paul ehling and christian heyerdahl-larsen · 2016-02-11 · paul ehling and christian...

CORRELATIONS

Paul Ehling and Christian Heyerdahl-Larsen

Documentos de Trabajo N.º 1413

2014

CORRELATIONS

Documentos de Trabajo. N.º 1413

2014

(*) We would like to thank Suleyman Basak, João Cocco, Ilan Cooper, Nam Huong Dau, Stephan Dieckmann, Giulia Di Nunno, Mike Gallmeyer, Francisco Gomes, João Comes, Trond Stølen Gustavsen, Burton Hollifeld, Philipp Illeditsch, Tom Lindstrøm, Thomas Maurer (CICF discussant), Iñaki Rodríguez Longarela (EFA discussant), Anna Pavlova, Lasse H. Pedersen, Richard Priestley, Skander Van den Heuvel (FIRS discussant), Costas Xiouros, Amir Yaron (AFA discussant), Knut Kristian Aase, Bernt Øksendal, and participants at a BI Brown Bag, the Workshop on Risk Measures and Stochastic Games with Applications to Finance and Economics at the Department of Mathematics of the University of Oslo, the Arne Ryde Workshop in Financial Economics at Lund University, the NHH-UiO Macro Workshop, Nordic Finance Network (NFN) Workshop, SIFR - Institute for Financial Research, Texas A&M, Fourth Annual Empirical Asset Pricing Retreat at the University of Amsterdam, Symposium on Stochastic Dynamic Models in Finance and Economics at the University of Southern Denmark, EFA 2007 meetings, European Meeting of the Econometric Society 2008, FIRS Conference 2009 Prague, a EIEF Brown Bag, AFA 20ll meetings, 4th Financial Risk International Forum in Paris 2011, a Banco de España internal seminar, a LBS Brown Bag, Bank of Canada, Johannes Gutenberg-Universitat Mainz and the 2013 China International Conference in Finance (CICF) in Shanghai for helpful comments and suggestions. We are grateful to Nam Huong Dau and Jing Yu for excellent research assistance. Paul Ehling thanks the Centre for Asset Pricing Research (CAPR) at BI for funding support. Part of this research was conducted while the first author was a Research Fellow at the Banco de España. The views expressed are those of the authors and should not be attributed to the Banco de España.(**) Department of Financial Economics, BI Norwegian Business School, Nydalsveien 37, 0442 Oslo, [email protected].(***) London Business School, Regent’s Park, London, NWl 4SA, [email protected].

Paul Ehling (**)

BI NORWEGIAN BUSINESS SCHOOL

Christian Heyerdahl-Larsen (***)

LONDON BUSINESS SCHOOL

CORRELATIONS (*)

The Working Paper Series seeks to disseminate original research in economics and fi nance. All papers have been anonymously refereed. By publishing these papers, the Banco de España aims to contribute to economic analysis and, in particular, to knowledge of the Spanish economy and its international environment.

The opinions and analyses in the Working Paper Series are the responsibility of the authors and, therefore, do not necessarily coincide with those of the Banco de España or the Eurosystem.

The Banco de España disseminates its main reports and most of its publications via the Internet at the following website: http://www.bde.es.

Reproduction for educational and non-commercial purposes is permitted provided that the source is acknowledged.

© BANCO DE ESPAÑA, Madrid, 2014

ISSN: 1579-8666 (on line)

Abstract

Correlations of equity securities have varied substantially over time and remain a source

of continuing policy debate. This paper studies stock market correlations in an equilibrium

model with heterogeneous risk aversion. In the model, preference heterogeneity causes

countercyclical variations in the volatility of aggregate risk aversion. At times of high volatility

of aggregate risk aversion, which is a common factor in returns, we see high correlations. The

calibrated model matches average industry return correlations and changes in correlations

from business cycle peaks to troughs, and replicates the cyclical dynamics of expected

excess returns and standard deviations. A proxy for model-implied aggregate risk aversion

jointly explains average industry correlations, expected excess returns, standard deviations

and turnover volatility in the data. We fi nd supportive evidence for the model’s prediction that

industries with low dividend-consumption correlation have low average return correlation

but experience disproportionate increases in return correlations in recessions.

Keywords: equity return correlations, heterogeneous risk aversion, volatility of turnover,

cyclical dynamics of stock price moments.

JEL classifi cation: G10, G11.

Resumen

Las correlaciones entre los títulos de renta variable han variado sustancialmente con el tiempo

y siguen siendo una fuente de constante debate de política económica. Este artículo estudia

las correlaciones bursátiles en un modelo de equilibrio con aversión al riesgo heterogénea. En

el modelo, la existencia de preferencias heterogéneas causa variaciones contracíclicas en la

volatilidad de la aversión al riesgo agregada. En episodios de elevada volatilidad de la aversión

al riesgo agregada, que es un hecho común en los rendimientos, se observa una correlación

muy alta. El modelo calibrado ajusta las correlaciones promedio de las rentabilidades de la

industria y los cambios en las correlaciones entre los picos y valles del ciclo económico, al

tiempo que replica la dinámica de los excesos de rentabilidad esperados y las desviaciones

estándar a lo largo del ciclo. Una proxy para la aversión al riesgo agregada implícita del modelo

explica conjuntamente las correlaciones medias de la industria, el exceso de los rendimientos

esperados, las desviaciones estándar y la volatilidad de los rendimientos observadas en los

datos. Finalmente, encontramos evidencia a favor de la predicción del modelo, según la cual

las industrias con baja correlación entre dividendo y consumo tienen una baja correlación en las

rentabilidades medias, pero experimentan subidas desproporcionadas en las correlaciones de

las rentabilidades durante las recesiones.

