Fin285a:Computer Simulations and Risk Assessment Section 6.2 Extreme Value Theory Daníelson, 9 (skim), skip 9.5
Overview Extreme value distributions Generalized Pareto distributions Tail shapes Using power laws with returns Time aggregation Chebyshev s inequality Summary Fall 2016: LeBaron Fin285a: 6.2 2 / 42
Extreme value distributions Generalized Pareto distributions Tail shapes Using power laws with returns Time aggregation Chebyshev s inequality Extreme value distributions Summary Fall 2016: LeBaron Fin285a: 6.2 3 / 42
Extreme value distributions R t daily returns x m,t = max(r t ) over month x m,t follows one of the following Gumbel (exponential decay - like Normal) Frechet (power law decay - fat tail) Wiebull (finite endpoint) Sort of like a central limit theorem Block maxima Key application: floods Fall 2016: LeBaron Fin285a: 6.2 4 / 42
Extreme value distributions Generalized Pareto distributions Example: fitgpareto.m Definition: Empirical CDF Example: fitgpareto.m Example: fitgpareto.m Tail shapes Generalized Pareto distributions Using power laws with returns Time aggregation Chebyshev s inequality Summary Fall 2016: LeBaron Fin285a: 6.2 5 / 42
Peaks over thresholds (POT) Distributions for all values exceeding some threshold Converge to the Generalized Pareto Distribution (GPD) Often more useful than distributions for maximums Fall 2016: LeBaron Fin285a: 6.2 6 / 42
Conditional tail distributions Note: This deals with right tails. F u (x) = Pr(X u x X > u) (6.2.1) As u the distribution of tail values approaches a Generalized Pareto Distribution (GPD). F u (x) = G ξ,β (x) = { 1 (1+ξ x β ) 1 ξ ξ 0 1 e (x β ) ξ = 0 (6.2.2) Fall 2016: LeBaron Fin285a: 6.2 7 / 42
Fitting a tail for stock returns Matlab has several functions to deal with Generalized Pareto Distributions. What should we do? 1.Adjust data Flip sign (around mean) Find threshold level t = 0.05 or other small level (tailprob in code) Threshold: u = quantile(ret, 1-t) Get right tail ret(ret>u)-u 2.Fit pareto parameters gpfit() Fall 2016: LeBaron Fin285a: 6.2 8 / 42
Finding extreme quantiles in tails 1.Choose small p,pr(x > x) = p 2.Convert this to probability in tail Pr(X > x X > u) = Pr((X > x)&(x > u)) Pr(X > u) (6.2.3) Pr((X > x)&(x > u)) = Pr(X > x X > u)pr(x > u) (6.2.4) 3.Since we are only interested in x > u, Pr(X > x) = Pr(X > x X > u)pr(x > u) (6.2.5) and Pr(X > x X > u) = Pr(X > x) Pr(X > u) (6.2.6) Fall 2016: LeBaron Fin285a: 6.2 9 / 42
Conditional CDF Pr(X > x X > u) = Pr(X x X > u) = 1 Pr(X > x) Pr(X > u) Pr(X > x) Pr(X > u) (6.2.7) Fall 2016: LeBaron Fin285a: 6.2 10 / 42
Picture for a basic tail region t = 0.05,u = 1.64 3 2 1 0 1 2 3 Define tail region as starting in the upper 0.05 of the distribution Assume normal distribution (just for example) For a normal, this would start at 1.64. Pr(X > 1.64) = 0.05 Fall 2016: LeBaron Fin285a: 6.2 11 / 42
Picture for a basic tail region t = 0.05,p = 0.02 3 2 1 0 1 2 3 Now we look at the point where Pr(X > x) = 0.02 For a normal this is x = 2.05 Would like to get this probability conditional on being > 1.64 = u Fall 2016: LeBaron Fin285a: 6.2 12 / 42
