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Coherent Measures of Financial Risk

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Coherent Measures of Financial Risk COHERENT MEASURES OF FINANCIAL RISK THE IMPORTANCE OF THINKING COHERENTLY Philippe De Brouwer Wednesday 9th November, Krakow´ SECTION 1 COHERENT RISK MEASURES PHILIPPE DE BROUWER INTRODUCTION NTRODUCTION HAT OHERENCE I :W C CASES BONDS MORE BONDS IS COHERENCE AND LEGISLATION LIMITS CONCLUSIONS RISK? c PROF.DR.PHILIPPE J.S. DE BROUWER 2/34 COHERENT RISK MEASURES PHILIPPE DE BROUWER INTRODUCTION COHERENCE CASES BONDS MORE BONDS LEGISLATION LIMITS CONCLUSIONS FIGURE 1 : Euclid Proposed 5 Axioms (or rather 3 + 2 definitions) in his “Elements” as foundation of Geometry. — see eg. (Heath 1909) c PROF.DR.PHILIPPE J.S. DE BROUWER 3/34 ALTERNATIVE COHERENT GEOMETRIES COHERENT RISK MEASURES PHILIPPE DE BROUWER INTRODUCTION COHERENCE CASES BONDS MORE BONDS LEGISLATION LIMITS CONCLUSIONS FIGURE 2 : Alternative coherent geometries. Where in Ecuclid’s geometry there is exactly one line parallel to line D and through point M, in Nikola Lobatchevski’s hypersphere there are an infinite number and in Bernhard Riemann’s sphere there are none. c PROF.DR.PHILIPPE J.S. DE BROUWER 4/34 THINKING ABOUT FINANCIAL RISK COHERENT RISK Idea Reference MEASURES PHILIPPE DE no risk, no rewards Ecclesiastes 11:1–6 (ca. 300 BCE) BROUWER diversify investment Ecclesiastes 11:1–2 (ca. 300 BCE) INTRODUCTION and Bernoulli (1738) COHERENCE TABLE 1 : Key ideas about investment risk CASES BONDS MORE BONDS LEGISLATION Risk Measure Reference LIMITS VAR CONCLUSIONS variance ( ) Fisher (1906), Marschak (1938) and Markowitz (1952) Value at Risk (VaR) Roy (1952) semi-variance (S) Markowitz (1991) Expected Shortfall (ES) Acerbi and Tasche (2002) and De Brouwer (2012) TABLE 2 : Normative theories and their risk measures implied. c PROF.DR.PHILIPPE J.S. DE BROUWER 5/34 DEFINITIONS OF RISK MEASURES I COHERENT RISK MEASURES PHILIPPE DE DEFINITION 1 (STANDARD DEVIATION /VARIANCE) BROUWER 2 INTRODUCTION VAR := variance = E[(X − E[X]) ] COHERENCE p σ := standard deviation = VAR CASES BONDS MORE BONDS LEGISLATION LIMITS CONCLUSIONS DEFINITION 2 (VALUE-AT-RISK) VaRα(P) := −(the best of the 100α% worst outcomes of P) c PROF.DR.PHILIPPE J.S. DE BROUWER 6/34 DEFINITIONS OF RISK MEASURES II COHERENT RISK MEASURES PHILIPPE DE BROUWER DEFINITION 3 (EXPECTED SHORTFALL) INTRODUCTION COHERENCE ES(α)(P) := −(average of the worst 100α% realizations) CASES BONDS MORE BONDS LEGISLATION LIMITS DEFINITION 4 (WORST EXPECTED LOSS) CONCLUSIONS WEL := Worst Expected Loss = −E[min(P)] c PROF.DR.PHILIPPE J.S. DE BROUWER 7/34 VISUALIZATION OF SOME RISK MEASURES COHERENT RISK MEASURES PHILIPPE DE BROUWER INTRODUCTION COHERENCE CASES BONDS MORE BONDS LEGISLATION LIMITS CONCLUSIONS FIGURE 3 : visualization of ES, VaR and σ. Note that WEL is not defined. c PROF.DR.PHILIPPE J.S. DE BROUWER 8/34 SECTION 2 COHERENT RISK MEASURES PHILIPPE DE BROUWER INTRODUCTION N XIOMATIC OHERENCE A A C CASES BONDS MORE BONDS APPROACH TO LEGISLATION LIMITS CONCLUSIONS FINANCIAL RISK c PROF.DR.PHILIPPE J.S. DE BROUWER 9/34 A SET OF AXIOMS PROPOSED BY ARTZNER,DELBAEN,EBER, AND HEATH (1997) COHERENT RISK MEASURES DEFINITION 