Use of crops and livestock futures contracts in portfolios: an analysis of feasibility

Use of crops and livestock futures contracts in portfolios: an analysis of feasibility

Fabio L. Mattos^I; Joaquim Bento de Souza Ferreira Filho^II

^IMaster in Applied Economics. Email:[e-mail]

^IIProfessor at ESALQ/USP. Piracicaba - SP. Brazil. 13418-900. B.O.Box 9. Email: [e-mail]

ABSTRACT

According to Portfolio Theory, by combining assets that show a correlation inferior to one (1) among their individual returns, it becomes possible to create portfolios that reduce risk without damaging expected return. Crop and livestock futures contracts and company stocks show such a characteristic, which signals potential benefits when forming portfolios combining these two types of assets. This investment strategy is not often utilized in Brazil. The purpose of our research was to assess whether such an asset combination is actually advantageous to those creating investment portfolios in the Brazilian market. Our evaluation used instruments of analysis developed by Markowitz in Portfolio Theory and data about the return from crop and livestock futures contracts and stocks. The data was gathered from the Brazilian Futures and Commodities Exchange (BM&F) and Brazil’s National Association of Open Market Institutions (ANDIMA) between July 1994 and December 1998. The results of this work showed that the combination of these two types of assets in investment portfolios can be an interesting portfolio management alternative.

Key words: portfolio, crops and livestock futures contracts, Markowitz

1. Introduction

The purpose of this work was to analyze the use of crop and livestock futures contracts to reduce stock portfolio risk and to determine the advantages and disadvantages of such a combination in Brazil.

Chance (1994) notes that the world’s financial markets went through a period of significant evolution in early 1980s that included an increase in the popularity of futures contracts as investment instruments. In the specific case of crop and livestock commodities, Chance (1994) gives the example of a portfolio manager who decides that the incorporation of soybeans in his asset portfolio could increase his return without significantly modifying his risk. Unfortunately, the acquisition of a physical product would incur various warehousing costs and involve difficulty in finding a buyer for the product in the future. A soybean futures market would solve these problems, since it allows such a manager to add the product to his portfolio (with expected advantages) without need of physical possession.

When emphasizing the significance of futures markets, Chance (1994) points out that their main advantage lies in the fact that the returns from commodities futures contracts, a distinct type of asset, show a low correlation with the returns from other types of assets. A correlation inferior to 1 between the returns from two assets means, roughly speaking, that the assets’ returns do not show the same behavior. The further this correlation is below one, the less similar will be the development of these two assets’ returns over time. If the return correlation is less than one, a decreasing return from one asset does not imply that the other asset’s return is also decreasing.

According to Portfolio Theory, by adding assets to a portfolio that show a return correlation inferior to one with the portfolio’s existing assets, it is possible to reduce the portfolio’s risk without damaging its expected return. It is worth pointing out that investors and portfolio mangers prefer more return and less risk (Markowitz, 1952).

In one study of the financial markets in the United States, France, Germany, Japan, Switzerland, and the United Kingdom between 1989 and 1998, Allen (1999) found that the correlations between the returns from future contracts, stocks, and bonds were close to zero. The same author prepared various portfolios combining futures contracts, stocks, and bonds from the markets under analysis and verified that such a combination increased the efficiency of the portfolios¹ in all situations studied. Works carried out by Bodie & Rosansky (1984), Lee et al. (1985), Herbst & McCormack (1986 e 1987), Elton et al. (1990), Irwin et al. (1993), Edwards & Park (1996), Lintner (1997) and Schneeweis & Spurgin (1997) proved that such strategy could be successfully adopted in various situations.

It was not possible to find analyses that assess the validity of this strategy in the Brazilian market. In addition, crop and livestock futures contracts are a very minor participant in the Brazilian financial market, accounting for about 1% of the Brazilian Futures and Commodities Exchange’s (BM&F) trading. Nevertheless, this type of study would be of a great value for the Brazilian economy as, by exposing the value of commodity futures contracts, it could develop alternative sources of agricultural funding, such as private investors and portfolio managers The strengthening of private investment in Brazilian agriculture is especially relevant under the current environment of tight government agricultural expenditures and reduced government incentives directed toward the support of agricultural markets.

The present work was designed to answer the question, is the combination of Brazilian equity assets with Brazilian crop and livestock commodity instruments advantageous when creating investment portfolios in Brazil? In other words, is there empirical evidence capable of proving the advantages of such a combination in Brazil?

