Risk Modeling

Chapter 20

What is "risk"?

Some outcomes, such as yields or prices, are not known with certainty. In risk modeling,we often assume that the distribution ofa parameter is known with certainty, even if a particular observation's value is not known with certainty. Risk modeling techniques aredesigned to give a robust solution to aproblem involving parameters with probability distributions.

Ways to deal with risk Ignore it. Not always a good option!Assume producers are risk neutral

and maximize expected returns, deal with risk only as it affects transitions from a one state of nature to the next over time. (A complicated topic.).

Assume producers respond to risk as well as expected returns.

Risk, Decision Making, and Recourse

Fundamental distinction between cases:

1) All decisions must be made now with uncertain outcomes resolved later, afterall random draws from the distribution havebeen taken.

2) Some decisions are made now, then aftersome uncertainties are resolved, other decisions are made later.

Examples1) Invest at the beginning of the year, discoverreturns at end of year with no intermediatebuying or selling decisions.

2) Invest at the beginning of the year, buyor sell during the year depending on stockprices.

Model TypesThe first type of model is very commonand is generally called a stochastic programming model.

The second type is called stochasticprogramming with recourse.

Aside: "Discounting" CoefficientsData for LP models are almost never certain.Rather than use the expected mean of astochastic coefficient, a conservative approachis taken in a "deterministic model." Objective function revenue coefficients may be deflated while cost coefficients are inflated. The difficulty with this approach is the resulting probability of the solution. Conservative estimates on all parameters can imply a highly unlikelyoutcome and yield overly conservative decisions.

Modeling Risk in the Decision Process

Put both the expected returns and some representation of risk in the objective function, with weights.

Maximize expected returns subject to a limit on variability.

Minimize variability subject to an income target.

Hybrids.

Quadratic Programming and Risk Modeling

Quadratic programming can be usedfor mean-variance analysis (Freund).

Max jcjXj - jkSjkXjXks.t. jaijXj bj for all j constraints

is a parameter representing degree of riskaversion. cj is expected return.The Sjk are elements of the variance-covariance matrix for the incomeof the alternatives.

Matrix Notation

Max C X - X'SXs.t AX b

(X non-negative)

Portfolio AnalysisThe model becomes:

Max jcjXj - jkSjkXjXks.t. jXj = 1

The Xj represent the way the total portfolio dollars should be divided amongalternative investments.

Markowitz's E-V Formulation

Min X'SX s.t. CX = AX b

is the income target. By running the modelwith various levels of , one can plot the"risk-efficient" frontier.

(S is the variance-covariance matrix with elements sjk)

Risk Theory and E-V ModelsDebate has raged since the introductionof E-V analysis in the 1950's on the conditions under which an E-V modelleads to choices equivalent to utilitymaximization. The general consensus isthat E-V is consistent with utility maximizationwhen either: 1) the underlying income distribution is normal and utility function is exponential, or 2) the underlying distributionssatisfy Meyer's location and scale restrictions.

Specifying "risk aversion"For the E-V specification (Freund model)one needs to specify a risk aversion parameter, . (Markowitz's specification avoids that need.) Some researchers have used historical data toestimate the risk parameter under theassumption that historical data reflectsrisk preferences. Others have subjectivelyelicited the risk aversion parameter. There is a vast body of research in this area.

Linear Approximation

Minimization Of Total Absolute Deviations(MOTAD).

The measure of risk is the absolute deviation.

Back up to Chapter 9

MOTAD is a form of "MAD" model, where"MAD" stands for "minimization of absolute deviations."

From 9.1

Minimize |e| = e+ + e-

s.t. Yi = bjXji + ej+ + ej-

The Yi are "constants" in this framework.

ExamplePrice of Oranges Quantity of Oranges Sold Quantity of Juice Sold

10 8 55 9 14 10 92 13 86 15 29 17 3

Fit a line explaining the price of orangesas a function of quantity of oranges andorange juice sold.

Model

| ei|s.t. Pricei = b0 + b1QOSi + b2 QJSi + ei

where ei and the bi are unrestricted in sign.

Tableau and Solution

e1p e1n e2p e2n e3p e3n e4p e4n e5p e5n e6p e6n b0 b1 b2 min RHS usedObj 1 1 1 1 1 1 1 1 1 1 1 1 11.277obs 1 1 -1 1 8 5 eq 10 10

2 1 -1 1 9 1 eq 5 53 1 -1 1 10 9 eq 4 44 1 -1 1 13 8 eq 2 25 1 -1 1 15 2 eq 6 66 1 -1 1 17 3 eq 9 9

answers 5.79 0 0 0 0 0 0 2.72 0 0 2.77 0 3.43 0.19 -0.15

General MOTAD ModelMax E(Z) = cjXj - F

st. (crj – cj)Xj + yr 0 yr = sM/2(and resource restrictions, as appropriate)

F= fixed costs. s= sample size and M is the mean absolute deviation. So we maxexpected profit subject to a limit ona measure of variance. yr is negativedeviation.

