One of the most widely used risk management measures is the so-called VaR. VaR is the maximum potential positive or negative return over a given time horizon given a prespecihed confidence level (1-a). For example, when a = 1 % the left tail VaR in daily commodity returns is the cutoff point in the distribution under which there is only 1 % probability of observing a more negative return and the right tail VaR is the cut-off point in the distribution above which there is only 1 % probability of observing a more positive return. Such measures can be quite useful in food import expenditure planning and hedging for food-insecure countries.

Suppose that the cumulative distribution function is denoted by F. The left tail VaRj_{own} is given by F^{-} (a) and the right tail VaR_{up} is given by F^{-1}(1-a). However, VaR does not provide any information with respect to the size of the returns at the extreme tails of the distribution. A popular risk management measure that remedies this shortcoming of VaR is the Expected Shortfall or conditional VaR. The Expected Shortfall is the conditional expectation of the return given that the return has exceeded VaR. The Expected Shortfall of the left tail (denoted as VaRd_{own}) is given by the expected value of the distribution truncated at the left, namely the lowest values, E(r|r < VATh^J, while the Expected Shortfall of the right tail (denoted as VaR_{up}) is given by the expected value of the distribution truncated at the right, namely the high values, E(r|r > VAR_{up}). If VaRd_{own }is negative then clearly the expected value of the truncated distribution is also negative.

In commodity markets, left tail risk management measures are useful for farmers, because they indicate the maximum loss from sales, while right tail risk management measures are useful for commodity users and policy makers, as they indicate the maximum cost of purchases (see also van Oordt et al. 2013). The exact values of Varj_{own} and Var_{up} depend on the confidence level a at which the truncation is made.

Table 9.3 reports daily VaRs and Expected Shortfalls for soya, maize and wheat using the parameter estimates from Sect. 3 and two probability levels (a = 1 %, and a= 0.1 %).

Table 9.3 Value-at-Risk and Expected Shortfall of soya, maize and wheat daily returns and 1 % and 0.1 % probability levels

Right tail VaR

Soya

Maize

Wheat

a= 1.00 %

3.15 %

4.45 %

4.57 %

a= 0.10 %

6.33 %

7.05 %

7.88 %

Right tail Expected Shortfall

Soya

Maize

Wheat

a= 1.00 %

4.52 %

5.56 %

5.99 %

a= 0.10 %

9.08 %

8.81 %

10.31 %

Left tail VaR

Soya

Maize

Wheat

a= 1.00 %

-3.59 %

-4.41 %

-5.44 %

a= 0.10 %

-6.73 %

-6.70 %

-10.00 %

Left tail Expected Shortfall

Soya

Maize

Wheat

a= 1.00 %

-4.94 %

-5.39 %

-7.39 %

a= 0.10 %

-9.26 %

-8.19 %

-13.60 %

Source: Authors' estimates

The right tail VaR of soya at 99 % confidence level is 3-15 %, indicating that there is a 1 % probability that the daily return to soybeans is above 3-15 %- The left tail VaR is -3-59 %, indicating that there is a 1 % probability that the daily return to soybeans is less than -3-59 %- The corresponding right tail Expected Shortfall is 4-52 % and the corresponding left tail Expected Shortfall is -4-94 %, and these are the expected gains and losses respectively during these infrequent times when the soybean prices are above or below the respective VaR values- The corresponding values of VaR in the maize and wheat markets are larger in absolute value, implying that the levels of daily gains or losses that can occur with 1 % probability or smaller are larger than those of soybeans- In other words, if we specify a given extreme value of a daily price gain such as 4 %, it is more probable that such a gain will happen in soybeans than in the maize and wheat markets- Nevertheless, the results indicate that there is sizable risk in tails of the soya, maize and wheat return distributions, implying that the frequencies of very high or very low prices are larger than what would be implied by simple normal distributions-

From a policy-making perspective it would be useful to examine if it is possible to forecast extreme positive returns at low frequencies- We define as extreme positive returns the monthly returns which are larger than one or two monthly standard deviations. We measure monthly historical standard deviations using daily returns within each month. The categorical binary variables which indicate the 1-sigma and 2-sigma price spikes are the following:

To predict extreme events we use as forecasting variables inventory data, hedging pressure and 3-month Treasury Bill. We search for commodity- specific forecasting variables of price spikes in maize, wheat and soybeans futures markets since many studies have identified significant linkages between inventory levels, uncertainty and agricultural commodity prices (Deaton and baroque 1992; Pietola et al. 2010; Cooke and Robles 2009; Tadesse et al. 2014; Triantafyllou et al. 2015; Zawojska 2010). Motivated by the relevant literature which links monetary factors and commodity prices (Frankel and Hardouvelis 1985; Frankel 1986; Frankel 2008; Gilbert 2010; Gordon and Rouwenhorst 2006) we add into our information variable set the level of the short-term interest rate (3-month US Treasury Bill rate).

