Managing a Portfolio of Weather Derivatives

Managing a Portfolio Managing a Portfolio of Weather Derivativesof Weather Derivatives

June 14, 2000June 14, 2000

beyond The BoxThe Box

ThinkingThinking

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BackgroundBackground

Portfolio Risk / ReturnPortfolio Risk / Return

Portfolio OptimizationPortfolio Optimization

OutlineOutline

Managing a Weather Derivative PortfolioManaging a Weather Derivative Portfolio

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Correlation of Underlying IndexesCorrelation of Underlying IndexesGlobal physical processes involve ocean Global physical processes involve ocean

and atmosphereand atmosphere

Indexes Not TradedIndexes Not TradedSpecial pricing considerationSpecial pricing consideration

Basis RiskBasis RiskPortfolio performance vs. hedged riskPortfolio performance vs. hedged risk

Why a portfolio of weather derivatives deserves Why a portfolio of weather derivatives deserves special attentionspecial attention


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Global climate variationGlobal climate variation


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Global climate variationGlobal climate variation


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Regional impactRegional impact


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Weather derivative: call, put, combinationsWeather derivative: call, put, combinations


BuyerBuyer

SellerSeller

Prem

ium

Prem

ium

Payo

utPa

yout

Weather Index

20 40 60 80 100

010

020

030

040

050

0

CallCall

PutPut

StrikeStrike StrikeStrike

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Pricing weather derivativePricing weather derivative


No-Arbitrage Model Not ApplicableNo-Arbitrage Model Not ApplicableUnderlying index not tradedUnderlying index not traded

Pure Statistical Method Not AdequatePure Statistical Method Not AdequateHistorical data characterized by large-scale trend Historical data characterized by large-scale trend

and variation (natural and instrumental)and variation (natural and instrumental)

Combining Dynamic / Actuarial ApproachesCombining Dynamic / Actuarial ApproachesProbability distribution of weather indexes based Probability distribution of weather indexes based

on dynamic/statistical seasonal predictionon dynamic/statistical seasonal prediction

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Probabilistic perspectiveProbabilistic perspective


0 20 40 60 80 100

0.0

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Weather Index

Dynamic/Statistical ModelDynamic/Statistical Model

Weather Index (W)Weather Index (W)

Payout P = f ( W )Payout P = f ( W )

Weather Index

20 40 60 80 100

010

020

030

040

050

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Payout ( P )Payout ( P )

0 50 100 150 200

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Payoff

1010

ReturnReturn R = ( R = ( prpr + + iiii - - PP - - e e )) PDF of P ---> PDF of RPDF of P ---> PDF of R

RiskRisk Return volatility: Return volatility: RR / / RR

Value at risk: VValue at risk: VRR

Risk-based Return: E ( R ) / VRisk-based Return: E ( R ) / VRR

A single weather derivative: risk / returnA single weather derivative: risk / return


-200 -150 -100 -50 0

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Return

ReturnReturn

1111

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Return

PDF of portfolio returnPDF of portfolio return


Portfolio ReturnPortfolio Return

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Return

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Return

Return of Individual Return of Individual Derivative: Derivative: RR11

Return of Individual Return of Individual Derivative: Derivative: RRNN . . . . . . RRii . . . .

No general analytical solutionNo general analytical solution

R = R = ( R ( Rii x x ii ) )

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Joint PDF of { Joint PDF of { WWii } }

Three alternative approachesThree alternative approaches Assuming normal distribution of { Assuming normal distribution of { WWii } } Pattern analysis (EOF / Principal component )Pattern analysis (EOF / Principal component ) Pattern-preserving simulationPattern-preserving simulation

Calculating the PDF of portfolio returnCalculating the PDF of portfolio return


Predicted PDF of individual Predicted PDF of individual WWii WWi i pattern of variationpattern of variation

Simulation based on derivative payout functionsSimulation based on derivative payout functions

Joint PDF of Joint PDF of { R{ Rii } }

PDF of PDF of RR

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Simulated Simulated weather indexesweather indexes

XX (M X N matrix)(M X N matrix)

M simulations, N indexesM simulations, N indexes

Simulation and pattern preservationSimulation and pattern preservation


Predicted PDF of individual Predicted PDF of individual WWii WWi i pattern of variationpattern of variation

Y = Y = transformation transformation (X, A)(X, A)such that such that the correlation matrix for Y = Athe correlation matrix for Y = Athe PDFs of the columns of Y = those of Xthe PDFs of the columns of Y = those of X

Correlation matrixCorrelation matrix

AA (N X N matrix) (N X N matrix)

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Simulation and pattern preservationSimulation and pattern preservation


Individual contract returnIndividual contract return

Z = g(Y, Z = g(Y, contract termscontract terms))(M X N matrix)(M X N matrix)

Portfolio returnPortfolio return

R = Z R = Z

(( N XN diagonal matrix containing positions of N XN diagonal matrix containing positions of individual contracts)individual contracts)

PDF of PDF of RR

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-200 -150 -100 -50 0

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Return

Portfolio risk / returnPortfolio risk / return


Expected Return: E ( R )Expected Return: E ( R )

RiskRisk Return volatility: Return volatility: RR / / RR

Value at risk: VValue at risk: VRR

Risk-based Return: E ( R ) / VRisk-based Return: E ( R ) / VRR

Other Measures of Risk / ReturnOther Measures of Risk / Return Depending on risk tolerance, Depending on risk tolerance,

financial objective, etc.financial objective, etc.

PDF of PDF of RR

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Atlanta

Chicago

CincinnatiNewYork

Dallas

Philadephia

Portland

Tucson

DesMoines

LasVegas

ExamplesExamples


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Hypothetical portfolioHypothetical portfolio


City StrikeOption Type

Risk Based Return (%)

Atlanta 509 call 5.49Chicago 233 put 7.44Cincinnati 363 call 5.86New York 362 put 8.78Dallas 681 call 6.86Philadelphia 351 put 9.26Portland 220 call 6.11Tucson 629 put 20.96Des Moines 389 call 4.04Las Vegas 691 put 5.18Portfolio 14.9

July 2000 Cooling Degree Day at $5,000/CDDJuly 2000 Cooling Degree Day at $5,000/CDD

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Hypothetical portfoliosHypothetical portfolios


City StrikeOption Type

Risk Based Return (%)

Chicago 233 put 7.44New York 362 put 8.78Philadelphia 351 put 9.26Portfolio 7.23

Chicago 233 put 7.44New York 448 call 5.11Philadelphia 351 put 9.26Portfolio 9.34

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Portfolio optimization: feasibilityPortfolio optimization: feasibility


Scientific understanding Scientific understanding of the behavior and of the behavior and pattern of the pattern of the underlying weather underlying weather indexesindexes

Inefficient marketInefficient market

Various purposes of Various purposes of using weather using weather derivativesderivatives

Significant opportunity to build Significant opportunity to build

a high return / low risk portfolioa high return / low risk portfolio

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Portfolio optimization: strategyPortfolio optimization: strategy


Existing PortfolioExisting Portfolio Market Bids / OffersMarket Bids / Offers

Buy / SellBuy / Sell

Portfolio Risk / Return OptimizerPortfolio Risk / Return Optimizer

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SummarySummary


Unique characteristics of weather Unique characteristics of weather derivatives requires special attention in derivatives requires special attention in pricing and portfolio managementpricing and portfolio management

Scientific understanding of the global Scientific understanding of the global climate variability makes feasible building an climate variability makes feasible building an optimal portfolio of weather derivatives with optimal portfolio of weather derivatives with high return and low riskhigh return and low risk

Managing a Portfolio of Weather Derivatives

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