Environmental Finance - Schwarzthal5 Pricing weather insurance 6 Pricing option with APARCH methods...

Environmental Finance

FRUNZA Marius CristianCarbon Finance - Course 2

Master 203/104

January 29, 2016

FRUNZA Marius Cristian Carbon Finance - Course 2 Master 203/104Environmental Finance January 29, 2016 1 / 47

Plan

1 MotivationGlobal warming

2 Environmental regulationExternalitiesPigouvian taxCoase’s property rights

3 European emissions marketsArchitectureA quest for efficiencyStatistical featuresModeling the emissions pricesForecasting capacity

4 Driving factors for CO2 prices

5 Bibliography


Motivation

Project and evaluation

1 Ivory Coast crisis and Cocoa prices

2 Influence of Weather Variability on the Orange Juice Prices

3 Valuation of forestry taking into account carbon price

4 Volatility of agricultural commodities

5 Pricing weather insurance

6 Pricing option with APARCH methods

7 Macro model for the LNG supply


Motivation Global warming

Kiribati: A country threaten by climate change

Figure : Kiribati



Rise of the sea level

Figure : Sea level



Rise of the temperature

Figure : Rise of the temperature



Global warming

Figure : Consensus scientific projections estimate that the average global temperature will riseby between 1.8C and 5.8C by the year 2100 if the current rate of increase in emissions is notcurbed. A figure of 2C is considered for the implementation of the Kyoto protocol.


Environmental regulation Externalities

Externalities

An externality occurs when the activity of one entity affects a third party in a waythat it is outside of the market mechanism. Externalities are referred to in themacroeconomic jargon as spill over effects

Positive externalities are benefits that are enjoyed by a third-party as aresult of an economic transaction. Third-parties include any individual,organisation, property owner, or resource that is indirectly affected. Whileindividuals who benefit from positive externalities without paying areconsidered to be free-riders, it may be in the interests of society to encouragefree-riders to consume goods which generate substantial external benefits.

Negative externalities are external costs that are supported by a thirdparty as a result of an economic transaction. In a transaction, the producerand consumer are the first and second parties, and third parties include anyindividual, organisation, property owner, or resource that is indirectlyaffected.



Externalities: Examples

Positive externalites

Healthcare system

Education / migration

Research and development

Negative externalites

Air pollution from industries

Noise pollution from a railroad station

Intense exploitation of resources

Currency devaluation

Social media impact and financial markets



Externalities: Examples

Company A: Gold mine exploration using cyanides and diverting them in ariver

Company B: A cattle farm placed on the same river producing Angus meat

Company A generates a negative externality for Company B. The negativeexternality generates inefficiency. How to restore the efficient allocation of thecosts?

Private solutions based on market mechanisms: Coase’s system ofpermits/allowances

Public solutions based on taxation mechanisms: Pigou’s tax



Dealing with negative externalities


Environmental regulation Pigouvian tax

Pigouvian taxesDefinitions

Pigouvian tax: A tax levied on each unit of a negative externality-generator’s output inan amount equal to the marginal damage at the efficient level of output. [Pigou(1932)]


Environmental regulation Pigouvian tax

Dealing with negative externalities

A tax would increase the marginal production cost of the gold miner(Company A), which would force him to reduce the production fromQ to Q*

What is the efficient level of Q*?

What is the right level of tax?

How have better negotiation power? Farmers vs Miners?

What we should tax? The amount of cyanides? The quantity ofwater? The sold gold?

What if the mining company has a monopole on the market?

What if the mining company relocates in no tax country? Regulatoryarbitrage


Environmental regulation Coase’s property rights

Putting in place property rights: Example

Figure : The miners would like to reduce their production of gold if they receivethe lost profit from Q-Q*. The farmer would agree to be exposed with somepollution if they receive the equivalent of their damage costs



Coase’s rights based solution

Hypothesis

Property rights are in place for each player

Perfect information, Profits and utility maximized, No transactions costs

Coase theorem

Where there are complete competitive markets with no transactions costs, anefficient set of inputs and outputs to and from production-optimal distributionwill be selected, regardless of how property rights are divided. When propertyrights are involved, parties naturally gravitate toward the most efficient andmutually beneficial outcome. [Coase(1974)]



Coase vs. PigouLets assume there are two polluters: Company A - the gold miner andCompany C - an ironworks company

Figure : A and C have different marginal production cost. A taxation approachwill induce an asymmetric situations



Cap and trade: a market based solutionAn authority issues water pollution rights which are allocated to Company A andCompany C

Figure : There are potential gains from trading right between A and C as theyhave different marginal cost.C can reduce more its pollution and sell the rights toAFRUNZA Marius Cristian Carbon Finance - Course 2 Master 203/104Environmental Finance January 29, 2016 17 / 47


Environmental regulation

Kyoto Protocol an international framework signed by the major economic countries(except US). Emission quotas were agreed by each participating country, with theintention of reducing their overall emissions by 5.2 percents of their 1990 levels by the endof 2012.

