The Risk Channel of Unconventional Monetary Policy Dejanir Silva - PowerPoint PPT Presentation

Firms • Linear technology: Y t = AK t • Law of motion of capital: dK t = g t dt + σ dZ t K t • Problem of the firm   � ∞ π s S t = max ( A − ι ( g s )) K s ds (1) E t   π t g ���� t � �� � q t K t dividends where ι ′ ( · ) , ι ′′ ( · ) > 0. 8 / 36

Firms • Linear technology: Y t = AK t • Law of motion of capital: dK t = g t dt + σ dZ t K t • Problem of the firm   � ∞ π s S t = max ( A − ι ( g s )) K s ds (1) E t   π t g ���� t � �� � q t K t dividends where ι ′ ( · ) , ι ′′ ( · ) > 0. 8 / 36

Firms • Linear technology: Y t = AK t • Law of motion of capital: dK t = g t dt + σ dZ t K t • Problem of the firm   � ∞ π s S t = max ( A − ι ( g s )) K s ds (1) E t   π t g ���� t � �� � q t K t dividends where ι ′ ( · ) , ι ′′ ( · ) > 0. • The SPD π t satisfies (determined in equilibrium): d π t = − r t dt − η t dZ t π t ���� ���� int. rate mkt. price of risk 8 / 36

Firms • Linear technology: Y t = AK t • Law of motion of capital: dK t = g t dt + σ dZ t K t • Problem of the firm   � ∞ π s S t = max ( A − ι ( g s )) K s ds (1) E t   π t g ���� t � �� � q t K t dividends where ι ′ ( · ) , ι ′′ ( · ) > 0. • The SPD π t satisfies (determined in equilibrium): d π t = − r t dt − η t dZ t π t ���� ���� int. rate mkt. price of risk 8 / 36

Active traders • Decision problem of active traders (bankers j = b , savers j = s ): V j = max ( c j ,α j ) U j ( c j ) (2) subject to n j , t ≥ 0 and � � dn j , t r t + α j , t ( µ R , t − r t ) − c j , t = dt + α j , t σ R , t dZ t n j , t n j , t 9 / 36

Active traders • Decision problem of active traders (bankers j = b , savers j = s ): V j = max ( c j ,α j ) U j ( c j ) (2) subject to n j , t ≥ 0 and � � dn j , t r t + α j , t ( µ R , t − r t ) − c j , t = dt + α j , t σ R , t dZ t n j , t n j , t 9 / 36

Active traders • Decision problem of active traders (bankers j = b , savers j = s ): V j = max ( c j ,α j ) U j ( c j ) (3) subject to n j , t ≥ 0 and � � dn j , t r t + α j , t ( µ R , t − r t ) − c j , t = dt + α j , t σ R , t dZ t n j , t n j , t � �� � ≡ σ j , t 9 / 36

Active traders • Decision problem of active traders (bankers j = b , savers j = s ): V j = max ( c j ,σ j ) U j ( c j ) (4) subject to n j , t ≥ 0 and � � dn j , t r t + α j , t ( µ R , t − r t ) − c j , t = dt + σ j , t dZ t n j , t n j , t 9 / 36

Active traders • Decision problem of active traders (bankers j = b , savers j = s ): V j = max ( c j ,σ j ) U j ( c j ) (5) subject to n j , t ≥ 0 and   � µ R , t − r t �   dn j , t − c j , t   =  r t + α j , t σ R , t  dt + σ j , t dZ t   n j , t σ R , t n j , t � �� � ≡ σ j , t � �� � η t 9 / 36

Active traders • Decision problem of active traders (bankers j = b , savers j = s ): V j = max ( c j ,σ j ) U j ( c j ) (6) subject to n j , t ≥ 0 and � � dn j , t r t + σ j , t η t − c j , t = dt + σ j , t dZ t n j , t n j , t 9 / 36

Active traders • Decision problem of active traders (bankers j = b , savers j = s ): V j = max ( c j ,σ j ) U j ( c j ) (6) subject to n j , t ≥ 0 and � � dn j , t r t + σ j , t η t − c j , t = dt + σ j , t dZ t n j , t n j , t • Preferences: continuous-time EZ • EIS ψ > 1 and risk aversion γ j • Savers are more risk averse than bankers: γ s > 1 > γ b Empirical studies on heterogeneous risk aversion 9 / 36

