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Lecture 8, Value-At-Risk and Expected Shortfall Value-at-Risk Normal VaR Approximate - Normal VaR t VaR Coherent risk measures Expected shortfall Quantitative Finance Lecture 8, Value-at-Risk and Expected Shortfall Adam Farago Department of Economics, University of Gothenburg 2019 Quantitative Finance, Lecture 8 1 / 56 Value-at-Risk Normal VaR Approximate - Normal VaR t VaR Coherent risk measures Expected shortfall Value-at-Risk Introduction Ways to measure market risk: Scenario-based risk measures, stress testing Factor sensitivity analysis Risk measures based on the loss distribution approach Value-at-Risk (VaR) Expected shortfall (ES) Quantitative Finance, Lecture 8 2 / 56 Value-at-Risk Normal VaR Approximate - Normal VaR t VaR Coherent risk measures Expected shortfall Value-at-Risk Introduction Ways to measure market risk: Scenario-based risk measures, stress testing Factor sensitivity analysis Risk measures based on the loss distribution approach Value-at-Risk (VaR) Expected shortfall (ES) Quantitative Finance, Lecture 8 2 / 56 Value-at-Risk Normal VaR Approximate - Normal VaR t VaR Coherent risk measures Expected shortfall Value-at-Risk Introduction So far we were interested in small changes of the underlying. But how bad things can get? What loss level is such that we are X% confident it will not be exceeded in N business days? VaR is certainly the risk measurement standard in the industry today. Regulators base the capital they require banks to keep on VaR. Mutual funds and hedge funds use it as a risk measure to report to clients. It captures an important aspect of risk in a single number. It is easy to understand. It asks the simple question: How bad can things get? Quantitative Finance, Lecture 8 3 / 56 Value-at-Risk Normal VaR Approximate - Normal VaR t VaR Coherent risk measures Expected shortfall Value-at-Risk Formalization Consider a fixed time period [t; t + ∆t] , where ∆t can be e.g 1 day, 10 days, one month, one year, etc. We can define the following objects: The change in the value of the portfolio during the period [t; t + ∆t] is ∆V ≡ Vt+∆t − Vt The return on the portfolio during the period [t; t + ∆t] is ∆V RV ≡ Vt The loss of the portfolio during the period [t; t + ∆t] is L ≡ −∆V = − (Vt+∆t − Vt ) The relative loss of the portfolio during the period [t; t + ∆t] is L RL ≡ = −RV Vt Quantitative Finance, Lecture 8 4 / 56 Value-at-Risk Normal VaR Approximate - Normal VaR t VaR Coherent risk measures Expected shortfall Value-at-Risk Formalization The distribution FL (x) of the random variable L is called the loss distribution (over the time period [t; t + ∆t]) It is defined as FL (x) = P (L ≤ x) FL (x) is the cumulative distribution function (cdf) of L. Be aware of the sign! L takes a positive value if the value of the portfolio goes down. Quantitative Finance, Lecture 8 5 / 56 Value-at-Risk Normal VaR Approximate - Normal VaR t VaR Coherent risk measures Expected shortfall Value-at-Risk Defintion \What loss level is such that we are X% confident it will not be exceeded in N business days?" Value-at-Risk (VaR) Given the loss L and a confidence level α 2 (0; 1), VaRα (L) is given by the smallest number x such that the probability that the loss exceeds x is not larger than 1 − α. That is, VaRα (L) ≡ inf fx 2 R : P (L ≥ x) ≤ 1 − αg Note that VaRα (L) = inf fx 2 R : 1 − P (L ≤ x) ≤ 1 − αg = inf fx 2 R : P (L ≤ x) ≥ αg = inf fx 2 R : FL (x) ≥ αg Quantitative Finance, Lecture 8 6 / 56 Value-at-Risk Normal VaR Approximate - Normal VaR t VaR Coherent risk measures Expected shortfall Value-at-Risk for discrete random variables 0.5 0.45 If α = 0:99, then 0.4 fx 2 R : FL(x) ≥ 0:99g = [6000; 1[ 0.3 so VaR (L)0:99 = 6000 0.2 0.2 0.2 If α = 0:97, then probability fx 2 R : FL(x) ≥ 0:97g = [4000; 1[ 0.1 0.06 0.06 so VaR (L) = 4000 0.03 0:97 0 1 If α = 0:95, then 0.8 fx 2 R : FL(x) ≥ 0:95g = [4000; 1[ so VaR (L)0:95 = 4000 0.6 cdf If α = 0:9, then 0.4 fx 2 R : FL(x) ≥ 0:9g = [2000; 1[ 0.2 so VaR (L)0:9 = 2000 0 -6000 -4000 -2000 0 2000 4000 6000 L (loss) Quantitative Finance, Lecture 8 7 / 56 Value-at-Risk Normal VaR Approximate - Normal VaR t VaR Coherent risk measures Expected shortfall Value-at-Risk for continuous random variables 10-4 For continuous random variables, 2 the definition simplifies. 