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A Mathematical Approach to Increasing the Long-term Wealth of an Agricultural Enterprise

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This study focuses on developing an agricultural investment model based upon proven financial investment portfolio techniques. The model can be used as a tool to diversify agricultural risk over the long-term by optimising the proportion of land allocated to each of the agricultural products, resulting in increased value of the agricultural enterprise. Sensitivity analysis allows the strategist to understand the impact that future prices, gross margins and land availability may have on the long-term sustainability of the farming enterprise.
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ORiON, Vol. 19, No. 1/2, pp.53-74 ISSN 0259-191-X
A MATHEMATICAL APPROACH TO INCREASING THE
LONG-TERM WEALTH OF AN AGRICULTURAL
ENTERPRISE1
PETER THERON
Graduate School of Business, University of Cape Town, Rondebosch 7701, South Africa
(petertheron@yahoo.com)
ROB VAN DEN HONERT
Department of Statistical Sciences, University of Cape Town, Rondebosch 7701, South Africa
(vandenhonert@optusnet.com.au)
ABSTRACT
This study focuses on developing an agricultural investment model based upon proven
financial investment portfolio techniques. The model can be used as a tool to diversify
agricultural risk over the long-term by optimising the proportion of land allocated to each of
the agricultural products, resulting in increased value of the agricultural enterprise.
Sensitivity analysis allows the strategist to understand the impact that future prices, gross
margins and land availability may have on the long-term sustainability of the farming
enterprise.
KEYWORDS
Agriculture, diversification, efficient frontier, expected return, long-term, risk, portfolio
model, gross margin, quadratic programming
INTRODUCTION
The traditional role of management is to increase the value of a firm. In order to successfully
increase value, management should focus on increasing profits by operating more efficiently,
driving down costs and increasing revenue. Analysis may lead to certain products being
discontinued and others adopted in order to maximise the firm’s profits. The volatility of
demand (and resulting price per unit) of these products may impose certain additional risks on
the business due to the uncertainty associated with these future prices. These risks usually
1 This paper is based on the first author’s winning entry in the 2001 ORSSA Student Project Competition. His
research project was entitled “Minimising Long-Term Agricultural Price Risk: A Quadratic Programming Model
based on the Markowitz Mean-Variance Approach to risk Minimisation”.
54
transform into higher costs of capital, as stakeholders demand higher rates of return or interest
to offset the higher risk that their investments may be exposed to. By attempting to diversify
the product portfolio, the firm may be able to reduce the portfolio risk.
Given that most agricultural products such as wheat, maize, coffee and meat are commodities,
and thus highly influenced by demand and supply (which leads to volatile price adjustments
in free market economies), it is important for the producers of these products to choose the
correct product range and weightings in order to maximise the overall profitability of the
business and minimise the risks associated with potential price movements over the long term.
In this paper we develop an optimisation model to advise managements of agribusinesses on
optimal agricultural product portfolios and their weightings. In the next section we develop
the model. Thereafter we describe how this model may be implemented on a spreadsheet. An
implementation case study is then discussed, followed by some observations on real world
applications of the tool.
THE MODEL
In general, the current value of a business is the present value of the sum of its future cash
flows (adjusted for capital expenditure). The Free Cash Flow to Firm approach is a popular
model for valuing a business and forms the basis of more complex models used by many
financial institutions (see, for example Damodaran (1996: 242)). Using this model
()
=
=+
=
t
1t
t
t
WACC1
FCFF
Firm of Value
where
FCFFt = free cash flow to the firm in year t
WACC = weighted average cost of capital.
With reference to the above formula, it can be seen that value can be increased by either
decreasing the weighted average cost of capital of the firm, or by increasing the free cash flow
to the firm. The most desirable effect would then be a combination of increased free cash
flow through growing profits, whilst at the same time driving down the weighted average cost
of capital by reducing various forms of risk within the business.
55
In March 1952, Harry M. Markowitz published his now-famous paper in the Journal of
Finance entitled "Portfolio Selection" (1952:77-91). In it, he demonstrated how to reduce the
standard deviation of returns on asset portfolios (i.e. portfolio risk) by selecting assets that do
not move in exactly the same way as each other. In the same article he laid down some basic
principles for establishing an advantageous relationship between risk and return.
Although this model was developed and applied with the optimisation of financial security
portfolios in mind, it is possible to apply this theory to a different field, that of optimising the
resources utilised for producing agricultural products. Depending on the risk profile of the
agricultural firm and the resulting strategies adopted by it, the quadratic programming model
developed here may be used to minimise risk and increase free cash flow to the agricultural
enterprise, and in so doing increase the potential value of the business.
