Research
Reports
Report 200302:
Portfolios
of Agricultural Market Advisory Services: How Much Diversification is
Enough?
April, 2003
Brian G. Stark, Silvina
M. Cabrini, Scott H. Irwin,
Darrel L. Good and Joao
MartinesFilho
[a]
Copyright 2003 by Brian G.
Stark, Silvina M. Cabrini, Scott H. Irwin, Darrel L. Good and Joao
MartinesFilho. All rights reserved. Readers may make verbatim copies
of this document for noncommercial purposes by any means, provided that
this copyright notice appears on all such copies.
DISCLAIMER
The advisory service
marketing recommendations used in this research represent the best efforts
of the AgMAS Project staff to accurately and fairly interpret the information
made available by each advisory service. In cases where a recommendation
is vague or unclear, some judgment is exercised as to whether or not to
include that particular recommendation or how to implement the recommendation.
Given that some recommendations are subject to interpretation, the possibility
is acknowledged that the AgMAS track record of recommendations for a given
program may differ from that stated by the advisory service, or from that
recorded by another subscriber. In addition, the net advisory prices
presented in this report may differ substantially from those computed
by an advisory service or another subscriber due to differences in simulation
assumptions, particularly with respect to the geographic location of production,
cash and forward contract prices, expected and actual yields, storage
charges and government programs.
This
material is based upon work supported by the Cooperative State Research,
Education and Extension Service, U.S. Department of Agriculture, under
Project Nos. 98EXCA30606 and 00521019626. Any opinions, findings,
conclusions, or recommendations expressed in this publication are those
of the authors and do not necessarily reflect the view of the U.S. Department
of Agriculture. Additional funding for the AgMAS Project has been provided
by the American Farm Bureau Foundation for Agriculture and Illinois
Council on Food and Agricultural Research.
Introduction
One of the most important
areas in agricultural farm management is the management of risks. Various
surveys conducted across the United States in the 1990's found that price
risk is one of farmers’ biggest management challenges (e.g., Patrick and
Ullerich, 1996; Coble et al., 1999; Norvell and Lattz, 1999). There
are many tools to assist farmers in price risk management. Patrick, Musser
and Eckman (1998) and Schroeder et al. (1998) reported that farmers specifically
viewed one of these tools, professional market advisory services, as an
important source of information in their efforts to manage price risk.
For a subscription fee, market advisory services offer specific advice
to farmers on how to market their commodities. It is often thought that
advisory services can process market information more rapidly and efficiently
than farmers to determine the most appropriate marketing decisions.
Despite this general
view, limited economic analysis has been done to test the true effectiveness
of advisory services. Gehrt and Good (1993) examined the returns for
corn and soybeans producers assuming they had followed the recommendations
of five advisory services over 198589 and compared returns against a
benchmark price. They concluded that there is some evidence that services
could beat the benchmark price. MartinesFilho (1996) analyzed the preharvest
recommendations of six advisory services for corn and soybeans over the
199194 production years. Slight evidence was found supporting the ability
of the services to generate a higher return than the compared benchmark.
In 1994, the Agricultural Market Advisory Services (AgMAS) Project was
initiated at the University of Illinois to expand research on market advisory
service performance. The AgMAS Project has monitored and evaluated about
25 advisory services each crop year and the empirical findings have been
disseminated through various AgMAS research reports. In the most recent
report, Irwin, MartinesFilho and Good (2002) presented results from the
evaluation of corn and soybeans advisory services over 19952000. When
both average price and risk are considered, only a small fraction of services
for corn and a moderate fraction for soybeans outperformed market benchmarks.
On the other hand, a majority of the services outperformed a farmer benchmark
for both crops.
The research reviewed
above examined the pricing performance of market advisory services only
on a standalone basis. In other words, individual advisory services
are evaluated against benchmark prices, without analyzing possible gains
from diversification among these services. In reality, farmers can choose
more than one advisory service and market a certain proportion of production
following the advice of each of the selected services. For example, a
farmer can choose two advisory services and market 50% of grain production
applying recommendations of one service and the other 50% applying recommendation
of another service. Furthermore, according to portfolio theory (Markowitz,
1952), a combination of advisory services may have greater risk/return
benefits compared to individual services or benchmarks. Portfolio theory
shows that a portfolio of advisory services may reduce price risk compared
to a single service and still obtain the same expected price. Therefore,
diversifying among several services may be a better alternative for farmers
compared to following an individual service. Farmers should be interested
in the magnitude of potential gains from diversification and on how many
advisory services should be included in the portfolio to capture risk
reduction benefits.
