asset quality dynamicserwan.marginalq.com/index_files/wp_files/slidesuga.pdf1998 10 35 158 208 387...

Asset Quality Dynamics

Dean Corbae and Erwan Quintin

Wisconsin School of Business

March 1, 2018

Corbae Quintin Asset Quality Dynamics

Motivation

I Different corporate liabilities display different cyclicalities(Jerman and Quadrini, 2011, Covas and Den Haan, 2011)

I Debt use tends to be pro-cyclical, except for the largestfirms

I Safe corporate debt, in fact, is acyclical, at best (Erel, Kimand Weisbach, 2012)

I We propose a model that is quantitatively consistent withthis fact . . .

I . . . and use it to quantify the impact of safe corporate debtmarkets on the US business cycle

Motivation

I Debt use tends to be pro-cyclical,

except for the largestfirms

Motivation

Methodological approach

I We lay out a macroeconomic model that is standard on thereal side but where the security space respondsendogenously to changes in fundamentals

I We do so by embedding Allen and Gale’s 1988 “OptimalSecurity Design” model into a dynamic environment

I Fixed point problem:1. Taking the contingent path of financial structures as given,

agents choose an optimal consumption policy2. This consumption path, in turn, determines agents’

willingness to pay for different securities3. Taking this willingness to pay as given, producers issue

menus of securities that maximize their profits4. The resulting financial structure must coincide with the

guess agents made in the first place

Methodological approach

I We lay out a macroeconomic model that is standard on thereal side but where the security space respondsendogenously to changes in fundamentals

I We do so by embedding Allen and Gale’s 1988 “OptimalSecurity Design” model into a dynamic environment

I Fixed point problem:1. Taking the contingent path of financial structures as given,

agents choose an optimal consumption policy2. This consumption path, in turn, determines agents’

willingness to pay for different securities3. Taking this willingness to pay as given, producers issue

menus of securities that maximize their profits4. The resulting financial structure must coincide with the

guess agents made in the first place

I Firms in Compustat, 1985-2016 (394,682 firm-year)I Exclude foreign, utility and financial firms (197,629)I Exclude missing or bad data (96,994)

