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Estimation of Weather and Energy Price Relationship

Estimation of Weather and Energy Price RelationshipCash Flow Distribution

Ilnaz AsadzadehDan Calistrate

December 10th, 2014

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Objective

The focus is on the daily, weather-driven Volumetric Risk component and howit relates to price risk, as opposed to Volumetric Risk. Cash flows for a hedgedload obligation are driven by two factors:

∆V : Volume (consumption) deviation from expected (i.e. fromweather-normal assumptions).

∆P: Price deviation from expected (i.e. from the forward monthly indexsettle).

CashFlow

Cash Flow = P∆V + V∆P + ∆P∆V .

However, the demand is mainly driven by weather factor so deviation oftemperatures from normal (weather surprise ∆W ) is being used as a proxy fordemand shocks.

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Algonquin Data

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Overview

Estimation ofWeather and

Price Relationship

Price andWeatherdeviation

relationship

LinearRegression

PolynomialFit

Auto-Regressive

Price andWeatherDeviationProduct

Distribution

NIG Dis-tribution

Copula

Result

Discussion

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Linear and Multilinear Relation

Linera Regression

The first model is based on scatter plot between weather and price deviation.

Linear Model

∆P(t) = a1 + a2∆W (t) + εt .

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Polynomial Fit

Another linear regression which we apply in order to improve the estimationbetween price deviation and weather deviation relation is polynomial of degreefour.

Polynomial Fit

∆P(t) = a0 + a1∆W (t) + a2∆W (t)2 + a3∆W (t)3 + a4∆W (t)4 + εt

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Plot of Result

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Autoregressive Model

There is a strong autocorrelation in price deviation data:

in order to remove the autocorrelation we apply autoregressive model of orderone:

AR(1) Model

∆Pt = a0 + a1∆Wt + a2∆Pt−1 + εt

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Summary

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Estimation ofWeather and

Price Relationship

Price andWeatherDeviationProduct

Distribution

NIG Dis-tribution

Copula

ResultDiscussion

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Cash Flow Distribution

∆P∆W Distribution

From Measuring Load Following Risk research the Load Following Unit Costis :

LFUC

LFUC := ∆W∆P/P0,

where P0 is the monthly hedging price for the delivery month. ∆P∆W is arandom quantity, the objective of this section is to find appropriate distributionof this term.

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Normal Inverse Gaussian Distribution

The NIG is continuous probability distribution which works with 4 moments ofdata.

PDF

X ∼ NIG(µ, α, β, δ)

pdf(x) =αδK1(α

√δ2 + (x − µ)2)

π√δ2 + (x − µ)2

exp(δγ + β(x − µ))

µ: Location Parameter

α: Tail heaviness

β: Asymmetry parameter (Skewness)

δ: Scale parameter

γ =√α2 − β2

K1 is a modified Bessel function of the second kind

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Parameter Estimation

First we apply Maximum likelihood estimation and then compare the resultswith Moment matching method.

Table: Four main moments

Mean STD Skewness KurtosisSample Moments 11.2887 49.7302 13.6332 326.6702

MLE 11.2879 70.3144 17.0439 497.8289Moment Matching 11.2829 49.7256 13.6377 326.8982

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Plots

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Plots

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Distribution of Cash-Flow Using Copula

One of the main drawbacks of previous methods is working with dependentdata set. Data that we are working on has linear and non-linear dependencyshown below. Rather than relying on potentially misleading historicalbehaviour, the aim is to generate data independently.

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Copula

Copulas allow us to construct models which go beyond the standard ones atthe level of dependence. Mathematically, a copula is a function which allows usto combine univariate distributions to obtain a joint distribution with aparticular dependence structure.

Copula

Let F (X ,Y ) be a joint distribution with margins FX and FY . Then there existsa function C : [0, 1]2 → [0, 1] such that

F (X ,Y ) = C(FX (x),FY (y))

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Non-Parametric Copula

We are able to to model marginal distribution and dependence structureseparately, so we assume NIG distribution for marginal distribution andnonparametric copula for the dependence part.

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Seasonal Mean Reverting process (SMR) for delta

In order to capture different spikes in the cash-flow, in our simulation weassume the 4th parameter in NIG distribution (δ) which is the scale parameteris random:

SMR delta

δ(t + 1) = aδ(t) + b(t) + σ(t)ε(t)

We can see the heat map of various quantities of delta

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Algorithm for Simulating Winter Cash Flow

Steps

Generate n independent uniform variates u1, . . . , un.

Let ku(v) = ∂C(u,v)∂u

and set x1 = u1.

For i = 2, . . . , n, set xi = k−1xi−1

(ui ).

The desired sample is x1, . . . , xn.

Where, ui ∈ [0, 1]. ku(v) is conditional density of copula given u. k−1xi−1

is theinverse of marginal distribution (Inverse of NIG distribution).

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Last Winter Cash Flow

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One Simulated Path

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Simulated CashFlow for Winter

With different paths for deltas, we simulate different paths for winter cash-flow

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January CashFlow Distribution

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Conclusion

The NIG distribution is a better estimate for the product distribution ofprice and weather deviation.

By using Copula the distribution is improved by removing all sorts ofdependency from the data.

By having updated data, one can find the cash-flow distribution for anytime of the year.

This method is computationally expensive, but more accurate thanprevious methods.

The more accurate estimates helps to drive in a consistent way the riskmitigation amounts (reduction in load following cost) from load followinghedges

The next step would be using number of financial deal types and physicalpeaking assets to mitigate risk.

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Thank You for Listening,

Questions?

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Result of Models

Linear regression

Estimate t value Pr(>|t|) Significant(Intercept) -0.1952 -2.1760 0.0297 True

∆W 0.1887 17.8478 1.1737× 10−66 True

Polynomial Fit

Estimate t value Pr(>|t|) Significant

(Intercept) -0.5956 -5.0064 5.9853× 10−7 True

∆Wt 0.1502 8.7929 2.8775× 10−18 True

∆W 2t 0.0032 1.7958 0.0727 False

∆W 3t 0.00022 3.5730 3.6052× 10−4 True

∆W 4t 0.0001 2.9809 0.0029 True

Table: The result of polynomial fit

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Result

Autoregressive Model

Estimate t value Pr(>|t|) Significant(Intercept) -0.0505 -0.8384 04019 False

∆Wt 0.0603 8.0310 1.5580e−15 True∆Pt−1 0.7355 51.9739 0 True

Maximum likelihood and Moment matching Result

Table: Result of estimation using MLE and Moment matching

µ α β γ

MLE 0.0236 0.0224 0.0224 3.7532Moment Matching 0.0218 0.0197 -0.0176 4.2330

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Plots

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