We obtain quarterly inventory data for maize, wheat and soybeans from the National Agricultural Statistics Service of the USA for the period 1990 till 2011. We construct monthly data for these variables from quarterly observations, using the method of polynomial interpolation. We take the monthly prices that make the best fit at the polynomial which is being created by the quarterly prices. We use the natural logarithm of these interpolated monthly levels of stocks for each monthly period. The hedging pressure is defined as the difference between the number of short and the number of long hedge positions in the futures markets relative to the total number of hedge positions by large (commercial) traders.

Weekly data for the number of short and long hedge positions for wheat, maize and soybeans futures were obtained from the US Commodity Futures Trading Commission. The data for the 3-month Treasury Bill rate were obtained from the Federal Reserve Bank of Saint Louis and cover the period from January 1990 through December 2011.

In Table 9.4 we present the results form a probit model that forecasts commodity price spikes using commodity-specific and macroeconomic factors- The multivariate probit model uses as explanatory variables the following:

SPIKE, which is the binary variable that indicates 1-sigma and 2-sigma price spikes given in equations (2) and (3) respectively.

INV, which is the inventory level.

HP, which is the hedging pressure.

RV, which is the monthly realized variance.

USTBILL, which is the 3-month US Treasury bill rate.

All the above variables are lagged by one month in the estimations.

Table 9.4 Probit regressions forecasting 1- and 2-sigma price spikes in the maize, wheat and soybeans market

Maize

Wheat

Soybeans

Panel A: 1-sigma price spikes

Const

Coef.

-1.711

0.844

0.253

t-stat

(-0.762)

(1.177)

(1.505)

INV

Coef.

0.095

-0.042

0.004

t-stat

(0.666)

(-0.791)

(0.190)

HP

Coef.

1.191

0.108

-0.065

t-stat

(1.982)

(0.566)

(-0.608)

RV

Coef.

-7.586

-0.870

-0.947

t-stat

(-3.222)

(-2.823)

(-3.120)

USTBILL

Coef.

-4.055

-1.038

-0.753

t-stat

(-0.865)

(-0.738)

(-0.571)

% Mc Fadden R^{2}

6.6

2.3

2.1

Panel B: 2-sigma price spikes

Maize

Wheat

Soybeans

Const

Coef.

-1.667

0.300

0.342

t-stat

(-0.559)

(0.714)

(1.902)

INV

Coef.

0.062

-0.015

-0.030

t-stat

(0.329)

(-0.514)

(-1.194)

HP

Coef.

-0.795

-0.009

0.017

t-stat

(-1.054)

(-0.115)

(0.237)

RV

Coef.

-9.640

-0.336

-0.593

t-stat

(-2.284)

(-2.312)

(-2.804)

USTBILL

Coef.

-4.914

-0.471

-1.758

t-stat

(-0.770)

(-0.879)

(-1.833)

% Mc Fadden R^{2}

7.6

0.2

3.8

Source: Authors' estimates

Table 9.4 indicates the results of the estimations. We observe that price spikes in the maize market are difficult to predict by macroeconomic or by commodity-specific factors. Inventory, hedging pressure and shortterm interest rate are insignificant predictors of extreme events (with the exception of hedging pressure in the 1-sigma price spike). We find a negative and statistically significant coefficient of lagged realized variance when we forecast 1-sigma and 2-sigma spikes one month ahead. The negative coefficient is somewhat odd, as it implies that the lower the past month market volatility, the higher is the likelihood of a price spike. The interpretation and economic justification of the negative coefficient of realized variance could be that low market volatility or uncertainty implies low expectations of a spike in the following month, and hence any unexpected news is likely to lead to overreaction and a spike.

The results of the wheat and soybeans markets are similar. Inventory, hedging pressure and short-term interest rate remain insignificant. The negative and statistically significant coefficient of realized variance may be interpreted as above. These results are in line with those of Vilkov and Xiao (2013), who examine the predictive power of equity market uncertainty on the occurrence of equity market price spikes. They report negative uncertainty coefficients as well, and they interpret this somehow odd result as an overreaction of equity investors. According to them, uncertainty in equity markets increases only after a market crash has occurred, thus it cannot act as an early warning signal of extreme returns. By our empirical analysis, we find that the same thing seems to hold for maize and wheat markets.

Unlike maize and wheat markets, in the soybeans market the shortterm interest rate is a significant predictor of 2-sigma price spikes with a negative coefficient. This result indicates that lax monetary policy (the reduction in short-term interest rates) may have contributed to the occurrence of more frequent extreme events in the soybeans market post 2003 (monetary-easing era). These results are in line with those of Gilbert (2010), Frankel (2008) and Frankel and Hardouvelis (1985), who find a negative relationship between the monetary policy stance and commodity price booms. Nevertheless, the above results highlight the difficulty of predicting extreme events in the agricultural commodity market.