Firm count details

AAA AA A BBB <BBB No rating

1986 13 64 159 108 258 2,040

1987 13 59 137 117 316 2,023

1988 13 60 136 100 291 1,996

1989 14 51 140 102 230 1,980

1990 14 54 129 107 255 2,098

1991 13 55 130 114 222 2,292

1992 13 55 131 117 225 2,526

1993 13 54 134 132 270 2,697

1994 13 49 136 141 283 2,863

1995 12 48 135 155 289 3,251

1996 12 45 146 176 331 3,282

1997 12 43 149 186 336 3,175

1998 10 35 158 208 387 3,244

1999 9 32 144 218 383 3,065

2000 9 27 136 223 406 2,739

2001 8 22 124 223 425 2,522

2002 7 20 115 219 465 2,516

2003 6 15 113 220 482 2,531

2004 6 13 115 210 504 2,466

2005 6 12 113 205 509 2,466

2006 6 11 113 201 495 2,399

2007 6 11 101 203 478 2,287

2008 6 12 93 200 454 2,226

2009 6 13 86 194 461 2,190

2010 5 14 83 197 448 2,166

2011 4 13 89 209 441 2,131

2012 4 13 87 205 453 2,198

2013 4 13 88 212 472 2,232

2014 4 15 90 211 488 2,095

2015 3 19 87 208 467 2,046

2016 3 20 76 209 456 1,9422

Frequency of default by rating, 1920-2010

AAA AA A BAA BA B CAA-C

Rating by size

20.00%

40.00%

60.00%

80.00%

100.00%

>AA >A >BBB

Large firms (>99%) Small firms

Cyclicality of safe corporate debt

Firm rating ≥ AA ≥ A ≥ BBB <BBB Allρ(D,Y ) 0.06 0.28∗ 0.29∗ 0.70∗∗∗ 0.55∗∗∗

ρ (E ,Y ) 0.07 −0.07 0.06 0.33∗∗ 0.27∗

HP filter

Cyclicality of safe corporate yields

1985-2006 1985-2016 1947-2016ρ(AAA yield,Y ) -0.0439 -0.2114 -0.3060

ρ(BAA-AAA spread,Y ) -0.2991 -0.3774 -0.6028

Related work

1. Allen and Gale (1988, 1991)2. Quadrini and Jerman (2011), Covas and Den Haan (2011),

Karabarbounis, Macnamara and McCord (2014), Eler et. al(2012), Khale and Stulz (2011)

3. Bernanke et. al. (2011)4. Brunnermeier and Sannikov (2012)

The model

I Time is discrete and infinite, one goodI Mass 1 of households who value consumption only,

time-separable CRRA preferencesI Large mass of producers characterized by

talent/productivity z ∼ µ

Producers

I Can activate a project by investing 1 unit of capital at thestart of the period

I Gross output isAtz1−αnαt + (1− δ)

I At is aggregate TFP and nt is labor inputI TFP follows a first-order markov process with transition GA

I Net operating income is:

Π(At ,wt ; z) ≡ maxn>0

Atz1−αnα − nwt

I Labor demand is:

n∗(At ,wt ; z) ≡ arg maxn>0

Producers

I Can activate a project by investing 1 unit of capital at thestart of the period

I Gross output isAtz1−αnαt + (1− δ)

I At is aggregate TFP and nt is labor inputI TFP follows a first-order markov process with transition GA

I Net operating income is:

Π(At ,wt ; z) ≡ maxn>0

I Labor demand is:

n∗(At ,wt ; z) ≡ arg maxn>0

Security markets

I Producers sell claims to households and to world marketsI Take as given the households’ willingness to pay qH

I Claims sold to world markets must be risk-freeI Participation in that market costs κ > 0 per periodI World markets pay qW

t per unit of risk-free claims at date tI qW is measurable with respect to the history of aggregate

shocks

Security markets

t per unit of risk-free claims at date t

I qW is measurable with respect to the history of aggregateshocks

Security markets

t per unit of risk-free claims at date tI qW is measurable with respect to the history of aggregate

shocks

Producer problem

MVt (z) ≡ maxbs≥0

bsqWt +

qHt (A)

[(Π(A,wt ; z) + 1− δ)− bs

−(1 + 1{bs>0}κ

subject to:

bs ≤ Π(At ,w ; z)

Debt vs. EBITDA

Producers issue risk-free debt if:(qW

t −∫

t (A)dA)

Π(At ,wt ; z) ≥ κ.

Producer problem

MVt (z) ≡ maxbs≥0

bsqWt +

qHt (A)

[(Π(A,wt ; z) + 1− δ)− bs

−(1 + 1{bs>0}κ

subject to:

bs ≤ Π(At ,w ; z)

Debt vs. EBITDA

Producers issue risk-free debt if:(qW

t −∫

t (A)dA)

Π(At ,wt ; z) ≥ κ.

Risky security prices and returns

I Securities sold by producers of type z pay

Π(A,wt ; z) + 1− δ − bst (z)

and sell for:∫A

qHt (A)

[(Π(A,wt ; z) + 1− δ)− bs

I Stochastic return on the same security is

rt (A; z) =Π(A,wt ; z) + 1− δ − bs

t (z)∫A qH

[(Π(A,wt ; z) + 1− δ)− bs

Risky security prices and returns

I Securities sold by producers of type z pay

Π(A,wt ; z) + 1− δ − bst (z)

and sell for:∫A

qHt (A)

[(Π(A,wt ; z) + 1− δ)− bs

I Stochastic return on the same security is

rt (A; z) =Π(A,wt ; z) + 1− δ − bs

t (z)∫A qH

[(Π(A,wt ; z) + 1− δ)− bs

Household problem

I Ht : possible histories of aggregate TFP shocks up to date tI Households assume a mapping

