# Robust Optimization-Based Commodity Portfolio Performance

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## Abstract

**:**

## 1. Introduction

## 2. Data and Methodology

#### 2.1. Commodity Sample and Returns Construction

#### 2.2. Robust Optimization

**r**. The expected value and the variance of the function $\mathrm{f}\left(w,r\right)$ are given by:

#### 2.3. Algorithms for Robust Optimization under Uncertainty

#### 2.4. Performance Metrics

## 3. Results

#### 3.1. Sectoral Risk and Returns Analysis

#### 3.2. Portfolio Risk and Returns

#### 3.3. Optimal Performance by Holding Period

## 4. Concluding Remarks

## Author Contributions

## Funding

## Conflicts of Interest

## References

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1 | In practice, there are many different definitions of robustness based on various mathematical formulations. See Kim et al. (2016) for an overview of several optimization methodologies. |

2 | It is worth noting that there is an elliptical constraint on the standard deviation of returns in Kim et al. (2017) that is not assumed in our robust estimation process which may impact on the discrepancy in findings. |

3 | Returns are always calculated on the same contract and we do not include the return on collateral associated with the futures contract in the calculation. |

4 | Asness et al. (2013) and Moskowitz et al. (2012) create a monthly series with the same procedure; specifically, to convert the daily returns to monthly returns the following formula is applied: ${\mathrm{R}}_{\mathrm{Month}}=\left(\prod _{\mathrm{i}\in \mathrm{day}}\left(\frac{{\mathrm{r}}_{\mathrm{i}}}{100}+1\right)-1\right)$. |

5 | In fact, when the past 24 months of return data are used in the robust portfolio optimization process, both the MV and CVaR portfolios further underperform the naïve buy-and-hold benchmark. |

**Figure 1.**Empirical Distributions of Monthly Commodity Futures’ Sector Returns. Notes: This figure shows the distribution of monthly returns for the five commodity sectors over the sample period from January 1986 to December 2018. “Foods” represents the foods and fibers sector, “Grains” represents the grains and oilseeds sector, and “Metals” represents the precious metals sector.

**Figure 2.**Cumulative Monthly Returns by Commodity Sector. Notes: This figure shows the cumulative monthly returns of the five commodity sectors used in this study for January 1986 to December 2018.

**Figure 3.**Performance of Equally Weighted and MV Optimization-based Commodity Portfolios. Notes: This figure shows the cumulative return for the naïve buy-and-hold benchmark portfolio relative to the mean-variance robust optimization-based portfolios over the sample period from January 1986 to December 2018. Equally weighted represents the naïve buy-and-hold benchmark portfolio. MV12, MV15, and MV18 represent the robust optimization-based mean-variance portfolios that use a 12-month, 15-month, and 18-month lookback period of returns, respectively, with a one-month portfolio holding period.

**Figure 4.**Performance of Equally Weighted and CVaR Optimization-based Commodity Portfolios. Notes: This figure shows the cumulative return for the naïve buy-and-hold benchmark portfolio relative to the robust optimization-based portfolios over the sample period from January 1986 to December 2018. Equally weighted represents the naïve buy-and-hold benchmark portfolio. CVaR12, CVaR15, and CVar18 represent the robust optimization-based conditional value-at-risk portfolios that use a 12-month, 15-month, and 18-month lookback period of returns, respectively, with a one-month portfolio holding period.

Sector | Commodities |
---|---|

Foods and Fibers | Cocoa, Coffee, Orange Juice, Sugar #11, Cotton, Lumber |

Grains and Oilseeds | Corn #2, Oats, Rough Rice #2, Soybeans, Soybean Meal, Soybean Oil, Wheat, Barley, Canola |

Livestock | Feeder Cattle, Live Cattle, Lean Hogs, Pork Bellies |

Energy | Crude Oil, Heating Oil #2, Unleaded Gas, Natural Gas, Propane |

Precious Metals | Copper, Gold, Palladium, Platinum, Silver |

Performance Metric | Description |
---|---|

Arithmetic mean | Reported as the average monthly return expressed as an annualized percentage. |

Standard deviation | Reported as the average monthly standard deviation expressed as an annualized percentage. |

