Quantifying Weather Risk Analysis

Size: px

Start display at page:

Download "Quantifying Weather Risk Analysis"

Barry McCoy
6 years ago
Views:

1 Quantifying Weather Risk Analysis Now that an index has been selected and calibrated, it can be used to conduct a more thorough risk analysis. The objective of such a risk analysis is to gain a better understanding of the financial impact weather can have on a particular agricultural activity. This means that one has to look at the probability distribution of risk to understand the frequency and potential magnitude of adverse weather events. To study the probability distribution of risk, one must study the distribution of the chosen and calibrated index. This section outlines three methods of increasing complexity that can be undertaken to perform a risk analysis: 1. Historical Burn Analysis (which is something everyone can do in a spreadsheet) 2. Historical Distribution Analysis (where one can fit a parametric or non-parametric distribution to the historical data to extend the historical distribution beyond the sample size you have) 3. Monte Carlo simulation approach The contract design webtool at the end of the course provides users with a simulation engine that allows the simulation of rainfall data and WRSI. 138 Module 5C Designing Index Based Weather Risk Management Programs

2 Quantifying Weather Risk Analysis Historical Burn Analysis (HBA) Historical Burn Analysis (HBA) is the simplest method of analyzing the weather risk of a particular exposure. It simply involves looking at historical values of the index, which may be based on raw, cleaned, and possibly detrended weather data. Advantages of HBA: Assuming that the data used to calculate the historical indexes are of good quality for the risk analysis, HBA can give a useful and intuitive first indication of the mean and range of possible weather events and their financial impact. HBA is simple and can be easily done using a spreadsheet. It involves calculating the average, standard deviation, maximum, minimum, and different percentiles of the historical index values to understand their distribution and the probability associated with different index values occurring. The ability to attach a confidence level to averages and standard deviations calculated from historical data is an advantage of HBA. These measures can be quantified, so that one can interpret the values in the context of the data sample used. The standard error in the average decreases as the number of years included in the average increases (an increase in a number of data points under consideration makes the results more robust). However, although weather stations often have 3040 years of historical data, the representative nature of older data for today s weather and climate should always be questioned. Before performing an HBA or, indeed, any distribution analysis, the time-series of historical index values should first be checked for any significant trends to ensure the data that is representative for current conditions. Disadvantages of HBA: The disadvantage of HBA is that it gives a limited view of possible index outcomes and may not capture the possible extremes. HBA is overly influenced by individual years in the historical dataset. Therefore, estimating the tails of the index distribution, that hold information on the most extreme events, is difficult. However, the largest historical value is always a good reality check when considering the possible variability of payouts. Figure: Historical Burn Analysis (HBA) Calculating the average, standard deviation, maximum, minimum and different percentiles of the historical index 139 Module 5C Designing Index Based Weather Risk Management Programs

3 HBA does not need specialized software to perform the analysis and it is often sufficient to understand the risk in question. For completeness, however, two more sophisticated approaches are also presented. 140 Module 5C Designing Index Based Weather Risk Management Programs

4 Quantifying Weather Risk Analysis Historical Distribution Analysis (HDA) Historical Distribution Analysis (HDA) involves determining the probability distribution that best fits the historical (possibly detrended) index data. Various standard tests and goodness-of-fit statistics can be used to pick the best distribution from a potential selection. See more notes on statistics below. Advantages of HDA: The advantage of the HDA method is that it can be more accurate than HBA for computing tails of the index distribution. It provides a good understanding of the likelihood and magnitude of extreme weather events and their financial impact. If index values are calculated from historical meteorological data, then ascertaining the probability distribution function of the index can give a better estimate of the expected and variable weather events that can lead to financial losses. Disadvantages of HDA: HDA assumes that the underlying distribution is a correct representation of the data. Therefore, fitting, and putting too much emphasis, on a distribution that does not capture the higher moments of variability can lead to underestimating or misunderstanding the variability and underlying financial risk. As a result, the process can be one of trial and error. Figure: Historical Distribution Analysis (HDA) Example 141 Module 5C Designing Index Based Weather Risk Management Programs

By determining the distribution of the index, the statistical properties of the index can be calculated either by simulation from the distribution (see next section) or analytically, depending on the

