1. 1 Lecture Plan 14 00 -15 00 Modelling Profit Distribution from Wind Production (Excel Case: Danish Wind Production and Spot Prices) Reasons for copula modelling (non-linearity between variables and different distributions of the various variables) Copulas – The concept and theory Calibration of a copula to data Simulation of from copula Application: Modelling jointly wind production and prices and hence the profit function from a wind power plant

2. 2 Volatility and Correlation are a commonly used measures in financial risk management together with some parametric joint distribution for the variables (e.g. multinormal or multi-t distribution) The correlation measure is based on the idea that there exist a linear dependence between the variables. Often there is a non-linear relationship between Y and X. In such a case correlation is misleading as a dependency measure! Reasons for Copula Modelling

3. 3 The marginal distribution for each variables (e.g. Y and X) might be different! In addition the tail dependence might be different (e.g. Y and X are more dependent when both variables are low compare to when both variables are high) Reasons for Copula Modelling

4. 4 The joint distribution of two i.i.d. random variables X and Y is the bivariate distribution function that gives the probabilities of both X and Y taking certain values at the same time In this lecture we build joint distributions of two or more variables by first specifying the marginals or “stand alone” distributions, and then using a copula to represent the association between the returns Copulas can be applied to any marginal distribution, and the marginal distributions can be different for each series Copula can capture the important property of asymmetric tail dependence! Reasons for Copula Modelling

5. 5 Example Case/Problem: Model the distribution of profit (hence expected risk/return) from wind production Gross Profit = Wind Production * Electricity Price

6. 6 Reasons for Copula Modelling Marginal Empirical distributions 2013 (hour 5-6 DK1):

7. 7 Reasons for Copula Modelling Scatter Plot Wind and Price 2013 (hour 5-6 DK1):

8. 8 Wind and Prices have very different marginal distributions. We can apply a given parametric model for each of them or simply capturing the distribution by the empirical/discrete distributions The dependence structure between wind and prices is non-linear The tail behavior is asymmetric (When wind is high and prices low we have less dependency than when wind is low and prices are high ) Reasons for Copula Modelling

9. 9 Copula is a very flexible tool for creating joint distributions and gives a function form that captures the observed behavior of the variables Different copulas will also create different joint distributions when applied to the same marginal distributions and can capture complex dependency structures Reasons for Copula Modelling

10. 10 Copulas – The Concept and Theory Sklar, A. (1959), "Fonctions de répartition à n dimensions et leurs marges", Publ. Inst. Statist. Univ. Paris 8: 229–231. Abe Sklar is still teaching at Illinois Institute of Technology

11. 11 Consider two random variables (e.g. wind and prices) X 1 and X 2 with continuous marginal distribution functions F 1 (x 1 ) and F 2 (x 2 ) and set u i =F i (x i ), i=1,2. Sklar’s theorem says that given any joint distribution function F(x 1,x 2 ), there is a unique copula function C: [0,1]x[0,1] [0,1] such that: Conversely, if C is a copula and F 1 (x 1 ) and F 2 (x 2 ) are distribution functions then F(x 1,x 2 ) given above defines a bivariate distribution with marginal distributions F 1 (x 1 ) and F 2 (x 2 ) Copulas – The Concept and Theory

12. 12 Differentiating the formula above with respect to x 1 and x 2 gives the joint density function f(x 1,x 2 ) in terms of the marginal density functions f 1 (x 1 ) and f 2 (x 2 ) Copulas – The Concept and Theory

13. 13 Examples of Copulas These are the most used copulas in financial modeling (we will focus on the Clayton Copula as an example): Normal copula Student t copula Normal mixture Clayton copula Gumbel

14. 14 Examples of a Copula Clayton Copulas

15. 15 Examples of a Copula Clayton Copulas α = 0.5 α = 1.0 α = 1.5 U 1 and U 2 are numbers in the interval (0,1) from a uniformally distributed variable The higher the alpha the higher the lower tail dependence

16. 16 Calibrating of a copula to data Correspondence between the alpha in Clayton Copulas and Kendall’s Tau It can be shown that Kendall’s tau (a measure of dependence in a dataset) has a direct relationship with the alpha parameter in the Clayton copula (higher alpha gives higher left tail dependence) Hence, in this case the copula depends on 1 parameter and we can calibrate this parameters using a sample estimate of Kendall’s tau from the dataset Copulas with more parameters must be estimated by maximum likelihood or similar techniques

17. 17 KENDALL’S TAU Suppose a sample contains n paired observations (x i,y i ) for i=1,2,…,n. Kendall’s Tau is calculated by comparing all possible pairs of observations { (x i,y i ), (x j,y j ) } for i≠j. Ordering does not matter, so the total number of pairs is: Count the number N C og concordant pairs and the number N D of disconcordant pairs. That is, the pairs are concordant if (x 1 -x 2 )(y 1 -y 2 ) > 0 and disconcordant if (x 1 -x 2 )(y 1 -y 2 ) < 0 Kendall’s Tau is given by: Calibrating of a copula to data

18. 18 Calibrating of a copula to data

19. 19 Calibrating of a copula to data

20. 20 Correspondence between alpha in the Clayton copula and Kendall’s tau from the data Clayton copula α = 2τ(1-τ) -1 In our example τ=-0.40 that makes α = 2*(-0.40)*(1-(-0.40)) -1 = -0.57 Calibrating of a copula to data

21. 21 Simulation with Copulas In simulation first simulate the dependence on uniform distributed variables and then we simulate the marginals by inversion to a given distribution A nice feature of copulas is that the distribution of the marginals can all be different and different from the copulas We will here use the empirical distribution of wind and prices

22. 22 Simulation with Copulas Step 1 Step one is to generate random numbers u 1 from a uniform (0,1) distribution This is done by the RAND() function in Excel

23. 23 Simulation with Copulas Step 2 Step two is to generate random numbers u 2 from a uniform (0,1) distribution that has a Clayton copula dependency with the numbers u 1 This is done by using a new random number from a uniform (0,1) distribution (here v) and the conditional Clayton density function

24. 24 Simulation with Copulas Step 2 We now have u 1 and u 2 Both have uniform (0,1) marginal distributions u 1 and u 2 have the depndency structure according to a Clayton copula with alpha = -0.57

25. 25 Simulation with Copulas Step 3 But we are not finished yet….We need Y (Wind) and X (Price) to follow their respective marginal distributions This is achieved by inverting the uniform (0,1) marginal distributions of u 1 and u 2 into the empirical distributions for Y and X Having the empirical marginal distribution (with the dependency structure) we can now find the profit distribution (wind*price)

26. 26 Simulation with Copulas Step 3

27. 27 Summary & Conclusion Energy Prices and Factors have complex (and different) marginal return distributions Energy Prices and Factors have various non-linear dependencies and various tail dependencies Simple parametric models with volatility and correlation as input will not capture true risk Modeling energy prices and factors with different marginals and copulas get the “right” calculation of the joint behaviour and risk

28. 28 Book Chapter Copulas for Energy Markets

29. 29 Exercise Perform a similar copula analysis for wind and prices in DK1 and DK2 at these hours: 5-6 9-10 16-17 22-23 How does the areas and hours affect the modelling of the profit function?