The relationship
between the number of portfolio components and portfolio risk has been
widely studied in the finance literature (e.g., Markowitz, 1959; Elton
and Gruber, 1977). It is well known that when stocks are randomlyselected
to construct equallyweighted portfolios, portfolio risk decreases as
the number of stocks increases. But as the number of stocks increases,
the decrease in risk from adding a new component diminishes to the point
that after several stocks have been added, the benefits of adding a new
component becomes very small. The same concepts can be applied to portfolios
of market advisory services. A farmer who follows a large number of randomlyselected
advisory services can expect to have more stable pricing performance than
a farmer who follows fewer services. But, the risk reduction gain from
following an additional service gets smaller as the portfolio size increases.
Hence, there is a tradeoff between the complexity of following a large
number of services and the risk reduction benefits from greater diversification.
The purpose of this
study is to analyze the relationship between size and risk for portfolios
of market advisory services in corn and soybeans. Data on market advisory
prices over 1995 to 2000 are obtained from Irwin, MartinesFilho and Good
(2002). Based on these prices, the risk for portfolios of 1 to 17 components
is estimated. The results provide information on the number of advisory
services that a farmer should follow in order to minimize price risk.
The
rest of this article is organized as follows: the next two sections explain
the mathematics of diversification and the analytical relationship between
risk and size for randomlyselected portfolios. Next, data and methodology
are described. Finally, results are presented and discussed.
Mathematical
Concepts Related to Portfolio Theory
Portfolio theory
shows how a combination of assets or, in this case, advisory services,
may represent a better alternative for farmers than individual services.
In order to fully understand how portfolio theory aids a farmer’s decision,
a few mathematical concepts need to be defined. The first concept behind
portfolio theory is expected return or, in the advisory services' case,
expected price. To begin, define _{
} as each of the possible net prices received
by advisory service i. Then, the expected price for advisory service
i is simply the weightedaverage of the prices received by the
advisory service over all possible k outcomes, with the weights
being the probability that a given net price _{
} occurs. The computation of expected net
price_{}for service i is:
where _{
} is the probability that the advisory service i
receives net price_{}.
The second mathematical
concept associated with portfolio theory is risk or, in the grain marketing
case, price risk. By comparing an advisory service's price received in
each crop year to the advisory service's expected price, a farmer can
determine the magnitude of risk associated with the advisory service.
If the advisory service consistently receives a price close to its expected
price over time, the advisory service is categorized as having little
risk. Similarly, if the advisory service has large and frequent deviations
from its expected price, the service is said to have high risk. However,
a measure is needed to quantify this price risk. Because this price risk
can be thought of as the dispersion of advisory service prices from periodtoperiod,
a statistical measure known as variance is used to compute the level of
risk. The variance of an individual advisory service, _{
}, is the weightedaverage of the squared deviations
between each possible price and the expected price of that advisory service:
Standard deviation
is the more common risk measure plotted in return/risk space, and it is
simply the square root of the variance. Consequently, the larger the
variance or standard deviation, the more unlikely the advisory service
will receive a price close to its expected price. Likewise, the smaller
the variance or standard deviation, the more likely the service will obtain
a price that is close to the expected price.
Another important
statistical concept in portfolio theory is covariance, which measures
the tendency of advisory service prices to move up or down together.
The covariance between advisory service i and j, _{
}, can be defined as:
where _{
}is the joint probability of prices _{
} for the i^{th} service and _{
} for the j^{th} service occurring.
However, covariance is often hard to interpret because it depends on the
units of measurement of the i^{th} and j^{th}
advisory service prices. To overcome this problem, a new statistic, termed
the correlation coefficient, is introduced.
The correlation between
advisory services is just a different way of presenting the covariance
for easier interpretation of the relationship that exists between the
advisory services in the portfolio. Correlation can be quantitatively
defined as:
where_{}is the correlation coefficient between the i^{th}
and j^{th} advisory service in the portfolio,_{} is the covariance between the i^{th }and
the j^{th} advisory service in the portfolio, _{
}and _{
}are the standard deviations of the i^{th}
and j^{th} advisory service in the portfolio, respectively.
The relevance of correlation in portfolio theory can be shown easily.
Correlation can take on any value from 1 to +1. A negative correlation
between two advisory services simply means that the corresponding prices
for those services tend to move in opposite directions. A positive correlation
between two advisory services means that the services' prices tend to
move in the same direction. From a risk standpoint, the most desirable
correlation coefficient is a value of 1. A perfectly negative relationship
between advisory services would mean that perfect diversification of risk
is achieved. The reason perfect diversification occurs with a correlation
of 1 is that the prices of the two advisory services always move in opposite
directions. When one service receives a low price, the other service
will always receive a high price, and vice versa. Therefore, price risk
is reduced as much as possible. However, the chance of this type of a
relationship is exceedingly small. It can be shown that any correlation
value less than +1 leads to a reduction in risk, when the comparison is
between an investment in a portfolio of advisory services compared to
an individual service. Only investing in one service means that a farmer
must accept the price risk associated with that service. However, if
two services have a correlation less than +1, one service's poor performance
is offset by the other service's performance. For this reason, price
risk can be successfully reduced with any correlation less than +1.
