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Portfolio Theory in Illiquid Markets A New

Formalism

Carlo Acerbi, Abaxbank Giacomo Scandolo,

Università di Firenze

Essex University, Feb 09

- Introduction
- Assets and Portfolios
- The Value function
- Risk Measures
- Conclusions

Introduction

Liquidity Risk the blind spot of Portfolio Theory

- Liquidity Risk (LR) is a key feature of

financial markets. Its also the dominant risk of

any big financial crisis which transforms

isolated troubles into systemic epidemics. - Portfolio Theory (PT) has never been

convincingly extended to general illiquid

markets. It does not provide a sufficiently rich

notation to formalize LR-related questions. - Even worse, PT, betrays hidden assumptions of

perfect market liquidity, and therefore doesnt

need be extended, but rewritten from scratch. - Our main goal is to define a formalism for a PT

where basic concepts (Asset, Portfolio, price,

value, risk, ...) are redefined in such a way to

describe illiquid markets. We will work on

general ground with no hypotheses. Hence, ours

will not be a model, but a pure formalism.

What is Liquidity Risk ?

LR a multi-faceted reality

- Depending on the context, LR, is usually thought

of as - Portfolio LR the risk that a portfolio runs

out of cash money necessary for future payments

(the treasurer point of view) - Market LR the risk of buying or selling on a

market which is shallow compared to the trade

size (the trader point of view) - Systemic LR the risk that the liquidity

circulating in the economy is dried up (the

policy-maker point of view) - LR is all these things together, not only one. An

appropriate formalism should not only describe

one of these aspects, but must lend itself to the

joint formulation of all these problems

A first fundamental observation

A portfolio is a set, not a sum...

- A Portfolio is essentially a collection of

assets. Yet, we typically do not use a set

notation - but an algebraic notation
- confusing the two concepts of portfolio and

portfolio value and concluding automatically

that the latter is the sum of its consituent

assets values.

- We have seen eq. (1) so many times that it may

appear harmless and even obvious, because its

equation (1) of any financial textbook. On the

contrary we will see that its not only wrong but

even nonsensical in illiquid markets. - (1) is the main taboo to break if we want to

build an appropriate formalism for LR.

A first fundamental observation (continued)

A portfolio is a set, not a sum...

- Giving up eq. (1) we have
- the value of a portfolio is no more necessarily a

linear function in the portfolio space - Assets and Portfolios are distinct entities. They

do not live in the same vector space (portfolios

are not mere linear combination of assets) - Portfolio p and Portfolio Value V(p) must be

kept distinct conceptually and notationally.

Second fundamental observation

Alan and Ben

- In a illiquid market it is fundamental to realize

that both the value and the risk of a given

portfolio may take on different values in the

hands of two distinct investors. - Ex let p be a portfolio of very illiquid bonds

with tenor 7-10 ys and face value 1 mln ,

quoting around par - Let Alan be an investor who (for some reason) may

afford to keep p until maturity - Let Ben be an investor who (for some reason) will

be obliged to liquidate periodically large

portions of p. - It is clear that
- For Alan the portfolio is worth more or less 1

mln . For Ben certainly much less because

liquidating he will face a liquidation cost. Ben

may very well be willing to sell it immediately

for much less than 1 mln - Clearly, for Alan the LR of p is zero, whereas

for Ben is very large.

Second fundamental observation (continued)

An hidden variable ?

- But then there is no value V(p) nor any risk

measure ?(p) depending only on the portfolio,

because otherwise they would have a definite

value and not a different value depending on the

investor. - These functions must therefore depend also on

some further hidden variable X, which PT does

not consider yet. We will have functions of the

kind VX(p) and ?X(p) - The variable X in our formalism will be said

Liquidity Policy (LP) and represents the

constraints (ALM, regulatory, ) to which the

portoflio is subject. - The value of p is higher (the LR of p is lower)

for Alan than for Ben because Alan is subject to

a less restrictive LP than Ben - The correct formalization of LP is the subtlest

ingredient of the formalism.

- Introduction
- Assets and Portfolios
- The Value function
- Risk Measures
- Conclusions

Asset

Just a list of market quotes (i.e. an order book)

- An Asset is a good exchanged in the market in

standardized units. At time t the market will

express a number of quotes (both bid and offer)

with relative maximum sizes

- The same information may be condensed in a

function Marginal Supply Demand Curve (MSDC), s

?m(s) defined as the last price m(s) hit in a

trade of size s contracts where conventionally

sgt0 represents a sale and slt0 a purchase of s

contracts.

