Spatial Risk Premium on Weather and Hedging Weather Exposure in Electricity

Transcription

1 and Hedging Weather Exposure in Electricity Wolfgang Karl Härdle Maria Osipenko Ladislaus von Bortkiewicz Chair of Statistics C.A.S.E. Centre for Applied Statistics and Economics School of Business and Economics Humboldt-Universität zu Berlin

2 5 Motivation 1-1 "Everybody talks about the weather but nobody does anything about it." Mark Twain

3 5 Motivation 1-2

4 5 Motivation 1-3 log electricity consumption in Kwt per capita cumulated average temperature in C Figure 1: Estimated dependency of log electricity consumption in Germany on cumulative average temperature: , effects of prices and income removed (from Akdeniz Duran et al., 211).

5 5 Motivation 1-4 Future Contracts on Temperature transfer weather related risks, number of agents limited, weather non-tradable, insurance nature, geographically separated markets, spatial relationship.

6 5 Motivation 1-5 Oslo Stockholm Essen Paris Berlin Prague Figure 2: Contracts on 11 cities in Europe are traded on the CME.

7 5 Motivation 1-6 Pricing Model Econometrics of temperature (Benth et al., 27) seasonal function Λ(t) = a + bt + k j=1 { c 2j 1 sin ( ) 2jπt + c 2j cos 365 p-dimensional Ornstein-Uhlenbeck process X(t): dx(t) = AX(t)dt + e p σ(t)db(t), ( )} 2jπt, 365 (1) B(t) is Brownian motion, e p pth column of I p, σ(t) seasonal variation.

8 5 Motivation 1-7 Pricing Model Price of a future on Cumulative Average Temperature index (CAT): (Benth et al., 27) F CAT (t,τ1,τ 2 ) = τ2 + + τ 1 Λ(u)du + a(t)x(t) (2) τ1 t τ2 τ 1 θ(u)σ(u)a(u)e p du θ(u)σ(u)e 1 A 1 [exp {A(τ 2 u)} I p ] e p du. with a(t) = e 1 A 1 [exp {A(τ 2 t)} exp {A(τ 1 t)}] MPR θ(t) unknown for specific locations.

9 5 Motivation 1-8 Pricing Temperature around the Globe θ(t) varies across locations. Connect θ(t) to a known risk factor! Spatial model to interpolate for other locations. How can FPCA help pricing an arbitrary location?

10 5 Motivation 1-9 Outline 1. Motivation 2. Spatial Model for Risk Premium Functional Principal Components (FPCA) Geographically Weighted Regression (GWR) 3. Empirical Risk Premia and Hedging Weather Exposure 4. Outlook

11 5 Spatial Model for Risk Premium 2-1 Risk Premium Given (2) the RP at t = τ 1 for ith location and jth contract (RP ij ): RP ij (t = τ 1, τ 2 ) = F CAT,ij (θ i, t = τ 1, τ 2 ) F CAT,ij (, t = τ 1, τ 2 ) + ε ij, = τ2 τ 1 θ i (u)σ ij (u)e 1 A 1 i [exp {A i (τ 2 u)} I p ] e p du + ε ij. F CAT,ij (, t = τ 1, τ 2 ) estimated price for jth contract in ith location, zero MPR, A i is the matrix of O-U-Process coefficients for the ith location and ε ij noise.

12 5 Spatial Model for Risk Premium 2-2 a functional regression set up with scalar response: w i (t) def = e 1 A 1 [exp {A(τ 2 t)} I p ] e p and θw i (t) = θ(t)w i (t): RP ij (τ 1, τ 2 ) = τ2 τ 1 θ i w (u)σ ij (u)du + ε ij. Regress RP on PC scores of σ(t) for dimension reduction. Note θ i w (t) contains location dependent parameters, need spatial setting.

13 5 Spatial Model for Risk Premium 2-3 Functional Principal Components 1. Decompose σ ij (t) = {σ ij (t) σ i (t)} + σ i (t), σ ij (t) variation curve for ith location and jth month, σ i (t) average curve for ith location. RP ij = τ2 τ 1 τ2 + τ 1 θ i w (u) σ i (u)du θw i (u) {σ ij (u) σ i (u)} du. }{{} FPCA for σ ij 2. Perform FPCA: derive scores for temperature variation

14 5 Spatial Model for Risk Premium 2-4 PC Scores PC scores for functions σ ij (t) i = 1,..., 9 (9 cities), j = 1,..., 7 (7 traded months): c ijk = τ2 τ 1 ξ ik (t) {σ ij (t) σ i (t)} dt, c ijk scores for K largest eigenvalues, ξ ik (t) orthonormal eigenfunctions of Cov{σ( )} operator. Collect scores capturing the variance in the data in matrix C. Parametrize the relationship to RP by geographically weighted regression.