Active traders • Decision problem of active traders (bankers j = b , savers j = s ): V j = max ( c j ,σ j ) U j ( c j ) (6) subject to n j , t ≥ 0 and � � dn j , t r t + σ j , t η t − c j , t = dt + σ j , t dZ t n j , t n j , t • Preferences: continuous-time EZ • EIS ψ > 1 and risk aversion γ j • Savers are more risk averse than bankers: γ s > 1 > γ b Empirical studies on heterogeneous risk aversion • Mortality risk ⇒ stationary distribution 9 / 36

Risk Concentration and Balance sheets Proposition (Risk Concentration) Suppose γ s > γ b , γ s − γ b small, then σ b , t − σ s , t > 0 . 10 / 36

Risk Concentration and Balance sheets Proposition (Risk Concentration) Suppose γ s > γ b , γ s − γ b small, then σ b , t − σ s , t > 0 . Riskless asset Riskless liability Net Worth Risky Claims Risky Claims Net Worth Bankers Savers 10 / 36

Passive Traders and the Central Bank • Passive traders invest on market portfolio ⇒ n p , t = α S t c p , t = α D t + T t (7) ���� ���� dividends transfers Evidence on passive behavior/limited participation 11 / 36

Passive Traders and the Central Bank • Passive traders invest on market portfolio ⇒ n p , t = α S t c p , t = α D t + T t (7) ���� ���� dividends transfers Evidence on passive behavior/limited participation • Central bank is subject to No-Ponzi condition and � � dn cb , t r t + σ cb , t η t − T t = dt + σ cb , t dZ t (8) n cb , t n cb , t 11 / 36

Passive Traders and the Central Bank • Passive traders invest on market portfolio ⇒ n p , t = α S t c p , t = α D t + T t (7) ���� ���� dividends transfers Evidence on passive behavior/limited participation • Central bank is subject to No-Ponzi condition and � � dn cb , t r t + σ cb , t η t − T t = dt + σ cb , t dZ t (8) n cb , t n cb , t 11 / 36

Passive Traders and the Central Bank • Passive traders invest on market portfolio ⇒ n p , t = α S t c p , t = α D t + T t (7) ���� ���� dividends transfers Evidence on passive behavior/limited participation • Central bank is subject to No-Ponzi condition and � � dn cb , t r t + σ cb , t η t − T t = dt + σ cb , t dZ t (8) n cb , t n cb , t 11 / 36

Passive Traders and the Central Bank • Passive traders invest on market portfolio ⇒ n p , t = α S t c p , t = α D t + T t (7) ���� ���� dividends transfers Evidence on passive behavior/limited participation • Central bank is subject to No-Ponzi condition and � � dn cb , t r t + σ cb , t η t − T t = dt + σ cb , t dZ t (8) n cb , t n cb , t 11 / 36

Passive Traders and the Central Bank • Passive traders invest on market portfolio ⇒ n p , t = α S t c p , t = α D t + T t (7) ���� ���� dividends transfers Evidence on passive behavior/limited participation • Central bank is subject to No-Ponzi condition and � � dn cb , t r t + σ cb , t η t − T t = dt + σ cb , t dZ t (8) n cb , t n cb , t ( σ cb , t , T t ) determined by policy rules σ cb , t = σ cb ( x t , w t ); T t = T ( x t , w t ) where ( x t , w t ) is the vector of state variables. 11 / 36

Aggregate State Variables and Market Clearing Define ( x t , w t ) as follows n b , t n cb , t x t = w t = n b , t + n s , t n b , t + n s , t + n cb , t 12 / 36

Aggregate State Variables and Market Clearing Define ( x t , w t ) as follows n b , t n cb , t x t = w t = n b , t + n s , t n b , t + n s , t + n cb , t 12 / 36

Aggregate State Variables and Market Clearing Define ( x t , w t ) as follows n b , t n cb , t x t = w t = n b , t + n s , t n b , t + n s , t + n cb , t 12 / 36

