The PROTEUS Commodity Risk Model

Transcription

1 Quantitative Research November 2013 Your Global Investment Authority Analytics The PROTEUS Commodity Risk Model Peter Matheos 1 Executive Vice President Masoud Sharif Senior Vice President Mahmoud Hajo Vice President PIMCO commodity exposures and the factor structure of the PROTEUS Commodity Risk Model Across multiple accounts and several billion dollars in notional exposure, PIMCO accounts invest in commodities through futures, options on futures, total return swaps on commodity futures and published commodity benchmarks. Through these positions we have exposure both to spot commodity pricing and to the term structure of commodity pricing. Specifically, commodity futures trade with expirations varying from one month to 10 years (with varying liquidity for different expirations and different commodities). Commodity exposure can be taken both in beta terms, via total return swaps, and in alpha terms, trading spreads between related commodities (e.g., different wheat contracts or crack spreads) and between points in the term structure and outright directional views on commodity appreciation (e.g., long gold as an implementation of a secular view on inflation or long term dollar devaluation). The PROTEUS Commodity Risk Model is PIMCO s internal model for commodity volatility, both in absolute terms and relative to the benchmark. This model builds risk estimates from the bottom up by using detailed information on volatilities across commodity markets and across maturities. It allows users easily to zoom in on detailed risk contributions and zoom out to get a broad view of top-down risk. One of the strengths of the model is the ability to better understand sources of tracking error in our commodity portfolios and assess the potential risk-return tradeoff of various alpha trades. For institutional investor use only: not for public distribution.

2 The commodity risk model represents commodity exposures through factors that cover futures and physical commodities held in PIMCO accounts and benchmarks. The commodities covered in the model 2 are listed in Table 4; as expected, given that they include (but are not limited to) all commodity futures in the Dow Jones UBS Commodity Index (DJUBS) and Goldman Sachs Commodity Index (GSCI), they cover the most economically significant commodity futures and positions that are significant exposures for PIMCO. For each commodity covered in the model, we use a term structure of constant maturity factors designed to capture variation in futures volatility associated with time to futures expiration. The key maturity points used in the term structure factors are monthly points for the first year, bi-monthly from months 12 to 36, and two annual points at years 4 and 5. PROTEUS supports multiple estimation methodologies for the factor covariance matrix used in computing risk estimates. In the tool which we have delivered to the PIMCO Commodities desk, we have provided equally weighted and exponentially decayed matrices calibrated to different risk estimation horizons. Throughout this article, to be consistent among the various figures presented, we use the factor covariance matrix estimated as of 27 September 2013, using five years of trailing daily returns, exponentially decayed so that returns have a half-life of one year, calibrated to a monthly risk horizon. 3 Risk analysis of the DJUBS and GS commodity indexes Table 1 presents, in its first four columns, the composition by commodity sector of the two most widely used commodity indexes, the DJUBS and the GSCI. While each is a broad commodity index, the composition between the two is quite different; the GSCI is dominated by the Energy sector, while DJUBS places more weight on agriculture and base and precious metals. As a result of the compositional differences, the return profiles of the two indexes are significantly different, and it is not reasonable to attempt to capture the risk across different commodity mandates with a single commodity risk factor. In the final three columns of Table 1 we present the Commodity Risk Model estimates of the total return volatility of each index and the relative return volatility between the two indexes in the Total row. Below the totals, by sector, we present a risk attribution that decomposes the risks by the broad sectors. While underlying this decomposition is highly granular decomposition of risk into individual securities, which are themselves exposed to commodity term structure factors, the risk output is highly intuitive and actionable, because the risk attribution is done in terms of the marginal risk sensitivities 4 of each sector. Based on our analysis, energy, where there is the largest difference in exposure between the indexes, contributes almost all of the tracking error volatility between the two indexes; while the aggregate risk of the DJUBS Commodity Index is far more evenly composed of risk contributions from the four largest sectors. Note that the large tracking error between the two benchmarks is not simply a function of the fact that the DJUBS Commodity Index is intrinsically less volatile than the GSCI. Even leveraged to have the same absolute risk, the tracking error is actually greater: the tracking error between the DJUBS THE PROTEUS FRAMEWORK FOR MULTI-FACTOR RISK MODELING PROTEUS was the name of a mythological Greek sea-god who continuously changed form while wrestling the hero Menelaus; we have named the new risk model architecture developed by Analytics after Proteus as a reminder of the ever changing nature of the risks in financial markets (and also out of sympathy for those who would find Menelaus hard to pronounce). PROTEUS provides integrated risk measurement across rates, credit spreads, Agency and non-agency spreads, EM spreads, equities, commodities and currencies. It is the production system for the firm s ex-ante portfolio risk, tracking error and Value-at-Risk. PROTEUS addresses the dual goals of providing robust, intuitive risk reporting and flexible, detailed factor structures that comprehensively identify investment risks to which PIMCO accounts are exposed. The system produces risk reporting at top (i.e., account, bogie and account-bogie-relative) level, sector or asset class level, and also at security level, so that users of the system are able to understand risk at broad scales and in terms of individual positions or trade ideas. 2 November 2013 quantitative research