Palabras clave: tenencias de liquidez, correlación rentabilidades bursátiles, volatilidad

del volumen de facturación, dinámica cíclica de los momentos del precio de las acciones,

aversión al riesgo heterogénea.

Códigos JEL: G10, G11.

BANCO DE ESPAÑA 7 DOCUMENTO DE TRABAJO N.º 1413


0.4 0.73 0.34


0.2 10

1 10

12.1

16.6 8.2 6.8


[0, T ]

N

dδi(t) = δi(t)(μδidt+ σ�

δidW (t)

), σδi ∈ RN , δi(0) > 0, i = 1, ..., N.

N ×N σδ

N × 1 μδ N

W (Ω,F , P, {Ft})

sδ(t) = (sδ1(t), sδ2(t), ..., sδN (t))� =

(δ1(t)

C(t),δ2(t)

C(t), ...,

δN(t)

C(t)

)�, C(t) =

N∑i=1

δi(t)

dsδi(t) = sδi(t)((μsi(t)− σsi(t)

�σC(t))dt+ σsi(t)

�dW (t)),

μsi(t) = μδi − sδ(t)�μδ, σsi(t) = σδi − σ�

δ sδ(t).

dC(t) = C(t)(μC(t)dt+ σC(t)

�dW (t)), μC(t) = sδ(t)

�μδ, σC(t) = σ�δ sδ(t).

[0, T ] Ω Fσ P F(.)

FT = FP


dSi(t) + δi(t)dt = Si(t)(μi(t)dt+ σi(t)

�dW (t)), Si(T ) = 0.

B

dB(t) = r(t)B(t)dt, B(0) = 1.

μi σi r

Uj (C,X) = E0

[∫ T

0

e−ρtuj(Cj(t), X(t))dt

],

uj (Cj(t), X(t)) =1

1− γjCj(t)

1−γjX(t)γj−η, ρ > 0, η ≤ min (γj) = γL

u Cj

X γ

u j = L

j = H η

x(t) = x(0)e−λt + λ

∫ t

0

e−λ(t−u)c(u)du, x(t) = log(X(t)), c(t) = log(C(t)).

λ x

ω = c − x


dω(t) = λ (ω(t)− ω(t)) dt+ σC(t)�dW (t), ω(t) =

μC(t)− 12σC(t)

�σC(t)

λ.

ω

u(C(t), X(t), t) = maxCL(t),CH(t)

{ae−ρt 1

1−γLCL(t)

1−γLX(t)γL−η

+(1− a) e−ρt 11−γH

CH(t)1−γHX(t)γH−η

}

CL(t) + CH(t) = C(t)

a L

CL(t) = f(t)C(t) CH(t) = (1− f(t))C(t),

f(t) =

(a

1− a

) 1γL

e

(γHγL

−1)ω(t)

(1− f(t))γHγL .

f

ω f ω

L

H


R (t) =1

f(t) 1γL

+ (1− f(t)) 1γH

P(t) = (1 + γL)

(R(t)

γL

)2

f(t) + (1 + γH)

(R(t)

γH

)2

(1− f(t)).

R γL γH

R(t) (1 +R(t)− P(t)) σC(t).

P

1+R

ξ(t) = e−ρt

(f(t)

f(0)

)−γL(C(t)

C(0)

)−1

e(1−γL)(ω(t)−ω(0))

r(t) = ρ+ ηλω(t) +R (t) (μC(t)− λω(t))− 1

2R (t)P(t)σC(t)

�σC(t).

θ(t) = R (t) σC(t).


R σC ω

σC

σi(t) =Et

[∫ T

tξ(u)δi(u)

([R(u)− η]λ

∫ u

te−λ(t−v)σC(v)dv

)du

]Et

[∫ T

tξ(u)δi(u)du

]

+Et

[∫ T

tξ(u)δi(u) (θ(t)− θ(u)) du

]Et

[∫ T

tξ(u)δi(u)du

] + σδi .

Rθ

σδi

Y π

Yj(t) =1

ξ(t)Et

[∫ T

t

ξ(u)Cj(u)du

]πj(t) =

(σ(t)�

)−1[θ(t)Yj(t) +

ψj(t)

ξ(t)

],

ψj(t) =

(1

γj− 1

)Et

[∫ T

t

ξ(u)Cj(u)R(u)

[σC(u)− λ

∫ u

t

e−λ(t−v)σC(v)dv

]du

]

+(1− η)Et

[∫ T

t

ξ(u)Cj(u)

[λ

∫ u

t


]du

].

ψj

t u


dR(t)

dRi(t) =dSi(t) + δi(t)dt

Si(t)= μi(t)dt+ σi(t)

�dW (t).

i

V art(dRi(t)) = ‖σi(t)‖2 dt, ‖σi(t)‖ =√σi(t)�σi(t).

i k k = 1, .., N

Covt (dRi(t), dRk(t)) = σi(t)�σk(t)dt i k

Corrt (dRi(t), dRk(t)) =σi(t)

�σk(t)

‖σi(t)‖ ‖σk(t)‖ .

dπj(t) = μπj(t)dt+ σπj

(t)�dW (t)

i jπi,j(t)

Si(t)σπi,j

Si

(t)

σπi,jSi

(t) = σπi,j(t)− σi(t).