Aggregate State Variables and Market Clearing Define ( x t , w t ) as follows n b , t n cb , t x t = w t = n b , t + n s , t n b , t + n s , t + n cb , t Market clearing condition for risk: x t σ b , t + (1 − x t ) σ s , t = ω r t ( σ + σ q , t ) � �� � net supply of risk 12 / 36

Aggregate State Variables and Market Clearing Define ( x t , w t ) as follows n b , t n cb , t x t = w t = n b , t + n s , t n b , t + n s , t + n cb , t Market clearing condition for risk: x t σ b , t + (1 − x t ) σ s , t = ω r t ( σ + σ q , t ) � �� � net supply of risk 12 / 36

Aggregate State Variables and Market Clearing Define ( x t , w t ) as follows n b , t n cb , t x t = w t = n b , t + n s , t n b , t + n s , t + n cb , t Market clearing condition for risk: x t σ b , t + (1 − x t ) σ s , t = ω r t ( σ + σ q , t ) � �� � net supply of risk share of risk held by active traders ω r t ≡ share of wealth held by active traders 12 / 36

Aggregate State Variables and Market Clearing Define ( x t , w t ) as follows n b , t n cb , t x t = w t = n b , t + n s , t n b , t + n s , t + n cb , t Market clearing condition for risk: x t σ b , t + (1 − x t ) σ s , t = ω r t ( σ + σ q , t ) � �� � net supply of risk share of risk held by active traders ω r t ≡ share of wealth held by active traders Market clearing condition for consumption: A − ι ( g t ) c s , t = ω d x t ˆ c b , t + (1 − x t )ˆ t q t 12 / 36

Aggregate State Variables and Market Clearing Define ( x t , w t ) as follows n b , t n cb , t x t = w t = n b , t + n s , t n b , t + n s , t + n cb , t Market clearing condition for risk: x t σ b , t + (1 − x t ) σ s , t = ω r t ( σ + σ q , t ) � �� � net supply of risk share of risk held by active traders ω r t ≡ share of wealth held by active traders Market clearing condition for consumption: A − ι ( g t ) c s , t = ω d x t ˆ c b , t + (1 − x t )ˆ t q t 12 / 36

Aggregate State Variables and Market Clearing Define ( x t , w t ) as follows n b , t n cb , t x t = w t = n b , t + n s , t n b , t + n s , t + n cb , t Market clearing condition for risk: x t σ b , t + (1 − x t ) σ s , t = ω r t ( σ + σ q , t ) � �� � net supply of risk share of risk held by active traders ω r t ≡ share of wealth held by active traders Market clearing condition for consumption: A − ι ( g t ) c s , t = ω d x t ˆ c b , t + (1 − x t )ˆ t q t t ≡ share of dividends consumed by active traders ω d share of wealth held by active traders Definition ( ω r t , ω d Definition equilibrium t ) 12 / 36

Unconventional Monetary Policy: Policy Rule 0.5 0.45 Banker’s share of wealth: x t 0.4 • Strong balance sheet ⇒ ω r = 1 0.35 • Weak balance sheet ⇒ ω r < 1 0.3 1- ω r 0.25 0.2 Unconventional ”Greenspan’s put” 0.15 • CB intervene in bad times 0.1 0.05 Transfers/low values of w 0 0 0.2 0.4 0.6 0.8 1 x 13 / 36

Outline Environment 1 Balance sheet recession and the risk channel 2 Risk concentration and financial stability 3 Exit strategies 4 Long-term bonds 5 Effectiveness of asset purchases 6 Conclusion 7 13 / 36

Two benchmarks 1) Homogeneous risk-aversion ( γ b = γ s ) : • No risk concentration σ b , t = σ s , t • Balanced growth path • No balance sheet recession 14 / 36

Two benchmarks 1) Homogeneous risk-aversion ( γ b = γ s ) : • No risk concentration σ b , t = σ s , t • Balanced growth path • No balance sheet recession 2) Full participation benchmark: No passive traders. Fix initial ( σ cb , T ) and consider ( σ ∗ cb , T ∗ ). • Savers exactly offset policy change: σ ∗ � � s , t n ∗ s , t − σ s , t n s , t = − σ ∗ cb , t n ∗ cb , t − σ cb , t n cb , t • Neutrality result: no changes in consumption, prices, and investment • Modigliani-Miller / Ricardian Equivalance type of result (see Wallace (1981)). 14 / 36

