A Call for Credit Policy
Dilip B Madan
Robert H Smith School of Business
University of Maryland
January 31, 2011
This short paper summarizes what has been learnt in a handful of recently
completed papers with numerous coauthors. I begin with Cherny and Madan
(2009, 2010) that gave us an understanding of how to construct the bid and
ask prices relevant to a theory of two price markets that I now call Conic Finance. In Conic Finance the market is modeled as a passive counterparty for all
economic agents. The market is de…ned by describing the set of zero cost cash
‡ows the market will accept. These cash ‡ows are modeled by a convex cone of
state contingent cash ‡ows that contains the nonnegative cash ‡ows. An application in Madan (2010) led to an understanding of who, how and why capital
requirements are to be established for limited liability entities with access to
modern derivative markets. More was learned when we applied these methods
to determine capital requirements for the major US banks as at the end of 2008
in Eberlein and Madan (2010). This paper de…ned the taxpayer put option for
the …rst time. The objective of credit policy as an arm of regulation in otherwise
free markets is to ensure that the value of this freely distributed put option is
kept within limits and is not allowed to get excessively valuable. Finally, Madan
and Schoutens (2010) rede…ne the corporate balance sheet for relevance to two
price markets, showing that equity capital is a poor measure of …nancial health
as it can be contaminated by the excessive value of the taxpayer put option and
must be further corrected for the two price haircuts on both sides of the balance
sheet, before we get a true measure of capital available. As a result agency
arguments designed to align CEO compensation with shareholders can end up
distorting …nancial health by activities seeking to maximizing the value of the
taxpayer put option. These considerations necessitate a cleaner de…nition of equity capital that is closer to the real equity stake and is measured as suggested
in Carr, Madan and Vicente Alvarez (2010) by the ask price of the business less
its bid price, where both are constructed without the limited liability in place.
Any business plan be it a hedge fund or a corporation has in place, before the
issuance of any securities related to ownership and the distribution of returns,
access to potential gains or receipts of funds that we call the assets A; and
potential losses or payments of cash ‡ows to other market participants that we
call the liabilities L; of the business. Both A; L are positive random variables
and the fate of the business is determined by the joint probability law of these
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random variables. It is useful to work in these terms as opposed to just the
probability law of the di¤erence as one can relate matters better to classical
corporate balance sheets by keeping both entities in mind. The promises to and
from the …rm trade in free markets with current market prices that we take to be
A0 and L0 : For purely …nancial …rms in the absence of arbitrage opportunities
these values must be in balance. More generally the …rm may create value and
the expectation is that A0 exceeds L0 ; but not by too much.
Now no balanced hedge fund with A0 = L0 ; or other corporate entity, can be
permitted to exist and be given a limited liability status if no funds are placed
at stake with a capacity to absorb potential losses. We denote the level of such
funds placed at stake as M and recognize that they will grow at the period
interest rate of r to M er at the end of the period. The risk being held by the
economy once the …rm is in existence and all the contracts are signed is the cash
‡ow
C = M er + A L:
By virtue of limited liability the value received by the …rm is the positive part
of this cash ‡ow or
+
V = (M er + A L) :
This …rm value is the payo¤ to a call option on A L and its value exceeds the
intrinsic value of M + A0 L0 by the value of the option to put losses back into
the economy with the cash ‡ow
P = ( M er
(A
+
L)) :
Eberlein and Madan (2010) have called this put option the taxpayer put, but
the name does not matter, what is clear is that the …rm holds such a put option.