The Markowitz model, adapted for use in the agricultural industry, can be written as:
Minimise Risk ijjij
n
i
n
j
iPP
ρσσ
∑∑
==11
Subject to:
Required adjusted price increase a
n
i
iii RrPg
=1
Productive land constraint 1
1
=
=
n
i
i
P
Category A constraint
=
x
a
aCP
1
1
Category B constraint 2
1
CP
y
b
b
=
Category C constraint 3
1
CP
z
c
c
=
where:
Pi = proportion of land allocated to agricultural product i
Pa = proportion of land allocated to category A products
Pb = proportion of land allocated to category B products
56
Pc = proportion of land allocated to category C products
n = total number of products under consideration.
x = total number of category A products under consideration where x n
y = total number of category B products under consideration where y n
z = total number of category C products under consideration where z n
σi = standard deviation of price returns of product i
ri = expected price growth of product i
gi = expected gross margin of product i
Ra = required adjusted return of the agricultural portfolio
C1 = portion of land available for cultivation of category A products
C2 = portion of land available for cultivation of category B products
C3 = portion of land available for cultivation of category C products
ρij = correlation of price returns between agricultural products i and j.
When comparing the assets and their returns of the security markets with the returns and
assets of a typical farm, some fundamental differences emerge, and thus certain assumptions
need to be made before the model may be used with any degree of success or reliability.
The returns on a security may be written as Rs = f (
P, Q0, c, t) where Rs is the return
generated by changes in the security price2
P over a certain time period t by a certain
quantity of securities Q0 initially invested. The small costs involved in commissions and
transactional costs c may be considered as insignificant. The only significant variable in the
function is
P. Thus for all intents and purposes, we could re-write the above equation as Rs
f (
P).
Let us compare this to the return generated by an agricultural product: Rp = f (
P,
Q,
VC,
FC, t) where the return generated by the agricultural product Rp is a function of the change in
product price between the time of planting and harvesting
P, and
Q represents the yield
(quantity) of the product (e.g. tons per hectare). The variable costs
VC and fixed costs FC of
producing this product must also be considered. The gross margin GM of each product is
defined as (P-VC)/P. The time period between planting and harvesting is denoted by t. By
assuming that FC is constant we could re-write the above equation as Rp f (
P,
Q,
VC).
57
If we assume that the yield of each product remains fairly constant over time, and recognising
that gross margin is a function of both price and variable costs, the return of the agricultural
product may be written as Rp f (
P,
GM).
Thus in the case of equity securities the return generated by assets is a function of a change in
price only (ignoring the effects of dividend yields), whilst the return generated by an
agricultural product is a function of a change in price and the marginal contribution of the
products produced.
Should the agricultural model only take the changes in prices
P into account, we would be
solving only part of the problem. The model would prove most accurate if the historical
changes in contribution margin were to be used. But if, over the long-term, the yields and
variable costs are known and considered to be fairly consistent (i.e. less volatile relative to
expected change in prices), and if this is consistent over the entire range of products, then the
expected change in price would be the major driver which would result in the contribution
being more volatile (i.e. the volatility of the contribution over time would be most dependant
on the volatility of the price). This future expected price growth of the agricultural product
must be weighted by its gross margin percentage to avoid biased allocations of land to
products with potential for high price growth, but with low gross margins. Given that the
model is an attempt to minimise risk over the long term, the following equation must then
hold: Rp E(Contribution) E(P.GM).
Note that this model is best suited for long-term strategic decision-making where land
utilisation is the major resource employed in producing the agricultural products. The model
could be used with some degree of success in the short-term if the costs3 of switching between
divesting product and investing product is low. For example, the switching costs between
barley and wheat would be considerably lower than the switching costs between an apple
orchard and a pear orchard.
Referring to the constraints in the adapted Markowitz model shown above, the historical
change in price of each agricultural product under consideration is weighted by its gross
margin contribution, resulting in an adjusted price increase metric. The model also allows for
2 We assume that the dividend yield has been built into the share price in the case of equity securities.
58
constraining several categories of products (we have indicated three such categories). For
example, an enterprise may have fertile land close to irrigation facilities which would be
suited to crops requiring a high degree of irrigation. Fertile land with no access to irrigation
facilities, on the other hand, may be restricted to certain crops, which may thrive under these
conditions (dry land crops). Specific soil composition may also limit land suited for orchards,
for example. The balance of productive land may be assumed to be available for products not
grouped into a specific category, for example livestock such as cattle and sheep. The
productive land constraint does not include land occupied by non-operating assets such as
residential housing, roads, sheds etc.
IMPLEMENTING THE MODEL IN A SPREADSHEET
A simple Microsoft Excel spreadsheet can be used to apply the model to real world
applications. Some VBA (Visual Basic for Applications) coding is used to streamline the
macro and perform multiple iterations through the Solver Add-in (which is part of the
Microsoft Excel package). For example, below is an extract of code that is used to plot the
efficient frontier.
'Solve Frontier values
SolverOk SetCell:="$O$4", MaxMinVal:=2, ValueOf:="0", ByChange:=Range("C2:C101")
For i = 1 To a
Range("R3").Value = e + (i * f)
SolverSolve UserFinish = False
Range(Cells(i + 19, 15), Cells(i + 19, 15)).Value = Range("O4").Value 'Risk
Range(Cells(i + 19, 16), Cells(i + 19, 16)).Value = Range("O3").Value 'Return
Next i
Initially, data is entered into a data template, which is divided into three sections (see
Appendix 1 for a snapshot view of an extract of the template). The first section of the
template requires the user to enter the periodic change in price of each product under review.
