Long-Term Financial Risks: the One-Dimensional Case Roger Kaufmann, RiskLab, ETH Z¨ urich RiskLab workshop June 7, 2001 joint work with Pierre Patie http:/ /www.risklab.ch/Papers.html#SLTFR
Long-Term Financial Risks: the One-Dimensional Case I Introduction II Models III Backtesting IV Conclusions c � 2001 (R. Kaufmann, RiskLab) 1
Aim of the Project One-dimensional: • Model one-year returns for stock indices, 10-year bonds and foreign exchange rates. Multi-dimensional: • Measurement of one-year financial risk of investment portfolios. c � 2001 (R. Kaufmann, RiskLab) 2
Risk Measures Definition 1. The value-at-risk VaR p at level p of the return R is VaR p ( R ) = − inf { x ∈ R | P [ R ≤ x ] ≥ p } , i.e. VaR is the negative of the p -quantile of R . Definition 2. The expected shortfall ES p at a level p is defined by ES p ( R ) = − E [ R | R < − VaR p ( R )] . We consider as risk measure the expected shortfall for the level p = 1%. The expected shortfall is a coherent risk measure in the sense of Artzner, Delbaen, Eber and Heath. In general, value-at-risk is not! c � 2001 (R. Kaufmann, RiskLab) 3
Value-at-Risk and Expected Shortfall Density of P&L distribution ES(5%) VaR(5%) -60 -40 -20 0 20 40 60 yearly returns in % c � 2001 (R. Kaufmann, RiskLab) 4
Problems 1. Which frequency do we use to fit models? • Are long datasets stationary? • What are the statistical restrictions? (lack of yearly returns) • How can we keep as much information as possible? 2. Do the properties of financial data change when we choose another time horizon? 3. What is the reliability of the time aggregation rule of each model if there is any such rule? 4. How can we compare different time horizons and models? c � 2001 (R. Kaufmann, RiskLab) 5
Model Comparison We fix a horizon h < 1 year, for which we can use our models. For the gap between h and 1 year, we use a scaling rule. scaling rule suitable model today h 1 year = 261 trading days c � 2001 (R. Kaufmann, RiskLab) 6
II Models • Random walk • GARCH(1,1) • Vector Error Correction Model ❀ Auto-regressive • Heavy-tailed distribution c � 2001 (R. Kaufmann, RiskLab) 7
Random Walk Model We assume that h -day log-returns are independent and normally dis- tributed: r h ( t ) iid ∼ N ( µ h , σ 2 h ) for t ∈ h N . We estimate the one-year expected shortfall at a level p by � � � σ 2 ( t ) � Φ( x p − � σ ( t )) � ES p � t = − exp µ ( t ) + − 1 � 2 p for t ≥ nh , where: x p : p -quantile of the standard normal distribution, Φ : cumulative standard normal distribution function, � n − 1 µ ( t ) = 261 µ h ( t ) = 1 µ h ( t ) , i =0 r h ( t − ih ) , � h � � n � 261 � n − 1 1 σ 2 µ h ( t − ih )) 2 . i =0 ( r h ( t − ih ) − � σ ( t ) = σ h ( t ) , � h ( t ) = � h � n − 1 c � 2001 (R. Kaufmann, RiskLab) 8
Generalized Autoregressive Conditional Heteroskedastic Model A GARCH(1,1) model with a Student- t distributed innovation process for centered h -day log-returns X h ( t ) = r h ( t ) − µ h is defined by for t ∈ h N , X h ( t ) = σ h ( t ) ǫ h ( t ) σ 2 h ( t ) = α 0 ,h + α 1 ,h X 2 h ( t − h ) + β 1 ,h σ 2 h ( t − h ) , ǫ h ( t ) iid E [ ǫ 2 where ∼ t ν h , E [ ǫ h ( t )] = 0 , h ( t )] = 1 . We estimate the one-year expected shortfall at a level p in 4 steps: 1. Fit the GARCH(1,1) process to the h -day log-returns using quasi maximum likelihood estimators (QMLE). α 1 , � 2. Apply the Drost–Nijman scaling rule to get the parameters � α 0 , � β 1 , � ν of the weak GARCH process for centered yearly log-returns X ( t ). c � 2001 (R. Kaufmann, RiskLab) 9
3. Forecast the yearly volatility � σ ( t ) using the following recursive rela- tion: σ 2 ( s +261 , t ) = ˆ µ ( t )) 2 +ˆ σ 2 ( s, t ) , α 1 ( r ( s ) − � s = t 0 , . . . , t − 261 , t ˆ α 0 +ˆ β 1 ˆ starting with n − 1 � σ 2 ( t 0 , t ) = 261 1 µ h ( t )) 2 , ˆ ( r h ( t − ih ) − � h n − 1 i =0 µ h ( t ) : QMLE for mean h -day log-return at time t, � µ ( t ) = 261 µ h ( t ) , � � h σ ( t ) = � σ ( t + 261 , t ) . � � � � p � � 1 4. ES p � t = − 0 exp µ ( t ) + � σ ( t ) x � dq − 1 , � ν,q p where x � ν,q is the q -quantile of a t � ν -distributed random variable with mean zero and variance one. c � 2001 (R. Kaufmann, RiskLab) 10