European Union Emission Trading Scheme (or EU ETS) commenced operation in January2005 with all 15 (now 25 of the 27) member states of the European Union participating.Its linked to Kyoto Protocol.

A fragmented approach in United States started with local policies and voluntary marketsI 1990: SO2 trading system under the framework of the Acid Rain Program and

Clean Air Act in the U.S. Under the program, which is essentially a cap-and-tradeemissions trading system, SO2 emissions were reduced by 50 percents from 1980levels by 2007.

I 1997: the State of Illinois adopted a trading program for volatile organic compoundsin most of the Chicago area, called the Emissions Reduction Market System

I 2003: U.S. corporations were able to trade CO2 emission allowances on the ChicagoClimate Exchange under a voluntary scheme

I 2007: the California Legislature under Governor Arnold Schwarzenegger agreedupon a flexible mechanisms in the form of project based offsets have been suggestedfor five main project types: manure management, forestry, building energy, SF6, andlandfill gas capture

I 2015: COP21, the US adhered to the global project



CO2 Emissions world wide

Level of CO2 emission per country in 2010

5 year growth of CO2 emission per country

25 year growth of CO2 emission per country


http://www.schwarzthal.com/Geo1.html




Cap and trade

Definitions

A cap and trade system is a market-based approach to controllingpollution that allows corporations or national governments to tradeemissions allowances under an overall cap, or limit, on thoseemissions.

Cap and trade is an environmental policy tool that delivers resultswith a mandatory cap on emissions while providing sources flexibilityin how they comply.

Cap and trade system allows players to exchange their depollutioncost curves.



Cap and trade for CO2 emissions

Figure : Let’s assume that two players have different depollution marginal costs.




Figure : If player 1 makes an extra reduction of emissions and sells it to player 2,both players have gains from this system.




Figure : The system of cost exchanges could be generalized to countries



Carbon markets

Carbon transactions can be grouped in two main categories:

Trades of emission allowances, such as, for example, AssignedAmount Units (AAUs) under the Kyoto Protocol, or allowances underthe EU Trading Scheme (EUAs). These allowances are created andallocated by a regulator, usually under a cap-and-trade regime;

Project-based transactions, that is, transactions in which the buyerparticipates in the financing of a project which reduces GHG emissionscompared which what would have happened otherwise, and getemission credits in return. Unlike allowance trading, a project-basedtransactions can occur even in the absence of a regulatory regime: anagreement between a buyer and a seller is sufficient.



What are CO2 permits rights?

An independent investor would perceive her money as working in aphysical mechanism that reduces the emissions.The allowances are imposed by governments in order to stimulate theindustries to reduce their emissions. If an industry has more allowancesthan actual emissions it will cash them out on the market. If it is short ofallowances it will fill the need by buying it on the market. This wouldmake an independent investor to perceive her money as being held in aregulatory paper traded between industries without having behind anyphysical mechanism.All these remarks make the modelling of carbon be particular and not beconsidered as a classical commodity.


European emissions markets Architecture

Architecture of EU ETS framework

Figure : EU ETS frameworkFRUNZA Marius Cristian Carbon Finance - Course 2 Master 203/104Environmental Finance January 29, 2016 26 / 47


Carbon markets: EUA prices

Figure : History of EUA prices



Convenience yieldThe convenience yield corresponds to dividend yield for stocks, thereby theprice of a forward contract is given by:

Ft,T = St · exp((rt,T − δt,T ) · (T − t)) (1)

Figure : Autocorrelation


European emissions markets A quest for efficiency

A quest for efficiency

The efficient market hypothesis is a common assumption in traditional financialeconomics, but as the European emission allowances market (EUA) is a directconsequence of a regulatory system commonly accepted by the market actors,some sources of market inefficiency do exist.

The new information is unequally diffused amongst market players;

The EUA is perceived mainly as a financial liability and has no intrinsic valuecapable of generating economic added value for an investor;

The EUA system is a dealer market.