The monthly product price changes (or monthly returns) are calculated by using the following
formula:
i+1 i
i+1
i
Pr ice in Month - Price in Month
Monthly Return = Price in Month
3 These costs would include the opportunity cost of capital, as it may take time to generate income.
59
An extract showing monthly returns can be seen in Figure 1 (note that areas shown with a
shaded background require input from the user).
Maize Wheat Lucerne Cattle Sheep Pigs Barley Sugar
0.0% 0.0% 0.0% 0.7% 0.0% -0.7% 0.0% 0.0%
0.0% 0.0% 0.0% 0.0% -9.5% -13.8% 0.0% 0.0%
0.0% 0.0% 17.6% -0.4% -5.6% -7.8% 0.0% 0.0%
54.7% 0.0% 0.0% 1.2% -4.3% 0.6% 0.0% -1.8%
0.0% 0.0% 0.0% -4.2% -2.1% -1.9% 0.0% 0.0%
0.0% 0.0% 8.8% -9.4% -1.6% -3.0% 0.0% 0.0%
0.0% 0.0% 0.0% -2.7% 6.9% 3.2% 0.0% 0.0%
0.0% 0.0% 0.0% 2.5% -4.1% 5.9% 0.0% 0.0%
0.0% 0.0% -23.5% -2.5% 5.0% 13.3% 0.0% 0.0%
0.0% 4.3% 0.0% 6.1% 0.9% -2.6% 7.2% 0.0%
0.0% 0.0% 0.0% 8.9% 13.8% 11.5% 0.0% 0.0%
Figure 1 - Data template: monthly returns for agricultural products
Monthly prices for this study were obtained from the South African Department of
Agriculture and can be manipulated into the correct format with the help of a spreadsheet.
Appendix 2 contains an extract of original product prices.
The second section of the data template allows the user to group certain products into
different categories, which will form constraints within the model. The expected annual price
growth and gross margin of each potential product are also entered into this template.
Historical average results for the expected price growth and gross margins may be used, but it
is advised that these be adjusted with future performance in mind; for example the potential
performance of wheat may be adversely affected by a move to wheat-free products by
consumers. In this case the future price growth of wheat should be deflated by a suitable
percentage. Any potential future variable costs associated with products under review should
also be accounted for by adjusting the gross margin percentages.
A column named “Current Land Utilization” requires the user to enter the area of land
currently occupied by each specific product. The units must be consistent throughout the
model; in this case hectares are used. An example of this section of the data template can be
seen in Figure 2.
60
Products Enter
Category
Expected
Price
Growth Gross
Margin
Current
Land
Utilization
[Ha]
1 Maize A 7.7% 5.0% -
2 Wheat A 5.6% 5.0% 1,331
3 Lucerne A 6.6% 5.0% -
4 Cattle 0.7% 13.0% 5,790
5 Sheep 2.9% 13.0% 3,083
6 Pigs 2.0% 13.0% -
7 Barley A 3.3% 5.0% 1,993
8 Sugar A 4.3% 34.0% 7,257
9 Apples B 6.2% 1.0% 145
10 Bananas C 4.8% 23.0% 400
11 Pears B 15.6% 1.0% 21
12 Avocados C 10.4% 23.0% 13
13 Citrus B 6.7% 1.0% 340
Figure 2 - Data template: expected price growth, gross margins and current land utilisation
for agricultural products
The third section of the template allows the user to enter labels for the three category
constraints and select whether the current land utilization should be less than or equal, or
equal to the Land Available Constraint specific to that category. This is shown in Figure 3.
The Required Adjusted Return is the minimum weighted sum of the returns on all the
products under review required by the agribusiness.
Description of Category
Current
Utilization
[Ha]
Land
Available
Constraint
[Ha]
A Irrigated Crops 10,581 15,000
B Drip Irrigated Orchards 506 1,000
C Dry Land Plantations 413 1,000
Balance 8,873 3,373
Required Adjusted Return 0.66%
Current Adjusted Return 0.66%
Total Av ailable Productiv e Land 20,373
Cycle Time (years) 10
Figure 3 - Category and land available constraints
The Balance quantities (current and available) are the difference between the Total Available
Productive Land and the sum of the three category totals. Should this balance be negative, the
61
spreadsheet will prompt the user to correct the mistake and will not attempt to solve the model
until all the entries have been checked for validity.
The Cycle Time is used to calculate a proxy value of the enterprise by using the Free Cash
Flow to Firm formula. A cycle may coincide with the productive life expectancy of a fruit
orchard4, for example.
The user also has the option of choosing to solve and plot the current and optimal positions of
the portfolio of agricultural products on an efficient frontier.
The macro generates a new sheet called “Model5, which contains the optimal solution and
the efficient frontier, if this has been selected. Appendix 3 contains a snapshot of the output
sheet. The model is dynamic in the sense that the user has the ability to change values in the
template and see the changes to the optimal solution of the model. “What-if” analysis can be
exercised this way. The model output will be described in more detail in the next section.
