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1 Slides for Risk Management VaR and Expected Shortfall Groll Seminar für Finanzökonometrie Prof. Mittnik, PhD 1 Introduction Value-at-Risk Expected Shortfall Model risk Multi-period / multi-asset case 2 Multi-period VaR and ES Excursion: Joint distributions Excursion: Sums over two random variables Linearity in joint normal distribution 3 Aggregation: simplifying assumptions Normally distributed returns 4 Properties of risk measures Groll Seminar für Finanzökonometrie Slides for Risk Management Prof. Mittnik, PhD 1 / 133 Introduction Notation Groll Seminar für Finanzökonometrie Slides for Risk Management Prof. Mittnik, PhD 2 / 133 Introduction From profits to losses risk often is defined as negative deviation of a given target payoff riskmanagement is mainly concerned with downsiderisk convention: focus on the distribution of losses instead of profits for prices denoted by P t, the random variable quantifying losses is given by L t+1 = P t+1 P t distribution of losses equals distribution of profits flipped at x-axis Groll Seminar für Finanzökonometrie Slides for Risk Management Prof. Mittnik, PhD 3 / 133 Groll Seminar für Finanzökonometrie Slides for Risk Management Prof. Mittnik, PhD 4 / 133

2 Quantification of risk Introduction Decomposing risk Introduction decisions concerned with managing, mitigating or hedging of risks have to be based on quantification of risk as basis of decision-making: regulatory purposes: capital buffer proportional to exposure to risk interior management decisions: freedom of daily traders restricted by capping allowed risk level corporate management: identification of key risk factors comparability information contained in loss distribution is mapped to scalar value: information reduction You are casino owner. 1 You only have one table of roulette, with one gambler, who bets 100 on number 12. He only plays one game, and while the odds of winning are 1:36, his payment in case of success will be 3500 only. With expected positive payoff, what is your risk? completely computable 2 Now assume that you have multiple gamblers per day. Although you have a pretty good record of the number of gamblers over the last year, you still have to make an estimate about the number of visitors today. What is your risk? additional risk due to estimation error 3 You have been owner of The Mirage Casino in Las Vegas. What was your biggest loss within the last years? Groll Seminar für Finanzökonometrie Slides for Risk Management Prof. Mittnik, PhD 5 / 133 Introduction Decomposing risk Groll Seminar für Finanzökonometrie Slides for Risk Management Prof. Mittnik, PhD 6 / 133 Introduction Risk measurement frameworks the closing of the show of Siegfried and Roy due to the attack of a tiger led to losses of hundreds of millions of dollars model risk notional-amount approach: weighted nominal value nominal value as substitute for outstanding amount at risk weighting factor representing riskiness of associated asset class as substitute for riskiness of individual asset component of standardized approach of Basel capital adequacy framework advantage: no individual risk assessment necessary - applicable even without empirical data weakness: diversification benefits and netting unconsidered, strong simplification Groll Seminar für Finanzökonometrie Slides for Risk Management Prof. Mittnik, PhD 7 / 133 Groll Seminar für Finanzökonometrie Slides for Risk Management Prof. Mittnik, PhD 8 / 133

3 Risk measurement frameworks Introduction Risk measurement frameworks Introduction scenario analysis: define possible future economic scenarios stock market crash of -20 percent in major economies, default of Greece government securities,... derive associated losses determine risk as specified quantile of scenario losses 5th largest loss, worst loss, protection against at least 90 percent of scenarios,... since scenarios are not accompanied by statements about likelihood of occurrence, probability dimension is completely left unconsidered scenario analysis can be conducted without any empirical data on the sole grounds of expert knowledge risk measures based on loss distribution: statistical quantities of asset value distribution function loss distribution incorporates all information about both probability and magnitude of losses includes diversification and netting effects usually relies on empirical data full information of distribution function reduced to charateristics of distribution for better comprehensibility examples: standard deviation, Value-at-Risk, Expected Shortfall, Lower Partial Moments standard deviation: symmetrically capturing positive and negative risks dilutes information about downsiderisk overall loss distribution inpracticable: approximate risk measure of overall loss distribution by aggregation of asset subgroup risk measures Groll Seminar für Finanzökonometrie Slides for Risk Management Prof. Mittnik, PhD 9 / 133 Value-at-Risk Value-at-Risk Groll Seminar für Finanzökonometrie Slides for Risk Management Prof. Mittnik, PhD 10 / 133 Value-at-Risk Loss distribution known Value-at-Risk The Value-at-Risk VaR at the confidence level α associated with a given loss distribution L is defined as the smallest value l that is not exceeded with probability higher than 1 α. That is, VaR α = inf {l R : P L > l 1 α} = inf {l R : F L l α}. typical values for α : α = 0.95, α = 0.99 or α = as a measure of location, VaR does not provide any information about the nature of losses beyond the VaR the losses incurred by investments held on a daily basis exceed the value given by VaR α only in 1 α 100 percent of days financial entity is protected in at least α-percent of days Groll Seminar für Finanzökonometrie Slides for Risk Management Prof. Mittnik, PhD 11 / 133 Groll Seminar für Finanzökonometrie Slides for Risk Management Prof. Mittnik, PhD 12 / 133

4 Loss distribution known Value-at-Risk Loss distribution known Value-at-Risk Groll Seminar für Finanzökonometrie Slides for Risk Management Prof. Mittnik, PhD 13 / 133 Value-at-Risk Estimation frameworks Groll Seminar für Finanzökonometrie Slides for Risk Management Prof. Mittnik, PhD 14 / 133 Value-at-Risk Properties of historical simulation in general: underlying loss distribution is not known two estimation methods for VaR: directly estimate the associated quantile of historical data estimate model for underlying loss distribution, and evaluate inverse cdf at required quantile derivation of VaR from a model for the loss distribution can be further decomposed: analytical solution for quantile Monte Carlo Simulation when analytic formulas are not available modelling the loss distribution inevitably entails model risk, which is concerned with possibly misleading results due to model misspecifications simulation study: examine properties of estimated sample quantiles assume t-distributed loss distribution with degrees-of-freedom parameter ν = 3 and mean shifted by : VaR 0.99 = 4.54 VaR = 5.84 VaR = estimate VaR for simulated samples of size 2500 approximately 10 years in trading days compare distribution of estimated VaR values with real value of applied underlying loss distribution even with sample size 2500, only 2.5 values occur above the quantile on average high mean squared errors mse Groll Seminar für Finanzökonometrie Slides for Risk Management Prof. Mittnik, PhD 15 / 133 Groll Seminar für Finanzökonometrie Slides for Risk Management Prof. Mittnik, PhD 16 / 133

