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Application of Portfolio Theory to Dairy Sire Selection1
Application of Portfolio Theory to Dairy Sire Selection 1
M. S C H N E E B E R G E R , 2 A. E. F R E E M A N , and M. D. BOEHLJE Departments of Animal Science and Economics Iowa State University Ames 50011
ABSTRACT
INTRODUCTION
Data for 285 Holstein sires from eight studs were used to determine the efficient set of portfolios or the expected income versus variance frontier, that is, the combination of sires that gave the smallest variance of income was found for each expected income. Expected income was measured by an index combining the sire's genetic superiority (in dollars) transmitted to his offspring and semen price. Variance of income increased with increasing expected income, and fewer bulls remained in the efficient set of portfolios as expected income and variance increased. The optimum solution was computed by an estimated utility function for dairymen. Because of the small negative weight for variance of income, three top-income sires were selected; if the weight for variance of income would have been larger, more bulls would have been in the optimum solution. For practical application, the expected income versus variance frontier can be determined with all sires available for artificial insemination, and dairymen can make their selections according to their individual weightings of expected income and variance of income.
Dairymen consider both expected income and variance of income (as a measure of risk) when selecting sires for use in artificial insemination (3). Expected income (E) is a function of the predicted differences for economically important traits, and variance of income (V) is a function of the repeatabilities associated with the predicted differences and the population variances of the corresponding predicted differences. The weighting of expected income vs variance of income differs from one individual to another and is expressed by a utility function (1). Portfolio theory (4) deals with the problem of finding the portfolio (the combination of available activities, e.g., for the stock market, the combination of stocks) that maximizes utility. Quadratic programming (2) is employed to solve the problem. The objective of this study was to apply portfolio theory to sire selection in dairy herds. Sires that maximize dairymen's utility, and proportions in which they are to be used, were determined. METHODS AND DATA
To find the expected income vs variance (E-V) frontier (minimum variance for each expected income) quadratic programming (QP) was applied. The QP problem can be written in matrix notation (2): Maximize U = p'x + £x'Qx
Received August 8, 1980. 1Journal Paper No. J-9941 of the Iowa Agriculture and Home Economics Experiment Station, Ames. Project 1053, contributing to North Central Regional Project, NC-2, Improvement of Dairy Cattle through Breeding. :Institute of Animal Production, Swiss Federal Institute of Technology, 8092 Zurich, Switzerland. Present address: Herdbook Office for Braunvieh, Chamerstr. 56, 6300 Zug, Switzerland. 1982 J Dairy Sci 65:404-409
Subject to A x <~ b x~>0
[11
where U is utility, x is the vector of solutions, p is a given vector with the expected incomes of the activities, Q is a given matrix of variances and covariances among the activities, A is a given coefficient matrix of constraints, b is a
404
405
PORTFOLIO THEORY AND SIRE SELECTION
Subject to
given vector of the right-hand sides of the constraints, and £ is the subjective weight for variance relative to e x p e c t e d income, which can vary according to dairymen's willingness to accept risk. A negative 1 indicates risk aversion, i.e., among portfolios with the same e x p e c t e d income, the one with the smallest variance is selected. A positive 1 belongs to the utility f u n c t i o n of a risk prone dairyman. The case where £ = 0 is risk neutrality.
xi~> 0
i= 1,2, 3,4. The solution is x1
=
.25, xu = .50, x3 = .25, x4 = 0 and thus
p ' x = 104, x ' Q x = 645, U = 91.
