Hedging Corporate Revenues
with Weather Derivatives:
A Case Study
Master of Science in Banking
and Finance - MBF
Master’s Thesis
Antoni Ferrer Garcia
Franz Sturzenegger
Université de Lausanne
Ecole des Hautes Etudes Commerciales
HEC - 2001
Abstract
This paper searches for the implications in the use of a new generation of financial
derivatives known as Weather Derivatives as a form of hedging future corporate
revenues. According to the US Department of Commerce about 22 per cent of
the US$ 9 trillion GDP in the United States is sensitive to weather. This figure
supports the growth in the market that started at the beginning of the 1997s. Likewise, it is estimated that already some 1,800 deals worth roughly US$ 3.5 billion
have been transacted in the U.S. An estimated 70 per cent of all businesses face
weather risk which extends across geographic and market borders. The current
weather derivatives market is still illiquid and several pricing models are being
used by financial institutions. On this paper we show the characteristics, pricing models and hedge strategies about such new contracts. Our case study has
been done within a multinational corporation that we will be here called XYZ to
preserve its confidentiality.
Acknowledgments
Before starting, we would like to thank all the people that with their support
and understanding have contributed to make this master’s thesis somehow better:
Mrs. F. Kafader of Kundendienst-Account, Swiss Météo, Prof. Dusan Isakov
(HEC - Genève), Mr. Jürg Trüb (Swiss Re-insurance), Robert Dischel, Melanie
Cao and Jason Wei.
Several institutions that have supported us with data, advice or knowledge
about the weather derivatives markets: Enron, Aquila Energy, AC Nielsen, Migros, and Koch Energy Trading.
Special thanks to company XYZ since it was their idea to write about this
topic. It is also theirs most of the data contained on this thesis. With their help
and that of Mr. Lagger and Mr. Silen, we started our research.
Special thanks, also, to professor Didier Cossin to direct our thesis and to give
academical support to such an interesting topic.
Last but not least, thanks to our family, girlfriend and friends - Manuel Kast,
Dr. Alexander Passow, Beatriz Rueda - that certainly have helped us during the
whole MBF Program and during the preparation of this thesis. This thesis is
dedicated to them.
1
Preface
In today’s financial markets, derivative instruments have certainly a granted
place on corporate risk management as a way to insure against or hedge business
hazards. Derivatives are financial instruments whose values depend on the value
of other securities known as the underlying. Those underlyings are often traded
assets such as stock, commodities, currencies, bonds but can also be non-traded
assets such as stock index. Futures and options are actively traded on major exchanges while forward and swap contracts are evenly traded outside exchange
by financial institutions in the over-the-counter market (OTC). Since the study of
Black and Scholes, ‘The Pricing of Options and Corporate Liabilities’ and Robert
Merton,‘The Theory of Rational Option Pricing’, we have seen an astonishing
growth in derivative markets and in the development of more complex instruments that simple plain-vanilla options, such as Asian Options, Lookback Options, Barrier Options, Catastrophic Bonds and others. Nowadays, a new class of
derivative securities has been created to offer corporate managers an instrument to
hedge their firms against climate conditions’ hazards. They are known as Weather
Derivatives and are designed to minimise or avoid the risks due to changes in
weather conditions.
On the other hand, several questions have been raised for why corporate managers should hedge their business and on what are the consequences of the use of
derivatives as a form to offset undesired risks. Sometimes, instead of using derivatives for hedging purposes, managers have traditionally used them to simply speculate1 in financial markets with the intention of profit from market discrepancies.
Nonetheless, an uncautious use of derivatives could lead to huge losses that might
have an impact not only in the company but be spreaded to the whole financial
markets.
1
Whereas hedgers want to avoid an exposure due to price movements, speculators wish to take
a position in the markets by betting that the price will go up or down.
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The objective of this research is twofold: i) first, present the explanations for
corporate hedging; the theory behind it and attempt to shed some light on the use
of Weather Derivatives as a form of hedging volumetric risks for corporate institutions and ii) second, a case study where weather derivatives are used to hedge
potential risks due to climatological effects on a company’s business. To achieve
these objectives, we conduct our case study with the help of a multinational company that has provided us with data on sales for juice and milk in Switzerland,
Germany, France and U.K. To keep confidentiality, we will call this company
“XYZ”.
For this reason, this paper is organised in three chapters. Chapter 1 introduces
the theory behind corporate hedging as well as the weather derivatives. Section 1
introduces the concepts of why corporations should hedge risks. Section 2 shows
the characteristics of the weather derivatives markets. Section 3 introduces the
impact of weather conditions in real businesses. Section 4 shows the methods of
forecasting weather. Section 5 introduces a literature survey of weather derivatives
models. Section 6 ends this chapter with mathematical applications and modelling
of weather derivatives as well as some extention of the model. Chapter 2 presents
the case study using data from company XYZ. Section 1 introduces the company’s business. Section 2 explains the correlations between sales and weather in
Switzerland and abroad. Section 3 presents the market exposure of XYZ’s sales
under climatological changes. Section 4 shows the instruments to hedge weather
risks. Section 5 contains the hedge strategy using weather derivatives. Chapter
3 concludes this study and propose further ideas about the topic. References and
appendices are presented at the end.
3
Contents
1 Weather Derivatives: A literature Review
1.1 Corporate Hedging . . . . . . . . . . . . . . . . . . . . . . .
1.2 Weather derivatives: Market and characteristics . . . . . . . .
1.3 Impact of weather conditions on the economy . . . . . . . . .
1.3.1 Extreme weather conditions and natural catastrophes .
1.3.2 Abnormal weather . . . . . . . . . . . . . . . . . . .
1.4 Forecasting the weather . . . . . . . . . . . . . . . . . . . . .
1.5 An analysis of weather models: A literature survey . . . . . .
1.5.1 Understanding the weather evolution models . . . . .
1.6 Mathematical applications / Modelling . . . . . . . . . . . . .
1.6.1 Modelling weather derivatives . . . . . . . . . . . . .
1.6.2 The “burn analysis” model . . . . . . . . . . . . . . .
1.6.3 Auto-correlated regression model. The case for Geneva
1.6.4 Extension to the model: Some US cases . . . . . . . .
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2 The Case Study
2.1 The company business . . . . . . . . . . . . . . . . . . . . . . .
2.2 Understanding the correlation between sales and weather . . . . .
2.2.1 The correlation of XYZ’s sales with the temperature . . .
2.2.2 Lagging data to make it more consistent . . . . . . . . . .
2.2.3 Country specificity . . . . . . . . . . . . . . . . . . . . .
2.3 Market exposure of XYZ’s sales under changing environmental
conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4 The hedge procedure . . . . . . . . . . . . . . . . . . . . . . . .
2.4.1 Natural hedging . . . . . . . . . . . . . . . . . . . . . . .
2.4.2 Weather Derivatives Instruments . . . . . . . . . . . . . .
2.5 Designing a hedge strategy using weather derivatives . . . . . . .
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3 Conclusion and further ideas
3.1 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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A Human Physiology
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B Exhibits
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Chapter 1
Weather Derivatives: A literature
Review
1.1
Corporate Hedging
Several financial economists have provided us with many theories as to why managers would hedge corporate revenues. Modigliani and Miller, according to their
seminal paper, pointed out the lack of need for corporate hedging. After this statement many academics have taken up the challenge of explaining why we see this
phenomenon. In the area of corporate hedging Clifford Smith, David Mayers and
René Stulz have certainly contributed by giving us some of the reasons. In 1982
Mayers and Smith published the earliest work on hedging corporation in their
Journal of Business article, “On the Corporate Demand for Insurance”. In this article the authors suggest seven possible explanations for why corporations would
insure their assets, even if their shareholders are well diversified. However the focus was on property and liabilities rather than in derivatives. Concerning the use
of derivatives as a hedge strategy the main reasons that make these instruments
attractive to hedge are:
1. Non-diversifiable stakeholders (employees, customers and suppliers) will
demand expensive terms in contracts with a risky firm since they would be
over-exposed to the fluctuation of cash flows without being offset by other
external non-correlated revenues.
2. The probability of costly bankruptcy can be reduced. Volatility-reduction
can be achieve by hedging and consequently increasing the recovery rate on
defaulted debt which leads to decrease bankruptcy costs.
6
3. Tax reasons:
Progressive corporate tax rates induce firms to smooth their profits.
Under this tax regime, companies would pay more taxes if their revenues are for example 30 and 70 than 50 and 50 in some years.
Limited or delayed deductibility of large losses, due to time-limits on
loss carry-backs and carry-forward and due to government’s abstention from participating in the firm’s losses.
Furthermore other articles have also provided explanations for corporate hedging:
Leverage motivation
- Over investment problem: Asset substitution(Jensen & Meckling 1976)
where debt creates an incentive to take risky projects since the debtholders will bear all the downside risk of a project. Equity can be seen as
a call option on the firms assets since shareholders are entitle to have
limited responsibility on the liabilities. Stockholders might exercise
their option to default when firm gets in trouble. Requiring hedging in
those bonds can reduce the firms’ incentive to increase risk and consequently reduce bondholders’ discount of those securities.
- Under investment problem: Debt overhang (Myers 1977) where investing in positive NPV projects imply transferring value from equityholders to debtholders because the latter have to share the profits but
not the costs. This creates an incentive to forgo positive NPV projects.
A bond covenant requiring hedge may act in favour of undertaking
positive NPV projects and consequently reducing the cost of debt.
Assymmetric information issues
- Costly external financing (Froot, Scharfstein and Stein 1993). External financing is costly because potential investors are less informed
about the project. Hedging adds value to the firm to the extent that it
helps ensure that a corporation has sufficient internal funds available
to take advantage of attractive investments opportunities in the future.
In order to maintain a cheap access to capital, corporations may need
a risk management program.
7
Although the use of derivatives in the corporate sector and its consequences
have been widely studied1 , numerous articles have criticised its use. As an example, the use of derivatives have caused spectacular losses for financial institutions
(Barings Bank), non-financial companies (Procter & Gamble, Landis & Gyr, etc)
and government institutions (Orange County), which put the whole financial markets in alert due to the value of the losses. Notice that those cases occurred by
simple speculation or a misunderstanding in the features of derivatives without
taking in consideration its implications in case of not been possible to market
margin calls.
1.2
Weather derivatives: Market and characteristics
According to the Department of Commerce about 22 per cent of the 9 trillion
USD gross domestic product in the United States is sensitive to weather. This
figure supports the growth in the market for weather derivatives that started at
the beginning of the 1997s and it is estimated that some of 1,800 deals worth
roughly $3.5 billion have been done in the USA. An estimated 70 per cent of
all businesses face weather risk which extends across geographic and markets
borders. Before 1997 utility companies managed their earnings stabilisation primarily through price hedging derivatives while volumetric risks were largely left
unhedged.
However, the recent deregulation in the energy sector increased competition
leading to hedge volumetric risks caused by unexpected weather conditions with
derivatives. Whilst the weather is still an uncontrollable variable, a new class of
financial instrument - weather derivatives - enable companies to have a more active approach to manage weather risk. Weather derivatives are nowadays not only
traded in over-the-counter (OTC) market. Standardised contracts are being traded
on the Chicago Mercantile Exchange (CME) which provides contracts based on
the temperature for many US cities. I-Wex.com, a LIFFE-backed (London International Financial Futures and Options Exchange) and Internet companies Wire
and Intelligent Financial System have joined forces to create an internet-based
weather derivatives exchange2 .
1
See, for example,recent surveys by Wharton School and Chase Manhattan Bank (1995) and
by Ernst and Young (1995)
2
Risk Magazine, March 2000
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A weather derivative or weather option is a financial instrument that has a
payoff derived from variables such as temperature, snowfall, humidity and rainfall. However, the industry has set up temperature as the common underlying
for those contracts. Unlike insurance and catastrophe linked-instruments, which
cover high-risk and low probability events, weather derivatives shield revenues
against low-risk and high probability events (such as mild winters). Temperature contracts are more specifically traded in what is called Heating Degree-Days
(HDD) or Cooling Degree-Days (CDD) defined on daily average temperatures.
The number of heating degree-days is the difference between 65 degrees Fahrenheit and the daily average temperature whilst the number of cooling degree-days is
the difference between the daily average temperature and 65 degrees Fahrenheit.
HDD and CDD can never be negative. Daily average temperatures are the arithmetic average of the minimum and maximum records in a midnight to midnight
basis. A more elegant description of HDD and CDD is done below:
HDD = max 65 degree Fahrenheit - daily average temperature, 0 CDD = max daily average temperature - 65 degree Fahrenheit, 0 Typical contracts are written on cumulative HDD/CDD structured by options,
futures, swaps and collars for a given period. HDD contracts last during the
November to March period whilst CDD for May to September. One could define
four basic elements in options or futures/swap contracts: i) the underlying variable HDD/CDD; ii) the accumulation period: a season or a month; iii) the specific
weather station that record the daily temperature and iv) the tick size assigned to
each HDD/CDD.
The world’s first exchange-traded, temperature-related weather derivatives,
which started trading on September 22 , 1999, on the Chicago Mercantile Exchange, remains the only major exchange where weather products are traded. The
CME introduced the electronic trading of weather derivatives on its Globex system with the intention of enlarging the size of the market and remove credit risk by
trading on weather contracts. CME contracts have attracted new participants and
increased liquidity in the weather derivatives market for a number of reasons:first,
it allows small transaction sizes which leads to increment the number of investors;
second, it provides price discovery since weather options and futures are quote
in real time and can be accessed by everyone; third, it ensures low trading costs
on the Globex system by using an electronic system that needs less personal to
operate; and fourth, it eliminates credit risk for participants which is bypassed to
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the clearing house system. Table 1.1 provides an example of contracts traded on
the Chicago Mercantile Exchange for HDD option for Atlanta and CDD futures
contract for Chicago.
