An Appraisal of Single Index Models for Risk Efficient Farm Planning

D.K.Clarke, P. H. Carter, R. L. Batterham and R. G •. Drynan

University of Sydney

Contributed Paper to be presented at the Thirty~Sixth Annual Conference of the

Australian Agricultural Economics Societv. ~ustralian National University, Canberra.

lL'-J~ II February. 1992

An Appraisal of Single Index Models for Risk Efficient Farm Planning

Introduction

In fannplanning situations choices generally have to be made amongst various combinations

of risky, often interrelated, activities. For example in stocking a property a fanner rarely has

to choose only cattle or only sheep. Some combination of these .risky choices is possible. The

generally accepted economic theory of decision making under .risk in such situations is .known as portfolio theory. In portfolio analysis the aim is to find the portfolio or the combination of activities that maximises the decision maker's utility function.

Modern ponfolio theory dates back to Markowitz (1952) who showed 'how the standard

deviation of a portfolio of stocks could be reduced by choosing stocks which do not "move

together". He then .presented the basic principles of portfolio construction illustrating the trade

off between risk and return, developing the notion of "efficientUportfolios.

At about the same time as Markowitz was pioneering portfolio analysis in securities, Heady

(1952) identified the role thatdiversification on fanus could play in reducing risk to achieve

stability in fannincomes.Whilst not developed under the title of portfolio selection, Heady's

concepts are remarkably similar to the independent work of Markowitz.

Markowitz showed that quadratic programming can be used to obtain a frontier of mean­

variance (E-V) efficient portfolios. Considerable difficulty was experienced with.the use of

the quadratic programming codes to solve portfolio selection (and other problems involving

the use of quadratic programming) until the early 19705 when the Rand Corpordtion code became available to researchers (see Takayama and Batterham ] 972),

The difficulties experienced with quadratic programming lead researchers interested in

ponfolio selection to seek alternative methods for identifying the set of efficientportfoHos. This work was initiated by Sharpe (196:1, 1970) who showed that a significant amount of the

variation in activities' returns could be captured via the relationship the returns of each activity

has with some common factor. The advantage of Sharp's single index models (as they are

now known) is that the risk measures used in a singJeindex model summarise activity and

ponfoliorisk with many fewer pammetersthan are necessary in the full variance,-covariance

matrix required in a general quadratic prog11lmming model. A funheradvantage of some

versions of the single index model is that a set of efficient portfolios can bedetennined using

linear programming. Linear programming codes were widely available, genemlly familiar to

researchers. and relatively reliable as compared to the quadratic programming codes.

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Index models became relatively widely used fotportfolio sele.ction, at least 1n the scholarly

literature (see Hanington 1987). Several authors have used them for farm planning purposes

(good examples are Collins and Barry 1986 and Turvey, Driver and Baker 1988).

Over the last decade there has been enonnous progress in personal computing technology.

This technology has been accepted, in part at least, asatool for fannplanning purposes. Reliable software forquudratic programming is .now widely available.· A \questionaddressed

in this paper is whether there is still a role to 00 played by index models in fattn planning.

Two versions of the Sharpe index models are used to estimate E-V efficient ponfo}iost for

two case studies offann planning in central and southern NSW. The portfolios fonned by the

index models are compared tothosepr¢duced from quadratic programming using the full

variance-covariance matrix. The problems associated with each of the methods is considered,

and the future usefulness of index models to farm planning is discussed.

Tbe Case Studies

The analysis is based on two sets of historical gross margins obtained for two mixed livestock

and cropping areas within N.S.W.lnthefirst case, the gross margins were based on price,

yield and. variable cost records for thepropeny "Bull Plain" near West Wyalong. Additional

data for activities considered in the farm planning exercise; but notpreviou~ly pursued on the

property were obtained from neighbouring properties that have similar resources and

productive capacity. Price figures were indexed to 1990 prices using an index of prices

received by fanners published by ABARE. Costs were assumed to be constant at 1990 levels.

Infonnation that could not be 'provided from the propertyrecoros or from neighbouring

propeIties was obtained from district records obtained from the local office of New South

Wales Agriculture & Fisheries.

In the second case study~ the gross margins were based on historical records of price; yield

and variable costs associated with a number offannactivities undertaken in the Orange

DistriclofN.S.W. The data relates to the pe.riod 1975 to 1989. For the cropping activities

considered, average yield data from the Shire of Weddin as published by the Australian

Bureau of StatisticS, served as .the basis of production variability over the period. The use of

Shire data will obviously reduce.estimates of the variability of yields, and bencegross

margins. as compared to the estimates derived of similar variables derived from fann level

data.