5 Value-at-Risk Distribution of estimated values Value-at-Risk Groll Seminar für Finanzökonometrie Slides for Risk Management Prof. Mittnik, PhD 17 / 133 Value-at-Risk Modelling the loss distribution Groll Seminar für Finanzökonometrie Slides for Risk Management Prof. Mittnik, PhD 18 / 133 Value-at-Risk Remarks introductory model: assume normally distributed loss distribution VaR normal distribution For given parameters µ L and σ VaR α can be calculated analytically by VaR α = µ L + σφ 1 α. Proof. P L VaR α = P L µ L + σφ 1 α L µl = P Φ 1 α σ = Φ Φ 1 α = α note: µ L in VaR α = µ L + σφ 1 α is the expectation of the loss distribution if µ denotes the expectation of the asset return, i.e. the expectation of the profit, then the formula has to be modified to VaR α = µ + σφ 1 α in practice, the assumption of normally distributed returns usually can be rejected both for loss distributions associated with credit and operational risk, as well as for loss distributions associated with market risk at high levels of confidence Groll Seminar für Finanzökonometrie Slides for Risk Management Prof. Mittnik, PhD 19 / 133 Groll Seminar für Finanzökonometrie Slides for Risk Management Prof. Mittnik, PhD 20 / 133

6 Expected Shortfall Expected Shortfall Expected Shortfall Expected Shortfall Definition The Expected Shortfall ES with confidence level α denotes the conditional expected loss, given that the realized loss is equal to or exceeds the corresponding value of VaR α : Expected Shortfall as expectation of conditional loss distribution: ES α = E [L L VaR α ]. given that we are in one of the 1 α 100 percent worst periods, how high is the loss that we have to expect? Groll Seminar für Finanzökonometrie Slides for Risk Management Prof. Mittnik, PhD 21 / 133 Expected Shortfall Additional information of ES Groll Seminar für Finanzökonometrie Slides for Risk Management Prof. Mittnik, PhD 22 / 133 Expected Shortfall Estimation frameworks ES contains information about nature of losses beyond the VaR : in general: underlying loss distribution is not known two estimation methods for ES: directly estimate the mean of all values greater than the associated quantile of historical data estimate model for underlying loss distribution, and calculate expectation of conditional loss distribution derivation of ES from a model for the loss distribution can be further decomposed: analytical calculation of quantile and expectation: involves integration Monte Carlo Simulation when analytic formulas are not available Groll Seminar für Finanzökonometrie Slides for Risk Management Prof. Mittnik, PhD 23 / 133 Groll Seminar für Finanzökonometrie Slides for Risk Management Prof. Mittnik, PhD 24 / 133

7 Properties of historical simulation Expected Shortfall ES under normal distribution Expected Shortfall high mean squared errors mse for Expected Shortfall at high confidence levels: ES for normally distributed losses Given that L N µ L, σ 2, the Expected Shortfall of L is given by ES α = µ L + σ φ Φ 1 α 1 α. Groll Seminar für Finanzökonometrie Slides for Risk Management Prof. Mittnik, PhD 25 / 133 Expected Shortfall Proof Groll Seminar für Finanzökonometrie Slides for Risk Management Prof. Mittnik, PhD 26 / 133 Expected Shortfall Proof ES α = E [L L VaR α ] = E [ L L µ L + σφ 1 α ] [ = E L L µ ] L Φ 1 α σ [ = µ L µ L + E L L µ ] L Φ 1 α σ [ = µ L + E L µ L L µ ] L Φ 1 α σ [ L µl = µ L + σe L µ ] L Φ 1 α σ σ = µ L + σe [ Y Y Φ 1 α ], with Y N 0, 1 Furthermore, P Y Φ 1 α = 1 P Y Φ 1 α = 1 Φ Φ 1 α = 1 α, so that the conditional density as the scaled version of the standard normal density function is given by φ Y Y Φ 1 α y = φ y 1 {y Φ 1 α} P Y Φ 1 α = φ y 1 {y Φ 1 α}. 1 α Groll Seminar für Finanzökonometrie Slides for Risk Management Prof. Mittnik, PhD 27 / 133 Groll Seminar für Finanzökonometrie Slides for Risk Management Prof. Mittnik, PhD 28 / 133

8 Proof Expected Shortfall Example: Meaning of VaR Expected Shortfall Hence, the integral can be calculated as with : E [ Y Y Φ 1 α ] = = φ y = 1 2π exp ˆ Φ 1 α ˆ Φ 1 α ˆ = 1 1 α y φ Y Y Φ 1 α y dy y φ y 1 α dy Φ 1 α y φ y dy = 1 1 α [ φ y] Φ 1 α = 1 1 α = φ Φ 1 α, 1 α y φ Φ 1 α 2y = y φ y 2 You have invested 500,000 in an investment fonds. The manager of the fonds tells you that the 99% Value-at-Risk for a time horizon of one year amounts to 5% of the portfolio value. Explain the information conveyed by this statement. Groll Seminar für Finanzökonometrie Slides for Risk Management Prof. Mittnik, PhD 29 / 133 Expected Shortfall Solution Groll Seminar für Finanzökonometrie Slides for Risk Management Prof. Mittnik, PhD 30 / 133 Expected Shortfall Example: discrete case for continuous loss distribution we have equality example possible discrete loss distribution: P L VaR α = 1 α transform relative statement about losses into absolute quantity VaR α = , 000 = 25, 000 pluggin into formula leads to P L 25, 000 = 0.01, interpretable as with probability 1% you will lose 25,000 or more a capital cushion of height VaR 0.99 = is sufficient in exactly 99% of the times for continuous distributions Groll Seminar für Finanzökonometrie Slides for Risk Management Prof. Mittnik, PhD 31 / 133 the capital cushion provided by VaR α would be sufficient even in 99.3% of the times interpretation of statement: with probability of maximal 1% you will lose 25,000 or more Groll Seminar für Finanzökonometrie Slides for Risk Management Prof. Mittnik, PhD 32 / 133