Geometric mean | Reported as the average monthly return expressed as an annualized percentage. |

Cumulative return | Reported as the portfolio return over the full sample period. |

Sample skewness | Reported as a monthly average. |

Sample excess kurtosis | Reported as a monthly average. |

Sharpe ratio | Reported as the average excess monthly return divided by the monthly standard deviation, where the risk-free rate is obtained from Ken French’s website. |

Tracking error | Reported as the monthly average standard deviation of the difference between a commodity portfolio returns and the value-weighted market index of returns from the Center for Research in Security Prices (CRSP). The CRSP market index return is obtained from Ken French’s website. |

Information ratio | Reported as the excess monthly return of a portfolio in excess of the CRSP value-weighted market index of returns divided by the tracking error. |

CAPM alpha | Reported as the average monthly alpha computed following Jensen (1968) expressed as an annualized percentage, where the market factor is obtained from Ken French’s website. |

CAPM beta | Reported as the average monthly beta computed following Sharpe (1964), where the market factor is obtained from Ken French’s website. |

Treynor ratio | Reported as the average excess monthly return divided by the monthly portfolio beta, where the risk-free rate is obtained from Ken French’s website. |

Sortino ratio | Reported as the average excess monthly return divided by the monthly standard deviation of negative asset returns, where the risk-free rate is obtained from Ken French’s website. |

Historical 95% VaR | Reported as the average expected 1-month loss with 95% certainty, based on historical returns. |

Normal 95% VaR | Reported as the average expected 1-month loss with 95% certainty, under normality. |

Historical 95% CVaR | Reported as the average expected 1-month loss beyond the VaR with 95% certainty, based on normality. |

Normal 95% CVaR | Reported as the average expected 1-month loss beyond the VaR with 95% certainty, based on historical returns. |

M-square | Reported as the average monthly return of a portfolio plus the product of the average monthly Sharpe ratio of the equally weighted benchmark and the average deviation of the standard deviation of the portfolio under consideration from the benchmark portfolio. |

Statistics | Foods | Grains | Livestock | Energy | Metals |
---|---|---|---|---|---|

Arithmetic Mean (%) | 1.0662 | 0.1996 | 3.4703 | 7.6406 | 7.2003 |

Standard Deviation (%) | 15.7008 | 19.1108 | 17.0492 | 31.3845 | 18.2431 |

Geometric Mean (%) | −0.1667 | −1.5985 | 1.9863 | 2.6406 | 5.4434 |

Cumulative Returns (%) | 35.0131 | 6.5806 | 112.7373 | 243.7192 | 230.1109 |

Sample Skewness | 0.1328 | 0.3862 | 0.0607 | 0.6909 | 0.0068 |

Sample Excess Kurtosis | 0.9356 | 3.1046 | 0.4567 | 3.0975 | 2.3497 |

Sharpe Ratio (%) | −0.0384 | −0.0446 | 0.0045 | 0.0390 | 0.0605 |

Tracking Error (%) | 0.0560 | 0.0648 | 0.0646 | 0.0971 | 0.0607 |

Information Ratio (%) | −0.1466 | −0.1377 | −0.0966 | −0.0302 | −0.0539 |

CAPM Alpha (%) | 0.2201 | 0.2008 | 0.0442 | 0.1818 | 0.2621 |

CAPM Beta | 0.0040 | 0.0008 | 0.0645 | 0.0339 | 0.0222 |

Treynor Ratio (%) | −0.1460 | −0.1362 | −0.0964 | −0.0301 | −0.0539 |

Sortino Ratio (%) | −0.0539 | −0.0630 | 0.0064 | 0.0605 | 0.0900 |

Historical 95% VaR | 6.5260 | 8.6736 | 7.9052 | 13.5137 | 8.0517 |

Normal 95% VaR | 7.3668 | 9.0577 | 7.8108 | 14.2868 | 8.0812 |

Historical 95% CVaR | 9.0241 | 12.3231 | 10.1816 | 17.8220 | 11.4665 |

Normal 95% CVaR | 9.2607 | 11.3630 | 9.8673 | 18.0726 | 10.2818 |

M-Square (%) | −0.0081 | −0.0084 | −0.0063 | −0.0048 | −0.0038 |

Statistics | EW | MV12 | MV15 | MV18 | CVaR12 | CVAR15 | CVaR18 |
---|---|---|---|---|---|---|---|