To answer this question, we analyzed the performance of one portfolio composed of stocks only and other portfolios composed of different combinations of stocks and crop and livestock futures contracts. This process was carried out in four phases: (a) analysis of the return/risk ratio of a stock portfolio and of Brazilian crop and livestock futures contracts; (b) assessment of correlation between the returns from futures contracts and from stocks, which signals the potential of diversifying a stock portfolio through the inclusion of commodities futures contracts; (c) comparison of the performance of all constructed portfolios; and (d) an assessment of the feasibility of creating efficient portfolios through the combination of stocks and crop and livestock futures contracts.

2. Methodology

The Ibovespa, an index that reflects the price behavior of the more traded stocks in the São Paulo Stock Exchange (Bovespa) was used to represent the all stock portfolio’s assets and the stock portion of all other portfolios. The Ibovespa was chosen as it answers for approximately 85% of total volume traded in the Brazilian stock market and is considered representative of the exchange’s average performance. In the case of crops and livestock futures contracts, we used coffee, cotton, corn, live cattle, and soybean contracts traded in the BM&F over the period under consideration.

Data used in this work refer to the period between July 1994 and December 1998: 54 months. This choice was made for two reasons: (a) prior to this period, few crop and livestock futures contracts were continuously traded, and (b) various asset price distortions between the mid 1980s and mid 1990s, which were the result of a sequence of economic stabilization programs. Within our study period, various sub-periods are also considered. The sub-periods were to analyze the results obtained by strategies based on crop and livestock futures contracts drawn up at the beginning of 1995, 1996, and 1997 and maintained until the end of period covered by this work (December 1998). We suppose here that investors review their investment approach at the beginning of each year, after the close of annual planning and accounting.

It is worth pointing out that only the Ibovespa stocks and coffee and live cattle contracts were being traded at the beginning of the study period. In the case of sugar and soybean futures contracts, we used 38 observations taken from November 1995 to December 1998; and for corn and cotton futures contracts, we used 25 observations from December 1996 to December 1998. All data were obtained from the National Association of Open Market Institutions (ANDIMA) and the BM&F.

The crop and livestock commodities futures market in Brazil accounts for only 1% of the total trade in the Brazilian futures market. From this 1%, only coffee and live cattle contracts show significant liquidity (although their liquidity is still far behind the liquidity of interest markets, the currency exchange, and the stock index). The other contracts, sugar, soybean, corn and cotton, form a low liquidity market, which may have important consequences on this study.

The evolution of futures contract prices may be influenced by the level of market liquidity; hence, the behavior of these prices may suffer alteration as significant changes occur in the liquidity level of such markets. The conclusion reached from the analysis of an illiquid commodity market may be drastically adjusted if data related to higher liquidity markets are used. Anyway, we opted to use all crops and livestock futures contracts traded over the period under consideration, as they represent the reality found by investors interested in this type of Brazilian market.

The analysis presented in this work was constructed in four phases. In the first phase, return rates from crop and livestock futures contracts and the Ibovespa stocks were calculated. The Ibovespa return rate is given by the variation of its value over time. Bawa and Chakrin (1979) point out that various authors have verified that financial asset prices are very similar when distributed lognormally, and such distributions are still often employed to model asset returns. Hence, the return rate adopted for this analysis was given by the natural logarithm of monthly asset value variation (Elton et al., 1990):

(1)

where

is the return rate from the Ibovespa in the month t;

is the Ibovespa quotation in the last day of month t; and

is the Ibovespa quotation in the last day of month t-1.

Similarly, the calculation of return rate for long position future contracts is given by (Bodie & Rosansky, 1984):

(2)

where

is the return rate from a long position in future contract j in month t;

is the settlement price of a future contract j on the last day of month t; and

is the settlement price of a future contract j on the last day of month t-1.

In the case of short positions, it is possible to adapt the same principle:

(3)

where

is the return rate from a short position in future contract j in month t;

is the settlement price of a future contract j on the last day of month t; and

is the settlement price of a future contract j on the last day of month t-1.

All settlement prices considered are related to the first maturity of the future contract, which is the date with the highest trading volume and open interest².