9 Example: Meaning of ES Expected Shortfall Solution Expected Shortfall The fondsmanager corrects himself. Instead of the Value-at-Risk, it is the Expected Shortfall that amounts to 5% of the portfolio value. How does this statement have to be interpreted? Which of both cases does imply the riskier portfolio? given that one of the 1% worst years occurs, the expected loss in this year will amount to 25,000 since always VaR α ES α, the first statement implies ES α 25, 000 the first statement implies the riskier portfolio Groll Seminar für Finanzökonometrie Slides for Risk Management Prof. Mittnik, PhD 33 / 133 Expected Shortfall Example: market risk Groll Seminar für Finanzökonometrie Slides for Risk Management Prof. Mittnik, PhD 34 / 133 Expected Shortfall estimating VaR for DAX Groll Seminar für Finanzökonometrie Slides for Risk Management Prof. Mittnik, PhD 35 / 133 Groll Seminar für Finanzökonometrie Slides for Risk Management Prof. Mittnik, PhD 36 / 133

10 Expected Shortfall Empirical distribution Expected Shortfall Groll Seminar für Finanzökonometrie Slides for Risk Management Prof. Mittnik, PhD 37 / 133 Expected Shortfall Historical simulation Groll Seminar für Finanzökonometrie Slides for Risk Management Prof. Mittnik, PhD 38 / 133 Expected Shortfall Historical simulation VaR 0.99 = , VaR = , VaR = ES 0.99 = , ES = , ES = Groll Seminar für Finanzökonometrie Slides for Risk Management Prof. Mittnik, PhD 39 / 133 Groll Seminar für Finanzökonometrie Slides for Risk Management Prof. Mittnik, PhD 40 / 133

11 Under normal distribution Expected Shortfall given estimated expectation for daily index returns, calculate estimated expected loss ˆµ L = ˆµ plugging estimated parameter values of normally distributed losses into formula VaR α = µ L + σφ 1 α, for α = 99% we get for VaR we get VaR 0.99 = ˆµ L + ˆσΦ = = VaR = = Groll Seminar für Finanzökonometrie Slides for Risk Management Prof. Mittnik, PhD 41 / 133 Model risk Performance: backtesting for Expected Shortfall, using we get and Expected Shortfall ES α = µ L + σ φ Φ 1 α 1 α ÊS 0.99 = φ Φ = φ = = , ÊS = φ = = Groll Seminar für Finanzökonometrie Slides for Risk Management Prof. Mittnik, PhD 42 / 133 Model risk Backtesting: interpretation, how good did VaR-calculations with normally distributed returns perform? backtesting VaR-calculations based on assumption of independent normally distributed losses generally leads to two patterns: percentage frequencies of VaR-exceedances are higher than the confidence levels specified: normal distribution assigns too less probability to large losses VaR-exceedances occur in clusters: given an exceedance of VaR today, the likelihood of an additional exceedance in the days following is larger than average clustered exceedances indicate violation of independence of losses over time clusters have to be captured through time series models Groll Seminar für Finanzökonometrie Slides for Risk Management Prof. Mittnik, PhD 43 / 133 Groll Seminar für Finanzökonometrie Slides for Risk Management Prof. Mittnik, PhD 44 / 133

12 Model risk Model risk Model risk Appropriateness of normal distribution given that returns in the real world were indeed generated by an underlying normal distribution, we could determine the risk inherent to the investment up to a small error arising from estimation errors however, returns of the real world are not normally distributed in addition to the risk deduced from the model, the model itself could be significantly different to the processes of the real world that are under consideration the risk of deviations of the specified model from the real world is called model risk the results of the backtesting procedure indicate substantial model risk involved in the framework of assumed normally distributed losses Groll Seminar für Finanzökonometrie Slides for Risk Management Prof. Mittnik, PhD 45 / 133 Model risk Appropriateness of normal distribution Groll Seminar für Finanzökonometrie Slides for Risk Management Prof. Mittnik, PhD 46 / 133 Model risk Appropriateness of normal distribution Groll Seminar für Finanzökonometrie Slides for Risk Management Prof. Mittnik, PhD 47 / 133 Groll Seminar für Finanzökonometrie Slides for Risk Management Prof. Mittnik, PhD 48 / 133

13 Student s t-distribution Model risk Student s t-distribution Model risk Groll Seminar für Finanzökonometrie Slides for Risk Management Prof. Mittnik, PhD 49 / 133 Model risk Student s t-distribution Groll Seminar für Finanzökonometrie Slides for Risk Management Prof. Mittnik, PhD 50 / 133 Model risk Comparing values VaR historical values normal assumption Student s t assumption ES historical values normal assumption Student s t assumption note: clusters in VaR-exceedances remain Groll Seminar für Finanzökonometrie Slides for Risk Management Prof. Mittnik, PhD 51 / 133 Groll Seminar für Finanzökonometrie Slides for Risk Management Prof. Mittnik, PhD 52 / 133

14 Comparing number of hits Model risk Model risk Model risk sample size: 2779 VaR 0.99 VaR VaR historical values frequency normal assumption frequency Student s t assumption frequency besides sophisticated modelling approaches, even Deutsche Bank seems to fail at VaR-estimation: VaR 0.99 note: exceedance frequencies for historical simulation equal predefined confidence level per definition overfitting Groll Seminar für Finanzökonometrie Slides for Risk Management Prof. Mittnik, PhD 53 / 133 Model risk Model risk Groll Seminar für Finanzökonometrie Slides for Risk Management Prof. Mittnik, PhD 54 / 133 Model risk Model risk Groll Seminar für Finanzökonometrie Slides for Risk Management Prof. Mittnik, PhD 55 / 133 Groll Seminar für Finanzökonometrie Slides for Risk Management Prof. Mittnik, PhD 56 / 133

