Valuation of Weather Derivatives 13 June 2001

1. Valuation of Weather Derivatives 13 June 2001

2. Temperature About a year ago, there was a rumour that Coca-Cola the soft drink company, had plans for a vending machine, which varies the price for a drink according to outside temperature. The company denied the plans but we can see the logic behind the idea. One of the main causes of uncertainties in cash flow is the weather. Temperature in this case. In the hot summers of 1976 and 1995, the machine could have charged a fortune A regulated power utility company must fix their price for electricity for a certain period. Prices can only be increased with permission of the regulators. Yet, power consumption will be low in warm winters. There is a source of uncertainty of incoming cash flow: temperature

3. Rain Agriculture companies have been aware of weather risk for a long time. The major climatic factor which influences crop growth is rainfall “Excess moisture continues to be a problem, particularly to laterseed crops.” “… but rainfall over the weekend brought haying to a standstill.” “Excess moisture has taken its toll in some areas as crops in low lying lands have drowned out…” “…heavy rains causing some localized crop damage”

4. Wind One source of alternative energy is on the rise: Wind farms are being built, not only in this country but also in Europe. Their exposure to lack of wind (and also an excess of wind) is obvious No wind, no electricity, no money

5. Weather Derivatives

6. Example One: Description • Swap • I shall pay you USD 5,000 for every degree Celsius the average temperature at Heathrow airport is below18 on any day in the observation period (1 Nov 2001 - 31 Mar 2002) • In exchange, you pay me a fixed amount • Of course, these two payments are netted and the difference paid at the end of the contract period • Netted pay-out is limited to USD 1,000,000

7. Example One: Description 2 • Option • As above, however, cash flow only occurs when the differential is positive (or negative) • Plus an up-front premium payment • Pay-outs limited to USD 1,000,000

8. Example One: Index • Heating Degree Days (HDD) • On each day of the observation period • Calculate the number of degrees the average temperature is below18 degrees (or other reference temperature as defined in contract) • Sum them up (and maybe round according to contract details) • Formula

9. Index Calculation: Numerical Example

10. Example One: Trade • Swap • Calculate Differential = Fixed swap level - HDD index • When differential is positive, receive USD 5,000* differential • When negative,pay USD 5,000* differential • Put Option • Calculate Differential = Fixed swap level - HDD index • When differential is positive, receive USD 5,000* differential • When negative, no cash flow • Up to the value of the limit

11. Example One: Payout Formula Swap Put Option

12. Example One: Underlying • Daily average temperature • or • The index (HDD) • A model is needed for either the temperature process or the index

13. Example Two: Description • Pay me USD 500,000 for every day in the observation period (1 Jul 2001 - 31 Aug 2001) on which rainfall is above 0.1 inch at Des Moines, Iowa, International Airport • In exchange fora fixed up-front premium payment (call option) • Pay-out limited to USD 2,500,000

14. Example Two: Index • Count the number of days in the observation period on which the rainfall is above the specified threshold

15. Index Calculation: Numerical Example

16. Example Two: Trade • Call Option • Call option with strike 0 and tick size USD 5,000 • Pay an up-front premium payment • Calculate Differential = rain index – strike • When differential is positive, receive USD 5,000* differential • When differential is zero, no cash flow • Maximum pay-out USD 2,500,000

17. Example Two: Underlying • Daily rainfall • or • The index • Again, a model is needed for either rainfall process or the index

18. Exotic Structures • Further complications and additions • Dual trigger event: temperature and rain • Multi-station trades: baskets or multi-station events • Event only counted when it occurs on two or three consecutive days

19. Models

20. Burn Rate Analysis • Clean, reconstruct and de-trend the data • Calculate the index (HDD, etc.) from historical annual observations • Calculate the resulting trade pay-off for every year • Calculate the average of the trade pay-offs • Discount back from settlement date to today • Add risk premium

21. Data Modification • Clean: fill in missing data and correct errors • Reconstruct: modify data to account for change of equipment, change of location • De-trend: remove trends due to global warming or urbanisation