15 5 Spatial Model for Risk Premium Regress the response RP ij at t = τ 1 on the PC scores (Ramsay & Silverman, 28): RP ij = β i + τ2 K τ 1 k=1 β ik ξ ik (t){σ ij (t) σ i (t)}dt + ε ij for ith location and jth month, with ε ij containing ε ij and the truncation error resulting from taking first K PC scores. Functional form of MPR: θ i w (t) = k β ikξ ik (t) Need spatial model for regression on PC scores. APPENDIX

16 5 Spatial Model for Risk Premium 2-6 Spatial Specification: GWR Why GWR (Fotheringham et al., 22)? distance based weights, nonstationarity over space, local nature of spatial dependence.

17 5 Spatial Model for Risk Premium 2-7 GWR: the Model W 1 2 i RP = W 1 2 i Cβ i + e i, e i vector of iid errors, RP = (RP 1,1, RP 2,1,..., RP n,1, RP 1,2,......, RP n,7 ), c 1,1,1... c 1,1,K C = c 2,1,1... c 1,1,K c n,7,1... c n,7,k n total number of locations K number of PC scores

18 5 Spatial Model for Risk Premium 2-8 GWR: the Model W i = diag(w i ), i = 1,..., n [ { w i = diag exp 1 ( ) } { 2 di1 2 h,..., exp 1 2 h = arg min h H 7n m=1 { RP m RP m (h)} 2, ( din h ) 2 }] with d il, l = 1,..., n euclidean distances to ith city, RP m (h) estimated RP without the mth value using h.,

19 5 Empirical Risk Premia and Hedging Weather Exposure 3-1 Temperature Data City First Date Last Date First F CAT Trade Berlin Essen Paris Stockholm Table 1: Average Temperatures without 29th February. Source Bloomberg and DWD.

20 5 Empirical Risk Premia and Hedging Weather Exposure 3-2 Volatility Functions σ i (t) Fit to data Seasonality Λ(t), AR(p) process with seasonally heteroscedastic errors, estimate RP ij, i = 1,..., 9 and j = 1,..., 7, estimate σ i (t), t [1, 365] using residual standard deviation for each day of year and smooth by Fourier series.

21 5 Empirical Risk Premia and Hedging Weather Exposure 3-3 Seasonality Berlin Essen

22 5 Empirical Risk Premia and Hedging Weather Exposure 3-4 Seasonality Paris Stockholm Figure 3: Daily average temperatures T (t) (blue) and seasonality Λ(t) (red).

23 5 Empirical Risk Premia and Hedging Weather Exposure 3-5 Essen estimate t.stat estimate t.stat estimate t.stat a b c c c c c c Table 2: Estimated Parameters of seasonality (1) for Essen,, Paris

24 5 Empirical Risk Premia and Hedging Weather Exposure 3-6 AR(p) Berlin estimate t.stat estimate t.stat estimate t.stat α α α α Table 3: Estimated Parameters of AR(p) for,, Berlin

25 5 Empirical Risk Premia and Hedging Weather Exposure 3-7 RP 5 Berlin Stockholm Figure 4: Average RP for traded locations computed according to (2)

26 5 Empirical Risk Premia and Hedging Weather Exposure 3-8 Seasonal Variation Berlin Jan Mar Nov Jan Jan Mar Nov Jan Jan Mar Nov Jan Essen Jan Mar Nov Jan Jan Mar Nov Jan Jan Mar Nov Jan

27 5 Empirical Risk Premia and Hedging Weather Exposure 3-9 Seasonal Variation Paris Stockholm Jan Mar Nov Jan Jan Mar Nov Jan Jan Mar Nov Jan Figure 5: Estimated σ(t) (blue) and smoothed by Fourier series (red).

28 5 Empirical Risk Premia and Hedging Weather Exposure 3-1 Eigenfunctions Berlin Essen

29 5 Empirical Risk Premia and Hedging Weather Exposure 3-11 Eigenfunctions Paris Stockholm Figure 6: FPCA weight functions: eigenfunction ξ 1 (solid), ξ 2 (dashed), ξ 3 (dotted).

30 5 Empirical Risk Premia and Hedging Weather Exposure 3-12 FPCA Scores Berlin Essen

31 5 Empirical Risk Premia and Hedging Weather Exposure 3-13 FPCA Scores Paris Stockholm Figure 7: FPCA scores c ij1 and c ij2

32 5 Empirical Risk Premia and Hedging Weather Exposure 3-14 Explained Proportion of Variance Berlin Proportion of Variance Proportion of Variance Proportion of Variance PC1 PC2 PC3 PC4 PC5 PC1 PC2 PC3 PC4 PC5 PC1 PC2 PC3 PC4 PC5 Essen Proportion of Variance Proportion of Variance Proportion of Variance PC1 PC2 PC3 PC4 PC5 PC1 PC2 PC3 PC4 PC5 PC1 PC2 PC3 PC4 PC5

33 5 Empirical Risk Premia and Hedging Weather Exposure 3-15 FPCA Scores Paris Stockholm Proportion of Variance Proportion of Variance Proportion of Variance PC1 PC2 PC3 PC4 PC5 PC1 PC2 PC3 PC4 PC5 PC1 PC2 PC3 PC4 PC5 Figure 8: Proportion of Variance explained by the corresponding PC.