πi,L(t)

Si(t)+

πi,H(t)

Si(t)= 1

πi,L(t)

Si(t)σπi,L

Si

(t) +πi,H(t)

Si(t)σπi,H

Si

(t) = 0.


i

RQVt

(d

(πi,L(t)

Si(t)

))=

√(πi,L(t)

Si(t)

)2

σπi,LSi

(t)�σπi,LSi

(t)

RQV i

RQV

RQV

RQV RQV

RQV

sδ1

L

f f

ω

μδ = 0.02 σδ = 0.05

1 50

L 0.5

1% η 1 λ 0.1

23.5 1890 2007

μδ = 0.02 σδ = 0.05

πi,L(t)

Si(t)σπi,L

Si

(t)


γ = 3 γ = 10

sδ1(0)

0 0.029

0.05

3 0.086 10

θ1(t) = γsδ1(t)σδ1 θ2(t) = γ (1− sδ1(t)) σδ2 .

γ

γ = 10

sδ1(0) = 0.5σδ


γ

log

− 3 log − 10

L log − 10

log − 3

0.05 0.2 log − 10 0.8

u(C,X) = log(CX

)u′(C,X) = 1

C X

log − 10


H

γ

3 0.05

log − 3 0.2

log − 3

γ = 3

log − 3

γ

0.4 0.6

log − 3 0.5 1.0 log − 10

T


H

RQV


10

1927 2009

1978

1978 2009

1927 2009

3.73%

6.48%

1927 2009

4.28%

Et

[dδiδi

]= Et

[dδjδj

]V art

[dδiδi

]= V art

[dδjδj

] Covt

[dδiδi

,dδjδj

]

V art[

dδiδi

] = ρδ i �= j

Covt [.] i, j = 1, .., 10 δ1(0) = δ2(0) = ... = δ10(0)

0.019970 0.029397


λ

log − 30 a

η γL 1

ρ

1.1%

6.6% 4.3%

1000.0451 0.1099

0.0070 0.00290.0153 0.0065


211 1927 2009

996 21.18%

ω

ω

0.1046 ω

0.8980 0.9126

ω 0.1046

100 1000

t t t + 36

0.42 0.18 0.410.68 −0.29


ω

ω

ω

∂R(t)/∂exp (ω(t)) ω

γH 10 15

ω


20 40 γH = 30

γL a L

ω 0.1 0.2

γH

γH = 10

3.7%

γH = 10

ω

ω

−0.01 0.53

[0, 0.5]

70% 30%


0 0.5

ω

0.9

i

ω i ω

i ω

ω

ω


−0.6031

−6.5560

0.1946


dCj(t) = Cj(t)(μCj

(t)dt+ σCj(t)�dW (t)

),

μCj(t) =

(R(t)

γj

)μC(t) +

(1− R(t)

γj

)λω(t)

+1

2

[(1 + γj)

(R(t)

γj

)− P(t)

](R(t)

γj

)σC(t)

�σC(t),

σCj(t) =

(R(t)

γj

)σC(t).

dπj(t) = μπj(t)dt+ σπj

(t)dW (t).

dt

N ×N σπj

πj(t) =(σ(t)�

)−1[θ(t)Yj(t) +

ψj(t)

ξ(t)

].

σ θ Yj ψj

ξ σπj(t)

ξ

dYj(t) =(r(t)Yj(t) + πj(t)

� (μ(t)− r(t)I)− Cj(t))dt+ πj(t)

�σ(t)dW (t).


σYj= πj(t)

�σ(t) θ(t) σ(t) ψj(t)

dθ(t) = μθ(t)dt+ σθ(t)dW (t),

dσi(t) = μσi(t)dt+ σσi

(t)dW (t),

dψj(t) = μψj(t)dt+ σψj

(t)dW (t),

σθ σσiσψj

N×N θ(t) = R(t)σC(t)

R(t) σC(t)

dR(t) = d

(−uCC (C(t), X(t), t)

uC (C(t), X(t), t)C(t)

)

= [.] dt+

((uCC (C(t), X(t), t)

uC (C(t), X(t), t)

)2

C(t)− uCC (C(t), X(t), t)

uC (C(t), X(t), t)− uCCC (C(t), X(t), t)

uC (C(t), X(t), t)C(t)

)

×C(t)σ�CdW (t)

= [.] dt+R(t) (1 +R(t)− P(t)) σ�CdW (t);

dσC(t) = d(σ�δ sδ(t)

)= σ�

δ dsδ(t)

= σ�δ Isδ(t) (νs(t)dt+ σs(t)dW (t))

= μσC(t)dt+ σσC

(t)dW (t)

μσC(t) = σ�

δ Isδ(t)νs(t) σσC(t) = σ�

δ Isδ(t)σs(t)

dsδ(t) = Isδ(t) (νs(t)dt+ σs(t)dW (t)) νs(t) =

⎡⎢⎢⎢⎢⎣

μs1(t)− σs1(t)�σC(t)

.

.

μsN (t)− σsN (t)�σC(t)

⎤⎥⎥⎥⎥⎦ ,

dθ(t) = d (R(t)σC(t))

= [.] dt+[R(t) (1 +R(t)− P(t)) σC(t)σC(t)

� +R(t)σσC(t)

]dW (t).


σi(t) = θ(t) +Qi(t)

ξ(t)Si(t)+ σδi Qi(t) = Et

[∫ T

t

ξ(u)δi(u)h(t, u)du

].

