Commodity Prices, Scarcity and Risk Premia

Commodity Prices, Scarcity and Risk Premia Saqib Khan Zeigham Khokher Timothy Simin ∗

March 2006 Abstract

We study how risk premia in commodity futures markets vary with scarcity as summarized in a dataset of inventories. We document 1) that futures risk premia increase during periods of scarcity when stocks are withdrawn from storage 2) that scarcity in the crude oil markets is a common risk factor across markets 3) a decreasing relationship between convenience yields and inventories except for crude oil, where the spot price plateaus on the upside and 4) that contingent claims models in current practice, which omit the interaction between inventories and futures risk premia under-report risk exposures. Our evidence is important for subsequent theoretical research, as it points out that inventories relate to risk premia in a manner distinct from that noted in prior research.

∗

Khan is at the Ivey School of Business, University of Western Ontario, 0N56 NCMRD Building, London,

ON, Canada N6A 3K7. Khokher is at the Ivey School of Business, University of Western Ontario, 2N54 NCMRD Building, London, ON, Canada N6A 3K7. Simin is with the Smeal School of Business, The Pennsylvania State University, 345 Business Building, University Park, PA 16802 USA. We would like to thank Hank Bessembinder, Neil Brisley, Murray Carlson, Wayne Ferson, Sheridan Titman and Jason Wei for comments. Special thanks are due to Walid Busaba and Jaime Casassus for extensive discussions. Thanks are also due to Maria Coronado (American Petroleum Institute), Kashif Rashid (Schlumberger) and Scott Byrne (NYMEX). Remaining errors are ours alone.

Economists have long reasoned that risk premia cause futures prices to deviate from expected spot prices. While theory clearly motivates the role of inventory levels as a determinant of commodity futures risk premia, a lack of reliable commodity inventory data has hindered empirical research. Consistent with the theory that inventories contain valuable information as indicators of scarcity, plain vanilla and digital options written on the weekly withdrawal of crude oil and natural gas stocks from inventory are now traded on the New York Mercantile Exchange. In this paper we explore the impact of scarcity upon futures risk premia using a dataset of inventories from among the most liquid commodity markets in the world; the crude oil, copper, gold and natural gas markets. 1 Supporting the hypothesis that futures risk premia increase during periods of scarcity, we document that inventories have forecast power for realized futures returns. We also find cross market effects that suggest scarcity in the oil market plays an important role across commodity markets.

Furthermore, we show that the current practice of

omitting the interaction between inventories and futures risk premia when modeling contingent claims is costly. In particular, we demonstrate how this omission causes current models of commodity prices to severely under-report risk exposures in Value at Risk applications. Finally, we provide evidence on the relation between inventories and convenience yields beyond the relation between spot prices and convenience yields suggested by the theory of storage. While the evidence documenting non-zero conditional futures risk premia is well known, the determinants of the premia remain a subject of debate. Brennan (1958) suggests that storage operators become increasingly risk averse as their stocks accumulate, increasing net hedging pressure, and hence increasing risk premia, but evidence on this positive relation between inventories and risk premia remains sparse. Inventories act as a state variable which will generally be lower during periods of relative scarcity, i.e. when stocks are withdrawn and prices are relatively high, we find that futures risk premia increase during periods of scarcity. Inventories retain their forecast power even after

1

For crude oil we collect U.S. inventories data from the American Petroleum Institute’s weekly bulletin-

U.S. Natural gas storage data is obtained from the Energy Information Administration’s weekly survey. The New York Mercantile Exchange (NYMEX) provided us with copper and gold data.

1

controlling for net hedging pressure for all commodities except copper.

While it is

plausible that speculators demand greater risk premia during conditions of relative scarcity, these results are somewhat surprising in light of Brennan’s (1958) prediction. 2 In addition to their own-market effects, crude oil and natural gas inventories also possess significant cross-market forecast power. In particular, crude oil (natural gas) inventories are negatively (positively) related to returns on the remaining commodities except gold. In a setting where risk premia covary with economic state variables, return predictability can be traced to forecastability of the corresponding factor premia. The evidence on the cross market impact of crude and natural gas inventories as predictive instruments, is consistent with these instruments being correlated to such factor premia. In particular, the evidence for crude oil suggests that scarcity in this market is a priced factor across a number of commodity markets.

Analogously, Chen, Roll and Ross (1986)

suggest oil price as a priced economic risk within equity markets. After verifying the statistical significance of supply variables for price dynamics, we turn to their economic impact. The contingent claims framework is a natural setting to demonstrate such effects, but extant models cannot be calibrated to our inventory data. We extend the model of Casassus and Collin-Dufresne (CCD 2005) in a parsimonious manner to include inventory as an explicit state variable that can impact risk premia. Calibrating both the CCD (2005) and the extended models to observed time series of forward prices, we then calculate the Values at Risk for an investment in the commodity implied by the two models over our sample period. The extended model generates prices that are, on average, consistent with those calibrated in CCD (2005). However, inventories have an additional impact upon risk premia that causes under-reporting of risk exposures and hence the capital required to cover exposures, sometimes by over 30%. The evidence reported here also speaks to Schwartz (1997), who suggests that commodity futures prices do not satisfy the standard no-arbitrage restriction and so cannot be viewed as “an asset in the usual sense.” A central insight of CCD (2005) that builds on

2

The evidence for natural gas is somewhat consistent with the logic of Brennan (1958) although natural gas

inventories may not track conditions of scarcity as closely as the remaining commodities given our sample for natural gas.

2

the evidence reported in Bessembinder et al. (1995) is that allowing convenience yields to depend upon the spot price resolves the noted paradox. In addition, CCD (2005) suggests that the negative relation between inventory and spot price underlies the convenience-yield spot price relationship. In our data inventories retain a marginal impact upon convenience yields beyond prices, an important finding for subsequent theoretical research. Citing the paucity of reliable inventory data, Fama and French (1988), resort to indirectly testing the theory of storage. In particular, they check whether convenience yields are more variable when positive to test the hypothesis that marginal convenience yields decline with inventory at a decreasing rate. They find supportive evidence even for non-agricultural commodities; a surprising result since declining marginal convenience yields classically explained backwardation only in agriculture commodities. We confirm the Fama and French tests using regressions that directly examine the inverse relationship between convenience yields and inventories. However, convenience yields plateau during periods of low supply/inventory in the crude oil market. This is consistent with both anecdotal and theoretic evidence of commodity prices being essentially capped by either the cost of a substitute good or government intervention. The remainder of this article is organized as follows. After describing the data in the first section, in the next section we revisit results relating convenience yields and inventories as a means of confirming our proxies of scarcity. In Section 3 we document how inventories are related to time varying risk premia. In Section 4, we illustrate the economic significance of this covariation by calibrating our data to a contingent claims model and examining the implications for risk management. The paper ends with a brief conclusion.

I. Data Description A. Futures Prices, Treasury Yields and Net Hedging Pressure Our data is from the copper (CO), crude oil (CL), gold (GC) and natural gas (NG) markets, a composition similar to that studied by Schwartz (1997), and covers the period

3

01:1995 to 05:2004. 3 For these markets weekly settlement prices for futures contracts traded on the New York Mercantile Exchange (NYMEX) are obtained from Bloomberg. We obtain the third, sixth and ninth closest to maturity futures price and, as is standard practice, we use the nearest-to-maturity contract to proxy for the spot price. The summary statistics for the prices are in Table I and are similar to those in previous studies. We use constant maturity Treasury yields to proxy for zero-coupon bonds. Net hedging pressure is defined as the number of short minus long hedge positions divided by total hedge positions. Net hedging pressure is calculated using Trader’s Commitment data from the Commodity Futures Trading Commission (CFTC). 4

B. Discretionary Commodity Inventories To capture relative scarcity of a commodity we require a proxy for economic inventory. For crude oil we use inventory data from the American Petroleum Institute’s weekly bulletin which contains current data on U.S. inventories of crude oil petroleum products. The natural “working” gas storage data is obtained from the Energy Information Administration. 5 Both the crude oil and natural gas data is published on Wednesday mornings, and are based on reports filed by energy companies operating in the United States. Because data aggregated across a sample of individual firms is not available for copper and gold, we use data on stocks held at the COMEX division of the New York Mercantile Exchange (NYMEX). From these daily data we compute a weekly average.

3

We include natural gas since this energy source differs from the more global crude oil market and is less

susceptible to monopolistic pressures. While inventory data is available for silver and heating oil, results for these commodities are qualitatively similar and available from the authors. 4

The CFTC requires traders to report the nature of their trading activity. It uses the nomenclature

commercials and non-commercials, which equates to a distinction between hedgers and speculators respectively. See de Roon et al. (2000) for an excellent discussion. 5

Working gas is the amount of gas in underground storage available for withdrawal. The actual amount of

gas in storage is larger than this amount. The difference called “base gas” must be present to maintain proper functioning of the storage facilities. Our data spans a period over which working gas in storage formed about 10 % of the total consumption, the discrepancy is made up by imported gas.

4

While it may be possible that squeezes and other temporary effects cause short term deviations between these measures and aggregate inventories, we require only that the two are sufficiently correlated over the long run. There is evidence to suggest that such proxies adequately capture long run variation in economic inventory. Nielsen and Schwartz (2004) find that copper stocks held at the London Mercantile Exchange serve as an adequate proxy and Khan (2005) finds that NYMEX stocks for copper and gold are related to the slope of the term structure of volatility. Although non-discretionary inventories may have convenience value, it is aggregate discretionary inventory that determines the trade-off between the commodities current and future consumption value. The focus of our analysis are stocks held in storage in excess of those already committed to production processes which we proxy for in two different ways.

For the energy commodities we assume that the periodic seasonal

component of the inventory time series represents committed inventory. We extract the discretionary component of inventory using the Seasonal Trend Loess (STL) seasonal adjustment technique developed by Cleveland, Cleveland, McRae, and Terpenning (1990). We choose the STL procedure over the ARIMA seasonal adjustment methods for several reasons. STL is better suited for use with higher frequency data that are not decomposable by standard methods such as the Census Bureau’s X12 ARIMA method. The STL method also provides trend and seasonal components that are robust to outliers and allows for decomposition of a time series with missing values. 6 For the metal commodities we use the stochastic detrending technique advocated by Campbell (1991) to remove the component of inventories committed to production. Brennan (1958) expresses inventory stocks as a percentage of a 13-month average to allow for long term growth in production and stocks.

