Topic 4: EV analysis, stochastic
dominance-decision analysis with preferences unknown
Shauna Phillips
School of Economics
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AREC3005 Agricultural Finance & Risk
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Dr Shauna Phillips (Unit Coordinator) Phone: 93517892
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Utility functions
› Difficulties exist in eliciting DM’s utility function.
› Recommendations may need to be made for a group of people.
› What other decision making tools are available?
› Decision analysis with preferences unknown: efficiency analysis:
– E,V analysis. Decisions can be based entirely on expected income and variance
that accompanies it.
– Stochastic dominance. Risky prospects/projects compared in terms of full distributions of outcomes (not just moments).
E,V analysis
› Efficiency criteria can be devised to allow some ranking of risky alternatives when specific utility functions unavailable.
› Set of available strategies for farmer can be partitioned into “efficient” and “inefficient” sets.
› Assume: individual’s preferences are consistent with assumptions made about the utility function, and, individual’s subjective probability distributions for outcomes are same as those assumed in the analysis. Variable under consideration is normally distributed.
E,V analysis(moments approach)
› Can focus on the probability distribution of the risky prospect in order to approximate utility derived as functions of mean and variance of the risky prospect.
› Utility is expressed in terms of moments (mean & variance) of risky prospect.
› Approach is equivalent to direct estimation of utility function under certain assumptions (normality- the normal distribution is completely specified by the mean and variance).
› The general form of the E,V utility function is:
– U(x) = U[E(x), V(x)]
where E(x) is the statistically expected value of y (U is in x ) – and V(x) is the statistical variance of y (U is in x )
Taylor series expansion
› The value of a function U(y) can be approximated in the region of its mean, E(x) = E :
› U(x) = U(E) + U(1) (E)(x-E) + U(2) (E)(x-E) 2 /2! + U(3) (E)(x-E) 3 /3! + ….
– U(k) (.) is the kth derivative of the utility function U(.) – n! is factorial n
› Take expectations and simplify:
› U(x) = U(E) + U(2) M2(x) /2! + U(3) M3(x) /3! + ….
– Mk(x) is the kth moment about the mean of x.
› Hence, the utility of the risky prospect x, is equal to the utility evaluated at the mean plus a series of products comprised of moments, corresponding derivatives of the U(.) function and inverse factorials.
› Furthermore, provided U(k) E /k! becomes small more quickly than MK(x) becomes large, the a series with only the first 2 terms is an adequate approximation.
› Also for a distribution where x is normal (can be described solely by mean and variance), the decision analysis using just these 2 moments is exact.
› E,V is relatively simple and deterministic approach
› E,V best regarded as an approximate rule
› Advantage of E,V approach requires only information on means and variances of the outcome distributions to establish some ordering of alternative actions.
E,V analysis
› Efficiency analysis requires assumptions about preferences (nature of utility function).
› For all DMs to whom the assumed preferences apply, various actions can be divided into an efficient and an inefficient set.
› The inefficient set contains actions that are dominated (less preferred) by the actions in the efficient set.
› Efficient set contains actions not so dominated.
› Optimal action for any DM will lie in the efficient set provided:
› 1. DM’s preferences consistent with assumptions used to derive set.
› 2. DM’s subjective probability distributions for outcomes are identical to those assumed.
E,V analysis
› Mean-variance or E,V efficiency rule based on proposition that, if E(A) ≥ E(B), and
› V(A) ≤ V(B), then A is preferred to B by all DMs whose preferences meet certain criteria.
› That is, some strategy is E,V efficient if no other strategy can be found that has greater expected payoff with a lower variance.
› Only those actions not dominated in E,V sense are in the efficiency set.
› Criteria on preferences are that DM prefers more to less of the consequences x, and is universally not risk preferring with respect to the level of x.
E,V analysis
› Trade off in analysis between restrictions placed on utility function, and power of selection criteria, i.e. the fewer restrictions placed on utility function, the more generally applicable the results will be, but the less powerful will be the criterion in selection between alternatives.
› Weak assumptions may have too many alternatives in the efficient set- too restrictive assumptions risks excluding too many actions for efficient set.
› The use of E,V analysis allows for groups of efficient strategies to be identified ( a reduction in the pool of options) but does not necessarily resolve the optimal choice.
E,V analysis – example (Source: Hardaker et al. (2004))
› Beef farm in semi-arid Australia – due to poor returns DM considering quitting beef and cropping instead.
› Several crops under consideration but must fit sound rotations to allow sustainability.
› Unpredictable seasonal weather and product prices.
› DM wants to cull the alternative actions to a smaller number for final choice.