5 (COHERENT RISK MEASURE) PHILIPPE DE BROUWER A function ρ : V 7! R is called a coherent risk measure if and only if INTRODUCTION COHERENCE 1 monotonous: 8X; Y 2 V : X ≤ Y ) ρ(X) ≥ ρ(Y) CASES 2 sub-additive: BONDS MORE BONDS 8X; Y; X + Y 2 : ρ(X + Y) ≤ ρ(X) + ρ(Y) LEGISLATION V LIMITS 3 positively homogeneous: CONCLUSIONS 8a > 0 and 8X; aX 2 V : ρ(aX) = aρ(X) 4 translation invariant: 8a > 0 and 8X 2 V : ρ(X + a) = ρ(X) − a Law-invariance under P: 8X; Y 2 V and 8t 2 R : P[X ≤ t] = P[Y ≤ t] ) ρ(X) = ρ(Y) c PROF.DR.PHILIPPE J.S. DE BROUWER 10/34 WHICH RISK MEASURE IS COHERENT? COHERENT RISK MEASURES PHILIPPE DE BROUWER INTRODUCTION • VAR (or volatility) is not coherent because it is not COHERENCE monotonous (trivial) CASES • VaR is not coherent, because it is not sub-additive BONDS MORE BONDS (Artzner, Delbaen, Eber, and Heath 1999) LEGISLATION LIMITS • ES is coherent (Pflug 2000) CONCLUSIONS • WEL is not usable because it is not Law-Invariant . but who should care? c PROF.DR.PHILIPPE J.S. DE BROUWER 11/34 SECTION 3 COHERENT RISK MEASURES PHILIPPE DE BROUWER INTRODUCTION COHERENCE CASES BONDS ASE TUDIES MORE BONDS C S LEGISLATION LIMITS CONCLUSIONS c PROF.DR.PHILIPPE J.S. DE BROUWER 12/34 CASE 1 ONE BOND COHERENT RISK MEASURES EXAMPLE 1 (ONE BOND) PHILIPPE DE Assume one bond with a 0.7% probability to default in one BROUWER year in all other cases it pays 105% in one year. What is the INTRODUCTION 1%VaR? COHERENCE CASES BONDS MORE BONDS LEGISLATION LIMITS CONCLUSIONS [A] The 1% VaR is −5% ) VaR spots no risk! c PROF.DR.PHILIPPE J.S. DE BROUWER 13/34 CASE 2 COHERENT RISK MEASURES EXAMPLE 2 (TWO INDEPENDENT BONDS) PHILIPPE DE BROUWER Consider two identical bonds with the same parameters, but independently distributed. What is the 1%VaR now? INTRODUCTION COHERENCE CASES BONDS MORE BONDS LEGISLATION LIMITS CONCLUSIONS [A] The 1% VaR of the diversified portfolio is 47.5%! c PROF.DR.PHILIPPE J.S. DE BROUWER 14/34 CASE 4 THE EVIL BANKER AND HIS CUSTOMERS COHERENT RISK MEASURES PHILIPPE DE EXAMPLE 2 (THE EVIL BANKER’S FIRST DILEMMA) BROUWER Consider an Evil Banker who has to compose a portfolio for INTRODUCTION his private client. If there is at least one default in the COHERENCE portfolio, then the banker will loose that client. CASES BONDS How can our banker minimize his work and maximize his MORE BONDS income? LEGISLATION LIMITS CONCLUSIONS [A] The Evil Banker should minimize the probability that at least one bond defaults. This is: N P[at least one default] = 1 − Q P[one default] = 1 − (0:7)N. n=1 The optimal value is hence N = 1. c PROF.DR.PHILIPPE J.S. DE BROUWER 15/34 CASE 5 THE EVIL BANKER AND BASEL III COHERENT RISK MEASURES PHILIPPE DE BROUWER EXAMPLE 2 (THE EVIL BANKER’S SECOND DILEMMA) INTRODUCTION Consider an Evil Banker who has to comply with Basel III, COHERENCE hence uses for assessing market risk VaR. Being Evil he does CASES not care about the size of a bailout. So how does he BONDS MORE BONDS minimize VaR? LEGISLATION LIMITS CONCLUSIONS [A] One bond is optimal. However, VaR only informs that there is 1% chance that the loss will be higher than the VaR. The Evil Banker does not care, but the society should care about the size of an eventual bailout. c PROF.DR.PHILIPPE J.S. DE