Since we are interested in how

convenience yields and futures prices respond to more immediate shocks, we remove committed inventories by subtracting off a trailing moving average of the previous four weekly lags. For both metal inventories we replaced six missing values over our sample period with the previous week’s data before extracting the stochastic trends. 6

Interested readers can review the March, 1990 issue of the Journal of Official Statistics (Vol. 6, No. 1,

1990) which is dedicated entirely to the STL procedure.

5

Figure 1 shows time series plots of the four inventory series. While the amount of seasonality in the natural gas inventories is obviously large, it is more difficult to visually ascertain any seasonality in the other three inventory series. To gauge the amount of seasonality we regress the levels of the inventories separately on 52 weekly and 12 monthly dummies. Panel A of Table II shows the number of significant dummy variables for each of these regressions. Using either weekly or monthly seasonal dummies, both crude and natural gas inventories exhibit a significant seasonal component. There is little evidence of a seasonal component in the metal inventory levels, supporting the use of a lagged moving average to extract the discretionary component of these inventories. Figure 2 presents the time series of inventories after removing the periodic seasonal component from the energy inventories and after removing the stochastic trends from the metals inventories. Panel B of Table II contains the results from the seasonal dummy regressions where we have replaced the inventories with their discretionary counterparts.

After adjustment, there is little evidence of any important seasonal

components using weekly or monthly dummies. 7 Summary statistics for the discretionary storage data are reported in Table III. Panel B reports the correlation coefficients between these series.

Except for the

correlation of discretionary gold inventories with those of copper and crude oil, the correlations are positive and range from 17 % to 50 %. The last row of Panel B notes that for copper, crude and gold, withdrawals of stocks from storage are significantly negatively correlated with the level of stocks held in storage. This indicates that inventory levels are generally lower during periods of relative scarcity (heightened demand) when stocks are

7

Our results are robust to the seasonal adjustment method and the amount of detrending. Applying the

multiplicative seasonal ARIMA model to the energy inventories and picking the best specification based on the AIC statistic does not qualitatively change the results of the univariate regressions of convenience yields on inventories we later present for either of the energies. Using the SARIMA adjusted inventories produces slightly larger t-statistics for crude and only modestly lower, but still significant results for natural gas. In the same regressions, increasing the number of lags used in detrending the metals also does not qualitatively change the results. Indeed, we find that increasing the number of lags used to detrend both copper and gold results in consistently higher t-statistics except for the longer maturities for gold. These results are available upon request.

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withdrawn from storage. Such an effect is least apparent for natural gas.

II. Convenience Yields While not the focus of our paper, we use previous results relating inventories and convenience yields to confirm that our proxies are capturing the relevant aspects of scarcity. In this section we use direct tests that incorporate inventories to replicate tests of the theory of storage that had previously been done by indirect methods. The confirmation of the indirect results gives us confidence that our discretionary inventories are sufficient proxies for scarcity. The notion of a convenience yield for commodities is intuitive. Those who hold stocks of a commodity receive benefits from their inventory because they are able to respond to supply and demand shocks. Inventories generate such benefits for holders of the commodity, but not for holders of the futures contract on the commodity. For a commodity in backwardation, the futures price falls below the spot price. The difference between the spot and futures price is the convenience yield. We follow Fama and French (1987) and calculate the convenience yield as the negative of the interest adjusted basis which is defined as

Ft ,T − S t St

− rt . Here Ft ,T is the futures price at time t maturing in T

months, S t is the spot price of the commodity at time t and rt is the matching maturity zero-coupon bond yield. As reported in Table IV the mean convenience yield is largest for crude oil, followed by copper and smallest for gold. Table IV column 4 shows that convenience yields are typically positive for the commodities we consider, the forward curve is downward sloping most often for crude oil and copper. However, the convenience yield is most variable for natural gas, which has a standard deviation that is often double that of other commodities, and least variable for gold. Fama and French (1987) argue that marginal convenience yields that decline at a decreasing rate are evidenced by convenience yields that are more variable when positive. Except for three month copper futures which nearly always have positive convenience yields and gold at the longest maturity, our results support the Fama and French (1987) conjecture. The p-values for an

F-test for equality of the variances are in the final column of Table IV.

7

The theory of storage predicts a negative relationship between convenience yields and inventories. In Table V we show the results from the following regression:

ct = a + bInvt + et.

(1)

Where ct is the convenience yield at time t and Invt is the discretionary component of inventory extracted as described above. The column Inv contains the slope coefficients of the inventory level and the columns labeled t are the t-statistics. 8 A linear trend has been removed from the copper and gold convenience yields and all the storage data are studentized to normalize the coefficient magnitudes. The convenience yield-inventory relationship is significant and negative for crude oil, copper, and natural gas, all of which are important inputs to production processes. For crude oil, up to 42% of the variation in the convenience yield can be explained by inventories. For gold, the relationship is only significant at the longest maturity where less than 3 % of the variation in the convenience yield can be traced to inventories. The weaker relation of convenience yields with inventories for gold is expected since gold is often held for investment purposes rather than as input to production. Overall, these results suggest that the convenience yield is highest when inventories are low, i.e. the benefit of holding inventories is greatest during periods of relative scarcity or heightened demand as predicted by the theory of storage. We also test the prediction that the convenience yield on inventory falls at a diminishing rate as inventory increases by regressing the convenience yield on inventory and its square. Table VI contains the results from the following regression:

ct = a + b1Invt + b1Invt2 + et.

(2)

We find significantly positive coefficients on the squared term for copper and natural gas, confirming the theory of storage prediction. Figure 3 illustrates this relationship in the 8

In the analysis that follows we employ Newey and West standard errors using 52 lags given weekly data;

our results are qualitatively similar at other lags.

8

data, showing that slope of the relationship is high when inventory is low. Consistent with the results of the linear specification, the results for gold are the weakest. Distinct from the copper, natural gas and gold markets, convenience yields in the crude oil market plateau during periods of low supply/inventory, which indicates an effective cap on the spot price of crude oil. 9

Carlson, Khokher and Titman (2006) develop a general

equilibrium model of exhaustible resources where commodity prices are essentially capped by the cost of a substitute good. Our evidence for crude oil constitutes (weak) support for their specification.

III. Forecastable Futures Returns and Scarcity A. Background When risk premia cause futures prices to deviate from expected spot prices the result is non-zero expected futures price changes which would otherwise be random. Keynes (1930) suggests that futures prices are downward biased estimates of the expected spot price and that this bias results in positive risk premia. 10 In this Keynesian view, risk premia serve as compensation to speculators for the insurance that they provide to hedgers. Cootner (1960) noted that risk premia can be positive or negative depending on the sign of the (net) demand for risk reduction.

In support of these predictions,

Bessembinder (1992) documents that futures returns increase with net hedging pressure (as defined above) and De Roon et al. (2000) show that such pressure can have effects across markets. A second aspect to risk premia outlined in Dusak (1973), Black (1976) and Breeden (1980), assumes that all claims can be marketed free of cost. In this setting, risk premia depend on the sensitivity of price changes to latent economic state variables. A hedge portfolio perfectly correlated with such a variable and uncorrelated with the others 9

Recent events also lend credence to such a cap, “The US and Europe are releasing more emergency crude

oil than refineries in the Gulf of Mexico can handle, reinforcing suspicions that governments are using the crisis triggered by Hurricane Katrina to cap record oil prices” – Financial Times (September 9, 2005). 10

Interestingly, Hardy (1940), held the opposite view, that futures markets represent the equivalent of a

casino in which speculators should be willing to pay for the “privilege” of gambling.

9

is expected to earn the factor premium. Non-zero expected futures returns stem from nonzero beta coefficients and the corresponding factor premia. 11 As is consistent with such models of time varying risk premia, Bessembinder and Chan (1992) document that instruments known to possess forecast power for equities also forecast futures returns. In addition, Bessembinder (1992) examines slope coefficients from regressions of futures returns on equity index returns. Brennan (1958) suggests that storage operators become increasingly risk averse as their stocks accumulate and beyond a critical level operators will seek to hedge their stocks. The increased hedging demand then elevates futures risk premia, consistent with the Keynesian view. Schwartz (1997) mentions the significance of covariance between the market price of risk and inventories, but there is little evidence on this relation. Summary statistics for futures returns are reported in Table VII. We use the procedure outlined in Bessembinder (1992) to construct continuous series of futures returns for the third, sixth and ninth closest to maturity contracts. 12 These return series quantify the experience of an investor who continuously holds the “ith” nearest to maturity contract. As reported in Bessembinder (1992), the mean unconditional futures returns for the return series are not statistically significant. 13 The mean return on natural gas is largest, but these returns also have the largest standard deviation. Mean returns on gold and copper are close to zero and the returns on gold are the least volatile. Except for gold, the standard deviation of these returns decreases along the term structure, which is consistent with a mean reverting component to prices. Table VII also documents the unconditional beta relative to the market, proxied by the S&P 500 index. De Roon et al. (2000) have found that for close to maturity contracts 11

In this setting predictability in returns can be traced to forecastability of factor premia.

12

Although the term “futures return” is a misnomer, it is standard. Thus percentage changes in future prices

are henceforth referred to as futures returns. All returns are calculated from futures prices on the same contract and not across contracts with different maturity dates. 13

In unreported analysis, we find realized futures returns for copper are significantly positively correlated to

those of gold (12-16%). Natural gas returns are similarly correlated with those of crude oil (21-25%). However, the remaining cross-correlations between realized returns, including those with returns on the S&P 500 index are statistically insignificant.

10

this relationship is insignificant, except for financial futures and the precious metals. The results in Table VII mirror these conclusions for longer maturity crude oil and natural gas, but copper and gold do have systematic risk components, respectively positive and negative. Finally, Table VII reports that net hedging pressure is a significant determinant of futures returns for all contracts. Net hedging pressure displays significant variation, but is positive on average, suggesting that producers are the predominant hedgers.

B. Futures Risk Premia, Scarcity and Withdrawals In this subsection, we study covariation between futures risk premia and scarcity.14 We examine whether realized futures returns are higher during periods of relative scarcity, when stocks are withdrawn from storage. To do so, we regress realized futures returns for the i ’th continuous futures return series, rti , on withdrawals from storage, ΔSt − L ,

rti = α i + bi ΔSt − L + ε ti .