St : Ht 7→ S

I . . . where St (ht ) = {qWt , rt (•, z) : z ≥ 0}

Household problem

maxbd≥0,ed≥0

E+∞∑t=0

βtU(ct )

subject to:

qWt bd

t (z)dµ(z) + ct = at (ht ) +

∫max {MVt (z),0}dµ(z),

at+1(ht ,A) =

t (z)rt (A, z)dµ(z)

+ bdt + wt (ht ,A), for all A ∈ A

where:{qW

t , rt (•, z) : z ≥ 0} = St (ht ),

Equilibrium

Prices, security menus and and decisions such that:1. Decision plans are optimal given prices;2.∫{z:MVt (z)≥0} n∗(A,wt ; z) = 1 for all A ∈ A;

3.∫{z:MVt (z)≥0} bs

t (z)dµ ≥ bdt ;

4. edt (z)rt (A, z) = Π(A,wt ; z) + 1− δ − bs(z) for all A ∈ A;

5. MVt (z) = bst (z)qW

t +∫A qH

[(Π(A,wt ; z) + 1− δ)−

bst (z)

]dA−

(1 + 1{bs

t >0}κ).

6. qHt (At ) =

βGA(At |At−1)U′(ct+1(ht ,At ))U′(ct )

Equilibrium

Prices, security menus and and decisions such that:1. Decision plans are optimal given prices;2.∫{z:MVt (z)≥0} n∗(A,wt ; z) = 1 for all A ∈ A;

3.∫{z:MVt (z)≥0} bs

t (z)dµ ≥ bdt ;

4. edt (z)rt (A, z) = Π(A,wt ; z) + 1− δ − bs(z) for all A ∈ A;

5. MVt (z) = bst (z)qW

t +∫A qH

[(Π(A,wt ; z) + 1− δ)−

bst (z)

]dA−

(1 + 1{bs

t >0}κ).

6. qHt (At ) =

βGA(At |At−1)U′(ct+1(ht ,At ))U′(ct )

Aggregation

Given capital K , labor N and exogenous TFP A, aggregateoutput is:

F (A,K ,N) = AE [z|z ≥ z(K )]1−α K 1−αNα.

GDP accounting:

ct+1+Kt+1−(1−δ)Kt +

t >0κdµ+bW

t −qWt+1bW

t+1 = F (At ,Kt ,Nt ).

Aggregation

Given capital K , labor N and exogenous TFP A, aggregateoutput is:

F (A,K ,N) = AE [z|z ≥ z(K )]1−α K 1−αNα.

GDP accounting:

ct+1+Kt+1−(1−δ)Kt +

t >0κdµ+bW

t −qWt+1bW

t+1 = F (At ,Kt ,Nt ).

Security markets

PropositionThe solution to the producer security design problem at a givendate t is fully described by two thresholds 0 ≤ z t ≤ zt such that:

1. Producers issue securities if and only if z ≥ z t ;2. bs

t (z) = 0 if z < zt ;3. bs

t (z) = Π(At ,wt ; z) if z > zt .

Proof: Producers issue risk-free debt if:(qW

t −∫

t (A)dA)

Π(At ,wt ; z) ≥ κ.

Security markets

PropositionThe solution to the producer security design problem at a givendate t is fully described by two thresholds 0 ≤ z t ≤ zt such that:

1. Producers issue securities if and only if z ≥ z t ;2. bs

t (z) = 0 if z < zt ;3. bs

t (z) = Π(At ,wt ; z) if z > zt .

Proof: Producers issue risk-free debt if:(qW

t −∫

t (A)dA)

Π(At ,wt ; z) ≥ κ.