3 Commodity Index levered to have an estimated absolute volatility equal to 1659, and the GSCI is 938 bps per annum. Correlations and basic factor analysis Given the heterogeneous economics of various commodities, it is probably not surprising that, as shown in Table 2, even within sectors commodities display relatively modest correlations. (In Table 2 we have reported natural gas separately from other energy commodities as its correlation to them is particularly weak.) The lack of a simple, coherent single factor underlying commodity returns underscores the need for a granular, comprehensive approach to risk measurement for commodities simple proxies are likely to miss embedded risks. Note that, aside from natural gas, the diagonal elements in Table 2 are not one because they are averages of individual commodity correlations within the sector. The fact that no single factor captures the structure of commodity risk can also be understood through principal component analysis of the covariance matrix among the commodities. Principal component analysis identifies ranked common factors, or weighted exposures among (in this case) commodities, which best explain variation in time series in the following sense: the first principal component is the weighted set of commodity exposures which captures the TABLE 1: WEIGHTS AND RISK ESTIMATES OF THE DJUBS AND GOLDMAN SACHS COMMODITY INDEXES Risk estimates based on 27 September 2013 covariance matrix with one year half-life exponential decay, monthly horizon risk estimates; index weights as of 30 September 2013 Notional weight Annualized risk estimates (basis points) Sector DJUBS GSCI Net DJUBS GSCI Tracking error Total 100% 100% 0% 1,293 1, Agriculture 29% 13% 16% Base metals 16% 6% 10% Energy 37% 73% -36% 514 1, Livestock 6% 5% 0% Precious metals 13% 3% 10% Source: PIMCO TABLE 2: AVERAGE CORRELATIONS AMONG six MONTH CONSTANT MATURITY FACTORS BY SECTOR Agriculture Base metals Energy (ex. natural gas) Natural gas Livestock Precious Metals Agriculture 0.35 Base metals Energy (excl. natural gas) Natural gas Livestock Precious metals Source: PIMCO The factor covariance matrix estimated as of 27 September 2013, using five years of trailing daily returns, exponentially decayed so that returns have a half-life of one year, calibrated to a monthly risk horizon. quantitative research November

4 greatest amount of volatility that can be explained for a fixed amount of exposure 5 to commodities; the second captures the greatest amount of volatility that can be explained after the first principal component has been accounted for, etc. Principal component analysis is widely used in finance for example, it is well known that the first three principal components of the variation in the term structure of interest rates capture well above 90% of total volatility across developed markets. By comparison to rates, a large number of principal components is required to achieve a high explanatory power for commodity return variation. The first principal component of the covariance matrix of commodity returns estimated at a monthly horizon, on an exponentially decayed basis with a half-life of one year, captures about 34% of the in-sample variability; together, the first three principal components capture about 59.5% of the variability. Figure 1 presents the three largest principal components of the factor model. The first principal component is essentially a parallel shift across commodities, with the notable exception of natural gas and livestock, which have almost zero weight in the first principal component. The second principal component captures the offset between industrial commodities and agriculture and natural gas 6. The third principal component captures a higher order offset between energy and industrial metals, and splits agricultural commodities. The relationship between time to expiry and commodity volatility A striking empirical regularity in the volatility of commodity futures is the so-called Samuelson Effect : for a given commodity contract, volatility declines as the time to futures expiration increases. Table 3 lists the volatilities of the one month and one year points for the commodities studied in this article, and the ratio of these volatilities. For those not familiar with these markets, the declines may appear striking: FIGURE 1: LARGEST PRINCIPAL COMPONENTS OF THE FACTOR MODEL First Three Principal Components 27 September 2013 (estimated from daily overlapping monthly horizon shocks, exponentially decayed matrix, one year half-life) Brent crude WTI crude oil Heating oil Gasoil Gasoline Natural gas Aluminum Lead Standard deviation (%, annualized) Nickel Copper Zinc Gold Silver White sugar Sugar Canola Soybean meal Kansas wheat Wheat Minnie wheat Matif wheat Soybeans Soybeans Corn Cocoa Coffee Cotton Live cattle Lean hogs Feeder cattle First principal component Second principal component Third principal component Source: PIMCO. As of 27 September November 2013 quantitative research

