Portfolio Theory and Risk Management in the presence of Illiquidity

Transcription

1 Portfolio Theory and Risk Management in the presence of Illiquidity A new formalism Carlo Acerbi Giacomo Scandolo, Università di Firenze Risk Europe - Frankfurt, Jun 5, 2009

2 1. Introduction 2. Assets and Portfolios 3. The Value function 4. Risk Measures 5. Conclusions

3 Introduction Liquidity Risk: the blind spot of Portfolio Theory Liquidity Risk (LR) is a key feature of financial markets. It s 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 doesn t 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: 1. Portfolio LR : the risk that a portfolio runs out of cash money necessary for future payments (the treasurer point of view ) 2. Market LR : the risk of buying or selling on a market which is shallow compared to the trade size (the trader point of view ) 3. 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 it s equation (1) of any financial textbook. On the contrary we will see that it s 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 Second fundamental observation The value of a portfolio is in some sense subjective 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 A. Let Alan be an investor who (for some reason) may afford to keep p until maturity B. Let Ben be an investor who (for some reason) will be obliged to liquidate periodically large portions of p. It is clear that 1. 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 $ 2. Clearly, for Alan the LR of p is zero, whereas for Ben is very large.

7 Second fundamental observation (continued) An hidden variable? But then there is no value V(p) nor any risk measure ρ(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 V X (p) and ρ 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.

8 1. Introduction 2. Assets and Portfolios 3. The Value function 4. Risk Measures 5. Conclusions

9 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 s>0 represents a sale and s<0 a purchase of s contracts.

10 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 (m + =m(0 + )) and the best offer (m - =m(0 - )). The MSDC is decreasing by construction. prices trade size

11 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 won t need them.

12 MSDC and trades A trade s proceeds The MSDC is a convenient notation to represent the proceeds from the sale of s contracts (or the cost of buying if s<0). In both cases we have eq. (2).

13 MSDC and trades (continued) A trade s proceeds

14 The Market What is the meaning of summing two assets? It is fundamental to realize that the sum A 1 +A 2 of two assets A 1 and A 2 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 A 1 and A 2. The dollar ($) is a very peculiar Asset that we will denote with A 0 whose MSDC is identically equal to 1$ for any value s R (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 = {A 0, A 1, A 2,...,A N } 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.

15 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 1. The sum of assets is nonsensical between formal objects because the sum of assets is undefined 2. The sum cannot be meant among asset values because assets does not have any value at all.

16 Portfolios Just collections of assets The definition of Portfolio is less surprising. In our notation it s the vector of positions in each market s assets. The sum of two portfolios p 1 +p 2 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.

17 1. Introduction 2. Assets and Portfolios 3. The Value function 4. Risk Measures 5. Conclusions

18 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

19 The operator U Don t 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

20 Useful definitions Some useful definitions We define liquidation cost the difference C=U-L We will adopt the following notation:

21 Some properties of L, U e C No hypotheses and yet a lot of properties One can show the following

22 Some properties of L, U e C An example: concavity of L(p)

23 Some properties of L, U e C An example: concavity of U(p)

24 Some properties of L, U e C An example: convexity of C(p)

25 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

26 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

27 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. We recognize in particular that L and U are two special cases of Value function with LP representing two extreme cases: be prepared to liquidate everything and don t be prepared to liquidate anything

28 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 Notice that also in this simple case the optimization problem is instead non-trivial. In particular, notice that V(p) is a NONLINEAR map on P(M)

29 The Value of a Portfolio: properties Two consistency checks The LP U is the least prudent of all. If we are really forced to switch from p to q* in order to comply LP, then the value does not drop, provided that q* is the optimal portfolio.

30 The Value of a Portfolio: properties The fundamental theorem The following result can be proved in complete generality

31 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.

32 An example of V(p)

33 The value of money The additional value of liquid cash Translational supervariance of V, formalizes that the injection of liquid cash doesn t 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.

34 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 Hadn t we had such a strong result, the applicability of this formalism for risk management purposes would have been very questionable.

35 Alan and Ben

36 Alan and Ben (follows)

37 Alan and Ben (follows)

38 Alan and Ben (follows)

39 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

40 Another example: a mutual fund (follows)

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

42 Another example: a mutual fund (follows)

43 1. Introduction 2. Assets and Portfolios 3. The Value function 4. Risk Measures 5. Conclusions

44 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 don t 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!

45 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.

46 A toy model Gaussian market. Exponential MSDCs, cash LP Assuming we get Market Risk is in the dynamics of a i. Liquidity Risk is in nonzero k i and in its randomness Let s study (a i, k i ) joint normal distributed in different cases

47 A toy model 0.03 without liquidity risk market risk only x 104 with liquidity risk - one factor msdcs 1 market risk and static liquidity risk x 104 with liquidity risk - two-independent-factor msdcs 1 market risk and independent random liquidity risk x 104 with liquidity risk - two-dependent-factor msdcs 1 0 market risk and correlated random liquidity risk x 104 with liquidity risk - two-dependent-factor msdcs and liquidity crises 1 0 market risk and correlated random liquidity risk + liq. shocks

48 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!...

49 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, we d like to recover fully Coherent Axioms in the limiting case when LR goes to zero (e.g. portfolios size << 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.

50 Coherency Axioms, revisited Taking Coherency Axioms seriously The argument against the axioms is false, because X in is ρ(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, V L (p) is no more a linear function and we may study the behavior of a coherent measure ρ L (X) = ρ(v L (p)) associated to our Value function Notice that now the Risk of a portfolio depends on the chosen LP!

51 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

52 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.

53 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.

54 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.

55 The liquid limit: coherency is back Back to Coherency when markets are liquid Every value function V L (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

56 1. Introduction 2. Assets and Portfolios 3. The Value function 4. Risk Measures 5. Conclusions

57 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.

58 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.

59 References C. Acerbi e G. Scandolo, Liquidity Risk and Coherent Measures of Risk, to appear on Quantitative Finance, C. Acerbi, in Pillar II in the New Basel Accord, RISK books, ed. A. Resti, 2009