Portfolio Theory in Illiquid Markets

1
Portfolio Theory in Illiquid Markets A New
Formalism
Carlo Acerbi, Abaxbank Giacomo Scandolo,
Università di Firenze
Essex University, Feb 09
2

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

3
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.

4
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

5
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.

6
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.

7
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.

8
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.

9

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

10
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.

11
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.

12
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.

13
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).

14
MSDC and trades (continued)
A trades proceeds
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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.

16
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.

17
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.

18

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

19
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

20
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

21
Useful definitions
Some useful definitions

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

22
Some properties of L, U e C
No hypotheses and yet a lot of properties
One can show the following
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Some properties of L, U e C
An example concavity of L(p)
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Some properties of L, U e C
An example concavity of U(p)
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Some properties of L, U e C
An example convexity of C(p)
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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

27
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

28
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.

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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

30
The Value of a Portfolio properties
Two consistency checks

• The LP U is the least prudent of all.

31
The Value of a Portfolio properties
The fundamental theorem

• The following result can be proved in complete
  generality

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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.

33
An example of V(p)
34
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.

35
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.

36
Alan and Ben
37
Alan and Ben (follows)
38
Alan and Ben (follows)
39
Alan and Ben (follows)
40
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

41
Another example a mutual fund (follows)
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Another example a mutual fund (follows)
concave ! (granularity at work)
43
Another example a mutual fund (follows)
44

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

45
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 !

46
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.

47
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

48
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
49
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 ! ...

50
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.

51
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 !

52
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
53
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.

54
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.

55
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.

56
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

57

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

58
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.

59
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.

60
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