How to Identify Extreme Returns
In this section we study the empirical behavior of the returns of three basic food commodity products and describe some simple statistical tools than can be used for modeling tail risk. The details of the empirical analysis are available from the authors on request. The commodities under consideration are soya, maize and wheat, which are the most important internationally traded food products. We use daily futures data for maize, wheat and soybeans which are obtained from the Chicago Board of Trade (CBOT). CBOT is the major global market for food commodities, and international cash prices for them in most markets are strongly related to CBOT futures prices. We use futures data instead of spot prices because futures markets are more liquid and provide more reliable data at high frequencies. The futures data for maize, wheat and soybeans cover the period from January 1990 to December 2011. We construct a single time series of data for each commodity using the nearby (close to maturity) futures contracts. We choose the 2month expiration as a fixed time for nearest maturity commodity futures, since the expiration dates on agricultural commodity futures are the 1st of March, May, July, September and December. If the nearby contract has less than 60 days to expiration, we replace it with the next contract, which always has more than 60 days to expiration. For example, when the nearest futures contract has 75 days to expiration, we keep it only for 15 days and then we change it with the next deferred contract, which by definition will have more than 60 days to expiration to get the best possible approximation of a fixed 2month horizon futures contract. From the time series data (F) we compute daily and monthly returns. The daily return at day d is defined as (Fj—Fd_j)/Fd_j and the monthly return at the end of month m is defined as (F_{m}—F_{m1})/F_{m}_j. The behavior of daily returns is of great interest to both farmers and importers who wish to hedge price risk using derivative products, while the behavior of lowerfrequency returns, such as monthly returns, is of great interest to policy makers who want to know longer term trends in commodity prices.
Daily futures prices of the three commodities show strong volatility. Prices of all three commodities are volatile and occasionally display large upward as well downward price swings (e.g., during the period 2007—2009). Table 9.1 reports the descriptive statistics of daily and monthly returns.
Table 9.1 Descriptive statistics of daily and monthly returns of soya, maize and wheat
Daily returns 
Soya 
Maize 
Wheat 
Mean 
0.02 % 
0.03 % 
0.03 % 
Stdev 
1.50 % 
1.66 % 
1.78 % 
Min 
12.19 % 
19.09 % 
9.31 % 
Max 
19.70 % 
22.56 % 
11.17 % 
Monthly returns 
Soya 
Maize 
Wheat 
Mean 
0.54 % 
0.67 % 
0.57 % 
Stdev 
7.01 % 
7.62 % 
8.42 % 
Min 
23.49 % 
22.80 % 
26.18 % 
Max 
21.37 % 
27.52 % 
44.44 % 
The data cover the period from January 1990 to December 2011 Source: Authors' estimates
The most volatile commodity is wheat with a standard deviation of daily (monthly) returns equal to 1.78 % (8.42 %) and the least volatile commodity is soya with a standard deviation of daily (monthly) returns equal to 1.50 % (7.01 %). However, the standard deviation of returns may be a misleading measure of risk when returns depart signihcantly from normality and the distribution is fattailed. Soya is the least volatile commodity according to the standard deviation criterion, but it has a maximum daily return which is 13.12 standard deviations above the mean and a minimum return which is 8.14 standard deviations below the mean. If the returns follow a normal distribution the probability of observing a return that is 13.12 standard deviations above the mean is 6.11 x 10^{38} and the probability of observing a return that is 8.14 standard deviations below the mean is 1.97 x 10^{15}. These are both very small values, implying the nonnormality of returns. Maize has a minimum return that is 11.52 standard deviations below the mean and a maximum return that is 13.57 standard deviations above the mean. Wheat has less extreme returns despite the fact that it is the most volatile commodity. The maximum return is 6.26 standard deviations above the mean and the minimum is 5.25 deviations below the mean. Still, if the returns follow a normal distribution, a 6sigma event is expected to happen once every 4 million years, which is again much too infrequent.
From the analysis of the QQ plot, which indicates the quantiles of the daily returns of each commodity (soya, maize and wheat) against the quantiles from a standardized normal distribution, the following inferences are drawn. Under perfect normality, points in the Q—Q plot should approximately lie on a straight line. From the results we observe that in all three commodities under consideration there are substantial deviations from normality in both tails of the distribution. The same inference is drawn if one plots the quantiles of the monthly returns of each commodity. This suggests that the actual price return distribution has many more frequent price spikes and price depressions compared to what would be obtained if the price returns followed a normal distribution.
The most standard approach in the literature to capture heavy tails in the empirical distributions is to use distributions that obey power laws. The tails of power law distributions diminish according to power and the rate of the tail decay is usually slower than the exponential that governs the normal distribution. The use of power laws dates to the end of the nineteenth century (Pareto 1896). Gabaix (2009) provides an excellent survey of power laws in economics and hnance. If the tail of the commodity return distribution obeys a power law then the probability that the commodity return r exceeds some large enough threshold x is given by
where C, Z >0. The C parameter is called the scale and the Z parameter is called the shape or tail index or power law exponent. The shape parameter determines the thickness of the tail of the distribution. In order to estimate the scale and shape parameters C and Z the commodity returns are first ordered in descending order from high to low, and then an estimate can be made of the shape parameter and a threshold parameter k that defines the point in the tail of the distribution, below which the distribution of returns is assumed to obey the power law in (1) (Hill 1975). The exponent of the left tail is estimated using the same procedure after multiplying the returns by 1. We estimate the tail index Z and the threshold parameter k using the method based on the goodness offit described by Clauset et al. (2007). The parameter estimates of the tail index for the right and left tails of soya, maize and wheat daily returns are reported in Table 9.2.
Soya has the lowest right and left tail indices, indicating that the tails of the price return distributions are the fattest among the three commodities, and therefore that soybeans is the commodity which is most exposed to (namely has most frequent) extreme returns at both tails of the distribution.
Table 9.2 Estimates of power law exponents using soya, maize and wheat daily returns
Soya 
Maize 
Wheat 

Power law exponent—right tail 
3.30 
5.01 
4.24 
Power law exponent—left tail 
3.67 
5.51 
3.78 
Source: Authors' estimates