1.5 Reason: The cdf of a continuous 1 random variable is continuous pdf and strictly increasing. 0.5 Therefore: 1- 0 1 VaR (L)α = fx 2 R : FL (x) = αg 0.8 −1 = FL (α) 0.6 cdf −1 0.4 where FL is the inverse cdf. 0.2 That, is the VaR is a quantile of the distribution. 0 -6000 -4000 -2000 0 2000 4000 6000 L (loss) Quantitative Finance, Lecture 8 8 / 56 Value-at-Risk Normal VaR Approximate - Normal VaR t VaR Coherent risk measures Expected shortfall Comments VaRα(L) will typically be a positive number, since it is connected to the loss-distribution. The VaR can also be reported in terms of returns. Because of the linearity of VaR, VaRα(L) VaRα (RL) = Vt where Vt denotes the current total value of the portfolio. −1 If we can find an analytical expression for the inverse function FL (y), then we can also find an analytical expression for the VaR. Quantitative Finance, Lecture 8 9 / 56 Value-at-Risk Normal VaR Approximate - Normal VaR t VaR Coherent risk measures Expected shortfall Comments VaR at different horizons The Value-at-Risk is defined for a certain time horizon (as L is defined over a a certain time period [t; t + ∆t]) If ∆t is one day, then we talk about a one-day VaR. If ∆t is h days, then we talk about a h-day VaR. In market risk, the typical horizon is 1 day, 5 days, or 10 days. Quantitative Finance, Lecture 8 10 / 56 Value-at-Risk Normal VaR Approximate - Normal VaR t VaR Coherent risk measures Expected shortfall Linearity of the Value-at-Risk Lineraity Let a 2 R and b > 0 constants. Then, VaRα (a + b · L) = a + b · VaRα (L) Proof: VaRα (a + bL) = inf fx 2 R : P (a + bL ≤ x) ≥ αg x − a = inf x 2 : P L ≤ ≥ α R b = inf fa + bx~ 2 R : P (L ≤ x~) ≥ αg = a + b inf fx~ 2 R : P (L ≤ x~) ≥ αg = a + bVaRα (L) Quantitative Finance, Lecture 8 11 / 56 Value-at-Risk Normal VaR Approximate - Normal VaR t VaR Coherent risk measures Expected shortfall Analytical VaR when L is normally distributed Let L be normally distributed with mean µ and variance σ2, that is, L ∼ N µ, σ2. Then x − µ F (x) = Φ L σ where Φ (z) is the cdf for a standard normal random variable, i.e., Z z 2 1 − y Φ(z) = p e 2 dy −∞ 2π −1 So FL (y) is obtained by solving for x in the equation y = FL (x) x − µ x − µ Φ = y , = Φ−1 (y) , x = µ + σΦ−1 (y) σ σ Quantitative Finance, Lecture 8 12 / 56 Value-at-Risk Normal VaR Approximate - Normal VaR t VaR Coherent risk measures Expected shortfall Analytical VaR when L is normally distributed So, the Value-at-Risk at confidence level α is −1 VaRα (L) = µ + σΦ (α) where Φ−1 (y) can be obtained from tables or in any software such as Matlab, Excel, Stata, etc. (in Matlab, write icdf('norm',y,0,1)). Some often used values are: α Φ−1 (α) 0.999 3.090 0.995 2.576 0.99 2.326 0.95 1.645 0.9 1.282 Quantitative Finance, Lecture 8 13 / 56 Value-at-Risk Normal VaR Approximate - Normal VaR t VaR Coherent risk measures Expected shortfall Example: portfolio of stocks Consider a portfolio of US stocks, which invests 10,000$ in each of the following five stocks (the total value of the portfolio is 50,000$): Apple, Coca-Cola, Exxon Mobil, General Electric, and Procter & Gamble. Let us calculate the 5-day VaR and ES with α = 0:95 using the assumption that the return on the five assets follow a multivariate normal distribution. Quantitative Finance, Lecture 8 14 / 56 Value-at-Risk Normal VaR Approximate - Normal VaR t VaR Coherent risk measures Expected shortfall Example: portfolio of stocks Multivariate normal returns Means, standard deviations, and correlations of the weekly (five-day) returns on these stocks using the sample Jan 2, 1986 to Jun 29, 2018: Mean Std. Dev. Correlations 1. 2. 3. 4. 5. 1. Coca-Cola 0.30 3.30 1.00 0.40 0.42 0.18 0.54 2. Exxon 0.26 3.00 1.00 0.47 0.18 0.38 3. GE 0.24 4.14 1.00 0.28 0.40 4. Apple 0.60 6.39 1.00 0.12 5. P&G 0.28 3.19 1.00 Means and standard deviations are expressed in % per five trading days. Let us assume that the five return series follow a multivariate normal distribution R ∼ N(µ, Σ) where µ is the mean vector from the above table and Σ is the variance-covariance matrix based on the above descriptives.
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