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# Retrospective evaluation of portfolio allocation in the energy sector

### Piotr Mirowski, New York University, 2007

The following computing project was done in December 2007 for the course "Quantitative Risk and Portfolio Management" taught by Prof. Attilio Meucci) at the Courant Institute of Mathematical Sciences. Starting from NYSE daily stock prices and Brent/Oklahoma oil spot prices, I conducted a retrospective portfolio allocation study on a 4-week prediction horizon, estimating missing prices (Expectation-Maximization), detecting outliers (Minimum Volume Ellipsoid), computing the Normal or Student-t distributions of invariants (weekly compounded returns), and performed a mean-variance optimal portfolio allocation.

### Methodology:

Each retrospective portfolio allocation estimation used all available data between two dates, with a weekly estimation interval and a 4-week prediction horizon. Weekly compounded returns were chosen as market invariants. As a result, several portfolio allocations were estimated, and the profit made on the respective investments computed thanks to available target data at the prediction horizon.

Estimation of missing data:
In most cases, days with missing Brent and Oklahoma spot prices and days with missing NYSE stock prices are two disjoint sets. Estimation of missing prices was done using the Expectation-Maximization algorithm, based on the conditional covariance with observed prices [1, chapter 4].

Outlier detection:
T = max(Tmcd, Tmve) outliers were removed using the Minimum Covariance Determinant and Minimum Volume Ellipsoid [1, chapter 4]. Outliers were removed from the invariants.

Estimation of the distribution of market invariants:
Maximum-Likelihood Normal distribution, shrinkage to a Normal distribution with a narrow covariance, and Maximum-Likelihood Student-t distributions were investigated (for the Student-t distribution, the ML scatter and location parameters were computed for each degree of freedom, and then the optimal (ML) degree of freedom selected. The distribution of invariants that was chosen for the projection was the alpha=0.1 shrinkage of the Normal distribution towards the grand mean and half of the grand covariance.

Optimal allocation:
Total-return and benchmark mean-variance optimal allocation w.r.t. the certainty equivalent satisfaction (with an exponential pessimistic xi=10 utility function). 3 benchmarks were chosen: equal-money in all securities, equal-money in Oil&Gas only, equal-money in Nuclear energy only.

Evaluation:
The content of each portfolio as well as retrospective profits (i.e. profits made on each allocation, using the actual securities prices at the investment horizon) was output for the total returns (optimized), nuclear benchmark-relative (optimized), nuclear benchmark (given), Oil&Gas benchmark-relative (optimized), Oil&Gas benchmark (given), diversified benchmark-relative (optimized) and diversified benchmark (given). Graphs showing stock and oil prices with missing and outlier data, marginal distributions of each security, and allocation riskreward, portfolio composition and certainty-equivalent were saved as PDF.

Results for an 4-week investment of USD 100 at the end of the estimation period:
Estimation 4-Jan-2000 through 1-Nov-2007, best profit on 29-Nov-2007:
nuclear benchmark profit USD 1.21

Estimation 4-Jan-2000 through 15-Mar-2003, best profit on 12-Apr-2003:
oil benchmark profit USD 2.1

Estimation 2-Sep-1998 through 20-Oct-2000, best profit on 17-Nov-2000:
nuclear benchmark profit USD 3.52

Estimation 1-Jun-1997 through 20-Aug-1998, best profit on 17-Sep-1998:
oil benchmark relative profit USD 3.52

Estimation 1-Jan-1991 through 1-Jun-1997, best profit on 29-Jun-1997:
oil benchmark relative profit USD 7.45

Estimation 1-Nov-1987 through 30-Jun-1990, best profit on 28-Jul-1990:
oil benchmark relative profit USD 10.25

References:
[1] Attilio Meucci, Risk and Asset Allocation, Springer, 2005.
[2] Stock market data from NYSE Euronext, http://www.nyse.com.
[3] John Carey, “Grabbing a Piece of the Nuclear Action”, Business Week, 27 December 2004.
[4] Stephen D. Simpson, “A Healthier Glow for Nuclear Power?”, Motley Fool, 8 June 2005.
[5] Rich Smith, “America Goes Nuclear”, Motley Fool, 26 September 2007.
[6] James L. Williams, “Oil Price History and Analysis”, http://www.wtrg.com/prices.htm.

### Data:

From [2], [3], [4], [5]:
1) Brent and Oklahoma oil spot prices
23 NYSE prices in 4 sectors:
2) Oil&Gas: Exxon-Mobil XOM, BP, Chevron-Texaco CVX, Total TOT, Royal Shell RDS-B, ENI S.p.A. E, ConocoPhillips
COP 3) Oilfield services: Schlumberger SLB, Haliburton HAL, Baker-Hughes BHI
4) Uranium extraction: RioTinto RTP, Billiton BHP, Cameco CCJ, USEC USU
5) Nuclear services: General Electric GE, Entergy/EDF International ETR, Southern Company SO, Constellation Energy CEG, Dominion Resources D, NRG, McDermott MDR, Exelon EXC
+ historical geo-political events affecting oil prices and likely to affect energy sectors [6].
PM_Project_Data.mat
NYSE stock and Brent/Oklahoma oil spot prices

### Project code (in Matlab):

The entire code is contained in a zip file: PiotrMirowski_QRPM_Code.zip.

PM_Project.m
PM_AllocateBenchmark.m
PM_AllocateTotalWealth.m
PM_DetectOutliers.m

PM_Display.m
PM_EvaluateInvestment.m
PM_GetFileNames.m
PM_Mahalanobis2.m
PM_MarketStudy.m
PM_MissingDataEM.m
PM_MLEstimate_Normal.m
PM_MLEstimate_Student.m
PM_ProjectNormal.m