E,V analysis – example (Source: Hardaker et al (2004))
Using past farming records for the area, and subjective judgements, the DM (or her advisor) can estimate the distributions of net returns from alternative rotations.
Using notation F to K these distributions are presented in the table below:
statistics
E,V analysis – example
› Apply E,V to actions F-K above, need to set out means and variances in E,V space.
› Assuming DMs are risk averse, the relevant E,V indifference or iso-utility curves will slope upward – more risk averse implies steeper curves.
› U1 < U2 < U3
E,V space: crop rotations & utility fns
E,V analysis - example
› Given the risk aversion expressed in the curves above, K is the best alternative (on highest indifference curve, U3).
› K has the most preferred combination
E(x) =$418 & V(x)=$6519.
› H has relatively high mean, but greatest variance.
› As assumed, degree of risk aversion unknown, we cannot in fact identify
the alternative that gives the highest E(U).
› Instead, we apply the E,V efficiency rule: an alternative is in the E,V efficient set if there is no other alternative that lies in its north-western quadrant.
E,V analysis - example
E,V analysis - example
› I, J and K are not dominated by any other alternatives.
› Hence, they form the E,V efficiency set.
› DM then can choose amongst these (unless more information can be elicited about farmer’s attitude to risk) based on other preferences.
› (E,S analysis - variant of E,V analysis but using standard deviations instead of means. Same criteria and procedures apply).
Applications of E,V analysis
› Portfolio analysis is an application of E, V
› E,V analysis can be used to make decisions about a mix of risky prospects eg investment portfolios or farm plans. The mean and variance of any mix involving qi (units or proportion) of prospect i are given by:
› E=Σi qi ei and V=Σ i Σj covijqi qj
› Where ei is expected return of prospect i, and covij is covariance of
returns of prospects i and j.
› With one or more constraints on the qi (eg land and labour constraints in a farm plan), the set of possible mixtures of prospects forms a convex set in E, V space.
› A risk averse DM will have indifference curves with positive gradients.
› A frontier comprises the E,V efficient set and can be generated by quadratic programming (beyond scope of AREC3005).
Portfolio analysis efficient frontier and feasible set
U2 Efficient frontier
Feasible set
Efficient frontier and feasible set
› Optimal mix will be a point on the north-western frontier of the set, eg. C.
› Efficient set here differs that in ordinary E,V analysis as it contains an infinite number of points representing different decisions about the combination of alternatives.
› We need to know the utility function to derive the iso-utility curves which is again problematic.
› More practical approach is to determine the efficiency set (by quadratic programming) and let the farmer choose from this set.
Application of portfolio analysis in Fijian electricity
Policy problem rising oils prices
› “High oil prices have had adverse macroeconomic implications for SIDS, with their impact being particularly detrimental in the electricity sector given widespread reliance on generators that operate on diesel and heavy fuel oil . “
› Among Pacific island countries, high oil prices have increased electricity prices, compromising the energy security of poorer households.
› Renewable energy technologies: potential contribution to affordable, reliable energy supply via fuel mix diversification- less reliance on fossil fuels such as oil and gas. In terms of electricity, renewable technology investments can “reduce the variability of generation costs” through diversification
Example: Fiji renewables energy sector
› Oil dependence of PICs – budgetary implications
› Economic impacts of renewable technology investments in Fiji’s electricity
› Model incorporates variability of output from different technologies
› What’s the optimal mix of electricity technologies in an electricity sector portfolio?
› Different approach to that of minimised levelised costs.
› Renewable energy technologies advocated as risk minimising measure against oil price volatility.
Fiji - fuel imports (source Dornan & Jotzo 2015)
Electricity generation by technology (source Dornan & Jotzo 2015)
Mean and variance of technology portfolio
Electricity grid with a high-risk , low-cost technology 1 (eg gas-fired power generation)
and a low-risk , high-cost technology 2 (eg solar-power)
X1 and X2 are t proportions of total generation equipment made up of technology 1 and technology 2, 𝐶1𝑒𝑥𝑝𝑒𝑐𝑡𝑒𝑑 and 𝐶2𝑒𝑥𝑝𝑒𝑐𝑡𝑒𝑑 are the mean generation costs.
Financial risk of generation portfolio, σp, is weighted average of the financial risk associated with the cost technologies, and correlation coefficient between the costs of the two technologies.
Portfolio effect for 2 technologies
Risk that actual generation costs will differ from expected generation costs in the future
Dornan & Jotzo (2015) results
› “Investments in low-cost, low-risk renewable technologies, such as geothermal, energy efficiency, biomass and bagasse technologies, can be expected to lower both generation costs and financial risk for the electricity grid in Fiji”.