(3)

Here, withdrawals from storage are lagged (L) and so within the information set before the forecast period commences. 15 The results of this regression are presented in Panel A of Table VIII.

The slope coefficient relating realized futures returns to withdrawals is

consistently positive for all return series. Further, this coefficient is statistically significant at the 2% level, for crude oil and natural gas for all of the considered maturities. For copper and gold, the results remain significant but at a lower (10%) significance level. The evidence that withdrawals from storage have forecast power for realized futures returns is consistent with the hypothesis that the relative current and future scarcity of the commodity impacts futures risk premia, i.e. speculators demand greater risk premia during 14

Futures premia are defined as the expected future spot price less the current futures price. Since the

expected future spot is not observed, the futures premium is also unobservable. An equivalent definition of futures premia can be stated in terms of expected futures returns. Expected futures returns are unobservable, realized futures returns are used which measure the futures risk premium with error. 15

We relax Brennan’s (1958) assumption of no time lag between sales out of stocks by suppliers of storage

and utilization by households since there is likely to be a lag and its length (L) may vary by market.

11

conditions of relative scarcity, at which time stocks are withdrawn from storage. As noted futures risk premia may be impacted by hedging demands and systematic risk. While neither the S&P index returns nor hedging pressure (due to a reporting lag) are predictive instruments, they can have a contemporaneous effect upon futures returns. Thus, a control regression will not take away from the forecast power of withdrawals, but will speak to the channel through which this predictability operates. After accounting for both of these effects, Table VIII documents that the results for copper become weaker, suggesting that for copper the forecast power of withdrawals may be traced to hedging demands. For the remaining commodities, Table VIII shows that relative scarcity remains a significant determinant of futures premia.

C. Futures Risk Premia, Scarcity and Inventory Levels The previous subsection documents covariation between futures risk premia and relative scarcity using withdrawals from storage. In this subsection, we examine such covariation using the level of stocks held in storage as suggested by Brennan (1958) and Schwartz (1997). To do so we regress the i ’th continuous futures return series, rti , on the (lagged) inventory level from all four markets. 16 We include each measure of hedging pressure since De Roon et al. (2000) present evidence of cross market hedging pressure effects, as well as the return on the S&P 500 index as additional regressors, 4

4

k =1

l =1

rti = α i + ∑ β i , k Invtk−1 + β SP rt SP + ∑ μ i ,l NHPt + eti .

(4)

Table IX documents the results of this regression for the sixth maturity, results for the remaining maturities are qualitatively similar. Realized futures returns are negatively related to the commodities own (lagged) inventory levels for crude oil and gold at the 1%

16

In unreported results we find that, using withdrawals as the instrumental variable yields negligible cross

effects on futures returns except for natural gas. Realized natural gas futures returns are significantly positively related to lagged withdrawals of stocks on crude oil.

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level. Point estimates for copper are negative but insignificant. 17 The negative own slope coefficients for copper, crude oil and gold are somewhat surprising, in light of Brennan (1958). Within these markets, the correlation between withdrawals from storage and inventory levels is negative and significant (Table III). This indicates that for these commodities, stocks are withdrawn from storage during periods of scarcity at which time inventory levels are generally low. Thus, the results of the previous subsection are consistent with the negative slope estimates reported here. The slope coefficient relating natural gas returns to its (lagged) own inventory levels is positive and significant at the 1% level. Table III documents a negative but insignificantly small correlation between natural gas withdrawals and its inventory level, indicating that natural gas inventories do not track conditions of scarcity as closely as the remaining commodities, we return to this issue in the next section. However, the noted positive slope coefficient is consistent with the logic of Brennan (1958). As noted above, a class of models predicts common variation in expected returns across markets that stems from sensitivity to shared economic factors.

Table IV

documents that in addition to their own-market effects, crude oil and natural gas inventories also have a cross-market impact.

In particular, crude oil (natural gas)

inventories are significantly negatively (positively) related to returns on the remaining commodities except gold. Gold inventories are cross-related to only natural gas returns (statistically significant at only this maturity), while copper inventories do not display cross market forecast power. If risk premia covary with economic state variables then predictability in returns comes from the forecastability of the corresponding factor premia. 18 The abilities of crude and natural gas inventories to predict across markets is consistent with correlation with factor premia. The cross market strength of natural gas suggests that while the positive own slope coefficient discussed above may stem, in part, 17

Consistent with the findings using withdrawals, omitting net hedging pressure in the joint regression

restores the significance of the slope coefficient of copper inventories for its own returns. 18

Within latent variable models risk premia covary with economic state variables that correspond to factor

premia, evidence of forecastability in returns is consistent with the instruments being correlated to either factor premia or beta coefficients. Bessembinder (1992) finds little variation in betas; if betas are assumed constant then predictability in returns will due to forecastability of factor premia.

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from hedging demands, a component of this forecast power may also stem from covariation with a priced common factor. Bessembinder and Chan (1992) document that consistent with a prediction of Fama (1991), the forecast power of instrumental variables from the equity and bond markets carries over to futures markets. They take this evidence to be consistent with latent variable models that predict time variation in risk premia. Reversing this logic to examine whether instrumental variables from the futures markets possess forecast power for equity markets, we find (weak) evidence in support of such forecastability. Returns on the S&P 500 index are related to crude oil and gold inventory levels but statistically significant only at the 10% level. While Chen, Roll and Ross (1986) suggest oil price as a priced economic risk within equity markets the evidence for crude oil here suggests scarcity in this market is a priced factor across a number of commodity markets as well.

IV. Empirical Evidence and Implications for Risk Management The previous section documented covariation between risk premia and scarcity, in this section we relate the economic significance of such covariation within a contingent claims setting. We find that inventories have an additional impact upon risk premia that is omitted from contingent claims models in current practice. When using the models to estimate Values at Risk this omission causes severe under-reporting of risk exposures. Furthermore, our evidence points out that recent extensions of the classic contingent claims models cannot be solely attributed to the theory of storage.

A. Scarcity, Risk Premia and Contingent Claims Analysis The indirect evidence of Fama and French (1988) on the theory of storage motivates recent extensions of the classic contingent claims models. 19

For instance,

following the literature on affine models of the term structure [see Duffie, Pan and 19

Such reduced form models assume an exogenous price process that is often calibrated to the term structure

of historical and current futures prices and then used for risk management, option valuation and real investments.

These recent extensions can have a dramatic economic impact, despite being calibrated

without inventories. Directly studying the impact of omitted inventory information is a logical step forward.

14

Singleton (2000) and the references therein] Casassus and Collin-Dufresne (CCD 2005) present a three factor model of commodity prices, nevertheless such extensions omit inventory as an explicit state variable. To explore the empirical content of our inventory data, we extend the model of CCD (2005) in a parsimonious manner to include inventory as a state variable that covaries with risk premia. 20 The extended model nests the model of CCD (2005), which in turn nests those of Brennan and Schwartz (1985), Schwartz (1997) and Schwartz and Smith (2000). The four economic factors in the extended model, { r (t ) , q (t ) , δ (t ) , X (t ) }, are the

risk free rate, inventory levels, convenience yields and the logarithm of the spot price, respectively.

The convenience yield, δ (t ) , can be decomposed into an autonomous

component, δˆ(t ) , and other components linearly related to the short rate, inventory and the spot price. Similarly, inventory has its own stochastic component but is also linearly related to interest rates,

q (t ) = qˆ (t ) + α rq r(t)

(5)

δ (t ) = δˆ(t ) + α r r(t) + α q q(t) + α X X(t) = δˆ (t ) + [ α r + α q α rq ]r(t) + α q qˆ (t ) + α X X(t).

(6)

The focus of our study is the representation, Wˆ (t ) = { r (t ) , qˆ (t ) , δˆ(t ) , X (t ) } and the dynamics of these economic variables under the physical measure (P) are, 21

20

dr (t ) = κ rP [ θ rP – r(t)] dt + σ r dZ r (t )

(7.1)

dqˆ (t ) = κ qˆP [ θ qˆP – qˆ (t)] dt + σ qˆ dZ qˆ (t )

(7.2)

dδˆ(t ) = κ δPˆ [ θ δPˆ – δˆ (t)] dt + σ δˆ dZ δˆ (t )

(7.3)

For ease of comparison, we strive for a parsimonious extension that remains similar in notation, price

dynamics and the treatment of risk premia. 21

In Appendix A, we derive the dynamics of these four economic factors, starting with their definition as

functions of a canonical four factor Gaussian state vector.

15

P dX (t ) = { κ Xr [ θ rP – r(t)] + κ XPqˆ [ θ qˆP – qˆ (t)] +

κ XPδˆ [ θ δPˆ – δˆ (t)] + κ XP [ θ XP – X(t)] } dt + σ X dZ X (t ) (7.4) where,

κ rP = κ rQ - β rr κ qˆP = κ qQˆ - β qˆqˆ κ δPˆ = κ δQˆ - β δˆδˆ

θ rP =

κ rQθ rQ + β 0 r κ rQ − β rr

θ qˆP =

κ θ + β 0 qˆ κ qQˆ − β qˆqˆ

θ δPˆ =

κ δQˆ θ δQˆ + β 0δˆ κ δQˆ − β δˆδˆ

κ XP = α X - β XX

θ XP = β 0 X −

σ x2 2

Q qˆ

P κ Xr = α r + α qα rq – 1 - β Xr

κ XPqˆ = α q - β Xqˆ

Q qˆ

κ XPδˆ = 1 - β Xδˆ

(7.5)

P − κ Xr θ rP − κ XPqˆθ qˆP − κ XPδˆθ δPˆ κ XP

The dynamics under the risk neutral measure (Q) are readily obtained by setting the risk premia parameters ( β 0 , β ) in the above system to zero. Within the above model risk premia are specified to be linear in the economic variables as follows, ⎛ Z rQ ⎞ ⎛ Z r ⎜ Q⎟ ⎜ ⎜Z ⎟ ⎜ Z d ⎜ qQˆ ⎟ =d ⎜ qˆ ⎜ Z δˆ ⎟ ⎜ Z δˆ ⎜ZQ ⎟ ⎜Z ⎝ X⎠ ⎝ X