BROUWER 16/34 CASE 6 MORE BONDS COHERENT RISK MEASURES PHILIPPE DE BROUWER INTRODUCTION COHERENCE EXAMPLE 3 (N INDEPENDENT BONDS) CASES BONDS Consider now an increasing number of independent bonds MORE BONDS LEGISLATION with the same parameters as in previous example. LIMITS Trace the risk surface. CONCLUSIONS c PROF.DR.PHILIPPE J.S. DE BROUWER 17/34 RISK IN FUNCTION OF DIVERSIFICATION CONVECITY (I) COHERENT RISK MEASURES PHILIPPE DE BROUWER INTRODUCTION COHERENCE CASES BONDS MORE BONDS LEGISLATION LIMITS CONCLUSIONS FIGURE 4 : ES and VaR in function of number of bonds. c PROF.DR.PHILIPPE J.S. DE BROUWER 18/34 INCOHERENT RISK MEASURES IN LEGISLATION COHERENT RISK MEASURES PHILIPPE DE BROUWER legislation “risk measure” result INTRODUCTION UCITS VaR and VAR non suitable assets COHERENCE Basel VaR crisis CASES BONDS Solvency VaR insolvency MORE BONDS LEGISLATION TABLE 3 : Law makers increasingly use non-coherent risk LIMITS measures in legislation, resulting in encouraging to take large CONCLUSIONS bets, ignore extreme risks and mislead investors. All building up to the next crisis . building up to the next global disaster. c PROF.DR.PHILIPPE J.S. DE BROUWER 19/34 YES, IT IS IMPORTANT! COHERENT RISK MEASURES PHILIPPE DE BROUWER INTRODUCTION COHERENCE • Internal • AI, Big CASES Combus- • Transistor BONDS tion Data & MORE BONDS • • Steam / / Laser / Algos LEGISLATION Engine • Engine Com- • Quantum LIMITS • Electric- puter Computing CONCLUSIONS ity 1770s–1857 •Internet • • Mag- Biotech 1940s–2008 • netism Nanotech 1870s–1929 2010s–? FIGURE 5 : A simplified model of science propelling welfare and economy, but leading to crisis situations. c PROF.DR.PHILIPPE J.S. DE BROUWER 20/34 SECTION 4 COHERENT RISK MEASURES PHILIPPE DE BROUWER INTRODUCTION HE IMITS OF OHERENCE T L C CASES BONDS MORE BONDS COHERENT RISK LEGISLATION LIMITS CONCLUSIONS MEASURES c PROF.DR.PHILIPPE J.S. DE BROUWER 21/34 THE LIMITS OF COHERENT RISK MEASURES LIQUIDITY COHERENT RISK MEASURES PHILIPPE DE BROUWER INTRODUCTION COHERENCE EXAMPLE 4 (ILLIQUID ASSETS) CASES Imagine that you hold twice the average daily volume in BONDS MORE BONDS stock X. Is it realistic to demand from a risk measure that it LEGISLATION LIMITS is positive homogeneous and hence that CONCLUSIONS 8a > 0 and 8X; aX 2 V : ρ(aX) = aρ(X) ? c PROF.DR.PHILIPPE J.S. DE BROUWER 22/34 THE LIMITS OF COHERENT RISK MEASURES NOT A REAL VALUED STOCHASTIC VARIABLE COHERENT RISK MEASURES PHILIPPE DE BROUWER INTRODUCTION COHERENCE EXAMPLE 5 (THIRSTY) CASES Imagine that you need to drink in order to cross the desert, BONDS MORE BONDS but you know that one of your five bottles is poisoned (of LEGISLATION LIMITS course you don’t know which one). What strategy do you CONCLUSIONS take to minimize risk? Diversify or Russian Roulette? c PROF.DR.PHILIPPE J.S. DE BROUWER 23/34 THE LIMITS OF COHERENT RISK MEASURES SYSTEMICITY COHERENT RISK MEASURES PHILIPPE DE BROUWER INTRODUCTION EXAMPLE 6 (BASEL II WITH ES?) COHERENCE Would it make sense to replace VaR in the capital CASES requirements for banks by ES? BONDS MORE BONDS LEGISLATION LIMITS [A] It would be a significant improvement, but would it also CONCLUSIONS not work systemic? (i.e.
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