Linear programming (LP) & quadratic programming (QP)
› If we have farm data, we can use quadratic programming to apply E,V analysis to the crop rotation data above.
- 1. Build a farm decision LP model
- 2. Incorporate risk into the model (QP)
- 3. re-estimate the model by varying values of expected farm returns to estimate the associated variances
- 4. Graph or tabulate E, V pairs
Basic idea of LP
› Farm decision LP model maximises an objective function (total gross margin (TGM) subject to resource constraints (land, labour, etc).
› Simple application of microeconomics, but linearity is assumed.
› Soybeans (g1) $91
› Wheat (g2) $110.7
› Corn (g3) $138.7
› Objective function: Maximise Z=91g1 +110.7g2 + 138.7g3
› thesolutionwilldeterminehowmuchofeachcrop(gi)isproducedto maximised TGM...NB decision variable (gi )values are the solution to this problem.
› Aim is to maximise the objective function subject to resource constraints.
Constraints: resource requirements
Farmer produces wheat, corn & soybeans using labour (family and hired) and machinery.
Three-month production period, 1,000hrs hired labour, 500 hrs family labour, 2,000 hrs machine time, 600 ha land.
Technical coefficients (amount of input requirement per unit of output) and constraints:
Available supply of limiting inputs
Constraint (RHS)
Family lab
LP constraints
- 1g1 +1.1g2 + 1.3g3 ≤ 1000 - 0.4g1 +0.4g2 + 0.4g3 ≤ 500 - 2.2g1 +2.8g2 + 3g3 ≤ 2000 - 1g1 +1g2 + 1g3 ≤ 600
hire labour (hours) family labour (hours) machinery (hours)
solution of LP to find E*
› Maximise Z = 91g1 +110.7g2 + 138.7g3 › s.t.
- 1g1 +1.1g2 + 1.3g3 ≤ 1000 - 0.4g1 +0.4g2 + 0.4g3 ≤ 500 - 2.2g1 +2.8g2 + 3g3 ≤ 2000 - 1g1 +1g2 + 1g3 ≤ 600
› Solution gives the risk neutral value for E*
Simplified 2 variable LP (g1 & g2) 2 constraints
› Maximise Z= 91g1 +110.7g2 › s.t.
- 1g1 +1.1g2 ≤ 1000 hire labour (hours) - 0.4g1 +0.4g2 ≤ 500 family labour (hours)
The feasible region (for simplified 2 variable LP)
family labour
feasible region satisfies all constraints
hire labour
The feasible region (for simplified 2 variable LP)
family labour
Optimal solution-some combo of gi
hire labour
Solver solution
Objective Cell (Max)
Only g3 is in solution – full specialisation
Variable Cells
Original Value
Original Value
Cell Value
Final Value
$J$8 vars g1
$K$8 vars g2
$L$8 vars g3
Constraints
$M$11 hired lab
$M$12 man labour
$M$13 machine time
$M$14 land
Final Value
$M$11<=$N$11
$M$12<=$N$12
$M$13<=$N$13
$M$14<=$N$14
Status Slack
Not Binding 220
Not Binding 260
Not Binding 200
initial solution of LP to find E*
Target Cell (Max)
$J$11 avgallresourcesshouldbedevotedtocorn
Original Value
Final Value
risk neutral value for E*
Adjustable Cells
soybeans wheat
Original Value
Final Value
304.4 83220
Parametric quadratic programming
› Can derive solutions for various levels of β.
› β is a link between objective function and utility function: β =0 generates
risk neutral case ( same as maximising net revenue)
› Resulting set of solutions from varying β can be presented to DM for choice, or unique optimum found by setting up the E,V frontier and finding tangency point with assumed (or elicited) indifference curve (derived from utility curve).
› Application of QP and the E,V criterion is consistent with the expected utility hypothesis under the following conditions:
› 1. DM’s utility function is quadratic
› 2. probability distribution of returns is normally distributed
QP: example (Source: Kaiser et al (2011))
› 3 month time lag between planting and harvest.
› Possible states of nature (e.g. low or high rainfall) are exogenous to the
› Farmer expects per unit profits to be equal to average profits from past experience.
› Soybeans (g1) $91
› Wheat (g2) $110.7
› Corn (g3) $138.7
Variance covariance matrix of returns
Variance-covariance matrix can be generated in spreadsheet:
› Farmer is risk averse, an optimal mix of commodities is needed to minimise total profit risk based on a minimum level of expected profit, E*.