⎧⎛ β 0 r ⎞ ⎟ ⎪⎜ ⎟ -1 ⎪⎜ β 0 qˆ ⎟ +Σ ⎨⎜ β ⎪⎜ 0δˆ ⎟ ⎟ ⎪⎜⎝ β 0 X ⎠ ⎩

⎞ ⎛ β rr ⎟ ⎜ ⎟ ⎜ β qˆr ⎟ + ⎜β ⎟ ⎜ δˆ r ⎟ ⎜β ⎠ ⎝ Xr

β rqˆ β qˆqˆ β δˆ qˆ β Xqˆ

β rδˆ β qˆδˆ β δˆ δˆ β Xδˆ

β rX β qˆX β δˆ X β XX

⎞⎛ r (t ) ⎞⎫ ⎛ σr ⎟⎜ ⎟⎪ ⎜ ⎟⎜ qˆ (t ) ⎟⎪ ⎜ σ qˆ ⎟⎜ δˆ (t ) ⎟⎬ ; Σ= diag ⎜ σ δ ⎟⎜ ⎟⎪ ⎜⎜ ⎟⎜ X (t ) ⎟⎪ ⎠⎭ ⎝σ X ⎠⎝

⎞ ⎟ ⎟ (8) ⎟ ⎟⎟ ⎠

We restrict all cross risk premia parameters except those involving the spot price to zero. 22 The results of the previous section document strong covariation between risk premia and scarcity which is reflected in β Xqˆ , extant models implicitly restrict this parameter to zero. We estimate the model using standard maximum likelihood techniques and with

22

This ensures that the autonomous component of the convenience yield remains independent of the

remaining factors under the physical measure, a restriction similar to that imposed in CCD (2005); many other models ignore such covariance.

16

weekly data from the four commodity markets and the U.S. treasury markets. 23 As noted in Table X most parameters are significant indicating that previous models do not capture the information content implicit in the inventory data. In particular, the point estimates for covariation between risk premia and inventory (β Xq) ) are negative for all commodities.

This evidence further confirms that risk premia do in fact increase during periods of relative scarcity (low inventory).

In the previous section, we confirmed that during

conditions of scarcity when stocks are withdrawn from storage, inventory levels will generally be low and risk premia high for all markets but natural gas. Since the link between withdrawals and inventory levels is weaker for natural gas, while realized futures returns remained significantly related to withdrawals the implied negative covariance with inventory levels was not apparent. In the current affine setting, the negative covariance between risk premia and inventory levels is apparent for all commodities, including natural gas. Fama and French (1988) show how a substantial increase in mean reversion can result from negative covariation between spot prices and risk premia. Analogously, mean reversion will also increase due to covariation between risk premia and relative scarcity. Extant contingent claims models do not consider this possibility, we show below that this omission is economically significant. Fama and French (1987) study covariation between risk premia and futures prices, however their results on such covariation are inconclusive.

CCD’s (2005) central

empirical findings are that risk premia covary negatively with prices and that convenience yields are positively related to prices and interest rates. In our sample, we find covariation between risk premia and prices ( β XX ) to be negative but insignificant for all commodities but gold. Consistent with our univariate results, point estimates for the convenience yieldinventory ( α q ) relationship are negative and significant for crude oil and natural gas. In addition, convenience yields remain positively related to interest rates ( α r ) and spot prices

23

We use observations on the second through the eighth closest to maturity futures contract over our sample

period to calibrate the model. For details of this standard calibration procedure, we refer the interested reader to Appendix E of CCD (2005).

17

( α X ) for all markets but gold. Bessembinder et al. (1995) study mean reversion in asset prices, arguing that a high convenience yield accompanying a high spot price is indicative of investors anticipating that prices will revert. CCD (2005) surmise that since “periods of scarcity e.g., low inventory, correspond to high spot prices” this underlies the convenience-yield spot price relationship. A subset of the factors that affect convenience yields also impact inventories and prices. Since inventories have a marginal impact upon convenience yields beyond prices, the convenience yield-price relationship ( α X ) is more complex then predicted by the theory of storage. The positive relationship between convenience yields and interest rates ( α r ) is well known empirically, but most equilibrium models of commodities, for example Carlson, Khokher and Titman (2006), do not speak to this effect. Nevertheless, CCD (2005) posits that this relationship is consistent with the theory of storage as higher interest rates induce lower inventories and hence higher convenience yields. The point estimates for α rq suggest that inventory levels do not decrease with interest rates for copper and crude oil. While gold inventories do decline with interest rates, convenience yields are not positively related to interest rates within this market. This evidence suggests direct variation in convenience yields with interest rates that is independent of storage. 24

B. Risk Management In the previous subsection we document that relative scarcity retained a marginal impact upon risk premia within the extended model. Negative co-variation between risk premia and inventories ( β Xqˆ ) impacts the drift of the extended model through its positive effect upon the reversion parameter, κ XPqˆ (see Equation 7.4). When inventory levels are above their long run mean this parameter decreases the drift of the price process and vice versa. Since contingent claims models in current practice ignore covariation between risk premia and scarcity they do not account for its effect on the drift of the price process, this 24

Since interest rates are correlated with economic activity, this evidence is consistent with Fama and

French’s (1988) suggestion that convenience yields vary across the business cycle.

18

creates a bias in the distribution of future spot prices implied by these models. For instance, when inventory levels are high our extended model implies a future spot price distribution with a lower mean than that implied by current models. In this case, current models will under-report the riskiness of a position in the spot asset and when inventory levels are high current models will over-report risk exposure. In this section, we report the economic significance of covariation between inventories and risk premia by documenting biases in risk exposures implied by models in current practice. These biases are both substantial and frequent within our empirically observed data. We quantify the impact of this bias using the Value at Risk (VaR) for a unit investment in the commodity over a given holding period. We use a standard experimental design from the literature (see, for example, Carlson, Khokher and Titman (2006)). 25 At each date in our sample the Value at Risk is calculated using parameter estimates from our extended model (Table X), this generates a time series of Value at Risk measures. We then re-calculate the Values at Risk using the model of CCD (2005). This experiment is repeated for quarterly holding periods up to a year. Since the model of CCD (2005) restricts β Xqˆ to zero, our experiment captures the effect of ignoring this covariance. 26 More specifically, defining St as the spot price at time t, we compute the Value at Risk as, St -V. Here, V is the price from the price distribution at the end of the holding period (time T) such that P(ST > V) = 95%. In other words, our Value at Risk measure is the difference between the current spot price of the commodity and the minimum price, with 95% probability, we expect to receive at the end of the holding period (time T). Thus, a positive Value at Risk is the dollar amount representing the “most” that an

25

Here we illustrate the economic impact of inventories instead of a horse race between models which

would require an out-of-sample analysis. CCD (2005) demonstrate biases relative to the Schwartz (1997) model by picking a point in their state space, the biases documented here occur with some frequency within the empirically observed data. 26

The implications for risk exposures is similar, when the extended model is compared to a restricted

version of itself that only sets

β Xqˆ

to equal zero.

19

investor in a unit of the commodity can lose, with 95% probability. 27 Figure 4 documents the under-reporting of Values at Risk by the model of CCD (2005). While the figure depicts results for a quarterly holding period, results at the other horizons are qualitatively similar. Under-reporting of the Value at Risk relative to our extended model is calculated as VaR(Extended) - VaR(CCD).

Here positive under

reporting numbers imply that the VaR(CCD) is lower than the “true” Value at Risk, meaning that the former yields a more optimistic view of the terminal price of the investment. Figure 4 expresses this under-reporting as a percentage of the “true” Value at Risk. The figure shows that the model of CCD (2005) can significantly under-report the risk of an investment relative to the extended model during conditions of relative abundance (i.e. high inventory levels). The under-reporting is most frequent for gold, but can exceed 30 % in each market.

V. Conclusions We use a dataset of inventories that has not been widely used to show that futures risk premia increase during periods of scarcity. This result is somewhat surprising in light of Brennan (1958). Contingent claims models in current practice omit such variation in risk premia and this causes them to severely under-report risk exposures during periods of relative abundance. The policy implications of the current under-reporting are clear, inadequate economic capital is being held to cover exposures to such commodities during periods when caution is perhaps most in order. Recent equilibrium models of commodity prices (e.g Carlson, Khokher and Titman (2006) or Routledge, Seppi and Spatt (2000)) assume risk neutrality and cannot speak to time variation in risk premia. Therefore, the evidence reported here provides useful guidance for subsequent theoretical research into the determinants of futures risk premia. A theoretical model analyzing the joint optimal production and storage decision in a 27

Given the Gaussian holding-period price distribution, the Value at Risk can be computed from the first

and second moments of the implied price distribution. The implied price distribution is obtained via the conditional characteristic function, by using standard results on Gaussian-affine models; see for example Duffie, Pan, and Singleton (2000).

20

setting similar to Carlson, Khokher and Titman (2006), but modified to incorporate risk premia, would be a natural extension of the current empirical study. Furthermore, the evidence of this study suggests that convenience yields are related to both interest rates and prices but that neither relation can be attributed solely to the theory of storage. Understanding the causes of these sources of mean reversion is another important avenue for theoretical research.