Household portfolio

Effectively, households invest in one security/portfolio whosestochastic payoff is

F (At ,Kt ,1) + (1− δ)Kt − bWt ,

and whose price at the start of the period is∫A

qHt (A)

[F (At ,Kt ,1) + (1− δ)Kt − bW

hence whose return, for all possible values of At is

rH(At ) =F (At ,Kt ,1) + (1− δ)Kt − bW

t∫A qH

[F (At ,Kt ,1) + (1− δ)Kt − bW

Recursive equilibrium

I Aggregate state is: θ = (a,A−1) ∈ IR+ ×A,I An equilibrium consists of the following objects:

1. g : Θ×A 7→ Θ2. K : Θ 7→ IR+

3. z × z : Θ 7→ IR2+

4. qH : Θ×A 7→ IR+

5. rH : Θ×A 7→ IR+

6. eH : Θ× IR+ 7→ IR+

7. c : Θ× IR+ 7→ IR+

8. w : Θ×A 7→ IR+

9. bW : Θ 7→ IR+

10. MV : Θ× IR+ 7→ IR+

11. V H : Θ× IR+ 7→ IR

Household value function

V H(θ,a) = maxeH>0,b>0

∫z≥z

MV (z)dz − eH − qW b)

V H (g(θ,A),a′(A))dG(A|A−1)

where, for all A ∈ A

a′(A) = b + eH rH(θ,A) + w(θ,A).

Allen-Gale condition

PropositionThe household value function V H is concave and differentiablein assets. Furthermore, for all possible values θ ∈ Θ of theaggregate state,

∂V (θ,a)

∂a= U

′(c(θ,a)) .

RCE conditions

1. K (θ) =∫

z≥z dµ

2. eH(θ,a) =∫A qH(θ,A)

[F (A,K (θ),1) + (1− δ)K (θ)− bW (θ)

z≥z MV (θ, z)dz = qW (θ)bW (θ) +∫A qH

[(Π(A,w(θ); z) + 1− δ)− bs(θ, z)

4. c(θ,a) = a +∫

z≥z MV (θ, z)dz − eH(θ,a)

5. a′(θ,A) = eH(θ,A)rH(θ,A) + w(θ,A)

6. qH(θ,A) =βG(A|A−1)U

′(c i (g(θ,A),a′(θ,A))

U′ (c(θ,a)for A ∈ A

7. Financial structure solves producer problems

Mapping from model to data

max {MVt (z),0}dµ(z)

⇒ Income of top managers andproprietors

2. Y ≡ F (A,K ,N)⇒ Value added by the private sector minusthe compensation of top managers and proprietors

max {MVt (z),0}dµ(z)⇒ Income of top managers andproprietors

2. Y ≡ F (A,K ,N)

⇒ Value added by the private sector minusthe compensation of top managers and proprietors

Parameters set to standard values

Parameter Description Valueβ Discount rate 0.95σ Utility curvature 2.00δ Depreciation rate 0.10α Labor share 0.60

Calibration

Parameter Value Target Data ModelTFP (A) {0.97,1.00,1.03} σ(log(Y )) 2.75% 2.51%

σ(log(z)) 0.40 RentsY 11.00% 10.80%

κ 0.0043 Ds

Y 3.11% 3.47%

Transition matrix for TFP:

0.47 0.53 0.000.28 0.44 0.280.00 0.53 0.47

Risk-free process:

qW = {0.972,0.968,0.976}

Calibration

σ(log(z)) 0.40 RentsY 11.00% 10.80%

κ 0.0043 Ds

Y 3.11% 3.47%

0.47 0.53 0.000.28 0.44 0.280.00 0.53 0.47

Risk-free process:

qW = {0.972,0.968,0.976}

Calibration

σ(log(z)) 0.40 RentsY 11.00% 10.80%

κ 0.0043 Ds

Y 3.11% 3.47%

0.47 0.53 0.000.28 0.44 0.280.00 0.53 0.47

Risk-free process:

qW = {0.972,0.968,0.976}

Calibration

σ(log(z)) 0.40 RentsY 11.00% 10.80%

κ 0.0043 Ds

Y 3.11% 3.47%

0.47 0.53 0.000.28 0.44 0.280.00 0.53 0.47

Risk-free process:

qW = {0.972,0.968,0.976}

Calibration

σ(log(z)) 0.40 RentsY 11.00% 10.80%

κ 0.0043 Ds

Y 3.11% 3.47%

0.47 0.53 0.000.28 0.44 0.280.00 0.53 0.47

Risk-free process:

qW = {0.972,0.968,0.976}

Calibration

σ(log(z)) 0.40 RentsY 11.00% 10.80%

κ 0.0043 Ds

Y 3.11% 3.47%

0.47 0.53 0.000.28 0.44 0.280.00 0.53 0.47

Risk-free process:

qW = {0.972,0.968,0.976}

Calibration

σ(log(z)) 0.40 RentsY 11.00% 10.81%

κ 0.0043 Ds

Y 3.11% 2.61%

0.47 0.53 0.000.28 0.44 0.280.00 0.53 0.47

Risk-free process:

qW = {0.972,0.968,0.976} ⇒ ρ

qW − 1,Y)

= −0.31

Basic horse race

Recall that producers issue risk-free debt if(qW

t −∑

qHt (A)

)Π(At ,wt ; z) ≥ κ.

Two sources of procyclicality for safe debt use:1. At is procyclical (Jerman-Quadrini effect)2. qW is procyclical

Two sources of countercyclicality:1. qH is procyclical2. wt is procyclical

Basic horse race

t −∑

qHt (A)

)Π(At ,wt ; z) ≥ κ.

Basic horse race

t −∑

qHt (A)

)Π(At ,wt ; z) ≥ κ.

RCE (1)

3 3.5 4 4.5 5 5.5 6 6.5 7

Assets

0.45Household state prices

qAhigh

3 3.5 4 4.5 5 5.5 6 6.5 7

Assets

5.5Capital stock

Medium

3 3.5 4 4.5 5 5.5 6 6.5 7

Assets

1.9Consumption

3 3.5 4 4.5 5 5.5 6 6.5 7

Assets

2.1Aggegrate output (F)

Low TFP

Med TFP

High TFP

RCE (2)