Carbon markets:EMH

Figure : Auto-correlation of carbon yields



Carbon markets:EMH

Figure : Auto-correlation of squared carbon yields


European emissions markets Statistical features

Carbon markets: Statistical featuresThus, the preliminary tests reject the null hypothesis that EUA daily returns arecharacterized by temporal independence and a Gaussian marginal distribution.The dataset moments for the daily returns are presented in Table 1 and testsresults are detailed in Table 2.

Moment Mean Variance Skewness KurtosisValue -0.0004 0.0008 -0.7502 14.8138

Table : The four moments of the dataset

Test p-value Statistic Critical valueBox-Ljung 0.0003 63.49 43.77

Jarque Berra 0.001 647 5.93Kolmogorov Smironov 0.000 0.08 0.04

Table : The Box-Ljung test rejects the null hypothesis of no serial correlation at a95% confidence level. Moreover, the Jarque-Bera and Kolmogorov-Smirnov testsreject the hypothesis of normality at a 95% confidence level



Carbon markets: QQplot

Figure : GBM Distribution adapted on EUA: on the left yields histogram and thefitted GBM; on the right QQ plot.



Carbon markets: QQplot

Figure : NIG Distribution adapted on EUA: on the left yields histogram and thefitted NIG; on the right QQ plot.


European emissions markets Modeling the emissions prices

GBM models calibrationThe results of the GBM calibration show that the historical volatility of the CO2 marketis around 47 % and that the null hypothesis of the drift cannot be rejected. Thus thismodeling appears very poor. The high volativity level is specific to the emmisionsmarket, whilst the negative trend is mainly due to the major market dowturn from theend of 2008 and the beginning of 2009.

Model GBM GBMMR GBMJ GBMMRJµ - 0.29 - -0.121 -

[-0.82, 0.25] - [-0.55, 0.31] -σ 0.46 0.46 0.28 0.27

[0.43, 0.48] [0.44, 0.49] [0.24, 0.32] [0.24, 0.32]β - 0.99 - 0.998

- [0.98, 1.005] - [0.989, 1.006]c - 0.009 - 0.007

- [-0.016,0.036] - [-0.011,0.031]µjump - - 2.28 2.23

- - [-0.35, 4.92] [-0.65, 4.87]σjump - - 0.82 0.82

- - [0.66,0.99] [0.641, 0.995]λ - - 0.20 0.20

- - [0.09, 0.29] [0.097, 0.302]BIC -3332 -3328 -3522 -3516NIC -2911 -2696 -2469 -2251



GH models estimationThe density of the NIG(α,β,µ, δ) distribution is equal to:

fNIG (x ;α;β;µ; δ) =δα · exp(δγ + β(x − µ))

π ·√δ2 + (x − µ)2

K1(α√δ2 + (x − µ)2).

The moments (mean, variance, skewness and kurtosis) are respectivelyequal to:

E(X ) = µ+ δβ

γ(2)

V(X ) = δα2

γ3(3)

S(X ) = 3β

α ·√δγ

(4)

E(K ) = 3 + 3(1 + 4(β

α)2)

1

δγ. (5)

Thus, the NIG distribution allows for behavior characterized by heavy tailsand strong asymmetries, depending on the parameters α, β and δ.FRUNZA Marius Cristian Carbon Finance - Course 2 Master 203/104Environmental Finance January 29, 2016 36 / 47


GH models estimation

Amongst the previous GH models the case of the NIG modeling provides the best fit interms of our model selection criterion. Compared to the jump modeling the GHdistribution captures more features of emissions prices and improves the figures of theBIC and NIC estimator.

Model α β µ δ BIC NICGH (λ =0.5) 41.65 3.33 -0.001 0.0076 -3546 -2914

[39.47, 43.84] [1.78, 4.88] [-0.002, -0.0005] [0.006, 0.009]GH (λ =0) 31.79 -3.08 -0.001 0.013 -3548 -2916

[26.09, 37.30] [-4.86, -1.31] [-0.002, -0.0004] [0.011, 0.014]GH (λ = -0.5) 23.32 -2.96 -0.001 0.0187 -3549 -2917

NIG [20.51, 26.13] [-4.19, -1.73] [-0.0019, -0.0005] [0.017, 0.019]GH (λ = -1) 14.48 -2.91 -0.001 0.025 -3549 -2917

[11.27, 17.69] [-4.31, -1.55] [-0.002, -0.0004] [0.023, 0.026GH (λ = -1.5) 5.30 -2.80 -0.001 0.0305 -3546 -2914