ILLUSTRATIVE RESULTS
We will illustrate the use of the model by introducing an example based on real price
information gathered from the South African Department of Agriculture over a sixty-month
period, and information submitted from a leading agricultural enterprise (which is listed on
the Johannesburg Securities Exchange). A variety of scenarios have been compiled in order to
demonstrate various rational and irrational long-term agribusiness strategies.
We will start with a base case where an established agricultural enterprise would like to map
its risk-return position relative to the optimal mix of products in order to minimise the
portfolio risk, given the current category constraints.
The established products as well as those under review, together with their expected returns
and gross margins are shown in Figure 4.
4 Most long-term agricultural products have limited “productive life spans” and have to be replaced periodically.
This model helps assess the optimal product for the next long-term cycle.
5 The macro will delete any sheets with the name “Model” already contained within the spreadsheet, before
generating a new solution.
62
Products Enter
Category
Expected
Price
Growth Gross
Margin
Current
Land
Utilization
[Ha]
1 Maize A 7.7% 5.0% -
2 Wheat A 5.6% 5.0% 1,331
3 Lucerne A 6.6% 5.0% -
4 Cattle 0.7% 13.0% 5,790
5 Sheep 2.9% 13.0% 3,083
6 Pigs 2.0% 13.0% -
7 Barley A 3.3% 5.0% 1,993
8 Sugar A 4.3% 34.0% 7,257
9 Apples B 6.2% 1.0% 145
10 Bananas C 4.8% 23.0% 400
11 Pears B 15.6% 1.0% 21
12 Avocados C 10.4% 23.0% 13
13 Citrus B 6.7% 1.0% 340
Figure 4 - Agricultural products considered, with expected returns and gross margins
The products have been grouped into specific categories, which are constrained due to the
area of suitable land available for producing that particular product. The cycle time of ten
years has also been entered into the template as shown in Figure 5.
Description of Category
Current
Utilization
[Ha]
Land
Available
Constraint
[Ha]
A Irrigated Crops 10,581 15,000
B Drip Irrigated Orchards 506 1,000
C Dry Land Plantations 413 1,000
Balance 8,873 3,373
Required Adjusted Return 0.66%
Current Adjusted Return 0.66%
Total Available Productive Land 20,373
Cycle Time (years) 10
Figure 5 - Input to category and land available constraints
With reference to the category constraints, it can be seen that the current land utilization of 10
581 Ha of Irrigated Crops is less than the potential 15 000 Ha available. The same is also true
for the other two category constraints, with the balance of land of 8 873 Ha currently
allocated to livestock.
63
The optimised model which minimises portfolio risk (the results are shown in Figure 6),
shows that it is possible to generate a reduction in risk of 29.9% with a 15.8% gain in
portfolio-adjusted return. Using a proxy for firm value based on the Free Cash Flow to Firm
model (see Appendix 4 for details of the proxy value calculation), a 20.5% improvement in
firm value can be attained, the main driver of this being the 16% reduction in portfolio risk. It
should also be noted that all three of the category constraints are binding, indicating that
further value might be gained should the firm invest in more production facilities suitable for
crops rather than livestock.
Model Results Optimal
Case Current
Case Para-
meters
Change
on
Current
Portfolio Adjusted Return 0.77% 0.66% 0.66% 15.75%
Portfolio Risk 4.35% 6.20% 29.85%
Irrigated Crops 15,000 10,581 15,000 41.76%
Drip Irrigated Orchards 1,000 506 1,000 97.63%
Dry Land Plantations 1,000 413 1,000 142.13%
Balance 3,373 8,873 3,373 61.99%
Total Productive Land 20,373 20,373 20,373
Proxy Value
Term 10
Cash Flow Factor 1.079452 1.06832 1.04%
Risk Factor 1.530255 1.82419 16.11%
Proxy Value 0.705406 0.58564 20.45%
Figure 6 - Optimised portfolio results
The risk-return efficient frontier is displayed in Figure 7. The optimal solution (Position C)
forms the base of the efficient frontier, which stretches up and to the right. Any combination
of adjusted return and risk on the efficient frontier would be a rational and in a sense optimal
one. Choosing a strategy that would result in the firm aiming for Postion B would be an
irrational one as a higher level of adjusted return may be realised from Position D for the
same level of portfolio risk. Adopting Position C may be referred to as a passive strategy, in
investment terms.
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Efficient Frontier
0.2%
0.4%
0.6%
0.8%
1.0%
1.2%
1.4%
4% 5% 6% 7% 8%
Portfolio Risk
Portfolio Adjusted Return
Figure 7 - Efficient frontier
The current portfolio of agricultural products yields an adjusted return and risk shown as
Position A. In order for the firm to move from Position A to Position C it would have to
reduce its investment in wheat, cattle, sheep, apples, bananas and avocados and increase its
investment in maize, lucerne, barley, sugar and citrus. The changes in products between
Points A and C are displayed graphically in
Figure 8. This would mean an increase in the proxy value of 21% (driven mainly by a
reduction in the risk factor) and of portfolio adjusted return of 16%, given a reduction of
portfolio risk of 30%.