Auto-regressive Model An AR(q) model with a normally distributed innovation process for the s ( t ) = s ( t ) − µt ( t ∈ h N ) is defined by drift-free h -day log-prices � q h � s ( t − ih ) + ǫ h ( t ) for t ∈ h N , t ≥ q n h, s ( t ) = � a h,i � i =1 ǫ h ( t ) iid ∼ N (0 , σ 2 where h ) . We estimate the one-year expected shortfall at a level p as follows: 1. Subtract the linear trend from h -day log-prices s h ( t ): µ = s ( t ) − s ( t − ( n − 1) h ) s ( t ) = s ( t ) − � � µt, � . ( n − 1) h 2. Fit the AR-process to the drift-free h -day log-prices: maximum likelihood estimation (MLE) ❀ � q h , � a h,i ( i = 1 , . . . , � q h ) , � σ h . c � 2001 (R. Kaufmann, RiskLab) 11
3. Forecast the one-year log-return using the following relation: µ ( t ) = 261 � µ + � m ( t ) , � � � 261 + 1 m ( t ) = � s ( t + kh ) − � � � s ( t ) , k = , 2 h where � s ( t + jh ) ( j = 1 , . . . , k ) are defined recursively: � � q h � � a h,i � � s ( t + jh ) = s ( t + ( j − i ) h ) , s ( u ) = � s ( u ) for u ≤ t. � � � � i =1 4. Forecast the yearly volatility using the following formulas: δ 0 = 1 , j � α h,i = 0 ∀ i > � δ j = α h,i δ j − i , � � q h , i =1 � � � k − 1 � � δ 2 σ ( t ) = � � σ h ( t ) j . � j =0 c � 2001 (R. Kaufmann, RiskLab) 12
5. � � � σ 2 ( t ) � Φ( x p − � � σ ( t )) ES p � t = − exp µ ( t ) + − 1 � , 2 p where: x p : p -quantile of the standard normal distribution, Φ : cumulative standard normal distribution function. c � 2001 (R. Kaufmann, RiskLab) 13
Heavy-Tailed Distribution We assume the h -day log-returns to be independent and identically distributed, further P [ r h < − x ] = x − α L ( x ) as x → ∞ , where α ∈ R + and L is a slowly varying function. By inverting this formula and using the scaling rule for heavy-tailed distributions (Feller’s theorem) we can derive estimates for the one- year expected shortfall at a level p : � � �� 261 k ( n, p ) � � p � 1 / � 1 α k ( n,p ) r k ( n,p ) ,n ES p � t = − 0 exp dq − 1 , p h n q where k ( n, p ) = ⌊ n ( p + 4 . 5% + h 2 1%) ⌋ , h = 1 , 5 , 22 , (heuristic choice) � � r i,n � � k 1 α k,n = 1 i =1 log (Hill estimator), � k r k,n r k,n is the k th order statistics, i.e. r 1 ,n ≤ r 2 ,n ≤ · · · ≤ r n,n . c � 2001 (R. Kaufmann, RiskLab) 14
III Backtesting • Backtesting description • Results: - foreign exchange rates - stock indices - 10-year bonds c � 2001 (R. Kaufmann, RiskLab) 15
Backtesting Measures Measure 1: Use values below the negative of the estimated value-at- VaR p � � � risk t : � t 1 p R t +1 − ( − � ES t ) 1 p { Rt +1 < − � t = t 0 VaR t } V ES = . � t 1 1 t = t 0 1 p { Rt +1 < − � VaR t } Measure 2: Use values below the “1 in 1/ p event” (for p = 1%: one in hundred event): � t 1 t = t 0 D t 1 { Dt<Dp } ES p V ES D t = R t +1 − ( − � = , t ) , � t 1 2 t = t 0 1 { Dt<Dp } where D p is the p -quantile of { D t } t 0 ≤ t ≤ t 1 . V ES = ( | V ES | + | V ES Combined measure: | ) / 2 . 1 2 Frequency of exceedance: � � � t 1 V freq = 1 t = t 0 1 { R t +1 < − � . p t 1 − t 0 +1 VaR t } c � 2001 (R. Kaufmann, RiskLab) 16
Backtesting Description Problem: Not enough yearly data for estimating model parameters and proceeding the backtesting! Solution: We use 4 stock samples (foreign exchange rates), each con- taining 16 years of data, we carry out the backtesting on each sample independently, then we aggregate the results. For each model, each intermediate horizon h and each sample we pro- ceed as follows: ES p 1. We estimate the yearly expected shortfall � t on a window of size N (e.g. N =2000 daily data). We use the n = ⌊ N/h ⌋ non-overlapping h -day log-returns for this estimation. 2. We compare the estimates with the following returns R t +1 using different measures. 3. We move the window by one, then we repeat steps 1 and 2 up to the end of the whole dataset. c � 2001 (R. Kaufmann, RiskLab) 17
Backtesting Results for the 1% One-Year Expected Shortfall averaged over 4 foreign exchange rates (DEM/CHF, GBP/CHF, USD/CHF, JPY/CHF) Freq ES VaR Model V ES V ES V ES V freq days 1 2 Optimal 0% 0% 0% 1% GARCH(1,1) 1 N/A N/A 3 . 9 0 . 0 5 2 . 2 1 . 9 2 . 6 0 . 3 22 0 . 8 0 . 7 − 0 . 9 0 . 7 65 2 . 2 2 . 1 − 2 . 4 1 . 2 261 11 . 4 − 6 . 0 − 16 . 8 6 . 3 c � 2001 (R. Kaufmann, RiskLab) 18
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