'*-There nrc many commerdal codes, but mOst researchers use MINOS (sec Murttlgh and Saunders 1983) or the Rand code

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Grain prices were based on Sydney retail feed ingredient bi-monthlyestimates provided by New South Wales Agriculture & Fisheries. Sheep enterprise returns reflected variability in wool production per head. wool prices, live weight sale and purchase prices. Livestock values were based upon infonnation in the monthly Homebush saleyard reports produced by New South Wales Agriculture & Fisheries and on the New South Wales Meat Industry Authority state-wide monthly averages. Statistics provided by the Australian Wool Corporation were used to fonnulateproductionestimates as well as wool price variability over theperiod~ Additional infonnation was collected from local district staff of New Soth Wales Agriculture & Fisheries and from a number ofproducers.

The Quadratic Programming Model

Quadratic programming was used to identify the set of E-V efficient fann plans, that is, plans for each given 1evelof the variance of gross .margin (V) such that no other combination of

activities exist() providing a greater expected gross margin (E); or conversely, plans for each given level of expected gross margin such thatn9othercombination of activities exists providing a lower level of variance of gross margin. The quadratic programming model (specified for the Rand QPprogram) can be expressed as follows:

Maximise

subject 10

and

where

z = c'x - A x'Qx

Ax,Sb

x~ 0

Z is the objective function

c is the vector of expected activity returns

Q is the variance-covariance matrix of activity returns

x is the vector of activity levels

A is a parameter which is varied to alter the relative weightsQn expected

return and variance of returns and so produce the E-V frontier

A isa matrix ofinput-output coefficients

b is the vector of resource constraints or requirements

Given the set ofE-V efficient fann plans, the acceptability of one particular plan wiUdepend

on the decision maker's preferences among various expected gross margins and the associated

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variance levels. The utility function of the decision maker describes these preferences, and the

optimal fann.plan (assuming utility depends only on E and V) is located on .the E-V frontier .at

the point of.tangency with the highest utilitycUlVe for the decision maker as illustrated. in

figure L Whileitis possible to identify the optimum point directly with quadratic programming by explicitly incorporatinginfonnation about the utility function into the

quadratic program specification, given the difficulties associated with expliCitly specifying an individual's utility function it is arguably preferable to allow the decision maker to select from

the set of efficient farms plans. Other considerations may therefore be taken into account in choosing the most desirable plan.

Expected return

E-V indifference curve .,/

E-Vefficient frontier

Standard deviation of return

Figure 1; Identification ofoptimalpol1folio from theE-Vefficient set

Data requirements/or the quadratic programmillg model

The quadraticprogrammingmodeJ requires knowledge of the mean gross margins for each

activity considered, ,and their corresponding variances and covariances. Asthesepammeters

are unknown, estimates are commonly based on the mean, and variance-<=ovariance figures

obtained from a sample of the population,using cross~sectional data of observed historical

gross margins. This method of estimation wasempJoyedin this .study. The mean gross

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margin estimates,and the varian .... e-covarianceestimates forilietwQ farms are outlined in appendix 1. No allowWlceis made for the, uncertainty in the estimates themselves. The models

were solved usingtheR&i~d QP progrdll1l11e on a Macintosh computer.

Single .Index ~1odels

Sharpe t ssingle index ,model (1963,1970) is based on the assumption that each activity '8

.retum,(Ri)is related to somccommon factor (RnJand to an independentrandomelement (ev,

as follows:

Fi!"rure 2 illustmtesthis relationship.

R. 1

cx. { 1

(1)

Figure 2; The relauQnshipbetween activity returns and the common factor

tn fannplanning situations; Ri in equation 1 represent') the gross mCU"gin for the ·'ith" activity; ~i aconstant;Bi the expected change in Riin re~ponse toa change .in Rm; Rm the common factor or index; and £j a disturbance tenn. The ordinary least squares regression statistics

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presented in .. appendix 2 show the derivation of the essential parameters fortbe two index mode1s used in this study~

According to Sharpe, the common factorRm :should be the single most important factor

influencingretums. In the work on index models in finance, this factor is the Hmarket portfolioretum"'. Oneorotber of the availab1e finlmcial indices is used as a proxyforilie mark,~tportfolio,retum. Relationships among~t in ties are derived from common

relationships witbtheindex. In fann planningO.>l1ins and Barry (1986) and TurveYt Driver and B~er (1988) used 'the return -on a reference fann ponfolio for Rm. However any factor

believed to explain a significant atnountof .thejolflt variation in activityretumscould be used If. for example, the set of feasible activities consisted Qnlyofdryland winter cropping a1tematives~ raiI1faUthroughout the growing season or subsoil moiStUre may weUprovide a good explanation of the variations in gross margins.