Arithmetic Mean (%) | 3.4571 | 4.7156 | 3.0853 | 2.2482 | 4.6876 | 2.3093 | 2.3355 |

Standard Deviation (%) | 11.4143 | 15.5863 | 15.8840 | 16.0667 | 15.5513 | 15.6294 | 16.0739 |

Geometric Mean (%) | 2.7800 | 3.4714 | 1.7792 | 0.9629 | 3.4493 | 1.0537 | 1.0478 |

Cumulative Returns (%) | 107.2094 | 145.4249 | 95.8390 | 70.1003 | 144.5779 | 71.9832 | 72.7921 |

Sample Skewness | −0.5860 | 0.6603 | −0.2534 | 0.8613 | 0.6664 | −0.2570 | 0.8595 |

Sample Excess Kurtosis | 3.1129 | 12.0746 | 6.7088 | 10.6254 | 12.1978 | 6.1192 | 10.7345 |

Sharpe Ratio (%) | 0.0095 | 0.0294 | 0.0003 | −0.0144 | 0.0290 | −0.0137 | −0.0129 |

Tracking Error (%) | 0.0477 | 0.0555 | 0.0561 | 0.0573 | 0.0555 | 0.0560 | 0.0573 |

Information Ratio (%) | −0.1181 | −0.0834 | −0.1057 | −0.1155 | −0.0838 | −0.1172 | −0.1142 |

CAPM Alpha (%) | 0.1852 | 0.2218 | 0.2231 | 0.2014 | 0.2205 | 0.2084 | 0.2014 |

CAPM Beta | 0.0153 | 0.0173 | 0.0114 | 0.0092 | 0.0173 | 0.0091 | 0.0096 |

Treynor Ratio (%) | −0.1192 | −0.0823 | −0.1059 | −0.1129 | −0.0827 | −0.1174 | −0.1117 |

Sortino Ratio (%) | 0.0129 | 0.0444 | 0.0004 | −0.0213 | 0.0437 | −0.0189 | −0.0190 |

Historical 95% VaR | 5.0998 | 5.8583 | 6.7128 | 6.4342 | 5.8908 | 6.6751 | 6.3910 |

Normal 95% VaR | 5.1362 | 7.0161 | 7.2886 | 7.4435 | 7.0017 | 7.2309 | 7.4398 |

Historical 95% CVaR | 7.7301 | 9.2222 | 10.6792 | 10.2852 | 9.2187 | 10.7611 | 10.2213 |

Normal 95% CVaR | 6.5130 | 8.8962 | 9.2046 | 9.3815 | 8.8776 | 9.1162 | 9.3787 |

M-Square (%) | −0.0055 | −0.0047 | −0.0059 | −0.0066 | −0.0047 | −0.0065 | −0.0065 |