5 By determining the distribution of the index, the statistical properties of the index can be calculated either by simulation from the distribution (see next section) or analytically, depending on the type of distribution chosen. Usually, software such as Excel [[AQ: are these program names correct?]] or a statistical programming package is needed to be able to determine whether a parametric or non-parametric distribution can represent the index well. HDA is discussed in more detail in module 7. Some notes on statistical measures and analysis Goodness of fit of a statistical model describes how well it fits a set of observations. Measures of goodness of fit typically summarize the discrepancy between observed values and the values expected under the model in question. There are many goodness-of-fit tests that are beyond the scope of this training course. Much information can be found on the internet (see Mathworld or Wikipedia ) or in statistics books. Such measures can be used in statistical hypothesis testing; some measures are described below: Kolmogorov-Smirnov test: used to test whether two samples are drawn from identical distributions Anderson-Darling test: used to test whether there is evidence that a given sample of data did not arise from a given probability distribution Pearson s chi-square test: used to test whether outcome frequencies follow a specified distribution Maximum likelihood estimation ( MLE ) is a popular statistical method used for fitting a mathematical model to some data. The modeling of real-world data, using estimation by maximum likelihood, offers 142 Module 5C Designing Index Based Weather Risk Management Programs

6 a way of tuning the free parameters of the model to provide a good fit. The method was pioneered by Sir R. A. Fisher between 1912 and For a fixed set of data and underlying probability model, maximum likelihood picks the values of the model parameters that make the data more likely than any other values of the parameters would make them. 143 Module 5C Designing Index Based Weather Risk Management Programs

7 Quantifying Weather Risk Analysis Monte Carlo Simulation The most complicated and sophisticated approach to estimating the true distribution of an underlying index is to use a simulation approach. This approach is only recommended to those that have access to and knowledge of statistical programming packages and have a good understanding of time-series analysis and Monte Carlo simulation. The philosophy of a simulation approach is that once a distribution is identified to represent an index, constraints associated with the length of the historical data records are no longer valid. Thus, thousands of realizations of the index can be simulated to estimate the index statistics to any arbitrary degree of statistical accuracy. Index values can be simulated statistically by drawing samples from the chosen distribution to generate large numbers (years) of artificial index values from which index statistics can be calculated. The method is particularly good for cumulative indices, such as cumulative seasonal rainfall and indices that depend on a weather parameter crossing some threshold (e.g., a three-phase deficit rainfall approach (Module 6, book C). In addition, it is valuable for contracts that depend on several weather variables, and where the correlation between these variables can be included in the simulation process. Simulating the underlying weather data used to construct the index directly, and then calculating the synthetic index values from that simulated data, proves to be the most complex way to analyze an index and the weather risk it represents. Nevertheless, this approach could offer some insight into the possible extremes of weather and, therefore, index variability that have not been observed in the historical data. A statistical model can be built for daily, dekadal, or monthly meteorological variables to create thousands of years of artificial daily, dekadal, or monthly data, or data aggregated over any time period as appropriate for the index. Index values can then be calculated from these data to create thousands of simulated index values from which the risk statistics can be calculated. Building daily simulation models that correctly capture the physical relationships between many meteorological variables at many sites poses significant scientific, mathematical, and programming challenges. As these models involve manipulating daily data, they tend to be much slower than the other methods outlined above and, if built correctly, do not offer more accuracy than the other approaches outlined. 144 Module 5C Designing Index Based Weather Risk Management Programs

8 Quantifying Weather Risk Analysis Further Reading In addition to the statistical resources given in the HDA section, there have been many other papers on quantifying financial weather risk. The Social Science Research Network in particular, at contains many papers and articles on aspects of weather insurance, including weather and index simulation models, detrending methods, and fitting distributions. The papers written by Dr. Stephen Jewson are particularly recommended and can be found on The recent publication, Weather Derivative Valuation: The Meteorological, Statistical, Financial and Mathematical Foundations (Jewson et al., 2005), contains an excellent summary of this work and the theoretical foundations for quantifying the financial impact of weather risk. 145 Module 5C Designing Index Based Weather Risk Management Programs