The disturbance teon ~i reflects factors unique lothe individual activity ·itselft iaclQrsunretated

to the level of the index. Its expected value is assumed to 00 zero.

The expecte(iretum toacdvity i is then:

Ei= {Xi + BjEm (2)

where Em equals the expected tev~~l of the index.

The risk associated with activity j~ measured by the variance ofretums (ciZRjl. can bedividC(l

.into two parts, syStematic (or market}risk,and non .. systematic(or unique) risk:

where

= V(6jRm ) + V{ej)

alu, is the variance of the common factor orlndex;

o2£i is the variance of the error tetmtand

V(*) also denotes variance.

(3)

The fIrst tenn isfue systematic risk and the second the risk unique to the panicularactivity. The systematic risk for a particular .activity depends ()nits B<;oefficientand the variation in .the

• • l\l' ",'i .... I'

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common factor. Covariancetenns do not appear in this fonnulabecause the unique factors are assumed independent of the index.

Thererums from .any twodiffe~ntactivities will be cor related because of their joint

dependence on the common single index:

COVij= Bi Bj cr2m (4)

The set of efficient fannplans can be calculated via parametric qua.dratic programming using

the variance-covariance matrix fonned with the variances and covariances defined above.

Although the model still has lobe solved as aquadraticprogrammingmodeJ,tbel'educed

infonnation requirements have given this model considerable appeal. In summary , given an

appropna.teilldex, a quadratic program based on Sharpe's index model requires the following

parameters for each activity:

1) Ei - its .expected return

2) Bj - the responsiveness of itsretums to changes in the level of the index (its Beta)

3) cr2Ei - its unique .or non-systematic risk.

Infonnation is also needed on a2m, the variance of the common factor.

The Diago/lal ft10del

A .particularcombination of n activities~at .levels Xi, i= 1. .. n. will have an expected return of

and variance

n Ep= L XiEj

i=l (5)

(6)

The form of the expression for the portfolio variance suggests that the quadratic ·programming

problem in nactivities with means and variance-covariance matrix defined by equations (2) to

(4) can be rewritten as one in n+lactivities with a diagonal variance-covariance matrix. That

is, jf an n+ 1 th activity is defined (via.an additional constrc.iint) as the beta weighted sum of the firstnactivities ( Xn+ 1 =L Xi 3i), tbe portfolio expected return is correctly calculated if this

activity is given a zero expected gross margin. and theportfoJio variance is properly calculated

jf the variance-covariancematrix is diagonal with the unique risks in the first n diagonal

positions and the variance of the index in the n+lthposition.

The d!agonalrepresentation is often preferred since the problem can be specified without prior

calculation of the systematic risks and covariances.

The Miltimise Portfolio Beta Model

Sharpe (1963) .arguedtbat wbena 'portlolio of activities is well diversified, .non-systematic

risk becomes relatively small and can be ignored as an approximation. thus reducing equation 6 to:

n 02p= (20 Xi .f3j)202m

i=1 (7)

Portfolio variance will then be minimized when portfolio beta (l:Xj60 is minimized. The

problem can therefore be solved using linear programming. The great advantages of the

model are the fewparruneters needed. tospecifytbe model, and the ability to solve the problem

using linear programming. Measures of each activi~"tsbeta and expected retumarethe only

infonnation needed toconstr'Uct .the model to derive the efficienlplans (information on the risk

or variance of the index would be needed to calculate the variances of the plans). The

minimize portfolio beta models in this paper weresolverl with a parametric LPpackage on a

standard MS-DOS personal computer.

Results

Case/aml J

The combmed E-V frontiers presented in figure 3 illustrate the results of the diagonal and minimize beta models along with the E-V frontier produced bythequadrauc programming

model using the full variance-covariance matrix. A summary of the results are provided in

appendix 3,and thefuU results are available from the authors. Thediagonal model provided:a

C10Sf approximation of the QP frontier over the whole range of portfolios. This is consistent

with the results of the studies of Collins and .Barry (1986) and Turvey 1 Driver and Baker (1988). In effect, a single index is able to capture most of the joint variation in activity returns.

The "frontier" produced by the minimize poltfoliobeta model however provided .a good

approximation only over the parts of the frontier where risk is relatively lightly weighted. The

unexpected, but feasible, .decline in risktbatoccurs as expected return for theeady portfolios

increases, indicates the significance of the substantial non-systematicrisk ignored by the

model. The importance of diversification to the minimize beta model is therefore highlighted.

