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Deriving the EV Frontier |
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Let us begin with the traditional portfolio
model. Assume that we want to
minimize the variance associated with attaining a given level of
income. To specify this problem we assume
a variance matrix: |
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In this initial formulation we find that the
optimum solution is x which yields a variance of 228.25. |
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GAMS Program |
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Sets |
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Tables |
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Parameters |
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Variables |
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Equations |
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Model Setup |
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Starting with the basic model of portfolio
choice: |
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Freund showed that the expected utility of a
normally distributed gamble given negative exponential preferences could be
written as |
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The Variance matrix for the problem is |
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The maximization problem can then be written as: |
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Using r
= 1/1250.0 we obtain an optimal solution under risk of |
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The objective function for this optimum solution
is 5,383.08. Putting r equal to
zero yields an objective function of 9,131.11 with a allocation of |
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Question:
How does the current solution compare to the risk averse
solution? Which crop makes the
greatest gain? Which crop has the
largest loss? Why? |
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A second point is that although the objective
function under risk aversion is 5,383.08, the expected income is
7207.24. What does this difference
manifest? |
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Quantifying Gains to Risk Diversification Using
Certainty Equivalence in a Mean-Variance Model: An Application to Florida Citrus |
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The traditional formulation of the mean-variance
rules begins with the negative exponential utility function: |
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Our discussion of Bussey indicated that this
expected utility can be rewritten under normality as |
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Hence, our tradition of maximizing |
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The implications of this objective function is
actually much broader, however.
Solving the negative exponential utility function for wealth |
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Other implications include the interpretation of
the shadow values of the constraint as changes in certainty
equivalence. For example, given the
original specification of the objective
function, the shadow values of the second land constraint is 34.73
and the shadow value of the first capital constraint is 93.98. These values are then the price of each
input under uncertainty |
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Moss, Charles B., Allen M. Featherstone, and
Timothy G. Baker. “Agricultural
Assets in an Efficient Multiperiod Investment Portfolio.” Agricultural Finance Review
49(1987): 82-94. |
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Historically, ownership of agricultural assets
has been dominated by farmer equity and debt capital. |
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The implication of this form of ownership are
increased variability in the return on equity to farmers |
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A direct manifestation of the unwillingness of
nonfarm investors to invest in agriculture can be seen in the unexplained
premium on farm assets in the Capital Asset Pricing Model. |
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This study examines whether autocorrelation in
the returns on farm assets versus other assets may explain the discrepancy. |
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Autocorrelation in farm returns refers to the
tendency of increased returns to persist over time. Mathematically: |
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Given this vector of returns, the problem is to
design the expected value/variance problem for holding a given portfolio of
assets over several periods.
Mathematically, this produces two problems: |
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Given the autoregressive structure of the
problem, what is the expected return? |
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A similar problem involves the variance
matrix. Using the autoregressive
estimation above, the variance matrix for the investment can be written as |
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Notes