The previous paragraphs
defined the concept of expected price and variance for individual services
and covariance for pairs of advisory services. The rest of this section
presents measures that characterize portfolios of services, that is, portfolio
expected price and portfolio variance. The expected portfolio price depends
on individual services’ expected prices and the weights of the services
in the portfolio. Since the net price for a portfolio of services is
a linear combination of individual service prices, the expected price
of a portfolio of services is the weightedaverage of the expected prices
of the advisory services in the portfolio:
where_{}is the expected price of the portfolio and_{} are the weights for each advisory service in the
portfolio, and the weights sum to one. The variance of a portfolio depends
not only on the individual variances of the portfolio components, but
also on the covariances between the services, and is defined as:
where_{}is the portfolio variance, and _{
} and _{
} are the weights of the i^{th} and
j^{th} advisory services in the portfolio respectively.
An example can help
in explaining how portfolio theory aids a farmer's decision to select
a portfolio of advisory services over one individual service. Suppose
a farmer has 100,000 bushels of corn to market and is considering two
advisory services, A and B. The hypothetical prices for both services
from 19952000 are presented in Table
1. If the farmer invests all bushels in A or B, and prices fluctuate
in the future as they have in the past, the expected price for either
service is $2.46/bushel (average price for 19952000). However, the risk
associated with each advisory service is such that price in a particular
year could be substantially different than what the farmer expects. On
the other hand, if the farmer invests half of the corn bushels in each
advisory service, the expected price would be the same, $2.46/bushel,
but, as shown in Figure 1, the price variability for this portfolio of
services will be less that the price variability for the individual services.
The reason risk is reduced with a portfolio of the two advisory services
versus just one service is due to the fact that the correlation between
their respective prices is less than +1. Figure
1 shows that their prices do not always move in the same direction
and by the same amount. Thus, diversification helps reduce the risk associated
with advisory services. Consequently, the correlation between advisory
service prices is important when constructing a portfolio of services.
Relationship
Between Portfolio Risk and Size
The previous section
presented the mathematics of portfolio diversification and explained why
a portfolio of services can have risk/return benefits when compared to
individual services. Based on this fact, a reasonable question to ask
is how much diversification is enough, or in other words, how many advisory
services should be included in a portfolio to minimize price risk. In
this study the relation between the risk and number components is analyzed
for “naively” diversified portfolios. “Naïve diversification” is a term
commonly used in the finance literature to refer to portfolios that are
constructed by randomlyselecting the stocks to be included and assigning
equal weight to each component.
Naïve diversification
is not necessarily the optimal way of constructing portfolios. For example,
the Markowitz portfolio selection model (1952) implies that the assets
to be included in a portfolio and their respective weights should be selected
to minimize portfolio variance for a given level of expected return.
Under this model, the composition of optimal portfolios is based on the
individual assets’ expected return, variance and correlations. Although
the Markowitz model is in theory a better approach, naïve diversification
is widely used in practice (e.g., Lhabitant and Learned, 2002). The reason
why this approach is so commonly applied is that naïve portfolios are
a very reasonable alternative when information on individual expected
returns, variance and correlations is limited, and therefore, the estimates
for these parameters may not be reliable. In this case, naïve diversification
is likely to be a safer way of constructing portfolios, since, as it will
be explained in the rest of this section, the risk and return of naïve
portfolios depends only on the average expected return, average variance
and average covariance for the set of assets to be included in the portfolio.
Averages of these parameters can be estimated more accurately than the
individual values, with this advantage being more important when data
available for the estimation is limited.
It is well known
that when stocks are randomlyselected and combined in equal proportions
in a portfolio, the risk of the portfolio declines as the number of different
stocks increases (e.g., Markowitz, 1959; Elton and Gruber, 1977). Or,
in other words, portfolios with a larger number of securities have lower
variance than portfolios with a smaller number of securities. In naïve
diversification, risk always decreases with portfolio size, but it goes
down at a decreasing rate as more stocks are added. At some point, the
diversification gain from adding another stock becomes negligible. Hence,
when decision makers select portfolio size they need to consider the tradeoff
between the decreased risk due to a more effective diversification and
the operational disadvantage and cost associated with managing a more
complicated portfolio.
The idea of naïve
diversification can be also applied to portfolios of market advisory services.
The basic idea is that a portfolio of size _{
} can be constructed by randomlyselecting _{
} advisory services from the set the services available
to the farmer and assigning equal weight of _{
}to each service (this means that the farmer applies
the recommendations from each advisor to _{
} of total production). Hence, it is useful to analyze
the risk level of naïve portfolios containing different numbers of advisory
services.