Finally, the return rate of a portfolio that combines stocks and futures contracts is given by the weighted average return of each of such assets (Markowitz, 1952 and 1959):

(4)

where

is the return of P portfolio in the month t;

is the participation of Ibovespa in the portfolio;

is the return of Ibovespa in the month t;

is the participation of future contract j in the portfolio;

is the return of future contract j in the month t; and

is the number of different future contracts in the portfolio.

Markowitz (1952 and 1959) demonstrated that the standard deviation of an asset’s return can be used to measure the asset holder’s exposure to uncertain change in the value of the asset: the asset’s "risk." The risk of each asset analyzed by this work was given by its deviation of return in relation to its average value, that is to say, by the standard deviation of its return:

(5)

where

is the standard deviation of asset j return;

is the return rate of asset j in the month t;

is the return expected for asset j; and

is the total number of days of the month.

In the case of a portfolio, the calculation of risk follows the same reasoning, i.e., the risk of a portfolio is given by standard deviation of its return. Hence, the standard deviation of portfolio returns is equivalent to (Markowitz, 1952 and 1959):

(6)

where

is the standard deviation of portfolio return;

is the square of standard deviation of asset i return composing such portfolio;

is the participation of asset i in the portfolio;

is the co-variance between the return of assets which compose the portfolio; and

is the number of portfolio assets.

Therefore, a portfolio’s risk depends not only on the deviation of the returns from its assets but also on the co-variance between the assets.

The second phase of this work was the determination of the correlation coefficient between the returns from stocks and the returns from crop and livestock futures contracts. Such a coefficient plays an important role in Portfolio Theory since a strategy of scientific investment diversification is only feasible if there is a combination of assets whose returns show a correlation inferior to 1 (Markowitz, 1952 and 1959).

The scientific diversification suggested by Markowitz consists of composing an investment portfolio with types of assets that respond differently to the same event. The basic principle is that the behavior followed by each asset’s return cannot be equal. Therefore, if a certain event lowers the return from portfolio asset X, the returns from other portfolio assets would not be similarly affected, offsetting the decreased return from X. It is important that a portfolio be composed of assets that show a correlation of returns inferior to 1 if risk is to be minimized.

To reduce the possibility that our sample coefficient estimates lead to the wrong conclusions, we made use of a statistical test to determine if the correlation could be equal to zero (Hoel, 1971). We tested the significance of correlation by assuming that it is null in population and by carrying out a t test (Hoel, 1971):

(7)

(8)

where

t is the statistics used in the test;

r is the sample correlation coefficient; and

n is the number of sample observations. Given a certain level of significance, the assumption of a null correlation cannot be rejected if the value calculated for t is lower than its critical value (listed).

The third phase of this work consisted of analyzing the performances of the portfolios that combined stocks with differing percentages of crop or livestock futures contracts and a portfolio composed entirely of stocks. This comparison was made as follows: from a portfolio composed only of stocks (100% Ibovespa), we started to gradually include a futures contract, allowing the participation of this contract to grow progressively while the participation of Ibovespa stocks diminished (1% futures contract and 99% Ibovespa, 2% futures contract and 98% Ibovespa, and so on until there was no stock in the portfolio, i.e. 100% futures contract).

This process gave rise to an assembly of points representing the returns from portfolios with various combinations of stocks and futures contracts. It was then possible to calculate the return and the risk of each portfolio according to equations (4) and (6). In order to properly assess the assembly of points, we applied a performance measure to the portfolios created. The measure chosen was the Sharpe Index (SI), which measures the return obtained above the risk free interest rate by unit of risk assumed by the investor (Tobin, 1958; Sharpe, 1966):

(9)

where

E(R_i) is the expected return of portfolio i;

R_F is the risk free rate; and

is the standard deviation of the return of portfolio i.

The Selic rate, which represents the return from Brazilian government bonds, was used to represent the risk free rate (Securato, 1999). The risk free rate for a given period was deemed as the simple arithmetic average of the bond return values during the period analyzed.

A positive Sharpe Index means that a certain asset offered a premium for the risk assumed. If the index is above 1, the premium offered was proportionally higher than the risk assumed. For values between 0 and 1, the index indicates that the premium offered was proportionally lower than risk assumed. Therefore, assets with higher Sharpe Index values are preferable than those with lower values. At the end of this phase, it was possible to verify the point at which the inclusion of futures contracts in stock portfolios was able to increment the portfolio return/risk ratio.

