Review of Agricultural Economics--Volume21, Number 1--Pages 99-111

Mark R. Manfredo and Raymond M. Leuthold Value-at-risk (VaR) determines the probability of a portfolio of assets losing a certain arnount in a given period at a particular level of confidence. Vahe-at-risk is receiving considerable attention in the finance literature for its use in reporting the risks of derivatives. This article provides a state-of-the-art review of VaR estimation techniques and empirical findings. The ability of VaR estimates to represent large losses varies among procedure, confidence levels, and data used. To date no consensus exists regarding the most appropriate estimation technique. Potential applications of VaR are suggested in the context of agricultural risk management.

alue-at-risk (VaR) measures are used to estimate the probability of a portfolio of assets losing more than a specified amount over a specified time period because of adverse movements in the underlying market factors of the portfolio. For example, a VaR measure of $1 miUion at the 95% level of confidence implies that overall portfolio losses will not exceed $1 million more than 5% of the time over a given holding period under normal market conditions (Jorion 1996, 1997; JP Morgan; Linsmeier and Pearson 1996, 1997; Mahoney). Currently, VaR is being touted as the state-of-the-art in measuring the risks associated with a portfolio of assets, especially derivatives positions. In essence, VaR estimates attempt to capture extreme events that occur in the lower tail of the portfolio's return distribution. The major advantage of VaR relative to more traditional risk measures is its focus on downside risk. Consequently, VaR is praised for being an intuitive measure of risk

V

9 Mark R. Manfredo is assistant professor, Morrison School ofAgribusiness and Resource Management, Arizona State University. 9 Raymond M. Leuthold is the T.A. Hieronymus Professor and director, Officefor Futures and Options Research, Department of Agricultural and Consumer Economics, University of Illinois at Urbana-Champaign.

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Value-at-Risk Analysis: A Review and the Potential for Agricultural Applications

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and for its ability to capture the risks of many different assets into a single, concise number. The recent explosion of interest in VaR stems from its use in risk disclosure and risk reporting. In the wake of several well-publicized derivatives debacles, such as the Barrings Bank failure, several regulatory bodies have recommended or mandated the reporting of VaR estimates by firms (i.e., large trading banks) that maintain large derivatives positions to provide a clear measure of the firm's downside risk potential. Notably, in January 1997 the Securifies and Exchange Commission (SEC) established rules for the quantitative and qualitative reporting of risks associated with market sensitive assets (i.e., derivatives positions) of reporting firms. VaR was one of three quantitative risk reporting methods approved for use in SEC disclosures (Linsmeier and Pearson 1997). Similarly, futures exchanges use VaR to measure the probability of default by clearing members (Fuhrman). Because of VaR's emphasis on downside risk, it is considered by many to be a more intuitive measure of risk and more easily understood by top-level managers and outside investors who might or might not be well trained in statistical methods. As a result of the interest in VaR, an entire industry has evolved that is devoted to the implementation and use of VaR, especially software designed to calculate the risk measure. Much of this can be attributed to JP Morgan's Risk Metrics system available on the Web (http: / / www.riskmetrics.com / rm / index.html). JP Morgan has attempted to position its estimation methodology as the industry standard for computing VaR. In addition, an entire Web page is devoted to all facets of the topic of VaR and has become a clearinghouse for research and discussion regarding the risk measure (http://www.GloriaMundi.org). Despite the obvious uses for risk disclosure purposes, VaR is also being suggested for firm-level risk management. VaR could be beneficial in making hedging decisions, managing cash flows, setting position limits, and overall portfolio selection and allocation. It also has several potential applications in agricultural economics. The AgRisk program developed by Ohio State University and the University of Illinois at Urbana-Champaign uses VaR analysis in determining the potential downside revenue from implementing (or not implementing) alternative preharvest marketing strategies for com, wheat, and soybeans (see http://www-agecon. ag.ohio-state.edu / agrisk/agrisk.htm). At a Commodity Futures Trading Commission (CFTC) hearing regarding the lifting of the ban on agricultural trade options for certain commodities, VaR was recommended for use in reporting the market risk associated with such contracts. 1 This, along with continued volatility of agricultural prices, implies several applications of VaR in agricultural economics. This review article presents the state-of-the-art concerning VaR, drawing mainly from the finance literature. A theoretical definition of VaR that links VaR to traditional measures of volatility is provided, followed by potential agricultural applications. Much of the current research and literature on VaR can be considered survey oriented, discussing the potential pros and cons of various estimation techniques, providing ana priori evaluation of how these models might perform under alternative portfolio conditions. Special attention is given to empirical evidence concerning the ability of VaR estimates to measure tail behavior. From this literature review, extensions of VaR for use in agricultural risk management and related research topics are presented.