Chicago CDD (Future contract)
Contract Size
Measuring Station
Contract Month
100 x the CME degree day index
O’Hara Airport (ORD)
12 consecutive calendar months
Minimum Tick Size
Regular Strike Price
1.00 (HDD/CDD) index point =$100
Not Applicable
Exercise
Final Settlement
Price
Initial Strike Range
Not Applicable
The Exchange will settle the contract to
the CME degree day index of the contract
month by EarthSat
Not Applicable
Position Limits
Trading Hours
10,000 futures contracts
3.45 p.m. To 3.15 p.m. (next day)
Atlanta HDD (Option contract)
One futures contract
Hartsfield Airport (ATL)
5 months (HDD): Nov–Mar
5 months (CDD): May–Sep
1.00 (HDD/CDD) index point =$100
HDD: 50 index points CDD:25 index
points
European Style
Not Applicable
150 HDD / 75 CDD index points up and
down from at-the-money
(Futures equivalents)
Same as futures
Table 1.1
Although CME weather derivatives contracts are traded for 11 cities in the US,
most of the deals are executed by OTC participants under ISDA (International
Swap and Derivatives Association) Master Agreements standards, that provide
tailor-made products to suit clients’ needs. ISDA’s standardised documentation
allows any firm to enter into a contract with another firm readily, if both have
derivatives market experience.
The first transaction in the weather derivatives market took place in 1997,
when Aquila Energy included an option in a weather contract. After that Aquila
Trading and Risk Management and Koch Supply and Trading have joined, transacting some 140 and 80 contracts, respectively. Enron Corp. has been active in
the market with some 70 deals since it announced a weather contract to Northeast
utility on the same year. Another important player has been Willis Risk Management, the London based risk management group, that until now has structured 92
weather option strategies. Others have also started to offer customised weather
hedge contracts such as Worldwide Weather Trading Co, TradeWeather.com, Natsource, Southern Co. TradeWeather.com is a New York based company servicing
the global weather derivatives market. It is an Internet system that includes automated order placement and execution, real-time quotes 24 hours per day. In
Europe, the market is also gaining amplitude with an increase in the number of
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deals executed by firms such as Swiss Re New Markets, Société Générale. Swiss
re-insurance has recently launched the sale of weather derivatives via its ELRiX
platform. ELRiX stands for Electronic Risk Exchange and forms the Swiss Re’s
electronic trading of standardised risk transfer products.
Weather derivatives structures commonly used are: i) cap - a call option; ii)
floor - a put option; iii) collar - a put and a call option, usually with little or no
premium; iv) swap - a derivative with a profit and loss profile of a futures contract
and v) digital option - an option that pays either a predetermined amount if a
certain temperature or degree day level is reached, or nothing at all in other case.
A business with weather exposure may choose to buy or sell a futures contract,
which is equivalently to a swap such that one counterparty gets paid if the degree
day over a specific period are greater than the strike level, and the other party gets
paid if the degree day over that period are less than the strike. A business may also
choose to write an option. A heating oil retailer may feel that if the winter is very
cold they will have high revenues - so they might sell an HDD call. If the winter is
very cold, the retailer can afford to write the option and pay out with higher than
normal revenues.
The weather risk market has a huge potential and the growing number of
deals has sparked interest among some of the world’s biggest corporation, banks,
brokers and insurers. However the market’s rise has not evolved as participants
hopped. Several issues have stopped the way of this recent market such as lack of
end-user demand and liquidity. A recent conference of the Weather Risk Management Association (WRMA) tried to shed some light on the future of the weather
derivatives market. An important issue mentioned there was the fact that to create
an effective commodity market to operate the main aspect is whether the data is
reasonably consistent and the market participants agree upon.
1.3
Impact of weather conditions on the economy
Almost all businesses are in certain way affected or exposed to weather conditions, sometimes in a cyclical way like the energy, gas, heating oil sector or in a
irregular way such as entertainment or leasure businesses and, as a consequence
the providers of those. Nonetheless, several businesses have performed their activities without taking in consideration meteorological conditions nor have built a
team of weather experts in the running of their deals. So, the question posed here
is why companies should care about weather events throughout the time. To answer this question first, let us have a look at the kinds of weather events that might
11
affect firms and then how weather derivatives can provide the right solutions for
the consequences of some weather events.
1.3.1 Extreme weather conditions and natural catastrophes
Natural catastrophes such as earthquakes, hurricanes, floods, and large scale fires3
have increased in terms of frequency and losses caused during the past twenty
years. One answer for this phenomenon might be attributed to climate changes,
specifically global warming. The expansion of cities with the consequent growth
of gas emission by industries and vehicles together with an increase of building
areas, which avoid the solar rays to be be absorbed, have certainly contributed to
shift upward the temperature level throughout this century. However, there is still
little agreement on the effects of the overall warming.
Global warming trend is also displayed by temperature data, principally in
cities that have had an increased population growth. Although this warming trend
is very small in terms of absolute values, it produces significant changes in the
temperature as one may observe today. Additionally, seasonality is a feature that
one observes when depicts temperature data over years. This seasonality is not
completely the same for different samples of past data even when we observe
very similar characteristics for all years.
Another reason for this is the climatic response to El Niño. El Niño, the
oceanic phenomenon of warming sea surface temperature in the Eastern Pacific,
is widely known to alter the patterns around the world. According to the National
Oceanic and Atmospheric Administration, the 1997-1998 El Niño coincided with
the highest land and ocean temperatures during this century. In addition to El
Niño, other recent weather phenomena have had a deep impact in the US economy. Though not defined as an El Niño, a less pronounced warm ocean phase
even led to 500 year floods in the Midwest during the summer of 1993. 1993,
the year of the March “Superstorm”, dumped feet of snow in the eastern US disrupting utility services cancelling schools and shutting down businesses on the
coast. A 1996 cold ocean phase led to record cold temperatures in the upper Midwest and included massive snow-melt flooding in the North Dakota the following
spring. These are disasters that have occurred since 1988 with at least one disaster
each year according to survey conducted in the U.S. by the National Climatic Data
3
The insurance industry defines a catastrophe as “an event which causes in excess of $5 million in insured property damage and effects a significant number of insurers and insurers”. See
Loubergé and Schlesinger (1999)
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Centre (NCDC), an agency responsible for monitoring and assessing the climate4 .
1.3.2 Abnormal weather
Less dramatic climate events might also generate important losses. Several industries are closely related to weather and even not so extreme events can cause huge
losses if they are abnormal and persistent during certain period of time. Viticulture
industry is extremely sensitive to the weather. Lack of sunshine and cool temperatures during the stages between pre-bloom and maturation do significantly affect
the quality of grapes, and consequently the vintage of the resulting wine. In 1998,
California’s production of wine grapes felt almost 30%. This was due to a rainy
and cold spring, followed by a very hot July and August. Higher-than-average
rainfall during the summer months can also be very expensive for wine makers as
this leads to the grapes rotting on the vines and delays the harvest. Brewing industry is also affected by changes of weather. Sales of beer drop during colder-than
normal summers, and although it is possible to estimate seasonal trading patterns,
long term forecasts are still notoriously unreliable. Without considering reduced
sales, brewers are also affected by colder summers due to the fact that the beer not
consumed has to be stored, increasing the overall expenses.
Another important industry that is directly related with weather is the construction industry and here the size of the contracts are generally high. In this
sector, heavy penalties can be imposed for works that are not concluded in the
schedule period and delays can soar the costs. Concrete needs to set at a certain
temperature to obtain its maximum strength, but if it is too hot or too cold, it
sets too quickly or too slowly respectively. High winds mean that workers cannot
work at heights, and crane use is banned due to safety regulations. Nevertheless,
the biggest forewarning to the construction industry is the rainfall followed by
freezing temperature.
Risk Professional Magazine5 has underlined the successfully effects in the
US entertainment-to-drinks conglomerates in smoothing their earnings by actively
managing weather exposures. The lucrative but volatile world of entertainment is
exposed to weather changes that can severely decrease the revenues. A long than
normal rainfall during the production of a movie would make the costs immense.
The same principle is applicable to theme parks where cloudy days could lead
4
A complete survey of extreme weather and climate events can be obtained on the following
web site : http://www.ncdc.noaa.gov/extremes.html
5
Risk Professional issue 2/6 July/August 2000
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to a significant decline in volume of visitants during the holidays’ season. Some
studies have been conducted in this field trying to find some evidence about the
relationship between weather and businesses. Ross (1984) presents the influences
of weather changes in the frozen orange juice concentrated production in the US.
More than 98 per cent of the production took place in central Florida region surrounding Orlando. The study suggests the interaction between prices and a truly
exogenous determinant of value, the weather. His empirical results show that cold
weather is bad for orange production. Orange trees cannot withstand freezing
temperatures that last for more than a few hours. Florida occasionally has freezing weather and the history of citrus production in the state has been marked by
famous freezes. In 1895, almost every orange tree in Florida was killed to the
ground, production declined by 97 per cent and 16 years passed before it had recovered to its previous level. Even a mild freeze will prompt the trees to drop
significant amounts of fruit.
1.4
Forecasting the weather
There are at least three methods of forecasting. In order of increasing complexity and sophistication, they are persistence, statistical and modelling. Persistence
leads us to say that tomorrow will be similar to today, that next month will continue the trend of last month (warmer than normal, for example). Persistence, by
itself, incorporates little about the dynamics of the environment. It can be compared to charting the behaviour of the stock market. This forecasting method can
be relatively accurate for a short period.
Statistical forecasting is an effort to match the patterns of the past to the
present. When past patterns fit, the inclination is to forecast that the future will be
similar to what happened in the pattern of the past. Unfortunately, nature rarely
repeats itself exactly and the number of variables is great. Statistical forecasting,
as persistence forecasting does neglects the dynamics of the environment.
Model forecasts incorporate the dynamics of the environment. They include
the current conditions and mathematical representation of the physical environmental process that influence weather. The environmental events are written as
differential equations that are then solved by numerical integration. In general, in
weather models, there is a correspondence between forecast reliability over time
- confidence declines with the increasing time to the forecasted future. There is
also an issue of geographic scale over which the model is integrated.
Many countries have meteorological services that record temperature, wind,
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humidity since the beginning of the century and the degree of confidence in the
measurement techniques are high enough to accept the data as a reliable source
to use for weather derivatives. Relevant questions arise about the impact of the
change in data collection in places where the degree of population growth could
alter the measure mechanism. When using data to provide a consistent pricing
model, one observes a controversy among the market participants in determining
the length of data that might incorporate all the significant factors that influence
the weather. A period of 10 to 20 years is considered relatively short to capture
the characteristics of the weather cycle. A longer period such as 50 years might
be more accurate in identifying those patterns. Specialists say that even a century
would not incorporate all factors due to the extreme complexity of the weather as
a changing phenomenon.
Pricing weather derivatives requires an historical database and application of
statistical methods for fitting distribution functions to data. In our research, historical data is available from the National Climate Data Centre (NCDC) a subsidiary
of the National Oceanic and Atmospheric Administration (NOAA) for the case of
US and Suisse Météo for Switzerland (similar offices exist for different European
countries). Defining the appropriate mean and standard deviation is the key challenge in simple-option pricing. As mentioned above the length of historical data
that should be used is a critical factor. This problem is well known among climate
researchers who have struggled to determine the Optimal Climate Normal (OCN),
or the optimal average time scale of previous years for determining the expected
value for this year. The National Centre for Environmental Prediction (NCEP)
runs an operational tool that is a simplified OCN calculation. This product examines whether the previous 10 years are a better climate predictor than the defined
“climate normal period”. Where the historical data indicates that the previous 10
years provides an improved estimate, this 10-year average is used. One of the
primary drivers that make the previous 10 years a better predictor than the period
from 1961-1990 is a large trend in urbanisation. Any city’s temperature that has
a strong warming trend will be better approximated using the most recent data 10
years than NCEP’s 30-year normal.
Actual sequence of weather near a measurement site will differ from the measurements at the site: weather is that variable and the contractual data requirement
is that specific. This leads to what is called basis risk: one or more weather stations do not exactly measure an enterprise’s exposure to weather. It can be that the
stations are too far from the exposure site, or that the exposure is integrated over
a region. Basis risk in weather options is poorly understood and difficult to quantify. The scale of weather impacts varies with local and regional sensitivities, from
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place to place in a region, and changes with time. To measure exposures beyond
the measurement site, hedgers will create a weighted group or a basket of sites
within a region. This provides a more representative dimension of weather exposure over that region. Basis risk arises from the reality that site-specific measurements and revenues shortfalls do not correlate perfectly. Although accumulating
values over a season and area averaging reduce a hedger’s basis risk, neither technique eliminates it completely. Basis risk is even more complex when more than
two factors affect weather sensitivity that also varies over time. Air temperature
and precipitation are not correlated when viewed as a concurrent and contiguous
time series of data, unless we apply an understanding of the weather patterns to
stratify the time series. All farmers intuitively understand this.
However, this is particularly true for the US market where the continental
distances may cause such differences in temperature values from place to place.
When analysing this in Europe, one concludes that the basis risk is relatively small
and might be discarded due to the high correlation the temperature presents within
countries.
Section 1.6 elaborates on the pricing of weather options. Weather option pricing relies on an accurate weather prediction model. In section 1.6.3 we present a
case study where a temperature forecast is done for the city of Geneva.
1.5
An analysis of weather models: A literature survey
The previous section lead us to conclude that business losses arising from extreme weather events cannot be hedged using weather derivatives contracts. Those
events are insured today by using a number of financial products that include
exchange-traded catastrophe options, OTC swaps and options, CAT bonds and,
CatEPuts. The recent issuance of a CAT bond related to an earthquake in Disneyland Japan is an example of these innovations in the field of catastrophe insurance
market.
The events related in the previous section as abnormal weather match the category of non-catastrophic events. That is when the use of weather derivatives trading is particular useful to hedge variability of revenues due to the influences of a
colder or warmer period. Those weather phenomena are in general easily measurable, can be independently verifiable and transparent be use in written contracts.
Weather derivatives contracts focus on the use of those characteristics by rely-
16
ing on a specific index together with selected locations on where they have been
written using reliable information sources. We present in the following sections
the stochastic process for weather derivatives as well as the market structure and
trading features of such contracts.
1.5.1 Understanding the weather evolution models
Everyone knows that to predict weather is a hazardous task because of the existence of multiple variables that govern the characteristics of the weather. However looking to the past, one might obtain precious information about possible
behaviour of the weather which is possible to assume as regular behaviour because changes in the weather seem to follow a cyclical pattern although with some
variability.