5 TABLE 3: SAMUELSON EFFECT: one YEAR HALF-LIFE, MONTHLY HORIZON ESTIMATES Commodity One month constant maturity factor volatility One year constant maturity factor volatility Vol Ratio (One year/ One month) Natural gas 35% 19% 55% White sugar 23% 15% 64% Cotton 28% 18% 64% Wheat 27% 20% 72% Lean hogs 17% 12% 73% Live cattle 9% 7% 75% Corn 31% 24% 77% Kansas wheat 26% 20% 78% Gasoline 23% 18% 79% Soybeans 23% 18% 79% Matif wheat 26% 21% 81% WTI crude oil 23% 18% 78% Minnie wheat 23% 19% 82% Brent crude 21% 17% 80% Feeder cattle 13% 10% 75% Cocoa 20% 16% 82% Gasoil 20% 16% 81% Heating oil 20% 17% 83% Canola 19% 14% 75% Coffee 26% 23% 89% Aluminum 19% 18% 94% Lead 27% 25% 95% Zinc 23% 22% 95% Nickel 26% 25% 98% Copper 21% 20% 98% Gold 18% 18% 100% Silver 35% 35% 100% SOURCE: PIMCO. As of 27 September the volatility of the one year natural gas futures price is around 54% less than the volatility one month out. The Samuelson effect may be thought of as a reflection of the fact that short term shocks to supply and demand (e.g., through variation in the business cycle) affect shorter horizon contracts more than long horizon contracts (either because the shocks are intrinsically short-term in nature, or because production or utilization capacity in the economy changes more slowly than information from shocks is reflected in spot prices). Thus it is not surprising that precious metals, which are less tied to economic production, appear in the table with essentially no Samuelson effect the same could be said for financial futures, for which there is similarly no Samuelson effect. Intuitively, the presence of a positive convenience yield, which indicates potential investor preferences for ownership (resulting in backwardation), is also consistent with the Samuelson effect, as shocks typically impact shorter settlement pricing more significantly as a result of their interaction with investor preferences. However, we note that, this intuition notwithstanding, the largest Samuelson effect in our data appears for natural gas. Because of the Samuelson effect, for accurate risk measurement it is important to capture the term structure of volatility for commodities. For example, a trade which attempts to capture a differential in the term structure of a commodity s roll 7 through a calendar spread is exposed to differential volatilities at the two maturities; positions on strips of consecutive monthly contracts will (for most commodities) be less volatile than the front contract in the strip. Risk horizon Risk management is performed at widely varying time scales, from overnight risk controls used in liquidity management to annual horizons used in client guidelines. Scaling high frequency volatility estimates with the square root of time to produce long horizon risk estimates is a widely used approximation which, in general, fails to capture risk at different horizons correctly because of mean reversion or serial quantitative research November

