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Expected Shortfall Chapter 4 Expected Shortfall Expected Shortfall (ES) is a risk measure computed by averating potential losses above a certain level given by the Value at Risk (VaR). It can be shown that the Expected Shortfall at the confidence level p coincides with the Tail Value at Risk (TVaR) defined as the average of losses suffered in the worst (1 − p)% of events. This chapter presents the concept of coherent risk measure, including Expected Shortfall and Tail Value at Risk (TVaR), together with experiments based on financial data sets. 4.1 Tail Value at Risk (TVaR) . 77 4.2 Conditional Tail Expectation (CTE) . 78 4.3 Expected Shortfall (ES) . 81 4.4 Market Data vs Gaussian Risk Measures . 88 Exercises . 97 4.1 Tail Value at Risk (TVaR) A natural shortcoming of Value at Risk is failing to provide information on p the behavior of probability distribution tails beyond VX . The next figure illustrates the limitations of Value at Risk, namely its inability to capture p † the properties of a probability distribution beyond VX . † “Value at Risk is like an airbag that works all the time, except when you have a car accident”. - D. Einhorn, hedge fund manager. 77 " This version: June 13, 2021 https://personal.ntu.edu.sg/nprivault/indext.html N. Privault 95% Fig. 4.1: Two distributions having the same Value at Risk VX = 2.145. The Tail Value at Risk (or Conditional Value at Risk) aims at providing a solution to the tail distribution problem observed with Value at Risk at the level p, by averaging over confidence levels ranging from p to 1. Definition 4.1. The Tail Value at Risk of a random variable X at the level p ∈ (0, 1) is defined by the average p 1 1 q TV := V dq. (4.1) X 1 − p wp X p Note that since the function p 7−→ VX is non-decreasing, we always have p 1 1 q 1 1 p p TVX = VX dq > VX dq = VX . 1 − p wp 1 − p wp 4.2 Conditional Tail Expectation (CTE) Recall that by Lemma 10.14, given an event A such that P(A) > 0, the conditional expectation of X : W −→ N given the event A satisfies 1 E[X | A] = E [X1 ] , P(A) A see Section 3.2 for an example. p Definition 4.2. Consider a random variable X such that P X > VX > 0. The Conditional Tail Expectation of X at the level p ∈ (0, 1) is the quantity p p 1 CTE := E X | X > V = E X1 p . X X p {X>VX } P X > VX The use of the strict inequality “>” in the definition of the Conditional Tail p Expectation allows us to avoid any dependence on P(X = V ), and to con- p X sider risky values strictly beyond VX . The Conditional Tail Expectation is 78 " This version: June 13, 2021 https://personal.ntu.edu.sg/nprivault/indext.html Notes on Financial Risk and Analytics also called Conditional Value at Risk (CVaR). Examples of Conditional Tail Expectations can be computed as in the fol- lowing R code. 1 library(quantmod) 2 getSymbols("^HSI",from="2013-06-01",to="2014-10-01",src="yahoo") returns <- as.vector(diff(log(Ad(`HSI`)))) 4 library(PerformanceAnalytics) var=VaR(returns, p=.95, method="historical") 6 cte=mean(returns[returns<as.numeric(var)],na.rm=TRUE) The next proposition shows by which amount the Conditional Tail Expecta- tion exceeds the Value at Risk. p p Proposition 4.3. For any p ∈ (0, 1] we have CTE > E[X] and CTE > p X X VX with, more precisely, p p p p + p CTEX = E X | X > VX = VX + E X − VX | X > VX . Proof. We have p 1 E X | X > V = E X1 p X p {X>VX } P X > VX 1 p p = E X − V 1 p + V E 1 p p X {X>VX } X {X>VX } P X > VX 1 p + p p = p E X − VX + VX P X > VX P X > VX p 1 p + = VX + p E X − VX P X > VX p p + p = VX + E X − VX X > VX . p See Exercise 4.2-(d) for a proof of CTEX > E[X]. p Next, we check that when P X = VX = 0, the Conditional Tail Expectation coincides with the Tail Value at Risk. Note that in this case we have PX > p VX > 0 by Proposition 3.9-(b). p p Proposition 4.4. Assume that P X = V = 0. Then we have CTE = p X X TVX , i.e. p p p 1 1 q p CTEX = E X | X > VX = E X | X > VX = VX dq = TVX . 