› Farmer measures profit risk by multiplying the variance-covariance matrix by the interaction of the individual “investments” in the “portfolio”
› Notation for individual “investments”:
› Let g1 be ha of soybeans produced
› Let g2 be ha of wheat produced
› Let g3 be ha of corn produced
› Then risk can be measured as:
› R = 32.6g1g1 + 154.6 g2g2 + 4546.2g3g3 -65g1g2 -101.7g1g3
- 65g2g1 +142.2g2g3 -101.7g3g1 + 142.2g3g2 › Variance – covariance matrix is symmetric (Vij=Vji ) so:
› R = 32.6g1g1 + 154.6 g2g2 + 4546.2g3g3
+2{-65g1g2 -101.7g1g3 +142.2g2g3}
› R = 32.6g1g1 + 154.6 g2g2 + 4546.2g3g3
-130g1g2 -203.4g1g3 +284.4g2g3
› Assume that farmer wants to minimise risk subject to the profit maximising risk neutral level of expected profit, E*:
› = 32.6g1g1 + 154.6 g2g2 + 4546.2g3g3 -130g1g2 -203.4g1g3 +284.4g2g3
91g1 +110.7g2 + 138.7g3 ≥ E* 1g1 +1.1g2 + 1.3g3 ≤ 1000
0.4g1 +0.4g2 + 0.4g3 ≤ 500 2.2g1 +2.8g2 + 3g3 ≤ 2000
1g1 +1g2 + 1g3 ≤ 600 gi ≥ 0
Incorporating an expected return constraint
› 91g1 +110.7g2 + 138.7g3 ≥ 38220 › 1g1 +1.1g2 + 1.3g3 ≤ 1000
› 0.4g1 +0.4g2 + 0.4g3 ≤ 500
› 2.2g1 +2.8g2 + 3g3 ≤ 2000
› 1g1+1g2 +1g3 ≤ 600 › gi ≥ 0
› Parametrically lowering E* in order to trace out the set of E-V efficient solutions of crop mixes based on alternative levels of risk is obtained.
› The range of feasibility in the Solver output on E* can be used in selecting new values of E* as it gives a new solution to the problem in the parametric analysis.
› Optimal strategy? 2 methods : iso-utility curves could be derived (but again, this requires knowledge of utility function) then pick tangency point E,V and highest iso-utility curve OR more practically, derive the E,V frontier and let the farmer choose based on some other preference.
› Solve profit max problem where risk is excluded to generate E*
› Then use E* in QP
› Parametrically lower E* in order to trace out the set of E-V efficient solutions in E, V space.
Qp initial solution of LP to find E*
› Maximise Z=91g1 +110.7g2 + 138.7g3 › s.t.
- 1g1 +1.1g2 + 1.3g3 ≤ 1000 - 0.4g1 +0.4g2 + 0.4g3 ≤ 500 - 2.2g1 +2.8g2 + 3g3 ≤ 2000 - 1g1 +1g2 + 1g3 ≤ 600
› Solution gives the risk neutral value for E*
E* ($), variance ($), activities (ha)
1,636,631,995
1,087,286,890
465,789,543
125,950,359
32,919,362
Risk neutral solution- all corn-highest profit, highest risk (st. dev approx half E*)
Last row is solution for most risk averse E*=$60,000, st. dev=$1686. This min risk level achieved via diversification. Emphasis on least risky crop, soybeans.
All 6 plans are E, V efficient (lowest possible risk for a give expected profit).
› Farmer then chooses from E, V set.
› Observations- trade-off between E & V, and “cost” of trade-off higher at extremes e.g. to increase E* from 80,000 to 83220 (4% increase), risk(SD) increases 22.7%. So, once farmer approaches the profit maximising level of E, the risk associated increases.
› Also, there exist 2 types of risk reducing strategies: switch from riskiest crop (corn) to less-risky wheat & soybeans OR diversify amongst all 3. Diversification is a risk-reducing strategy due to the negative covariance between several crop combinations - e.g. wheat and soybeans, corn and soybeans.
› Dual prices have same interpretation in QP as in LP. e.g. dual for E* constraint is $188743-if the farmer were to decrease E* to $83219 then the minimum variance would decrease by $188743. The trade-off between E & V will become smaller as E is lowered (diminishing marginal returns to risk).
› Dual on land constraint is -20,723,200: if we increase land by 1 ha, total variance would fall by $ 20,723,200.
QP alternative formulation
› We could instead formulate this problem to maximise E, subject to an acceptable level of minimum risk. This would yield an identical set of solutions in terms of E, V efficien
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