FINANCE
/ Solvency constraints and dynamics of prices of risky assetsJEL Classification: G 11 –
G 18
G 18 / 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.
Author: Patrick Artus
Secretary: Laurence Sanchez-Garrido
Document de travail n° 01 - 2 -
Document de travail n° 01 - 2 -
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.
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 supply[1] 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 increased[2], 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 – 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 , they receive a noisy signal from the future income of the risky asset.
The risky asset has a price in period; at date, investors receive the signal from the future income of the risky asset:
(1) and is independent from.
In principle, has the following distribution of probability:
(2)
Investors therefore calculate the conditional expectation (in view of the information available in) and the conditional variance of (the future income of the risky asset) in compliance with:
(3)
There are similar investors. They maximise an expected utility function written as expectation – variance; they each hold 1 to invest, and allocate to the risky asset, by carrying out:
(4)
where measures the degree of risk aversion.
As a result, we have:
(5)
is the part of total wealth 1 invested in risky assets at date; the number of risky assets each investor wants to hold is thus
We denote the number of risky assets offered at date
We write:
(6)
The supply of risky assets is affected by a hazard at which is a white noise; is independent from and
The equilibrium of the market of the risky asset at date is therefore written:
(7)
or, for the equilibrium price.
(8)
where and are given by (3), or alternatively:
(9)
The equilibrium price of the risky asset, rises in line with the signal received from the future income of asset, and decreases in line with the supply of risky assets in period .
The income of each investor is:
(10)
(7) shows that
hence
(11)
where is given by (9) and where
or alternatively
(12)
Hence the unconditional expectation of the investor’s income:
(13)
The unconditional variance of the investor’s income is, by omitting the higher order terms[3]:
(14)
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 held by each one of these investors.
The maximum amount that can be invested in risky assets is:
(15)
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)
the equilibrium of the market of the risky asset is written:
(17)
Or alternatively:
(18)
We denote:
(19)
the equilibrium price at date without any constraint, of the market of the risky asset.
If:
(20)
The equilibrium remains the equilibrium
However, if the inverse inequality of (20) is verified, then the equilibrium of the market of the risky asset is written:
(21)
In:
(22)
The equilibrium can be represented graphically as follows, when the constraint (16) is active, from:
(21’)
For the sake of simplicity, we suppose that (21’) has a solution. If this is the case, there are two solutions.
In because of (22).
There are thus two cases (if there are two solutions):
(a) Case 1
(23)
The equilibrium is then graphically represented:
Diagram 1
Equilibrium with equilibrium price lower than
There are then two equilibria and with
(b) Case 2
(24)
We then have:
Diagram 2
Equilibrium with an equilibrium price higher than
There are two equilibria and with
In case 1, at the “normal” equilibrium price of the risky asset, , 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, there is excess supply of the risky asset. If the price declines from, one heads towards the equilibrium of Diagram 1.
In case 2, in 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, 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 to.
- 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:
(25)
[active constraint
case (1) occurs]
We denote such as:
(26A)
and such as:
(26B)
We suppose that.
Case 1 occurs when:
(27)
Case 2 occurs when:
(28)
[active constraint]
or:
(29)
Since if (the proportion of investors who can face constraints) is 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 [is low] we have and we are still in the divergent case 2. We suppose that
We can therefore see that:
· if the signal of the future income of the risky asset is good, and we are in the case without any constraint;
· if the signal is bad, but not very bad, , and we are in case 1 with a lower equilibrium price, ;
· if the signal is very bad, 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 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
For, only the investors who do not face constraints are found in the market, and the equilibrium (which is found in is given by:
(30)
Or:
(30’)
which differs from the normal equilibrium price (9) by a higher risk premium (divided by
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.
To recap:
(a) If (20) is verified, the equilibrium is given by (20) is rewritten (with :
(31)
and in this case:
(32)
(b) If (25) is verified (intermediary values of, we have:
(33)
and in this case, we have such that:
(34)
(c) If, then, as we saw, the equilibrium price is lower than and only the investors who do not face constraints trade:
(35)
To make matters even simpler, let us suppose that the intermediary case (b) is either very unlikely or even disappears.
We then have only the following solution:
(36A)
[when]
and
(36B)
In the first case (36A), all the investors have the following income:
(37A)
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 have the following income:
(37B)
We denote:
(38)
We are in the first case when; and in the second case when
The income of investors changes in line with:
(39)
The solvency constraint normally enables investors to avoid losses due to an excessively negative value of defined by (39).
When would then need to have a high probability of being low.
When then.
If the signal is precise, is low and In this case, If the signal is imprecise, can be high, is high.
In this case which is not necessarily low if the hazard 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;
- 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.