15 Multi-period / multi-asset case Multi-period case Multi-period / multi-asset case given only information about VaRα A of random variable A and VaRα B of random variable B, there is in general no sufficient information to calculate VaR for a function of both: VaR f A,B α g VaR A α, VaR B α in such cases, in order to calculate VaRα f A,B, we have to derive the distribution of f A, B first despite the marginal distributions of the constituting parts, the transformed distribution under f is affected by the way that the margins are related with each other: the dependence structure between individual assets is crucial to the determination of VaR f A,B α as multi-period returns can be calculated as simple sum of sub-period returns in the logarithmic case, we aim to model VaR f A,B α = VaR A+B α even though our object of interest relates to a simple sum of random variables, easy analytical solutions apply only in the very restricted cases where summation preserves the distribution: A, B and A + B have to be of the same distribution this property is fulfilled for the case of jointly normally distributed random variables Groll Seminar für Finanzökonometrie Slides for Risk Management Prof. Mittnik, PhD 57 / 133 Multi-period / multi-asset case Multi-asset case Groll Seminar für Finanzökonometrie Slides for Risk Management Prof. Mittnik, PhD 58 / 133 Multi-period VaR and ES Excursion: Joint distributions Origins of dependency while portfolio returns can be calculated as weighted sum of individual assets for discrete returns, such an easy relation does not exist for the case of logarithmic returns discrete case: logarithmic case: r P = w 1 r 1 + w 2 r 2 P = ln 1 + r P = ln 1 + w 1 r 1 + w 2 r 2 = ln 1 + w 1 [exp ln 1 + r 1 1] + w 2 [exp ln 1 + r 2 1] = ln w 1 exp + w 2 exp 1 Groll Seminar für Finanzökonometrie Slides for Risk Management Prof. Mittnik, PhD 59 / direct influence: smoking health recession probabilities of default both directions: wealth education common underlying influence: gender income, shoe size economic fundamentals BMW, Daimler Groll Seminar für Finanzökonometrie Slides for Risk Management Prof. Mittnik, PhD 60 / 133

16 Independence Multi-period VaR and ES Excursion: Joint distributions Multi-period VaR and ES Example: independent dice Excursion: Joint distributions two random variables X and Y are called stochastically independent if P X = x, Y = y = P X = x P Y = y for all x, y the occurrence of one event makes it neither more nor less probable that the other occurs: P X = x Y = y = P X = x, Y = y P Y = y P X = x P Y = y P Y = y indep. = = P X = x because of independency, joint probability is given by product: P X = 5, Y = 4 indep. = P X = 5 P Y = 4 = = 1 36 joint distribution given by P X = i, Y = j indep. = P X = i P Y = j = = 1 36, for all i, j {1,..., 6} knowledge of the realization of Y does not provide additional information about the distribution of X Groll Seminar für Finanzökonometrie Slides for Risk Management Prof. Mittnik, PhD 61 / 133 Multi-period VaR and ES Excursion: Joint distributions Example: independent dice because of independency, realization of die 1 does not provide additional information about occurrence of die 2: P X = x Y = 5 indep. = P X = x conditional distribution: relative distribution of probabilities red row has to be scaled up unconditional distribution of die 2 is equal to the conditional distribution given die 1 Groll Seminar für Finanzökonometrie Slides for Risk Management Prof. Mittnik, PhD 62 / 133 Multi-period VaR and ES Excursion: Joint distributions Example: unconditional distribution given the joint distribution, the unconditional marginal probabilities are given by 6 P X = x = P X = x, Y = i marginal distributions hence are obtained by summation along the appropriate direction: Groll Seminar für Finanzökonometrie Slides for Risk Management Prof. Mittnik, PhD 63 / 133 Groll Seminar für Finanzökonometrie Slides for Risk Management Prof. Mittnik, PhD 64 / 133

17 Multi-period VaR and ES Example: independent firms Excursion: Joint distributions Multi-period VaR and ES Example: dependent firms Excursion: Joint distributions firms A and B, both with possible performances good, moderate and bad each performance occurs with equal probability 1 3 joint distribution for the case of independency: firm A costumer of firm B: demand for good of firm B depending on financial condition of A good financial condition A high demand high income for B increased likelihood of good financial condition of firm B: f A = 1 3 δ {a=good} δ {a=mod.} δ {a=bad} f B = 1 2 δ {b=a} δ {b=good}1 {a good} δ {b=mod.}1 {a mod.} δ {b=bad}1 {a bad} Groll Seminar für Finanzökonometrie Slides for Risk Management Prof. Mittnik, PhD 65 / 133 Multi-period VaR and ES Excursion: Joint distributions Example: common influence Groll Seminar für Finanzökonometrie Slides for Risk Management Prof. Mittnik, PhD 66 / 133 Multi-period VaR and ES Excursion: Joint distributions Example: common influence firm A and firm B supplier of firm C: demand for goods of firm A and B depending on financial condition of C : f C = 1 3 δ {c=good} δ {c=mod.} δ {c=bad} f A = 3 4 δ {a=c} δ {a=good}1 {c good} δ {a=mod.}1 {c mod.} δ {a=bad}1 {c bad} f B = 3 4 δ {b=c} δ {b=good}1 {c good} δ {b=mod.}1 {c mod.} δ {b=bad}1 {c bad} get common distribution of firm A and B: P A = g, B = g = P A = g, B = g C = g + P A = g, B = g C = m + P A = g, B = g C = b = = P A = g, B = m = P A = g, B = m C = g + P A = g, B = m C = m + P A = g, B = m C = b = = Groll Seminar für Finanzökonometrie Slides for Risk Management Prof. Mittnik, PhD 67 / 133 Groll Seminar für Finanzökonometrie Slides for Risk Management Prof. Mittnik, PhD 68 / 133

18 Multi-period VaR and ES Example: asymmetric dependency Excursion: Joint distributions Empirical Multi-period VaR and ES Excursion: Joint distributions two competitive firms with common economic fundamentals: in times of bad economic conditions: both firms tend to perform bad in times of good economic conditions: due to competition, a prospering competitor most likely comes at the expense of other firms in the sector dependence during bad economic conditions stronger than in times of booming market Groll Seminar für Finanzökonometrie Slides for Risk Management Prof. Mittnik, PhD 69 / 133 Multi-period VaR and ES Excursion: Joint distributions Bivariate normal distribution Groll Seminar für Finanzökonometrie Slides for Risk Management Prof. Mittnik, PhD 70 / 133 Multi-period VaR and ES Excursion: Joint distributions Bivariate normal distribution f x, y = ρ = 0.2 : [ ] 1 exp 1 x µx 2 + y µ 2πσ X σ Y 1 ρ 2 21 ρ 2 σ X 2 Y 2 2ρx µ σ Y 2 X y µ Y σ X σ Y ρ = 0.8 : Groll Seminar für Finanzökonometrie Slides for Risk Management Prof. Mittnik, PhD 71 / 133 Groll Seminar für Finanzökonometrie Slides for Risk Management Prof. Mittnik, PhD 72 / 133