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The ability of the indexto.explain a significantamountof the variance in gross margins is also

necessary to obtain reasonable results for the minimize portfolio beta model. Oats makes up the majority of the early portfolios for the fust farm. The R2 statistic for oats is zero to two

decimal places. The index: therefore ,has little ability to predict the variance for a portfolio

containing only the activity oats. It is not until the activities barley and yearlings (R20fO.02

and 0.11 respectively) enter the solutions and oats is forced out, that the portfolios .approach the frontiergenerated~y the fuU quadtatic programmingmodel.

Expected return

200000

150000

100000

50000 +

+

+ +

+

+ Full QPportfolioselection model Minimise portfolio .Beta model(LP)

Diagonal QP model

O~~~~--~-r--~~--~~--~~r--r~

o 10000 20000 30000 40000 50000 60000

St~ndard deviation of return

Figure 3: Efficient plans derived under 3 different models for case fann 1

Casefann2

The efficient portfolios from the three models are graphed in E-V space in figure 4 with a

fuller summary of the results provided in appendix 3. The results exhibit analogous trends to

those of the fust case study. The diagonal model is again superior to the minimize beta

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portfolio. The quality of the approximations improve significantly for the higher expected return portfolios.

At fn"st glance the three models appear to provide similar results. The surprising, yet feasible

nearlinear trend in the E-V frontier exhibited by the full variance-covariancemooel is also

observed in the index models. Both index models produce portfolios within two .standard deviations of the actual frontier. However, a closer examination suggests that the result may have little generality. The minimize portfolio beta solutions are generally dominated by one

activity ,merino wethers, which according to the single index model, has a zero systematic

risk. It also happens to have a high expected return. It completely dominates the solutions up to an expected return of $90,000. It also has relatively low unique risk making itan attractive

activity in all three models.

Expected return

100000

80000

60000

40000

20000

Full QPportfolio selection model

MinimiseportfoJio Beta model (LP)

Diagonal QP model

o·~------~----~--~------~----~--~ o 5000 10000 15000 20000

Standard deviation of return

Figure 4: Efficient plans derived under 3 different models for cac;e fann 2

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Discussion

In both case studies the diagonal.single index quadratic programming model gives a better

.approximation to the E-V frontier produced by the full quadratic programming portfolio

selection model than does the minimise portfolio beta linear programming model. This result is somewhat different to that of the Turvey, Driver and Baker (1988) study where the diagonal single indexquadraecprogr'c:UllIlliugand the minimize portfolio beta models provided similar

approximations of the E-V frontier producedbylhe full variance-covariance quadratic progranuning model. Inspection ofequations (6) and (7) sqggests that the two index models are like! y to give closer solutions when the unique risks .are small relative to systematic risk.

The solutions would also be closer lfthe constraints on activity choice (or other features of the

problem) are such to ensure that LXi Bi is large relative to any individual Xj.

Previous authors have claimed several advantages of .the index models. Theseadvantclges can be classified as computational advantages and datacollectionlmode! specification advantages. The computational advantages include the smalJercapacity computer and less computer processing time (Collins and8any, 1986) and the simplicity ofthernodels. Advances in personal computingtf'..chnology and the availability of good quadratic programming codes bas meant that running ageneralquadraticprogramming:portfolio selection model is no more

difficult than running the diagonal. model. Theminin1izebetamodel is easit~r to specify and

solve, but this study has thrown some doubt on the usefulness ·ofthe solutions it generates, .at

least in the two case studies.

The advantages associated with the fewer parameters needed to construct the index models are

largely irrelevant to farm planning situations. In contrast to the situation for.financialmarkets, there.are no published beta and. residual.error estimates available for agricultural activities.

Thus, for fann planing purposes, the same raw data that is needed to construct a full quadrdtic

programming portfolio selection model is also needed to estimate the beta coefficients and

residual error variances essential for the diagonal and minimize portfolio beta models. Indeed, more infonnation .may be needed since data .on the Inovements in the common factor must also

be used.

The index models also have a disadvantage. Even if these models can identify and select plans

which are similar to the efficient plans under the full variance-<ovariance matrix, because the. true variance-covariance matrix is approximated to varying degrees in the index.morlels, the models proyide misieading estimates of the real risks of the selected plans. This would be

particularly so for the minimize portfolio beta model.

