6 January 2009
FINANCE
No. 01
Solvency constraints and dynamics of
prices of risky assets
Abstract
We analyze, from a theoretical point of view, the effect on the
equilibrium of the market of a risky financial asset of the introduction of
a class of investors facing a solvency constraint, like the one put in place
by the “Solvency” regulation of institutional investors. These investors
face a limit in their capability of investing in risky assets if the prices of
these assets decline to an excessive extent, as this reduces their capital.
We show that the equilibrium price of the risky asset can, in this
situation, be lower than the one obtained without the solvency constraint,
and can even decline without limit.
We also question the relevance of the solvency constraint.
JEL Classification: G 11 –
G 18
G 18 Patrick Artus
Author:
Secretary: Laurence Sanchez-Garrido
Contraintes de solvabilité et dynamique
des prix des actifs risqués
Résumé
Nous analysons d’un point de vue théorique l’effet de la présence sur le marché d’un actif risqué d’une
catégorie d’investisseurs confrontés à une contrainte de solvabilité, inspirée de la réglementation
« Solvency » des investisseurs institutionnels. Ces investisseurs peuvent être limités dans leur capacité
d’achat d’actifs risqués si le prix des actifs risqués tombe à un niveau trop bas et que ceci réduit leur capital.
On montre alors que le prix d’équilibre de l’actif risqué peut se placer à un niveau inférieur au prix qui se
réalise sans contrainte, ou même peut chuter sans limite.
Nous nous interrogeons aussi sur la pertinence de la contrainte de solvabilité.
Document de travail n° 01
-2-
INTRODUCTION
An abundant literature has studied the destabilising effects of some rules put in place in financial markets.
We are referring to the capital ratios banks have to comply with (so-called “Basel” ratios).
Basel 1 ratios (link between demanded capital and the amount of risky assets held) have a pro-cyclical effect,
since banks face an active constraint in terms of capital required during recessions, when they incur losses
and are affected by a reduction in their shareholders’ equity, and accordingly reduce the credit supply1 to a
noticeable extent as well as the amount of risky assets they hold.
Basel 2 ratios (Basel Committee (1999–2001)) introduced a link between the capital requirement for banks
and the rating (either by an agency or carried out internally) of borrowers. Since a differentiation is drawn
between the capital required according to the level of risk of borrowers, the incentive to lend to risky
borrowers found with Basel 1 ratios has to disappear, but the pro-cyclical effect is further increased2, since
the rating deteriorates during recessions, and this increases the intensity of the minimum capital constraint.
Next, we have margin calls, i.e. the fact that investors who use debt leverage have to increase the guarantee
deposits they have with lenders when the prices of risky assets they hold decline.
Chowdry-Nanda (1988) show how, in this context, there are multiple equilibria, and furthermore show the
difference between an initial decline in asset prices linked to fundamentals or linked to a market hazard. In
order to avoid instability, they either propose very high margin rates, or the introduction of caps on asset
prices. Bernardo-Welch (2004) highlight the fact that investors carry out “fire sales” when they realize they
could be forced to sell assets before their prices recover. Ewerhart-Valla (2007) show that the fall in the
equilibrium prices of assets due to forced sell-offs leads to investors defaulting although initial losses are
low.
Brunnermeier-Pedersen (2007) start off from the fact that there are multiple equilibria which they interpret as
a discontinuity in liquidity, and draw a distinction between the case where prices drop because the
fundamentals deteriorate and the one where they fall because liquidity decreases or volatility increases. Like
Chordia-Sarkar-Subrahmanyam (2005) they show that forced sales result in the crisis spreading from one
asset market to others. Artus (2008A) shows that the fact there are margin calls leads to multiple equilibria,
all the more easily as investors anticipate future margin calls.
Brunnermeier-Pedersen (2005) go further since they introduce a manipulation of forced sales. Some traders
sell in order to trigger a crisis affecting other traders and be able to buy assets from them at very low prices.
We analyse in this paper a similar issue with regard to the solvency rules imposed on investors (Solvency
and Solvency II regulations for institutional investors). The authorised amount of risky assets investors can
hold is lowered when their shareholders’ equity diminishes following a decline in the prices of the risky
assets they hold. This introduces a growing relationship between the capacity to hold risky assets and the
prices of these assets.
We analyse the effect on the equilibrium of the market of risky assets resulting from the fact that investors
face this kind of constraint as well as the pertinence of the introduction of this type of constraint.
1
Blum-Hellwig (1995), Thakor (1996), Ferri-Liu-Bernanke-Lown (1991), Berger-Udell (1994), Berger-Kashyap-Scalise (1995), Hancock-LaingWilcox (1995), Peek-Rosengreen (1995), Borio-Furfine-Lowe (2001), Gordy-Howells (2004); Kashyap-Stein (2004).
2
Monfort-Mulder (2000), Ferri-Liu-Majnoni (2000), Altman-Saunders (2001); the effect of capital ratios increased by the fact that rating agencies
themselves behave in a pro-cyclical manner, Artus (2004), Artus (2008B).
Document de travail n° 01
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1 – Our initial model
In this initial model, we consider only one kind of investor; i.e. investors who build a portfolio made up of
two assets: a risky asset, with an uncertain future income; and a risk-free asset, for which they postulate
interest rates equal to 0 for the sake of simplicity. They do not face any further constraint. At date t ,
they receive a noisy signal from the future income of the risky asset.
The risky asset has a price Pt in period t ; at date t , investors receive the signal S t from the future income
Rt 1 of the risky asset:


(1) S t  Rt 1   t ;  t  N 0, Z 2 and is independent from Rt 1 .
In principle, Rt 1 has the following distribution of probability:

(2) Rt 1  N R, 2

Investors therefore calculate the conditional expectation (in view of the information available in t ) and the
conditional variance of Rt 1 (the future income of the risky asset) in compliance with:

2
Z2
R 2
 E t Rt 1   S t 2

 Z2
 Z2
(3) 
2 2
Var R    Z
t 1

2  Z2
There are N similar investors. They maximise an expected utility function written as expectation –
variance; they each hold 1 to invest, and allocate  t to the risky asset, by carrying out:


(4) Max E t   t 1 

 
 
Rt 1  Pt
Pt
 1

 R 
  1   t    Vart   t t 1 
 2
Pt 



where  measures the degree of risk aversion.
As a result, we have:
(5)  t  Pt
Et Rt 1   Pt 
Vart Rt 1 
 t is the part of total wealth 1 invested in risky assets at date t ; the number of risky assets each investor
wants to hold is thus
t
Pt
.
We denote  t the number of risky assets offered at date t.
Document de travail n° 01
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We write:

(6) t    t ; t  N 0,  2

The supply of risky assets is affected by a hazard at  t which is a white noise;  t is independent from  t
and Rt .
The equilibrium of the market of the risky asset at date t is therefore written:
(7) N
t
Pt
 t
or, for the equilibrium price.
(8) Pt  Et Rt 1  
   t Vart Rt 1 
N
where Et Rt 1  and Vart Rt 1  are given by (3), or alternatively:
(9) Pt  S t
2
2  Z2
R
Z2
   t


2  Z2  N
 2Z 2




2
2
  Z
The equilibrium price of the risky asset, Pt , rises in line with the signal received from the future income of
asset, S t , and decreases in line with the supply of risky assets in period t,    t .
The income of each investor is:

R P
(10) U t  t 1  t 1 t
Pt

(7) shows that  t 

  1  t 

Pt   t 
N
hence
(11) U t  1 
where Pt is given by (9) and where
S t  Rt 1   t
  t
N
Rt 1  Pt 
or alternatively
  t
(12) U t  1  
 N
Document de travail n° 01
  t
2
 2Z 2
  Z2