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Figure 1. Illustration of value-at-risk f(w)

w (doUars)

1-c

Theoretical Constructs of VaR Jorion (1996, 1997) defines VaR for a general class of distributions such that endof-period portfolio value is W = W0(1 + R), where W0 is the initial investment and R is the rate of return on the portfolio. Subsequently, Jorion (1996, 1997)defines critical end-of-period portfolio value as W*, where W* = W0(1 + R*), W0 is the initial investment, and R* is a critical level of portfolio return associated with a predetermined level of confidence (c).2 Therefore, W* can be thought of as the endof-period portfolio value when the worst possible portfolio retum (R*) is realized, a return that one is unlikely to encounter more than (1 - c) percent of the time under normal market conditions. For a specified confidence level (c) a n d a general distribution of future portfolio value f(W), Jorion (1996, 1997) defines VaR as (1)

f(W) dw

1 - c= -ov

thus isolating the area in the left tail of the distribution (figure 1). This area is associated with losses that are greater than or equal to the loss associated with confidence level (c), representing the downside risk, or VaR, of the portfolio. By assuming the general distribution of portfolio value, flW) in equation (1) is the standard normal distributionfle), where e --- (0,1). Normalizing (R* - ~), Jorion (1996, 1997) defines a normal deviate (a) as (2)

-~

=

-IR*lfr

Associating the normal deviate (a) with R*, Jorion (1996, 1997) shows that

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Y

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(3)

Review of Agricultural Economics

1 - c =

fw*

f(W)

-o~

dw =

f-lml

al(R) dR =

-co

f-~,

.f(~) & .

-oo

(4)

V a R , = WooL(r

where W0 is defined as before to be initial portfolio vahe, ~ the normal deviate associated with (1 - c), and (r the standard deviation of portfolio retums. To find the VaR of a portfolio, one needs to multiply the estimated cr by the relevant ~ and initial investment. Obviously, under the assumption of normality, the only true unknown is the estimate of cr. Therefore, the problem becomes one of forecasting the volatility and correlations between individual assets and subsequently portfolio volatility. V a R in Agriculture A direct application of VaR in agriculture, one that closely parallels examples often found in the finance literature, is that of an agribusiness firm involved with the procurement and processing of agricultural commodities. For example, a food processing firm might purchase a wide range of agricultural commodities as inputs. In an attempt to reduce the firm's overall costs, the risk manager could incorporate forward contracts or futures and/or options positions in managing the risks assodated with these input prices. The risk manager might maintain a trading book that contains several different cash, forwarcl contracts, futures, and options positions. Accordingly, the risk manager can examine the VaR of this portfolio at a particular level of confidence (e.g., 95%) over a specified time period, assessing the magnitude of a potential large loss in its value. Following equation (4), suppose that the trading book portfolio is worth $1 million on day t (W0) and that the risk manager is interested in the VaRat the 95% level of confidence (o~ = 1.65) over the next day. The risk manager estimates the one-day standard deviation of portfolio returns (cr) to be 4?/o. Therefore, the VaR of this portfolio is simply $1,000,000 x 1.65 x 0.04 = $66,000, which is a loss that is likely to be exceeded only 5% of the time (figure 1). By incorporating the use of VaR forecasts, the risk manager can take actions (e.g., alter forward, futures, or options positions) to reduce the downside risk exposure, or VaR, of the trading book. An agricultural production example of how VaR can be used is a cattle feeding operation. A cattle feeder is exposed most notably to the variability of the price of cattle (the output) and the prices of feeder cattle and com (the major inputs). 3 The cattle feeder's profit margin is the difference between revenue generated from the sale of fed cattle less the costs incurred for the purchase of feeder cattle and com.