[1.70, 8.90] [-4.30, -1.55] [-0.002, -0.0004] [0.0298, 0.0311]GH (λ = -2) 3.60 -3.60 -0.0015 0.0377 -3539 -2907

[1.90, 5.26] [-5.26, -1.93] [-0.0022, -0.0006] [0.0367, 0.0386]


European emissions markets Forecasting capacity

Comparing distributions

We estimated the statistical features of three models (GBM, GBMJ and NIG) on anin-sample basis for 100, 250 and 500 days ”rolling windows” and compared the modelsusing Vuong’s test([Vuong(1989)] and [Clarke(2003)]). If we consider two competingmodels f and g, Vuong tests the null hypothesis that two models (nested or non-nested)are close to the actual model against the alternative that one model is closer than theother to the true model using the statistic:

Λn = nE(log(f (θn))− log(g(φn)))− [(p

nlog(n)− (

q

nlog(n)] (6)

where p and q are the number of estimated coefficients in models f and g for a sampleof length n. Under the null hypothesis the test statistic is such that:

Z =Λn√nωn→ N(0, 1) (7)

with ωn defined by setting ω equal to the standard deviation of the pointwiselog-likelihood ratios. Thus, with non-nested test, model f is preferred with significancelevel α (0.05), if the statistic Z exceeds the positive (1− α) -quantile of the standardnormal deviation.



Comparing distributions

Alternative vs null model / GBMJ vs GBM NIG vs GBM NIG vs GBMJ

Window length

100 days 11.6% 17.4% 16.8%

250 days 32.1% 62.4% 61.6%

500 days 99.9% 99.9% 99.8%

Table : BIC ratio test results: Percentage of subsamples where the alternativemodel outperforms the null model.



Comparing distributions forecasts

Figure : Evolution of the BIC for the GBM, GBMJ and NIG models estimated on a 100 daysmoving window; Period 1: GBMJ outperforms the GBM model; Period 2: NIG outperformsboth the GBMJ and the GBM models.



Comparing distributions forecasts: Tails focus

We tested also the stability of our results by comparing the density forecasts via a weightedlikelihood ratio test, introduced by [Amisano and Giacomini(2007)], adapted for non-nestedmodels. In our case we estimate few models (GBM, GBMJ and NIG) on sample xi∗ withi∗=1,2,...,m consecutive returns and we test the density forecast capability of our models on aout-of-the-sample xi with i=m+1,m+2,...,m+n. Here m is the length of in-the-sample windowand n is the length of out-of-the sample window. We repeat the testing procedure on a rollingwindow scheme. Let Ω be the total sample size. The first n-step ahead density forecast tests areproduced at time m, using for model parameters estimation data indexed 1,...,m and comparedwith the sample indexed m+1,...,m+n. The estimation window is then rolled forward one stepand the second set of model parameters are obtained using observations 2,...,m+1 andcompared to the sample indexed m+2,...,m+n+1. This framework is thus iterated and the lastdensity estimations are obtained utilizing observations Ω-m-n+1,...Ω-n, and they are comparedto the sample Ω-n+1,...,Ω.The test is useful for comparing weighted averages of the logarithmic scores, thereby allowingfor a greater weight on particular regions of the distribution. We considered the followingpositive and bounded weight function:

For assessing the core of the distribution w(x) = 1n1x<K∗σ(x)

For assessing the tails of the distribution w(x) = 1n1x≥K∗σ(x), σ being the standard

deviation of the sample and K a constant representing the cut-off point between the coreand the tail.




The weighted likelihood ratio is defined for a given function w(•) and two alternativeconditional densities forecasts f and g, on the sample xi as following:

WΛm,n,i = w(xi )(log(f (xi , θm))− log(g(xi , φm)), (8)

where i=m+1,m+2,...,m+n and θm and φm are parameters of f and g, estimated on the samplexi∗ with i∗ = 1, ...,m. This statistic is defined as:

Zm,n,i =E(WΛm,n,i )√

nσn(9)

where E(WΛm,n,i ) = n−1∑m+n

i=m+1 WΛm,i and σn is a consistent estimator of the asymptotic

variance var(√nE(WΛm,n,i )).