Picture for a basic tail region t = 0.05,p = 0.02 3 2 1 Pr(X > 2.05 X > 1.64) = 0 1 2 3 Pr(X > 2.05) Pr(X > 1.64) = 0.02 0.05 = 0.4 (6.2.8) The upper 0.02 tail of the entire distribution is the upper 0.4 tail of the conditional distribution beyond the u threshold. Fall 2016: LeBaron Fin285a: 6.2 13 / 42
Now convert to quantiles in this example Pr(X > 2.05 X > 1.64) = 0.02 0.05 = 0.4 Pr(X 2.05 X > 1.64) = 1 Pr(X > 2.05 X > 1.64) = 1 0.02 0.05 = 0.6 Now here s what we can do: 1.Adjust the desired tail probability and subtract it from 1, α = 1 p t 2.Use this to find the q α quantile on the GPD distribution using the inverse CDF (matlab knows this function) 3.Add this to the threshold, u. 4.This will be our extreme return at probability p. 5.Finally, flip the sign around the mean. On to matlab... Fall 2016: LeBaron Fin285a: 6.2 14 / 42
Data transformations The Generalized Pareto distribution starts at zero, and only covers positive numbers It can only give you a CDF of Pr(X x) for x 0 Since we are interested in negative returns we need to transform them twice 1.Flip them around the mean, R t = (R t R t ), Rt = mean(r t ) (6.2.9) 2.Then subtract u, ˆR t = R t u Fall 2016: LeBaron Fin285a: 6.2 15 / 42
Example: fitgpareto.m First, manipulate returns, and estimate Pareto params, % threshold probability level tailprob = 0.05; % this is t in the notes % Reverse sign on returns and remove the mean % This lets us look at the right tail mret = mean(ret1); adjustret = -1*(ret1-mret); % u = quantile(ret,1-t) threshold return level uthresh = quantile(adjustret,1-tailprob); % u in notes % limit to only returns beyond threshold tailret = adjustret(adjustret>uthresh); % take off threshold so low value is zero tailret = tailret-uthresh; % estimate pareto params pparams = gpfit(tailret); Fall 2016: LeBaron Fin285a: 6.2 16 / 42
Definition: Empirical CDF CDF for actual data. F e (x) = Pr(x i x) = #x i x N Matlab example: % find the fraction of xref<=y datacdf = empcdf(y,xref); % y can be a vector % often use with self datacdf = empcdf(y,y); % this gives the empirical quantile on each entry y % converts all values to uniform [0,1] - very useful Compare to theoretical model, GPD. (6.2.10) Fall 2016: LeBaron Fin285a: 6.2 17 / 42
Example: fitgpareto.m Second, plot empirical and theoretical CDF s. % cdf from data datacdf = empcdf(tailret, tailret); % cdf from pareto gpdcdf = gpcdf(tailret,pparams(1),pparams(2),0); % convert to Prob(X>x X>u) datapgreater = 1-dataCDF; gpdpgreater = 1-gpdCDF; % mutiply by Prob(X>u) to get true tail, Prob(X>x) datapgreater = datapgreater * tailprob; gpdpgreater = gpdpgreater * tailprob; % remember to add threshold for return in plotting loglog(tailret+uthresh,datapgreater, *b ); hold on loglog(tailret+uthresh,gpdpgreater, *r ); Fall 2016: LeBaron Fin285a: 6.2 18 / 42
Example: fitgpareto.m Third, get tail returns at probs. probs = [0.01 0.005 0.001 0.0001 0.00005 0.00001]; % get theoretical quantiles % remember to adjust for conditional tail prob adjustex = gpinv(1-probs/tailprob, pparams(1), pparams(2),0); % first add threshold adjustex = adjustex + uthresh; % flip sign and add mean % these are now back to regular returns adjustex = -1*adjustEx + mret; Let s go run this code now. Fall 2016: LeBaron Fin285a: 6.2 19 / 42