**Table 5.**Performance Metrics by Holding Period for MV Optimization Portfolio using a 12-Month Lookback Period.

Statistics | HP1 | HP3 | HP6 | HP9 | HP12 |
---|---|---|---|---|---|

Arithmetic Mean (%) | 4.7156 | 3.2670 | 3.2149 | 2.9777 | 2.8339 |

Standard Deviation (%) | 15.5863 | 16.2129 | 16.1022 | 15.9408 | 16.4617 |

Geometric Mean (%) | 3.4714 | 1.9352 | 1.9033 | 1.6893 | 1.4789 |

Cumulative Returns (%) | 145.4249 | 101.3997 | 99.8060 | 92.5416 | 88.1293 |

Sample Skewness | 0.6603 | 0.5784 | 0.5969 | 0.4159 | 0.7399 |

Sample Excess Kurtosis | 12.0746 | 10.4462 | 10.0213 | 7.6699 | 6.2305 |

Sharpe Ratio (%) | 0.0294 | 0.0034 | 0.0025 | −0.0016 | −0.0040 |

Tracking Error (%) | 0.0555 | 0.0566 | 0.0561 | 0.0582 | 0.0589 |

Information Ratio (%) | −0.0834 | −0.1023 | −0.1040 | −0.1035 | −0.1042 |

CAPM Alpha (%) | 0.2218 | 0.2333 | 0.2404 | 0.1644 | 0.1804 |

CAPM Beta | 0.0173 | 0.0115 | 0.0110 | 0.0149 | 0.0129 |

Treynor Ratio (%) | −0.0823 | −0.1008 | −0.1025 | −0.1024 | −0.1026 |

Sortino Ratio (%) | 0.0444 | 0.0050 | 0.0037 | −0.0024 | −0.0061 |

Historical 95% VaR | 5.8583 | 6.5522 | 6.9092 | 6.5516 | 6.7393 |

Normal 95% VaR | 7.0161 | 7.4301 | 7.3817 | 7.3243 | 7.5833 |

Historical 95% CVaR | 9.2222 | 9.9578 | 10.2224 | 9.9530 | 9.9973 |

Normal 95% CVaR | 8.8962 | 9.3858 | 9.3241 | 9.2472 | 9.5690 |

M-Square (%) | −0.0047 | −0.0058 | −0.0058 | −0.0060 | −0.0061 |

**Table 6.**Performance Metrics by Holding Period for CVaR Optimization Portfolio using 12-Month Lookback Period.

Statistics | HP1 | HP3 | HP6 | HP9 | HP12 |
---|---|---|---|---|---|

Arithmetic Mean (%) | 4.6878 | 3.1452 | 3.1313 | 2.9732 | 2.8119 |

Standard Deviation (%) | 15.5513 | 16.1615 | 16.0486 | 15.9299 | 16.3438 |

Geometric Mean (%) | 3.4495 | 1.8229 | 1.8290 | 1.6876 | 1.4750 |

Cumulative Returns (%) | 144.5836 | 97.6744 | 97.2473 | 92.4031 | 87.4524 |

Sample Skewness | 0.6665 | 0.5738 | 0.5911 | 0.4486 | 0.6995 |

Sample Excess Kurtosis | 12.1978 | 10.5528 | 10.1023 | 7.8684 | 5.8570 |

Sharpe Ratio (%) | 0.0290 | 0.0013 | 0.0011 | −0.0017 | −0.0044 |

Tracking Error (%) | 0.0555 | 0.0564 | 0.0559 | 0.0582 | 0.0586 |

Information Ratio (%) | −0.0838 | −0.1044 | −0.1055 | −0.1036 | −0.1051 |

CAPM Alpha (%) | 0.2205 | 0.2345 | 0.2417 | 0.1643 | 0.1807 |

CAPM Beta | 0.0173 | 0.0110 | 0.0106 | 0.0149 | 0.0128 |

Treynor Ratio (%) | −0.0827 | −0.1028 | −0.1039 | −0.1024 | −0.1035 |

Sortino Ratio (%) | 0.0437 | 0.0019 | 0.0016 | −0.0025 | −0.0067 |

Historical 95% VaR | 5.8908 | 6.5674 | 6.9211 | 6.5630 | 6.7529 |

Normal 95% VaR | 7.0017 | 7.4156 | 7.3631 | 7.3195 | 7.5291 |

Historical 95% CVaR | 9.2186 | 9.9688 | 10.2255 | 9.9368 | 9.9135 |

Normal 95% CVaR | 8.8776 | 9.3651 | 9.2990 | 9.2411 | 9.5006 |

M–Square (%) | −0.0047 | −0.0059 | −0.0059 | −0.0060 | −0.0061 |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Adhikari, R.; Putnam, K.J.; Panta, H.
Robust Optimization-Based Commodity Portfolio Performance. *Int. J. Financial Stud.* **2020**, *8*, 54.
https://doi.org/10.3390/ijfs8030054

**AMA Style**

Adhikari R, Putnam KJ, Panta H.
Robust Optimization-Based Commodity Portfolio Performance. *International Journal of Financial Studies*. 2020; 8(3):54.
https://doi.org/10.3390/ijfs8030054

**Chicago/Turabian Style**

Adhikari, Ramesh, Kyle J. Putnam, and Humnath Panta.
2020. "Robust Optimization-Based Commodity Portfolio Performance" *International Journal of Financial Studies* 8, no. 3: 54.
https://doi.org/10.3390/ijfs8030054