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To summarise,there seem to be no major computational advantages of the single index

models over the full quadratic programming portfolio selection model, the raw data requirements are the same,andcL1taprocessing requires about the same effort in each model. Thus .thereseems little pointin using the index. models for fann planning.

List of References

Collins, R. A.and Barry, P. J. (1986) 'Risk analysis with single-index portfolio ,models~an

application to fannplanning' American Journal of Agricultural Economics 68(1), 152-61.

Harrington, D. R. (1987) Modern Portfolio TheOlY, TheCapitai AssetPricir.s Model alld Arbitrage Theory: A User's Guide, Prentice-HalI, Englewood Cliffs.

HeadYt E. O. (1952) Economics of Agricultural Productioll alldResource Use, Prentice-Hall, Englewood Cliffs.

Markowhz,H. M. (1952) 'Portfolio selectiont Jourllal o/Finance?, 77-91.

Markowitz, H. M. (1959) Portfolio Selection. Efficient Diversification oflnvestmellfs, Yale

University Press, New Haven.

Munagh, B. A. and Saunders, M. A. (1983) 'MINOS 5.0 User's Guide', Department of

Operations Research, Stanford University.

Sharpe, W. F. (1963) .~ A simplified model for portfolio analysis' Management Science 9(2),

277-293.

Sharpe, W. F. (1970) Portfolio Theory alld Capital Markets, McGraw-HilI, New York.

Takayama, T. and Batterham, R. L. (1972) 'Portfolio selection and resource allocation for

financial and agricultural finns with the Rand QP360 quadratic programming code' ,

Department of Agricultural Economics, University of Illinois, Agricultural Economics

Research Report 117.

Turvey, C. G., Driver, H. C. and Baker, T. G. (1988) 'Systematic and non systematic risk in

fann portfolio selection' American Ji)Urllal of Agricultural Economics 70(4),831-36.

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Appendix 1

Calculation of Farm Activities Risk Using an Index of Historical Averages Case 1

Activity gl'OSS margins MARKET (Average

WHEAT OATS BARLEY FPf.AS WOOL 1STX' of put tetUn:1lJ)

YEARLS 282.47 t.81.'28 '201.34 92~24 23.01 43.868 269.5 156.2 117:7$ 144.01 191.34 .65.23 25.92 41.5835 197.65 93.3 185.88 53.71 158.49 199~23 28.54 40.049 303.79 1'38.5 1:29.24 55.89 131.61 131.76 24.98 35.1045 322.26 US:7 220.07 74.99 137.89 126.97 26.74 34.56 264.57 126.5 361~59 92.91 166.0t 2U.1! 41.00 57~125S 277.03 .73.5 201.77 IOl.Sf 168.94 tS3.04 5\.61 67.3t1 203.56 135.7 199.20 168 .. 27 212.85 101.33 37.19 37.7165 226.11 1.40.4 123.53 l28.93 171.38 87.64 1.8.41 27.632 238,66 113:7

.Mean 202.39 11 1.50 111.09 116.19 30.82 42.71 255.90 132.9.5 Var 5675.76 1958.55 6S7.69 5960.59 96.82 1.33.25 1635.66 496.14

Variance - Covariance Matrix WlW;\T OATS 8ARWY FPF.AS WOOL 1 STX \'EARLS MARKET

\\-'HEAT 6385.23 337.98 28.l.85 3713.73 343.50 516.1) 672.12 175(1.073 OATS 331.98 2203.36 1114;01 .2146.09 ·38.61 ·)1.56 .1210.65 44.06368 BARl.E.Y 281.$5 1174,01 739.90 ·955.71 28.1S 34.69 -690.38 87;50082 FPltAS 3713.73 -2146,09 -955.11 6105.66 352;65 J29.81 2064.73 1431.826 WOOL 343.50 .38,61 28 •. 15 352.65 108.92 111.72 .150.22 108.0147 lSTX 516.11 ·lLS6 34.69 329~g.1 111.72 149.90 ·149.47 140.1721 YEAR1_C; 672.[2 ·1210.65 ·690.38 2064.13 ·150.22 -149,41 1840.1.1 il39.4639 .MARKEr 17,s0.07 44:06 87.50 1437.83 108.01 140.17 339.45 558.1594

Betas

Beta Single Inde" Estimate of Risk

Diagonal model residual Acti'llti- Beta(i)·SctII(i)"' Var (mark.t.t) WlIEA 3.135 5487.24 897.98 OATS 0.019 3,47 2199.88 :BARLEY 0.157 13.71 726.17 Fl'EAS 2.576 3703,866 30GL79 WOOL 0.194 20;90 88.02 ISTX 0.251 35.203 114.70 YEARI.s 0.608 206.45 1633.65 .MARKh'T t~OOo 55815