Qi

Qi(t) = Et

[∫ T

0


]−

∫ t

0


= QMi (t)−

∫ t

0


QMi Qi

dQMi (t) = σQi

(t)dW (t) σQi(t)� = Et

[∫ T

0

Dt

(ξ(u)δi(u)h(t, u)

�) du] ,σQi

(t) N ×N

Dt

(ξ(u)δi(u)h(t, u)

�) = δi(u)h(t, u)�Dtξ(u) + ξ(u)h(t, u)�Dtδi(u) + ξ(u)δi(u)Dth(t, u)

�

= ξ(u)δi(u)h(t, u)h(t, u)� + ξ(u)δi(u)σδih(t, u)

�

+ξ(u)δi(u)g(t, u).

d (ξ(t)Gi(t)) = ξ(t)Si(t) (σi(t)− θ(t))� dW (t)

dσi(t) = [.] dt+(σθ(t) + σQi

(t) + (θ(t)− σi(t)) (θ(t)− σi(t))�)dW (t)

= [.] dt+ σσi(t)dW (t).

ψj

ψj(t) = Et

[∫ T

0

ξ(u)Cj(u) (h(t, u) +Hj(t, u)) du

].


σψj(t)� = Et

[∫ T

0

Dt (ξ(u)Cj(u) (h(t, u) +Hj(t, u)))

]

= Et

[∫ T

t

{ξ(u)Cj(u)h(t, u)h(t, u)

� + ξ(u)Cj(u)gj(t, u)}du

]

+Et

[∫ T

t

{ξ(u)Cj(u)h(t, u)Hj(t, u)

� + ξ(u)Cj(u)Gj(t, u)}du

].

f


a 0.020.074

γHγL

aρη

λ

BA

NC

O D

E E

SP

AÑ

A40

DO

CU

ME

NT

O D

E T

RA

BA

JO

N.º 1

413

r(t) μM(t)−r(t)M


ω ω

ω


r(t)μM(t) − r(t)

M


ω


r(t)μM(t) − r(t) M

γHγL

aη

BA

NC

O D

E E

SP

AÑ

A45

DO

CU

ME

NT

O D

E T

RA

BA

JO

N.º 1

413

ω

βcorr βstdev βexr ρδi,Ci

ρδi,Cβcorr

βstdev

βexr

ρδi,Cβcorr

βstdev

βexr


βcorr βstdev βexr ρδ,C

R2

BA

NC

O D

E E

SP

AÑ

A47

DO

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NT

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N.º 1

413

Cor

rela

tions

1930 1940 1950 1960 1970 1980 1990 2000 2010

0.4

0.5

0.6

0.7

0.8

0.9

Exc

ess

Ret

urns

1930 1940 1950 1960 1970 1980 1990 2000 2010

−0.4

−0.2

0

0.2

0.4

Sta

ndar

d D

evia

tion

1930 1940 1950 1960 1970 1980 1990 2000 20100.1

0.2

0.3

0.4

0.5

0.6

Qua

drat

ic V

aria

tion

Turn

over

1930 1940 1950 1960 1970 1980 1990 2000 20100.1

0.15

0.2

0.25

0.3

0.35

0.4

BA

NC

O D

E E

SP

AÑ

A48

DO

CU

ME

NT

O D

E T

RA

BA

JO

N.º 1

413

sδ1(0) = δ1(0)/ (δ1(0) + δ2(0))ω L

log − log 3 − 3 10 − 10 3 − 310− 10 log− 3 log− 10

log− 10 ρ0.01 λ 0.10.02 0.05 T 50

0 0.2 0.4 0.6 0.8 10

0.05

0.1

0.15

0.2

sδ1

(0)

Cor

r 0(dR

1(0),d

R2(0

))

Log −−− Log, 3 −−− 3, 10 −−− 10

Log −−− Log3 −−− 310 −−− 10

00.5

1

−0.50

0.51

0.7

0.75

0.8

0.85

0.9

0.95

sδ1

(0)

3 −−− 3

ω(0)C

orr 0(d

R1(0

),dR

2(0))

00.5

1

−0.50

0.51

0.94

0.96

0.98

1

sδ1

(0)

10 −−− 10

ω(0)

Cor

r 0(dR

1(0),d

R2(0

))

00.5

1

0

0.5

1

0

0.1

0.2

0.3

0.4

sδ1

(0)

Log −−− 3

f(0)

Cor

r 0(dR

1(0),d

R2(0

))

00.5

1

0

0.5

1

0

0.2

0.4

0.6

0.8

sδ1

(0)

Log −−− 10

f(0)

Cor

r 0(dR

1(0),d

R2(0

))

0

0.5

1

−0.50

0.51

0.4

0.6

0.8

1

sδ1

(0)

Log −−− 10

ω(0)

Cor

r 0(dR

1(0),d

R2(0

))

BA

NC

O D

E E

SP

AÑ

A49

DO

CU

ME

NT

O D

E T

RA

BA

JO

N.º 1

413

H R−∂R

∂C

γL = γH = R (t)H

C(0) γL = log γH = 3

0 5 100.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

C(0)

CH

(0) /

C(0

)

0 5 100

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

C(0)

−∂ (C

H(0

) / C

(0))

/ ∂

C(0

)

0 5 101

1.5

2

2.5

3

C(0)

R(0

)

0 5 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

C(0)

−∂ R

(0) /

∂ C

(0)

0 5 100

0.05

0.1

0.15

0.2

0.25

0.3

0.35

C(0)