As a consequence the …rm cannot be the one to set the level of funds M
supporting the business as its incentive is to maximize the value of the embedded
put by choosing the maximum strike of zero. So who should set the value M: It
is clear that the limited liability is granted by the government and the economy
has to hold the risk C: So the government on behalf of the economy should
set the value M: One could argue and some do, that creditor counterparties
with the liability L will force the …rm to hold an acceptable level of funds for
them to be willing counterparties. But these parties can be small and diverse
with insu¢ cient market power to enforce capital levels on the …rm. In my view
the same motivation that requires the approval of a business prospectus by the
stock exchange before stocks can be issued to the investing public necessitates
the need for a credit monitoring authority that attempts to ensure su¢ cient
levels of reserves in the form of M:
Before we ask and prescribe how M is to be set, given that we understand
who determines M and why M is to be determined it is useful to view the
balance sheet at this point. In a traditional balance sheet we will have on the
asset side M + A0 and on the liability side L0 + J; where J is the equity. For
a balanced business with A0 = L0 we have M = J and the cash reserve is also
the …rm’s equity. The policy determining M then determines equity and the
leverage (M + A)=J > 1:
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However the traditional balance sheet is incorrect as J is actually the value
of a call option with payout V and includes the value of the put option with
payout P: We actually have on the asset side M + A0 + P0 and the liability
side is then L0 + J: Equity seen as J can be vastly overstated when actions
are taken lead to a high value for P0 : The moral of this paragraph is beware
of traditional equity as a measure of capital as it contains the contamination
of P0 : Our interest is M or for a balanced fund the magnitude of J P0 or
equity stripped of the value of the taxpayer put. In general this value would be
M + A0 L0 and one is valuing assets and liabilities at the law of one price.
Accounting for two price laws one must further take the haircut of A0 b and
a L0 as assets are marked down to the bid price b while liabilities are marked
up to the ask price a: Equity net of the taxpayer put and the two haircuts is
M + b a or risk adjusted assets less risk adjusted liabilities. This is the correct
capital computation. With these warnings we may focus attention on how M
is to be determined.
For this prescription we observe that the economy must hold the risk C and
M should be …xed so as to make this risk acceptable to the economy. Fortunately
for us these questions were answered in the abstract by Artzner, Delbaen, Eber
and Heath (1999) who characterized the set of acceptable risks. This set is
de…ned by a convex cone containing the nonnegative cash ‡ows and must take
the following form. The cash ‡ow C is acceptable just if
E Q [C]
0; all Q 2 M
where M is a convex set of probability measures. When the decision of acceptability just depends on the probability law of the random variable in question
Cherny and Madan (2009) showed that one may test for acceptability by ensuring that one has a positive expectation after a concave distortion of the
distribution function. Hence for F (c) the distribution function of C and
a
concave distribution function on the unit interval one must ensure that
Z 1
cd (F (c)) 0:
1
Madan (2009) then shows that
M=
e
r
Z
1
xd (FA
L (x));
1
where FA L is the distribution function of A L: The suggested distortion
is termed minmaxvar in Cherny and Madan (2009) and is given by a one
parameter family
(x) = 1
1
1
x 1+
1+
:
This distortion combines two ways of forming worst cases, one of which is the
expectation of the minimum of independent draws for integral : The higher
the value of the more stressed the expectation and the higher the required
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capital. Of course we do not wish to over capitalize and one may choose to be
the smallest level that counteracts the adverse risk incentives built into equity
values by the limited liability put. Madan (2009) determines this by ensuring
that the derivative of required capital with respect to extraneous risk dominates
the derivative of equity value. A value of between a quarter and unity is quite
adequate.