1 − p wp (4.2) U Proof. By Lemma 3.13 we construct X as X = VX where U is uniformly distributed on [0, 1], with 79 " This version: June 13, 2021 https://personal.ntu.edu.sg/nprivault/indext.html N. Privault U p p U > p =⇒ VX > VX =⇒ X > VX , and p U p X > VX =⇒ VX > VX =⇒ U > p. p Since P X = VX = 0 we find that, with probability 1, U p p p U > p ⇐⇒ U > p ⇐⇒ VX > VX ⇐⇒ X > VX ⇐⇒ X > VX , hence p p CTEX = E X | X > VX U U p = E VX | VX > VX U = E VX | U > p 1 U 1 = E VX {U p} P(U > p) > 1 1 q = V dq. 1 − p wp X The next figure shows the location of Value at Risk and Conditional Tail Expectation on a data set. Note that the sign of the data has been changed according to Proposition 3.10. Fig. 4.2: Value at Risk and Conditional Tail Expectation. The Conditional Tail Expectation of a Gaussian N (µ, σ2) random variable is computed in the next proposition. Proposition 4.5. Gaussian CTE. Given X 'N (µ, σ2) we have p σ p σ −(V p)2 2 CTE = µ + φ(V ) = µ + √ e Z / , (4.3) X (1 − p) Z (1 − p) 2π 80 " This version: June 13, 2021 https://personal.ntu.edu.sg/nprivault/indext.html Notes on Financial Risk and Analytics p where VZ is the Value at Risk of Z 'N (0, 1) at the level p and 1 2 φ(z) = √ e−z /2, x ∈ R, 2π is the standard normal probability density function. p p Proof. Using the relation P X > VX = P X > VX = 1 − p, cf. Proposi- tion 3.9, we have p p CTEX = TVX p = E X | X > VX 1 = E X1 p p {X>VX } P X > VX 1 ∞ 2 2 dx = xe−(x−µ) /(2σ ) √ wV p 2 1 − p X 2πσ µ ∞ 2 2 dx 1 ∞ 2 2 dx = e−(x−µ) /(2σ ) √ + (x − µ)e−(x−µ) /(2σ ) √ wV p 2 wV p 2 1 − p X 2πσ 1 − p X 2πσ 2 µ p σ h 2 2 i∞ = P(X V ) + √ −e−(x−µ) /(2σ ) > X 2 p 1 − p (1 − p) 2πσ VX 2 σ −((V p −µ) σ)2 2 = µ + √ e X / / (1 − p) 2πσ2 σ −(V p)2 2 = µ + √ e Z / (1 − p) 2π σ p = µ + φ(V ), (1 − p) Z p p due to the rescaling relation VX = µ + σqZ , cf. (3.7). 4.3 Expected Shortfall (ES) p There are several variants for the definition of the Expected Shortfall ESX . Next is a frequently used definition. p Definition 4.6. The Expected Shortfall ESX of a random variable X at the level p ∈ (0, 1) is defined by p p 1 p 1 ESX := VX + E X − VX {X V p } . (4.4) 1 − p > X We also have p p p P X > VX p p ES = V + E X − V X V X X 1 − p X > X 81 " This version: June 13, 2021 https://personal.ntu.edu.sg/nprivault/indext.html N. Privault p 1 1 p 1 = VX + E X {X V p } − VX E {X V p } 1 − p > X > X p 1 1 p p = VX + E X {X V p } − VX P X > VX 1 − p > X V p 1 1 X p = E X {X V p } + 1 − p − P X > VX , 1 − p > X 1 − p and 1 1 p p p E X {X V p } = E X | X > VX = TVX if P X = VX = 0, 1 − p > X p ES = X p 1 V p p 1 p X E X {X V } + 1 − p − P X > VX if P X = VX > 0, 1 − p > X 1 − p as shown in the next proposition. Note that by Proposition 3.9-(b) we have p p P X > VX > 0 when P X = VX = 0. p p Proposition 4.7. When P X = V = 0 the Expected Shortfall ES co- X p X incides with the Conditional Tail Expectation CTE and the Tail Value at p X Risk TVX , i.e., we have p p p p ESX = E X | X > VX = E X | X > VX = TVX . p Proof. By Proposition 3.9, when P X = VX = 0 we have p p p p = P(X 6 VX ) and 1 − p = P(X > VX ) = P(X > VX ), hence p 1 1 ESX = E X {X V p } 1 − p > X 1 1 = E X {X>V p } 1 − p X 1 = E X1 p p {X>VX } P X > VX p = E X | X > VX p = TVX , by Proposition 4.4. p When P X = VX = 0, we also have V p p 1 1 p X p ESX = E X {X V p } + VX − P X > VX 1 − p > X 1 − p V p 1 1 X p = E X {X>V p } + 1 − p − P X > VX .
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