PRIN Conference Stochastic Methods in Finance Torino - July, 2008 A new approach for valuing a portfolio of illiquid assets Giacomo Scandolo - Università di Firenze Carlo Acerbi - AbaxBank Milano
Liquidity risk 1/23 What is liquidity risk? Treasurer s answer: the risk of running short of cash Trader s answer: the risk of trading in illiquid markets, i.e. markets where exchanging assets for cash may be difficult or uncertain Central Bank s answer: the risk of concentration of cash among few economic agents Setting a precise mathematical framework is not easy
Outline 2/23 The theoretical framework Portfolios and Marginal Supply-Demand Curves Liquidation value vs. usual mark-to-market value Liquidity policies and general mark-to-market values Coherent/convex risk measures and liquidity risk Some numerical examples
Portfolios 3/23 It is possible to trade in N 1 illiquid assets cash, which is by definition the only liquidity risk-free asset We define A portfolio is a vector p R N+1 p0 is the amount of cash p = (p1,..., p N ) is the assets position pn is the number of assets of type n
Liquid/Illiquid markets 4/23 Perfectly liquid market (S0(t) 1) Sn(t) is the unique price, at time t, for selling/buying a unit of asset n; this price does not depend on the size of the trade V (p, t) = p0 + N n=1 pnsn(t) is linear Illiquid markets (S0(t) 1) Sn(t) = Sn(t, x) will depend on the size x R (x > 0 is a sale) of the trade V (p, t) need not be linear anymore. A first idea is: V (p, t) = p0 + N n=1 pnsn(t, pn) But this is not the only sensible notion of value
Marginal Supply-Demand Curves - 1 5/23 Some basic definitions A Marginal Supply-Demand Curve (msdc) is a decreasing function m : R (0, + ) m + = m(0+) and m = m(0 ) are the best bid (sell) and ask (buy) prices. Of course m + m. Let x be the size of the transaction (x > 0 sale, x < 0 purchase). The unit price is and the proceeds are S(x) = 1 x x 0 P (x) = xs(x) = m(y) dy > 0 x 0 m(y) dy 0 Same setting as in Cetin-Jarrow-Protter 2005, but our focus is on m.
a typical MSDC 108 106 Bids 104 102 100 98 Asks 96-20 -15-10 -5 0 5 10 15 20
Marginal Supply-Demand Curves - 2 6/23 We can also allow for (care is needed with the details): Assets which are not securities (e.g. swaps) and can display negative (marginal) prices: m : R R Securities with finite depth market: m(x) = + for x << 0 and/or m(x) = 0 for x >> 0 Swaps with finite depth market: m(x) = + for x << 0 and/or m(x) = for x >> 0
Two simple notions of value - 1 7/23 Given m = (m1,..., m N ) a vector of msdc. Let p R N+1 be a portfolio. The Liquidation Value of p is L(p) = p0 + N n=1 pnsn(pn) = p0 + N n=1 p n 0 m n(x) dx The Usual Mark-to-Market Value of p is U(p) = p0 + pnm + n + as if only the best bid and ask would matter. pn>0 pn<0 pnm n Note that U(p) L(p) for any p.
Two simple notions of value - 2 8/23 Some properties of L and U: concavity. both L and U are concave (but not linear) additivity. L is subadditive (L(p + q) L(p) + L(q)) whenever p and q are concordant (pnqn 0 for n 1); it is superadditive for discordant portfolios U is always superadditive, and it is additive for concordant portfolios scaling if λ 1 L(λp) λl(p) U(λp) = λu(p)
Two simple notions of value - 3 9/23 Liquidation Mark-to-Market Value (L): measure of the portfolio value as if we are forced to entirely liquidate it (so, liquidity risk is a big concern) Usual MtM Value (U): measure of the portfolio value as if we don t have to liquidate even a small part of it (so, liquidity risk is not a concern) Our aim is to introduce notions of value between the two extreme cases. Whether and what to liquidate is a need that may vary.
Acceptable portfolios - 1 10/23 First we give a notion of acceptability for a portfolio: A liquidity policy is a convex and closed subset L R N+1 such that 1. p L implies p + a L for any a 0 (adding cash cannot worsen the liquidity properties of a portfolio) 2. (p0, p ) L implies (p0, 0 ) L (if a portfolio is acceptable, its cash component is acceptable as well) L collects the portfolios whose liquidity risk is not a concern and thus may be valued through U
Acceptable portfolios - 2 11/23 Examples of liquidity policies: 1. L = R N+1 Every portfolio is acceptable: no need to liquidate (this will lead to U) 2. L = {p : p = 0 } Only pure-cash portfolios are acceptable: need to entirely liquidate p (this will lead to L) 3. L = {p : p0 a} (a 0 fixed) This is a typical requirement imposed by the ALM of an institution 4. other examples may be based on bounds on concentration...