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From the equality in equation (3), Jorion (1997, p. 89) states that "the problem of Value-at-Risk is equivalent to finding the deviate ~ such that the area to the left of it is equal to 1 - c." From the cumulative standard normal distribution, the confidence level c associated with the normal deviate oLcan be found. For example, at the 95% confidence level, 1 - c = 5%. Therefore, the associated c~ corresponding to the lower 5% of the normal distribution is equal to 1.65. Jorion (1996, p. 49) notes that equation (3) provides an illustrative linkage that shows that "VaR may be found in terms of portfolio value (W*), cutoff return (R*), or normal deviate (a)." Therefore, VaR under the assumption of normality is

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VaR Estimation Techniques used to generate VaR measures are not novel. VaR calculations are synonymous with forecasting the volatility of a portfolio over a particular holding period, paying special attention to the lower tail of the probability distribution. Two major classes of estimation procedures for VaR have received the most attention: parametric and full-valuation procedures. Parametric procedures determine estimates of volatility and correlations under the assumption of normality. Both the numerical examples presented in the previous section incorporate parametric procedures because they utilize explicit estimates of portfolio volatility. Full-valuation procedures attempt to model the entire empirical retum or revenue distribution. Despite the array of procedures available, much debate exists over the best method. Table I provides a summary and comparison of the principal methodologies. P a r a m e t r i c Estimates o f VaR Parametric procedures for developing VaR measures are often referred to as variance/covariance methods (Linsmeier and Pearson 1996, 1997) because of the emphasis placed on developing forecasts of volatility and correlations. Historical volatility estimates (Boudoukh, Richardson, and Whitelaw; Hend¡ Jorion 1996, 1997), implied volatility from options prices (Hopper; Jorion 1996, 1997; Mahoney), various conditional time-series models (Duffie and Pan; Hendricks; Hopper; Jorion 1996, 1997; JP Morgan), and regime-switching models (Duffie and Pan; Venkataraman) have been suggested for use in developing VaR estimates. Much of the interest in parametric models of VaR has been motivated by the efforts of JP Morgan and the dissemination of its Risk Metrics methodology for developing estimates of

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At any given time, the cattle feeder can observe the value of this cash feeding margin, forecast its volatility, and subsequently calculate the VaR of the cattle feeding rnargin. Similar to the example presented previously for a food processing firm, suppose that a cattle feeder observes the cattle feeding margin (W0) durŸ a particular week to be $50 per hundredweight (cwt). Furthermore, the cattle feeder has estimated the weekly standard deviation of the cattle feeding margin ((r) to be 11% and is interested in the VaR of the cattle feeding margin at the 95% level (~ = 1.65) over the next week. Again, using equation (4) the VaR of the cattle feeding margin is (figure 1) $50/cwt x 1.65 x 0.11 = $9.08/cwt. Therefore, a cattle feeder should expect to realize no more than a $9.08/cwt decrease in the value of the feeding margin over a one-week time frame with a 95% level of confidence. In general, using VaRas a risk management tool means examining the VaR associated with the exposure of risky assets prior to and after the implementation of risk management strategies. In the case of a cattle feeding enterprise, the manager can reduce the variability of the cash feeding margin, and subsequently the VaR, by implementing any number of risk management strategies, including futures and options contracts. There might also be times when a cattle feeder is content with a specific VaR number and willing to accept the subsequent feeding risks, implementing no risk management strategy. These two examples illustrate potential applications of VaR estimates in the context of agricultural prices and market risk management. Other situations and potential applications of VaR are discussed at the end of this artide.

Variance/Covariance

Historical Simulation

Source: Adapted from Linsmeier and Pearson (1996).

Easy to explain to senior manage- No. Yes. ment? Produces misleading value at risk Yes, except that alternative correlations Yes. estimates when recent past is or standard deviations may be used. atypical? Easy to perform "What if" anaEasily able to examine alternative asNo. lyses to examine effect of altersumptions about correlations or stannative assumptions? dard deviations. Unable to examine alternative assumptions about the distribution of market factors, i.e., distributions other than the normal.

Yes.

Yes, except that alternative estimates of parameters may be used.

No, except for relatively small portfolios. No.

Yes, for portfolios restricted to instruments and currencies covered by available off-the-shelf software. Otherwise moderately to extremely difficult to implement.

Yes, regardless of the options content of the portfolio.

Monte Carlo Simulation

Full-Valuation Procedures

No, except when computed using a Yes, regardless of the short holding period for portfolios options content of with limited or moderate options the portfolio. content. Easy to implement? Yes, for portfolios restricted to instruYes, for portfolios for ments and currencies covered by which data on the available off-the-shelf software. Otherpast values of the wise reasonably easy to moderately market factors are difficult to implement, depending on available. the complexity of the instruments and availability of data. Computations performed quickly? Yes. Yes.

Able to capture the risks of portfolios which include options?

Factor

Parametric

Table 1. Comparison of value-at-risk methodologies

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