6 FIGURE 2: HORIZON IMPACT ON RISK ESTIMATES Ratio of quarterly horizon to daily horizon volatility estimates for six month factor Ratio of quarterly horizon to daily horizon volatility estimates for nine month - three month spread position Commodity Volatility ratio Commodity Volatility ratio Lean hogs 0.81 Silver 0.22 Cocoa 0.82 Gold 0.25 Zinc 0.83 Copper 0.36 Natural gas 0.84 Feeder cattle 0.52 Aluminum 0.87 Matif wheat 0.65 Lead 0.87 Cocoa 0.68 Canola 0.89 Soybeans 0.69 Copper 0.89 Lead 0.71 Gold 0.90 Zinc 0.72 Live cattle 0.90 Natural gas 0.76 Copper 0.90 Minnie wheat 0.81 Nickel 0.90 Soybeans 0.82 WTI crude oil 0.91 Brent crude 0.83 Feeder cattle 0.93 WTI crude oil 0.83 Heating oil 0.95 Heating oil 0.84 Coffee 0.96 Gasoline 0.86 Soybeans 0.96 Cotton 0.92 Corn 0.97 Aluminum 0.93 Brent crude 1.00 Live cattle 0.94 Gasoline 1.00 Corn 1.01 Silver 1.00 Wheat 1.01 Gasoil 1.01 Canola 1.01 Sugar 1.02 Gasoil 1.01 Wheat 1.03 Soybean meal 1.02 Soybean meal 1.03 Lean hogs 1.08 White sugar 1.07 Copper 1.10 Kansas wheat 1.07 White sugar 1.11 Minnie wheat 1.08 Nickel 1.15 Cotton 1.13 Sugar 1.16 Soybeans 1.19 Kansas wheat 1.19 Matif wheat 1.21 Coffee 1.20 Source: PIMCO. As of 27 September Volatility ratios from covariance estimates Q-Q plots of volatility ratios Volatility ratio distribution from Monte Carlo simulation Vol ratio 6m- Cov Matrix Vol ratio sprd (9m-3m) Cov Matrix Source: PIMCO. As of 27 September 2013 Reference line (y=x) correlation in returns and because of effects like the Samuelson effect (which causes a single position held for a long time to experience higher volatility as the time to settlement decays) 8. To capture the impact of horizon on volatility, we have calibrated the model to factor series estimated at daily, weekly, monthly, and quarterly horizons. Figure 2 presents two examples of tests for these dynamics. The first table in Figure 2 presents the ratio of quarterly horizon volatility estimates to daily horizon volatility estimates across commodities. The Q-Q plot of this data (the blue diamonds in the chart) is consistent with no mean reversion and perhaps mild serial correlation for the commodities at the bottom of the table (Matif wheat, cotton, soybeans, etc.). The second table presents volatility ratios for nine month three month spreads across commodities; these appear to be significant in aggregate. Each volatility ratio is, in effect, a comparison between two volatilities. Among many such comparisons, even if the true volatilities were equal, some comparisons would appear to be statistically significant, simply because of 6 November 2013 quantitative research

7 sampling noise. The Q-Q plot is a scatter plot of, on the x-axis, the percentiles of a theoretical distribution for volatility ratios that would result if there were no difference between the two volatilities in the numerator and denominator of the volatility ratio, and on the y-axis, the realized volatility ratios from the data. For the volatility ratios from the data, it is possible a priori that the two volatilities are equal across commodities, or that they are unequal. If they were equal across commodities, the scatter plot would lie perfectly on the line representing equality between the theoretical and realized distributions. For the six month constant maturity factors (the blue diamonds), the data do lie on the line, for the most part. However, for the spread trades, the data deviate from the theoretical distribution. Measurement of the constant maturity factors and factor exposures The constant maturity factors are measured at daily, weekly, monthly, and quarterly horizons by computing the return, on a given date, to a position in two futures which are closest to the constant maturity point, each weighted by its relative closeness to the constant maturity point, e.g., the constant maturity daily factor return for the six month Brent Oil factor for 11 May 2012 was computed as the weighted average of the return on the COX2 contract (expiring 1 November 2012) which is -0.60% and COZ2 contract (expiring 1 December 2012) which is -0.58%. The 6m return associated with the 6m point will be -0.59% 9. Weekly, monthly, or quarterly series are computed similarly, holding the two contracts at the starting weights for the indicated horizon. points. Thus the two exposures will sum to one; as the futures contract passes from exactly eight months to expiration to exactly seven months to expiration, its exposure to the two factors will pass from full exposure to the eight month constant maturity point to the seven month constant maturity point. Consider a given futures contract, say, the December 2013 WTI contract. One might think it natural to estimate volatility for the December contract rather than use the constant maturity machinery described above. However, as time passes, the time to maturity of the contract decays. We cannot construct risk model factors calibrated to the December contract easily, because the volatility of the December contract varies per the Samuelson effect as the time to expiration of the contract declines, and this effect is dominant in terms of price volatility: the front contract tends to be the most volatile, regardless of the settlement month. Conversely, each futures contract is exposed to the two constant maturity points which border the expiration of the contract. For example, the March 2013 Brent Crude futures (ticker COH3) settles on 13 February On 1 July 2012, when there will be approximately seven and one-half months to its settlement, it will be exposed to the seven month and the eight month factors, with exposures computed as the number of days from settlement to the constant maturity point divided by the number of days between the two constant maturity quantitative research November