Cor

r 0(dR

1(0),d

R2(0

))

HeterogeneousHomogeneous

0 5 101.5

2

2.5

3

3.5

4

4.5

5x 10−3

C(0)

E0(d

RM

(0)−

r(0)

)

0 5 100.034

0.036

0.038

0.04

0.042

0.044

0.046

C(0)

Std

ev0(d

RM

(0))

0 5 100

0.002

0.004

0.006

0.008

0.01

C(0)

RQ

V0((

dπM

,H(0

))/S

M(0

))

BA

NC

O D

E E

SP

AÑ

A50

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N.º 1

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H γL = log γH = 3

00.20.40.60.81

−0.50

0.51

0.4

0.5

0.6

0.7

sδ1

(0)ω(0)

Cor

r 0(dR

1(0),d

R2(0

))

00.20.40.60.81

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2

3

4

5

6

7

8

x 10−3

sδ1

(0)ω(0)

E0dR

M(0

)−r(

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−0.50

0.51

2

4

6

8

x 10−3

sδ1

(0)ω(0)

Var

0(dR

M(0

))

00.20.40.60.81

−0.50

0.51

1

1.5

2

2.5

3

x 10−3

sδ1

(0)ω(0)

RQ

V0((

dπ1,

H(0

))/S

1(0))

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N.º 1

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ω(t) R(t)∂R(t)/∂exp (ω(t))

Cor

rela

tions

1930 1940 1950 1960 1970 1980 1990 2000

0.5

0.6

0.7

0.8

0.9 DataImplied by Heterogeneous Investor Model

Sta

ndar

d D

evia

tions

1930 1940 1950 1960 1970 1980 1990 20000.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5DataImplied by Heterogeneous Investor Model

Hab

its

1930 1940 1950 1960 1970 1980 1990 2000 2010−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

ω(t)R

isk

Ave

rsio

n D

ynam

ics

1930 1940 1950 1960 1970 1980 1990 2000 2010−120

−100

−80

−60

−40

−20

0

20

R(t)∂ R(t) / ∂ exp(ω(t))

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ω0 0.5

(0, 0.5)11 10 10

0 0.05 0.1 0.15 0.2 0.25 0.30.4

0.6

0.8

1

ω

Ret

urn

corr

elat

ions

0 0.05 0.1 0.15 0.2 0.25 0.30.1

0.2

0.3

0.4

0.5

ω

Ret

urn

stan

dard

dev

iatio

ns

0 0.05 0.1 0.15 0.2 0.25 0.30

0.1

0.2

0.3

0.4

ω

Exp

ecte

d ex

cess

retu

rns

ρδ,C = 0

ρδ,C = 0.5


∗ †

∗

†


maxCj

E

[∫ T

0

e−ρt 1

1− γjCj(t)

1−γjX(t)γj−ηdt

]

E

[∫ T

0

ξ(t)Cj(t)dt

]≤ fY,j(0)E

[∫ T

0

ξ(t)C(t)dt

],

fY,j(0) =Yj(0)

YL(0)+YH(0)j

Cj(t) =(yje

ρtX(t)η−γjξ(t))− 1

γj ,


yj > 0

E

[∫ T

0

ξ(t)(yje

ρtX(t)η−γjξ(t))− 1

γj dt

]= fY,j(0)E

[∫ T

0

ξ(t)C(t)dt

],

u(C(t), X(t), t) = maxCL(t),CH(t)

{ae−ρt 1

1−γLCL(t)

1−γLX(t)γj−η

+(1− a) e−ρt 11−γH

CH(t)1−γHX(t)γj−η

}

s.t.

CL(t) + CH(t) = C(t).

ae−ρt

(CL(t)

X(t)

)−γL

X(t)−η = (1− a) e−ρt

(CH(t)

X(t)

)−γH

X(t)−η.

f(t) = CL(t)C(t)

L

f(t) =

(a

1− a

) 1γL

e

(γHγL

−1)ω(t)

(1− f(t))γHγL .

R(t) = −uCC(C(t), X(t), t)

uC(C(t), X(t), t)C(t),

uC uCC

R

a uL,C(CL, X(t), t) = (1− a) uH,C(CH , X(t), t).


uC(C(t), X(t), t) = a uL,C(CL, X(t), t)∂CL

∂C+ (1− a) uH,C(CH , X(t), t)

∂CH

∂C

= a uL,C(CL, X(t), t)

(∂CL

∂C+

∂CH

∂C

)= a uL,C(CL, X(t), t),

uCC(C(t), X(t), t) = a uL,CC(CL, X(t), t)∂CL

∂C.

j

Aj(t) = −uj,CC(Cj(t), X(t), t)

uj,C(Cj(t), X(t), t).

A(t) = −uCC(C(t), X(t), t)

uC(C(t), X(t), t)

= −a uL,CC(CL(t), X(t), t)

a uL,C(CL(t), X(t), t)

∂CL

∂C

= AL(t)∂CL

∂C.

∂CL

∂C= A(t)

AL(t)∂CH

∂C= A(t)

AH(t)∂CL

∂C+ ∂CH

∂C= 1

A(t)

AL(t)+

A(t)

AH(t)= 1,

A(t) =

(1

AL(t)+

1

AH(t)

)−1

.


R(t) = A(t)C(t)

R(t) = A(t)C(t)

=

(1

AL(t)+

1

AH(t)

)−1

C(t)

=

(CL

C(t)γL+

CH

C(t)γH

)−1

=

(1

γLf(t) +

1

γH(1− f(t))

)−1

.