3 3.5 4 4.5 5 5.5 6 6.5 7

Assets

1Household willingness to pay for risk-free claims

3 3.5 4 4.5 5 5.5 6 6.5 7

Assets

20Securitization cutoffs

3 3.5 4 4.5 5 5.5 6 6.5 7

Assets

1Fraction of establishments that issue risk-free debt

3 3.5 4 4.5 5 5.5 6 6.5 7

Assets

3.5Volume of risk-free debt

Sample paths

Cyclical properties of key model variables

Moment Data Modelρ(DS,Y ) 0.06 -0.06

ρ(ES,Y ) 0.07 -0.05

ρ(E ,Y ) 0.27 0.21

ρ (D + E ,Y ) 0.62 0.57

ρ(E(rH)− rF ,Y

)-0.25 -0.04

ρ(E(rH)− rF , I

)-0.37 -0.27

ρ(E(rH)− rF ,C

)-0.54

ρ(ES,Y ) 0.07 -0.05

ρ(E ,Y ) 0.27 0.21

ρ (D + E ,Y ) 0.62 0.57

ρ(E(rH)− rF ,Y

)-0.25 -0.04

ρ(E(rH)− rF , I

)-0.37 -0.27

ρ(E(rH)− rF ,C

)-0.54

ρ(ES,Y ) 0.07 -0.05

ρ(E ,Y ) 0.27 0.21

ρ (D + E ,Y ) 0.62 0.57

ρ(E(rH)− rF ,Y

)-0.25 -0.04

ρ(E(rH)− rF , I

)-0.37 -0.27

ρ(E(rH)− rF ,C

)-0.54

ρ(ES,Y ) 0.07 -0.05

ρ(E ,Y ) 0.27 0.21

ρ (D + E ,Y ) 0.62 0.57

ρ(E(rH)− rF ,Y

)-0.25 -0.04

ρ(E(rH)− rF , I

)-0.37 -0.27

ρ(E(rH)− rF ,C

)-0.54

ρ(ES,Y ) 0.07 -0.05

ρ(E ,Y ) 0.27 0.21

ρ (D + E ,Y ) 0.62 0.57

ρ(E(rH)− rF ,Y

)-0.25 -0.04

ρ(E(rH)− rF , I

)-0.37 -0.27

ρ(E(rH)− rF ,C

)-0.54

ρ(ES,Y ) 0.07 -0.05

ρ(E ,Y ) 0.27 0.21

ρ (D + E ,Y ) 0.62 0.57

ρ(E(rH)− rF ,Y

)-0.25 -0.04

ρ(E(rH)− rF , I

)-0.37 -0.27

ρ(E(rH)− rF ,C

)-0.54

ρ(ES,Y ) 0.07 -0.05

ρ(E ,Y ) 0.27 0.21

ρ (D + E ,Y ) 0.62 0.57

ρ(E(rH)− rF ,Y

)-0.25 -0.04

ρ(E(rH)− rF , I

)-0.37 -0.27

ρ(E(rH)− rF ,C

)-0.54

Tranching cost and capital formation

3 3.5 4 4.5 5 5.5 6 6.5 71.5

BenchmarkHigh kappa

Low safe yields and capital formation

3 3.5 4 4.5 5 5.5 6 6.5 72

BenchmarkLow safe yields

Safe debt markets and the business cycle

Benchmark High κ Low safe yieldsMean Ds

Y 3.47% 0.00% 17.55%

Mean Y 1.9540 -0.01% -0.03%std(log(Y ) 0.0250 -1.60% -0.40%

Mean Cons 1.5840 +0.23% -0.81%std(log(C)) 0.0148 +6.48% +0.00%

Mean Assets 5.1825 +1.40% -5.53%std(log(a)) 0.0275 -16.73% +17.09%

Mean K 3.6656 +0.05% -0.05%std(log(K )) 0.0271 -2.21% -7.75%

Y 3.47% 0.00% 17.55%

Mean Y 1.9540 -0.01% -0.03%std(log(Y ) 0.0250 -1.60% -0.40%

Mean K 3.6656 +0.05% -0.05%std(log(K )) 0.0271 -2.21% -7.75%

Y 3.47% 0.00% 17.55%

Mean Y 1.9540 -0.01% -0.03%std(log(Y ) 0.0250 -1.60% -0.40%

Mean K 3.6656 +0.05% -0.05%std(log(K )) 0.0271 -2.21% -7.75%

Y 3.47% 0.00% 17.55%

Mean Y 1.9540 -0.01% -0.03%std(log(Y ) 0.0250 -1.60% -0.40%

Mean K 3.6656 +0.05% -0.05%std(log(K )) 0.0271 -2.21% -7.75%

Y 3.47% 0.00% 17.55%

Mean Y 1.9540 -0.01% -0.03%std(log(Y ) 0.0250 -1.60% -0.40%

Mean K 3.6656 +0.05% -0.05%std(log(K )) 0.0271 -2.21% -7.75%

Summary

1. A dynamic model of costly security creation can accountfor the acyclicality of safe corporate debt issues

2. Access to safe debt markets has little effect on the level ofoutput but helps reduce consumption volatility

3. Exogenous, permanent reductions in safe yields havelimited effects on the level of GDP because the reduction ininterest rates depresses household wealth accumulation

Cyclicality of safe corporate debt (HP filter)

Firm rating ≥ AA ≥ A ≥ BBB <BBB Allρ(D,Y ) 0.22 0.23 0.28 0.67∗∗∗ 0.47∗∗∗

ρ (E ,Y ) 0.07 −0.16 0.09 0.31 0.31

Safe debt vs. EBITDA

100000

200000

300000

400000

500000

600000

1985 1990 1995 2000 2005 2010 2015

Debt (US$, million) EBITDA (US$, million)

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