With reference to
Figure 8, the practical implications of reducing land available for sheep and increasing land
available for sugar plantations would entail the ploughing of land, planting of sugar cane and
the installation of irrigation infrastructure. The proceeds from the sale of the sheep may
contribute towards the development of the sugar plantation6. In essence, sheep (which falls
into the ‘balance of productive land’ category) currently occupy land available for irrigation
(sugar forming part of this category).
6 This would form part of the feasibility study of replacing sheep with sugar and may include immediate
financing and resource constraints. This short term detailed analysis is important, but does not form part of the
discussion of this paper, which is focused more on the long-term implications.
Current
Position (A)
Position
(B)
Efficient Frontier
Position
(C)
Position
(D)
Position
(E)
65
Maize
Whe a t
Lucerne
Cat tle
Sheep
Pig s
Bar l ey
Sugar
Apples
Bananas
Pears
Avocados
Cit r us
Position A
Position C
-
1,000
2,000
3,000
4,000
5,000
6,000
7,000
8,000
9,000
Figure 8 - Land allocations for current portfolio and the minimum risk portfolio
Land allocations amongst all products for Positions A to E on the efficient frontier are given
in Appendix 5. Assume that the firm decides to take a more aggressive approach and makes a
10-year strategic decision to position itself on the efficient frontier at Position D in Figure 7.
This would mean striving for a higher adjusted return by accepting a higher degree of risk.
The results of this scenario appear in Figure 9.
Model Results Optimal
Case Current
Case Para-
meters
Change
on
Current
Portfolio Adjusted Return 0.82% 0.66% 0.66% 24.27%
Portfolio Risk 4.39% 6.20% 4.39% 29.10%
Irrigated Crops 15,000 10,581 15,000 41.76%
Drip Irrigated Orchards 1,000 506 1,000 97.63%
Dry Land Plantations 1,000 413 1,000 142.13%
Balance 3,373 8,873 3,373 61.99%
Total Productive Land 20,373 20,373 20,373
Proxy Value
Term 10
Cash Flow Factor 1.085521 1.06832 1.61%
Risk Factor 1.537146 1.82419 15.74%
Proxy Value 0.706193 0.58564 20.59%
Figure 9 - Optimised solution at Position D
66
As may be expected there is a significant increase in portfolio-adjusted return (24.3%) from
the status quo portfolio, with a reduction of 29.1% in portfolio risk relative to the current
position.
Having discussed the situation where a firm is constrained by its land availability and would
like to optimise its strategic (but passive) position, we will now consider the scenario where
the firm decides to improve its long-term position by acquiring more productive land. Assume
that on analysing the previous scenarios, the agribusiness decided to acquire an additional
5,000 Ha of fields fit for growing crops requiring irrigation and would like to view the effect
that this acquisition would have on the business over the cycle period of 10 years. The
expected change in risk/return space is depicted in Figure 10.
0.2%
0.4%
0.6%
0.8%
1.0%
1.2%
1.4%
3.5% 4.5% 5.5% 6.5% 7.5%
Portfolio Risk
Portfolio Adjusted Return
Figure 10 - Shift of the efficient frontier
From Figure 10 it can be seen that the efficient frontier has shifted left and slightly upwards
compared to the ‘current’ frontier. In effect, by acquiring and developing an additional 5,000
Ha of land available for irrigation, the agribusiness has been able to reduce its portfolio risk a
further 11% with a slight gain of 1.32% on adjusted return, this effectively leading to an
additional increase of 4.75% of the proxy value (see Appendix 6 for a summary of these
values). Figure 11 displays the optimal area allocated to each product under the base scenario,
Current
Position
Current
Frontier
Expanded
Frontier
Before land
acquisition
After land
acquisition
67
and the case of the area expanded by 5,000 Ha. Figure 11 displays the optimal area allocated
to each product under the base scenario, and the case of the area expanded by 5,000 Ha.
Maize
Wheat
Lucerne
Cattle
Sheep
Pigs
Barley
Sugar
Apples
Bananas
Pears
Avocados
Citrus
Base
Expanded
2,000
4,000
6,000
8,000
10,000
12,000
Figure 11- Land allocations for expanded and base scenarios
Barley and Sugar take up more than 90% of the new 5,000 Ha expansion, occupying
respectively 26.3% and 41.7% of the total land available in the expanded enterprise.
PRACTICAL USE OF THE MODEL
It should be recognised that some products (such as fruit, coffee etc) have a considerable lead-
time to yield. Thus year-to-year switching by an agribusiness would usually not be practical
as the switching costs would be high (e.g. capital costs associated with replanting a plantation,
vineyard or orchard; a number of years of no yield before maturity of the crop etc). This is in
sharp contrast to the case of financial securities, where switching costs are relatively low
(small commissions and transaction costs), and yield is continuous.