95 4
A parametric procedure (5) can be used to find the efficient set of portfolios, i.e., for each e x p e c t e d income, the p o r t f o l i o with the smallest variance V is found. The o p t i m u m solution for a specific utility f u n c t i o n is included in the efficient set. The efficient set also is called the E-V ( e x p e c t e d income-variance) frontier. The parametric procedure is
L61J
Minimize V = x ' Q x
An Example The m e t h o d is illustrated with a small example. F o u r sires with the following e x p e c t e d incomes and variances were chosen as activities:
P=
X I + X 2 + X 3 + X 4 ~< 1
Subject to p ' x =/3 Q=
0 0 0
897 0 0
0 6132 0
[2]
Ax 0
344
E x p e c t e d i n c o m e is c o m p u t e d f r o m the predicted differences for economically i m p o r t a n t traits, and variances are obtained f r o m repeatabilities of the sires' predicted differences as will be shown in the description of the application to sires in artificial insemination. The sires are taken to be unrelated; thus, Q is diagonal. The relative weight for variance was chosen as £ = - . 0 2 . The objective is to find the p o r t f o l i o (i.e., the c o m b i n a t i o n of the f o u r sires) that maximizes utility U = p ' x .02x'Qx. The ith e l e m e n t of the solution vector x, xi, is the p r o p o r t i o n of cows in the herd that is to be bred to the ith sire. The constraints
[41
x, p, b, A, and Q are as in [1]. The t3 is varied according to the specified problem, f r o m a lower b o u n d to the m a x i m u m value in p. It suffices to find the solutions for the values at which the basis changes, i.e., the points in the E-V frontier at which an activity (in this particular application, a sire) enters or leaves the solution. The solutions b e t w e e n the d e t e r m i n e d
TABLE 1. Points on the expected income vs. variance frontier at which a sire enters or leaves the solution for the example with four sires. Solution
4
x~
x~
x3
x4
~
x'Qx
U
(i.e., not m o r e than all cows in the herd can be bred) and xi 1> 0 (i = 1, 2, 3, 4) were introduced. The problem, therefore, can be written
o .49 .42 o
as
0
0 .37 .43 .59 0
0 .08 .15 .41 1.00
0 .06 0 0 0
0 91 97 114 140
0 318 404 1350 6132
0 85 89 87 17
Xi < 1 i=l
x' = [x I x 2 x a x 4 ] = vector of solutions. Maximize
= p'x, where p is the vector of expected incomes.
U = 84x 1 + 95x2 + 140x3 + 61x4 - 1 2 x ~ - 18x~ - 123x~ - 6 9 x ~
Q = Variance-covariance matrix of the incomes. [31
U =p'x - .O2x'Qx = utility. Journal of Dairy Science Vol. 65, No. 3, 1982
406
SCHNEEBERGER ET AL.
/
Application to Sires in Artificial Insemination
//u
~0
U
100
U
110
c~
'
216
410
1120
EXPECTEO
1'40
,'6°
INCOME
The objective of this study was to find the E-V frontier and the optimum solution in an application of portfolio analysis to dairy-sire selection. Data from 285 Holstein sires from eight bull studs in the United States were used. Means of standard deviations and m i n i m u m and maximum values of Predicted Differences, Repeatabilities, and semen prices for the 285 bulls are given in Table 2. Predicted difference for dollars (PD$) was computed by using the price of $.2293 per kg milk with 3.5% fat and fat differential of $.0282 per 1% increase of fat content per kg milk. Thus,
Figure 1. Expected income vs. variance frontier with points at which a sire enters or leaves the solution and iso-utility (U) curves for the example with 4 sires. points are obtained by linear interpolation. For the example, 13 was varied from 0 to 140, the maximum p. Solutions at the points of the E-V frontier at which the basis changes are in Table 1. The U column gives the figures for utility U = p'x - .02x'Qx that is to be maximized. The maximum is in the neighborhood of U=89. In Figure 1, the E-V frontier is plotted with expected income, p'x, o n the abscissa, and standard deviation (SD) of income, (x'Qx). s, on the ordinate. The optimum solution is determined by plotting iso-utility curves and finding the point of tangency with the E-V frontier. The iso-utility curves for U = 90, U = 100, and U = 110 are plotted in Figure 1, and the optimum solution (the point of tangency) is at expected income = 104, SD of income = 25, and U = 91. The point of tangency also can be found algebraically by the procedure given in (1).
PD$ = .1306PDM + 2.82PDF
[5]
where PDM is predicted difference for milk (kg), and PDF is predicted difference for fat (kg). An index was constructed to estimate the expected income that a sire transmits to a daughter and her offspring. It was assumed that six units of semen are necessary to obtain a milking cow, three lactations are completed per cow, and each cow has a daughter born as her second calf. This approximates the situation where a cow replaces herself with a female calf in a herd of constant size. Factors used to convert mature equivalent to actual production in first, second, and third lactation were .81, .89, and .96. Three offspring generations of the sire's daughter are considered. The index construction is summarized in Table 3. Income and semen cost are discounted to the year of first calving by a discount rate of 10%. The sum of the coefficients of discounted income, from Table 3, is 3.76. If feed costs are .4 of gross
TABLE 2. Means, standard deviations, minimum and maximum Predicted Differences (PD), Repeatabilities, and semen prices of 285 Holstein sires.