8 Conclusion We have described the PROTEUS Commodity Risk Model and explored some of the factor dynamics it exposes. As expected by theory and practitioners, we find a significant term structure of volatility, or Samuelson effect, across commodities, and also that confirmatory evidence through principal component analysis that a large factor structure is warranted for commodities. While there is no significant evidence of serial correlation or mean reversion in commodity pricing, we find that horizon effects materially impact risk analysis through increased cross commodity correlations and through the variation of the steepness of the Samuelson curve for the same commodity at different horizons. The model is specified to be comprehensive for PIMCO s commodity exposures. Appendix TABLE 4: COMMODITIES COVERED IN THE MODEL Factors (futures ticker) Agriculture Wheat (W)* Soybeans (S)* Kansas wheat (KW) Soybeans oil (BO)* Matif wheat (CA)* Soybean meal (SM)* Cocoa (QC) Cocoa (CC)* Coffee (KC)* Corn (C )* Corn TGE (JC) White sugar (QW)* Minnie wheat (MW)* Soybeans TGE (JS) Coffee rob LIFFE (DF) SB (Sugar)* Canola (RS)* Cotton (CT)* Rapeseed LIFFE (IJ) Rough rice (RR) Red bean (JE) Base metals Nickel (LN)* Zinx (LX)* Lead (LL)* Copper (LP)* Aluminum (LA)* Copper (HG)* Energy Brent (CO)* WTI (CL)* Nat gas (NG)* Heating oil (HO)* Gasoline (XB)* Gasoil (QS)* Fuel oil (LR) Livestock FC (Feeder cattle)* Live cattle (LC)* Lean hogs (LH)* Precious metals Gold (GC)* Silver (SI)* Platinum (PL) Palladium (PA) *Commodity included in the set of 31 commodities used for the study results below see footnote 3. Works Cited Gorton, G., Hayashi, F., & Rouwenhorst, K. G. (2008). The Fundamentals of Commodity Returns. Yale International Center for Finance. 8 November 2013 quantitative research

9 1 We would like to thank Georgi Vassilev, Sebastien Page, Ravi Mattu, Vadim Yasnov, Vasant Naik and Lutz Schloegl for useful discussion and edits to the article. 2 Table 4 covers the full set of factors in the model. The results presented in the rest of the paper are for a smaller preliminary version of the model, comprising the 31 commodities marked with an asterisk in the table. 3 By this we mean we have computed the volatility in terms of covariance matrices computed with overlapping monthly constant maturity return series, computed as described in the final section of the article. 4 Marginal risk decompositions are also referred to as risk decompositions in terms of the Euler rule, an algebraic property of covariance based risk estimates which results in the total risk reconciling to the weight-scaled first-derivative contributions. 5 The exposure is fixed across all principal components in the sense that, across commodities, the sum of the squared exposures sums to a constant. 6 We are grateful to Mihir Worah, Nic Johnson and Greg Sharenow for this characterization. 7 A commodity futures price curve for which futures prices decline with longer time to contract settlement is said to manifest normal backwardation; as with the Samuelson effect, this pattern is typical of many, though not all, commodities. For the case of normal backwardation, if the term structure of the commodity curve does not change, then long positions will earn a positive roll as time to settlement decreases. The case where the futures price curve increases with time to settlement is known as contango. 8 E.g., commodities which are close but imperfect substitutes should have long term relationships that are relatively stable compared to daily variation because over long horizons the price difference between substitute commodities should be bounded by transportation and costs related to substitution. Common factors to economic activity and inflation will also lead to relatively high long horizon correlations. 9 Computed as (20/30)x(-0.60%)+(10/30)x(-0.58%). quantitative research November

10 Past performance is not a guarantee or a reliable indicator of future results. Investing in the bond market is subject to certain risks including market, interest-rate, issuer, credit, and inflation risk; investments may be worth more or less than the original cost when redeemed. Commodities contain heightened risk including market, political, regulatory, and natural conditions, and may not be suitable for all investors. Derivatives and commodity-linked derivatives may involve certain costs and risks, such as liquidity, interest rate, market, credit, management and the risk that a position could not be closed when most advantageous. Commodity-linked derivative instruments may involve additional costs and risks such as changes in commodity index volatility or factors affecting a particular industry or commodity, such as drought, floods, weather, livestock disease, embargoes, tariffs and international economic, political and regulatory developments. Investing in derivatives could lose more than the amount invested. Investors should consult their investment professional prior to making an investment decision. This material contains hypothetical results based on a Monte Carlo simulation. No representation is being made that any account, product, or strategy will or is likely to achieve profits, losses, or results similar to those shown. Hypothetical or simulated performance results have several inherent limitations. Unlike an actual performance record, simulated results do not represent actual performance and are generally prepared with the benefit of hindsight. There are frequently sharp differences between simulated performance results and the actual results subsequently achieved by any particular account, product, or strategy. In addition, since trades have not actually been executed, simulated results cannot account for the impact of certain market risks such as lack of liquidity. 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