The model presented here thus focuses on long-term agricultural investment through
development of a new piece of land or through a one-off restructuring of existing farming
assets. The potential products that would be considered for analysis would be those that are
suited to the climatic conditions and soil types available on the relevant agricultural land, and
68
grouped into relevant categories (for example wheat, oats and barley may form a category
called “winter grains”, given that they thrive in similar climatic conditions and soil types).
This approach may offer real practical advantages in decision support. A South African
agribusiness indicated that it intends to test the model on certain long-term, but well-defined
decisions.
CONCLUSIONS
By using this model and running various scenarios, it is possible for an agribusiness to
quantify its strategic position in risk, return and value terms. This allows the decision makers
to develop the enterprise by aiming for a rational position along the efficient frontier and
developing and acquiring productive land that optimises this position over the long term.
By embarking on a strategy which moves the enterprise’s position closer to the efficient
frontier, the agribusiness is likely to reduce its adjusted portfolio risk whilst increasing return,
which leads to an increase in overall value of the enterprise.
The benefits of this model lie in its ability to use the relationships between prices of
agricultural commodities with each other (i.e. the magnitude of covariance), and thus the
ability to reduce risk by selecting the best mix of products over the long-term, to help the user
make a reasonable assessment of what the optimal mix should be from a strategic point of
view.
Given the inherent nature of the various products under consideration and the way their prices
move in relation with one another, it is unlikely that the covariance between them will change
significantly from one long-term cycle to the other. In other words, once the optimal portfolio
has been identified and initiative taken to transform the agribusiness from current status to
that shown by the model to be optimal (within the bounds of the agribusiness’s risk policy), it
is unlikely that the agribusiness will have to make huge capital investments every few years in
order to remain optimal unless there are drastic price adjustments or changes in gross
contribution margins.
In essence, this model yields the most meaningful results when new acquisitions or
expansions are considered (off the base of a currently optimal product mix) or where the
69
agribusiness has reached the end of its products’ productive life cycle and is considering what
the best next step over the next long-term cycle should be. The economic cost implications of
switching should always be balanced against the long-term strategic advantages when the
model suggests agribusiness transformation.
ACKNOWLEDGEMENTS
The authors wish to thank Mr Bruce Roberts of Crookes Brothers Ltd for his input and
comments on the model.
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71
APPENDICES
Appendix 1 – Extract of the template used to enter data into the model
Products Enter
Category
Expected
Price
Growth Gross
Margin
Current
Land
Utilization
[Ha] Description of
Category
Current
Utilization
[Ha]
Land
Available
[Ha] Maize Wheat Lucerne Cattle
1 Mai ze A 7.7% 5. 0% A I rrigation Crops 3,324 4,500 0.0% 0.0% 0.0% 0.7%
2 Wheat A 5.6% 5.0% 1,331 B Dry Land Crops 7,257 8,000 0.0% 0.0% 0. 0% 0.0%
3 Lucerne B 6.6% 5.0% C Orchards 919 2,000 0. 0% 0.0% 17.6% -0.4%
4 Cattle 0.7% 13.0% 5,790 Balance 8,873 5,873 54.7% 0.0% 0.0% 1.2%
5 Sheep 2.9% 13.0% 3,083 0.0% 0.0% 0. 0% -4.2%
6 Pi gs 2.0% 13.0% Required Adjusted Return 1. 00% 0.0% 0.0% 8.8% -9.4%
7 Barley A 3.3% 5.0% 1, 993 Current Adjusted Return 0.66% 0.0% 0.0% 0. 0% -2. 7%
8 Sugar B 4.3% 34.0% 7,25 7 0.0% 0.0% 0. 0% 2.5%
9 Apples C 6.2% 1. 0% 145 Total Avail able Producti ve Land 20,373 0.0% 0.0% -23.5% -2.5%
10 Bananas C 4.8% 23.0% 400 Cycle Time (years) 10 0.0% 4.3% 0. 0% 6.1%
11 Pears C 15.6% 1.0% 21 0.0% 0.0% 0. 0% 8.9%
12 Avocados C 10.4% 23.0% 13 0.0% 0.0% -3.5% -3.7%
