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The purpose of this paper is to estimate the “market price of risk” (MPR) for energy commodities, the ratio of expected return to standard deviation. The MPR sign determines whether energy forward prices are upward- or downward-biased predictors of expected spot prices. We estimate MPRs using spot and futures prices, while accounting for the Samuelson effect. We find long-term MPRs generally positive and short-term negative, consistent with positive energy betas and hedging, respectively. In spot electricity markets, MPRs in Day-Ahead Prices agree with short-dated futures. Our results relate risk premia to informed hedging decisions, and futures prices to forecast/expected prices.

The purpose of this paper is to determine the magnitude and sign of the commodity “market price of risk” (MPR) in energy markets. Defining the commodity market price of risk as compensation per unit standard deviation,1 equation(1) View the MathML sourceλF≡μσ, Turn MathJax on permits us to determine whether forward prices are upward- or downward-biased predictors of future spot prices. Whereas the market price of risk is assumed positive in financial markets (participants require a premium for bearing risk), its sign in commodity markets could be negative. The examination of the market price of risk has been performed in both financial (equity/bond) and commodity markets: 1. In equity markets, the estimation of the market price of risk — there denoted also the “Sharpe ratio” — is an enduring empirical and practical phenomenon. Researchers have addressed both the magnitude as well as the possible time variation in that variable. More recent estimates were provided in the AFA Presidential Address of George Constantinides (2002). As is well-known, in (positive “beta”) equity markets, no-arbitrage future prices are downward-biased.2 2. There is a significant debate on the question of whether forward prices in energy markets are biased or unbiased predictors of future expected prices. The empirical work dating back to Houthakker (1957) and Chang (1985), and more recently Fama and French (1987) and Bessembinder (1992), showed that in financial and mature commodity futures markets risk premia in general satisfy the integrated-market model, which predicts that risk premia are proportional to the covariance of the futures return with the return on the market portfolio. On the theoretical side, the model by Hirshleifer (1988) assumed a cost for speculators to participate in the market, related the risk premium to the number of speculators, and found that risk premia possess an additional positive component due to cost. More recently, Routledge et al. (2001) and Bessembinder and Lemmon (2002) have related risk premia to volatility of price changes, risk of price spikes and uncertainty in quantity demanded. Empirical work has been performed by Dincerler and Ronn (2001), who use mean-reverting spot prices to obtain a − 2.73 estimate of the MPR, and Doran and Ronn (2003), who consider the commodity market price of risk in the context of the market price of volatility risk. Finally, the paper by Longstaff and Wang (2004) analyzed daily and hourly electricity-price data, and computed the forward premium, defined as the difference between a one-day forward contract and the realized electricity spot price: “…the overall mean of the forward premia is 59 cents, but is not statistically significant. Despite this, there is clear evidence of significant forward premia when one looks at the results for the individual hours. In particular, the mean forward premium is statistically significant for 10 of the 24 hours. …The individual mean forward premia are often very large. For example, the mean forward premia for 6 p.m. and 7 p.m. are $5.41 and $5.44, respectively. In terms of the average spot prices for these hours, these premia represent percentage premia of 12.8 and 13.8 respectively.” While our results are broadly consistent with those of Longstaff and Wang, we do note some differences. First, in computing the market price of risk (rather than the forward premium), we find a statistically significant market price of risk for these PJM electricity contracts, which translates into a significant positive forward premium proportionately related to volatility. The statistical significance of our day-ahead MPR results may in part be attributable to our modeling a constant MPR — consistent with the motivation we have provided for such modeling — whereas Longstaff and Wang model the constancy of the forward premium. Further, whereas Longstaff and Wang analyze Day-Ahead Prices and thus clearly describe a short-term effect, our other analyses consider long-dated PJM futures contracts and hence extend beyond their short-term results. Consider first the case for crude-oil. If the Capital Asset Pricing Model applies to commodities, and if oil's CAPM “beta” is negative, then the market price of risk for commodities is negative and we would expect forward prices to be upward-biased predictors of expected spot prices: equation(2) F>E(ST),F>E(ST), Turn MathJax on where E(ST) is the expected price at the maturity date of the forward contract F. Such a situation would be expected to prevail if the developed world is a net consumer, not producer, of crude-oil, and is therefore “averse” to higher crude-oil prices and willing to pay a risk premium to avoid such higher prices. In fact, were we to calculate a regression estimate of beta for crude-oil including the turbulent ‘70’s and ‘80’s, we would indeed find such a negative beta: As oil prices rose in the ‘70’s, stock markets declined; as oil prices fell in the ‘80’s, stock markets rose. 3 In electricity, the argument over a negative beta is less obvious. Although electricity prices clearly spike upwards, electricity is entirely domestically produced and thus may have a positive beta: That is, its prices may rise as a growing economy increases demand. Thus, the objective of this research is to address the magnitude and sign of the commodity market price of risk in electricity and natural-gas prices. There are several implications to the work we propose: 1. On the academic side, we seek to understand the relationship between forward prices and expected prices as an important factor in understanding the energy markets and their relationship to the other physical and financial markets. 