9 Worked Example 3: Risk Analysis Going back to the example of the bank, consider whether or not to lend to farmers growing a rain-fed crop introduced in Worked Example 2. A credit officer at the bank has already identified that cumulative seasonal rainfall is a good indicator of the rainfall-related production deviations of the crop in the area and has conducted the regression analysis previously described. He used the 10 years of crop production data that he has for the area and the corresponding 10 years of cumulative seasonal rainfall data. To recap, he has determined that the gross revenue of a farmer growing one hectare of this crop can be estimated using cumulative seasonal rainfall as an index: Estimated Revenue = * CSR i.e., for every one millimeter change in cumulative seasonal rainfall from average, the farmers expected revenue reduces or increases by $0.43 for each hectare of the crop he or she cultivates. From this, the credit officer knows he or she has to account for the $200 a farmer, on average, needs for household expenses to estimate whether or not he will be in a position financially, at harvest time, to pay back the input and credit costs he or she will incur per hectare ($260). The credit officer uses the equation above to convert the cumulative seasonal rainfall totals from the past 10 years into farmer revenue estimates, which are given in the Credit Officer s Worksheet file. 146 Module 5C Designing Index Based Weather Risk Management Programs

10 Worked Example 3: Risk Analysis Credit Officer's Observations The credit officer uses the estimated revenue equation to convert the cumulative seasonal rainfall totals from the past 10 years into farmer revenue estimates, which are given in the credit officer s worksheet file. He or she observes that: On average, a farmer s estimated revenue, minus household expenses, is $281 per hectare, with a standard deviation of $51. The worst year was 2004, where the estimated revenue, minus expenses, was $175 He knows in that in 2004, many farmers, already receiving loans in the area, had problems repaying them in full Therefore, 2004 was a bad year for his bank. The credit officer wants to know how often these events occur. Should his bank expand lending for this crop to a new set of farmers in this area? The credit officer has 40 years of rainfall data from the weather station and therefore 40 cumulative seasonal rainfall values. Worked Example Figure 3a. 40 years of Rainfall Data 147 Module 5C Designing Index Based Weather Risk Management Programs

Worked Example 3: Risk Analysis Credit Officer's Analysis The credit officer again uses the estimated revenue equation to convert the rainfall data (the 40 values) into 40 estimated revenue values

11 Worked Example 3: Risk Analysis Credit Officer's Analysis The credit officer again uses the estimated revenue equation to convert the rainfall data (the 40 values) into 40 estimated revenue values and then recalculates the statistics of the data. He finds the mean expected revenue, minus expenses, is $292; the standard deviation is $43. The average is now higher, but: The worst year is still 2004 The second worst year is 1983 (where the estimated revenue, minus expenses, would have been $187) The third worst year is 1978 From the data, the Credit Officer can infer that years where a farmer s estimated revenue, minus expenses, is less than $260 happen on average 12.5 percent of the time (i.e., once every eight years). The credit officer also knows than 2004 has been the worst year recorded in 40 years. However, 87.5 percent of the time, farmers should be making plenty of profit on the crop to be able to save some money as a contingency to cover those bad years. Worked Example Figure 3b. Statistics from Farmer Revenue Data 148 Module 5C Designing Index Based Weather Risk Management Programs

12 Worked Example 3: Risk Analysis Credit Officer's Conclusions The credit officer conservatively assumes that every time a farmer has less than $260 in estimated revenue, minus expenses, he will default in full (i.e., he conservatively estimates the rainfall related default risk is 12.5 percent. The credit officer knows rainfall is the biggest risk to his customers. The credit officer reviews the information that he has in his credit tables for other customers with similar profiles who do not have rainfall risk. He then estimates the additional interest rate he should charge these farmers for undertaking this more risky exercise. The credit officer decides a 30 percent interest rate over nine months as the appropriate charge to cover the risk of weather-related default. After looking at the rainfall record, he decides that he is happy to scale-up his lending to farmers growing this crop in this area at this interest rate. 149 Module 5C Designing Index Based Weather Risk Management Programs

Weather Derivatives-Pricing and Design Issue in India

Weather Derivatives-Pricing and Design Issue in India By Anshul Anand & Surendra Mahadik Introduction: In the last decades, agricultural policies of certain countries focused on adopting and strengthening