34 5 Empirical Risk Premia and Hedging Weather Exposure 3-16 GWR Estimation Results Berlin Essen 5 5 5

35 5 Empirical Risk Premia and Hedging Weather Exposure 3-17 GWR Estimation Results Paris Stockholm Figure 9: RP (red) and fitted values with 95% CI (blue) returned by GWR.

36 5 Empirical Risk Premia and Hedging Weather Exposure 3-18 City β β 1 β 2 β 3 Rloc Berlin Essen Paris Stockholm Leipzig Table 4: Estimated Parameters of GWR (h =4.98). indicate significance on 1% level, on 5% and on 1%. Dummy variables for north and south sea coast cities omitted here. Weights for Leipzig (.13,,.42,.24,.2,,.5,.7,.7).

37 5 Empirical Risk Premia and Hedging Weather Exposure 3-19 Leave-One-Out Forecast for ust City F CAT (2181, 21831) FCAT (2181, 21831) Berlin Essen Paris Stockholm Table 5: Observed and predicted F CAT prices for ust 21 by leaving the location to predict out of the data for the GWR model calibration.

38 5 Empirical Risk Premia and Hedging Weather Exposure 3-2 Example: Hedging weather risk in electricity demand An electricity provider in Leipzig transfers risk via CAT futures. What RP one would pay for F CAT in ust 21? Berlin EssenLeipzig Paris Stockholm

39 5 Empirical Risk Premia and Hedging Weather Exposure 3-21 Out-of-Sample Forecast: Leipzig Jan Mar Nov Jan Figure 1: Λ t, σ t for Leipzig.

40 5 Empirical Risk Premia and Hedging Weather Exposure 3-22 Out-of-Sample Forecast: Leipzig Figure 11: ξ and the resulting forecast for RP for Leipzig vs RP of Berlin.

41 5 Empirical Risk Premia and Hedging Weather Exposure 3-23 F CAT (ˆθ, 2181, 21831) = F CAT (, 2181, 21831) + RP = = 536. log electricity consumption in Kwt per capita F_cat(2181,21831)= cumulated average temperature in C

42 5 Empirical Risk Premia and Hedging Weather Exposure 3-24 Example: Hedging weather risk in electricity demand c marginal costs of meeting additional log demand of 1% per person b estimated marginal effects of 1 C CAT on log demand starting from threshold F CAT α number of WD hold, p tick value of WD (for traded futures in Europe: 2EUR) hedging, s.t. exposure benefits cb(cat F CAT ) αp(cat F CAT ) cb = αp

43 5 Empirical Risk Premia and Hedging Weather Exposure 3-25 Example: Hedging weather risk in electricity demand parameter value units c 1, EUR per 1% log Kwt/pP b.16 1% log Kwt/pP and 1 C CAT p 2 EUR per 1 C CAT α 8 contracts long Table 6: An elementary example of a hedging strategy.

44 5 Empirical Risk Premia and Hedging Weather Exposure 3-26 Appendix RP ij = τ2 τ 1 θ i w (u) σ i (u)du } {{ } β i = β i + m k τ2 + τ 1 θ i w (u) }{{} m β imξ im (u) τ2 β im c ijk ξ ik (u)ξ im (u)du τ } 1 {{}, k m, = 1, k = m {σ ij (u) σ i (u)} du }{{} k c ijk ξ ik (u) = β i + k β ik c ijk. BACK

45 5 Empirical Risk Premia and Hedging Weather Exposure 3-27 Literature E. Akdeniz Duran and W.K. Härdle and M. Osipenko Difference based ridge and Liu type estimators in semiparametric regression models Working Paper SFB649, , submitted to Journal of Multivariate Analysis. F.E. Benth and J.S. Benth and S. Koekebakker Putting a Price on Temperature Scandinavian Journal of Statistics 34: , 27 A. Fotheringham and C. Brudson and M. Charlton Geographically Weigthed Regression: the Analysis of Spatially Varying Relationships John Willey & Sohns, 22.

46 5 Empirical Risk Premia and Hedging Weather Exposure 3-28 Literature W.K. Härdle and B.López Cabrera Implied Market Price of Weather Risk Working Paper SFB649, 29-1, submitted to Applied Mathematical Finance W.K. Härdle and M. Osipenko Derivatives and Hedging Weather Exposure in Electricity Working Paper SFB649, , submitted to Energy Journal. B. Øksendal Stochastic Differential Equations Springer, New York.

47 5 Empirical Risk Premia and Hedging Weather Exposure 3-29 Literature F. Perez-Gonzalez and H. Yun Risk Management and Firm Value: Evidence from Weather Derivatives Available at SSRN: J.O. Ramsay, B.W. Silverman Functional Data Analysis Springer Verlag, Heidelberg, 28

48 and Hedging Weather Exposure in Electricity Wolfgang Karl Härdle Maria Osipenko Ladislaus von Bortkiewicz Chair of Statistics C.A.S.E. Centre for Applied Statistics and Economics School of Business and Economics Humboldt-Universität zu Berlin 5 Berlin Stockholm