One could mention wind, precipitation, humidity, snow, temperature and so
on as the variables that constitute ‘the weather’. The scope of this work is more
precise and we only analyse the influences of the temperature as a weather risk.
Likewise, when using the concept of weather derivatives here we are assuming
that we are referring to temperature derivatives, and specifically heating degree
days (HDD) when we analyse a winter season. On the other hand, there can also
be cooling degree days (CDD) in a summer season.
Let us first see important characteristics when modelling weather derivatives.
1. Following the same approach used by Black and Scholes for pricing option
contracts one should start by modelling the behaviour of the temperature
index HDD or CDD. Let us use a geometric Brownian motion for this purpose.
Definition 1 Defining the probability space ( ), where de
fines the set of states of nature, the filtration of information avail
able at time and the statistical probability measure, the dynamics
governing the stochastic process of the average daily temperature can be set up by the following differential equation:
(1.1)
where the drift may be a mean-reverting process to capture
seasonal cyclical patterns, and a volatility that might be considered
not constant.
17
Special attention should be considered when dealing with the parameter .
To obtain the standard deviation, we need to set up a time-window frame in
order to calculate it. This could present problems when data is not available
in certain regions leading to a bias in the value of . Another point is the
consistency that past pattern will be repeated in the future and thus, making
it a reliable parameter.
As we have shown in previous section about the behaviour of the weather,
there seems to be a certain volatility over time due to an intrinsic feature
as well as due to human-made. Nonetheless, in principle it may be feasible
admit that is either a deterministic function of the time or a stochastic
pattern but depending only on the current value of the temperature index,
i.e. .
The following step should be to construct a riskless portfolio by using the
Fundamental Theorem of Finance 6 which would yield a risk-free rate. This
gives us the partial differential equation (PDE) for a call option:
'
"!
"!$#&%
!
!$#)#
(
%
+*
*-,
%
*.,
,
(1.2)
with the corresponding boundary condition given by
!
/
+0214365 %
87
*"9
(1.3)
In the case of stock option and even interest-sensitive securities the construction of such portfolio is feasible since the underlying is traded. However, for the case of weather derivatives, the same argument is no more valid
because the weather is not traded meaning that one does not have the underlying security such as stock, Treasure bills and so forth to build a riskless
portfolio.
2. Having in mind that weather options are written on cumulative HDD or
CDD we can conclude that in fact we are working with a type of exotic
option namely Asian-type derivatives.
6
The arbitrage theorem gives the formal conditions under which“arbitrage” profits can or cannot exist.
18
Definition 2 Asian option, or average rate option, is an option whose
payoff at maturity depends on the average price of the underlying instrument during all or part of the option life, rather than the price of
the underlying asset on maturity date.
Assuming that the price driven under the risk-adjusted probability : by the
dynamics described below:
;
<%
=8
%
(1.4)
and also assuming that the number of values whose average is computed is
large enough to allow the representation of the average > over[0, ] by
the integral,
?
'
@
%
A
CB
B
(1.5)
the value of an Asian call option at time t is expressed, by arbitrage argu?
D
ments, as
EGF5IH"JLKMONPJ
021436
*
9
S
!
RQ
(1.6)
T
S
where stands for the strike price of the option. Although the PDE for
Asian options is the same as Black and Scholes, the boundary conditions
are different. Let U denote the value of an Asian option and % the price of
the underlying asset. We introduce > as,
?
>
(1.7)
TV
where denotes the maturity of the option.
So, we need to solve the following PDE,
W
W
W
U
XU
(
%
W
W
XU
%
<%
W
W
U
%
%
W
U?
Y
U
+*
(1.8)
with the appropriate boundary conditions. This leads to a more complex
mathematical formulation than the classical plain-vanilla options style. Asian
options are generally priced through numerical methods or Monte Carlo
simulation.
3. An alternative methodology would be find an equivalent martingale measure where we do not exploit PDEs implied by arbitrage-free portfolios.
19
This procedure tries to find a “synthetic” probability
comes a martingale as shown below:
D
]
+E\ H"JLKMONPJ
RQ
0214365 %
N
Y7
[
Z
*"9
under which
%
be-
(1.9)
Definition 3 The Girsanov theorem establishes that if we have a Wiener
process
then multiplying the probability distribution of this process
by ^ we can obtain a new Wiener process Z with probability distri[
bution Z in such way the two process are related to each other through
Z
`_
(1.10)
Under this approach, once determine the equivalent probability we can discount equation 1.9 by the risk free rate in order to obtain its value.
Thus, modelling the weather process does not seem a straightforward task according to the above difficulties. Another approach proposed by Cao and Wei
(2000), is to model daily temperature evolution using a discrete, autoregressive
model using a generalised Lucas model of 1978 framework to include the weather
as a fundamental variable in the economy. They concluded that weather derivatives present a zero market price of risk.
For instance, some market-makers have incorporate elements of the actuarial
approach to price weather derivatives. Roughly, one might consider as :
1. Determine the average for temperature or an index (HDD/CDD) for the next
years
Calculate the expected mean for the temperature or an index (HDD/CDD)
using the following approaches:
– average of the last 10 years or another period; this may lead to
certain differences depending in the period of data used as proxy
for the relevant data gathering;
– linear regression models
– autoregressive methods that can incorporate autocorrelations in
temperatures changes beyond lag one;
– meteorological forecast that evaluate how the weather evolve using climate variables (wind, pression, temperature, humidity, etc)
2. Determine the volatility of the parameters
20
Determine the distribution function based:
– on historic data that might consider volatility constant and deterministic
– on forecast models, relying in more abroad parameters
3. Determining the risk premium
Integrate pay-out function multiplied with probability forecast
Special attention has to be taken when dealing with historical data. Basically,
models formulated using historical data perform relatively bad when are evaluated
out of the sample since the length of data is an important factor. An approach used
by Dischel to overcome this problem is to look back into meteorological record
only with the objective to obtain the volatility of weather needed to drive a model.
Monte Carlo simulation are than used to calculate the value of the option.
Another important point to consider is the availability of data to perform such
models. Meteorological files in many countries are rich with long and accurate
weather records. These files can be obtained very easily and most of the time
downloadable from the Internet. Therefore, if having the appropriate data to work,
one might expect that weather option price would be found directly from a statistical analysis of measured data.
Black and Scholes critic
Initially, some players in the weather derivatives market applied a classic BlackScholes option valuation model. But this approach is inappropriate in this filed.
First, because the underlying instruments - in this case, temperature - are not tradable. Second, it is also impossible to create a risk-neutral portfolio buy combining
positions in derivatives and in the underlying, through a delta hedging strategy for
the same reason.
Black and Scholes is also inappropriate for another reason. Weather options
accumulate value over a strike period. every day of colder-than-normal weather
over a term of an option might add to the total payout at expiry. This accumulation feature is similar in Asian-style options described above. Stochastic option
models, which allows to stochastic volatility paramether, can be formulated for
weather derivatives and though eliminating the strong assumption that volatility is
constant through time.
For this reason methods based on analysis of potential historical payouts of
the option became widely used. One of its advantages are simplicity and speed.
21
In our study we used a similar approach - the burn analysis method- described in
details below.
1.6
Mathematical applications / Modelling
In this section we will present some useful approaches to forecast the weather
and modelling weather derivatives. A “burn analysis” model is presented with an
illustrative example using data from Las Vegas (US). Pay-offs of a call and a put
weather derivative options are then calculated.
1.6.1 Modelling weather derivatives
Because of the inherent properties of the traded underlying in weather options
(CDD, HDD, rainfall, amount of snow, wind. . . ) these instruments cannot be
priced as other derivatives. Weather financial instruments’ underlying is not traded.
As a consequence, a risk-free portfolio cannot be constructed. Traditional option
valuation as Black and Scholes cannot be applied to price this kind of instruments.
What actual market participants are doing is basically to model weather, especially temperature. This is also the approach we undertake in our study. The
concept is rather straightforward: because of weather follows a mean-reverting
process, what will happen in the future shall to be an average of what happened
in the past, or at least, past data should be an excellent predictor of future weather
for those periods of time.
Biggest market players, observe the cumulative amount of both heating and
cooling degree days in a given season and in a given location. Averaging them
across all years, they get “normal” amount of HDD or CDD which is used to
price the weather contract for the following season. This basic approach is often
completed with some kind of prediction. Naturally, the predictive component
remains a secret for those companies pricing models. This method is called the
burn analysis.
We will first show that this approach is totally dependent on the amount of
data we have. That is, it does make a difference the number of seasons we use to
average the data available. Even temperature is completely stationary with constant mean and most the time variance, it is possible that the cumulative amount of
CDD/HDD varies during the years due to the fact that daily data does not present
such stationarity and after a cool winter usually comes a warm summer.
22
It is hard to say which is the right number of years to use when working with
weather data. In principle the longer history the more effects we capture. On the
other hand, it seems to be the convention of using between 10 and 20 years, since
temperature of near future seems to be a better predictor, due to the urbanisation
and heating affects
In our study we also have used two more approaches. The first one is a normal
regression. Knowing that temperature follows a mean-reverting process we have
obtained a reasonable good regression model for prediction. The second one is
running a Monte-Carlo simulation once we have captured the historical distribution of weather data to generate the distribution of possible weather. This method
allows to take the probability of the distribution we want to hedge as if it was a
VAR model.
1.6.2 The “burn analysis” model
Modelling the price of a weather option for the places our business is, presents
more than one difficulty that will have to be sorted out. To present the model,
we have chosen Las Vegas. Las Vegas is one of the cities for which the Chicago
Mercantile Exchange trades standardised weather options.
We consider the case of a hedger wanting to hedge cumulative degree days
(CDD). This option is used in a summer season. The buyer of a call will be
hedging against a warmer than normal summer, whereas the buyer of a put, or
the shorter of the call, will be buying protection against a decrease in summer
temperature.
Using maximum and minimum daily temperatures of Las Vegas (from 1/1/1959
to 31/12/1997), the average temperature is calculated using the midpoint between
both. This midpoint is what the market considers as the day’s temperature. From
our point of view this is not very consistent. We would recommend to use a
weighted average or to use more than two observations in a day.
Using an econometric software yearly averages have been
calculated for the
#
summer season, considered in out OTC option, from June 1 to September 30 .
From this yearly season average, we have obtained the cumulative CDD for everyone of the seasons. The mean and standard deviation of the CDD series are
then calculated. Following the convention in standardised market, the calls strike
price is set equal to the mean plus 1/2 of the standard deviation. Similarly, the
put’s strike is calculated as the mean minus 1/2 of the standard deviation. Hence,
23
Strike call: 2643.60
Strike put: 2466.34
Fig. B.1 (in the appendix) shows yearly CDD and call strike price.
At this point, the firm has to buy a number of contracts that allow it to hedge
the monetary exposure it wants to hedge for every CDD. In this example we assume $1 for every CDD.
The pay out of a call is obtained by differentiating the actual accumulation of
CDD with the strike. As usual, the option expires worthless if the total CDD has
fallen below the strike:
D
bab143
Ddcec
8%
VfgS
H4hi*
(1.11)
It is rational to assume that what will happen in a future period is an average of
what happened in previous periods. This reasonable assumption is only relatively
validated.
Historical data can be used to predict a normal behaviour of CDD for a long
future period. Because the mean-reverting pattern of temperature, if we take a long
period to hedge, warmer years will compensate cooler ones, and the prediction
will be accurate enough.
However, when it is matter to predict only one year, there can exist a notorious difference between different lengths of periods we use to average our data.
Furthermore, when the amount hedged increases, i.e. we hedge $10,000 instead
of $1 for every CDD, the whole pricing structure becomes very sensitive to the
prediction.
Fig. B.2 shows the different yearly payments that a call option would have
had during all period. The triangles indicate the payments with values on the right
axe. In Table , we have taken different averages. It is shown how the price of the
weather options changes, when the hedger believes that future temperature will be
an average of the only 5 years, or 10 years, or the whole history.
Average Pay-offs
last 5 years
last 10 years
last 20 years
10 years from 1993
120.85
42.74
47.57
83.27
133.39
per year hedged
Table 1.2
The same experience is repeated for a put option with similar results (Fig. B.3):
24
Average Pay-offs
last 5 years
last 10 years
last 20 years
10 years from 1993
32.60 per year hedged
0
0
17.40
59.59
Table 1.3
To avoid this problem, a polynomial can be fitted to predict the likely pattern
of next year expected CDD. In our case, the fitted pattern is:
j
+*lkm*4*"nLoV3qp
*rkts4u
(
3wv
yx
kmuzo
(
3
x
*rktsz{4oV3
(
u
x4x
kOn
(1.12)
Using the polynomial we can recalculate the prediction the model would have
done. The overall result is quite accurate: mean is almost the same as the actual value, 2545 CDD as an average, instead of the actual one of 2554. Due to
the smooth introduced by fitting the polynomial, the standard deviation decreases
from 177.26 to 77.66, when we use the polynomial to predict. As we can see
in the appendix (Fig. B.4), most of all differences are strongly significant. Once
more, the method is good when we are trying to assess the number of CDD for a
long period hedge (more than 4 years), but not for forecasting just one year.
1.6.3 Auto-correlated regression model. The case for Geneva
In the following an OTC CDD weather option is priced for the city of Geneva. Using maximum daily temperatures of Coitrin Airport (from 1/1/1959 to 31/12/1999),
the average maximum temperature is calculated.
Another deviation we introduce to the standard contract is the season. Since
Geneva# is a relatively# cold place, it is enough to hedge a summer season from
June 1 to August 31 only, instead of hedging until the end of September.
Although this model is relatively widespread, there is an inherent consistent
flaw: we always use historic data (temperatures from the past) to price an instrument alive in the future. Hence, it calls for a way of prediction for future
temperature. This task is nothing else but weather forecast. Of course, predict
the weather (or the temperature!) for a specific day next year is quite a hazardous
duty.
What we have done is a mix of both strategies in two steps. We have firstly
taken past data not to price a weather option but to predict the temperature. Once
temperature is predicted, we use the prediction to price the weather instrument.
25
The pricing part is done as we have done in the Burn Analysis section (1.6.2).