Rt 1  R   2 2  t 
 2
 2
2
N
 Z
  Z2
   Z




-5-
Hence the unconditional expectation of the investor’s income:
(13) E Ut   1 

1
 2Z 2

 2  2
2
2
2
N  Z

The unconditional variance of the investor’s income is, by omitting the higher order terms3:
2 
2
     Z
(14) Var U t    
 N     2  Z 2

2
 2  2
  

 2  Z 2


2
 2 4 2   2 Z 2
 Z 

2  2
2

N

  Z
2
 2 
 




2 – Constraint on the amount of risky assets held linked to capital
We suppose that a proportion  of investors is accordingly restricted in terms of its purchases of risky
assets by the capital K held by each one of these investors.
The maximum amount that can be invested in risky assets is:

(15)   K Pt  P

If    , the investor can invest  in risky assets,  is the demand (5) for risky asset in the previous
section.
If    , investors facing constraints (a proportion  ) invest  and not  in risky assets. (15) is a simple
way to describe the solvency constraints: the quantity of risky asset the investor can hold varies in line
with the price of this asset; when the price of this asset falls, the investor loses some of its shareholders’
equity, and can hold fewer risky assets.
Thus if:


(16) K Pt  P  Pt
Et Rt 1   Pt
Vart Rt 1 
the equilibrium of the market of the risky asset is written:
(17) N

Pt
 N 1   

Pt
 t
Or alternatively:
(18) NK
3
The variances of R, and 
Document de travail n° 01
Pt  P
E R   Pt
 N 1    t t 1
 t
Pt
Vart Rt 1 
2
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We denote:
(19) Pt *  S t
2
2  Z2
R
Z2
2  Z2
 t
  2Z 2
N 2  Z2
the equilibrium price at date t, without any constraint, of the market of the risky asset.
If:
 Pt *  P  E t Rt 1   Pt *  t


 P* 



Var
R
N
t
t

1
t


(20) K 
The equilibrium remains the equilibrium Pt  Pt * .
However, if the inverse inequality of (20) is verified, then the equilibrium of the market of the risky asset is
written:
(21) NK
Pt  P
Pt
 N 1   
Et Rt 1   Pt
 t
Vart Rt 1 
In Pt*  Pt* :
(22)
Et Rt 1   Pt t
P P

 K t
Vart Rt 1 
N
Pt
The equilibrium can be represented graphically as follows, when the constraint (16) is active, from:
(21’) 1   
Et Rt 1   Pt t
P  P t
P

 K t

 K  K
Vart Rt 1 
N
Pt
N
Pt
For the sake of simplicity, we suppose that (21’) has a solution. If this is the case, there are two
solutions.
In Pt  Pt * , 1   
Et Rt 1   Pt  t
P P

 K t
because of (22).
Vart Rt 1 
N
Pt
There are thus two cases (if there are two solutions):
(a) Case 1
(23)
Document de travail n° 01
1 
P
 K * 2
Vart Rt 1 
Pt
-7-
The equilibrium is then graphically represented:
Diagram 1
Equilibrium with equilibrium price lower than Pt*

N
 K
PP
P
1    ER  P
VarR
1    
N
P0
P1
Pt
Pt*
There are then two equilibria P0 and P1 with P0  P1  Pt*
(b) Case 2
(24)
Document de travail n° 01
1 
P
 K * 2
Vart Rt 1 
Pt
-8-
We then have:
Diagram 2
Equilibrium with an equilibrium price higher than Pt*