Marginal Supply-Demand Curve

An Asset may be identified with its MSDC

- An MSDC contains all the info on the market

prices relative to an Asset at time t. The

prices closest to the origin are the best bid

(mm(0)) and the best offer (m-m(0-)). The

MSDC is decreasing by construction.

Marginal Supply-Demand Curve (continued)

Asset and MSDC formal definition

- Note that the MSDC is not defined in the origin.

So, we do NOT define any MID-PRICE. Mid prices do

not exist on the market. They are pure

abstractions and we wont need them.

MSDC and trades

A trades proceeds

- The MSDC is a convenient notation to represent

the proceeds from the sale of s contracts (or the

cost of buying if slt0). In both cases we have eq.

(2).

MSDC and trades (continued)

A trades proceeds

The Market

What is the meaning of summing two assets ?

- It is fundamental to realize that the sum A1A2

of two assets A1 and A2 in general is not

defined. In fact, the market does not quotes bids

and asks on any collecion of assets, and

therefore ther is no MSDC in general for a

package containing both contracts A1 and A2. - The dollar () is a very peculiar Asset that we

will denote with A0 whose MSDC is identically

equal to 1 for any value s?? (m(s)1). It

corresponds to cash money and not to generic debt

instruments which instead will be represented by

some nontrivial MSDCs - The Market is a collection ofAssets M A0, A1,

A2, ... ,AN containing always also the dollar.

For what we said above, the market does not

exhibit any structure of linear space, because

the sum of two distinct assets is undefined.

Criticism to eq. (1)

Simply a nonsensical formula

- We are now in a position to realize that
- does not make any sense at all in an illiquid

market

- The sum of assets is nonsensical between formal

objects because the sum of assets is undefined - The sum cannot be meant among asset values

because assets does not have any value at all.

Portfolios

Just collections of assets

- The definition of Portfolio is less surprising.

In our notation its the vector of positions in

each markets assets.

- The sum of two portfolios p1p2 is a perfectly

legitimate sum between vectors. The space P of

all portfolios has a natural structure of vector

space. - We denote portfolios in boldface (p, q, ).

Cash-only portfolios a(a, 0,0) will be denoted

as scalars instead. - It remains to understand how to define the

Value of a portfolio. There is no more trivial

answer to this question.

- Introduction
- Assets and Portfolios
- The Value function
- Risk Measures
- Conclusions

The Liquidation operator L

Be prepared to liquidate everything

- The proceeds L(p) of a sudden liquidation of a

portfolio p define the operator L

- This may be thought of as an (extremely

conservative !) mark-to-market policy - This MtM policy may be necessary if the

portfolio, for some reason, could be forced to

sudden total liquidations. - This is a first attempt to give a meaning to the

value of p

The operator U

Dont be prepared to liquidate anything

- If on the contrary we mark long positions to

their best bid and short positions to best

ask, we obtain the following

- This is a widespread mark-to-market policy for

p, but not at all prudential, because it does not

probe at all the market depth. - This MtM policy may be sufficient only when it

is certain (for some reason) that the portfolio

will never face partial liquidations

Useful definitions

Some useful definitions

- We define liquidation cost the difference CU-L
- We will adopt the following notation

Some properties of L, U e C

No hypotheses and yet a lot of properties

One can show the following

Some properties of L, U e C

An example concavity of L(p)

Some properties of L, U e C

An example concavity of U(p)

Some properties of L, U e C

An example convexity of C(p)

The Liquidity Policy (LP)

The set of constraints of a portfolio

- To define a general concept of Mark-to-Market, we

introduce the following

- The idea when we mark a portfolio, we must

consider the constraints we could be forced to

satisfy in the future - These constraints will never breached if we add

cash or reduce all illiquid positions

proportionally

The Liquidity Policy (LP)

The set of constraints of a portfolio

- The definition of LP is not so abstract as it

may seem. We have examples in everyday finance - ALM constraints
- Risk management limits
- Investment policies
- Margin limits
- Basel II

The Value of a Portfolio

A liquidity sensitive Mark-to-Market

- The key definition of the formalism is
- The value of p is the maximum possible U(q) with

q being attainable from p and compliant with

the LP.

The Value of a Portfolio

Example cash liquidity policies

- A typical example of LP is the following a bank

estimates a minimum amount of cash to keep for

future payments - The corresponding LP is said a cash liquidity

policy

The Value of a Portfolio properties

Two consistency checks

- The LP U is the least prudent of all.