Net worth multipliers and demand for risk Net worth multipliers: V b ( n b , t ; x t , w t ) = ( ζ t n b , t ) 1 − γ b V s ( n s , t ; x t , w t ) = ( ξ t n s , t ) 1 − γ s 1 − γ b 1 − γ s • Net worth multiplier ζ t ( x t , w t ) (determined in equilibrium) d ζ t = µ ζ, t dt + σ ζ, t dZ t ζ t 15 / 36

Net worth multipliers and demand for risk Net worth multipliers: V b ( n b , t ; x t , w t ) = ( ζ t n b , t ) 1 − γ b V s ( n s , t ; x t , w t ) = ( ξ t n s , t ) 1 − γ s 1 − γ b 1 − γ s • Net worth multiplier ζ t ( x t , w t ) (determined in equilibrium) d ζ t = µ ζ, t dt + σ ζ, t dZ t ζ t 15 / 36

Net worth multipliers and demand for risk Net worth multipliers: V b ( n b , t ; x t , w t ) = ( ζ t n b , t ) 1 − γ b V s ( n s , t ; x t , w t ) = ( ξ t n s , t ) 1 − γ s 1 − γ b 1 − γ s • Net worth multiplier ζ t ( x t , w t ) (determined in equilibrium) d ζ t = µ ζ, t dt + σ ζ, t dZ t ζ t Demand for risk: η t + 1 − γ b σ b , t = ; σ ζ, t γ b γ b ���� � �� � myopic hedging 15 / 36

Net worth multipliers and demand for risk Net worth multipliers: V b ( n b , t ; x t , w t ) = ( ζ t n b , t ) 1 − γ b V s ( n s , t ; x t , w t ) = ( ξ t n s , t ) 1 − γ s 1 − γ b 1 − γ s • Net worth multiplier ζ t ( x t , w t ) (determined in equilibrium) d ζ t = µ ζ, t dt + σ ζ, t dZ t ζ t Demand for risk: η t + 1 − γ b • Myopic demand: current returns σ b , t = ; σ ζ, t γ b γ b ���� � �� � myopic hedging 15 / 36

Net worth multipliers and demand for risk Net worth multipliers: V b ( n b , t ; x t , w t ) = ( ζ t n b , t ) 1 − γ b V s ( n s , t ; x t , w t ) = ( ξ t n s , t ) 1 − γ s 1 − γ b 1 − γ s • Net worth multiplier ζ t ( x t , w t ) (determined in equilibrium) d ζ t = µ ζ, t dt + σ ζ, t dZ t ζ t Demand for risk: η t + 1 − γ b • Myopic demand: current returns σ b , t = ; σ ζ, t γ b γ b • Hedging demand: cyclicality of returns ���� � �� � myopic hedging 15 / 36

Market price of risk   � �  1 − γ b σ ζ, t + (1 − x t )1 − γ s   ω r   η t = γ t t ( σ + σ q , t ) − x t σ ξ, t   γ b γ s ���� � �� �  agg. risk aversion � �� � net supply of risk avg. hedging demand where � x t � − 1 + 1 − x t γ t ≡ γ b γ s 16 / 36

Market price of risk   � �  1 − γ b σ ζ, t + (1 − x t )1 − γ s   ω r   η t = γ t t ( σ + σ q , t ) − x t σ ξ, t   γ b γ s ���� � �� �  agg. risk aversion � �� � net supply of risk avg. hedging demand where � x t � − 1 + 1 − x t γ t ≡ γ b γ s 16 / 36

Market price of risk   � �  1 − γ b σ ζ, t + (1 − x t )1 − γ s    ω r  η t = γ t t ( σ + σ q , t ) − x t σ ξ, t   γ b γ s ���� � �� �  agg. risk aversion � �� � net supply of risk avg. hedging demand where � x t � − 1 + 1 − x t γ t ≡ γ b γ s Riskless asset Net Worth Riskless liability Deposits Risky Claims Risky Claims Risky Claims Net Worth Bankers Savers 16 / 36