N
 K
PP
P
1    ER  P
VarR
1    
N
Pt*
P0
P1
Pt
There are two equilibria P0 and P1 with Pt*  P0  P1
In case 1, at the “normal” equilibrium price of the risky asset, Pt* , a decline in prices increases to a
greater extent demand for the risky asset for investors who do not face constraints than it reduces
demand for risky asset for investors facing constraints.
In Pt* , there is excess supply of the risky asset. If the price declines from Pt* , one heads towards the
equilibrium P1 of Diagram 1.
In case 2, in Pt* a decline in prices increases to a lesser extent demand for the risky asset for investors
who do not face constraints than it reduces demand for risky asset for investors facing constraints.
When the price drops from Pt* , the excess supply of the risky asset increases, and the price diverges
downwards.
We therefore have:
- a case (case 1) where the investor’s solvency constraint (16) drives down the equilibrium price of
the risky asset from Pt* to P1 .
Document de travail n° 01
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- a case (case 2) where this constraint leads to a bottomless drop in the equilibrium price of the
risky asset.
Case 1 occurs (cf. (23)) when:

Pt *  P Et Rt 1   Pt *

(contrainte active)
K
Vart Rt 1 
Pt *

(25) 
1 
K P 
cas (1) se réalise 
*2




Var
R
P
t
t

1
t

[active constraint
case (1) occurs]
We denote P̂t such as:
(26A) K
Pˆt  P Et Rt 1   Pˆt

Vart Rt 1 
Pˆt
~
and Pt such as:
P
1 
(26B) K ~ 2 
Vart Rt 1 
Pt
~
We suppose that P  Pt  Pˆt .
Case 1 occurs when:
~
(27) Pt  Pt*  Pˆt
Case 2 occurs when:

Pt *  P Et Rt 1   Pt *
contrainte active 

K
Vart Rt 1 
Pt *

(28) 
1 
K P 
*
2

Vart Rt 1 
Pt

[active constraint]
or:
~
(29) Pt*  Pt  Pˆt
Document de travail n° 01
- 10 -
~  KP Vart Rt 1   2 ~
ˆ
Since Pt  
 , Pt  Pt if  (the proportion of investors who can face constraints) is
1




1
not too high. If  is high, the market cannot be rebalanced by changes in demand for the risky asset among
~
investors who do not face constraints 1   est petit , [is low] we have Pt  Pˆt and we are still in the
~
divergent case 2. We suppose that P  Pt  Pˆt .
We can therefore see that:
 if the signal S t of the future income of the risky asset is good, Pt*  Pˆt and we are in the case
without any constraint;
~
 if the signal S t is bad, but not very bad, Pt  Pt*  Pˆt , and we are in case 1 with a lower
equilibrium price, P1  Pt* ;
~
 if the signal S t is very bad, Pt*  Pt  Pˆt we are in case 2 and the equilibrium price drops towards
0. If the investors affected by the solvency constraint cannot take up a short position, the price then
declines only to P where their demand for the risky asset vanishes.
Let us suppose that this constraint of a lack of short positions operates. Case 2 is then graphically
represented as follows:
Diagram 3
Case 2 with a lower equilibrium price than P


N
N
 K
PP
P
Pt
P2
P
Pt
*
P0
P1
1    E R   P
Var R 
Document de travail n° 01
- 11 -
For Pt  P , only the investors who do not face constraints are found in the market, and the equilibrium
(which is found in P2  is given by:
(30) 1   
Et Rt 1   Pt t

Vart Rt 1 
N
Or:
(30’) Pt  Et Rt 1   
t  2 Z 2
1
2
2
N   Z 1 
which differs from the normal equilibrium price (9) by a higher risk premium (divided by 1   .
3 – Interest of the constraint
Here, for the sake of simplicity, we are going to suppose that the supply of the asset is not
random t   ;  t  0 .
To recap:
(a) If (20) is verified, the equilibrium is given by Pt  Pt* . (20) is rewritten (with  t  0 :
(31) Pt * 
KP
 Pˆ

K 
N
and in this case:
(32) Pt  Pt*  Rt 1   t 
2
2  Z2
R
Z2

2  Z2

 2Z 2
N 2  Z2


(b) If (25) is verified (intermediary values of Pt * , we have:
 *
KP
 Pˆ
 Pt 


K 
N




 2Z 2
(33) 