SD PD milk (kg) PD fat (kg) PD $ Repeatability (%) Semen price ($)
411.48 9.98 81.85 69.37 8.25
Journal of Dairy Science Vol. 65, No. 3, 1982
151.15 6.55 33.89 20.29 19.16
Minimum value
Maximum value
10.89 -7.71 --20.34 21.00 2.00
980.23 29.94 198.97 99.00 250.00
PORTFOLIO THEORY AND SIRE-SELECTION
407
TABLE 3. Estimation of the expected income of a sire transmitted to a daughter and her offspring. Year
Actual income
Discounted income
-3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 11
-6.P a 0 0 .81 PD$ b .89 PD$ .96 PD$ .5 (.81) PD$ .5 (.89) PD$ .5 (.96) PD$ .25 (.81)PD$ .25 (.89) PD$ .25 (.96) PD$ .125 (.81) PD$ .125 (.89) PD$ .125 (.96) PD$
-7.99P 0 0 .81 PD$~ .81 PD$ t .79 PD$ .30 PD$ .30 PD$ I .30 PD$ .11 PD$~ .11 PD$ I .11 PD$ .04 PD$ .04 PD$ I .04 PD$
production of daughter 1st offspring generation 2 nd offspring generation 3rd offspring generation
ap = price for one unit of semen. bpD$ = predicted difference dollars.
i n c o m e , t h e n t h e c o e f f i c i e n t o f PD$ is 2.26. E x p e c t e d i n c o m e , E, f o r a sire is, t h e r e f o r e ,
a i D S is t h e p o p u l a t i o n v a r i a n c e o f P D $ obt a i n e d as
E = 2.26 P D $ -- 7.99 P
OID s = ( . 1 3 0 6 ) 20"ID M + (2.82) 2 a i D F
[6]
+ 2(.1306)(2.82)rpDM,PDFOPDMOPD F. [8] w h e r e P is price for a u n i t fo semen. V a r i a n c e o f i n c o m e , V, f o r a sire was c a l c u l a t e d as V = ( 2 . 2 6 ) 2 ( l - - R ) OID $
[71
w h e r e R is r e p e a t a b i l i t y o f t h e sire's p r o o f , a n d
O~D M a n d a i D E are t h e p o p u l a t i o n variances of P D M a n d P D F , a n d rpDM,PD F is t h e correlat i o n b e t w e e n t h e t w o PD's. E s t i m a t e s for OIDM, OIDF, a n d rpDM,PD F were o b t a i n e d f r o m 1035 bulls s t u d i e d b y W. E. V i n s o n (Virginia P o l y t e c h n i c I n s t i t u t e a n d S t a t e University, personal c o m m u n i c a t i o n , 1979), so
16
aiD S = (.1306) 2 (253) 2 + (2.82) 2 (8.56) 2 + 2(.1306)(2.82)(.787)(253)(8.56) = 2930 Sires were a s s u m e d t o b e u n r e l a t e d . To obtain the optimum solution, a utility f u n c t i o n e s t i m a t e d f r o m d a t a d e s c r i b e d in (3) was used.
15 tJ
O:
d
:
U = E -- . 0 2 V 7 ~SI0
i50
200
250
EXPECTED
INCOME
3100 ($]
31S0
½00
Figure 2. Expected income vs. variance (E-V) frontier for 285 Holstein sires. The numbers of the points on the E-V frontier correspond with the numbers in Table 4.
[9]
w h e r e U is utility, E is e x p e c t e d i n c o m e , a n d V is variance of i n c o m e . T h e c o n s t r a i n t s i n t r o d u c e d i n t o t h e QP m o d e l were 285 xi = 1 and x i > 0 i=l for i = 1. . . . .
285,
[10]
Journal of Dairy Science Vol. 65, No. 3, 1982
408
SCHNEEBERGER ET AL.