The
derivation of the analytical relationship between portfolio risk and size
was presented by Elton and Gruber (1977). When the portfolio weights
are equal (_{}, for all i), the portfolio variance
equation (6) can be written as:
In
naïve diversification, the portfolio components are selected at random
from the set of available services, hence, there are several different
possible combinations of advisory services for each portfolio size, all
occurring with the same probability. Consider, for example, the case
where four services are available to the farmer (A, B, C and D), and the
farmer decides to follow the recommendations of two (_{}). Naïve diversification implies that the farmer
will follow any of the six possible two service portfolios: AB, AC, AD,
BC, BD or CD. The risk measure that characterizes naïve portfolios of
size _{
} is the expected variance. Expected portfolio variance
is just the average portfolio variance among all possible combinations
of the available services in sets of _{
} and is computed as:
where
_{
} is the expected variance of a portfolio of N
advisory services, _{
}is the average variance for all available advisory
services and _{
}is the average covariance between all pairs of services.
Note that these measures are averages along the whole set of services
available to the farmer. Rearranging equation (8):
Finally,
another way to write equation (9) is:
The previous analytical
expressions not only allows one to quickly determine the expected portfolio
variance for any portfolio size, but also shows which factors affect portfolio
risk. Equations (9) and (10) show that portfolio variance declines as
portfolio size _{
} increases. Specifically, as _{
} increases, the effect of the difference between
average variance and covariance decreases, and portfolio variance becomes
close to the average covariance for very large_{}.
It
is not unreasonable to think that, instead of applying equation (10),
simulation could be used to compute expected portfolio variance. In fact,
several research studies determine the expected portfolio variance using
simulation instead of applying the exact formula (e.g., Evans and Archer,
1968; Billingsley and Chance, 1996; Lhabitant and Learned, 2002). In
the case of market advisory portfolios, the procedure under simulation
would be to first randomlyselect _{
} services and then compute the portfolio variance
according to equation (7), then repeat the procedure a large number of
iterations, and finally compute the average portfolio variance among all
iterations, which is the estimate for the expected portfolio variance.
This procedure can be done for all the values of _{
} to be evaluated. This simulation approach will
give an approximation of the exact expected portfolio variance, with the
approximation more accurate the higher the number of iterations. As pointed
out by Elton and Gruber (1977), simulation is a powerful tool to be used
in the cases where the exact formula for the desired computation does
not exist. However, since in the present case the formula to compute
the exact result exists, applying equation (9) is preferred, because it
is more accurate and simpler.
The
financial literature includes numerous studies analyzing the relationship
between portfolio size and risk (e.g., Evans and Archer, 1968; Wagner
and Lau, 1971; Elton and Gruber, 1977; Lloyd, Hand and Modani, 1981; Bird
and Tippett, 1986; Statman, 1987; Newbould and Poon, 1993; Billingsley
and Chance, 1996; O’Neal, 1997; Henker, 1998; Henker and Martin, 1998;
Lhabitant and Learned, 2002). The next section presents a brief description
for several of these studies. The reviewed publications include “classic”
articles on risk versus size for US stock portfolios, a study employing
Australian data, an article evaluating portfolios of commodity trading
advisors and an article analyzing hedge funds portfolios. These studies
provide a representative sampling of the results available in the empirical
literature on naïve diversification. The last two articles are of particular
interest because diversification among commodity trading advisors and
hedge funds managers would appear to be closely related with the topic
of the current study.
Previous
Empirical Studies
Evans
and Archer’s classic study (1968) was the first attempt to measure the
relationship between the number of assets in a portfolio and portfolio
risk. They employed a simulation approach to construct naïve portfolios
of stocks listed in the S&P500 Index, and found that the standard
deviation of 5 and 10 stock portfolios was only 15% and 7% higher, respectively,
than the minimum possible standard deviation (the minimum possible standard
deviation corresponds to portfolio that includes all available assets).
In contrast, the expected standard deviation of a one stock portfolio
was 72% higher than the minimum possible. It was concluded that there
was probably no economic justification for increasing portfolio sizes
beyond 10 securities. Elton and Gruber (1977) employed the analytical
relationship presented in equation (9) and (10) to compute the expected
variance for naïve portfolios of stocks listed on the New York Stock Exchange.
They found that a one stock portfolio had an expected standard deviation
157% higher than the minimum risk portfolio, and that 50 stocks were needed
to have an expected standard deviation 11% higher than the minimum possible.
In contrast to Evans and Archer’s results, portfolios of 6 and 10 stocks
had standard deviations 93% and 56% higher than the minimum risk portfolio.
Elton and Gruber (1977) pointed out that even though it was true that
the first 10 or 12 stocks provided most of the advantages from diversification,
there were still significant risk reduction beyond adding 12 to 15 stocks.