For the valuation of securities issued by the …rm we take up the theory of two
price markets. In this perspective developed in Cherny and Madan (2010), the
market is viewed as a counterparty taking up zero cost cash ‡ows acceptable to
it and proposed by market participants. Applying concave distortions to de…ne
acceptable cash ‡ows leads to bid prices for cash ‡ows X as
Z 1
b(X) =
xd (FX (x))
1
where FX is the distribution function of the random variable X: The corresponding ask price is
Z 1
a(X) =
xd (1 FX ( x)) :
1
Madan and Schoutens (2010) go on to model di¤erent markets using di¤erent cones after showing that arbitrage is excluded between markets provided
the supporting measures have a nonempty intersection. They introduce a two
parameter family of cones of the form
;
(x) = 1
1
1
1+
x 1+
where the parameter measures loss aversion while measures the absence
of gain enticement in the market. All concave distortions have a nonempty
intersection for their supporting measures and arbitrage is excluded. They mark
all assets at bid prices and liabilities at ask prices with the di¤erence held as
reserves against potentially unfavorable unwinds. They show that regulatory
capital requirements as set by a credit monitoring authority as described here
can lead to conservative equity markets not funding the …rm and so there are
situations when all equity …rms fail to exist. They also show that existence
shopping is not possible between debt and equity provided the debt distortion
is more conservative than the equity distortion. Optimal debt levels arise out
of clientele e¤ects re‡ected by di¤erent cones in the absence of tax advantages
to debt. The …nal balance sheet re‡ects the value of the taxpayer put and the
distorted value of equity along with the right level of capital related to cleaning
out equity of its contamination. Interestingly it is observed in Eberlein, Gehrig
and Madan (2010) that when one’s own debt is valued at the ask price anomalous
pro…ts on one’s own credit deterioration disappear, theoretically. Such gyrations
in corporate pro…ts apologized for by every CEO are an accidental consequence
of one price markets.
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Eberlein and Madan (2010) show two further contaminations related to the
taxpayer put. First debt holders hold this put collectively with equity holders
and this leads to a lack of interest on their part in monitoring corporate risk
levels. Secondly as equity is a call option with the negative of cash reserves as
the strike, cash plays a special role in the economy by helping to hold up stock
prices. CEO’s paid in stock then inherit incentives to hold cash and maximize
risk taking. The advent of stock based CEO compensation may then have
coincided with adverse corporate management heavily engaged in maximizing
the value of the taxpayer put. There is considerable anecdotal evidence of such
actions prior to the onset of the …nancial crisis.
Regulators have to monitor not just capital levels for the …rm, but the same
issues arise at the level of individual structures especially when they are set up as
special purpose vehicles. Carr, Madan and Vicente Alvarez (2010) take up these
matters at a more micro level. They use the theory of two price markets to de…ne
capital on a trade as the di¤erence between the ask and the bid price. Formally
the ask price is the required capital for o¤ loading a risk on the economy, but
credit is given for the bid price with the di¤erence to held as reserves. The up
front pro…t is de…ned as the mid quote less the risk neutral expectation. The
structure of claims with positive and negative pro…ts is described. Competitive
pressures to lower ask prices and raise bid prices leads to capital conservation
as a corporate objective in the design of hedges and other related activities. It
is shown the maximization of expected utility can lead to an ine¢ cient use of
available capital while adjustments to delta hedging taking account of gamma
levels can economize capital.
References
[1] Artzner, P., F. Delbaen, J. Eber, and D. Heath, (1999), “De…nition of coherent measures of risk,” Mathematical Finance 9, 203-228.
[2] Cherny, A. and D. B. Madan (2009), “New measures of performance evaluation,” Review of Financial Studies, 22, 2571-2606.
[3] Cherny, A., and D. B. Madan (2010), “Markets as a counterparty: An introduction to conic …nance," International Journal of Theoretical and Applied
Finance, 13, 1149-1177.
[4] Carr, P., D. B. Madan and J. J. Vicente Alvarez (2010), “Markets, pro…ts,
capital, leverage and returns,” Working paper Robert H. Smith School of
Business.
[5] Eberlein, E., T. Gehrig and D. Madan (2010), “Pricing to Acceptability:
With applications to valuing one’s own credit risk,” Working paper Robert
H. Smith School of Business.
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[6] Eberlein, E. and D. B. Madan (2010), “Unbounded liabilities, capital reserve
requirements and the taxpayer put option,”Working paper, Robert H. Smith
School of Business.
[7] Madan, D. B. (2009), “Capital requirements, acceptable risks and pro…ts,”
Quantitative Finance, 7, 767-773.
[8] Madan, D. B., and W. Schoutens (2010), “Conic …nance and the corporate balance sheet,” forthcoming International Journal of Theoretical and
Applied Finance.
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