Secondly, temperature prediction has been done using lagged autocorrelation regressions. The results have proved to be surprisingly consistent. In the following
we give detail of this methodology. In order to prove the consistency mentioned
before, we have taken daily data for Geneva from 1959 to 1999. To check the
accuracy of the prediction with real data, we have sampled out until 1998. Then,
we have applied the regression model to predict daily temperature for 1999.
For predictions for 90 days, correlation between true temperature and predicted one has attained a value of 0.8977 (t-stat=26.329). More interesting is to
observe that with the best regression model for whole year, the correlation coefficient has only decreased to 0.8960 (t-stat=52.93). Adjusted R-squared decreased
from 0.7288 for the 90 days model to 0.7250 for the one year prediction.
As a consequence, and because it is more interesting to predict one year forward rather than only 90 days (although is a season period) we will explain the
first. The regression used to forecast is only based upon the two principal criteria
used in weather: past data and mean reversion. Hence, we have run an ARCH (3)
model using as variables a one-year lag and five-year lag of the daily temperatures
in Geneva. As instrumental variables we have used a couple more: one proxying
for medium-term history and the one-day lag. The ARCH variables are going to
help us to predict future volatility. One and five-year lags -the variables-, with
positive z-statistics, very high and significant. Adjusted R-squared equals 0.725.
Variable
C
FIVEYLAG
ONEYLAG
C
ARCH(1)
ARCH(2)
ARCH(3)
GENHIST
GENAVG1
R-squared
Adjusted R-squared
S.E. of regression
Sum squared resid
Log likelihood
Durbin-Watson stat
Coefficient
17.99245
0.42698
0.416073
617.4891
0.623547
-0.014718
0.027757
-1.20979
0.162732
0.725133
0.725011
37.43159
17899358
-62891.81
0.43148
Std. Error
0.334415
0.004532
0.004593
24.97543
0.01412
0.012081
0.00886
0.271284
0.275514
Mean dependent var
S.D. dependent var
Akaike info criterion
Schwarz criterion
F-statistic
Prob(F-statistic)
Table 1.4
26
t-Statistic
53.80283
94.20857
90.59138
24.72386
44.16187
-1.21823
3.132952
-4.459502
0.590647
98.37484
71.38072
9.846801
9.845801
4213.82
0.000000
Prob
0.0000
0.0000
0.0000
0
0
0.2231
0.0017
0
0.5548
We use these coefficients and the past history to predict temperatures for 1999.
In the Fig. 1.1, the predicted and the actual time series are depicted for 1999. It is
self-evident the fitness of the prediction. Correlation coefficient among both, only
for the predicted season of 1999, equals 0.896 (t-stat: 52.93).
Figure 1.1: Predicted and actual temperatures
The next step is to know something about the errors of the model. We have
taken the errors (true value minus the predicted one), and looked at their distribution. We cannot reject the hypothesis that errors follow a normal distribution since
its skewness and kurtosis are normally behaved and Jarque-Bera coefficient does
not exceed the critical value 5.99. Having normally distributed errors is a very
nice property because it implies that although we have huge departures from the
true data (almost 9 | C upside and downside), in the aggregate positive deviations
will off-set negative ones. Fig. 1.2 gives an idea of the statistics for errors done in
the forecast.
Temperature prediction is accurate enough to be used for pricing. Assuming
the predicted temperature as the true one, we would apply burn analysis as it is
presented in section 1.6.2.
27
Figure 1.2: Descriptive statistics for errors
1.6.4 Extension to the model: Some US cases
Most of the time when dealing with weather derivatives, they are not as standardised as in the previous section. As explained above, firms have to sign OTC
contracts with intermediaries that close the transaction with other counterparts
with the opposite exposure. In such cases, it is in the interest of the firm that is
going to pay for the hedge, to construct the deal which most accurately hedge for
its purposes.
In standardised contracts, pay-offs are calculated taking the daily difference
between the average temperature of that day and 65 | F, or 18.33 | C. Let us imagine
a place with continental weather, where temperature takes extreme values between
day and night. In such a place, the midpoint temperature would be affected by the
very low temperature registered at 6 a.m. -usually the coldest time in a day-. This
low record biases the correlation since the consumption for the goods which are
attractive when it is hot is still high.
Long term series for maximum temperatures are difficult to find, mainly for
non-US cities. Although there is not any inherent reason for the following, usually,
maximum temperatures series follow very similar path as average temperatures.
Correlations between both are high and they have similar histograms. Because of
the way to calculate the average we have higher standard deviations for maximum
temperature series. In the appendix, we present some examples from US cities
from which daily maximum and average temperatures are available. Fig. B.5
(Appendix) presents seasonal means for maximum and average daily temperatures in Las Vegas. Both series present correlation of 0.87. Fig. B.6, shows the
same relation in La Guardia (NY). Correlation between maximum and averages
temperatures is 0.97. In Fig. B.7, we have San Antonio with correlation of 0.84.
28
Finally, Fig. B.8 shows a correlation of 0.95 between the same series in Seattle.
These results imply that in the case weather data cannot be obtained for maximums and minimum temperature (those would be needed in the case we had to
design a weather hedge contract for CDD and HDD respectively), we could still
hedge using average temperature series, provided both series follow the same stationary distribution. If this is the case, the hedger just must bear in mind that he
has to adjust the hedge by a factor that will take into account the difference in
temperature between the average and the maximums.
Unfortunately, the feature for similar distributions does not always fulfil. Using simulation software, we have simulated standard distributions for maximum
and average temperatures for the five American cities studied. Interestingly, the
results have been diverse. In the case of Chicago, simulating a distribution has not
been possible due to the disparity of data, or irregularities in distribution, as for
example negative temperatures (Fahrenheit!) even in maximum temperatures. For
Seattle, both distributions fitted as gamma distributions. Once simulating the temperatures, distributions even both being of type gamma, showed slightly different
shapes, as it can be seen in Fig. B.9 and Fig. B.10. Las Vegas, is a border-line case.
Maximum temperatures have been best fitted by a Weibull distribution whereas
average temperatures depending on the statistic used best fitted by a Weibull or a
log-normal distribution. Results after simulations are in Fig. B.11 and Fig. B.12.
On the other hand, la Guardia, is a perfect for best correlation between both
time series. A Beta distribution is the one that has best fitted for both the maximum
temperatures and average temperatures. Run the simulation the only difference
noticeable between two distributions is just the mean and standard deviation which
is logically higher in maximums. Fig. B.13 and Fig. B.14.
Lastly, San Antonio, is again a case for which maximum and average temperatures seem to follow exact distributions. So, we conclude that for this type
of places, using maximum temperatures instead of average or conversely, cannot introduce major biases in the study. Fig. B.15 and Fig. B.16 are proof of the
simulation trials.
As part of our case study we present in chapter 2 a detailed explanation about
weather behaviour, forecasting and modelling with real data we gather from company XYZ.
29
Chapter 2
The Case Study
2.1
The company business
The previous chapter presented some insights about the features of corporate
hedging as well as a review about weather derivatives and its developments in
today’s financial markets. The second part of this study will focus on an empirical analysis of weather derivatives within a corporate environment. An empirical
study is conducted with a database provided by company XYZ on sales of juices
and milk for Switzerland, U.K., Germany and France for the period 1992 to 1998.
Sales are reported monthly and because of the high level of production certain
considerations have to be taken when products are not immediately sold to consumers; inventories of XYZ products are significant and marketing promotions
are cyclical. In the following sections, we first analyse the data to understand the
correlation between sales and weather. Then, we analyse the exposure of XYZ’s
sales in case of environmental changes and propose a hedge strategy to XYZ using
weather derivatives. Finally we conclude this study.
2.2
Understanding the correlation between sales and
weather
Out of the results of this section, conclusions can be extracted which will be used
thereafter to determine which is the optimal hedge. Consistency on correlation
between sales and weather is strictly dependent on the accuracy of data. Sometimes results are extracted or predicted using intuition or simple statistical analysis
30
as a guidance rather than very complex evaluation models. In most of the cases,
econometricians achieve a high level of confidence results when they use long
term series, which are completely stationary, objective, equally observed, equally
measured and other properties that may look more appropriate when treating with
data. This is the case for studies underlying interest rates, GDP and stocks.
Temperature is always measured in the same way, automatically reported,
never affected by any monetary or any other variable of other nature. However,
this is not the case for sales. In most of the cases, companies have a very good
record of sales, but this record is on a yearly or in a quarterly basis, in a monetary
measure which differs across countries, rather aggregate across the time. To conduct a good weather hedging project, sales data should be recorded day to day, in
a unitary basis and product by product; a feature that makes it easier to take into
account influences of consumer tastes, or fashion. For example, a study can yield
a result such that carton bricks used to package liquids are not under the influence
of temperature. Once we can differentiate the bricks used to package fruit juices
and the ones used to package milk, the result might change. It might show that
milk containers are indeed independent on temperature whether fruit juices are
totally dependent.
The data used in this study is either annual, which can not be used to assess
anything related to weather or contained enormous jumps, missing observations
and changes in the unit of measure. Consequently, from all data available, we have
extracted the best time series. In other words, as opposed to weather data, finding
long and consistent time series of XYZ’s products sales has been a hazardous task.
The company keeps better record of those that are more popular and consequently
market leaders. Therefore, our study only takes into account these products and
not others that - even they might be more correlated with weather -, do not have
sales record as good as it would be desired.
2.2.1 The correlation of XYZ’s sales with the temperature
The analysis starts by Switzerland and afterwards is extended to other countries.
The first striking observation we find in Switzerland is the fact that milk presents
a strong seasonality and just slightly less volatility than juice. We conduct an
analysis of the evolution of juice and milk from data gathered from January 1992
to December 1998 included. Fig. 2.1 and Fig. 2.2 depict their main statistics.
In this section, the analysis of correlation between temperature and XYZ’s
sales is performed. However, we need an amount of interpretation because the
relationship between both variables is not straightforward. The main problem
31
Figure 2.1: Statistics for Swiss
juice sales
Figure 2.2: Statistics for Swiss
milk sales
here is that, when we deal with data i.e. temperature and income (sales), both
time series are reported very differently. Temperatures are reported daily and
instantaneously. In addition, it is an objective measure that does not depend on
exchange rates or any other variable. On the other hand, sales are in the best of the
cases- reported in a monthly bases. Unlike temperature they are reported under
accounting standards and are restricted to other criteria such as payment terms and
exchange rates.
Fig.2.3 shows the pattern followed by monthly average temperature in Switzerland and sales of fruit juices in the Swiss market for period 1992-1998. It is obvious that between both time series it exists a certain level of correlation. Monthly
average temperature has been calculated as the average for the 28-31 days in a
month of the midpoint temperatures between the daily maximum and daily minimum in Bern, Zurich and Geneva. Fig.2.4 is a regression line fitted on the scattered plot between the same two time series.
Correlation coefficient between sales and temperature is 0.21 (t-stat: 2.14),
roughly significant.
We argue that the definition of temperature as the monthly average of intradaily average temperature can be consistently improved. Namely, because of
the nature of the product sold by XYZ, intradaily maximum temperature should be
a much better approximation for the variable “temperature”. Fruit juices are most
of the time what in marketing is called an impulse product: sales of fruit juices
can be boosted when the weather is hot and consumers feel that need something to
drink. Maximum temperatures capture much better this effect since average temperatures are severely biased by the intra-daily minimum temperature. In terms of
hedging, taking into account maximum temperatures instead of average is more
efficient for two reasons. First, the hedger has to rely upon the variable which influences the most the product is wanted to hedge. Second, maximum temperatures
32
Figure 2.3: Time series for juice sales and monthly average
temperatures in Switzerland
Figure 2.4: Regression line fitted on the scattered plot between the same two time series
33
Figure 2.5: Time series for juice sales and monthly average
of maximum temperatures in Switzerland
-by definition- will hit the strike price of the weather option much less times than
average temperatures. However, maximum temperatures (Std.deviation: 7.563)
are more volatile than average (Std.deviation: 6.54) which will make the option
more expensive. Moreover, average temperature and maximum temperature are
highly correlated: 0.9932 (t-stat: 109)
Fig.2.5 depicts juice sales and maximum temperatures defined as the monthly
average of daily maximum temperatures in a month in the cities of Bern, Zurich
and Geneva. Correlation coefficient between maximum temperature and juice
sales is now 0.2526 (t-stat: 2.6458). Fig.2.6 is a scatter plot with a positive slope
regression line showing the positive relationship between sales and temperature.
As a consequence of the previous, in the following section we strictly base ourselves on maximum temperatures since they prove to be -a priori- more correlated
with sales.
In the case of milk sales, as it is obvious, the figures are not correlated with
weather. Namely, the correlation is negative though not significant 0.09 (t-stat:
-0.78) with average temperature and 0.07 (t-stat: -0.61) for maximum temperatures. This result implies first that milk is not correlated with temperature since
milk consumption does not only depend on weather as it is not an “impulse” product, but also consumed to feed children or to ingest the required vitamins for an
34
Figure 2.6: Regression line fitted on the scattered plot between juice and temperature in Switzerland
adult. Second, there is a slight bias towards consuming more milk when it is cold
although the correlation coefficient is not significantly negative. This is supported
by conventional wisdom, also. Finally, it is important to notice that the correlation coefficient is smaller when the regression is done with maximum temperatures. This is due to the fact that milk tends to correlate better with cold weather.
Maximum temperatures do not take into account minimum temperatures whereas
average do.
Fig.2.7 and Fig.2.8 show the relationship existing between milk and average
temperature
2.2.2 Lagging data to make it more consistent
The fact that temperature data is reported daily while sales data is only reported
monthly introduces a major data interpretation problem into our study. A monthly
observation on sales is a too large observation. It cannot capture the changes
in weather that take place during that month. Results, therefore, will be biased
because colder days in a month may or may not compensate hot days, but these
effects cannot be regressed with revenues because sales can only be quantified in a
35
Figure 2.7: Time series for milk sales and monthly average
temperatures in Switzerland
Figure 2.8: Regression line fitted on the scattered plot between milk and average temperature in Switzerland
36
monthly basis. There is another bias effect, the fact that sales reports are somehow
delayed or differently reported across the time.