Case 2 Activity gross margins

Obs WflHAT OATS BARLEY lRmC CANOLA UJPlNS WETIlERS MWSRMER 2XIJ.lJ1S YEARl.s MARKhl I 272.00 194.(10 72.70 17200 331.00 86.10 44.60 21;90 34.80 30.10 125.92 2 '24200 2M.OO 114.00 234,00 267,,00 It 1.00 46.80 22.20 53.30 21.50 131 •. 88 l .$7 • .20 17.90 .45.00 17UlO 17.80 $6.60 48.70 32.30 58.70 23.40 46.86 4 544.00 69.00 25.50 198,00 342.00 152.00 46,30 39.20 64 • .50 43,60 152:.41 S 23500 5600 3310 lS.60 116.00 ·63.00 39.50 2990 52.30 50.40 56.54 6 14.5.00 t08.00 3.42 26200 -48.00 -36.00 33.10 26.20 $4,6/\ 69.00 61.73 7 381100 164,00 96.90 248.00 -121.00 195.00 36.40 ;Z1.80 48./.1 .. 51,.00 113.49 8 .70,00 .6300 -1.58.00 ·67.00 .132.00 .. 121.00 39.60 19.30 22.40 37.00 -49.27 9 512.00 148.00 100.00 35700 166.00 236.00 34.20 26.30 46,80 68,50 169,48

to 188.00 80.30 IS.60 J8!i.(10 80AO 336.00 40AO 31.90 39.110 $6.10 10S.35 U 253.00 to·tOO 4020 346.00 139,00 274.00 31.00 29.70 37.10 54.80 130.88 t2 In.on 12~tOO Hi 60 19500 236.00 136.00 35,20 25.50 54.70 30.40 99.04 13 14600 98J!0 6.16 164,00 150.00 123.00 60.50 33.60 55,00 23.10 g6,04 14 209.00 147.00 8330 208.00 266;00 t90,00 51.90 31.10 51.20 28.80 126;63 15 131.00 9170 3230 176Jlll 19100 131.00 40.00 30.50 4570 31.00 90.22

Mean 22.561 103.45 .29.16 19097 133.55 122A5 4.1.8.8 28.5) 47.98 41.25 96.48 V .. 2406721 442011 4251.28 1074),61 21)9849 14261.67 59.36 24.68 10693 24396 2646.67

Variance· Covariance .Matrix

WHEAT OATS BARU:."Y TRmc CANOI.A l..t.1P1NS WETIlERS MWSRMER '2XU·LBS YF.ARIS MAIU(ET

WHf..AT OATS BARl.EY 'nunc CANOI..A UJPINS WETItER.~ MWSRMER 2XB.LBS ),EARlS MARKET 257$6. 10 5683.76 7577.80 9892,66 lOU1.34 9.553.34 .168.03 290.24 627.00 11)61.J4 11l41.55 5683.76 471583 4264.50 4857.65 5250.25 3~53.S0 2.00 -51.38 114.34 -25.412874.50 75n 80 4264.50 456137 4914.22 525447 4513,86 .1 u~s 33.66 253.06 U8.813153.98 9892,66 4851.65 4914.22 IlSIlOI .41JO.5~ 9390.23 ·225.64 102 "to 34$16 619;79 455183

HH II 34 525(1'25 5254,47 4130.54 22712.67 5987,06 3nso 189,89 503,48 .880.91 5365,66 955334 3&5),50 4573869390,23 59.8706 U'280.36 ·31\.25 25706 tlJ.Ol 301.5949211 'III ·168.03 2;00 ·11,95 ·11S.fi4397.80 ·3 IUS 63,60 l·DO 21A1 ·9~;69 ·310 19014 -51.3$ 3366 I02.7() J89,89 251.06 14.70 '26,45 :34.22 192 89.9~ 621.00 174.34 25:1.06 345,16 S03A8 lutU 21.41 34.'22 t 14.57 *20.63 2l!U6

106t 14 ·.2541 118 81 61979·880.91 30159 ·92.69 192 ·20.63 261.38 134,50 104L5S 2874.50 ;U5398 455383 536566 492818 ·310 8995 2)816 134.50 283572

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Betas

Activity .. WHEAT OATS BARI.E.Y TRmC CANOl.A UJPINS WblfmRS MWSRMER lXB.f..BS YF.ARLS MARKIIT

Bela 2..18 UU 111 1.61 U9 1.14 O~(lO 0.03 0,08 0 .. 05 1.00

Single Indt~~timale of Risk 8eta{i}"'BetA(il '" Var (m.rk~)