19 Multi-period VaR and ES Conditional distributions Excursion: Joint distributions Multi-period VaR and ES Conditional distributions Excursion: Joint distributions distribution of Y conditional on X = 0 compared with distribution conditional on X = 2: ρ = 0.8: the information conveyed by the known realization of X increases with increasing dependency between the variables Groll Seminar für Finanzökonometrie Slides for Risk Management Prof. Mittnik, PhD 73 / 133 Multi-period VaR and ES Excursion: Joint distributions Interpretation Groll Seminar für Finanzökonometrie Slides for Risk Management Prof. Mittnik, PhD 74 / 133 Multi-period VaR and ES Excursion: Joint distributions Marginal distributions given jointly normally distributed variables X and Y, you can think about X as being a linear transformation of Y, up to some normally distributed noise term ɛ : marginal distributions are obtained by integrating out with respect to the other dimension X = c 1 Y c 2 + c 3 ɛ, with proof will follow further down c 1 = σ X σ Y ρ c 2 = µ Y c 3 = σ X 1 ρ 2 Groll Seminar für Finanzökonometrie Slides for Risk Management Prof. Mittnik, PhD 75 / 133 Groll Seminar für Finanzökonometrie Slides for Risk Management Prof. Mittnik, PhD 76 / 133

20 Multi-period VaR and ES Asymmetric dependency Excursion: Joint distributions Covariance Multi-period VaR and ES Excursion: Joint distributions there exist joint bivariate distributions with normally distributed margins that can not be generated from a bivariate normal distribution Covariance The covariance of two random variables X and Y is defined as E [X E [X ] Y E [Y ]] = E [XY ] E [X ] E [Y ]. captures tendency of variables X and Y to jointly take on values above the expectation given that Cov X, Y = 0, the random variables X and Y are called uncorrelated [ CovX, X = E [X E [X ] X E [X ]] = E X E [X ] 2] = V [X ] Groll Seminar für Finanzökonometrie Slides for Risk Management Prof. Mittnik, PhD 77 / 133 Multi-period VaR and ES Excursion: Joint distributions Covariance Groll Seminar für Finanzökonometrie Slides for Risk Management Prof. Mittnik, PhD 78 / 133 Multi-period VaR and ES Excursion: Joint distributions Covariance Groll Seminar für Finanzökonometrie Slides for Risk Management Prof. Mittnik, PhD 79 / 133 Groll Seminar für Finanzökonometrie Slides for Risk Management Prof. Mittnik, PhD 80 / 133

21 Multi-period VaR and ES 1 % initialize parameters 2 mu1 = 0; 3 mu2 = 0; 4 sigma1 = 1; 5 sigma2 = 1.5; 6 rho = 0.2; 7 n = 10000; Excursion: Joint distributions Covariance Multi-period VaR and ES Excursion: Joint distributions 8 9 % simulate data 10 data = mvnrnd [ mu1 mu2 ],[ sigma1 ^2 rho * sigma1 * sigma2 ; rho * sigma1 * sigma2 sigma2 ^2],n; 11 data = data :,1.* data :,2 ; 12 hist data,40 Groll Seminar für Finanzökonometrie Slides for Risk Management Prof. Mittnik, PhD 81 / 133 Multi-period VaR and ES Excursion: Joint distributions Covariance under linear transformation Groll Seminar für Finanzökonometrie Slides for Risk Management Prof. Mittnik, PhD 81 / 133 Multi-period VaR and ES Excursion: Joint distributions Linear Correlation Cov ax + b, cy + d = E [ax + b E [ax + b] cy + d E [cy + d]] = E [ax E [ax ] + b E [b] cy E [cy ] + d E [d]] = E [a X E [X ] c Y E [Y ]] = ac E [X E [X ] Y E [Y ]] = ac Cov X, Y Linear correlation The linear correlation coefficient between two random variables X and Y is defined as ρx, Y = CovX, Y, σx 2 σ2 Y where CovX, Y denotes the covariance between X and Y, and σ 2 X, σ2 Y denote the variances of X and Y. in the elliptical world, any given distribution can be completely described by its margins and its correlation coefficient. Groll Seminar für Finanzökonometrie Slides for Risk Management Prof. Mittnik, PhD 82 / 133 Groll Seminar für Finanzökonometrie Slides for Risk Management Prof. Mittnik, PhD 83 / 133

22 Multi-period VaR and ES Jointly normally distributed variables Excursion: Sums over two random variables Multi-period VaR and ES Jointly normally distributed variables Excursion: Sums over two random variables all two-dimensional points on red line result in the value 4 after summation for example: 4 + 0, 3 + 1, , or Groll Seminar für Finanzökonometrie Slides for Risk Management Prof. Mittnik, PhD 84 / 133 Multi-period VaR and ES Excursion: Sums over two random variables Jointly normally distributed variables Groll Seminar für Finanzökonometrie Slides for Risk Management Prof. Mittnik, PhD 85 / 133 Multi-period VaR and ES Excursion: Sums over two random variables Distribution of sum of variables approximate distribution of X + Y : counting the number of simulated values between the lines gives estimator for relative frequency of a summation value between 4 and 4.2 dividing two-dimensional space into series of line segments leads to approximation of distribution of new random variable X + Y Groll Seminar für Finanzökonometrie Slides for Risk Management Prof. Mittnik, PhD 86 / 133 Groll Seminar für Finanzökonometrie Slides for Risk Management Prof. Mittnik, PhD 87 / 133

23 Effects of correlation Multi-period VaR and ES Excursion: Sums over two random variables Effects of correlation Multi-period VaR and ES Excursion: Sums over two random variables increasing correlation leads to higher probability of joint large positive or large negative realizations joint large realizations of same sign lead to high absolute values after summation: increasing probability in the tails small variances in case of negative correlations display benefits of diversification Groll Seminar für Finanzökonometrie Slides for Risk Management Prof. Mittnik, PhD 88 / 133 Multi-period VaR and ES Excursion: Sums over two random variables Moments of sums of variables Theorem Given random variables X and Y of arbitrary distribution with existing first and second moments, the first and second moment of the summed up random variable Z = X + Y are given by and E [X + Y ] = E [X ] + E [Y ], V X + Y = V X + V Y + 2Cov X, Y. In general, for more than 2 variables, it holds: [ n ] n E X i = E [X i ], n V X i = n V X i + n Cov X i, X j. Groll Seminar für Finanzökonometrie Slides for Risk Management Prof. Mittnik, PhD 90 / 133 i j Groll Seminar für Finanzökonometrie Slides for Risk Management Prof. Mittnik, PhD 89 / 133 Multi-period VaR and ES Excursion: Sums over two random variables Proof: linearity in joint normal distribution [ V X + Y = E X + Y E [X + Y 2]] [ = E X + Y 2] E [X + Y ] 2 = E [ X 2 + 2XY + Y 2] E [X ] + E [Y ] 2 = E [ X 2] E [X ] 2 + E [ Y 2] E [Y ] 2 + 2E [XY ] 2E [X ] E [Y ] = V X + V Y + 2 E [XY ] E [X ] E [Y ] = V X + V Y + 2CovX, Y Groll Seminar für Finanzökonometrie Slides for Risk Management Prof. Mittnik, PhD 91 / 133