13 Citrus C 6.7% 1.0% 340 Creat e and solve a new Model 0.0% 0.0% 0. 0% 10.3%
14 Trace the Ef ficient Frontier 0.0% 0.0% 0.0% 0.9%
15 0.0% 0.0% -1.8% -2.7%
16 2.6% 0.0% 0. 0% -6.8%
17 0.0% 0.0% 0. 0% -5.8%
18 0.0% 0.0% 4. 2% -0.9%
19 0.0% 0.0% 0. 0% 16.7%
20 0.0% 0.0% 0. 0% -6.1%
21 0.0% 0.0% -15.7% 5.1%
22 0.0% 15.5% 0. 0% 1.9%
Appendix 2 – An extract of monthly agricultural product prices
Source: The South African Department of Agriculture
Month Maize
(R/ton) Wheat
(R/ton) Lucerne
(R/ton) Cattle
(c/ton) Sheep
(c/ton) Pigs
(c/ton) Barley
(R/ton)
Jan-95 387.02 754.90 388.33 802.60 1235.70 617.80 671.79
Feb-95 387.02 754.90 388.33 808.10 1236.30 613.70 671.79
Mar-95 387.02 754.90 388.33 801.20 1119.00 528.80 671.79
Apr-95 387.02 754.90 456.67 797.60 1056.10 487.70 671.79
May-95 598.62 754.90 456.67 807.00 1011.00 490.80 671.79
Jun-95 598.62 754.90 456.67 773.50 990.00 481.50 671.79
Jul-95 598.62 754.90 496.67 701.00 974.20 466.90 671.79
Aug-95 598.62 754.90 496.67 682.30 1041.50 481.80 671.79
Sep-95 598.62 754.90 496.67 699.60 999.10 510.40 671.79
Oct-95 598.62 754.90 380.00 682.40 1049.20 578.20 671.79
Nov-95 598.62 787.58 380.00 723.80 1058.80 562.90 720.11
Dec-95 598.62 787.58 380.00 788.40 1205.00 627.40 720.11
72
Appendix 3 – Extract (shown in three parts) of the output screen contained in the sheet
called “Model”
Products Cat Optimal
Weights Current
Weights Expected
Return Average
Return Standard
Deviation Gross
Margin Adjusted
Return Optimal
Area [Ha] Current
Area [Ha] Variance
Area [Ha]
Maize A (0.00%) 7.70% 0.64% 7.60% 5.00% 0.39% (0) (0)
Wheat A (0.00%) 5.32% 5.60% 0.39% 3.15% 5.00% 0.28% (0) 1,331 (1,331)
Lucerne A 6.60% 0.55% 5.94% 5.00% 0.33%
Cattle (0.00%) 23.16% 0.70% 0.15% 4.22% 13.00% 0.09% (0) 5,790 (5,790)
Sheep 90.00% 12.33% 2.90% 0.32% 5.49% 13.00% 0.38% 22,501 3,083 19,418
Pigs 2.00% 0.91% 7.10% 13.00% 0.26%
Barley A (0.00%) 7.97% 3.30% 0.19% 1.66% 5.00% 0.17% (0) 1,993 (1,993)
Sugar A 8.00% 29.03% 4.30% 0.27% 1.64% 34.00% 1.46% 2,001 7,257 (5,256)
Apples B (0.00%) 0.58% 6.20% 0.60% 9.72% 1.00% 0.06% (0) 145 (145)
Bananas C (0.00%) 1.60% 4.80% 0.49% 18.49% 23.00% 1.10% (0) 400 (400)
Pears B (0.00%) 0.08% 15.60% 1.80% 14.70% 1.00% 0.16% (0) 21 (21)
Avocados C 2.00% 0.05% 10.40% 1.03% 21.35% 23.00% 2.39% 500 13 487
Citrus B 1.36% 6.70% 0.56% 14.28% 1.00% 0.07% 340 (340)
Model Results Optimal
Case Current
Case Para-
meters
Change
on
Current
Portfolio Adjusted Return 0.50% 0.54% 1.00% 6.69%
Portfolio Risk 17.25% 5.05% 241.66%
Irrigated Crops 2,000 10,581 2,000 81.10%
Drip Irrigated Orchards (0) 506 1,000 100.10%
Dry Land Plantations 500 413 = 500 21.07%
Balance 22,501 13,500 21,500 66.67%
Total Productive Land 25,000 25,000 25,000
V
alue
Term 10
FCFF 1.051575 1.05537 0.36%
Risk 4.910742 1.63652 200.07%
Proxy Value 0.214138 0.64489 66.79%
Maize Wheat Lucerne Cattle Sheep Pigs Barley
Maize 0.00569 -0.00002 -0.00007 -0.00002 -0.00056 -0.00040 -0.00001
Wheat -0.00002 0.00098 -0.00002 0.00007 0.00013 0.00034 0.00028
Lucerne -0.00007 -0.00002 0.00348 0.00014 -0.00026 -0.00111 -0.00001
Cattle -0.00002 0.00007 0.00014 0.00176 0.00027 0.00084 0.00005
Sheep -0.00056 0.00013 -0.00026 0.00027 0.00297 0.00112 -0.00002
Pigs -0.00040 0.00034 -0.00111 0.00084 0.00112 0.00496 0.00003
Barley -0.00001 0.00028 -0.00001 0.00005 -0.00002 0.00003 0.00027
Sugar -0.00032 -0.00001 -0.00000 -0.00008 0.00016 0.00008 -0.00001
Apples 0.00116 0.00017 -0.00077 -0.00003 0.00115 0.00282 0.00007
Bananas -0.00233 -0.00022 0.00079 0.00162 0.00074 0.00181 -0.00023
Pears 0.00136 0.00033 -0.00092 -0.00016 0.00188 0.00459 0.00024
Avocados -0.00110 0.00036 -0.00135 0.00119 0.00080 0.00519 0.00032
Citrus -0.00236 0.00035 -0.00199 0.00056 -0.00086 0.00294 0.00009
73
Appendix 4 – Details of proxy value calculation
Cash Flow Factor
Proxy Value = Risk Factor
()
()
t
p
t
p
1+R
Proxy Value =
1+σ
where:
R
p = Adjusted Portfolio Return
σp = Portfolio Risk
t = Cycle time
Appendix 5 – Detailed results for the base case
Point A Point B Point C Point D Point E
Maize A 1,008 1,370 1,190 1,609
Wheat A 1,331 0 879
Lucerne B 1,660 48 802 -
Cattle 5,790 3,512 3,298 3,435 2,942
Sheep 3,083 1,814 2,075 1,938 2,431
Pigs 47 0 -
Barley A 1,993 3,992 3,630 3,810 2,512
Sugar B 7,257 7,340 8,952 8,198 9,000