22. Burn Rate Analysis : Numerical Example

23. Burn Rate Analysis : Pros and Cons • Positive • Simple to implement • Easy to understand • Suitable for portfolio valuation • Negative • Low probability events ignored or unreliably estimated • General problem is that in such a low sample, a single observation can strongly influence the option value • Unclear selection and calculation of risk parameters

24. Stochastic Models for Weather Paths • Choose a stochastic model for daily observations • Estimate the model parameters using weather data • Simulate a weather path • Calculate index and option value for this path (=simulated observation) • Repeat many times and calculate mean • Discount back to today

25. Stochastic Models: Examples • Auto-Regression (AR): use the weighted sum of previous days’ temperatures for the estimation of next day’s temperature • Mean-reverting diffusion (MR): a Markov model with pull-back to the mean • Cao-Wei: a variation of AR • Bob Dischel: a variation of MR

26. Estimators • Moment estimator • find parameters based on observed and model moments, • e.g. mean and standard deviation • Maximum-likelihood estimator • find parameters which are most likely, based on observed data

27. Example: Auto-regression Equation

28. Example: Mean-reverting Diffusion • Equation • or • alternatively

29. Stochastic Models: Pros and Cons • Positive • One (model) fits all (indexes) • Positive/negative • Portfolio risk management possible but may have associated problems (multi-location correlation) • Negative • Slow or inaccurate (in particular for low-event contracts) • Harder to implement

30. Weather Forecasts

31. Long-range Weather Forecasts • So far, the approach has been to base an estimation of the future on past observations • With weather, can we do better and use forecasts? • And how could we integrate them into pricing models?

32. Ensemble Forecasting - Probability Forecasts • Ensemble forecasting has been developed over the past decade and has lead to a large increase in the accuracy and usability of medium-range forecasts • The technique is now being applied to longer range forecasts • Ensemble forecasting uses the sensitivity of forecasts to the initial conditions to provide probability forecasts Ensemble Forecasting By perturbating the initial conditions of a forecast and re-running it multiple times, a range of forecasts for the same time period are produced. These can be used to produce probability distributions of possible outcomes.

33. Climate Prediction Center Ensemble Forecasts • They have been found to be reliable in El Nino and La Nina years for predicting the US climate, but have a tendency to be too cold • CPC predicts whether temperature and rainfall will be above, below or near normal for 30 and 90 day periods • But El Nino does not effect Europe so this approach cannot be used

34. Forecast Format • Forecasts are expressed as the probability anomaly of the observation • Three classes - above - near - or below normal • Forecast probability anomaly is the difference between the actual forecast probability of the verifying observation falling in a given category and its climatological value of 33.3% • Mean temperature and total precipitation

35. Forecast Format: Example

36. Integration of Forecasts into Pricing Models • Adjust model parameters so that model probability distribution matches forecast • Read our paper for more details

37. Summary and Conclusions

38. Theoretical Considerations of Modelling Approaches • Arbitrage-free pricing approach (risk-neutral valuation) as used in standard derivatives industry • price = risk-neutral value (+profit margin) • Actuarial approach • price = statistical value + risk premium (+ profit margin) • Actuarial approach currently used in weather derivatives industry and insurance • Shadow-price approach

39. Summary • A market for weather derivatives is being established • Typical examples of currently traded products • Untradable weather risk will probably remain, in particular extremes. They will remain insurance cases • Two modelling approaches, both have their strengths and weaknesses • Forecasts • Actuarial approach

40. Two Challenges Ahead • Transition to arbitrage-free approach (when markets become liquid, take future contracts and back out a risk-free temperature curve, take options and work out a volatility surface) • Portfolio management of mixed books (e.g. power and weather), to give you an idea of that challenge, even portfolio management of weather books is not fully solved

41. References

42. References • M. Cao and Wei, J.: Equilibrium Valuation of Weather Derivatives, Working Paper. • B. Dischel: At Last: A Model for Weather Risk, EPRM Mar 1999. • G. Considine: Introduction to Weather Derivatives, Working Paper. • I. Nelken: Weather Derivatives – Pricing and Hedging, Working Paper. • L. Zeng: Weather Derivatives and Weather Insurance: Bulletin of the American Meteorological Society, Vol. 81, No.9, Sep 2000.

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