TABLE 4. Expected income vs. variance (E-V) frontier for 285 Holstein sires. Point on E-V frontier in Figure 1
No. of sires in solution
Expected income ($)
Variance of income ($2)
Utility U=E -- .02V
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
274 263 236 191 161 128 104 82 67 56 44 34 24 13 3 1
156.31 167.77 181.20 195.94 205.93 217.85 228.05 236.43 242.54 251.22 258.42 267.20 282.68 306.56 349.18 372.05
5.57 6.42 7.92 10.64 13.53 18.45 24.52 31.39 38.03 50.48 64.09 86.52 155.22 389.87 1332.83 3290.30
156.19 167.64 181.04 195.73 205.66 217.48 227.56 235.81 241.78 250.21 257.14 265.47 279.58 298.76 322.53 306.24
where x i is the proportion of cows bred to the iTM sire. The first constraint means that all cows in the herd are to be bred. R ESU LTS A N D D ISCUSSlON
The E-V frontier is plotted in Figure 2, and solutions at the points marked on the E-V frontier are listed in Table 4. Numbers of sires in the solutions decrease with increasing expected income. The sires with the smallest income drop out of the solution as expected income increases. The point at the upper end of the E-V frontier consists only of the top-income bull. Variance of the income increases with increasing expected income. In the stock market, activities with high expected incomes often are associated with high risks or variances of income. Smaller variance results from choosing activities with smaller expected income and from diversification, i.e., from choosing a large number of activities. With dairy sires, it is not necessarily true that high-income sires have large variances of income (i.e., small repeatabilities) and that low income sires have small variances (i.e., large repeatabilities). A reduction of variance (and, thus, risk) results from diversification by choosing a large number of bulls. This means that poorer bulls need to be included, thus, expected income decreases. The optimum solution was found algebraiJournal of Dairy Science Vol. 65, No. 3, 1982
cally (1) for the utility function U = E - - . 0 2 V and is in Table 5. The three bulls with the highest expected incomes are in the optimum solution. These three bulls have high expected income because of high PD$ and low semen price. The optimum solution is close to the upper end of the E-V frontier. This is because of the small negative weight for risk or variance, compared with the weight for expected income. But the income-maximizing solution of only the topincome bull was not the optimum solution because the utility function, U, includes a negative weight for variance of income which encourages diversification. A decision maker with a more risk-averse behavior (i.e., with a larger negative weight for variance of income than used in U) would obtain an optimum solution with smaller expected income and smaller variance of income; he would diversify and use more bulls. A less riskaverse decision maker would choose a solution with larger expected income and larger variance, and would use fewer bulls. A practical applicaton of portfolio analysis to dairy sire selection can be obtained by computing the E-V frontier using all sires that are available to dairymen. Dairymen then would select individual optimum solutions among the solutions on the E-V frontier according to their individual utility functions. Additional constraints can be incorporated easily into the QP
PORTFOLIO THEORY AND SIRE SELECTION
409
TABLE 5. Optimum solution of the quadratic programming problem with 285 Holstein sires for U = E - .02V.
Proportion of cows bred
E ($)
V ($2)
.61 .36 .03 Weighted mean
372 330 297 355
3290 2692 3889 1572
PD$
PD milk (kg)
PD fat (kg)
R (%)
Semen price ($)
189 164 149 179
980 756 681 890
22 23 21 22
78 82 74 79
7 5 5 6
U = utility. E = expected income. V = variance of income. PD -- predicted difference. R = repeatability.
p r o b l e m , e.g., the m i n i m u m p r o p o r t i o n o f c o w s to be b r e d to y o u n g sires, t h e m a x i m u m p r o p o r t i o n t o be b r e d to a particular sire, and o t h e r c o n s t r a i n t s reflecting t h e d a i r y m e n ' s personal p r e f e r e n c e s . Also, if sires are related, covariances b e t w e e n sires can be c o n s i d e r e d in t h e Q m a t r i x o f t h e QP p r o b l e m [ 1 ] .
2
3
4 REFERENCES
1 Anderson, J. R., J. L. Dillon, and T. B. Hardaker.
5
1977. Agricultural decision analysis. 1st ed. Iowa State University Press, Ames. Pfaffenberger, R. C., and D. A. Walker. 1976. Mathematiczl programming for economics and business. Iowa State University Press, Ames. Schneeberger, M., A. E. Freeman, and M. D. Boehlje. 1981. Estimation of a utility function from semen purchases from Holstein sires. J. Dairy Sci. 64:1713. Sharpe, W. F. 1970. Portfolio theory and capital markets. McGraw-Hill, New York. Wolfe, P. 1959. The simplex method for quadratic programming. Econometrica 27:382.