Notice there is only one error term

In this case, we are only concerned withnegative deviations. If the total deviation in agiven observation is positive, then that y takes the value 0 and the ge constraint is satisfied without an error term.

MOTAD Model Example

From Anderson, Dillon, and HardakerAgricultural Decision AnalysisIowa State University Press 1977

3 crops and annual returns.

year crop 1 crop 2 Crop 31 99.8 68.3 112.72 133.3 130.4 238.43 142.7 33.3 93.94 154.3 74.4 83.25 11.4 25.4 109.7

Net Returns

Means: crop 1 – 108.3 crop 2 – 66.36

crop 3 – 127.58

Deviations from the Mean

year crop 1 crop 2 Crop 31 -8.5 1.94 -14.862 25 64.04 110.843 34.4 -33.06 -33.664 46 8.04 -44.365 -96.9 -40.96 -17.86

Deviations from Mean

Other InformationThe farmer has 12 ha of cropland, notmore than 8 ha can be sown in total of crops 1 and 3. There is $400 of working capital available and crop 1 takes $30, crop 2 takes $20, and crop 3 takes $40. There are 80 hours of labor. Crop 1 needs5, crop 2 needs 5, and crop 3 needs 8.

We can ignore fixed costs in modeling and subtract them from the obj later.

The Modelcrop 1 crop 2 crop 3 y1 y2 y3 y4 y5 type RHS

obj 108.3 66.36 127.58 max land 1 1 1 le 12rest1&3 1 1 le 8capital 30 20 40 le 400labor 5 5 8 le 80r1 -8.5 1.94 -14.88 1 ge 0r2 25 64.04 110.82 1 ge 0r3 34.4 -33.06 -33.68 1 ge 0r4 46 8.04 -44.38 1 ge 0r5 -96.9 -40.96 -17.88 1 ge 0risklim 1 1 1 1 1 le lambda

The X's and e's are all non-negative. Startingwith lambda of 0 and increasing it, we getthe risk-income frontier.

Target MOTAD

A modification developed by Lauren Tauer(see hand-out) to make MOTAD modelsmore theoretically robust.

Target MOTAD ModelMax E(Z) = cjXj

st. T- cjXj – yr 0 pryr (and resource restrictions, as appropriate)

T is the target level of return. Pr is the probability of the state occurring.

Target MOTAD Formulation of3-Crop Model

crop 1 crop 2 crop 3 y1 y2 y3 y4 y5 type RHSobj 108.3 66.36 127.58 max land 1 1 1 le 12rest1&3 1 1 le 8capital 30 20 40 le 400labor 5 5 8 le 80r1 -99.8 -68.3 -112.7 -1 le -Tr2 -133.3 -130.4 -238.4 -1 le -Tr3 -142.7 -33.3 -93.9 -1 le -Tr4 -154.3 -74.4 -83.2 -1 le -Tr5 -11.4 -25.4 -109.7 -1 le -Trisklim 0.2 0.2 0.2 0.2 0.2 le lambda

Note that it is the income, not the deviation fromthe mean income, in r1-r5.

Safety-First Models

Max CjXj

s.t jaijXj bi for all i

jckjXj S for all k

Where S is some "safety" level of income.

3-Crop Model – No Risk

crop 1 crop 2 crop 3 type RHS Usedobj 108.3 66.36 127.58 max 1260land 1 1 1 le 12 12rest1&3 1 1 le 8 8capital 30 20 40 le 400 386.7labor 5 5 8 le 80 80answers 1.333333 4 6.666667

Safety-First Solutioncrop 1 crop 2 crop 3 type RHS Used

obj 108.3 66.36 127.58 max 1248land 1 1 1 le 12 11.63rest1&3 1 1 le 8 8capital 30 20 40 le 400 385.4labor 5 5 8 le 80 80r1 99.8 68.3 112.7 ge 900 1140r2 133.3 130.4 238.4 ge 900 2305r3 142.7 33.3 93.9 ge 900 906.5r4 154.3 74.4 83.2 ge 900 985.7r5 11.4 25.4 109.7 ge 900 900answers 0.708885 3.625331 7.291115

We have lowered our expected return in orderto satisfy our $900 safety constraints.

When the RHS is "Risky"

Chance-constrained programming dealswith uncertain RHS values.

Our constraint becomes:

jaijXj bi - Zbi

resource use must be less than or equal to average resource availability less the standard deviation times a critical value which arises from the probability level.

Finding ZZ can be found: a) by making an assumption about the form of the probability distribution of b, e.g. that it is normal, and using values for the lower tail from a standard normal probability table;

or b) by uisng Chebyshev's inequality:

Z = (1-)-0.5

Example

We want an 87.5% probability of satisfying the constraint. Using the normal distribution we would set Z = 1.14.Using Chebyshev's inequality, we would set Z = 2.83.

The Chebyshev boundary may be toolarge for many problems.