The standalone and the portfolio risk of the Rogers energy commodity index

This paper tackles the rather recent weekly period from January 18, 2005 to February 28, 2018, encompassing 523 observations. The portfolio is constructed from the perspective either of a US investor or of a Lebanese one, since the US dollar foreign exchange rate was pegged during the above whole period. The portfolio consists of an investment in the US S & P 500 stock market index and in three Rogers international commodity indexes: agricultural, energy, and metals. The purpose of the paper is to estimate the diversification benefits of the energy commodity index. These benefits arise from the fall in the volatility of the investment portfolio when it is compared to an investment in the energy index only, or in the S & P 500 only. The procedure follows the seminal approach of Markowitz. The inputs of the model are the variance/covariance matrix, the average log returns, and the condition that all investment shares should sum up to 1. The outputs, obtained by matrix manipulation, are the optimal investment shares in the four assets, the volatilities of the optimal portfolios, the characteristics of the efficient frontier, the relation between portfolio shares and the expected, or required return, and finally, the predicted Capital Market Line (CML). The evidence shows that, by holding a portfolio composed of the above four assets, the volatilities are substantially reduced. Moreover, and since short sales are allowed in the model, all optimal investment shares in the energy commodity asset are negative, meaning that in the optimal portfolios the positions in the energy index are short positions. The paper points to the significantly high relative riskiness of the energy index, as a stand-alone asset, or as an aggressive and speculative investment on the CML, and to the substantial portfolio benefits of shorting this index.


Introduction
There is plenty of research on the relation between commodity prices and inflation.In this vein, authors have adopted the overshooting notion of commodity prices.Moreover there is a vast literature on the forecasting ability of commodity prices, and on the relation between commodity prices, the money supply, and the stock markets.There is little interest on the portfolio behavior of commodities, except maybe in finding out whether commodities are in general hedges or diversifiers.This paper stands within this niche.A portfolio of only commodity indexes, and a broad stock market index is constructed.The statistical properties of this portfolio are studied, with a special stress on the additional portfolio impacts of the energy index on the stock market.Always positive as long as the expected return is positive.

SIX commodity futures contracts
The share in the agriculture index is zero for a value for  ෨  of 0.00267, or 13.36%, in annualized terms.This means that for practically reasonable values for  ෨ the share in this index is also positive, and is rarely shorted.
The energy share is zero for  ෨  = 0.0006473, or 3.24% in annualized terms.Any value for  ෨  higher that this rate produces a negative share.Since the Minimum Variance Portfolio (MVP) has a return of 0.00091893, or 4.60% in annualized terms, the share in the energy index is never positive.
The Negative Energy Share The variable on the x-axis is the variance of the portfolio, and the variable on the y-axis is the expected return.The efficient frontier has the same shape as found in all finance textbooks.It is a parabola convex to the y-axis.
The vertex is the Minimum Variance Portfolio (MVP), which has a return of 0.00091893, or 4.60% in annualized terms, and a variance of 0.0003871, or around 1.94% in annualized terms.The annualized standard deviation is 13.9%.Moving from the vertex of the MVP rightward on the curve one draws the efficient frontier, which provides for all portfolios that dominate all others either in terms of expected return, or in terms of variance.

The Efficient Frontier (3)
The three assets, energy index, agriculture index, and the average return of the portfolio, are characterized by optimal variances that lie on the lower or inefficient part of the efficient frontier.Assuming that the optimal substitute for these three assets is to match their variances, the three optimal portfolios should have respectively expected returns of 0.00537 (26.85%), 0.00262 (13.1%), and 0.00230 (11.5%), with the annualized returns in parentheses.These figures give the extent by which the return on each one of these three assets must increase to become optimal, i.e. on the efficient frontier.
A Flatter CML with a Higher Risk-free Rate

The Tangency Portfolio
Exhibit 2 portrays the efficient frontier together with the tangency portfolio.This portfolio depends on the assumed risk-free rate.If one chooses a rate of 1.56% per annum, which is the in-sample average Eurodollar rate, one gets the straight line in Exhibit 2. Exhibit 3 presents the tangency portfolios from two assumptions of the risk-less rate: 1.56% as above, and 3.80%, as estimated from long historical data.The second one is flatter as expected.These tangency portfolios were obtained by maximizing the ratio: where  ෨  is the random expected return,   2 is the portfolio variance, and where  is the fixed riskfree rate.The first tangency line, which happens to be the Capital Market Line (CML) for that specific risk-less return, has an average return of 0.00248 (12.90% annualized), a figure which is quite close to the return on a well-diversified portfolio of common stocks (Brealey et al., 2017).The fact that the data in this paper result in reasonable values for the tangency portfolio is testimony to the soundness of the model.The second CML with the second estimate of the risk-free rate, carries an average return of 0.00282 (14.66% annualized), which is still reasonable for the US financial markets.The two estimated slopes of the CML produce two estimates of the market variance: 0.000604 (3.14% annualized variance) and 0.000709 (3.69% annualized variance).These two variances represent the estimates of the variance of the market portfolio.In terms of standard deviations, the figures become 17.72% and 19.20% respectively.Again such estimates are quite close to the standard deviation of a well-diversified portfolio of common stocks in the US financial markets.

CONCLUSION
• The energy index has the highest standalone risk, both economically and statistically with an annualized standard deviation of 32.8%.The next in line is the metals index with an annualized standard deviation of only 21.2%.• An annualized return of at least 27.9% is required to hold long the energy index.This required return can reach 41.9% with different assumptions.• A highly leveraged and aggressive position is needed to hold the energy index, e.g. the investor must borrow some three times her wealth.• The energy share in any portfolio is always negative, i.e. the energy index is always shorted.• It is unclear whether shorting the energy index to this extent is practicable and feasible.