Journal of Dairy Science Vol. 65, No. 3, 1982
M. S C H N E E B E R G E R , 2 A. E. F R E E M A N , and M. D. BOEHLJE Departments of Animal Science and Economics Iowa State University Ames 50011
ABSTRACT
INTRODUCTION
Data for 285 Holstein sires from eight studs were used to determine the efficient set of portfolios or the expected income versus variance frontier, that is, the combination of sires that gave the smallest variance of income was found for each expected income. Expected income was measured by an index combining the sire's genetic superiority (in dollars) transmitted to his offspring and semen price. Variance of income increased with increasing expected income, and fewer bulls remained in the efficient set of portfolios as expected income and variance increased. The optimum solution was computed by an estimated utility function for dairymen. Because of the small negative weight for variance of income, three top-income sires were selected; if the weight for variance of income would have been larger, more bulls would have been in the optimum solution. For practical application, the expected income versus variance frontier can be determined with all sires available for artificial insemination, and dairymen can make their selections according to their individual weightings of expected income and variance of income.
Dairymen consider both expected income and variance of income (as a measure of risk) when selecting sires for use in artificial insemination (3). Expected income (E) is a function of the predicted differences for economically important traits, and variance of income (V) is a function of the repeatabilities associated with the predicted differences and the population variances of the corresponding predicted differences. The weighting of expected income vs variance of income differs from one individual to another and is expressed by a utility function (1). Portfolio theory (4) deals with the problem of finding the portfolio (the combination of available activities, e.g., for the stock market, the combination of stocks) that maximizes utility. Quadratic programming (2) is employed to solve the problem. The objective of this study was to apply portfolio theory to sire selection in dairy herds. Sires that maximize dairymen's utility, and proportions in which they are to be used, were determined. METHODS AND DATA
To find the expected income vs variance (E-V) frontier (minimum variance for each expected income) quadratic programming (QP) was applied. The QP problem can be written in matrix notation (2): Maximize U = p'x + £x'Qx
Received August 8, 1980. 1Journal Paper No. J-9941 of the Iowa Agriculture and Home Economics Experiment Station, Ames. Project 1053, contributing to North Central Regional Project, NC-2, Improvement of Dairy Cattle through Breeding. :Institute of Animal Production, Swiss Federal Institute of Technology, 8092 Zurich, Switzerland. Present address: Herdbook Office for Braunvieh, Chamerstr. 56, 6300 Zug, Switzerland. 1982 J Dairy Sci 65:404-409
Subject to A x <~ b x~>0
[11
where U is utility, x is the vector of solutions, p is a given vector with the expected incomes of the activities, Q is a given matrix of variances and covariances among the activities, A is a given coefficient matrix of constraints, b is a
404
405
PORTFOLIO THEORY AND SIRE SELECTION
Subject to
given vector of the right-hand sides of the constraints, and £ is the subjective weight for variance relative to e x p e c t e d income, which can vary according to dairymen's willingness to accept risk. A negative 1 indicates risk aversion, i.e., among portfolios with the same e x p e c t e d income, the one with the smallest variance is selected. A positive 1 belongs to the utility f u n c t i o n of a risk prone dairyman. The case where £ = 0 is risk neutrality.
xi~> 0
i= 1,2, 3,4. The solution is x1
=
.25, xu = .50, x3 = .25, x4 = 0 and thus
p ' x = 104, x ' Q x = 645, U = 91.