The way to handle this issue is by doing a lead/lag study. We have studied
correlation coefficients supported by using regression significance tests lagging
one of both variables in order to detect if the effects of weather are delayed or
foreseen with respect to sales. We have also changed the period for which monthly
average maximum temperature is calculated. Both methodologies have led to
meaningful results which rely on further interpretation.
As previously explained, in the following, when we refer to temperature,
monthly average of maximum temperatures are used.
Lead/lag study
Temperature series have been delayed (lead) and brought to the present (lag) from
one to six periods of one month each. For example, lead 1 means that we are re'
gressing juice sales of month } with the temperature of month } . If this
relationship presents the highest correlation coefficient, we would say that temperature is leading juice by one month or juice is lagging temperature. In such
a case, we would claim that juice is dependent on temperature since the former
reacts according to the latter.
Table 2.1 shows correlation coefficients with their t-stats for all the lag variables. Max-T is the contemporaneous variable. Table 2.2 contains correlation
coefficients for leading variables.
Temp
Juice
T-Stat
Maxlag6
-0.31
-2.47
Maxlag5
0.07
0.64
Maxlag4
0.33
3.59
Maxlag3
0.55
7.50
Maxlag2
0.69
11.12
Maxlag1
0.51
6.65
MaxT
0.26
2.76
Table 2.1
Temp
Juice
T-Stat
MaxT
0.26
2.76
Maxlead1
-0.03
-0.26
Maxlead2
-0.37
-2.83
Maxlead3
-0.52
-3.81
Maxlead4
-0.62
-4.39
Maxlead5
-0.52
-3.82
Maxlead6
-0.27
-2.16
Table 2.2
Results looking at Table 2.1 and Table 2.2 seem contradictory. On the one
hand it is appreciated that contemporaneous correlation coefficient is one of lowest. This fact confirms our hypothesis that there is an important chronological
37
issue underlying our data. The variable with highest correlation is the Lag-2 followed by Lag-3, both with t-statistics for their correlations far above the rest. On
the other hand, we can appreciate that almost all coefficients (except for Lead-1)
are significant, being difficult to draw a conclusion.
To solve the previous issue we have performed cross-variable regressions, taking group variables and dummying for the rest. The regression with the strongest
explanatory power seems to indicate that Lag-2 and Lead-4 are the most powerful
variables when used to predict juice sales. Results of regression are in Table 2.3.
Dependent variable is juice fruit sales. Another regression with higher R-squared
(accounts for explanatory power taking into account the number of variables introduced) is presented in Table 2.4. In this regression, however, the only significant
variable is the second lag of temperature.
Variable
MAXLAG2
MAXLEAD4
R-squared
Adjusted R-squared
S.E. of regression
Sum squared resid
Log likelihood
Durbin-Watson stat
Coefficient
0.903768
0.325246
0.489416
0.482698
4.482117
1526.792
-226.6716
1.914687
Std. Error
0.039026
0.039547
Mean dependent var
S.D. dependent var
Akaike info criterion
Schwarz criterion
F-statistic
Prob(F-statistic)
t-Statistic
23.15788
8.224298
16.73093
6.231764
5.863374
5.923803
72.84920
0.000000
Prob
0.0000
0.0000
Coefficient
-0.26378
0.977727
0.010485
0.230223
6.843038
0.549176
0.521433
4.132521
1110.053
-196.0541
1.998306
Std. Error
0.141058
0.231522
0.253062
0.219335
4.252555
Mean dependent var
S.D. dependent var
Akaike info criterion
Schwarz criterion
F-statistic
Prob(F-statistic)
t-Statistic
-1.87006
4.223049
0.041432
1.049636
1.609159
16.30407
5.973707
5.744402
5.905008
19.79514
0.000000
Prob
0.0660
0.0001
0.9671
0.2978
0.1124
Table 2.3
Variable
SWT
SWTLAG2
SWATLEAD
SWATLEAD3
C
R-squared
Adjusted R-squared
S.E. of regression
Sum squared resid
Log likelihood
Durbin-Watson stat
Table 2.4
38
These results are really surprising. If we take Lag-2 as the true temperature
variable we should accept that temperature is lagging juice sales. In other words,
revenues look to be independent of the weather. Lead-4 presents similar difficulty:
it is hard to believe that it takes 4 months for sales to react (or to be reported)
when there is a change in temperature. XYZ marketing division disagreed with
the Lead-4 issue.
Sales at XYZ are instantaneously reported. However, they gave strong support to the fact that temperature could be lagging juice sales by two months (Lag2). According to them, fruit juices manufacturers anticipate weather. Basically,
weather has patterns that repeat themselves across years. Juice producers anticipate these seasons and systematically buy the same amount of XYZ product.
If Lag-2 is the true variable relating weather and juice sales in econometric
terms, we can conclude that there is no true relationship between both. At least
there cannot be any “causality” effect. However, we argue against this theory.
First of all, just by looking at the data, it is evident that juice sales and temperature
evolve somehow together. In addition, if it was true that juice producers anticipate
weather and they buy XYZ product beforehand, when it comes a cold summer, not
all packages would be sold and next XYZ product order would be reduced.
Another possible explanation is that changes in daily maximum temperature
can not be captured by monthly data of sales. In the next step, we have defined
monthly temperature in a different way: We have considered monthly maximum
temperature the average between the 15 day of a month to the 15 of the following month. In this case, we leave a fifteen-day lapsus that helps to adjust the
fortnight chronological effect. In this case, we consider that the average tempera'
ture for month } , is the average temperature from 15 of } to 15 of } . When
we correlate this temperature series with juice sales (including the leads and lags),
the results obtained are the following. Table 2.5 and Table 2.6 present the results.
Temp
Juice
T-Stat
Fift
0.65
10.01
Fiflag1
0.64
9.76
Fiflag2
0.43
5.10
Fiflag3
0.19
1.92
Fiflag4
-0.21
-1.75
Fiflag5
-0.46
-3.43
Fiflag6
-0.58
-4.15
Table 2.5
Temp
Juice
T-Stat
Fift
0.65
10.01
Fiflead1
0.41
4.79
Fiflead2
0.09
0.893
Fiflead3
-0.18
-1.50
Table 2.6
39
Fifleaad4
-0.46
-3.48
Fiflead5
-0.58
-4.19
Fiflead6
-0.56
-4.06
Figure 2.9: Time series for juice sales and monthly average
temperatures from 15th in Switzerland
As tables show, the highest correlation coefficient is the most recent. It equals
0.65 (t-stat: 10.01). The positive sign means that the hotter the weather the more
juice is being sold in the market. This is a very important result: the contemporaneous coefficient shows a strong and significant correlation with weather, which
implies that juice sales are influenced by weather.
Consistent with this result, the next significant variable would be the first lag.
Since temperature is now, shifted backwards by 15 days - by definition of the
variable -, correlation relies on the next month temperature as there is only half
a month of delay. Fig.2.9 and Fig.2.10 shows strong correlation between juice
sales and contemporary temperatures defined as of from mid-month to the next
mid-month.
Finally, we have checked the result if we define the monthly average as of the
average from twentieth day in a month to the following twenty. Results have been
disappointing.
In this case, contemporaneous correlation coefficient is non-significant, the
most significant is the second lag of temperature.
Correlation results between twentieth day in a month to the following one are
presented on Table 2.7 and Table 2.8.
40
Figure 2.10: Regression line fitted on the scattered plot between juice and average temperature in Switzerland
Temp
Juice
T-Stat
Twth
0.12
1.12
Twlag1
0.41
4.81
Twlag2
0.64
9.66
Twlag3
0.64
9.64
Twlag4
0.45
5.56
Twlead3
-0.57
-4.14
Twlead4
-0.58
-4-21
Twlag5
0.21
2.10
Twlag6
-0.17
-1.47
Table 2.7
Temp
Juice
T-Stat
Twth
0.12
1.12
Twlead1
-0.19
-1.57
Twlead2
-0.48
-0.56
Twlead5
-0.46
-3.43
Twlead6
-0.15
-1.25
Table 2.8
2.2.3 Country specificity
In this section we examine the main characteristics of weather for the rest of countries emphasised on this research: France, Germany, U.K. and Italy. The focus
is set on two points: Firstly, we observe specific characteristics of every country
weather. In the following sub-section, we explain the interactions and correlations
between the weather of each country. The latter will allow us to predict whether
it exists natural hedging due to negative correlation in the weather in Europe, i.e.
41
to which extent is sunny in Italy while is raining in Switzerland. Secondly, examine for every country the effects of the mean reverting pattern, whether abnormal
weather conditions in a specific season are compensated when looking at the entire
season.
In the US, where weather derivatives are traded in a standardised market, the
instruments traded have underlying for eleven specific cities. Nowadays, there
are weather options traded for cities like Atlanta, Chicago, Cincinnati, New York,
Dallas, Philadelphia, Portland, Tucson, Des Moines... Therefore, the geographical
share of a business market is rather important. In most of the cases, firms are not
running businesses in only one specific location. In our opinion, this is the main
reason why the market for weather options has not been fully developed as some
market makers and analysts predicted when this industry was born in 1997. This
brings us to the problem of ‘basis risk’ explained before (see section 1.4). For
farmers with crops based in a specific region, or skiing resorts, weather derivatives
are very useful. However, for those businesses that produce, sell and offer services
in other locations, either nationally or internationally, weather options may be
more difficult use and to price.
The reminder of this section is a review of the main trends and relations of
weather in different regions. Let us now take a look at the weather dynamics in
different cities.
France
We use monthly data of average temperatures for Paris, Bordeaux, Lille and Marseille. Fig.2.11 shows the wavy mean reverting pattern of the weather in the four
locations:
The weather in the four cities present strong degree of correlation. This feature
is relevant to examine the natural hedging and it will therefore be discussed in the
following sub-section. Table 5.1 presents the correlation matrix for the French
cities, which it shows to be high. Again, these results have a strong implication:
up to now, when we use the temperature of a country for studying its behaviour
to implement a hedging strategy or to measure the correlation with sales, we will
use the mean of all cities with data available or even just one of the cities knowing
that the bias introduced is small due to the strong correlation within all cities. To
be sure that the bias we introduced to the data by doing so is the minimum as
possible, we approach as follows: when we take the temperatures of only one city,
the ones taken will be those of the biggest city, where by assumption most of the
consumption of company XYZ is concentrated.
42
Figure 2.11: Monthly average temperature from January
1989 to December 1998 in some cities in France
Paris
Marseille
Lille
Bordeaux
Paris
1
0.977885
0.995398
0.986574
Marseille
Lille
0.977885 0.995398
1
0.969289
0.969289
1
0.980798 0.975993
Bordeaux
0.986574
0.980798
0.975993
1
Table 2.9
Looking at the data by months we appreciate a bell shape, whose highest
point lies in August. Traditionally for these cities, the coldest month is January,
followed by December. After it comes February, then November and so forth.
Fig. 2.12 is an average of the monthly temperatures. It is important not to confuse this bell-shape with a normal distribution. Weather, especially temperatures,
is usually not normally distributed: distributions are rather symmetric around the
mean; however kurtosis is usually substantially lower than three, meaning that
tails are flater than in a normal distribution. In other words, average temperatures
do not reach extreme values (in our case, monthly average). Jarque-Bera coefficient is slightly larger than the rejection value of 5.99, meaning that the hypothesis
of normality cannot be accepted. Histogram for French monthly temperatures is
depicted below (Fig. 2.13)
As seen on section 1.5.1, the weather presents a mean reverting pattern drifting to its historical mean as it is shown in the fig.2.11. For the period studied, (the
43
Figure 2.12: Average
temperature in France
monthly
Figure 2.13: Statistics for average
France temperature
period that have an impact in XYZ sales), the year averages remain unevenly distributed around the mean: we cannot extract the conclusion that after a single cold
year it comes a warmer year. On the other hand it is clear that after some cooling
season will arrive sooner or later a warming one. This result is also extremely
important: it means that firms hedging long terms bear much less risk than firms
hedging only one season. One easy way to prove the fact that sooner or later, a
cool season will compensate a warm one is to look at autocorrelation. The correlogram for temperatures presents a reversion of the sign. Exhibit 1 (Appendix)
shows the mean reversion pattern.
As we can see on Exhibit 1, all coefficients are significant. Looking at the
stars in the first column, we realise of the seasonality every 12 months.
We have run a similar study
taking seasonal averages. Summer season is un#
#
derstood to be
from
June,
1
to September 30 . Winter is from November 1 #
to March 31 . As also mentioned before, we do not have evidence about how
summers are correlated with winter seasons to compensate for the mean reversion:
What it is possible is to lag both time series for winter and summer temperatures to see if their correlation presents some kind of lead/lag characteristics. We
have obtained the following result:
Correlation
Summer/Winter
Summer lead +2
-0.059705
Summer lead +1
-0.851630
0
0.328065
0.298831
Summer lag +1
Summer lag +2
0.143310
Table 2.10
44
According to these correlations, summer is negatively correlated with one season difference with winter. Now we have empirical evidence that after a hot summer is possible to have a cold winter.
Germany
In Fig. 2.14 we present the average temperature for 4 cities in Germany. The study
has been done following the same procedures as the one used for France.
Figure 2.14: Monthly average temperature from January
1989 to December 1998 in some cities in Germany
Temperatures for a few German cities show very close and almost similar
paths. Just for consistency we calculate correlations. Results are extremely high,
reaching almost 1 for most of the cases (displayed in table 2.11). Once the tendency line is fitted across the temperatures, a given trend cannot be inferred. Also,
there is not any apparent heating at all.
Looking at its distribution the hypothesis of normality is also rejected, t-stat
for kurtosis is 2.41, meaning that the difference with the t-stat to its normal value
of three it significantly differs from zero. Jarque-Bera also exceeds the critical
value of 5.99. Carefully checking Fig. 2.16, there is evidence for Germany that
temperatures tend to be more evenly distributed around the mean of 13.9 | C, one
year up and the following down and so forth. As a consequence we will have large
and significant values for the autocorrelation. (Exhibit 2 in appendix)
45
Berlin
Frankfurt
Köln
Leipzig
Berlin
Frankfurt
1
0.989888
0.989888
1
0.992175 0.995963
0.997915 0.991942
Köln
Leipzig
0.992175 0.997915
0.995963 0.991942
1
0.993401
0.993401
1
Table 2.11
Figure 2.15: Yearly average
temperature in Germany
Figure 2.16: Statistics for average
Germany temperature
The seasonal study for Germany yields an interesting correlation between
summer and winter. Correlation between seasons is low and positive 0.34.