Apples C 145 203 74 156 -
Bananas C 400 93 308 206 428
Pears C 21 57 22 -
Avocados C 13 1 407
Citrus C 340 647 618 615 166
Irrigated Crops A 3,324 5,000 5,000 5,000 5,000
Drip Irrigated Orchards B 7,257 9,000 9,000 9,000 9,000
Dry Land Plantations C 919 1,000 1,000 1,000 1,000
Balance 8,873 5,373 5,373 5,373 5,373
20,373 20,373 20,373 20,373 20,373
74
Appendix 6 - Shift of the Efficient Frontier
Model
Results Current
Case Base
Case Expanded
Case Base on
Current Expanded
on Current Expanded
on Base
Portfolio Adjusted Return 0.66% 0.77% 0.78% 15.75% 17.28% 1.32%
Portfolio Risk 6.20% 4.35% 3.87% -29.85% -37.49% -10.88%
Irrigated Crops 10,581 15,000 20,000 41.76% 89.02% 33.33%
Drip Irrigated Orchards 506 1,000 1,000 97.63% 97.63% 0.00%
Dry Land Plantations 413 1,000 1,000 142.13% 142.13% 0.00%
Balance 8,873 3,373 3,373 -61.99% -61.99% 0.00%
Total Productive Land 20,373 20,373 25,373 0.00% 24.54% 24.54%
Proxy Value
Term 10 10 0.00%
Cash Flow Factor 1.068318 1.079452 1.0805378 1.04% 1.14% 0.10%
Risk Factor 1.824194 1.530255 1.4622874 -16.11% -19.84% -4.44%
Proxy Value 0.585638 0.705406 0.7389366 20.45% 26.18% 4.75%
Maize 1,423 1,726
Wheat 1,331 21.30%
Lucerne 829 1,003 -100.00% -100.00%
Cattle 5,790 1,978 1,954 20.90%
Sheep 3,083 1,395 1,419 -65.84% -66.25% -1.20%
Pigs -54.75% -53.98% 1.70%
Barley 1,993 4,629 6,678
Sugar 7,257 8,119 10,593 132.25% 235.07% 44.27%
Apples 145 210 168 11.88% 45.98% 30.48%
Bananas 400 770 799 44.99% 15.70% -20.20%
Pears 21 90 38 92.46% 99.77% 3.80%
Avocados 13 230 201 330.42% 81.89% -57.74%
Citrus 340 699 794 1670.39% 1445.48% -12.70%
105.70% 133.54% 13.53%
Product Area
Article
Abstract Currently pastoralists in Australia view native mammal species as one of many variables that impact, usually negatively, on their productivity and therefore profitability. This does not necessarily have to be the case. The species with the largest impact, kangaroos, have a value that could be incorporated into their income stream as a method of both reducing risk and increasing biodiversity, and therefore increasing resilience. An investigation of the idea of optimally allocating stocking rates using techniques analogous to classical portfolio selection optimization is conducted. Using historical pricing data for beef, wool, and kangaroo meat, an efficient frontier is formed to analyze the best scenario dependent on an investors risk aversion. It is shown that there is a clear opportunity for pastoralists to benefit economically by the inclusion of kangaroos in a mixed-grazing strategy for Australian rangelands.
Article
Full-text available
In the Swartland region farmers do not plant wheat exclusively. There are a lot of reasons for this. The main reason is that farmers who plant only one crop will end up with a situation where they will have to buy so much fertiliser that they would not be able to make any money. Every crop influences the crops on the same land later on. It is up to the farmer to decide what the influence will be. That means it is up to the farmer to decide what crop to plant on what land. The farmer ends up rotating a certain number of crops on his land. This rotation of crops is called a rotary crop system. In this situation arises the problem of what sequence of crops should be planted to ensure an optimal income to the farmer without exhausting the land. The problem could be solved by means of linear programming (LP). This problem, however, seem to get very large as the number of crops as well as the number of years over which the problem is solved is increased. By assuming that the influence of crops are only for three years and by restricting the number of years over which the problem is solved the problem is greatly reduced. If we look at the dual of the problem we find a further reduction. The solution of the dual problem also leads to the formulation of strategies. If we formulate the problem by means of the above mentioned strategies the problem reduces to a linear programming problem with only on constraint (which is the knapsack problem). The solution of this knapsack problem with help of a little game theory is then used in a computer program to assist farmers in deciding which crops to plant.