PA(t)

PA(t) = −uCCC(C(t), X(t), t)

uCC(C(t), X(t), t).

j

PAj (t) = −uj,CCC(Cj(t), X(t), t)

uj,CC(Cj(t), X(t), t).

uCCC(C(t), X(t), t)

uCCC(C(t), X(t), t) =∂2 (a uL,C(CL(t), X(t), t))

∂C2

=∂ (a uL,CC(CL(t), X(t), t)) ∂CL(t)

∂C

∂C

= a uL,CCC(CL(t), X(t), t)

(∂CL(t)

∂C

)2

+a uL,CC(CL(t), X(t), t)∂2CL(t)

∂C2.

uCCC(C(t), X(t), t) = (1− a) uH,CCC(CH(t), X(t), t)

(∂CH(t)

∂C

)2

+a uH,CC(CH(t), X(t), t)∂2CH(t)

∂C2.

∂CL(t)

∂CPA(t) = −∂2CL(t)

∂C2+ PA

L (t)

(∂CL(t)

∂C

)2


∂CH(t)

∂CPA(t) = −∂2CH(t)

∂C2+ PA

H(t)

(∂CH(t)

∂C

)2

.

∂2CL(t)∂C2 + ∂2CH(t)

∂C2 = 0

PA(t) = PAL (t)

( A(t)

AL(t)

)2

+ PAH(t)

( A(t)

AH(t)

)2

.

P(t) = PA(t)C(t)

= (1 + γL)

(R(t)

γL

)2

f(t) + (1 + γH)

(R(t)

γH

)2

(1− f(t)).

R(t) = A(t)C(t)

dR(t) = ...dt+

(C(t)

∂A(t)

∂ωσC(t) +A(t)C(t)σC(t)

)�dW (t)

= ...dt+R(t) (1 +R(t)− P(t)) σC(t)�dW (t) = ...dt+ σR(t)�dW (t).

C

X

dCj(t) =∂Cj(t)

∂CdC(t) +

∂Cj(t)

∂XdX(t) +

1

2

∂2Cj(t)

∂C2(dC(t))2 .

∂Cj(t)

∂C

∂Cj(t)

∂X

∂2Cj(t)

∂C2

∂Cj(t)

∂C=

A(t)

Aj(t),

∂2Cj(t)

∂C2= PA

j (t)

( A(t)

Aj(t)

)2

− PA(t)

( A(t)

Aj(t)

).

f

f eω=C(t)X(t)


∂CL(t)

∂X=

∂f(t)C(t)

∂X

= C(t)∂f(t)

∂X+ f(t)

∂C(t)

∂X

= C(t)∂f(t)

∂ω

∂ω

∂X

= −CL(t)

(R(t)

γL− 1

)1

X(t),

∂ω∂X

= − 1X(t)

∂f(t)∂ω

= f(t)(

R(t)γL

− 1)

∂CH(t)

∂X= −CH(t)

(R(t)

γH− 1

)1

X(t).

C X

ξ(t) =uC(C(t), X(t), t)

uC(C(0), X(0), 0).

dξ(t)

ξ(t)− (

r(t)dt+ θ(t)�dW (t)).

uC(C(t), X(t), t)

duC(C(t), X(t), t) = uCt(C(t), X(t), t)dt+ uCC(C(t), X(t), t)dC(t) + uCX(C(t), X(t), t)dX(t)

+1

2uCCC(C(t), X(t), t) (dC(t))2

= (uCC(C(t), X(t), t)C(t)μC(t) + uCX(C(t), X(t), t)X(t)λω(t)) dt

+

(1

2uCCC(C(t), X(t), t)C(t)2σC(t)

�σC(t) + uCt(C(t), X(t), t)

)dt

+uCC(C(t), X(t), t)C(t)σ�CdW (t).

uCC(C(t), X(t), t)


uCt(C(t), X(t), t) uCX(C(t), X(t), t)

uCt(C(t), X(t), t) = −ρuC(C(t), X(t), t).

uCX(C(t), X(t), t)

uCX(C(t), X(t), t) =∂a uL,C(CL(t), X(t), t)

∂X= (γL − η) a uL,C(CL(t), X(t), t)X(t)−1

+γLC(t)

CL(t)f ′(t)a uL,C(CL(t), X(t), t)X(t)−1

=

(γL − η + γL

C(t)

CL(t)

[ A(t)

AL(t)− f(t)

])a uL,C(CL(t), X(t), t)X(t)−1

= (R(t)− η)uC(C(t), X(t), t)X(t)−1.

f(t) = f (ω(t)) f ′(t) = df(ω(t))dω

∂CL(t)∂C(t)

= ∂f(t)C(t)∂C(t)

= f(t) + C(t)f ′(t) ∂ω(t)∂C(t)

= f(t) + f ′(t) ∂CL(t)∂C(t)

= A(t)AL(t)

uCCC(C(t), X(t), t) = uC(C(t), X(t), t)R(t)P(t)1

C(t)2.