It is important to focus on the lead/lag correlations. The correlogram for Germany has a different signal for every lag, alternatively changing from positive to
negative. In other words, it could be the case that after a hot summer, there is a
tendency for a cold winter or a cold summer. It is interesting to see that the largest
coefficient is the one for a leading summer in two seasons (Table 2.12).
Summer/Winter
Correlation
Summer lead +2
0.7466587
-0.7152580
Summer lead +1
0
0.3059276
Summer lag +1
0.1724454
Summer lag +2
-0.3387920
Table 2.12
United Kingdom
The United Kingdom follows the same pattern as the previous countries studied.
The following table shows the matrix correlation of four cities (Bristol, Manch46
ester, New Castle and London Heatrow Airport) in the U.K. Agian, we can concluded that temperatures are very correlated in the U.K.
The analysis of the temperature lead us to the same conclusion as reported to
France and Germany, but those results are not reported here.
Bristol
Manchester
Bristol
1
0.995118
Manchester
0.995118
1
New Castle
0.985027 0.988767
London Heat. 0.995640 0.993720
New Castle
0.985027
0.988767
1
0.986947
London Heat.
0.995640
0.993720
0.986947
1
Table 2.13
Italy
In Italy due to problems to obtain quality data the study has been restricted to
only two cities and for a shorter period of time. The cities chosen are Milano and
Napoli, that have a very different climate. The result are consistent with the ones
observed until now. Correlation between the two is 0.9536.
The results obtained for the studied European countries lead us to conclude
that there is a strong correlation between the temperatures of these countries. Natural hedge inside countries is therefore rejected.
2.3
Market exposure of XYZ’s sales under changing
environmental conditions
Company XYZ’s sales experiment fluctuations across time and they are not dependent on macro-economic data. To test this, we study the cross-correlation of
Swiss fruit juice sales with three macro-economic variables for Switzerland: the
rate of call money, the Swiss quarterly GDP and the interbank offering interest
rate. Table 2.14 shows that there is not correlation among all these variables (tstat is not significant). Assuming that distribution capacity of the firm remains
invariable, at the end of the producing chain of soft drinks and fruit juices, there
is only one major variable that can explain this finding: temperature. Changes
in temperature introduce a seasonal pattern on XYZ’s sales that can be explained
basically by two factors. The first, are changes in consumer demands. Due to a
47
warmer or colder weather, end consumers may change their preferences towards
drink consumption. The second is, drinks and fruit juices producers anticipate
this behaviour and concentrate all the bottling/packaging around the period when
consumption is highest. This fact could explain why for several products we only
have observations of sales once a year in the data available.
Call Money Rate Swiss GDP
Juice
0.15
0.04
t-stat
1.14
0.30
Interbank Offer. Rate
0.12
0.92
Table 2.14
It is worth to recall that XYZ sells primary material that will be used for
the production of packages, bottles and closures in appropriated machines for its
clients. So, its exposure to weather variation is influenced by the quantity of the
primary material that it sells and will be directly filled with the end product (juice,
ice-cream, soft drink, etc.). In this way, it depends on how much its clients produce to establish the rate of growth which depends on factors such as price of raw
material, market share, economic cycle and seasonal pattern. A firm assuming a
seasonal pattern for the weather can find itself under a turmoil when an unpredictable event such as El Niño might abnormally influence temperature and cause
a dramatic decline in the demand of goods and consequently in the production.
This situation is realistic for products that have a very short period of consumption, for example fruit juices and ice-cream but not so much for those with longer
periods of consumption such as frozen foods, mineral water and milk. Another
point worth mentioning is the fact that in general terms we are more interested
in possible changes in the weather during the summer or winter seasons rather
than in spring or autumn. The main reason is the significant impact of these two
seasons on the revenues as it is in that period they have maximum and minimum
sales for certain products.
Of course, there is also a matter of fashion regarded as independent from
weather. That is, from year to year, different final products (drinks) are differently advertised. Sometimes, new products appear in the market, as for example
ice tea irrupted into the market being a great success whereas ice-tea has never become as popular. Fashion also affects XYZ’s revenues, but not in great magnitude
since when fashion changes, consumers substitute they drinks, but XYZ products
still have the scope to remain invariable to their change. Furthermore, there is not
a way to control this variable. In the following section, we assume that XYZ’s
market share is big enough and is not affected by these trends.
48
Going backwards to the market exposure, it is self-evident that there are numerous rules that govern “XYZ-PF-product” and “XYZ-C-product” sales, as in
every other business; namely competition and preferences of consumers for other
brands not packaged by XYZ. Nevertheless, for the purpose of this study, we focus on the environmental conditions, the weather (proxied by average monthly
temperature and maximum daily temperature in some cases).
There is evidence, supported by previous studies (Enron, Migros), that icecream, soft drinks and dilutables (such as sirops and packaged sodas) are strongly
positively correlated with temperature. Support for this is the fact that some of
these products can only be found in supermarkets during the summer season. Following, with a much smaller correlation, there are fruit juices. Although common wisdom shows evidence that milk is independent of the temperature or perhaps correlated with coldness, the production of milk follows a regular sinusoidal
shape, attaining its maximum in April/May. Hence, still independent of temperature, milk may present a strong seasonality for “XYZ-C-product”. A final consideration for milk is that due to its non-durable character, it is easy to assume that
packaging for final consumption is regular and constant across all the year.
Not all revenue volatility comes from the demand side. A second source of
uncertainty is related to the supply side. Producers of natural fruit juices have
exposure to weather, especially to natural weather catastrophes. However, the
effect of this for XYZ’s sales is unclear and not so evident. Let us take the example
of the orange juice producer: if the juice producer does grow her own orange crop,
she will collect a smaller crop than expected, if the weather has been bad for that
year. But she can still buy oranges on the open market, perhaps incurring to some
extra costs, but she can keep the volume of orange juice produced unchanged. If
this happens, she will still need the same amount of packaging. Given the case
that our producer buys regularly the oranges from outside contracted farmers (as
it is usually the case), a similar reasoning applies. As a consequence, it is difficult
to quantify the relation with the number of cartons sold to package fruit juices and
temperature.
Taking into account the effect of inventories is a crucial point. Intuitively, we
might think in the exposure of XYZ products by looking at the total amount of
sales to the end-consumers reported by all the retailers because everything that
XYZ produce will be sold to the end-consumer sooner or later. However, XYZ
does not sell directly to retailers, which could represent a straight relationship
between XYZ sales and end-consumer’s consumption. In fact, XYZ sells just a
very small fraction of its products to its distributors that supply directly to endconsumer and all the rest is sent to distributors that store preform products that
49
Figure 2.17: Monthly average temperature and maximum
monthly temperature
will farther be sold to retailers. The latter might store their preforms and keep
them between certain level (buffers) which could not reflect the sales of XYZ together with the sales of the end-consumers. Distributors have different warehouse
policies ending to change the correlation of end products and XYZ production
even if demand is considerably increased during an immediate warm front. Thus,
understand the methodology used by distributors to store and maintain their level
of products is essentially to find how much XYZ sales are related to changes in
demand and the correlation them due to changes caused by the weather.
Finally, we elaborate a deeper analysis on the maximum and average temperatures. Both maximum and monthly average temperatures correlate almost in the
same way with XYZs revenues, specially whis taken the second-lag which has
been demonstrated to be the most correlated explanatory variable for both series.
That is to say, if XYZ products are weather sensitive, meaning by that they are
mostly consumed when the weather is hot, revenues should correlate best with
maximum temperatures instead of average. However, this is not true because the
correlation -in Switzerland at least- between average and maximum temperatures
for one month period is higher than 0.99, as it is shown in Fig.2.17.
Nevertheless, for the hedge strategy it does make a difference. Even if we
consider average or maximum temperature, the strike will be set accordingly, since
the pay-off of the weather option depends on the number of times the temperature
hits the strike. An option written on the maximum temperature will hit the strike
more times than another written on the average temperature.
50
2.4
The hedge procedure
In the following sections, we will describe the first steps in the analysis of the
hedging strategy of weather derivatives. First, we will have a look in the existence
or not of natural hedging and explain the different weather derivative instruments.
At the end, we suggest a hedging strategy.
2.4.1 Natural hedging
The golden rule for a risk manager is to have diversified risks in order to avoid major losses. Due to a turmoil of any nature, a company may have a large amount of
losses. However, if the business is conveniently diversified across regions, products, currencies, a particular loss would not mean an automatic bankruptcy of the
firm. In the case of XYZ and because of the purpose of our study we will threat
diversification from a geographical point of view. What is more important than
just diversifying is making sure that the different area in which the diversification is structured are not correlated among them: it is of little use for an investor
in emerging markets to diversify across different emerging markets government
bonds, given that when there is a shock in one of these countries the risk quickly
spills to other countries.
Weather hedging is a similar case. If the company has not spreaded it sales
across countries with uncorrelated climates, then, the company is not naturally
hedged and will have to spend more on weather hedging. On the other hand, if
climates of company’s market are completely uncorrelated, positive and negative
shocks in each of them will compensate. Finally, it is the case of negatively correlated weather. In such a case, the presence of a negative shock, for example an
exceptionally cold summer for a soft drink producer, is offset by an extraordinary
warm summer in other region. The feature of negative correlation is rare, mainly
when we consider the same period of time for both regions.
Findings for company XYZ is that because of its global presence, it should
have attained this level of “climatic diversification”. Yet, there are two reasons
why we can not conclude and continue investigating. The first, is that sales are
in a 75% concentrated in Europe, where simple intuition tells us that weather is
not extremely different. Fig. 2.18 is a distribution of geographical distribution of
sales.
The second is the illiquidity of the market for weather derivatives. The difficulty to find counterparts in other continents than North America and Europe,
makes technically impossible the use of weather options in a global view. In the
51
Figure 2.18: XYZ market segment
following we assume a scenario containing the five countries for which the emphasis is put on this study.
Assuming that weather, -specifically temperature- affects end consumers’ tastes,
a company can be naturally hedged only by two means. The first, is the one just
seen above: no correlation across different countries temperatures. The second
way can be fulfilled by two different ways: i) different products are sold in different climatic areas: even if temperatures were correlated between this areas, the
sales are still uncorrelated; ii) tastes change differently in different markets due to
‘cultural reasons’. An example for that is the market for ice-creams. In southern
Europe countries such as Spain, Italy, Greece, ice-creams are not longer consumed in winter/cold seasons whereas in Scandinavia ice-cream is still consumed
in a smaller extent.
Data availability and the context of the study pushes us to assume that for the
five countries studied the share of products sold is similar enough and consumer
preferences are the same and remain constant through the period of the study.
Therefore, we focus the investigation on natural hedging upon the first source i.e.
correlation across different countries’ weather. Further investigation about the
second one is pure marketing analysis and can be done using data from external
sources as Nielsen AC, Zenith, etc.
Correlation across different cities in the same country
The starting point was to assess the correlation between different cities in the same
country. The result was that for the five countries studied, the weather in its cities
is strongly correlated. Hence, there is not natural hedge for the company XYZ for
52
the fact of having a diversified market within these countries.
For Switzerland, a glance to the graphic of the temperature series (average
daily temperatures for Geneva, Bern and Zurich from 1959 to 1999) shows the
large correlation existing between the three. Fig. 2.19 is a representation of the
data sample from 1994 to 1999. It is possible to see the close track of temperatures
in Switzerland. The correlation matrix yields:
Zurich
Geneva
Zurich
1
0.972455
Geneva 0.972455
1
Bern
0.989265 0.983690
Bern
0.989265
0.983690
1
Table 2.15 - (t-stats significants at 0.01)
Figure 2.19: Monthly average temperature for some cities in
Switzerland
Correlation tables presented below are the final proof of very high correlation.
One interesting feature is the stationarity of weather pattern. If we calculate the
same correlation matrix for shorter samples, we obtain almost the same results.
The following correlation matrices have been taken from 1990-1999 and 19971999 respectively:
53
Zurich
Geneva
Bern
Zurich
1
0.9975997 0.988685
Geneva 0.975997
1
0.988448
Bern
0.988685 0.988448
1
Table 2.16 - (Data from 1990 to 1999)
Zurich
Geneva
Zurich
1
0.976079
Geneva 0.976079
1
Bern
0.988061 0.989164
Bern
0.988061
0.989164
1
Table 2.17 - (Data from 1997 to 1999)
In France, Fig. 2.20 shows a close path between temperatures for four different
cities. In France, it is possible to see more divergence than in Switzerland. This is
explained by the size of the country-France- which allows for warmer cities than
Switzerland. However, correlation is still exceptionally high among them.
Correlation across different countries
We have just shown how there is not natural hedge inside countries. Now we
repeat the same procedure across the five countries. In case the weather be non
correlated, or even negatively correlated, there would exist natural hedging. Previous results must give the first clue that weather across European countries is
not independent. In fact, there are cities like Paris closer to cities abroad (Geneva,
London, Frankfurt) than to cities inside the country like Marseille. Indeed, we can
see in Fig. 2.20 that all countries temperatures evolve together. For more detail
the correlation matrix is:
AvFrance AvGermany AvSwiss
AvUK
AvFrance
1
0.984036
0.990034 0.984123
AvGermany 0.984036
1
0.984476 0.972476
AvSwiss
0.990034
0.984476
1
0.967101
AvUK
0.984123
0.972476
0.967101
1
Table 2.18 - (all t-stats being higher than 59)
54
Figure 2.20: Average temperature between different countries
To be more consistent with the correlation study we have performed a deseasonalisation of the data. Indeed, temperature is affected by seasons. To take the
effect of seasonality out of the data, we have divided every temperature observation by the average of temperature of that month for the whole sample of data.
This monthly average accounts for the “season” and it is calculated as the average
of a particular month (the “season”) of every year during all the temperature time
series (1989-1998). Then we have subtracted 1 to have the data on basis 0.
~g€
g€
ƒ‚
„m…4† „
‡
'
ˆ=‰
Fig.2.21 depicts on a basis of 0, the detrended series for four countries. On
basis 0, temperatures show their departure above or below the “seasonal” average.