95 4
A parametric procedure (5) can be used to find the efficient set of portfolios, i.e., for each e x p e c t e d income, the p o r t f o l i o with the smallest variance V is found. The o p t i m u m solution for a specific utility f u n c t i o n is included in the efficient set. The efficient set also is called the E-V ( e x p e c t e d income-variance) frontier. The parametric procedure is
L61J
Minimize V = x ' Q x
An Example The m e t h o d is illustrated with a small example. F o u r sires with the following e x p e c t e d incomes and variances were chosen as activities:
P=
X I + X 2 + X 3 + X 4 ~< 1
Subject to p ' x =/3 Q=
0 0 0
897 0 0
0 6132 0
[2]
Ax 0
344
E x p e c t e d i n c o m e is c o m p u t e d f r o m the predicted differences for economically i m p o r t a n t traits, and variances are obtained f r o m repeatabilities of the sires' predicted differences as will be shown in the description of the application to sires in artificial insemination. The sires are taken to be unrelated; thus, Q is diagonal. The relative weight for variance was chosen as £ = - . 0 2 . The objective is to find the p o r t f o l i o (i.e., the c o m b i n a t i o n of the f o u r sires) that maximizes utility U = p ' x .02x'Qx. The ith e l e m e n t of the solution vector x, xi, is the p r o p o r t i o n of cows in the herd that is to be bred to the ith sire. The constraints
[41
x, p, b, A, and Q are as in [1]. The t3 is varied according to the specified problem, f r o m a lower b o u n d to the m a x i m u m value in p. It suffices to find the solutions for the values at which the basis changes, i.e., the points in the E-V frontier at which an activity (in this particular application, a sire) enters or leaves the solution. The solutions b e t w e e n the d e t e r m i n e d
TABLE 1. Points on the expected income vs. variance frontier at which a sire enters or leaves the solution for the example with four sires. Solution
4
x~
x~
x3
x4
~
x'Qx
U
(i.e., not m o r e than all cows in the herd can be bred) and xi 1> 0 (i = 1, 2, 3, 4) were introduced. The problem, therefore, can be written
o .49 .42 o
as
0
0 .37 .43 .59 0
0 .08 .15 .41 1.00
0 .06 0 0 0
0 91 97 114 140
0 318 404 1350 6132
0 85 89 87 17
Xi < 1 i=l
x' = [x I x 2 x a x 4 ] = vector of solutions. Maximize
= p'x, where p is the vector of expected incomes.
U = 84x 1 + 95x2 + 140x3 + 61x4 - 1 2 x ~ - 18x~ - 123x~ - 6 9 x ~
Q = Variance-covariance matrix of the incomes. [31
U =p'x - .O2x'Qx = utility. Journal of Dairy Science Vol. 65, No. 3, 1982
406
SCHNEEBERGER ET AL.
/
Application to Sires in Artificial Insemination
//u
~0
U
100
U
110
c~
'
216
410
1120
EXPECTEO
1'40
,'6°
INCOME
The objective of this study was to find the E-V frontier and the optimum solution in an application of portfolio analysis to dairy-sire selection. Data from 285 Holstein sires from eight bull studs in the United States were used. Means of standard deviations and m i n i m u m and maximum values of Predicted Differences, Repeatabilities, and semen prices for the 285 bulls are given in Table 2. Predicted difference for dollars (PD$) was computed by using the price of $.2293 per kg milk with 3.5% fat and fat differential of $.0282 per 1% increase of fat content per kg milk. Thus,
Figure 1. Expected income vs. variance frontier with points at which a sire enters or leaves the solution and iso-utility (U) curves for the example with 4 sires. points are obtained by linear interpolation. For the example, 13 was varied from 0 to 140, the maximum p. Solutions at the points of the E-V frontier at which the basis changes are in Table 1. The U column gives the figures for utility U = p'x - .02x'Qx that is to be maximized. The maximum is in the neighborhood of U=89. In Figure 1, the E-V frontier is plotted with expected income, p'x, o n the abscissa, and standard deviation (SD) of income, (x'Qx). s, on the ordinate. The optimum solution is determined by plotting iso-utility curves and finding the point of tangency with the E-V frontier. The iso-utility curves for U = 90, U = 100, and U = 110 are plotted in Figure 1, and the optimum solution (the point of tangency) is at expected income = 104, SD of income = 25, and U = 91. The point of tangency also can be found algebraically by the procedure given in (1).
PD$ = .1306PDM + 2.82PDF
[5]
where PDM is predicted difference for milk (kg), and PDF is predicted difference for fat (kg). An index was constructed to estimate the expected income that a sire transmits to a daughter and her offspring. It was assumed that six units of semen are necessary to obtain a milking cow, three lactations are completed per cow, and each cow has a daughter born as her second calf. This approximates the situation where a cow replaces herself with a female calf in a herd of constant size. Factors used to convert mature equivalent to actual production in first, second, and third lactation were .81, .89, and .96. Three offspring generations of the sire's daughter are considered. The index construction is summarized in Table 3. Income and semen cost are discounted to the year of first calving by a discount rate of 10%. The sum of the coefficients of discounted income, from Table 3, is 3.76. If feed costs are .4 of gross
TABLE 2. Means, standard deviations, minimum and maximum Predicted Differences (PD), Repeatabilities, and semen prices of 285 Holstein sires.