24 Multi-period VaR and ES Moments of sums of variables Excursion: Sums over two random variables Multi-period VaR and ES Linearity in joint normal distribution Proof: linearity underlying joint normal distribution calculation of V X + Y requires knowledge of the covariance of X and Y however, more detailed information about the dependence structure of X and Y is not required for linear functions in general: also: E [ax + by + c] = ae [X ] + be [Y ] + c V ax + by + c = a 2 V X + b 2 V Y + 2ab Cov X, Y CovX +Y, K +L = CovX, K+CovX, L+CovY, K+CovY, L define random variable Z as Z := σ X σ Y ρ Y µ Y + µ X + σ X 1 ρ 2 ɛ, then the expectation is given by ɛ N 0, 1 E [Z] = σ X ρe [Y ] σ X ρµ Y + µ X + σ X 1 ρ σ Y σ 2 E [ɛ] Y = µ Y σx σ Y ρ σ X σ Y ρ = µ X + µ X Groll Seminar für Finanzökonometrie Slides for Risk Management Prof. Mittnik, PhD 92 / 133 Multi-period VaR and ES Linearity in joint normal distribution Groll Seminar für Finanzökonometrie Slides for Risk Management Prof. Mittnik, PhD 93 / 133 Multi-period VaR and ES Linearity in joint normal distribution the variance is given by V Z Y = ɛ σx V ρy + V σ X 1 ρ σ 2 ɛ Y = σ2 X σy 2 σy 2 ρ2 + σx 2 1 ρ 2 1 = σx 2 ρ ρ 2 = σ 2 X being the sum of two independent normally distributed random variables, Z is normally distributed itself, so that we get Z N µ X, σ 2 X Z X for the conditional distribution of Z given Y = y we get Z Y =y N µ X + σ X 2 ρ y µ Y, σ X 1 ρ 2 σ Y it remains to show to get Z Y =y X Y =y Z, Y X, Y Groll Seminar für Finanzökonometrie Slides for Risk Management Prof. Mittnik, PhD 94 / 133 Groll Seminar für Finanzökonometrie Slides for Risk Management Prof. Mittnik, PhD 95 / 133

25 Multi-period VaR and ES Conditional normal distribution I Linearity in joint normal distribution Multi-period VaR and ES Conditional normal distribution II Linearity in joint normal distribution for jointly normally distributed random variables X and Y, conditional marginal distributions remain normally distributed: f x, y f x y = f y = 1 2πσ X σ Y 1 ρ 2 exp 1 2πσY 1 21 ρ 2! = 1 exp 2πσ [ x µx 2 σ 2 X x µ2 2σ 2 + y µ Y 2 σ 2 Y exp y µ Y 2 2σY 2 2ρx µ X y µ Y σ X σ Y ] 1 2πσ! = 1 2πσ X σ Y 1 ρ 2 1 2πσY 2π = 2πσ X 1 ρ 2 1 = 2πσX 1 ρ 2 σ = σ X 1 ρ 2 Groll Seminar für Finanzökonometrie Slides for Risk Management Prof. Mittnik, PhD 96 / 133 Multi-period VaR and ES Linearity in joint normal distribution Conditional normal distribution III Groll Seminar für Finanzökonometrie Slides for Risk Management Prof. Mittnik, PhD 97 / 133 Multi-period VaR and ES Linearity in joint normal distribution Conditional normal distribution IV exp x µ2 2σ 2 x µ2 σ 2 [ exp 1 x µx 2! 21 ρ = 2 σ X 2 exp = + y µ Y 2 2ρx µ σ Y 2 X y µ Y y µ Y 2 2σ 2 Y σ X σ Y ] [ 1 x µx 2 + y µ Y 2 2ρ x µ X y µ Y 1 ρ 2 σx 2 σy 2 σ X σ Y + y µ Y 2 σ 2 Y x µ2 1 = σ 2 1 ρ 2 σx ρ 2 σx 2 x µ X 2 + σ2 X σ 2 Y y µ Y 2 2ρ x µ X y µ Y σ X σ Y σ2 X σ 2 Y ] y µ Y 2 1 ρ 2 x µ 2 = x µ X ρ 2 σ 2 X x µ 2 = x µ X σ X ρ y µ Y σ Y µ =µ X + σ X σ Y ρ y µ Y ] [, σ 2 X [ µx for X, Y N 2 µ Y ρ σ X σ Y Y, X is distributed according to X N σy 2 2 y µ Y 2 2ρ x µ X y µ Y σ X σ Y ρ σ X σ Y σ 2 Y µ X + σ 2 X ρ y µ Y, σ X 1 ρ 2 σ Y ], given the realization of Groll Seminar für Finanzökonometrie Slides for Risk Management Prof. Mittnik, PhD 98 / 133 Groll Seminar für Finanzökonometrie Slides for Risk Management Prof. Mittnik, PhD 99 / 133

26 Multi-period VaR and ES Jointly normally distributed variables Linearity in joint normal distribution Remarks Multi-period VaR and ES Linearity in joint normal distribution Theorem Given jointly normally distributed univariate random variables X N µ X, σ 2 X and Y N µy, σ 2 Y, the deduced random vector Z := X + Y is also normally distributed, with parameters and That is, µ Z = µ X + µ Y σ Z = V X + V Y + 2Cov X, Y. X + Y N µ X + µ Y, V X + V Y + 2Cov X, Y. note: the bivariate random vector X, Y has to be distributed according to a bivariate normal distribution, i.e. X, Y N 2 µ, Σ given that X N µ X, σ 2 X and Y N µy, σ 2 Y, with dependence structure different to the one implicitly given by a bivariate normal distribution, the requirements of the theorem are not fulfilled in general, with deviating dependence structure we can only infer knowledge about first and second moments of the distribution of Z = X + Y, but we are not able to deduce the shape of the distribution Groll Seminar für Finanzökonometrie Slides for Risk Management Prof. Mittnik, PhD 100 / 133 Multi-period VaR and ES Linearity in joint normal distribution Convolution with asymmetric dependence Groll Seminar für Finanzökonometrie Slides for Risk Management Prof. Mittnik, PhD 101 / 133 Aggregation: simplifying assumptions Independence over time univariate normally distributed random vectors X N µ X, σ 2 X and Y N µ Y, σ 2 Y, linked by asymmetric dependence structure with stronger dependence for negative results than for positive results approximation for distribution of X + Y : Assumption The return of any given period shall be independent of the returns of previous periods: P rt log [a, b], t+k [c, d] = P rt log [a, b] P rt log [c, d], for all k Z, a, b, c, d R. note: X + Y does not follow a normal distribution! X + Y N µ, σ 2 Groll Seminar für Finanzökonometrie Slides for Risk Management Prof. Mittnik, PhD 102 / 133 Groll Seminar für Finanzökonometrie Slides for Risk Management Prof. Mittnik, PhD 103 / 133