Extreme value distributions Generalized Pareto distributions Tail shapes Using power laws with returns Time aggregation Chebyshev s inequality Tail shapes Summary Fall 2016: LeBaron Fin285a: 6.2 20 / 42
Generalized Pareto again F u (x) = G ξ,β (x) = { 1 (1+ξ x β ) 1 ξ ξ 0 1 e (x β ) ξ = 0 Most cases for stocks are ξ 0. In this case, as x β, Pr(X u x X > u) = F u (x) 1 A 1 x α α = 1/ξ (6.2.11) Pr(X u > x X > u) = 1 F u (x) A 1 x α (6.2.12) Pr(X u > x X > u)pr(x > u) = Pr(X > x) (6.2.13) α is known as the tail index. Pr(X > x) Ax α (6.2.14) Fall 2016: LeBaron Fin285a: 6.2 21 / 42
Generalized Pareto again Pr(X > x) Ax α α = 1/ξ For symmetric (around zero) returns we would also have Pr(R < x) A x α (6.2.15) What does this say? A bunch of interesting/important results. Fall 2016: LeBaron Fin285a: 6.2 22 / 42
Tail facts Pr(R < x) A x α 1.Estimate Generalized Pareto parameters (and use α = 1/ξ) 2.Plotting/linearity: log(pr(r < x)) = log(a) αlog( x ) 3.Scaling: Pr(R < 2x) = A 2x α = A2 α x α = 2 α Pr(R < x) 4.Moments: (don t underestimate this fact) E(R m ) = m α Fall 2016: LeBaron Fin285a: 6.2 23 / 42
Another tail fact: Student-t The student-t distribution with ν degrees of freedom has a power law tail with α = ν. This is also why the variance, E(R 2 ), doesn t exist for ν 2. Kurtosis, E(R 4 ), doesn t exist for ν 4. Fall 2016: LeBaron Fin285a: 6.2 24 / 42
Another tail fact: Normal distributions What about a normal distribution? For the normal, α = All moments exist for a normal This corresponds to the special case of ξ = 0 for the Generalized Pareto If that parameter is near zero, then it is likely your data is normal As we know, most higher frequency financial data does not look normal Fall 2016: LeBaron Fin285a: 6.2 25 / 42
Yet Another tail fact: Expected shortfall This only works out in the tail (small p), but it can be useful. ES(p) VaR(p) 1 1 ξ = 1 1 (1/α) (6.2.16) Fall 2016: LeBaron Fin285a: 6.2 26 / 42
Dow return left tail (EV, α = 3.5) 10-3 Prob(X<ret) 10-4 10-5 Data EV T(3) T(4) -10-1 Left Tail Return Fall 2016: LeBaron Fin285a: 6.2 27 / 42
Dow return left tail with normal 10 0 10-20 Data EV T(3) T(4) Normal 10-40 Prob(X<ret) 10-60 10-80 10-100 10-120 -10 0-10 -1-10 -2 Left Tail Return Fall 2016: LeBaron Fin285a: 6.2 28 / 42
Extreme value distributions Generalized Pareto distributions Tail shapes Using power laws with returns Time aggregation Chebyshev s inequality Using power laws with returns Summary Fall 2016: LeBaron Fin285a: 6.2 29 / 42
Estimate just α Estimate α Hill estimator (9.3.2) See also, LeBaron, Robust properties of stock return tails Picture/least squares (next slide) Big problem: determine tail region See 9.3.3 (fig 9.3) Bias: extreme tail Variance: less extreme tail Fall 2016: LeBaron Fin285a: 6.2 30 / 42
Least squares estimator Determine tail! Get x i values in tail log(pr(r < x)) = log(a) αlog( x ) Estimate (Pr(R < x i ), x i ) from data Take logs Estimate (A,α) using ordinary least squares regression Fall 2016: LeBaron Fin285a: 6.2 31 / 42
Extreme value distributions Generalized Pareto distributions Tail shapes Using power laws with returns Time aggregation Chebyshev s inequality Time aggregation Summary Fall 2016: LeBaron Fin285a: 6.2 32 / 42