SD PD milk (kg) PD fat (kg) PD $ Repeatability (%) Semen price ($)
411.48 9.98 81.85 69.37 8.25
Journal of Dairy Science Vol. 65, No. 3, 1982
151.15 6.55 33.89 20.29 19.16
Minimum value
Maximum value
10.89 -7.71 --20.34 21.00 2.00
980.23 29.94 198.97 99.00 250.00
PORTFOLIO THEORY AND SIRE-SELECTION
407
TABLE 3. Estimation of the expected income of a sire transmitted to a daughter and her offspring. Year
Actual income
Discounted income
-3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 11
-6.P a 0 0 .81 PD$ b .89 PD$ .96 PD$ .5 (.81) PD$ .5 (.89) PD$ .5 (.96) PD$ .25 (.81)PD$ .25 (.89) PD$ .25 (.96) PD$ .125 (.81) PD$ .125 (.89) PD$ .125 (.96) PD$
-7.99P 0 0 .81 PD$~ .81 PD$ t .79 PD$ .30 PD$ .30 PD$ I .30 PD$ .11 PD$~ .11 PD$ I .11 PD$ .04 PD$ .04 PD$ I .04 PD$
production of daughter 1st offspring generation 2 nd offspring generation 3rd offspring generation
ap = price for one unit of semen. bpD$ = predicted difference dollars.
i n c o m e , t h e n t h e c o e f f i c i e n t o f PD$ is 2.26. E x p e c t e d i n c o m e , E, f o r a sire is, t h e r e f o r e ,
a i D S is t h e p o p u l a t i o n v a r i a n c e o f P D $ obt a i n e d as
E = 2.26 P D $ -- 7.99 P
OID s = ( . 1 3 0 6 ) 20"ID M + (2.82) 2 a i D F
[6]
+ 2(.1306)(2.82)rpDM,PDFOPDMOPD F. [8] w h e r e P is price for a u n i t fo semen. V a r i a n c e o f i n c o m e , V, f o r a sire was c a l c u l a t e d as V = ( 2 . 2 6 ) 2 ( l - - R ) OID $
[71
w h e r e R is r e p e a t a b i l i t y o f t h e sire's p r o o f , a n d
O~D M a n d a i D E are t h e p o p u l a t i o n variances of P D M a n d P D F , a n d rpDM,PD F is t h e correlat i o n b e t w e e n t h e t w o PD's. E s t i m a t e s for OIDM, OIDF, a n d rpDM,PD F were o b t a i n e d f r o m 1035 bulls s t u d i e d b y W. E. V i n s o n (Virginia P o l y t e c h n i c I n s t i t u t e a n d S t a t e University, personal c o m m u n i c a t i o n , 1979), so
16
aiD S = (.1306) 2 (253) 2 + (2.82) 2 (8.56) 2 + 2(.1306)(2.82)(.787)(253)(8.56) = 2930 Sires were a s s u m e d t o b e u n r e l a t e d . To obtain the optimum solution, a utility f u n c t i o n e s t i m a t e d f r o m d a t a d e s c r i b e d in (3) was used.
15 tJ
O:
d
:
U = E -- . 0 2 V 7 ~SI0
i50
200
250
EXPECTED
INCOME
3100 ($]
31S0
½00
Figure 2. Expected income vs. variance (E-V) frontier for 285 Holstein sires. The numbers of the points on the E-V frontier correspond with the numbers in Table 4.
[9]
w h e r e U is utility, E is e x p e c t e d i n c o m e , a n d V is variance of i n c o m e . T h e c o n s t r a i n t s i n t r o d u c e d i n t o t h e QP m o d e l were 285 xi = 1 and x i > 0 i=l for i = 1. . . . .
285,
[10]
Journal of Dairy Science Vol. 65, No. 3, 1982
408
SCHNEEBERGER ET AL.