27 Aggregation: simplifying assumptions Consequences Aggregation: simplifying assumptions Multi-period moments consequences of assumption of independence over time combined with case 1: arbitrary return distribution moments of multi-period returns can be derived from moments of one-period returns: square-root-of-time scaling for standard deviation multi-period return distribution is unknown: for some important risk measures like VaR or ES no analytical solution exists case 2: normally distributed returns moments of multi-period returns can be derived from moments of one-period returns: square-root-of-time scaling for standard deviation multi-period returns follow normal distribution: VaR and ES can be derived according to square-root-of-time scaling expectation: independence unnecessary [ [ ] n 1 ] n 1 E t,t+n 1 = E = variance: V t,t+n 1 = V n 1 i=0 n 1 = V i=0 standard deviation: σ t,t+n 1 = i=0 t+i t+i t+i n 1 = i=0 i=0 V + 0 = nσ 2 V [ E t+i t+i ] n 1 = µ = nµ i=0 n 1 + Cov i j t,t+n 1 = nσ 2 = nσ t+i, t+j Groll Seminar für Finanzökonometrie Slides for Risk Management Prof. Mittnik, PhD 104 / 133 Aggregation: simplifying assumptions Normally distributed returns Distribution of multi-period returns Groll Seminar für Finanzökonometrie Slides for Risk Management Prof. Mittnik, PhD 105 / 133 Aggregation: simplifying assumptions Normally distributed returns Multi-period VaR assumption: rt log N µ, σ 2 consequences: random vector t, t+k follows a bivariate normal distribution with zero correlation because of assumed independence [ ] [ ] rt log, µ σ 2 0 t+k N 2, µ 0 σ 2 as a sum of components of a multi-dimensional normally distributed random vector, multi-period returns are normally distributed themselves using formulas for multi-period moments we get t,t+n 1 N nµ, nσ 2 notation: [ ] µ n := E t,t+n 1 = nµ σ n := σ t,t+n 1 = nσ VaR α n := VaR α t,t+n 1 rewriting VaR α for multi-period returns as function of one-period VaR α : VaR n α = µ n + σ n Φ 1 α = nµ + nσφ 1 α = nµ + nµ nµ + nσφ 1 α = n n µ + n µ + σφ 1 α = n n µ + nvar α rt log Groll Seminar für Finanzökonometrie Slides for Risk Management Prof. Mittnik, PhD 106 / 133 Groll Seminar für Finanzökonometrie Slides for Risk Management Prof. Mittnik, PhD 107 / 133

28 Aggregation: simplifying assumptions Multi-period VaR Normally distributed returns Multi-period ES Aggregation: simplifying assumptions Normally distributed returns furthermore, for the case of µ = 0 we get VaR α n = nσφ 1 α = nvar α this is known as the square-root-of-time scaling requirements: rt log returns are independent through time: no autocorrelation returns are normally distributed with zero mean: t N 0, σ 2 ES n α = µ n + σ n φ Φ 1 α 1 α = nµ + nσ φ Φ 1 α = n n µ + n = n n µ + nes α 1 α µ + σ φ Φ 1 α 1 α again, for µ = 0 the square-root-of-time scaling applies: ES n α = nes α Groll Seminar für Finanzökonometrie Slides for Risk Management Prof. Mittnik, PhD 108 / 133 Aggregation: simplifying assumptions Normally distributed returns Example: market risk Groll Seminar für Finanzökonometrie Slides for Risk Management Prof. Mittnik, PhD 109 / 133 Aggregation: simplifying assumptions Normally distributed returns extending DAX example, with parameters of normal distribution fitted to real world data given by ˆµ = and ˆσ = calculate multi-period VaR and ES for 5 and 10 periods using multi-period formulas for VaR and ES: VaR α n = n n µ + nvar α rt log = VaR α ES n α = n n µ + nes α = ES α using previously calculated values, for 5-day returns we get : VaR = for 10-day returns we get: = = ES = = VaR = ES = = Groll Seminar für Finanzökonometrie Slides for Risk Management Prof. Mittnik, PhD 110 / 133 Groll Seminar für Finanzökonometrie Slides for Risk Management Prof. Mittnik, PhD 111 / 133

29 Aggregation: simplifying assumptions Example: multi-period portfolio loss Normally distributed returns Model setup Aggregation: simplifying assumptions Normally distributed returns let S t,i denote the price of stock i at time t given λ i shares of stock i, the portfolio value in t is given by P t = one-day portfolio loss: λ i S t,i L t+1 = P t+1 P t target variable: n-day cumulated portfolio loss for periods {t, t + 1,..., t + n}: L t,t+n = P t+n P t capture uncertainty by modelling logarithmic returns = log S t+1 log S t as random variables t consequence: instead of directly modelling the distribution of our target variable, our model treats it as function of stochastic risk factors, and tries to model the distribution of the risk factors L t,t+n = f t,..., t+n 1 flexibility: changes in target variable portfolio changes do not require re-modelling of the stochastic part at the core of the model Groll Seminar für Finanzökonometrie Slides for Risk Management Prof. Mittnik, PhD 112 / 133 Aggregation: simplifying assumptions Normally distributed returns Function of risk factors Groll Seminar für Finanzökonometrie Slides for Risk Management Prof. Mittnik, PhD 113 / 133 Aggregation: simplifying assumptions Normally distributed returns Function of risk factors L t,t+n = P t+n P t = λ i S t+n,i = = = λ i S t,i λ i S t+n,i S t,i St+n,i λ i S t,i 1 S t,i St+n,i λ i S t,i exp log 1 S t,i = = = = g St+n,i λ i S t,i exp log 1 S t,i λ i S t,i exp t,t+n,i 1 n λ i S t,i exp k=0 t+k,i 1 t,1, t+1,1,..., t+n,1, t,2,..., t,d,... t+n,d target variable is non-linear function of risk factors: non-linearity arises from non-linear portfolio aggregation in logarithmic world Groll Seminar für Finanzökonometrie Slides for Risk Management Prof. Mittnik, PhD 114 / 133 Groll Seminar für Finanzökonometrie Slides for Risk Management Prof. Mittnik, PhD 115 / 133