Attainable portfolios 12/23 1. Start with a portfolio p, which need not be acceptable 2. Make it acceptable by liquidating the assets (sub)position q R N r = p q + L(0, q ) = (p0 + L(0, q ), p q ) L 3. Find the best way to do this, maximizing the Usual MtM value U(r) 4. Note that L is used in 2. and U in 3.: in 2. we care about liquidity risk, in 3. we don t as r L
A general definition of value 13/23 Having fixed a liquidity policy L we can define the associated MtM Value (sup = ) V L (p) = sup{u(r) : r = p q + L(0, q ) L, q R N } r R N+1 is optimal if V L (p) = U(r ). It is immediate to see that V L (r ) = U(r ) = V L (p) (there is no change in value passing from p to r ) The set over which U (concave) is maximized is convex. Thus The optimization program defining V L is always convex.
Some examples 14/23 1. If L = R N+1, then V L (p) = U(p) 2. If L = {p : p = 0 }, then V L (p) = L(p) 3. If L = {p : p0 a}, then V L (p) = sup{u(p q ) + L(0, q ) : L(0, q ) a p0, q R N+1 } which is not trivial (and non-linear)
Some properties 15/23 If L L, then V L V L. Thus, V L (p) U(p) L For any L, V L is concave and translational supervariant V L (p + a) V L (p) + a a 0
Computation of the value 16/23 As the problem defining V L is convex, many fast algorithms are available An analytical solution is sometime easy. Assume: {p : p0 a} mi continuous and strictly decreasing i Then if p0 a (p L) then r = p and V L (p) = U(p) if p0 < 0 then r i = m 1 i where λ is determined by L(r ) = p0 a. ( m i(0) 1 + λ )
Coherent risk measures - 1 17/23 Coherent risk measures (CRM) ρ : L R (L space of r.v.) are characterized by (Artzner-Delbaen-Eber-Heath-98) 1. Translation invariance: ρ(x + c) = ρ(x) c c R; 2. Monotonicity: ρ(x) ρ(y ) whenever X Y 3. Positive homogeneity: ρ(λx) = λρ(x) λ 0 4. Subadditivity: ρ(x + Y ) ρ(x) + ρ(y ). Axioms 3 and 4 do not seem to take into account liquidity risk: if I double my portfolio, its risk should more than double in many cases. They were replaced (Follmer-Schied02, Frittelli-Rosazza02) by the weaker axiom of convexity.
Coherent risk measures - 2 18/23 In our opinion, CRM are appropriate to deal with liquidity risk. The key point is that: If I double my portfolio... means p 2p, not X 2X The relation between p and its value X is not linear. We define risk measures defined directly on portfolios R = R(p) that are not necessarily positively homogeneous or subadditive.
Risk measures for portfolios 19/23 Given a liquidity policy L a probability space (Ω, F, P ) describing randomness up to T > 0 a coherent risk measure defined on some L L 0 (Ω, F, P ) the random future msdc: (mi(x, T )) for any i, where for any x, mi(x, T ) is a r.v. for any ω, x mi(x, T )(ω) is decreasing We compute V L (p) = V L (p, T )(ω) for any ω (it is a r.v.) and set R L (p) = ρ(v L (p))
Properties 20/23 Some properties (for general L and ρ) 1. R L is convex 2. R L is translational subvariant: R L (p + c) R L (p) c 3. R L is in general not homogeneous, nor subadditive 4. specific properties for R L may be derived from properties of V L (and coherency of ρ) 5. no monotonicity property can be introduced for R
A numerical example - 1 21/23 Consider (T is fixed) mi(x) = αi exp{ βix}, where, Ai > 0 and βi 0 are r.v. There can be Market risk only: αi jointly lognormal, βi = 0 Market and non-random liquidity risk: αi jointly lognormal, βi > 0 non-random Market and independent random liquidity risk: (αi, βi)i jointly lognormal, with αi βi Market and correlated random liquidity risk: (αi, βi)i jointly lognormal, with αi and βi negatively correlated Market and correlated random liquidity risk with shocks: (αi, βi)i jointly lognormal, with αi and βi negatively correlated, βi = βi + εi
A numerical example - 2 22/23 For a given portfolio p and L = {q : q0 a}, in any of the 5 previous situations we: set I = 10, αi and βi id. distr. for different i we perform 100k simulations of (mi(x))i for any outcome of the simulation we compute V L (p) we repeat for different inputs (p, a, mean, variances and correlations of αi and βi) A typical outcome is:
Conclusion 23/23 Messages: Liquidity risk arises when msdc are ignored Liquidity risk can be captured by a redefinition of the concept of value, which depends on a liquidity policy Coherent risk measures are perfectly adequate to deal with liquidity risk To do: study possible realistic (yet analytically tractable) stochastic models for a msdc (many studies of the bid-ask spread in the literature) portfolio optimization with liquidity risk