21

References Bessembinder, Hendrik, 1992, Systematic Risk, Hedging Pressure, and Risk Premiums in Futures Markets, Review of Financial Studies 5, 637-667. Bessembinder, Hendrik, Jay Coughenour, Paul Seguin, and Margaret Smoller, 1995, Mean Reversion in Equilibrium Asset Prices: Evidence from the Futures Term Structure, Journal of Finance 50, 361-375. Black. Fischer, 1976, The Pricing of Commodity Contracts, Journal of Financial Economics 3, 167-179. Brennan, Michael, 1958, The Supply of Storage, American Economic Review 48 (March) 50-72. Brennan, Michael, Eduardo Schwartz, 1985, Evaluating Natural Resource Investments, Journal of Business 20, 135-157. Carlson, Murray, Zeigham Khokher, Sheridan Titman, 2006, Equilibrium Exhaustible Resource Price Dynamics, forthcoming Journal of Finance. Casassus, Jaime and Pierre Collin-Dufresne, 2005, Stochastic Convenience Yield Implied from Commodity Futures and Interest Rates, Journal of Finance 60, 2283-2331. Chen, Nai-fu, Richard Roll, and Stephen Ross, 1986, Economic Forces and the Stock Market, Journal of Business, 59, 383-403. Cleveland, R. B., William. S. Cleveland, J.E. McRae, and I. Terpenning, 1990, STL: A Seasonal-Trend Decomposition Procedure Based on Loess. Journal of Official Statistics, 6, 3–73. De Roon, Frans A., Theo E. Nijman, and Chris Veld, 2000, Hedging Pressure Effects in Futures Markets, Journal of Finance 55, 1437-1456. Duffie, Darrell, Jun Pan, Kenneth Singleton, 2000, Transform Analysis and Asset Pricing for Affine Jump-diffusions, Econometrica 68, 1343-1376. Fama, Eugene and Kenneth French, 1987, Commodity Futures Prices: Some Evidence on Forecast Power, Premiums and the Theory of Storage, Journal of Business 60, 55-73. Fama, Eugene and Kenneth French, 1988, Business Cycles and the Behavior of Metals Prices, Journal of Finance 43, 1075-1093. Ferson, Wayne and Campbell Harvey, 1993, The Risk and Predictability of International Equity Returns, Review of Financial Studies 6, 527-566.

22

Kaldor, Nicholas, 1939, Speculation and Economic Stability, Review of Economic Studies, VII, 1-27. Khan, Saqib A., 2006, Evidence on the Samuelson Hypothesis, working paper, Ivey Business School, The University of Western Ontario. Litzenberger, Robert, Nir Rabinowitz,, 1995, Backwardation in Oil Futures Markets: Theory and Empirical Evidence, Vol . 50, No. 5, pg. 1517-1545. Ng, Victor, Craig Pirrong,, Fundamentals and Volatility: Storage, Spreads, and the Dynamics of Metals Prices, 1994, Journal of Business, vol. 67, no. 2, pg. 203-230. Nielsen, Martin and Eduardo Schwartz, Theory of Storage and the Pricing of Commodity Claims, 2004, Review of Derivatives Research, 7, 5-24. Routledge, Bryan, Duane Seppi, and Chester Spatt, 2000, Equilibrium Forward Curves for Commodities, Journal of Finance 55, 1297-1338. Schwartz, Eduardo, 1997, The Stochastic Behavior of Commodity Prices: Implications for Valuation and Hedging, Journal of Finance 52,923-973. Schwartz, Eduardo and James Smith, 2000, Short-Term Variations and Long-Term Dynamics in Commodity Prices, Management Science 46, 893-911. Working, Holbrook, 1948, Theory of the Inverse Carrying Charge in Futures Markets, Journal of Farm Economics 30 (February) 1-28.

23

Table I: Futures Contract Prices Weekly settlement prices for copper (CO), crude oil (CL), gold (GC) and natural gas (NG) futures contracts traded on the New York Mercantile Exchange (NYMEX) obtained from Bloomberg over the period 01:1995 to 05:2004, 491 observations. The data are the third, sixth and ninth closest to maturity futures price and, as is standard practice we use the nearest-to-maturity contract to proxy for the spot price.

Crude (1) Crude (3) Crude (6) Crude (9) Copper (1) Copper (3) Copper (6) Copper (9) Gold (1) Gold (3) Gold (6) Gold (9) Natural Gas (1) Natural Gas (3) Natural Gas (6) Natural Gas (9)

Mean 23.67 23.15 22.45 21.91 90.38 90.03 89.28 88.63 326.15 329.62 334.91 340.44 3.27 3.27 3.22 3.20

Median 22.92 22.50 21.54 20.75 81.35 81.80 82.40 82.75 315.10 317.30 320.50 323.40 2.65 2.68 2.62 2.55

Std. Dev. 6.44 5.92 5.26 4.77 22.36 21.20 19.05 17.33 48.10 48.34 49.05 49.97 1.56 1.44 1.34 1.33

Min 11.16 11.78 12.20 12.59 60.40 60.85 62.40 62.35 253.00 254.20 257.30 261.70 1.34 1.45 1.55 1.64

Max 41.50 41.00 39.06 37.32 144.15 140.75 134.85 132.05 427.30 429.30 432.20 435.80 9.98 8.29 6.89 7.13

Skew 0.25 0.29 0.36 0.43 0.87 0.87 0.84 0.82 0.30 0.30 0.31 0.33 1.23 0.91 0.79 0.90

Kurtosis 2.35 2.38 2.47 2.59 2.47 2.53 2.51 2.53 1.64 1.64 1.63 1.63 4.21 2.73 2.35 2.64

24

Table II: Inventories on Seasonal Dummies The table contains the number of significant dummy variables, measured as having a t-statistic greater than 1.96 in absolute value, from regressing each commodity inventory time series on a set of weekly dummy variables (left column) and on a set of monthly dummy variables (right column). Panel A contains the number of significant seasonal dummy variables using the raw inventory series. Panel B contains the number of significant seasonal dummy variables using the inventories adjusted to remove the component committed to production as described in the text.

CL CO GC NG CL CO GC NG

Significant Weekly Dummies Significant Monthly Dummies Panel A: Pre-Adjustment 7 6 1 1 1 1 12 6 Panel B: Post Adjustment 1 1 0 0 1 0 1 1 Table III: Discretionary Commodity Inventories

Summary statistics for the inventories of crude oil, copper, gold and natural gas markets over the period 01:1995 to 05:2004. Crude oil inventory data is from the American Petroleum Institute’s weekly bulletin, natural “working” gas storage data is obtained from the Energy Information Administration. Gold and Copper data are from the New York Mercantile Exchange. These data are reported daily so we use the average level of stocks over the previous week, computed each Wednesday. Panel B contains correlations between the discretionary inventory series. The final row labelled Withdrawals documents the correlation between withdrawals of stocks and the level of stocks held in storage. Only asterisked correlations are not statistically different from zero.

N Mean Median Std. Dev. Kurtosis Skewness

CO 487 631.4 677.5 7471.6 1.08 -0.46

CO CL GC NG Withdrawals

1 0.305 -0.176 0.214 -0.872

CL 491 306783.8 307814.5 17948.8 -0.74 0.04 Panel B: Correlations 1 -0.121 0.504 -0.110

GC 487 14505.6 4416.5 114259.3 1.57 -0.18

NG 491 2050.9 2062.6 260.4 -0.66 0.09

1 0.171 -0.760

1 -0.060*

25

Table IV: Commodity Convenience Yields Summary statistics for the convenience yields of the commodities. The convenience yield is the negative of the interest adjusted basis, defined as

Ft ,T − Pt Pt

− rt . The columns % Pos., Stddev + and Stddev – contain the

percentage of the sample when the convenience yield is positive, the standard deviation of the positive convenience yields and the standard deviation of negative convenience yields. The final column contains the ratio of variances F-test with the null H0: σ2(+)=σ2(-). All numbers are reported as percentages.

Crude (3) Crude (6) Crude (9) Copper (3) Copper (6) Copper (9) Gold (3) Gold (6) Gold (9) Natural Gas (3) Natural Gas (6) Natural Gas (9)

Avg. Std. Dev. % Pos Stddev (+) 5.7 4.3 93.3 3.8 8.0 7.2 85.3 5.3 9.8 9.3 84.1 6.3 4.2 2.8 99.0 2.8 4.6 4.5 88.6 4.3 5.0 5.9 74.7 5.7 3.0 1.4 100.0 1.4 1.5 0.9 92.5 0.8 -0.2 0.7 40.3 0.5 2.4 9.5 62.9 7.8 2.6 15.7 53.8 12.7 2.8 16.9 50.1 12.5

Stddev (-) 2.6 4.2 5.4 2.4 1.0 0.4 NA 0.1 0.5 6.6 8.3 7.9

F-test (0.003) (0.010) (0.050) (0.444) (0.000) (0.000) NA (0.000) (0.897) (0.011) (0.000) (0.000)

26

Table V: Regressions of Convenience Yields on Discretionary Inventory Levels This table contains the results from regressing convenience yields on the discretionary component of inventory levels. A linear trend has been removed from the copper and gold convenience yields. All storage data are studentized. The column Inv contains the slope coefficients of the inventory level. The column labeled t are the t-statistics calculated using Newey and West standard errors with a 52 lags.

Inv -1.76 -4.30 -6.09 -0.31 -1.06 -1.60 -0.10 -0.09 -0.10 -4.39 -8.39 -10.59

Crude (3) Crude (6) Crude (9) Copper (3) Copper (6) Copper (9) Gold (3) Gold (6) Gold (9) Natural Gas (3) Natural Gas (6) Natural Gas (9)

AdjR2 0.17 0.35 0.42 0.04 0.16 0.18 0.02 0.03 0.02 0.21 0.28 0.39

t -2.18 -3.36 -3.73 -2.84 -4.29 -3.38 -1.29 -1.95 -2.24 -4.34 -4.89 -6.40

Table VI: Regressions of Convenience Yields on Inventory Levels and Inventory Levels Squared This table contains the results from regressing convenience yields on the discretionary inventory levels and discretionary inventory levels squared. A linear trend has been removed from the copper and gold convenience yields. All storage data are studentized. The column Inv (Inv2) contains the slope coefficients of the inventory level (inventory level squared). The columns labeled t are the t-statistics calculated using Newey and West standard errors with a 52 lags.

Crude (3) Crude (6) Crude (9) Copper (3) Copper (6) Copper (9) Gold (3) Gold (6) Gold (9) Natural Gas (3) Natural Gas (6) Natural Gas (9)

Inv -1.68 -4.17 -5.92 -0.34 -0.98 -1.46 -0.10 -0.09 -0.09 -4.59 -8.61 -10.67

t -2.74 -4.66 -5.57 -3.24 -5.59 -4.64 -1.30 -1.86 -1.77 -4.55 -5.08 -6.13

Inv2 -0.92 -1.69 -2.34 -0.08 0.16 0.32 0.00 0.02 0.03 2.37 2.56 0.78

t -2.09 -2.61 -3.10 -1.65 2.14 2.36 -0.14 0.88 1.08 3.92 2.12 0.54

AdjR2 0.22 0.41 0.49 0.04 0.17 0.19 0.01 0.03 0.02 0.29 0.32 0.39

27

Table VII: Futures Returns Weekly (Wednesday to Tuesday) future returns are calculated as the percentage changes in future prices. All returns are calculated from futures prices on the same contract and never across contracts with different maturity dates. For all commodities we use the 3, 6, and 9 month futures. S&P is the weekly return on the Standard and Poor’s 500 index. The Avg. column contains the average return, column three (t) is the t-statistic from a test of the mean return being zero. Column four contains the standard deviation of each return series. The last four columns report the slope parameters and tstatistics from regressing futures returns (β) and net hedging pressure (μ) on the S&P 500. The tstatistics are calculated using Newey and West standard errors with a 52 lags.