The limit of this statistic for a sample sufficiently large is the normal standard distributionN(0,1). A level α test rejects the null hypothesis of equal performance of forecasts f and gwhenever Z > zα/2, where zα/2 is the (1− zα/2) -quantile of a standard normal distribution.In case of rejection, one could choose f if WΛm,n,i is positive and g if WΛm,n,i is negative.




Figure : Weighted likelihood ratio test comparing Gaussian and GBMJ models: It appearsthat the GBM provides better results when studying the core distribution forecasts for 80.5% ofthe samples while the GBMJ is more appropriate for the tail foreacast, the test being relevantfor 64.1% of the samples.




Figure : Weighted likelihood ratio test comparing Gaussian and NIG models: NIG modelprovides better tail forecasts, for 63.3% of the samples. When forecasting the core density, thetwo models (NIG and GBM) are close, the Gaussian model being better for only 24.1% of thestudied samples.


Driving factors for CO2 prices

Driving factors for CO2 pricesAn Arbitrage Pricing Theory model is developed by [Ross(1973)] in order to show theimpact of the factors previously identified on the CO2 price evolution, and showcalibration results with several noises. The model supposes that a risky asset returnfollows a factor structure and has the following representation:The model has the following representation:

r = r + β1f1 + β2f2 + . . .+ βk fk + ε (10)

where

r : The expected return of Carbon Allowance prices

βi : The factor loading of factor i

ε : The idosyncratic component

E(fi ) = 0 ∀iE(ε) = 0

Described by [Ross(1973)] and based on the underlying hypothesis that the markets areefficient the APT model assumes a Gaussian distribution of the residuals. Given theatypical nature of the CO2, the assumptions of the APT model are in some casesbroken. Hence the residuals do not follow a Normal distribution and the dependenciesare not stationary over the time. In order to overpass this issue different distributions areused to replace the classic Gaussian modelling for residuals. Amongst the candidatefunctions t-Student, GED and Normal Inverse Gaussian (NIG) retained our attention fortheir capacity to take in account heavy tails



Carbon markets

The Table ?? synthesis the results of our static calibration over the considered datasetusing a daily timestep, for different residual distributions. The discriminator element isthe log likelihood: the higher it is, the best is the modelling. It appears that the level ofdependencies of CO2 price are generally close for the different models. But the degree offitness depends highly of the chosen model. Hence the t-Student distribution forresiduals captures well the behavior of the residuals and offers a good explanatory ratiofor fundamental factors.The Ljung-Box test of residuals autocorrelation show no presence of persistence at 99percent of significance for all the distributions.

Factors/Residuals Gaussian GED T-Student NIGOil 0.131 0.103 0.122 0.116

[0.055 0.205] [-0.149 0.352] [0.070 0.174] [0.063 0.169]Dark Spread 0.262 0.2115 0.261 0.260

[0.231 0.349] [0.014 0.468] [0.221 0.304] [0.221 0.302]Spark Spread 0.002 0.0020 0.002 0.002

[-0.001 0.003] [-0.009 0.033] [0.000 0.004] [0.001 0.004]Equity 0.12 0.157 0.160 0.168

[0.017 0.242] [-0.213 0.543] [0.088 0.247] [0.097 0.250]

R2 0.24 0.24 0.23 0.24Log Likelihood 2054 2087 2092 2094



Conclusions

1 The carbon finance is a new market which emerged due toenvironmental regulations

2 The cap and trade system allows players to optimize their greenhousegas reduction strategy

3 Asymmetry, fat-tails and persistence shall be taken into account inorder to provide a reasonable model for carbon prices

4 Oil, gas, equity and power are the main drivers of the carbon prices


Bibliography

Amisano, G., Giacomini, R., 2007. Comparing density forecasts viaweighted likelihood ratio tests. Journal of Business & EconomicStatistics 25 (2), 177–190.

Clarke, K. A., 2003. A simple distribution-free test for nonnestedhypotheses. Department of Political Science, University of Rochester,Harkness Hall.

Coase, R. H., 1974. Choice of the institutional framework: Acomment, the. JL & Econ. 17, 493.

Pigou, A. C., 1932. The economics of welfare, 1920. McMillan&Co.,London.

Ross, S., 1973. The arbitrage theory of capital asset pricing. RodneyL. White Center for Financial Research, University of Pennsylvania,The Wharton School.

Vuong, Q., 1989. Likelihood ratio tests for model selection andnon-nested hypotheses. Econometrica: Journal of the EconometricSociety, 307–333.


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