In the fourth phase of this work, the feasibility of building efficient portfolios through a combination of stocks and crop or livestock futures contracts was assessed by means of an algorithm developed by Markowitz (1952 e 1959). An efficient portfolio is one that shows the lowest possible level of risk for a given return or the highest possible return for a given level of risk (Markowitz, 1952 and 1959). It is assumed that investors try to minimize investment risk at a certain level of expected return or maximize expected return given a certain level of risk. Therefore, for the case of N assets, we have:

minimize

subject to

e

(10)

or then

maximize

subject to

e

(11)

At the end of this phase, we could verify if it is possible to generate efficient investment frontiers through a combination of stocks and crop and livestock futures contracts in portfolios. It was also possible to verify if such combinations would be able to produce more efficient portfolios than portfolios made up of only the Ibovespa stocks.

3. Results

Analyses of the various rates of return for the period between July 1994 and December 1998 verified that the Ibovespa tended to produce higher returns than the both long and short crop and livestock futures contracts, returns from which were very modest (Chart 1). In addition, analysis of the futures contracts’ returns over the period reveals that better opportunities for gain were verified in specific sub-periods, i.e., there was not a consistent return from these assets over the entire period, July 1994 to December 1998. Therefore, an investor would not be motivated to search for crop and livestock futures contracts if his purpose is higher portfolio returns. As the return from a portfolio is the weighted average of the returns from the assets composing the portfolio, the combination of futures contracts and stocks in investment portfolios during the period analyzed would present a return inferior to those a portfolio composed of only the Ibovespa stocks.

In relation to the risk of each asset, it was possible to verify that crop and livestock futures contracts were riskier than the Ibovespa over the majority of the sub-periods analyzed (Chart 2). The Ibovespa showed higher standard deviation of returns values than most futures contracts. Anyway, all assets revealed a high associated risk, suggesting that a strategy of investment diversification could be attempted as a way of reducing the total risk assumed by the investor.

The relatively high risk associated with the assets analyzed tends to reduce their Sharpe Index values, which in the case of futures contracts is aggravated by their low rate of return (which ultimately turn these indexes negative in the majority of sub-periods). The Ibovespa is distinguished for presenting higher Sharpe Indices in the majority of cases studied. Therefore, as was suggested in prior analyses, the Sharpe Index shows that the futures contracts are less attractive assets if taken on an individual basis.

The results from the second phase of this work show that sample correlation coefficients between returns from the Ibovespa and from each of the crop and livestock futures contracts always have values close to zero. The results from statistical verification of these values, with a level of significance equal to 0.01, do not allow the rejection of the null assumption of no correlation between their the two types of assets’ returns (Charts 3 and 4). Therefore, it is possible to accept that the correlation coefficient between the Ibovespa’s returns and the returns from each of the six futures contracts is equal to zero, confirming the feasibility of a scientific investment diversification strategy based on the combination of stocks and a crop or livestock futures contract.

In the third phase of this work, we analyzed the effect the addition of a futures contract had on the return and risk of an all stock portfolio. In most cases, as the futures contract’s participation in the portfolio increased, the portfolio’s risk and return diminished. In 75% of the situations analyzed, the allocation that gave highest return was that with all funds invested in the Ibovespa stocks (Charts 5 and 6)³. The addition of a futures contract reduced risk in all but two situations, the inclusion of a long position in cotton in 1997 and a long position in coffee in 1996.

As overall return and risk were generally reduced by the inclusion of a futures contract in a previously all stock investment portfolio, it was necessary to use the Sharpe Index to assess if the lower portfolio risk was sufficient to offset the loss of return. It was verified that in 73% of the situations analyzed, the portfolios that showed higher Sharpe Index values were those composed of stocks alone (Charts 5 and 6). This means that the inclusion of a futures contract in an all stock portfolio tends to diminish the Sharpe Index, making it impossible to affirm that the reduction of risk was sufficient to offset the loss of portfolio return in most cases. However, it is worth pointing out that in a not insignificant 27% of the cases studied, the portfolio with the higher Sharpe Index was composed of stocks and a futures contract.