Crude (3) Crude (6) Crude (9) Copper (3) Copper (6) Copper (9) Gold (3) Gold (6) Gold (9) Natural Gas (3) Natural Gas (6) Natural Gas (9) S&P

Avg. 0.2 0.2 0.1 0.0 0.0 0.0 0.0 0.0 0.0 0.3 0.3 0.3 0.2

t 0.95 1.11 1.20 -0.09 -0.04 -0.02 0.05 -0.01 -0.07 0.98 1.53 1.63 1.65

Std. Dev. 3.9 3.1 2.7 3.0 2.6 2.5 1.8 1.8 1.8 6.6 4.3 3.9 2.4

β 0.03 0.04 0.04 0.16 0.16 0.15 -0.07 -0.06 -0.05 -0.05 0.09 0.03

t 0.30 0.50 0.54 3.25 3.47 3.52 -2.22 -2.00 -1.80 -0.44 1.30 0.64

µ 0.10 0.08 0.07 0.02 0.02 0.02 0.01 0.01 0.01 0.11 0.08 0.04

t 5.66 5.47 5.00 2.72 2.69 3.22 4.79 5.06 5.30 4.07 3.47 1.63

28

Table VIII: Regressions of Futures Returns on Withdrawals, S&P, and Net Hedging Pressure

This table contains the results from regressing realized futures returns on lagged withdrawals (ΔS ) levels (Panel A) and on lagged withdrawals (ΔS ) levels, the return on the S&P, and net hedging pressure (NHP) in (Panel B). The inventory data is seasonally adjusted as described in the text. Storage withdrawals for the Crude series are lagged once while the remaining storage withdrawals are lagged twice. The columns labeled t are the t-statistics calculated using Newey and West standard errors with a 52 lags.

Crude (3) Crude (6) Crude (9) Copper (3) Copper (6) Copper (9) Gold (3) Gold (6) Gold (9) Natural Gas (3) Natural Gas (6) Natural Gas (9)

ΔS 0.31 0.23 0.21 0.18 0.17 0.15 0.10 0.10 0.09 0.63 0.61 0.60

Panel A t 2.75 2.76 3.12 1.78 2.00 1.84 1.82 1.78 1.71 2.39 3.97 4.24

2

AdjR 0.004 0.003 0.004 0.002 0.002 0.001 0.001 0.001 0.001 0.007 0.019 0.021

ΔS 0.28 0.20 0.20 0.08 0.08 0.04 0.15 0.15 0.14 0.51 0.54 0.58

t 2.49 2.53 2.95 0.71 0.90 0.52 2.79 2.76 2.70 1.97 4.11 4.61

S&P 4.87 5.50 5.33 16.24 15.76 14.79 -6.62 -5.72 -5.01 -5.52 8.63 4.07

Panel B t 0.57 0.77 0.81 3.28 3.48 3.52 -2.40 -2.15 -1.92 -0.49 1.27 0.76

NHP 9.76 8.14 6.65 1.63 1.41 1.62 0.86 0.87 0.88 9.66 6.58 2.04

t 5.57 5.38 4.92 2.27 2.31 2.78 5.74 5.97 6.15 3.68 3.27 1.01

AdjR2 0.03 0.03 0.03 0.02 0.03 0.03 0.03 0.02 0.02 0.01 0.03 0.02

29

Table IX: Regressions of Futures Returns on Inventories, S&P, and Net Hedging Pressure This table contains the results from the regression of realized futures returns (maturity 6) on all lagged inventory levels, the return on the S&P, and all net hedging pressures (NHP). The inventory data is seasonally adjusted as described in the text. Inventories for the Crude series are lagged once while the remaining inventories are lagged twice. The columns contain the t-statistics using Newey and West standard errors with 52 lags.

Inventory Copper (6) Crude (6) Gold (6) Natural Gas (6)

CO -1.08 0.52 -0.31 -0.40

CL -3.55 -2.24 1.36 -2.94

GC -0.42 -0.54 -2.28 2.42

Net Hedging Pressure NG 4.48 2.05 0.17 2.26

SP 3.34 0.61 -1.53 2.08

CO 1.77 0.92 0.41 -0.69

CL 1.07 3.66 1.25 0.43

GC -0.74 -1.65 3.72 -1.25

NG -0.75 0.54 -0.24 2.71

AdjR2 0.04 0.02 0.02 0.04

30

Table X: Maximum-Likelihood Parameter Estimates for Crude Oil, Natural Gas, Copper and Gold Maximum-Likelihood parameter estimates and standard errors of the extended model for crude oil (CL), copper (CO), gold (GC), and natural gas (NG) using weekly prices, interest rate, and inventory data from 01/03/1995 - 05/31/2004. CO CL GC NG Coeff. Std. Err. Coeff. Std. Err. Coeff. Std. Err. Coeff. Std. Err. κ rQ 0.041 0.003 0.041 0.003 0.042 0.004 0.041 0.003

κ qQˆ

0.215

3.748

6.524

2.025

15.000

6.298

3.235

1.037

κ δQˆ

1.739

0.062

2.514

0.042

1.373

0.060

1.959

0.119

0.091 0.828

0.090 0.169

2.340 4.229

0.619 0.885

-1.809 -0.672

0.540 0.114

-0.725 2.571

2.115 4.500

αX

-0.196 0.356

0.685 0.011

-0.842 0.292

0.258 0.014

-0.032 -0.005

0.054 0.005

-1.689 0.147

0.933 0.085

θ rQ

0.101

0.007

0.101

0.007

0.100

0.007

0.101

0.007

θ

0.579

9.242

0.065

0.040

-0.060

0.958

0.465

0.216

-1.553 0.002

0.109 0.005

-0.890 0.001

0.058 0.004

0.036 0.000

0.042 0.004

0.246 0.000

0.638 0.004

-0.043

0.308

-0.385

0.242

10.001

14.300

-1.270

0.716

-24.538 -10.036 -0.087

1.772 1.456 0.092

0.057 1.730 -0.062

0.320 1.018 0.071

0.089 2.305 -0.048

0.110 1.127 0.082

0.830 2.221 -0.046

1.134 0.895 0.079

-3.358

3.841

4.813

2.037

1.055

6.375

2.680

1.048

-15.395

1.093

-0.198

0.359

-2.650

0.545

-1.975

0.477

-2.606

2.783

-4.985

3.960

-2.189

1.572

-8.767

5.035

-13.151

5.818

-2.058

1.313

-0.495

0.256

-2.478

0.987

-6.814

0.677

0.683

0.285

3.951

0.954

-0.380

0.205

σr

-0.060 0.009

0.180 0.000

-0.194 0.009

0.255 0.000

-0.359 0.009

0.200 0.000

-0.357 0.009

0.248 0.000

σ qˆ

0.021

0.000

0.100

0.002

0.611

0.008

0.126

0.003

σ δˆ

0.448

0.011

0.578

0.011

0.053

0.001

1.685

0.090

0.276

0.004

0.371

0.005

0.127

0.001

0.712

0.012

0.609

0.018

0.794

0.011

0.280

0.024

0.891

0.014

0.035

0.052

0.075

0.037

0.028

0.041

-0.048

0.069

0.064

0.028

0.025

0.024

0.002

0.036

-0.163

0.024

-0.070

0.047

-0.247

0.057

0.002

0.026

0.136

0.149

0.023 0.143 0.555 0.982

0.025 0.025 0.004 0.002

-0.014 0.068 0.741 0.982

0.028 0.022 0.006 0.002

0.207 -0.115 0.761 0.982

0.029 0.027 0.005 0.002

-0.076 -0.032 0.840 0.982

0.041 0.025 0.004 0.002

α rqˆ

αr αq

Q qˆ

θ δQˆ

β 0r

β 0 qˆ β 0δˆ

β0X β rr

β qˆqˆ β δˆδˆ β Xr β Xqˆ β Xδˆ β XX

σX ρδˆX ρ δˆδˆ ρ qˆX

ρ rqˆ ρ rδˆ

ρ rX ρF ρP

31

CL Invent ory/100

CO Inventory/100

4000

4500

3500

4000 3500

3000

3000

2500

2500 2000

2000

1500

1500

1000

1000

500

500

0 19950104

5000

19971231

20001227

20031224

GC Invent ory/1000

0 19950104

3500

4500

19971231

20001227

20031224

NG Invent ory

3000

4000 2500

3500 3000

2000

2500 1500

2000 1500

1000

1000

500

500 0 19950104

19971231

20001227

20031224

0 19950104

19971231

20001227

20031224

Figure 1: Unadjusted time series of the commodity inventories The figure contains the raw inventory time series for light crude oil (CL), copper (CO), gold (GC), and natural gas (NG).

32

CL Invent ory/100

CO Inventory/10 3000

4000 3500

2000

3000 1000

2500 2000

0 19950104

1500

500

5000

20001227

20031224

-1000

1000

0 19950104

19971231

-2000 19971231

20001227

20031224

GC Inventory/100

-3000

3000

4000

NG Invent ory

2500

3000 2000

2000

1000 1500

0 19950104 -1000

19971231

20001227

20031224

1000

-2000 -3000 -4000 -5000

500 0 19950104

19971231

20001227

20031224

Figure 2: Discretionary Inventories The figure contains the discretionary inventory time series for light crude oil (CL), copper (CO), gold (GC), and natural gas (NG). For the crude oil and natural gas data discretionary inventory is proxied by removing a seasonal component as described in the text. For the copper and gold data discretionary inventory is proxied by removing a 4 week lagged moving average.

33

Figure 3: Convenience Yields and Inventories Each panel contains the scatter plot of convenience yield against inventory. Column one contains the convenience yields for each commodity of the three month contract. The fitted line is a second order polynomial fit of the data.