A strong correlation is still evident on this figure. Previous correlation coefficient
studies have been performed obtaining high t-stats for those coefficients.
France detrend
Germany detrend
Swiss detrend
Germany detrend
0.80 (110.12)
Table 2.19
55
Swiss detrend
UK detrend
0.762 (94.35) 0.833 (123.55)
0.7371 (86.8) 0.787 (103.22)
0.549 (49.41)
Figure 2.21: Detrended temperatures series for 4 countries
Results of this detrended are presented in Table 2.19 (t-stats in parenthesis).
All these results show strong similarity between countries in Europe. As a consequence, we can claim evidence of the non-existence of natural hedge in XYZ
market.
2.4.2 Weather Derivatives Instruments
The basic trade inherent in weather related risk management products is indexed
on the Heating Degree Day (HDD), a widely used measure for the relative “coolness” of the weather in a given region during a specified period of time. HDDs are
calculated using temperature data provided by the National Weather Service1 . The
weather risk management product class includes caps, floors, collars, and swaps
with payoffs defined as a specified currency sum multiplied by differences between the HDD level specified in the contract (i.e. the strike) and the actual HDD
level which occurred during the contract period. Weather is ever changing and
unpredictable and it is not necessary to accurately predict the weather to protect
the business from the weather. The key is to find counterparties who are better
able to absorb weather related risks.
Before starting to design a hedge strategy, one might introduce the description
of the diverse weather derivatives contracts traded by the market, a brief view in
1
A complete relation of the national weather services in each country can be obtained from the
World Meteorological Organisation (WMO)-http://www.wmo.ch
56
the pricing methods that participants have applied and some techniques to hedge
with weather derivatives.
Swaps
Swaps are privately negotiated financial contracts in which two parties agree to
exchange, or “swap”, specific price risk exposure over a predetermined period
of time. They are OTC instruments that can be customised to meet a particular
set of needs. Swaps allow for strategies designed to protect against market price
fluctuations. There is no “cost” for swap. When used in connection with floating
price energy contracts, swaps afford a buyer and seller protection from adverse
price movements, in exchange for giving up the ability to capitalise on benefit
price movements. There are no standardised swap transactions. Nonetheless,
most transactions involve an exchange of periodic payments between two parties,
with one side paying a fixed price and the other side paying a variable price. Specific terms of swap agreements - including the fixed of the weather index and its
floating price reference, the terms of the contract, and the quantity to be hedged
- are established by the two parties involved, and can vary, subject to their specific needs and objectives. A swap contract is settled in cash, usually against as
agreed-upon market price index, and is customised as to volume, timing, location,
seasonality, and swing.
D
Swap Payoff =
a
f
}
C021436 %Š87
*
X
0
f
}
‹021436 %Š`7
Figure 2.22: Swap Pay-off
57
*
Œ
!
Caps and floors
Caps and floors are options which provide the right, but not the obligation, to
enter into a long or short position at a specified price. Caps and floors are similar
to swaps since they provide price protection at a predetermined level. However,
caps and floors are different from swaps in that allow producers and end users to
benefit from favourable price changes.
The buyer of the cap or floor pays an up-front cash premium for this price
protection. With caps and floor purchases, all risks are predefined; the premium
paid for the option will always be the maximum “loss” or “cost” incurred by the
buyer. Caps are sometimes referred as “call options”.
Figure 2.23: HDD Floor Pay-off representation
They provide full protection from rising prices. In addition, caps allow end
users to benefit fully from decreases in the relevant index (HDD or CDD).
D
Cap Payoff =
a
f
C021436 %Š87
}
*
X
Floors, on the other hand, are referred as “put options”.For this premium, the
buyer minimises exposure to adverse price movements while retaining the ability
to capitalise fully on advantageous of an upswing of the considering index (HDD
or CDD). An example is the situation faced by the heating gas providers during a
mild winter. A HDD floor will off-set losses due to the decrease in heating use.
Floor Payoff =
a
f
}
C021z36 7%
*
X
58
!
Collars
Collars provide price protection by limiting extreme market movements, forcing
price to move within a defined range. Costless collars are partially “paid for” by
giving up a portion of a favourable price change. No cash premium is involved for
costless or fair priced collars. Collars offer floor protection on commodity sales
prices. In exchange, one has to give up some potential to benefit from favourable
price moves by selling a cap. If index prices move within the specified collar
or range of commodity prices, one will sell the commodity at prevailing market
prices and no payments are made. However, if the index price falls below the
collars’ lower limit, the one will be reimbursed for the shortfall. Correspondingly,
if the index price for the commodity exceeds the collars’ upper limit or if it falls
below the collar’s lower limit, then one must pay the difference. In many ways,
collars are similar to swaps, but the former allows for greater flexibility through
some market responsiveness.
Figure 2.24: Collar Pay-off
D
Collar Payoff = a
f
}
)ab1436 %Š87Ž
*
X
0
f
}
‹021436 %Š87‘
*
Œ
!
Digital Options
Digital options, sometimes referred as “binary options” are options with discontinuous payoffs, which pay a predetermined amount if a certain temperature or
degree day level is reached, or nothing at all. A simple example of a digital option
is a cash-or-nothing call. This pays off nothing if the underlying price ends up
below the strike price at time T and pays a fixed amount, Q, if it ends up above the
strike price. A cash-or-nothing put is defined analogously to a cash-or-nothing
59
call. It pays off Q if the underlying level is below the strike price and nothing if it
is above the strike price.
Figure 2.25: Digital Option Pay-off
Another type of binary option is an asset-or-nothing call. This pays off nothing if the underlying level ends up below the strike price and pays an amount
equal to the underlying level itself if it ends up above the strike price. An assetor-nothing put pays off nothing if the underlying level ends up above the strike
price and an amount equal to the underlying level if it ends up below the strike
price. Fig. 2.25 depicts the pay off of a digital call option.
Compounded option
Compounded options are options on options. There are four main types of compounded options: a call on a call , a put on a call , a put on a call and a put on a
put. Compounded options have two strike prices and two exercise dates. These
exotic options provide the opportunity to enter in a weather derivative contract on
a specified date paying a small up-front fee and avoiding possible future risks.
Compounded options are not so often dealed in the weather derivatives market
but they represent an alternative approach to be hedge within certain characteristics
60
2.5
Designing a hedge strategy using weather derivatives
When still difficult to attribute full causality to the weather, for most data of drinks
studied, distribution of sales across the year is highly correlated to temperature.
After adjusting the time series for some delay basically due to the accounting
method and building-up of inventories by drink producers and focusing on fruit
juices since it is the most reliable data we have obtained-, correlation is +0.70
(t-stat: 11.57)
Figure 2.26: Average maximum temperature in a month in Switzerland versus
swiss sales of fruit juices. In this graphic we can appreciate the strong correlation
between both
In the graph Fig.2.26, apart from the strong correlation, it is possible to see
a downward trend on juice sales. When we design the hedge strategy, we will
account for this negative trend to isolate the effect of temperature on sales.
When hedging, it is very important to bear two points in mind:
1. Season: although OTC weather derivatives can be tailored to user needs,
standardised weather options are traded and thought to hedge seasons. The
seasons mostly hedged are summer (May-September, three months or one
month during that period) and winter (November - March, three months or
one month during that period). Mathematics are not very important here.
61
It is evident from Fig.2.26 that it is in summer (i.e. during hot periods)
when fruit juices are mostly consumed. That means that the sensitive season is summer: a hotter than normal summer can bring extra revenues and
conversely, a cold summer could put the company, or a particular product
under financial distress. For the same reason, it is better to correlate the
revenues with maximum temperature instead of average temperatures since
maximum temperature takes better into account the factor that triggers increased consumption rather than average temperature that is influenced by
minimum temperatures intra-day.
By no means the previous idea means that the company does not need to
care about winters. However, winters are cold by nature, and to hedge winters when the product has proved to be summer sensitive not only reflects
a lack of product strategy but also could be extremely expensive unless a
threshold is detected.
2. Threshold: we could distinguish two different thresholds: temperature and
amount of sales. A temperature threshold exists when for temperatures below a certain value, sales decrease dramatically i.e. putting the company
or the product in high risk. A threshold in sales is defined as the minimum
level of production - independent on temperature and other factors - that
the company needs to keep selling to repay fixed costs or another financial
target set by the financial department.
If there exists strong evidence of both these thresholds, then to hedge during
the season when the product is not sensitive might pay-off. That is because
the hedger knows exactly what is the temperature that would put the company under bankruptcy. As we show in the burn analysis (section 1.6.2), it is
possible to calculate the probability of having such temperature and hence
the price of the weather option hedging against this phenomenon could be
inferred. Comparing the benefits with the costs (price of the weather hedge
option) would determine the final decision for the corporate hedger.
Going back to fruit juice sales, the results are the following:
The warmer the weather gets, the more fruit juices is sold. Furthermore,
the relationship between temperature and sales is not linear: sales tend to
increase exponentially with weather.
Fig. 2.27 and Fig. 2.28 give strong evidence on a natural sales floor at 10
units (Mio. units). Fig. 2.27 is a scattered diagram where each dot relates
62
the amount of juice sold -on y axe- and the maximum temperature -x axe.
We have fit a regression line among these points with R-Sq. = 0.50. The
upside-sloped regression line indicates that the warmer it is, more juice is
sold. It is very important to notice that the relationship is not linear and the
existence of a floor when temperatures are very cold i.e. natural hedge.
Fig. 2.28 ranks every single observation by temperature. For each point, we
have its associate sales point. We can distinguish here a two-tiered pattern:
for hot temperatures correlation is very strong, whereas for temperatures
below 12 | C,sales seem to remain rather constant, showing existence of the
natural floor.
It is possible to appreciate a two-tiered pattern when sales are related to
temperature. Fig. 2.28 yields good understanding on that: for temperatures
higher than 15 | C approximately, fruit juice sales, although very volatile, are
strongly correlated with the temperature. However, for colder temperatures,
a floor at about 10 units exist. This is due to the fact that, there always exists
a residual amount of consumption which is not dependant to weather but to
other factors such as fashion or some wealthy attributes as for example the
habit to drink a glass of orange juice every morning.
Even when it is extremely cold, the sales’ volume remains positive. Fig. 2.29
discriminates the sales in two groups: very hot weather (defined as the
months when average temperature is year mean plus one standard deviation) and very cold (mean minus one standard deviation). We have found
that 23% of 8-year period sales take place in the first group whereas only 3%
take place in the coldest time. However, these findings are biased because
of the existence of more hot than cold months.
Discriminating the groups by the 5 coldest months and the 5 hottest in the
same 1992-1998 period Fig. 2.30-, we realise that 5% of the sales during
that period take place in the 5 hottest months, but a non-negligible 3% do
take place in the 5 coldest. This fact, confirms the existence of the minimum
natural floor in fruit juice sales.
The previous findings indicate that hedging in winter is not necessary, since
temperatures falling below ca. 13 | C still have a positive amount of sales associated.
As a consequence, we structure the weather hedge taking solely the summer
season. Fig. 2.27 shows that the relationship between temperature and sales is
63
Figure 2.27: Scattered diagram where each blue dot relates the amount of juice
sold and maximum temperature; R- Sq.=0.50
rather linear. Figure 2.31, 2.32, 2.33 and 2.34 depict monthly average maximum
temperatures and fruit juices sales in Switzerland just for the summer season, considered here (because of Swiss characteristics) from June to September. Careful
study of the graphs would suggest to take 18 | C as the temperature to hedge. Of
course, the ‘strike’ that is picked to hedge depends on the understanding of the
business and the risk aversion or availability to pay for the hedge. Let us pick for
our example 18 | C.
To have a better understanding of which is the effect of weather and to account
for the need to hedge, it is useful to study the cumulative effects of the time series
during a time period.
As explained in our first draft presentation, we study the effect of weather on
sales by looking at the correlation between the excess temperature and the potential excess of sales it can generate during a whole hedging season. In our case, the
hedging season are summers (i.e. from June to September). After using mathematical techniques to detrend the pattern of juice sales and normalise temperature
in a year period, we are able to construct the relevant graphic. (See Fig. 2.35)
In the previous graphic, excess temperature and excess juice have a correlation
coefficient of 0.1342 (t-stat, 1.31) which is non-significant.
The following step is to take the effect of the sole season we are going to hedge
i.e. summer.
By just taking into account summer season, correlation coefficient increases
64
Figure 2.28: Natural threshold in sales
to 0.2463 (t-stat, 1.4746), still insignificant. Although correlation coefficient has
increased from 0.1342 to 0.2463 just by taking summers, t-stat for that measure
remains as low because the decrease in the number of observations.
The important point here, is to notice that payments for the hedged season are
according to the behaviour of all the season and not just for some days. At the end
of the season the buyer of a floor in CDD receive the amount for the accumulation
of CDD for all the season. Hence, what we should be concerned is on the total
cumulative effect of temperature and sales for the whole season (from 1st June to
30th September). Fig.2.37 represents both measure for every year. At this point,
correlation coefficient between the temperature and the sales has soared to 0.9037,
the t-stat being 6.5144 and hence extremely significant.
Figure 2.29: Volume of sales
Figure 2.30: Volume of sales
65
Figure 2.31: Average temperature
Figure 2.32: Average temperature
In fig. 2.39 excess juice is defined as actual sales of juice in a month minus the
detrended average sales for that month. Excess temperature is defined as the average maximum temperature in a month minus the average of maximum temperature
s of that month during 10 years. An ideal weather sensitive situation would imply
that excess temperature is perfectly correlated with excess juice sales.
Regarding fig. 2.38 we have now taken the same data as in Fig. 2.37, but only
for the observations of June, July, August and September. Both excess temperature and excess juice are represented by the y-axe whereas x-axe does only account
for the number of summer months from 1992-1998 period.
In Fig. 2.39 the boxes are the addition of excess of temperature for the period
June-September in a year. The dark boxes are the addition of the excess juice for
the same four summer months in a year. In this case,(looking at only one season),
we find a stronger correlation of the cumulative effect of weather on juice sales.
