unstable banking regimes. Interests of investors not necessarily aligned with those of workers.
The belief function In models with multiple equilibria, the belief parameter can be anything and still be consistent with rational expectations. Nothing in theory dictates θ being purely exogenous. We could in principle assume any belief function θ = B(x), where x includes model parameters and endogenous variables. We have experimented with two belief functions, one which implies θ a decreasing function of µ (Stein) and one which implies θ a decreasing function of µ (Cochrane).
When reach-for-yield promotes instability
Extensions Model liquidation costs more realistically (OTC market structure, disruption in payment system,...). Introduce a role for bank capital (equity), study Basel III regulations (capital adequacy ratio). Consider multibank model with interbank market. Multibank model across two countries, with and without a common currency.