TABLE 4. Expected income vs. variance (E-V) frontier for 285 Holstein sires. Point on E-V frontier in Figure 1
No. of sires in solution
Expected income ($)
Variance of income ($2)
Utility U=E -- .02V
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
274 263 236 191 161 128 104 82 67 56 44 34 24 13 3 1
156.31 167.77 181.20 195.94 205.93 217.85 228.05 236.43 242.54 251.22 258.42 267.20 282.68 306.56 349.18 372.05
5.57 6.42 7.92 10.64 13.53 18.45 24.52 31.39 38.03 50.48 64.09 86.52 155.22 389.87 1332.83 3290.30
156.19 167.64 181.04 195.73 205.66 217.48 227.56 235.81 241.78 250.21 257.14 265.47 279.58 298.76 322.53 306.24
where x i is the proportion of cows bred to the iTM sire. The first constraint means that all cows in the herd are to be bred. R ESU LTS A N D D ISCUSSlON
The E-V frontier is plotted in Figure 2, and solutions at the points marked on the E-V frontier are listed in Table 4. Numbers of sires in the solutions decrease with increasing expected income. The sires with the smallest income drop out of the solution as expected income increases. The point at the upper end of the E-V frontier consists only of the top-income bull. Variance of the income increases with increasing expected income. In the stock market, activities with high expected incomes often are associated with high risks or variances of income. Smaller variance results from choosing activities with smaller expected income and from diversification, i.e., from choosing a large number of activities. With dairy sires, it is not necessarily true that high-income sires have large variances of income (i.e., small repeatabilities) and that low income sires have small variances (i.e., large repeatabilities). A reduction of variance (and, thus, risk) results from diversification by choosing a large number of bulls. This means that poorer bulls need to be included, thus, expected income decreases. The optimum solution was found algebraiJournal of Dairy Science Vol. 65, No. 3, 1982
cally (1) for the utility function U = E - - . 0 2 V and is in Table 5. The three bulls with the highest expected incomes are in the optimum solution. These three bulls have high expected income because of high PD$ and low semen price. The optimum solution is close to the upper end of the E-V frontier. This is because of the small negative weight for risk or variance, compared with the weight for expected income. But the income-maximizing solution of only the topincome bull was not the optimum solution because the utility function, U, includes a negative weight for variance of income which encourages diversification. A decision maker with a more risk-averse behavior (i.e., with a larger negative weight for variance of income than used in U) would obtain an optimum solution with smaller expected income and smaller variance of income; he would diversify and use more bulls. A less riskaverse decision maker would choose a solution with larger expected income and larger variance, and would use fewer bulls. A practical applicaton of portfolio analysis to dairy sire selection can be obtained by computing the E-V frontier using all sires that are available to dairymen. Dairymen then would select individual optimum solutions among the solutions on the E-V frontier according to their individual utility functions. Additional constraints can be incorporated easily into the QP
PORTFOLIO THEORY AND SIRE SELECTION
409
TABLE 5. Optimum solution of the quadratic programming problem with 285 Holstein sires for U = E - .02V.
Proportion of cows bred
E ($)
V ($2)
.61 .36 .03 Weighted mean
372 330 297 355
3290 2692 3889 1572
PD$
PD milk (kg)
PD fat (kg)
R (%)
Semen price ($)
189 164 149 179
980 756 681 890
22 23 21 22
78 82 74 79
7 5 5 6
U = utility. E = expected income. V = variance of income. PD -- predicted difference. R = repeatability.
p r o b l e m , e.g., the m i n i m u m p r o p o r t i o n o f c o w s to be b r e d to y o u n g sires, t h e m a x i m u m p r o p o r t i o n t o be b r e d to a particular sire, and o t h e r c o n s t r a i n t s reflecting t h e d a i r y m e n ' s personal p r e f e r e n c e s . Also, if sires are related, covariances b e t w e e n sires can be c o n s i d e r e d in t h e Q m a t r i x o f t h e QP p r o b l e m [ 1 ] .
2
3
4 REFERENCES
1 Anderson, J. R., J. L. Dillon, and T. B. Hardaker.
5
1977. Agricultural decision analysis. 1st ed. Iowa State University Press, Ames. Pfaffenberger, R. C., and D. A. Walker. 1976. Mathematiczl programming for economics and business. Iowa State University Press, Ames. Schneeberger, M., A. E. Freeman, and M. D. Boehlje. 1981. Estimation of a utility function from semen purchases from Holstein sires. J. Dairy Sci. 64:1713. Sharpe, W. F. 1970. Portfolio theory and capital markets. McGraw-Hill, New York. Wolfe, P. 1959. The simplex method for quadratic programming. Econometrica 27:382.
Journal of Dairy Science Vol. 65, No. 3, 1982