Market price of risk   � �  1 − γ b σ ζ, t + (1 − x t )1 − γ s   ω r   η t = γ t t ( σ + σ q , t ) − x t σ ξ, t   γ b γ s ���� � �� �  agg. risk aversion � �� � net supply of risk avg. hedging demand where � x t � − 1 + 1 − x t γ t ≡ γ b γ s Riskless asset Net Worth Riskless liability Risky Claims Risky Claims Net Worth Bankers Savers 16 / 36

Market price of risk   � �  1 − γ b σ ζ, t + (1 − x t )1 − γ s   ω r   η t = γ t t ( σ + σ q , t ) − x t σ ξ, t   γ b γ s ���� � �� �  agg. risk aversion � �� � net supply of risk avg. hedging demand where � x t � − 1 + 1 − x t γ t ≡ γ b γ s Riskless asset Net Worth Risky Claims Riskless liability Risky Claims Net Worth Bankers Savers 16 / 36

Market price of risk   � �  1 − γ b σ ζ, t + (1 − x t )1 − γ s   ω r   η t = γ t t ( σ + σ q , t ) − x t σ ξ, t   γ b γ s ���� � �� �  agg. risk aversion � �� � net supply of risk avg. hedging demand where � x t � − 1 + 1 − x t γ t ≡ γ b γ s Riskless asset Net Worth Risky Claims Riskless liability Risky Claims Net Worth Bankers Savers 16 / 36

Market price of risk   � �  1 − γ b σ ζ, t + (1 − x t )1 − γ s   ω r   η t = γ t t ( σ + σ q , t ) − x t σ ξ, t   γ b γ s ���� � �� �  agg. risk aversion � �� � net supply of risk avg. hedging demand where � x t � − 1 + 1 − x t γ t ≡ γ b γ s Riskless asset Net Worth Risky Claims Riskless liability Risky Claims Net Worth Bankers Savers 16 / 36

Balance sheet recession A − ι ( g t ) ι ′ ( g t ) = q t q t = r t + ( σ + σ q , t ) η t − µ S , t 17 / 36

Balance sheet recession A − ι ( g t ) ι ′ ( g t ) = q t q t = r t + ( σ + σ q , t ) η t − µ S , t 17 / 36

Balance sheet recession A − ι ( g t ) ι ′ ( g t ) = q t q t = r t + ( σ + σ q , t ) η t − µ S , t 17 / 36

Balance sheet recession A − ι ( g t ) ι ′ ( g t ) = q t q t = r t + ( σ + σ q , t ) η t − µ S , t Market price of risk Price of capital 1 0.95 0.8 0.9 0.6 0.85 log(q) η 0.4 0.8 0.2 0.75 0 0.7 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 x x Numerical solution 17 / 36

Balance sheet recession A − ι ( g t ) ι ′ ( g t ) = q t q t = r t + ( σ + σ q , t ) η t − µ S , t Market price of risk Price of capital 1 0.95 0.8 0.9 0.6 0.85 log(q) η 0.4 0.8 0.2 0.75 0 0.7 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 x Laissez-faire Central bank x Numerical solution 17 / 36

Effect on Interest Rates Interest rate 3 Weak balance sheet of bankers: 2.5 • High aggregate risk aversion 2 • Precautionary savings 1.5 r 1 0.5 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x 18 / 36

Effect on Interest Rates Interest rate 3 Weak balance sheet of bankers: 2.5 • High aggregate risk aversion 2 • Precautionary savings 1.5 r Effect of asset purchases: 1 • Precautionary savings • Intertemporal substitution 0.5 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x 18 / 36

Taking stock: crises vs normal times Balance sheet recession • Weak balance sheet of bankers ⇒ high risk premium • High risk premium ⇒ low price of risky asset • EIS > 1 important for this result 19 / 36

Taking stock: crises vs normal times Balance sheet recession • Weak balance sheet of bankers ⇒ high risk premium • High risk premium ⇒ low price of risky asset • EIS > 1 important for this result Unconventional monetary policy • Asset purchases ⇒ drop in market price of risk / rise in interest rates • Crises ⇒ risk premium dominates • Normal times ⇒ interest rate dominates 19 / 36

Outline Environment 1 Balance sheet recession and the risk channel 2 Risk concentration and financial stability 3 Exit strategies 4 Long-term bonds 5 Effectiveness of asset purchases 6 Conclusion 7 19 / 36

