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Portfolio Theory in Illiquid Markets

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Title: 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
15
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
23
Some properties of L, U e C
An example concavity of L(p)
24
Some properties of L, U e C
An example concavity of U(p)
25
Some properties of L, U e C
An example convexity of C(p)
26
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.

29
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

32
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)
42
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
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