30 Aggregation: simplifying assumptions Simplification for dimension of time Normally distributed returns Aggregation: simplifying assumptions Simplification for dimension of time Normally distributed returns assuming normally distributed daily returns rt log as well as independence of daily returns over time, we know that multi-period returns t,t+n = n i=0 t+i have to be normally distributed with parameters and µ n = nµ σ n = nσ the input parameters can be reduced to L t,t+n = λ i S t,i exp t,t+n,i 1 = h t,t+n,1,..., t,t+n,d non-linearity still holds because of non-linear portfolio aggregation Groll Seminar für Finanzökonometrie Slides for Risk Management Prof. Mittnik, PhD 116 / 133 Aggregation: simplifying assumptions Normally distributed returns Application: real world data Groll Seminar für Finanzökonometrie Slides for Risk Management Prof. Mittnik, PhD 117 / 133 Aggregation: simplifying assumptions Normally distributed returns Asset dependency estimating parameters of a normal distribution for historical daily returns of BMW and Daimler for the period from to we get and µ B = , σ B = µ D = σ D = assuming independence over time, the parameters of 3-day returns are given by µ B 3 = , σ B 3 = and µ D 3 = , σ D 3 = so far, the marginal distribution of individual one-period returns has been specified, as well as the distribution of multi-period returns through the assumption of independence over time however, besides the marginal distributions, in order to make derivations of the model, we also have to specify the dependence structure between different assets once the dependence structure has been specified, simulating from the complete two-dimensional distribution and plugging into function h gives Monte Carlo solution of the target variable Groll Seminar für Finanzökonometrie Slides for Risk Management Prof. Mittnik, PhD 118 / 133 Groll Seminar für Finanzökonometrie Slides for Risk Management Prof. Mittnik, PhD 119 / 133

31 Linearization Aggregation: simplifying assumptions Normally distributed returns Linearization Aggregation: simplifying assumptions Normally distributed returns in order to eliminate non-linearity, approximate function by linear function f x + t = f x + f x t denoting Z t := log S t, makes t expressable with risk factors: P t+1 = = = = f λ i S t+1,i λ i exp log S t+1,i λ i exp Z t,i + t+1,i Z t + t+1 Groll Seminar für Finanzökonometrie Slides for Risk Management Prof. Mittnik, PhD 120 / 133 Aggregation: simplifying assumptions Normally distributed returns Linearization linearization of one-period portfolio loss: L t+1 = P t+1 P t λ i S t,i + = λ i S t,i t+1,i λ i S t,i λ i S t,i t+1,i = a 1 t+1,1 + a 2 t+1, a d t+1,d linearization of 3-period portfolio loss: L t,t+2 λ i S t,i t,t+2,i function f u = d λ iexp u i has to be approximated by differentiation differentiating with respect to the single coordinate i: f f u = u i Z t + t+1 d λ iexp u i = λ i exp u i u i f Z t + f Z t t+1 = f Z t + = = λ i exp Z t,i + λ i S t,i + f Z t t+1,i Z t,i λ i S t,i t+1,i λ i exp Z t,i t+1,i Groll Seminar für Finanzökonometrie Slides for Risk Management Prof. Mittnik, PhD 121 / 133 Aggregation: simplifying assumptions Normally distributed returns now that non-linearities have been removed, make use of fact that linear function of normally distributed returns is still normally distributed assuming the dependence structure between daily returns of BMW and Daimler to be symmetric, joint returns will follow a bivariate normal distribution, and 3-day returns of BMW and Daimler also follow a joint normal distribution given the covariance of daily returns, the covariance of 3-day returns can be calculated according to Cov r B t,t+2, r D t,t+2 = Cov = r B t + r B t+1 + r B t+2, r D t + r D t+1 + r D t+2 2 Cov i,j=0 rt+i B, rt+j D = Cov rt B, rt D + Cov = 3Cov rt B, rt D rt+1, B rt+1 D + Cov rt+2, B rt+2 D Groll Seminar für Finanzökonometrie Slides for Risk Management Prof. Mittnik, PhD 122 / 133 Groll Seminar für Finanzökonometrie Slides for Risk Management Prof. Mittnik, PhD 123 / 133

32 Application Aggregation: simplifying assumptions Normally distributed returns Aggregation: simplifying assumptions Recapturing involved assumptions Normally distributed returns with estimated correlation ˆρ = , the covariance becomes Ĉov rt B, rt D = ˆρ σ B σ D = = and Ĉov rt,t+2, B rt,t+2 D = simulating from two-dimensional normal distribution, and plugging into function h will give a simple and fast approximation of the distribution of the target variable as the target variable is a linear function of jointly normally distributed risk factors, it has to be normally distributed itself: hence, an analytical solution is possible individual daily logarithmic returns follow normal distribution returns are independent over time non-linear function for target variable has been approximated by linearization dependence structure according to joint normal distribution Groll Seminar für Finanzökonometrie Slides for Risk Management Prof. Mittnik, PhD 124 / 133 Properties of risk measures Example: coherence Groll Seminar für Finanzökonometrie Slides for Risk Management Prof. Mittnik, PhD 125 / 133 Properties of risk measures Example Consider a portfolio consisting of d = 100 corporate bonds. The probability of default shall be 0.5% for each firm, with occurrence of default independently of each other. Given no default occurs, the value of the associated bond increases from x t = 100 this year to x t+1 = 102 next year, while the value decreases to 0 in the event of default. Calculate VaR 0.99 for a portfolio A consisting of 100 shares of one single given corporate, as well as for a portfolio B, which consists of one share of each of the 100 different corporate bonds. Interpret the results. What does that mean for VaR as a risk measure, and what can be said about Expected Shortfall with regard to this feature? setting: d = 100 different corporate bonds, each with values given by t t + 1 value probability defaults are independent of each other Groll Seminar für Finanzökonometrie Slides for Risk Management Prof. Mittnik, PhD 126 / 133 Groll Seminar für Finanzökonometrie Slides for Risk Management Prof. Mittnik, PhD 127 / 133

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