Bird and Tippett (1986) measured the advantages of naïve diversification
using Australian stock data, also employing the analytical relationship
between portfolio risk and size. Results show that individual stocks
had, on average, a standard deviation 144% higher than the minimum risk
portfolio. Portfolios of 5 and 10 stocks had 47% and 23% higher standard
deviations than the minimum risk portfolio, and more than 25 stocks were
needed to obtain a standard deviation only 10% higher than the minimum.
They concluded that portfolios of 10 stocks could not be considered welldiversified.
Statman (1987) conducted a marginal analysis using Elton and Gruber’s
(1977) empirical results. The basic idea was that diversification should
be increased as long as the marginal benefits exceed the marginal costs.
Statman expressed both risk benefits and costs of holding portfolios in
units of expected return and concluded that naïve portfolios should include
at least 30 to 40 stocks to be considered welldiversified.
Billingsley
and Chance (1996) conducted a simulation analysis of the optimal number
of commodity futures trading advisors (CTAs) in a portfolio. They found
that individual CTAs had, on average, 84% higher standard deviations than
the portfolio including all 120 available CTAs. Portfolios of 3 and 11
CTAs had about 30% and 10% higher standard deviations, respectively, than
the 120 CTA portfolio. They reported that less than 10 CTAs were needed
to capture most of the naïve diversification benefits. In a more recent
study, Lhabitant and Learned (2002) computed by simulation the expected
variance for hedge fund portfolios of different sizes. They recommended
including 5 to 10 hedge funds to eliminate 75% of diversifiable risk.
The
obvious conclusion that can be drawn from this brief review is that there
is not a unique optimal portfolio size. Different characteristics of
the assets to be included in a portfolio, as well as the cost related
to include a new component, will determine the reasonable portfolio size
in each situation. Recall that the relation between size and risk reduction
benefits depends mainly on the difference between the average variance
and the average covariance (equation 10). On one hand, when the difference
is low (the average correlation coefficient will be relatively high) the
possible benefits of naïve diversification will be small and portfolios
with a few assets capture almost all diversification benefits. This seems
to be the case with portfolios of CTAs and hedge funds (Billingsley and
Chance, 1996; Lhabitant and Learned, 2002). This seems also to be the
case with the Evan and Archer (1968) study, although several other publications
obtained quite different results for stocks portfolios. On the other
hand, when the difference between the average variance and average covariance
is high, the potential benefits from naïve diversification are higher
and may still be considerable after many assets have been included in
the portfolio. This last case appears to be the situation in the studies
by Elton and Gruber (1977) and Bird and Tippett (1986).
Data and
Methodology
Data on corn and
soybean net advisory prices and corn/soybean revenue from 1995 through
2000 are drawn from Irwin, Martines Filho and Good (2002). The sample
consists of the 17 advisory programs that were followed by the AgMAS Project
in each of these six crop years. The term “advisory program” is used
because several advisory services have more than one distinct marketing
program. Recommendations of individual marketing advisory programs collected
by the AgMAS Project over these years were used to compute a net price
that would be received by a farmer in central Illinois that sells the
grain based on the recommendations of each program. Details on the computations
can be found in Irwin, MartinesFilho and Good (2002). The analysis is
applied not only for corn and soybean prices individually, but also for
corn/soybean revenue because many subscribers to advisory services produce
and market both corn and soybeans. A cornsoybean rotation practice where
each crop is planted on half of the farmland is very common among central
Illinois farmers. The peracre revenue for each commodity is found by
multiplying the net advisory price for each market advisory program by
the corn or soybean yield for each year. A simple average of the two
per acre revenues is then taken to determine the total revenue obtained
from this practice, which is called “50/50 revenue” here.
Estimates
of expected prices (revenue), variances and covariances are needed to
compute the portfolio expected price (revenue) and portfolio variance.
Since the sample size is small compared to the number of parameters to
be estimated, the single index model (SIM) proposed by Sharpe (1963) is
employed in the estimation. The Sharpe model presents a way to reduce
the number of parameters to be estimated and, therefore, becomes a preferred
approach compared to traditional sample estimates when the available data
set is small (Frankfurter, Phillips and Seagle, 1976). This approach
is based on the simplifying assumption that net prices and revenues for
the different market advisory programs are related only through common
relationship with some index, in this case, a market benchmark, according
to the following linear model:
where _{
} is the price for advisory program i in
year t, _{
}is the component of advisory program i's
price that is independent of market performance, _{
}is the expected change in price of advisory program
i relative to the change in price of the market index, _{
}is the price for the market index in year t,
and _{
} is the random component of _{
} which has expected value of zero.[1] A key assumption of the SIM is that
error term for the different services, _{
}and_{}, are independent of each other. This assumption
basically means that the only reason advisory program prices vary together
is due to comovement with the market index.