The Value of a Portfolio properties

The fundamental theorem

- The following result can be proved in complete

generality

Granularity

The other side of the diversification principle

- Concavity of V means
- the value of a blend of portfolios is larger

than the blend of the portfolios values - This is a new kind of diversification principle

- It works on values and not on risks .
- This diversification benefit is related only to

the granularity reduction at current time and has

nothing to do with the correlation of its assets

future dynamics.

An example of V(p)

The value of money

The additional value of liquid cash

- Translational supervariance of V, formalizes

that the injection of liquid cash doesnt just

increase nominal value, but also improves the

liquidity and this must reflect in a further

added value that our formalism indeed detects. - This is essentially the old adage a single

added dollar may be worth million dollars (on

the edge of a liquidity crisis) - Notice that all these observations are

qualitatively well known to practitioners. The

novelty is that they find here a correct

quantitative formalization for the first time.

The optimization problem

A non-trivial but non-serious problem

- The optimization problem hidden in the Value is

nontrivial, but computationally straightforward,

because it can be shown to be a convex problem.

- The problem rarely admits analytical solutions,

but numerically it is always solvable by convex

optimization methods which are generally fast

also for large portfolios - Hadnt we had such a strong result, the

applicability of this formalism for risk

management purposes would have been very

questionable.

Alan and Ben

Alan and Ben (follows)

Alan and Ben (follows)

Alan and Ben (follows)

Another example a mutual fund

The NaV of a fund depends on the liquidity of the

assets

- Mutual fund fixed income high rated european

financial floaters. Data from 10/12/2007

Another example a mutual fund (follows)

Another example a mutual fund (follows)

concave ! (granularity at work)

Another example a mutual fund (follows)

- Introduction
- Assets and Portfolios
- The Value function
- Risk Measures
- Conclusions

Do we really need liquidity risk measures ?

... or RL is a problem of good accounting ?

- Risk measures are essentially statistics of

future values of a portfolio under chosen

probabilistic assumptions. Our formalism already

incorporates the effect of liquidity in the

value. - Therefore, the use of common risk measures

(stdev, VaR, ...) in this formalism turns out to

be already appropriate for measuring market and

liquidity risk together (and inextricably so) - In our opinion in PT what was really missing was

not some liquidity risk measure, but rather an

appropriate accounting method for general

illiquid markets. - It must be observed that now, modeling the

market dynamics is much more complex, because now

we need to model the joint dynamics of all MSDCs.

The number of degrees of freedom is enormously

higher. - But dont blame the formalism ! This is really

the additional complexity that real markets do

have. There are infinitely more ways to hurt

yourself in a illiquid market than in a ideally

liquid one !

No hypotheses

We made no hypotheses. These are up to you.

- The formalism does not contain any hypotheses and

is therefore totally general. - To study risks, it is necessary however to

introduce a specific model and therefore to make

probabilistic assumptions on market dynamics. - The hypotheses on the dynamics of MSDCs may be

the most various. These are up to you.

A toy model

Gaussian market. Exponential MSDCs, cash LP

- Assuming
- we get
- Market Risk is in the dynamics of ai.
- Liquidity Risk is in nonzero ki and in its

randomness - Lets study (ai , ki) joint normal distributed in

different cases

A toy model

market risk only

market risk and static liquidity risk

market risk and independent random liquidity risk

market risk and correlated random liquidity risk

market risk and correlated random liquidity risk

liq. shocks

A 10 ys old puzzle

Coherency Axioms are they incompatible with LR ?

Coherent Measures of Risk (Artzner et al. 1997)

have always been criticised for not been

appropriate to account for liquidity

risk In particular Axioms (PH) and (S)

seem manifestly incompatible. The argument goes

- but if I double an illiquid portfolio risk may

become more than double as much ! ...

Convex Measures of Risk ?

A possible solution changing the axioms

Convex Measures of Risk (Heath, Föllmer et al.,

Frittelli et al., Carr et al.) were introduced to

weaken the axioms replacing (PH) and (S) by a

single weaker axiom

- This approach has the drawback of giving up (PH)

e (S) even in the case of liquid portfolios (e.g.

when they are tiny), when these axioms are

considered correct. - More generally, wed like to recover fully

Coherent Axioms in the limiting case when LR goes

to zero (e.g. portfolios size ltlt market depth).

But within Convex Measures this is not ensured. - Actually, after 10 ys Convex Measures did not

provide any concrete tool for financial risk

management yet.