Risk Concentration and Financial Stability So far: • Effect on crises and normal times • What about probability of crises? • Concerns about reach-for-yield 20 / 36

Risk Concentration and Financial Stability So far: • Effect on crises and normal times • What about probability of crises? • Concerns about reach-for-yield Endogenous risk • Concentration of risk on bankers is endogenous • Risk concentration ⇒ endogenous volatility • Endogenous volatility ⇒ stationary distribution 20 / 36

Myopic and Hedging Demands η t + 1 − γ b σ b , t = ; σ ζ, t γ b γ b ���� � �� � myopic hedging 21 / 36

Myopic and Hedging Demands η t + 1 − γ b σ b , t = ; σ ζ, t γ b γ b ���� � �� � myopic hedging 21 / 36

Myopic and Hedging Demands η t + 1 − γ b σ b , t = ; σ ζ, t γ b γ b ���� � �� � myopic hedging 21 / 36

Myopic and Hedging Demands η t + 1 − γ b σ b , t = ; σ ζ, t γ b γ b ���� � �� � myopic hedging 21 / 36

Myopic and Hedging Demands η t + 1 − γ b • Sensitivity to η t : decreasing in γ b . σ b , t = ; σ ζ, t γ b γ b ���� � �� � myopic hedging Leverage Myopic component Hedging component 15 15 0 -0.5 -1 Hedging component Myopic component 10 10 σ b / ( σ + σ q ) -1.5 -2 5 5 -2.5 -3 0 0 -3.5 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 x x x 21 / 36

Myopic and Hedging Demands η t + 1 − γ b • Sensitivity to η t : decreasing in γ b . σ b , t = ; σ ζ, t γ b γ b ���� � �� � • γ b < 1, σ ζ, t < 0 ⇒ negative hedging dem. myopic hedging Leverage Myopic component Hedging component 15 15 0 -0.5 -1 Hedging component Myopic component 10 10 σ b / ( σ + σ q ) -1.5 -2 5 5 -2.5 -3 0 0 -3.5 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 x x x 21 / 36

UMP and Risk Concentration σ b , t = η t + 1 − γ b σ s , t = η t + 1 − γ s σ ζ, t ; σ ξ, t ; γ b γ b γ s γ s Risk concentration Myopic component Hedging component 0.8 0.8 0 0.7 0.7 -0.05 0.6 0.6 Hedgind component Myopic component 0.5 0.5 -0.1 σ b - σ s 0.4 0.4 -0.15 0.3 0.3 0.2 0.2 -0.2 0.1 0.1 0 0 -0.25 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 x x x 22 / 36

Hedging and return-sensitivity effects Two opposing forces: UMP reduces counter-cyclicality of returns Bankers increase hedging demand Risk more concentrated on bankers (hedging) 23 / 36

Hedging and return-sensitivity effects Two opposing forces: UMP reduces counter-cyclicality UMP reduces current returns of returns Bankers reduce Bankers increase myopic demand hedging demand Risk more concentrated on bankers Risk less concentrated on bankers (hedging) (return-sensitivity) 23 / 36

Hedging and return-sensitivity effects Two opposing forces: UMP reduces counter-cyclicality UMP reduces current returns of returns Bankers reduce Bankers increase myopic demand hedging demand Risk more concentrated on bankers Risk less concentrated on bankers (hedging) (return-sensitivity) Why does risk concentration fall? (intuition) • Hedging effect relies on drop of endogenous volatility • If hedging effect dominates ⇒ increase in risk concentration • Endogenous volatility would increase: contradiction 23 / 36

Endogenous volatility σ q , t = q x , t σ x , t + q w , t σ x , t = x t (1 − x t ) ( σ b , t − σ s , t ) σ w , t q t q t � �� � risk concentration 24 / 36

Endogenous volatility σ q , t = q x , t σ x , t + q w , t σ x , t = x t (1 − x t ) ( σ b , t − σ s , t ) σ w , t q t q t � �� � risk concentration 24 / 36

Endogenous volatility σ q , t = q x , t σ x , t + q w , t σ x , t = x t (1 − x t ) ( σ b , t − σ s , t ) σ w , t q t q t � �� � risk concentration 24 / 36