34

Copper

Crude Oil 100 Under Reporting (%)

Under Reporting (%)

100 50 0 -50 -100

0

0.02 Inventory Gold

-50

0

0.1 0.2 Inventory Natural Gas

0.3

100 Under Reporting (%)

Under Reporting (%)

0

-100

0.04

100 50 0 -50 -100

50

0

0.5 Inventory

1

50 0 -50 -100

0

0.2

0.4 Inventory

0.6

Figure 4: Under Reporting of Values at Risk Each panel contains the scatter plot of the under reporting in Values at Risk over a quarterly holding period against the inventory level. Under-reporting of the Value at Risk relative to our extended model is calculated as VaR(Extended) - VaR(CCD). Here positive under reporting numbers imply that the VaR(CCD) is lower than the VaR(Extended) and under-reporting is expressed as a percentage of VaR(Extended). The figure shows that durring conditions of relative abundance (high inventory levels) the model of CCD (2005) severely under reports the Value at Risk.

35

Appendix A: The Model A.1 Risk Neutral Dynamics We start by defining the logarithm of the spot price S(t) as:

X(t) = log S(t) ≡ φ0 + φYT Y(t)

(A1)

where, φ0 is a constant, φY is a 4x1 vector and YT(t) = {Y1(t), Y2(t), Y3(t), Y4(t)} is a canonical four factor Gaussian state vector. Y(t) is assumed to follow a Gaussian diffusion process under the risk neutral measure Q:

dY(t) ≡ -κQ Y(t) dt + dZQ(t)

(A2)

here, κQ is 4x4 lower triangular mean reversion matrix and dZQ is a 4x1 vector of independent Brownian motions. As is standard in the commodities literature, the risk free rate, r(t), follows a one factor Gaussian process and we specify inventory, q(t), to be linear in the canonical variables Y1 and Y2:

r(t) ≡ ψ0 + ψ1 Y1(t)

(A3)

q(t) ≡ π0 + π1 Y1(t) + π2 Y2(t)

(A4)

It is well known, but shown in Appendix B for completeness, that a no-arbitrage restriction together with equation (A3) delivers the convenience yield process, δ(t):

δ(t) = ψ0 – ½ φYT φY + ψ1 Y1(t) + φYT κQ Y(t).

(A5)

Since the four economic factors, { r (t ) , q (t ) , δ (t ) , X (t ) }, are linear in the state vector, Y(t), their dynamics can be derived from Ito’s Lemma. 28 Using standard techniques (see Proposition 1) the convenience yield, δ (t ) , can be decomposed into an autonomous component, δˆ(t ) , and other components linearly related to the short rate, inventory and the spot price. Similarly, inventory has its own stochastic component but is also linearly related to interest rates:

q (t ) = qˆ (t ) + α rq r(t) δ (t ) = δˆ(t ) + α r r(t) + α q q(t) + α X X(t) 28

(A6)

It is then immediate that each factor enters into the drift of both the convenience yield and the price

process. On the other hand, models in current practice artificially restrict inventory level dependence in the drift of the convenience yield and the price process to zero.

36

= δˆ (t ) + [ α r + α q α rq ]r(t) + α q qˆ (t ) + α X X(t).

(A7)

The focus of our study is the representation, Wˆ (t ) = { r (t ) , qˆ (t ) , δˆ(t ) , X (t ) }. Equation (A6) decomposes inventory into a component that is related to interest rates through α rq

and a separate autonomous component, qˆ (t ) . It is plausible that

elevated interest rates increase the cost of holding stocks and so lower inventory levels; this leads to a negative α rq . It is also possible that storage operators increase their stocks during periods of heightened economic activity/interest rates, to alleviate the effect of demand (supply) shocks. In this case α rq can be positive. Most equilibrium models of commodity prices, for example Routledge, Seppi and Spatt (2000), do not speak to this effect, so ours is a natural setting to empirically investigate this question. The theory of storage predicts that convenience yields are inversely related to inventories. Within Equation (A7), such a relationship will be reflected in a negative α q . The specification of the convenience yield in Equation (A7) also allows price level dependence in the convenience yield through the coefficient α X . As shown below, a positive α X induces mean reversion in the spot price. Bessembinder et al. (1995) study mean reversion in asset prices, arguing that a high convenience yield accompanying a high spot price is indicative of investors anticipating that prices will revert. CCD (2005) formalizes this effect and surmises that since “periods of scarcity e.g., low inventory, correspond to high spot prices” the theory of storage underlies the positive convenienceyield spot price relationship. In addition, CCD (2005) reports a positive relationship between convenience yields and interest rates. They posit that this relationship is “consistent with the theory of storage” because during high interest rate regimes inventories decline; this results in higher convenience yields. Equation (A7) also allows convenience yields to be affected by the inventory-interest rate relationship ( α rq ) through the term, α qα rq . Within Equation (A7), a negative inventory-interest rate relationship ( α rq ) combined with a negative convenience yield-inventory relationship ( α q ), can indeed make the effect of an interest rate shock upon convenience yields ( α qα rq ) positive.

37

If interest rates impact convenience yields independently of inventories, this effect will be captured in the coefficient α r . For example, changes in the elasticity of supply around business cycle peaks can also affect convenience yields [Fama and French (1988)]. Since interest rates are (imperfectly) correlated with the business cycle they may have an effect upon convenience yields that is independent of the inventory-interest rate relationship. A subset of the factors that affect convenience yields also impact interest rates, inventories and prices. Since we jointly specify the relationship between convenience yields and each of interest rates/inventory/prices, we can estimate the marginal significance of each effect. If inventories have a marginal impact upon convenience yields beyond prices (interest rates), then ignoring inventory-level dependence in the convenience yield, as do extant models, will mis-specify mean reversion. 29 Neither our univariate regressions nor any of the extant models can disentangle these effects, but the model estimated here can. The economic representation we study, Wˆ (t ) = { r (t ) , qˆ (t ) , δˆ(t ) , X (t ) }, is related to the latent variables through Equations (A1), (A3), (A4) and (A5) and an application of Ito’s lemma yields its dynamics. These results are captured in the following proposition:

Proposition 1: The model of futures prices can be represented as:

dr (t ) = κ rQ [ θ rQ – r (t ) ] dt + σ r dZ rQ (t )

(P1.1)

dqˆ (t ) = κ qQˆ [ θ qˆQ – qˆ (t ) ] dt + σ qˆ dZ qQˆ (t )

(P1.2)

dδˆ(t ) = κ δQˆ [ θ δQˆ – δˆ(t ) ] dt + σ δˆ dZ δQˆ (t )

(P1.3)

dX (t ) = { ( α r + α qα rq – 1) [ θ rQ – r (t ) ] + α q [ θ qˆQ – qˆ (t ) ]

29

For instance, if the positive relation between convenience yields and spot prices is entirely due to

inventories, then in this joint estimation

αq

will not be significant. However, if the convenience yield-spot

price relation operates only partially through the inventory channel then the convenience yield-inventory relationship may also be significant.

38

+ [ θ δQˆ – δˆ (t)] + α X [ θ XQ – X (t ) ] } dt + σ X dZ XQ (t )

(P1.4)

where,

θ XQ = [(1 - α r - α qα rq ) θ rQ – α q θ qˆQ – θ δQˆ – ½ σX2] / α X dZ qQˆ (t ) dZ δQˆ (t ) = ρ qˆδˆ dt

dZ rQ (t ) dZ qQˆ (t ) = ρ rqˆ dt

dZ qQˆ (t ) dZ XQ (t ) = ρ qˆX dt

dZ rQ (t ) dZ δQˆ (t ) = ρ rδˆ dt

dZ δQˆ (t ) dZ XQ (t ) = ρ δˆX dt .

dZ rQ (t ) dZ XQ (t ) = ρ rX dt

Proof: The proof follows immediately by specifying α r , α q and α rq in terms of the canonical parameters such that δˆ and qˆ are autonomous processes followed by an application of Ito’s lemma. Details are provided in the Appendix. In this representation, each of { r (t ) , qˆ (t ) , δˆ(t ) } represent autonomous processes, characterized by their speeds of mean reversion, long-term levels to which they revert and their volatilities. In contrast, the spot price process is mean reverting, but its drift depends upon the remaining autonomous components. This proposition shows that our extension, naturally, nests the model of CCD (2005). Further, since CCD (2005) nests several earlier specifications eg. Brennan and Schwartz (1985), Gibson and Schwartz (1990), Schwartz (1997) and Schwartz and Smith (2000), then all of the above are nested within our extension. For instance, restricting α q = 0 yields the model of CCD (2005) and Schwartz’s (1997) model 3 is obtained by setting

α q = α r = α X = 0. A.2 Risk Premia The above setting is specified under the risk neutral measure, but in order to explain the historical time series of prices a specification of risk premia is required. We allow risk premia to be affine in the economic variables. This specification can be written in terms of the rotated Brownian motion basis as:

39

⎛ Z rQ ⎞ ⎛ Z r ⎜ Q⎟ ⎜ ⎜Z ⎟ ⎜ Z d ⎜ qQˆ ⎟ =d ⎜ qˆ ⎜ Z δˆ ⎟ ⎜ Z δˆ ⎜ZQ ⎟ ⎜Z ⎝ X⎠ ⎝ X

⎧⎛ β 0 r ⎞ ⎟ ⎪⎜ ⎟ -1 ⎪⎜ β 0 qˆ ⎟ +Σ ⎨⎜ β ⎪⎜ 0δˆ ⎟ ⎟ ⎪⎜⎝ β 0 X ⎠ ⎩

⎞ ⎛ β rr ⎟ ⎜ ⎟ ⎜ β qˆr ⎟ + ⎜β ⎟ ⎜ δˆ r ⎟ ⎜β ⎠ ⎝ Xr

β rqˆ β qˆqˆ β δˆ qˆ β Xqˆ

β rδˆ β qˆδˆ β δˆ δˆ β Xδˆ

β rX β qˆX β δˆ X β XX

⎞⎛ r (t ) ⎞⎫ ⎛ σr ⎟⎜ ⎟⎪ ⎜ ⎟⎜ qˆ (t ) ⎟⎪ ⎜ σ qˆ ; Σ = diag ⎟⎜ δˆ (t ) ⎟⎬ ⎜ σδ ⎟⎜ ⎟⎪ ⎜⎜ ⎟⎜ X (t ) ⎟⎪ ⎠⎭ ⎝σ X ⎠⎝

⎞ ⎟ ⎟ ⎟ ⎟⎟ ⎠

(A8)

We restrict all cross risk premia terms except those involving the spot price to zero. This ensures that the autonomous component of the convenience yield remains independent of the remaining factors under the physical measure. This assumption is similar to the restriction imposed in CCD (2005) while many other models ignore such covariance altogether.