Tail properties and sums X = X 1 +X 2 (6.2.17) Pr(X i < x) = A i x α (6.2.18) Fall 2016: LeBaron Fin285a: 6.2 33 / 42
Tail properties and sums X = X 1 +X 2 (6.2.17) Pr(X i < x) = A i x α (6.2.18) Pr(X < x) = A x α (6.2.19) Fall 2016: LeBaron Fin285a: 6.2 33 / 42
Tail properties and sums X = X 1 +X 2 (6.2.17) Pr(X i < x) = A i x α (6.2.18) See Thm 9.1, case 1 Pr(X < x) = A x α (6.2.19) Think about log returns and horizons r t (2) = r t (1)+r t 1 (1) Trick: estimate power laws with higher frequency data Fall 2016: LeBaron Fin285a: 6.2 33 / 42
VaR and time scaling Thm 9.2 (page 178) VaR scales as T 1/α, T is time horizon If α = 4, then VaR scales as T 0.25 VaR 10 = 10 0.25 VaR 1 For normal returns VaR scales with σ, the standard deviation This goes as Tσ 1 VaR 10 = 10 0.5 VaR 1 Power law VaR increases more slowly Fall 2016: LeBaron Fin285a: 6.2 34 / 42
Danielson table 9.1 VaR for starting value of 100. Tail index is about 4.3. VaR p-level 1% 0.1% 0.05% Extreme value (1 day) 1.5 2.5 3.0 Extreme value (10 day) 2.5 4.3 5.1 Normal (1 day) 1.4 1.9 2.0 Normal (10 day) 4.5 5.9 6.3 Note: EVT gives larger values at 1 day, but scales more slowly to 10 days. 10 1 4.3 < 10 1 2 Square-root of two law 10 1 2 is a conservative risk measure here. This is all pretty far out in the tail. Fall 2016: LeBaron Fin285a: 6.2 35 / 42
Extreme value distributions Generalized Pareto distributions Tail shapes Using power laws with returns Time aggregation Chebyshev s inequality Chebyshev s inequality Summary Fall 2016: LeBaron Fin285a: 6.2 36 / 42
Chebyshev s inequality Probability bound that holds for any distribution Very conservative risk measure Pr( X µ nσ) 1 n 2 (6.2.20) p = Pr(X µ nσ) 1 n 2 (6.2.21) n = 1 p (6.2.22) Software: ChebyshevComp.m Fall 2016: LeBaron Fin285a: 6.2 37 / 42
Extreme value distributions Generalized Pareto distributions Tail shapes Using power laws with returns Time aggregation Chebyshev s inequality Summary Summary Fall 2016: LeBaron Fin285a: 6.2 38 / 42
Extreme value useful features Can improve tail probability estimates Useful when samples are Small or, You want to go far out into the tail Caution: Tail index (α) can be difficult to estimate Can be combined with empirical bootstrap methods Use Generalized Pareto in tail and empirical density in middle of distribution Fall 2016: LeBaron Fin285a: 6.2 39 / 42
Several methods Generalized Pareto Student-t Power-law Chebyshev (very conservative) Fall 2016: LeBaron Fin285a: 6.2 40 / 42
Complicated?? This all seemed rather complicated There is a lot of fancy math behind this Why did we do this? Fall 2016: LeBaron Fin285a: 6.2 41 / 42
Complicated?? This all seemed rather complicated There is a lot of fancy math behind this Why did we do this? Beyond fancy math, these distributions (power law, Pareto, extreme value) can be good approximations in the tail Probabilities for extreme risks Fall 2016: LeBaron Fin285a: 6.2 41 / 42
Overview Extreme value distributions Generalized Pareto distributions Tail shapes Using power laws with returns Time aggregation Chebyshev s inequality Summary Fall 2016: LeBaron Fin285a: 6.2 42 / 42