When the SIM is used
to estimate the comovement of the advisory programs relative to a market
index, the previous regression model is run with advisory program prices
and market benchmark prices. The regression estimates are used to obtain
parameter estimates that approximate the expected price and variance for
each program and the covariance between each pair of programs. The estimated
expected price of advisory program i _{
} is:
where _{
}is
the estimate of the independent component of advisory program i's
price,_{} is the estimate of the slope coefficient obtained
from the regression of advisory program i's prices on the market
index price, and _{
}is the sample average benchmark price. The value
for expected price for a program obtained under the SIM is equal to the
sample average price for that program. The estimated variance of advisory
program i _{
} is:
where_{}is the estimated variance of the market index, and_{}is the estimated error variance of the regression
with advisory program i and the market index. The covariance between
two advisory programs is:
where_{}is the estimated covariance between advisory program
i and advisory program j, and _{
} and _{
}are the estimated slope coefficients from advisory
program i's and j’s respectively. It is easy to see how
SIM reduces the amount of data needed for estimating parameters and helps
to provide more efficient estimates. Each advisory program's regression
with the index generates three parameter estimates, which leads to a total
of 51 (3*17) estimates, instead of 153 (17 variances and 136 covariances)
when using the traditional estimation procedure.
Tables
2 and 3 present some of the SIM estimation results. Table
2 shows the expected prices and standard deviation for each advisory
program for corn, soybeans and 50/50 revenue. Corn advisory prices range
from $2.20/bushel to $2.76/bushel, with an average of $2.38/bushel Soybean
advisory prices range from $5.85/bushel to $6.80/bushel, with an average
of $6.19/bushel. Revenue ranges from $304/acre to $358/acre, with an
average of $316/acre. Table
3 presents the average correlation between each program and the rest
of the programs. The values in this table show that, in general, advisory
prices are highly correlated with each other, and this is mainly due to
their correlation with the market benchmark price. The average correlation
between programs is higher for soybean advisory price (0.75), in the middle
for corn advisory price (0.71) and lower for 50/50 revenue (0.61). But
there are some exceptions. For instance, Allendale (futures only) and
Brock (hedge) for corn and AgResource and Brock (hedge) for soybeans have
very low average correlation with other programs.
Based on the individual
SIM estimates for corn, soybeans and 50/50 revenue, the estimated average
variance and average covariance among all 17 programs are computed. Then,
the estimated expected variance for portfolios of 1 to 17 programs is
calculated using equation (9). The expected variance results for the
different number of programs in the portfolio are reported in terms of
total and marginal gains from increasing portfolio size, as well as plots
of risk vs. size.
It is important to
remark that, because advisory programs are randomlyselected to construct
portfolios in this study, the results do not depend directly on estimates
for individual advisory programs, but on averages among all programs.
Note that the expected price (revenue) and price (revenue) variance for
naïve portfolios are computed based only on the estimates for average
price, average variance and average covariance.
Results
Initially, results
are focused on the risk characteristics of naïve portfolios of advisory
programs, rather than expected price (revenue). The reason is that, under
the assumption that advisory program costs are not related to portfolio
size; random selection of advisory programs for equallyweighted portfolios
restricts the expected price (revenue) to be equal to the average expected
price (revenue) among all programs. In other words, the expected net
price (revenue) will not change when the number of programs in the portfolio
increases. The expected portfolio prices (revenue) are the averages presented
in Table 2: $2.38/bushel
for corn price, $6.19/bushel for soybean price and $316/acre for 50/50
revenue. The assumption that advisory program costs are not related to
portfolio size will be relaxed later in this section.
Tables
4, 5 and 6 present average standard deviations for naïve portfolios
versus the number of programs in portfolios for corn, soybeans and 50/50
advisory revenue, respectively. Starting at the left of the each table,
the first standard deviation value is the expected standard deviation
of a portfolio of one program. This corresponds to the case where the
farmer selects, at random, one program among the 17 and follows the recommendation
of only that program. In the case of corn, the expected standard deviation
for one program portfolios is $0.44/bushel, for soybeans this value is
$0.75/bushel and $34.61/acre for 50/50 revenue. These expected portfolio
standard deviation values equal the average standard deviation among all
programs (the average standard deviation presented in Table
2). Note that this is easy to check in equation (9), since if _{
} the second term of the equation cancels out.
The
portfolio standard deviations presented in Tables
4, 5 and 6 show that when the number of programs in the portfolio
increases the portfolio standard deviation decreases. However, the third
column in each table reveals that the marginal decrease in risk is lower
each time another program is added. For example, in the corn case, when
a second program is added to the portfolio the expected standard deviation
decreases by $0.035/bushel, when a third program is added $0.012/bushel,
a fourth program $0.006/bushel. The decrease in standard deviation by
adding a program is lower for larger size portfolios, and after several
programs have been added in the portfolio, adding another one has only
a very small risk reduction effect. For example, in soybeans, the difference
in standard deviation between portfolios of 16 and 17 programs is only
$0.0004/bushel. The negative sloped lines in the three panels of Figure
2 provide a visual perspective on the relationship between portfolio risk
and size.