17485.32 2913.82 3501:96 7312:9l

D,'gonalmt>del ~idual 8300.98

Detrended Gross Margins Data

10152.72 $564.63

0.00 2.85

16.18 6,311

2835,72,

Appendix 2 Case 1

1822.01 10SlAl 4198.10

.t2S5!M4 67JS.72

63.60 23;59 97.79

255.00

Year 1982 1983 1984 .1985 1986 J987 1988 1989 1990

Whcu~ 282.47 117.78 185.88 129.24 220.07 361.59 201.77 199.20 ] 23.53

BilrJey 278.91 249.52 197.28 151.00 131.89 146.62 110.15 154.67 93.81

011($ 246.21 1.92.71 86.18 72.l2 74.99 76.68 71.04

119.57 64,00

FPcas 92.24

.65.23 199.23 131.76 126.87 218.71 153.04 101.33 87.64

Wool 23.01 25.92 2854 24~98 26.74 41 .. 00 51.61 37.19 18.41

Yeads 269.50 197.65 303.79 322.26 264.51 277.03 203.56 226.11 238.66

1ST)( 32.09 32.75 34.16 32.1.6 34.56 60.07 73.20 46.55 39.41

Estimation of Beta Coefficients and Error Terms Regfession Output: Constant Sid Err of Y Est RSqunrcd No. of Obscrvation~ Degrees ofFrcedorn X Coefficicnl(s) Std Err of cocc.

Wheat ,.214044

32.04 0.86

9 7

3.14 0.48

Barley 1,s0.24 28.81

0.02 9 7

0.16 0.43

Detrended Gross Margins Data Year

1975 1976 1977 1978 1979 1980 1981 1982 1983 1984 1985 1986 1987 1988 1989

Wheat 272.00 242.00 57.20

544.00 235.00 145.00 388.00 ·70.00 512.00 188.00 253.00 132.00 146.00 209.00 Il1.00

Oats 194.00 207.00

17.90 69.00 56.00

108.00 164.00 ~63.00 148.00 80.30

104;00 119 •. 00 98.80

147.00 91.70

Burley 72.70

114.00 .45 •. 00 25.50 33.10

3,42 96.90

~1,s8.00 100.00

15.60 40.20 16.60 6.16

83.30 32.30

Tritical 172.00 234.00 171.00 198.00 15.60

262.00 248.00 ·67.00 357.00 185.00 346.00 195.00 164.00 208.00 176.00

Oats 100.99 50 • .14

{l,OO 9 7

O.O~ 0.7:;

FPeas ·226.31

58.57 0.55

9 7

2 .. 58

Case 2

Canola 331.00 267.00

1'1.80 342.00 116.00 .48.00

~12.1.00 ~132.00 166.00 80.40

139.00 236.00 150.00 266.00 193.00

LUI ins 8~\. 10

111.00 86.60

152.00 ·63.00 ~36.00 195.00

-121.00 236.00 336.1.)0 274.00 136.00 123.0.0 190.00 131.00

0 •. 88

Wethers 44.60 46.80 48.70 46.30 39.50 33.10 36.40 39.60 34.20 40.40 31.00 35.20 60.50 51.90 40.00

Wool 5.09

10.03 0.19

9 7

0.19 0.15

SRMER 21.90 22.20 32.30 39.20 29.90 26.20 27.80 19.30 26.30 31.90 29.70 25S0 33.80 31. to 30.50

Market 156.24 93.29

138:.53 1.18.69 126.53 173.48 135.68 140.38 113.74

Yea Is 175.05 43.21

0.11 9 7

0.61 0.65

2XBLBS 34.80 53.30 58.70 64.50 52.30 54.60 48.80 22.40 46.80 39.80 37.10 54.70 55.00 51.20 45.70

ISTX 9.38

11.45 0 .. 23

9 7

0.25 0.17

YEARLS 30.10 2l.S0 23.40 43.60 50.40 69.00 51.00 37.00 6850 56.10 54.80 30.40 23.10 28.80 31.00

M41rket 125.92 131.88 46.86

152.41 56.54 61.73

1.13.49 -49.27 169.48 105.35 L30.88 99~04 86;04

126.63 90.22

-15-

:Estimation ·of Beta Coefficlentsand Error Terms Jt¢gfes~i()n OUtput! Conslan( StdErrof Y E$t RSqullrcd No. of Observations Degrees of Freedom X Coefficient(s) StdErrof Coer.