There is a caveat worth noting: use of the Sharpe Index alone may lead to distorted conclusions when return from the asset under consideration is negative or inferior to the free-of-risk interest rate, for example, the portfolio formed by stock and a cotton futures contract in 1998. In that year, the average return from long and short positions in cotton and the Ibovespa were, –0.28%, +0.28% and –3.39% respectively. The risk of these assets in the period was 4.33% for the futures contracts and 19.26% for the Ibovespa. A simple observation of these numbers would suggest that the option that includes a cotton futures contract is the more advantageous as it revealed higher return and inferior risk than the Ibovespa. Nevertheless, the Sharpe Index signaled the Ibovespa as the best investment alternative. This type of situation occurred in very few cases analyzed, i.e., it appears that this problem does not compromise this work’s general conclusions. Therefore, by means of the exercise carried out in this phase, it was verified that the inclusion of a crop or livestock futures contract would reduce an all stock portfolio’s risk but not sufficiently to offset the loss of return arising from such strategy.

In the fourth phase of this work, an algorithm developed by Markowitz (1952 & 1959) was employed to verify the feasibility of building efficient frontiers by merging Ibovespa stocks with various combinations of six crop and livestock futures contracts. The basic idea was to determine if portfolios combining stocks and futures contracts would be more efficient than portfolios composed only of stocks. We tested possible combinations of stocks and both long and short futures contracts for six commodities over the period between July 1994 and December 1998, which was again divided into various sub-periods. The six commodities were coffee, corn, cotton, live cattle, soybeans, sugar, and coffee. In total, we tried to build 272 efficient frontiers of which 230 attempts succeeded (it was not possible to find a solution for the other 42 combinations using Markowitz’s algorithm).

Among the 230 efficient frontiers built⁴, 114 showed combined portfolios that were more efficient than the Ibovespa stocks. This suggests that diversified portfolios could be created that show a higher return and the same risk or the same return and lower risk than the Ibovespa. For example, in 1997, a portfolio composed of long positions in live cattle (28.640%), long positions in coffee (38.785%), short positions in corn (6.141%), and the Ibovespa (26.434%) would have obtained an average return equal to that of the Ibovespa (3.1%) while the risk from this portfolio would have been 6.9% as compared with 12.4% from the Ibovespa alone.

It is worth pointing out that portfolios diversified with assets other than those employed in this study could show a better efficiency than an Ibovespa portfolio during the period of study. In addition, we call attention to the fact that the Ibovespa stocks cannot be deemed to comprise an efficient portfolio (Nakamura, 1998). Therefore, the determination that portfolios made up of a combination of Ibovespa stocks and crop and livestock futures contracts are more efficient than Ibovespa stocks alone, does not mean that this combination is the only mix of assets with this characteristic.

4. Conclusions

The results of tests carried out according to our proposed methodology allow us to reach two general conclusions regarding the use of crops and livestock futures contracts in stock portfolios: (a) from a theoretical basis, there are advantages in such strategy since it can be assumed that the correlation between Ibovespa returns and returns from each of six future contracts is null; and (b) there is empirical evidence showing that in certain situations there are advantages to be gained by combining crop and livestock futures contracts with stocks in investment portfolios, but these advantages did not appear on a regular basis over the period analyzed by this study.

It was also possible to verify that over the period analyzed crop and livestock futures contracts were not good investment alternatives to the Ibovespa stocks; however, these contracts offered benefits when combined with the Ibovespa stocks in an investment portfolio. The analysis of portfolios combining the Ibovespa and a futures contract proved that this strategy can effectively reduce portfolio risk, although this risk reduction does not always offset the accompanying reduced portfolio return. In addition, we found that it would be possible to build portfolios that combined stocks and crop and livestock futures contracts that were more efficient than a portfolio made up the Ibovespa stocks alone. If, in certain situations, benefits can be obtained by adding crop and livestock futures contracts to stock portfolios, we ask why this strategy is not so used in the Brazilian financial market.

One problem that may reduce the use of crop and livestock futures contracts in investment portfolios is this market’s low liquidity, which creates price formation difficulties and reduces investor flexibility. With few agents active in the Brazilian commodity futures market, prices are more sensitive to specific trading moves: manipulation. Illiquidity reduces investor flexibility when it comes time to trade: can the trade be made. Of course, no investor wants to encounter either of these market problems. The reduced liquidity still existing in this market also impedes investment by fund managers, who find that the crop and livestock futures contracts market is insufficient for their portfolio demands.

The growth of Brazil’s crop and livestock futures market is further hindered by specific characteristics that complicate investment in this market. As opposed to stock transactions, where the great investor concern is price, futures market investors must also focus on contract maturity dates and operational expenses.