2. On the managerial side, understanding that same relationship can assist managers in making more informed hedging decisions. Consider the case of a company's management wishing to utilize futures contracts in their optimal corporate-wide hedge ratio. One optimization approach considers the trade-off between expected cash-flow maximization and risk minimization. If hedging is “costly” — in the sense that futures contracts are biased predictors of expected spot prices — then in determining the magnitude of its hedging, management would naturally consider the attendant implications such hedging entails for the corporation's expected cash flows. In order to factor in the latter, however, the corporation must have an estimate of the expected price change on the futures contracts. 3. In the energy industry, many firms' economic/structural desks produce estimates of expected, or forecast, prices, especially for maturities beyond the liquid set of futures contracts.4 Managerial users of such prices should recognize they are distinct from forward prices: The use of structural prices in the valuation of real options must be tempered by explicit recognition that the market price of risk for commodities need not be zero. Consequently, (a) Forecast prices need not be continuous extensions of the longest-maturity traded forward prices. (b) For present-value calculations, structural prices should be discounted at the risk-adjusted rate of return; equivalently, such prices may be converted to forward prices (using the market price of risk), and then discounted at the risk-free rate of interest. In futures, we explicitly account for the Samuelson (1965) “term structure of volatility” (TSOV), that is, the volatility of futures contracts varies inversely with their time-to-maturity. We incorporate this TSOV into both one- and two-factor models. One-factor models can capture only long-term effects since most of a futures contract's data is available for dates long before maturity. In the one-factor case, we find that both electricity and gas markets MPRs are positive. In contrast, two-factor models can account for different MPRs for long- and short-term factors. We found that long-term MPRs are in agreement with one-factor models and short-term MPRs are generally negative for electricity prices and positive for gas prices. The statistical power of these tests is limited due to an insufficient number of price observations close to maturity. The approach in this paper is based on our plausible behavioral assumption that the market price of risk, the compensation per unit standard deviation, is to a first-order modeling approximation a constant. Under this constancy assumption, as volatility increases — due to the contract approaching its maturity date, or considering a seasonal futures contract for a peak delivery month — the risk premium commensurately increases: μt = λ · σt. Moreover, we permit the market price of risk to have a seasonal component which allows the market to exhibit greater or lesser sensitivity to volatility in the peak` and off-peak seasons. Our data permit the investigation of short-term effects using spot prices of electricity. We examine the relationship between Day-Ahead Prices and Real-Time Prices. Since Day-Ahead Prices can be viewed as prices of a one-day forward contract, we argue that short-term MPRs obtained this way should be compared to the short-term MPRs obtained from the application of a two-factor model to futures prices. We found that this is indeed the case. These observations support the Bessembinder and Lemmon (2002) model, which predicts that when the distribution of spot power prices becomes positively skewed, short forward positions incur large losses, since upward spikes in spot prices are frequent, and the equilibrium forward price is bid up to compensate for the skewness in the spot-price distribution. The distribution of short-term futures is significantly skewed for seasons with the greatest variability in the demand for power, and we observed negative short-term MPRs in those seasons. In contrast, long-term futures prices have reduced skewness and a positive MPR. The paper is now organized as follows. Section 2 provides the theoretical model, whereas Section 3 provides empirical results using maximum likelihood estimation methods. Section 4 then considers pooling the estimators to enhance statistical significance. Section 5 estimates the market price of risk in Day-Ahead electricity prices in the Pennsylvania–New Jersey–Maryland (PJM) area. Section 6 reports the empirical results, and Section 7 concludes.

This paper addressed the magnitude and sign of the commodity “market price of risk” (MPR) for electricity, oil and natural-gas prices. Noting the importance of understanding the commodity risk premium in making informed hedging decisions, we noted that the sign of this MPR determines whether forward prices in energy are upward- or downward-biased predictors of future expected spot prices: A positive (negative) MPR corresponds to the case Fb(N)E(ST). We estimated that risk premium by estimating the drift term in spot and forward prices. In futures prices, we explicitly account for the Samuelson effect “term structure of volatility.” In spot prices of electricity, we examine the relationship between Day-Ahead Prices and Real-Time Prices. We found that in domestic electricity markets the MPR is negative, when short-term horizons are considered. This result coincided with the results by Longstaff and Wang (2004) and supports hedging pressure conjecture in Bessembinder and Lemmon (2002) model. In contrast to Longstaff and Wang (2004) we investigated not only short-term risk premium but also long term and found that hedging pressure is reduced when longer time horizons are considered. The analysis of gas and oil prices, especially the longer-maturity contracts which dominate our sample, provided further support to the Bessembinder and Lemmon (2002) model that more mature markets should not have hedging pressures as they contain numerous non-industry participants. Finally we found that contrary to the intuition that in fairly new EEX market the short-term MPR is expected to be negative, actual risk premium is positive for both short and long time horizons. This could be explained by the market design, structure of the contracts or early participation of outside of industry investors. To pinpoint actual reasons require further investigation of this market, which lies beyond the scope of the current work. We see this work as part of an on-going attempt to understand better the relationship between energy markets and other physical and financial markets, for incorporating the risk premium in making informed hedging decisions in industry, and for relating futures prices to the forecast prices produced by industry.