Proposition 2: For the specification of risk premia outlined in equation (A8) with the restriction outlined above then the dynamics of the economic variables under the physical measure are:

dr (t ) = κ rP [ θ rP – r(t)] dt + σ r dZ r (t )

(P2.1)

dqˆ (t ) = κ qˆP [ θ qˆP – qˆ (t)] dt + σ qˆ dZ qˆ (t )

(P2.2)

dδˆ(t ) = κ δPˆ [ θ δPˆ – δˆ (t)] dt + σ δˆ dZ δˆ (t )

(P2.3)

P dX (t ) = { κ Xr [ θ rP – r(t)] + κ XPqˆ [ θ qˆP – qˆ (t)] +

κ XPδˆ [ θ δPˆ – δˆ (t)] + κ XP [ θ XP – X(t)] } dt + σ X dZ X (t )

(P2.4)

where,

κ rP = κ rQ - β rr κ qˆP = κ qQˆ - β qˆqˆ κ δPˆ = κ δQˆ - β δˆδˆ

θ rP =

κ rQθ rQ + β 0 r κ rQ − β rr

θ qˆP =

κ qQˆ θ qˆQ + β 0 qˆ κ qQˆ − β qˆqˆ

θ δPˆ =

κ δQˆ θ δQˆ + β 0δˆ κ δQˆ − β δˆδˆ

κ XP = α X - β XX

P κ Xr = α r + α qα rq – 1 - β Xr

κ XPqˆ = α q - β Xqˆ κ

P Xδˆ

= 1 - β Xδˆ

θ = β0X − P X

σ x2 2

P − κ Xr θ rP − κ XPqˆθ qˆP − κ XPδˆθ δPˆ κ XP

(P2.5)

40

Appendix B: The no arbitrage restriction implies that:

EtQ [dS (t )] = [r (t ) − δ (t )]S (t )dt

(B1)

Rearranging we get:

δ (t ) = r (t ) −

1 Q ⎡ dS (t ) ⎤ Et ⎢ dt ⎣ S ⎥⎦

(B2)

Substituting the values of r(t) from equation (A3) we get:

δ (t ) = ψ 0 +ψ 1Y1 (t ) −

1 Q ⎡ dS (t ) ⎤ Et ⎢ dt ⎣ S ⎥⎦

(B3)

From the properties of log-normal distribution we get: 1 ⎡ dS (t ) ⎤ EtQ ⎢ = EtQ [dX (t )] + Vt Q [dX (t )] ⎥ 2 ⎣ S ⎦ Where V is the variance of X. Substituting it in above equation we get: 1 1 δ (t ) = ψ 0 +ψ 1Y1 (t ) − {EtQ [dX (t )] + Vt Q [dX (t )]} dt 2

(B4)

(B5)

Now from equation (A1) we have X (t ) = φ0 + φY Y (t ) dX (t ) = φY dY (t )

[

dX (t ) = φY − κ QY (t )dt + dZ Q

]

(B6)

dX (t ) = −φY κ QY (t )dt + φY dZ Q

So we get

E t [dX ( t ) ] = −φ Y κ Q Y ( t ) dt T

(B7)

δ (t ) = ψ 0 − φY T φY + ψ 1Y1 + φY κ QY (t )

(B8)

V [dX ] = −φ Y φ Y dt Substituting above and re-arranging we get: 1 2

41

Appendix C: Proof of Proposition 1:

From equation (A5) we have:

δ (t ) = δˆ(t ) + α r r (t ) + α q q(t ) + α X X (t ) δˆ(t ) = δ (t ) − α r r (t ) + α q q(t ) + α X X (t ) 1 2 −α qπ 2Y2 (t ) − α X φ0 − α X φ1Y1(t ) − α X φ2Y2 (t ) − α X φ3Y3 (t ) − α X φ4Y4 (t )

δˆ(t ) = ψ 0 − φ T φ + ψ 1Y1(t ) + φ T κ QY (t ) − α rψ 0 − α rψ 1Y1(t ) − α qπ 0 − α qπ 1Y1(t ) (C1)

1 δˆ(t ) = ψ 0 − φ T φ − α rψ 0 − α qπ 0 − α X φ0 2 +ψ 1Y1(t ) − α rψ 1Y1(t ) − α qπ 1Y1(t )

−α qπ 2Y2 (t ) +φ T κ QY (t ) − α X φ Y (t ) separating the constant terms and the terms involving the Y’s we define: 1 2

ηˆ 0 = ψ 0 − φ T φ − α rψ 0 − α q π 0 − α X φ 0 ⎡ηˆ1 ⎤ ⎡(1 − α r )ψ 1 − α q π 1 ⎤ ⎢ηˆ ⎥ ⎢ ⎥ − α qπ 2 ⎥ ηˆY = ⎢ 2 ⎥ = ⎢ ⎥ ⎢ηˆ3 ⎥ ⎢ φTκ Q ⎥ ⎢ ⎥ ⎢ −α XφT ⎦ ⎣ηˆ 4 ⎦ ⎣

(C2)

From above we can write:

δˆ(t ) = ηˆ0 + ηˆY Y (t )

(C3)

Also

q (t ) = qˆ (t ) + α rq r (t ) qˆ (t ) = q(t ) − α rq r (t ) = π 0 + π 1Y1 + π 2Y2 − α rq (ψ 0 + ψ 1Y1 )

{[

= π 0 − α rqψ 0 + π 1 − α rqψ 1 π 2

(C4)

]

0 0 Y (t )

42

We transform the state vector as , Wˆ (t ) = { r (t ) , qˆ (t ) , δˆ(t ) , X (t ) }, In matrix form it can be written as:

Wˆ (t ) = νˆ + LYˆ (t )

(C5)

⎛ ψ0 ⎞ ⎜ ⎟ q ⎜π 0 − α r ψ 0 ⎟ υ= ⎜ ⎟ η0 ⎜ ⎟ ⎜ ⎟ φ 0 ⎝ ⎠

(C6)

0 0 0⎞ ⎛ ψ1 ⎜ ⎟ q ⎜π 1 − α r ψ 1 π 2 0 0 ⎟ L= ⎜ η1 η 2 η3 η 4 ⎟ ⎜ ⎟ ⎜ ⎟ ϕ ϕ ϕ ϕ 1 2 3 4⎠ ⎝

(C7)

Where υ and L are as follows:

and

Applying Itô’s lemma to equation P2 we get d Wˆ = Lˆ κQ Lˆ -1 (υ – Wˆ (t)) dt + Lˆ dZQ

(C8)

Comparing Lˆ κQ Lˆ -1 with the equivalent matrix in the latent model, we get the values of the parameters of economic representation in terms of the parameters of the latent model.

43

κ rQ = κ 11 ;θ rQ = ψ 0 ;σ r = ψ 1 ;κ qQˆ = κ 22 ;σ q = θ qq =

π 22κ 212 +π 2 ; (κ 11 − κ 22 )2 2

( π 1κ 22 − π 1κ 22 − π 2κ 21 ) + π 0ψ 1 (κ 11 − κ 22 ) ψ 1 (κ 11 − κ 22 )

(C9)

κ δQˆ = −κ 33 ;σ X = φ T φ Proof of Proposition 2:

For the canonical representation we specify risk premia as in Casassus and CollinDufresne (2005):

dZ Q (t ) = dZ (t ) + ( β 0 + β1Y (t ))dt

(C10)

where:

⎡ β rr ⎡ β0 r ⎤ ⎢β ⎢β ⎥ ⎢ qrˆ 0 qˆ ⎥ ⎢ and β1 = ⎢ β0 = β ⎢ β 0δˆ ⎥ ⎢ δˆr ⎢ ⎥ ⎢ β Xr ⎢⎣ β 0 X ⎥⎦ ⎣

β rqˆ β qqˆ ˆ

β rδˆ β qˆδˆ

βδˆqˆ

βδδˆ ˆ

β Xqˆ

β X δˆr

β rX ⎤ ⎥ β qX ˆ ⎥

βδˆ X ⎥

(C11)

⎥ β XX ⎥⎦

Substituting in equation P1.3 we get: dWˆ (t ) = Lˆ κ Q Lˆ−1 (νˆ − Wˆ (t ))dt + Lˆ dZ (t ) + Lˆ ( β 0Y + β1Y Y (t ))dt

(C12)

Substituting the value of Y(t) from equation (C5) dWˆ (t ) = Lˆ κ Q Lˆ−1 (νˆ − Wˆ (t ))dt + Lˆ dZ (t ) + ( Lˆ β 0Y − Lˆ β1Y Lˆ−1νˆ + Lˆ β1Y Lˆ−1Wˆ (t ))dt (C13)

Comparing with equation (C10) the risk premia parameters for the economic

44

representation are: ⎡ β 0r ⎤ ⎢β ⎥ β 0 = ⎢ 0 qˆ ⎥ = Lˆ β 0Y − Lˆ β1Y Lˆ−1νˆ ⎢ β 0δˆ ⎥ ⎥ ⎢ ⎣β 0 X ⎦

⎡ β rr ⎢β qˆ r β1 = ⎢ ⎢ β δˆr ⎢ ⎢⎣ β Xr

β rqˆ β qˆqˆ β δˆqˆ β Xqˆ

β rδˆ β qˆδˆ β δˆδˆ β Xδˆr

β rX ⎤ β qˆX ⎥⎥ = Lˆ β 1Y Lˆ−1 β δˆX ⎥ ⎥ β XX ⎥⎦

(C14)

and ∑ −1 = Lˆ LˆT

(C15)

dZ Q = dZ + ∑ _ 1 ( β 0 + β1Wˆ (t ))

(C16)

Comparing (C8) and (C13) we get

45