The portfolio of
all 17 advisory programs has the lowest risk level among the naïve portfolios
selected from this set of 17 programs. The expected standard deviation
values for 17 programs portfolios are $0.368/bushel, $0.652/bushel and
26.74 $/bushel for corn, soybeans and 50/50 revenue, respectively. The
difference in standard deviation between 1 and 17 programs is $0.0684/bushel,
$0.0992/bushel and $7.8670/acre, respectively. These values are the total
possible reduction in risk through naive diversification among the 17
programs. This risk reduction can also be expressed as a percentage of
the risk of one program portfolios. For example, the standard deviation
reduction from 1 to 17 programs is 16 % of the expected standard deviation
for following a single program (one program portfolios) in corn, 13 %
in soybeans and 23 % with 50/50 revenue. Note that this percentage is
greatest for 50/50 revenue, where the average correlation between programs
is lowest (0.61, Table
3). The percentage is lowest for soybeans, where the average correlation
is largest (0.75, Table
3), and it is in the middle for corn, where the average correlation
has a value in between the other two (0.71, Table
3). Recall from equation (10) that the diversification effect depends
on the difference between the average variance and average covariance.
When correlation is close to one, the average variance and covariance
are close, and the potential benefits from naïve diversification are small.
Figure
3 presents the relationship between the portfolio variance and the
number of programs in the portfolio. The shape of the curves is exactly
the same as in Figure
2, since the variance is just the standard deviation squared. The
purpose of presenting Figure 3 is to show how the portfolio variance approaches
the average covariance when the number of portfolio components increases.
This shows that the lowest possible risk level of randomlyselected portfolios
is determined by the average covariance between the portfolio components.
Adding more and more programs will make portfolio variance become almost
equal to the average covariance between them. Note that equation (10)
also indicates this fact: as _{
} increases the first terms approaches the value
of zero and the portfolio variance becomes the average covariance.
Comparing these results
with the results from other studies it is evident that the possible gains
through naïve diversification are relatively low in the case of advisory
programs. This is because, on average, advisory prices are highly correlated.
The last column of Tables
4, 5 and 6 present the ratio between the risk of a portfolio of size
_{
} and the minimum possible risk. The minimum risk
considered here corresponds to the square root of the average covariance,
which, as was mentioned before, measures the portfolio risk for very large_{}. Other authors define minimum risk as the expected
standard deviation of the portfolio containing all available assets in
the data set, but this definition does not seem to be appropriated for
this study, where the number of available programs is only 17, which is
a relatively low portfolio size. One can argue that there may be still
gains from diversification beyond 17 programs and hence, the square root
of the average covariance seems to be better measure of minimum risk for
naïve portfolios. The expected standard deviation for one program portfolios
is only 20%, 16% and 32% greater than the minimum risk for corn, soybeans
and 50/50 revenue respectively. These percentages are very low compared,
for example, with the 157% and 144% differences in risk between one stock
and all stocks portfolios reported by Elton and Gruber (1977) and Bird
and Tippett (1986), respectively. The last columns of Tables
4, 5 and 6 also show that three program portfolios have only 7%, 6%
and 12% higher risk than the minimum possible standard deviation for corn,
soybeans and 50/50 revenue, respectively. These portfolio sizes are very
small compared to Elton and Gruber’s (1977) results, where 50 stocks were
needed for a portfolio standard deviation 11% higher than the risk of
a portfolio including all available stocks. The naïve diversification
benefits for market advisory programs are more similar to those for portfolios
of CTAs and hedge funds (Billingsley and Chance, 1996; Lhabitant and Learned,
2002), where the authors recommend including less than 10 components in
the portfolios.
The results presented
so far indicate that the possible risk reduction benefits from naïve diversification
among market advisory programs are relatively small, and it is possible
to gain most of the risk reduction from diversification holding small
portfolios. Beyond a portfolio size of four or five the benefits from
adding another program are very small, and the disadvantages of managing
a more complicated portfolio may exceed the risk reduction benefits.
More specifically, a complete analysis of naïve diversification benefits
should also consider the cost associated with holding portfolios of different
sizes. For portfolios of advisory programs, this issue is more important
compared to stock portfolios, since there is a subscription fee associated
with each program, so portfolio costs unambiguously increase with size.
The average subscription
cost for the 17 programs between 1995 and 2000 was $304 per year (Irwin,
Martines Filho and Good, 2002). Based on this average value, the second
column of Table 7
presents the subscription cost for portfolios of 1 to 17 programs.[2] Note that portfolios with many programs
are expensive. For example, a portfolio of all 17 programs costs $5,168/year.