Low liquidity and somewhat complicated trading strategies are two of possibly numerous possible explanations for the low demand for crop and live stock futures contracts in the Brazilian financial market. A more comprehensive analysis of the issue will be incumbent upon future works.

• ALLEN, M. Diversifying a typical investment portfolio with managed futures: country overviews. RXR Capital Asset Management, 1999.
• BAWA, V.S.; CHAKRIN, L.M. Optimal portfólio choice and equilibrium in a lognormal securities market. In: ELTON, E.J.; GRUBER, M.J. (Ed.) Portfolio theory, 25 years after North-Holland Publishing Company, 1979, p. 47-62.
• BODIE, Z.; ROSANSKY, V.I. Risk and return in commodity futures. In: PECK, A.E. (Ed.) Selected writings on futures markets: research directions in commodity markets, 1970-1980. Board of Trade of the City of Chicago, 1984. Section 2, cap.5, p.159-176.
• CHANCE, D.M. Managed futures and their role in investments portfolios The Research Foundation of the Institute of Chartered Financial Analysts, 1994.
• EDWARDS, F.R.; PARK, J.M. Do managed futures make good investments?. The Journal of Futures Markets, v. 16, n. 5, p. 475-517, 1996.
• ELTON, E.J.; GRUBER, M.J.; RENTZLER, J. The performance of publicly offered commodity funds. Financial Analysts Journal, July August, 1990.
• HERBST, A.F.; McCORMACK, J.P. An examination of the risk-return characteristics of portfolios combining commodity futures contracts with common stocks Working paper, Center for the Study of Futures Markets, Columbia University, January 1986.
• HERBST, A.F.; McCORMACK, J.P. A further examination of the risk-return characteristics of portfolios combining commodity futures contracts with common stocks Working paper, Center for the Study of Futures Markets, Columbia University, January 1987.
• HOEL, P.G. Introduction to mathematical statistics 4^th ed. John Wiley & Sons, 1971.
• IRWIN, S.H.; KRUKEMYER, T.R.; ZULAUF, C.R. Investment performance of public commodity pools: 1979-1990. The Journal of Futures Markets, v.13, n.7, p.799-820, 1993.
• LEE, C.F.; LEUTHOLD, R.M.; CORDIER, J.E. The stock market and the commodity futures market: diversification and arbitrage potential. Financial Analysts Journal, p. 53-60, July-August 1985.
• LINTNER, J. The potential role of managed commodity-financial futures accounts (and/or funds) in portfolio of stocks and bonds. In: PETERS, C.; WARWICK, B. (Ed.) The handbook of managed futures: performance, evaluation and analysis. Rev. ed. Irwin Professional Publishing, 1997. Part one, chapter 5, p. 99-138.
• MARKOWITZ, H.M. Portfolio selection. The Journal of Finance, v.VII, n.1, p.77-91, March 1952.
• MARKOWITZ, H.M. Portfólio selection: efficient diversification of investments. New York. John Wiley & Sons, 1959.
• NAKAMURA, W. T. Eficiência da carteira teórica do índice Bovespa no contexto da moderna teoria de carteiras São Paulo, 1998. 344 p. Tese (Doutorado) Faculdade de Economia, Administração e Contabilidade, Universidade de São Paulo.
• SCHNEEWEIS, T.; SPURGIN, R. The benefits of managed futures. In: PETERS, C.; WARWICK, B. (Ed.) The handbook of managed futures: performance, evaluation and analysis. Rev. ed. Irwin Professional Publishing, 1997. Part one, chapter 2, p. 17-48.
• SECURATO, J.R. Cálculo financeiro das tesourarias: bancos e empresas. São Paulo: Saint Paul, 1999.
• SHARPE, W. Mutual fund performance. Journal of Business, p.119-138, January 1966.
• TOBIN, J. Liquidity preference as behavior towards risk. The Review of Economic Studies, n.67, p.65-86, February 1958.

1

Allen (1999) analyzes the efficiency of a portfolio by means of its return/risk ratio, i.e., when the return from a portfolio is higher, its risk is higher.

2

The sum of all long positions (or all short positions) in the futures market.

3

We show only the results for addition of cotton and corn contracts due to space considerations. The results of the other contracts may be obtained from the authors upon request.

4

The results referring to the composition of these portfolios are not included in this work due to space constraints. They can be obtained directly with authors, upon request.

Publication Dates

• Publication in this collection
  15 July 2003
• Date of issue
  Mar 2003