66
Figure 2.33: Average temperature
Figure 2.34: Average temperature
As argued before, we will in the following design the hedge strategy for the
revenues. We will assume that XYZ gets one dollar for every unit of fruit juice
sold. The objective here is to hedge against summer maximum temperatures
falling below 18 | C in Switzerland.
For that the company has three basic strategies:
Buy a put: the company pays the price of the option (the fee) to receive
a payment at the end of the season for the accumulation of degrees falling
below 18 | C.
Short a call: in such a risky strategy, XYZ would earn the proceeds of shorting the option (in order to earn the premium) and pay the counterpart for all
cumulative degrees in the summer exceeding, let us say, 24 | C. Technically,
this is not a hedge, but it is a way to benefit from the sensibility of corporate
revenues to weather
67
Figure 2.35: Excess temperature vs excess juice
Enter in a swap contract: in such a strategy, XYZ makes payments for degree days exceeding a certain amount and receives payments for degree days
falling below the same amount
It helps a lot to see what happened historically to figure out the behaviour of
such strategies. Fig.2.37 presents the accumulation of CDD for summers between
1992 and 1999. Leaving the strike at 18 | C, Fig.2.38 depicts the accumulation of
HDD for the same period.
If XYZ, wants to hedge against falling below 18 | C, that is the same as to hedge
on any positive amount of HDD. Looking at Fig.2.39 and assuming that the owner
of the weather option would receive $1 for every HDD, we can have an idea of the
payments received every season. Comparing the amount received with the fees
paid up-front, it is the option of the hedger whether to hedge or not and at a which
strike.
In the case of buying the put in HDD in 1999, the hedge would have been
expensive, because the HDD for 1999 were lower than historically average. The
prediction model we have used to predict weather predicted 52 HDD for 1999.
Every option trader/writer has its own models, but the fees of the option would
have been on that size, much higher than they actually were. If someone had
68
Figure 2.36: Excess temperature vs. excess juice only during the summer
bought that option, the odds were to have lost money on that since cumulative
HDD for 1999 were only 37 HDD.
The second strategy, short a call in CDD, is extremely risky, as all short positions in call options. The reason why is because, - even the company on one hand
is getting the up-front fee of the option and has the revenues naturally ’floored’ in
their lower level -, the weather could become unusually hot. In this case, the payments due could increase extraordinarily and not be compensated by the increase
in production due to the hot weather. As we see in Fig.2.37, XYZ would have
had to pay for 507 CDD, whereas the fees received would have been calculated
upon a milder weather than expected and hence the company loosing money in
this strategy too. It is important to bear in mind, that options are instruments to
trade volatility. With hedging strategies the company is worse off with increases
of temperature volatility and the converse.
A mid-way strategy between both is entering a swap contract. With a swap
contract, the company pays when temperatures are high and hence receives higher
revenues and receive payments when temperatures are low. For company XYZ, a
swap does not seem a very intelligent option since we have seen that revenues are
naturally ’floored’ anyway and temperature can become extreme.
69
Figure 2.37: Excess juice is defined as actual sales of juice in a month minus the
detrended average sales for that month
Figure 2.38: Accumulation of CDD for summers between 1992 and 1999
70
Figure 2.39: Accumulation of HDD between 1992 and 1999
71
Chapter 3
Conclusion and further ideas
3.1
Conclusion
The first part of this study is an overview of weather derivatives. Weather derivatives, or weather options, are used for hedging corporate revenues in those businesses that are weather sensitive. Because of the cost of bankruptcy, the nondiversifiable role of some stakeholders and for tax purposes, volatility in revenues
decreases the value of a company. Weather options, when they are correctly used,
contribute to hedge part of these risks so enhancing the value of a company.
Then, market characteristics for weather options are described. Economy is
strongly dependant on weather. 22% of the 9 trillion USD gross domestic product in the United States is sensitive to weather. In this environment, market for
weather derivatives started in 1997 and more than 1800 deals worth more than
$3.5 billion had already been transacted at the end of 2000. Agriculture, tourism,
utilities, etc are industrial sectors that can take advantage of weather derivatives.
Differently as in catastrophe insurance, where the amount received is lump-sum,
weather derivatives allow to hedge linearly once a pre-defined exposure is determined.
Weather options are priced using forecasts on the weather. Weather, namely
temperature, is a mean-reverting process. Hence, the most common pricing method
assumes that futures temperature follows a pattern based on historical temperature. This does not necessarily means that temperature remains constant during
time. But, forecasting methods, based on historical weather variables can be performed with a level of accuracy statistically significant. Section 1.6 presented an
ARCH (3) model for temperature forecast in Geneva. Once the temperature is
72
forecasted, an assumption of the cumulative Cooling or Heating Degree Days for
a season is done. The last step is to put a price to the predicted pattern. The
inappropriate use of the Black and Scholes model to price weather derivatives
was also discussed. The simple assumption of constructing a riskless portfolio
cannot be done as the underlying - temperature - is not trade. It is worth notice
that CDD or HDD are cumulative variables and thus, weather options have to
be handle as Asian Options . Burn analysis method replicates historical data to
price the amount of CDD/HDD that want to be hedged. Basically, the pricing
is determined by setting the revenue that will be earned in case that meteorological conditions trigger the strike of the option (i.e. strike is set as a number of
CDD/HDD that want to be hedged). The revenue depends on the financials of
every company/product.
The second main chapter of the thesis discusses about a real case study. We
have worked together with company XYZ with real data on their market. Because
of the nature of XYZ activity we have taken maximum temperatures (as opposed
to average temperature) for the length of the case study.
Firstly, XYZ business appears to be sensitive to weather. But this is a very
tricky point. Cause-effect mix together making difficult to assess whether i)when
it is hot, XYZ sells more product or ii)- independently of weather - it is just during the hot seasons that most of XYZ product is sold (i.e. by chance). Conversations with management in XYZ and more sophisticated analysis lead us to
think that XYZ is truly correlated with weather. This analysis consists of lagging
sales/revenues time series 15 days and performing the correlation analysis again.
In this second study, we obtain - for similar levels of significance - a high correlated contemporaneous correlation (revenues/maximum temperature) coefficient,
whereas in the first case, the highest correlation coefficient is obtained when sales
are lagging temperature by two months, a fact that seems quite difficult to explain.
The correlation between the temperature of countries were XYZ has a large
markets are almost perfectly correlated. Even accounting for deseasonalised data
(i.e. takes off the effect of seasons when comparing the temperatures of different
countries), t-statistics for correlation coefficients are for some countries higher
than 100 and not smaller than 49.41. Hence, XYZ is not naturally hedged. ‘Naturally means here, that there is not a diversification in temperature by selling to
different countries since these temperatures evolve so closely together.
Still, XYZs revenues are weather sensitive: Sales increase with hot temperatures. However, - this is the most important finding of our research - when temperatures get sensibly cold, sales do not decrease in the same proportion. In other
words, there exists a natural threshold in sales volume. Even weather becomes
73
extremely cold, there is always a minimum amount sold. An explanation for this
fact is that XYZ products can be considered by some consumers as ‘healthy. As
a consequence, it is not bought by an impulse or instinct triggered by hot weather
any more, but by the desire of consuming ‘health care. If this residual threshold
during the cold period proves to be profitable enough for the company to keep on
business, the hedge with weather derivatives is of scarce use.
For confidentiality reasons, XYZ has not disclosed the relevant financial data
to assess this extent.
It is straightforward to mention that when we depict the sales together with
temperature in a graphic (Fig.2.27), the shape of a fitted line is very similar to
a long position in a call option. Therefore, if any derivatives strategy should be
applied in the company, that would be to sell a weather option. However, it is
well known how dangerous a short position in a call option is. We would not
recommend that to a company we had to advise. The company is naturally hedged,
not because of weather characteristics but due to the product they sell. This is our
recommendation to XYZ.
74
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77
Appendix A
Human Physiology
Physiology- Liquid requirements for the human body
The amount of water (liquid) in a 70 Kg. male adult is about 40 litres or 57%
of his total weight. In the case of a recent born child, this amount can increase to
a 75% or decrease around the elderly age. For obese people, the amount of water
in their body can be reduced as to 45%. Continuing analysing the case of a 70
Kg. male adult, the daily optimal intake of liquid is 2300 ml per day. Two thirds
of this intake is direct drink of water or other beverages; the rest being taken by
the liquid embedded in meals or oxygenation in the hydrogen existing in the food
(150/200 ml).
On the other hand, the human body also loses liquid. at 20 | C, 1400 ml of the
2300 ml entered in the organism, are lost by means of urine, 200 ml by sweating
and other excrements. The 700 missing are lost by common evaporation from the
respiratory apparatus or the skin. However, in very hot circumstances or during
exercise, the sweating volume increases from 1.5-2 litres per hour. All the liquid
lost is usually replenished before 24 hours. The table below relates the liquid lost
according to environmental temperature (data in millilitres).
Insensitive lost-skin
Insensitive lost-respir.
Urine
Perspiration
Excrements
Total
Normal T. 20| C
350
350
1400
100
100
2300
Hot weather( ’ 30| C)
350
250
1200
1500
100
3300
78
Continued exercise
350
650
500
5000
100
6600
Figure A.1: Corporal Temperature
The Fig. A.1 illustrates the relationship between the corporal temperature and
atmospheric temperature, for ranges between 0 | C and 75 | C. Naturally, the exact
shape of the line will vary according to wind, humidity and air conditions.
For our study concerns, we would need an exact relationship of the liquid
needs per every degree of temperature. Unfortunately, there are no consistent
studies that relate both figures. Indeed, there are several studies that try to provide
results but all differ among themselves. However, several facts might bring us into
conclusion:
The adult human body needs at least 2 litres of liquid per day for his/her
ordinary life needs
An adult perspiring due to hot weather or tough and continued exercise can
lose up to 6 litres that will be replenished within 24 hours
An adult individual starts sweating at 25.5 | C when humidity is at 60% or at
30 | C when humidity is 30%
Apart from the 2 minimum litres and adult individual needs to replenish
In this point, which is the essential one, is where we have encountered the
most divergent issue. Of the two basic sources consulted, the results prove to be
consistently different. We state both and take them as a range within it lies the true
value. Also, note that environmental conditions affect such studies dramatically.
79
For every 2 | C of increased temperature, 1000 ml of liquid, which makes
an average of 500 ml. every 1 | C, according to Farreras Rozman (Intern
medicine, volume II, 8 edition)
200 ml. for every 1 | C, according to Harisson (Principles of intern medicine)
Our personal conclusion of this analysis, is that human beings need to increase
in a linear way their consumption of liquids after a triggering barrier lying at 2530 | C, depending on humidity. Since our study focus in European countries near
seaside or lakes, we assume that humidity is on the higher tier, hence we establish
the triggering temperature of increased thirst at 25 | C
80
Appendix B
Exhibits
Exhibit 1: Correlogram for temperatures - France
S
-lag
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
Auto correlation Partial correlation
0.772
0.772
0.420
-0.439
-0.023
-0.462
-473
-0.491
-0.724
-0.068
-0.825
-0.334
-0.713
-0.254
-0.410
-0.060
-0.006
0.031
0.412
0.064
0.728
0.179
0.828
0.044
0.687
-0.060
0.377
0.030
0.434
0.054
-0.418
0.050
-668
0.040
-0.758
-0.012
81
Q-Stat
72.818
94.479
94.547
122.53
188.78
275.56
363.97
385.93
362.67
385.10
455.78
548.01
612.20
613.98
694.53
656.37
719.36
801.24
Exhibit 1: Correlogram for temperatures - France (Continuation)
S
-lag
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
Auto correlation Partial correlation
-0.642
-0.051
-0.375
-0.077
-0.017
-0.108
0.386
0.138
0.640
-0.144
0.736
0.011
0.613
-0.164
0.318
0.013
-0.033
-0.082
-0.368
0.048
-0.596
-0.071
-0.665
0.012
-0.547
0.109
-0.310
-0.043
0.027
0.046
82
Q-Stat
860.57
881.07
881.11
903.25
964.69
1046.8
1104.3
1119.9
1120.1
1141.5
1198.3
1269.9
1318.9
1334.9
1335.0
Exhibit 2: Correlogram for temperatures - Germany
S
-lag
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
Auto correlation Partial correlation
0.792
0.792
0.433
-0.522
-0.017
-0.464
0.460
-0.436
-0.754
-0.224
-0.859
-0.282
-0.733
-0.158
-0.410
-0.060
0.015
0.0160
0.443
0.121
0.748
0.137
0.840
0.051
0.693
-0.056
0.373
0.035
-0.045
-0.015
0.434
0.054
-0.709
-0.036
-0.793
-0.023
-0.666
-0.054
-0.383
-0.170
0
-0.135
0.405
0.097
0.660
-0.177
0.749
-0.022
0.626
-0.092
0.329
-0.104
-0.019
0.085
-0.373
-0.034
-0.617
0.013
-0.685
0.054
-0.574
0.106
-0.315
0.001
0.039
0.131
0.370
0.017
83
Q-Stat
77.217
100.51
100.55
127.27
199.69
294.43
363.97
385.93
385.96
412.04
487.16
582.72
648.47
667.70
667.99
694.53
765.95
856.26
920.48
941.94
941.94
966.47
1032.3
1117.7
1178.2
1195.1
1195.1
1217.3
1278.6
1355
1409.2
1425.8
1426
1449.3
Figure B.1: CDD and Call Strike Price
Figure B.2: CDD Call Strike Price and Pay-out
84
Figure B.3: CDD and Put Strike Price and Pay-out
Figure B.4: CDD adjusted using a polynomial equation
85
Figure B.5: San Antonio temperature and statistics
Figure B.6: La Guardia temperature and statistics
86
Figure B.7: San Francisco temperature and statistics
Figure B.8: Seattle temperature and statistics
87
Figure B.9: Simulation for Seattle - Max. temperature
Figure B.10: Simulation for Seattle - Avg. temperature
88
Figure B.11: Simulation for Las Vegas - Max. temperature
Figure B.12: Simulation for Las Vegas - Avg. temperature
89
Figure B.13: Simulation for La Guardia - Max. temperature
Figure B.14: Simulation for La Guardia - Avg. temperature
90
Figure B.15: Simulation for San Antonio - Max. temperature
Figure B.16: Simulation for San Antonio - Avg. temperature
91