K

P

2 Z2
 Pt *  
1 






Document de travail n° 01






1
2
~
P
- 12 -
and in this case, we have Pt  P1t such that:
(34) 1    
Et Rt 1   P1t

 Z
2  Z2
2
2


N
 K
P1t  P
P1t
~
(c) If Pt*  P , then, as we saw, the equilibrium price is lower than P , and only the investors who do not face
constraints trade:
2
 2Z 2 1
(35) Pt  Rt 1   t  2
R 2
 
 Z2
  Z 2 N  2  Z 2 1 
Z2

To make matters even simpler, let us suppose that the intermediary case (b) is either very unlikely or
~
even P  Pˆ disappears.


We then have only the following solution:

2
Z2
  2Z 2

R


 Pt  Rt 1   1  2
2


Z2

Z2 N 2 Z2
(36A) 
2
Z2
  2Z 2
quand R    

R


 Pˆ
t 1
t

2 Z2
2 Z2 N 2 Z2
[when]
and

2
Z2
  2Z 2
1


P

R



R

 2
 t
t 1
1
2
2
2
2
2
N   Z 1 

 Z
 Z
(36B) 
2
2
Z
  2Z 2
quand R    

R


 Pˆ
t 1
1

2  Z2
2  Z2 N 2  Z2
In the first case (36A), all the investors have the following income:
(37A) U t  1 

N

Rt 1  Pt   1   
Z2
N  2  Z
Rt 1  R  
2
2
2 Z
 
2 t
 2 Z 2 
N  2  Z 2 


In the second case (36B):
 investors facing constraints (a proportion   cannot buy the risky asset, and they have income 1 (they
hold only the risk-free asset);
 investors who do not face constraints (a proportion 1    have the following income:
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(37B) U t  1 
1  Z2
2
  2Z 2
1 




R

R



 2
t 1
2
2
2
2 t
2

N 1    Z
N   Z 1   
 Z

We denote:
(38) V 
 2Z 2
2


 2Z 2
 Pˆ  R  

N 2  Z2





We are in the first case when Rt 1  R   t  V ; and in the second case when Rt 1  R   t  V .
The income of investors changes in line with:
(39) X 
Z2
2 Z2
Rt 1  R   t    t
The solvency constraint normally enables investors to avoid losses due to an excessively negative value
of X , defined by (39).
When Rt 1  R   t  V , X would then need to have a high probability of being low.
When Rt 1  R   t  V , then X 
Z2
2  Z2
V  t .
If the signal is precise,  t is low and Z 2  0. In this case, X  0. If the signal is imprecise,  t can be
high, Z 2 is high.
In this case X  V   t , which is not necessarily low if the hazard  t is negative and high.
A solvency constraint linking investors’ capacity to hold risky assets at the present prices of these
assets is meaningful only if investors receive a precise enough signal from the future income of the
risky asset, by consequence, if in reality the asset is not very risky.
CONCLUSION
We have introduced a category of investors whose capacity to hold risky assets is growing (in comparison
with the detrimental effect on their shareholders’ equity) in line with the present equilibrium price of the
risky asset.
We then showed that:
- if the signal from the future income of the risky asset is good, these investors do not face constraints
since the equilibrium price of the risky asset is high;
- if the signal from the future income of the risky asset is bad, the equilibrium price is lowered;
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- if it is very bad, the equilibrium price can drop endlessly. This is because, in this case, the rise in the
demand for the risky asset among investors who do not face constraints due to a decline in the
equilibrium price of this asset is smaller than the contraction in demand for the risky asset among
investors facing constraints due to this very same decline in the price of the asset.
We also showed that this kind of solvency constraint is meaningful, i.e. protects investors facing constraints
from the risk of a very negative value of their profits (of their portfolio net of its purchase value) only if the
signal received from the future income of the risky asset is precise enough.
If the signal is imprecise, regulations by consequence lead to the risk of a meltdown in prices of risky assets
without any offsetting advantage.
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