C(t)

X(t)

duC(C(t), X(t), t)

uC(C(t), X(t), t)= −

(ρ+ ηλω(t) +R (t) (μC(t)− λω(t))− 1

2R (t)P(t)σC(t)

�σC(t)

)dt

−R(t)σC(t)�dW (t).

r(t) = ρ+ ηλω(t) +R (t) (μC(t)− λω(t))− 1

2R (t)P(t)σC(t)

�σC(t)

θ(t) = R(t)σC(t).


i = 1, ..., N

Si(t) =1

ξ(t)Et

[∫ T

t

ξ(u)δi(u)du

]

=1

ξ(t)

(MSi

(t)−∫ t

0

ξ(u)δi(u)du

),

MSi(t)

MSi(t) = Et

[∫ T

0

ξ(u)δi(u)du

].

MSi(t)

MSi(t) = MSi

(0) +

∫ t

0

φSi(u)�dW (u),

φSi(t) = Et

[∫ T

t

Dt (ξ(u)δi(u)) du

].

σi = θ(u) +Et

[∫ T

tDt (ξ(u)δi(u)) du

]Et

[∫ T

tξ(u)δi(u)du

] .

Yj =1

ξ(t)Et

[∫ T

t

ξ(u)Cj(u)du

].

πj(t) =(σ(t)�

)−1[θ(t)Yj(t) +

ψj(t)

ξ(t)

],


ψj(t)

ψj(t) = Et

[∫ T

0

Dt (ξ(u)Cj(u)) du

].

u > t u, t ∈ [0, T ] DtF (u) = (D1,tF (u), ..., DN,tF (u))�

Dk,tF (u) k = 1, .., N

Dtδi(u) DtC(u) DtX(u) Dtω(u) Dtsδi(u)

DtσC(u) Dt

(C(u)σC(u)

�) Dtξ(u) DtCj(u) DtA(u) DtR(u) Dtθ(u) Dt (Dt log (ξ(u)))�

Dt (Dtlog (Cj(u)))�

i

Dk,tδi(u) = δi(u)σδi,k .

Dtδi(u) = δi(u)σδi .

ψj(t) Et

[∫ T

0ξ(u)Cj(u)du

]


DtC(u) = Dt

N∑i=1

δi(u)

=N∑i=1

Dtδi(u)

=N∑i=1

δi(u)σδi

= C(u)N∑i=1

sδi(u)σδi

= C(u)σC(u).

X(u)

DtX(u) = X(u)Dt logX(u)

= X(u)λ

∫ u

t

e−λ(t−v)Dtc(v)dv

= X(u)λ

∫ u

t

e−λ(t−v)σC(v)dv.

Dtω(u) = Dt (logC(u)− logX(u))

= σC(u)− λ

∫ u

t


Dtsδi(u) = Dt

(δi(u)

C(u)

)

=Dtδi(u)

C(u)− δi(u)DtC(u)

C(u)2

=δi(u)

C(u)σδi −

δi(u)C(u)σC(u)

C(u)2

=δi(u)

C(u)(σδi − σC(u))

= sδi(u)σsi(u),


DtσC(u)� = Dt

((σ�δ sδ(u)

)�)= Dt

(sδ(u)

�σδ

)= Dt

(N∑i=1

sδi(t)σ�δi

)

=N∑i=1

(Dtsδi(t)) σ�δi

=N∑i=1

sδi(u)σsi(u)σ�δi.

Dt

(C(u)σC(u)

�) = Dt

N∑i=1

(δi(u)σ

�δi

)

=N∑i=1

(Dtδi(u)) σ�δi

=N∑i=1

δi(u)σδiσ�δi

= C(u)N∑i=1

sδi(u)σδiσ�δi.

Dtξ(u) = Dt (uC (C(u), X(u), u))

= uCC (C(u), X(u), u)DtC(u) + uCX (C(u), X(u), u)DtX(u)

= uCC (C(u), X(u), u)C(u)σC(u) + uCX (C(u), X(u), u)X(u)λ

∫ u

t


= uC (C(u), X(u), u)

(−R(u)σC(u) + (R(u)− η)λ

∫ u

t


)= ξ(u)h(t, u)

h(t, u) = −θ(u) + (R(u)− η)λ

∫ u

t



DtCj(u) =∂Cj(u)

∂CDtC(u) +

∂Cj(u)

∂XDtX(u)

= CjHj(t, u),

Hj(t, u) =

(R(u)

γj

)σC(u)−

[(R(u)

γj

)− 1

]λ

∫ u

t


Hj(t, u) = Dtlog (Cj(u))

DtA(u) =∂A(u)

∂CDtC(u) +

∂A(u)

∂XDtX(u)

=

((uCC (C(u), X(u), u)

uC (C(u), X(u), u)

)2

− uCCC (C(u), X(u), u)

uC (C(u), X(u), u)

)DtC(u)

+

(uCC (C(u), X(u), u)

uC (C(u), X(u), u)2uCX (C(u), X(u), u)− uCCX (C(u), X(u), u)

uC (C(u), X(u), u)

)DtX(u)

= A(u)×[(R(u)− P(u))

(σC(u)− λ

∫ u

t


)− λ

∫ u

t


].

DtR(u) = Dt (A(u)C(u))

= C(u)DtA(u) +A(u)DtC(u)

= R(u)

[(1 +R(u)− P(u))

(σC(u)− λ

∫ u

t


)].

Dtθ(u)� = Dt

(R(u)σC(u)�)

= R(u)

[(1 +R(u)− P(u))

(σC(u)− λ

∫ u

t


)]σC(u)

�

+R(u)N∑i=1

sδi(u)σsi(u)σ�δi.

h(t, u) = Dt log (ξ(u))


h(t, u)

Dth(t, u)� = Dt

(−θ(u) + (R(u)− η)λ

∫ u

t


)= −R(u) (1 +R(u)− P(u))

×(σC(u)− λ

∫ u

t


)(σC(u)− λ

∫ u

t


)�

−R(u)N∑i=1

(sδi(u)σsi(u)σ

�δi+

(η

R(u)− 1

)λ

∫ u

t

e−λ(t−v)sδi(v)σsi(v)σ�δidv

)

= g(t, u).