Wheal Oats Barley TrilicllJ CanDIa Lupins Wei hers *l3.96 5.6473 .. 78.15 36.038 .49.01 -45.22 4.1.986

94.549 44.296 33.682 67.239 116.3 85.043 8.216 0.67(.1 0.6153 0.7691 0.6353 0.447 0.5605 5.00£·05

15 1S 15 15 15 15 IS 13 13 13 13 13 13 13

ZASl2 1.0137 1.1122 1.6059 1 .. 8922 1.7379 ·0.001 0.4745 0.2223 0.169 0.3375 0.5837 0.4268 0.041.5

Appendix 3 EV frontiers generated by the three models

Case 1

SRMER 2XBLBS YEARlS ~5.446 40.S57 36 .• 671 5.0407 .10.262 16.572 0.1079 0.1465 0.;0244

15 15 15 13 13 13

0.0317 0.0769 0.0474 0 .. 0253 0.0515 0.0832

QPE(TGM) SO TOM MPB.E(TGM) SDTGM DiagE(TGM) SDTGM .116.32

64276.21 64593.64 68746.91 73061.46 86079.06 88436.13 88459.59 89246.09 89287.20 91862.65 92088.U 9.2988.11 95769.61 96782.60 96846.38

102186,42 109.573.29 1~7.l4.20 1 t.0577.66 113043.60 113370.09 120425.55 125103.22 131528 . .15 141919.77 143125.30 143522.70 153966.15 160238.41 161.118;08 151301.83 155732.12 l59959,42 16U18.08

7.63 4214.08 4235.21 4515.87 48.07043 5687.1.0 5855.24 5857.03 5919.32 5922.69 6147S4 617.0.97 6281.25 6638.58 6774.86 6784.05 7730~32 9278,01 9308.95 9502.46

10063.26 10141.35 11953,47 l3.641..86 16879.64 23605.81 24589.12 24952.69 39984.97 53084.54 55154,49 32363.74 39993.10 49038.14 55154.49

0.00 0.00 17648.79 7430.13 32446.43 14419.97 39598.05 18024.96 55912.79 267.15.44 63227.83 24339.38 90609.93 1548J.20

110542.30 137.08.97 117381.8.0 13258.66 119975.60 13112.44 120J4 tOO 13.084.19 140097.7.0 22600.65 140481.90 22922.82 153964.9.0 39985..87 160228.70 53l>66.62 16J 116.3.0 55.154.36

4t78 63665.88 64761.48 78900.08 78953.93 80747.37 81494~98 82436~57 82548.99 85082.8.0 85581.89 8745.1.09 92694.36

115354.89 116852.53 119153.81 124508.68 137640.20 140137.26 141781.64 1.42145 .. 98 153284.48 1 534l3.98 159693.43 160H6.31 161109J9 16U 18.08

3.28 5005.42 5025.48 6156,30 6160.60 6181.25 6170.20 6186.99 6189.04 6393.41 ()36Q.93 65J3.26 694L17

13735.99 14219.21 14682.64 16616.54 21284.55 231490$7 24035.38 24386.40 39136.94 39286.45 521.18.90 52866.31 55152.31 55154.36

QP.E(TGM) 0.00

52.45 562.50 562~50

4548.66 4670,95 4851.48 5336.21 5385.46 5800.10 5956.40 8003.00 9532.32 9939.38

10186.12 11993.90 12177.52 19522.53 24441.52 31894.14 31998.74 32797.92 32907.54

SDTGM 0.00 5.12

54.92 54.92

444.1.1 4S6~55

476.33 531.24 536~82 583.80 601.50 833,3.6

1006.61 1054.57 1085.09 1311.34 1334.34 2254.93 2945.93 4157.32 4175.13 43 11.9) 4330.76

-16-

Case 2 MPB £(TOM) SO TGM

0.00 0.00 31008.46 6027.07 34453.856696.66 89687A 1 17944 .11 90050.36 18019.55 90179.61. 17849~76 93794.70 19548~74

DiagF..(TGM) 0.00

34.91 939.99

8565.71 9497.90

10242.24 12895.08 14019.98 16514.23 37491.63 40821.16 42863.79 47322~72

48540.42 49931.37 50498.06 55479.72 55566.65 60779.60 87307.97 87724.59 88804.33 93794.66

SDTGM

5.18 139~69

1274.03 1461.45 1612.98 2064.49 2253.87 2675.20 6217.78 :6863.90 7260.39 .8129.29 8347.81 8639.55 8757.41 9763.44 9795.18

10792.95 16026.80 15807.28 15368.28 19548.14