Coherency Axioms, revisited

Taking Coherency Axioms seriously

- The argument against the axioms is false,

because X in is ?(X) not a portfolio, but a

portfolio value. The argument tacitly relies on

the assumption that p ?V(p) is a linear function

(our taboo). - Abandoning this argument we do not see any other

reason why we should give up the coherency axioms - Now, VL(p) is no more a linear function and we

may study the behavior of a coherent measure

?L(X) ?(VL(p)) associated to our Value function

- Notice that now the Risk of a portfolio

depends on the chosen LP !

Coherency Axioms, revisited

Convexity and subvariance of CPRM

- The following result is completely general and

solves the puzzle

- Therefore, convexity of risk measures in the

space of portfolios is a result in our formalism

and not a new axiom ! - Translational subvariance is very clear. The

injection of cash reduces the risk more than its

nominal value, because it improves the portfolio

liquidity.

notice translational invariance had not been

criticized in the theory of convex meaures

S and PH of CPRMs

For axioms (PH) and (S) results depend on the

liquidity policy

- Example the case of the MtM procedure L

- We see that in this case scaling the portfolio,

risk scales more than proportionally. - Subadditivity wrt portfolios is generally lost,

but it remains valid for discordant portfolios.

For a liquidity policy of type L this is

intuitive.

S and PH of CPRMs

Axioms (PH) and (S) in the case of cash liquidity

policies

- The result for cash liquidity policies is much

more surprising at first sight

- We notice that as we scale the portfolio, the

risk increases less than proportionally. This may

seem strange, but it is in fact very reasonable

for policies with a fixed b - Subadditivity which in this case is in general no

more true, holds for concordant portfolios.

S and PH of CPRMs

Axioms (PH) and (S) in the case of cash liquidity

policies... intuition restored

- If we scale also the cash amount of the liquidity

policy we obtain what we expect

- We notice that if we scale the portfolio AND the

liquidity policy, the risk increases more than

proportionally.

The liquid limit coherency is back

Back to Coherency when markets are liquid

- Every value function VL(p) behaves exactly as

U(p) when there is no LR, namely when - The portfolio size is negligible wrt the market

depth - The assets MSDCs are flat
- Investors have no liquidity constraints
- Therefore, U may be thought of as a tool to

probe the liquid limit of the formalism. It is

important to see that CPRMs with U satisfy

formally all coherency axioms

- Introduction
- Assets and Portfolios
- The Value function
- Risk Measures
- Conclusions

Conclusions

- We have described a general formalism for

portfolio theory in illiquid markets, based on

the observation that Assets and Portfolios are

concepts to be kept distinct, which do not live

in the same vector space and for which distinct

categories apply. Assets have prices but not

values and Portfolios have values but not prices.

- We recover the standard portfolio theory

formalism in the limit of liquid markets. - The formalism is completely free from

hypotheses. Hypotheses are needed for any

implementation, in particular, for the stochastic

modeling of MSDCs. - The Value function of a Portfolio turns out to

depend from a new concept of liquidity policy,

which relates the evaluation of the portfolio to

the liquidity needs it has to sustain with its

cashflows. This is shown to be always a concave

map on the space of portfolios. We interpret this

as a granularity effect, which is the liquidity

side of the diversification principle.

Conclusions

- The Value function hides a high dimensional

optimization problem. This could generally

represent a serious obstacle for implementation

in general cases. Fortunately this is not the

case, because the problem can be shown to be

convex in general. - Analytically tractable cases are described, but

in general this formalism requires numerical

convex optimization methods. - This formalism can be adopted with any portfolio

risk measure. However, when used with Coherent

Measures of Risk, this formalism solves a

longstanding puzzle. CMR were criticized for not

being compatible with liquidity risk. In this

formalism one sees that the criticism was not

correct. The axioms of coherence need no change

for encompassing liquidity risk. It was necessary

to devise a new accounting method and not a new

class of risk measures. - The formalism naturally defines Coherent

Portfolio Risk Measures which are induced on the

space of portfolios by the choice of a CMR and a

liquidity policy. CPRMs turn out to be convex and

translational supervariant. The properties of

subadditivity and positive homogeneity do not

hold anymore on CPRMs and their deformed version

is shown to depend on the chosen liquidity

policy.

References

- C. Acerbi e G. Scandolo, Liquidity Risk and

Coherent Measures of Risk, to appear on

Quantitative Finance, 2008. - C. Acerbi, in Pillar II in the New Basel

Accord, RISK books, ed. A. Resti, 2008