DtHj(t, u)�

DtHj(t, u)� = Dt

((R(u)

γj

)σC(u)−

[(R(u)

γj

)− 1

]λ

∫ u

t


)�

=R(u)

γj(1 +R(u)− P(u))

×(σC(u)− λ

∫ u

t


)(σC(u)− λ

∫ u

t


)�

+R(u)

γj

(N∑i=1

(sδi(u)σsi(u)σ

�δi+

(γj

R(u)− 1

)λ

∫ u

t

e−λ(t−v)sδi(v)σsi(v)σ�δidv

))

= Gj(t, u).

U (c,X; γ) = E0

[∫ T

0

e−ρtu(c(t), X(t); γ)dt

].

u (c(t), X(t); γ) =1

1− γc(t)1−γX(t)γ−η,


γ

supc(C(t),X(t),t;γ)

∫ ∞

1

a (γ)c (C(t), X(t), t; γ)1−γ

1− γX(t)γ−ηdγ

s.t.∫ ∞

1

c (C(t), X(t), t; γ) ≤ C(t)

a (γ)

γ

c(t; γ) = f(t; γ)C(t)

f

f(t; γ) = a (γ)1γ e−ω(t)y (ω(t))−

1γ ,

y (ω(t))

∫ ∞

1

a (γ)1γ e−ω(t)y (ω(t))−

1γ dγ = 1.

R (t) =

(∫ ∞

1

1

γf (t; γ) dγ

)−1

.

P(t) =

∫ ∞

1

(1 + γ)

(R(t)

γ

)2

f (t; γ) dγ.


r(t) = ρ+R (t)μC(t) + (1−R(t))λω(t)− 1

2R (t)P(t)σC(t)

�σC(t).

θ(t) = R (t) σC(t).

a(γ)

a = 1 b = 10.0038

rv

rvadj = 1 + rv ∗ (γmax − 1) γmax = 183.9705 γ 183.9705

γ 1 a = 1 b γMax

30 − log

−0.05 0.35 ω 0.01

ω

30 − log

30 − log

sδ1 = sδ2 = 0.5

sδ1 = sδ2 = 0.5


sδ1 = sδ2 = 0.5

sδ1 = sδ2 = 0.5

sδ1 = sδ2 = 0.5

sδ1 = sδ2 = 0.5

sδ1 = sδ2 = 0.5

sδ1 = sδ2 = 0.5

λ λ

0.1 0.2 0.05 3− log 7

10.5 12 18 ω


14.06% 40.95%

ω


Corrt(Et(dRM(t)− r(t)), ω(t)) < 0,

ω

βt =Covt(Et(RM,t+1 − rt), ωt)

V art(ωt),


ω

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N.º 1

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γHγL

aρ

λ


r(t)μM(t)− 1

2σ2M(t) M


ω

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N.º 1

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ω 30 − log


ω

ω


ω

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N.º 1

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ω30− log

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N.º 1

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ω 30− log

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N.º 1

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ω 30− log


a = 1 b = 10.0038 0 1rvadj = 1 + rv ∗ (γmax − 1) γmax = 183.9705 γ

183.9705 γ 1

0 20 40 60 80 100 120 140γ

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N.º 1

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ω30 − log

ω 30 − log

0 0.1 0.2 0.3ω(0)

0 0.1 0.2 0.30

5

10

15

20

25

30

ω(0)

R(0)

Beta30−Log

0 0.1 0.2 0.35

10

15

20

25

30

35

40

P(0)

ω(0)

Beta30−Log

BA

NC

O D

E E

SP

AÑ

A87

DO

CU

ME

NT

O D

E T

RA

BA

JO

N.º 1

413

C(0) γL = log γH = 3

0 2 4 6 8 100

0.05

0.1

0.15

0.2

0.25

0.3

C(0)

Cor

rela

tion

Cor

r 0(dR

1(0),d

R2(0

))

μD=0.01

μD=0.02

μD=0.03

0 2 4 6 8 100

0.05

0.1

0.15

0.2

0.25

C(0)

Cor

rela

tion

Cor

r 0(dR

1(0),d

R2(0

))

σD=0.02

σD=0.05

σD=0.08

0 2 4 6 8 100

0.05

0.1

0.15

0.2

0.25

0.3

0.35

C(0)

Cor

rela

tion

Cor

r 0(dR

1(0),d

R2(0

))

T=20T=50T=100

0 2 4 6 8 100

0.1

0.2

0.3

0.4

0.5

C(0)

Cor

rela

tion

Cor

r 0(dR

1(0),d

R2(0

))

ρδ=0

ρδ=0.3

BA

NC

O D

E E

SP

AÑ

A88

DO

CU

ME

NT

O D

E T

RA

BA

JO

N.º 1

413

ω(0) γL = log γH = 3

−0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.70.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

ω(0)

Cor

rela

tion

Cor

r 0(dR

1(0),d

R2(0

))

λ=0.05λ=0.1λ=0.2


−0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

pd−r

atio

ω(0)

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Unidad de Servicios AuxiliaresAlcalá, 48 - 28014 Madrid

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