Because farm costs are commonly expressed in dollars per acre, Table 7
also shows the subscriptions cost per acre and the net 50/50 per acre
revenue for two farm sizes: 500 acres and 1,000 acres. Net revenues were
computed by subtracting the per acre subscription cost from the expected
50/50 revenue. Note, for instance, that a five program portfolio costs
$3.04/acre for a 500 acre farm and $1.52/acre for a 1,000 acre farm.
These costs are economically nontrivial, particularly relative to average
returns to farm operator management, labor and capital in Illinois, typically
about $50 per acre for grain farms (Lattz, Cagley and Raab, 2001). Given
these results, it is not unreasonable, then, for a farmer to choose portfolios
with fewer than four or five programs, since with the first two or three
programs a farmer can get most of the benefits from diversification at
a lower cost. For example, if a farmer follows two randomlyselected
programs, the expected portfolio standard deviation for 50/50 revenue
is only 14.7% ($30.68/$26.74) higher that the minimum standard deviation
and captures 50% (15.47% / 22.73%) of the total possible gains from naïve
diversification. Finally, it is important to emphasize that the cost
of implementing, monitoring and managing the marketing strategies recommended
by advisory programs was not accounted for in the analysis. Such costs
are difficult to measure, but are likely to be substantial (Tomek and
Peterson, 2001), adding further to the disadvantage of managing advisory
service portfolios of greater size.
Summary
and Conclusions
Agricultural market
advisory services offer specific advice to farmers on how to market their
commodities. Farmers can subscribe to one or more of these programs and
follow their advice as a way of managing price risk. According to portfolio
theory, a combination of these programs may have risk/return benefits
compared to individual programs. This report evaluates the potential
risk reduction gains from naïve diversification (equalweighting) among
market advisory programs. In particular, this study analyses the relationship
between the risk and number of components for naïve portfolios using data
for 17 market advisory programs obtained from the AgMAS Project at the
University of Illinois. Corn and soybean net advisory prices, as well
as combined corn/soybean revenue, are examined in this study.
The expected standard
deviation for portfolios of 1 to 17 advisory programs was computed using
the analytical relationship between risk and size that is derived from
the classical formula for portfolio variance. Results for corn and soybeans
advisory prices and 50/50 revenue are reported in terms of total and marginal
gains from increasing portfolio size, as well as plots of risk versus
size.
Results show that
increasing the number of programs reduces portfolio expected standard
deviation, but the marginal decrease in risk from adding a new program
decreases rapidly with portfolio size. For example, in the corn case,
the expected standard deviation of a one program portfolio is $0.437/bushel,
when a second program is added the expected standard deviation decreases
by $0.035/bushel, when a third program is added $0.012/bushel, a fourth
program $0.006/bushel.
The total standard
deviation reduction through naïve diversification is relatively small
compared to the results obtained in previous studies for stock portfolios,
and this is mainly because advisory prices, on average, are highly correlated.
A one program portfolio has 20%, 16% and 32% higher standard deviation
than the minimum risk naïve portfolio for corn, soybeans and 50/50 revenue,
respectively. Moreover, most risk reduction benefits are achieved with
small portfolios. For instance, a four program portfolio has only 5%,
4% and 9% higher risk than the minimum risk naïve portfolio for corn,
soybeans and 50/50 revenue, respectively. Based on these results, there
does not appear to be strong justification for farmers adopting portfolios
with large numbers of advisory services. Farmers may well choose portfolios
with as few as two or three programs, since the relatively high total
subscription costs associated with larger portfolios can be avoided while
obtaining most of the benefits from diversification. For example, if
a farmer follows two randomlyselected programs, the expected portfolio
standard deviation for 50/50 revenue is only 14.7% higher that the minimum
standard deviation and 50% of the total possible gains from naïve diversification
are captured.
For a more complete
analysis of the possible benefits from diversification among advisory
services, it is necessary to evaluate portfolios constructed using optimization
models. Under this approach, an efficient set of optimal portfolios of
market advisory programs is constructed by minimizing portfolio variance
for each level of expected net price or revenue. The portfolio components
and weights are selected based on each program’s expected prices, variances
and covariances, not just on averages of these parameters as is the case
with this study. The main difficulty in optimal portfolios is obtaining
good estimators for these values from the available data.
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Endnotes
[a]
Brian G. Stark is a former Graduate Research Assistant for the AgMAS Project
in the Department of Agricultural and Consumer Economics at the University
of Illinois at UrbanaChampaign. Silvina M. Cabrini is a Graduate Research
Assistant for the AgMAS Project in the Department of Agricultural and
Consumer Economics at the University of Illinois at UrbanaChampaign.
Scott H. Irwin and Darrel L. Good are Professors in the Department of
Agricultural and Consumer Economics at the University of Illinois at UrbanaChampaign.
Joao MartinesFilho is Manager of the AgMAS and farmdoc Projects
in the Department of Agricultural and Consumer Economics at the University
of Illinois at UrbanaChampaign.
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