APPENDIX: RISK AVERSION AND INDIFFERENCE CURVES
5.5 THREE-ASSET PORTFOLIO ANALYSIS
Portfolio analysis requires the input statistics listed in Table 5.4 to analyze a three- security portfolio.
TABLE 5.4 Inputs Necessary for Three-Security Portfolio Analysis E
ri
σi2 σij
E(r1) σ12 σ12
E(r2) σ22 σ23
E(r3) σ32 σ31
TABLE 5.5 Return and Variance of Three Assets and Their Covariances Company i Investment
weight (wi)
Expected returnE(ri)
Variance σi2
Covariance σij
Excelon 1 w1 2.07% 48.20 σ12=7.82
IH 2 w2 0.21% 16.34 σ23=0.99
Citi-Guys 3 w3 1.01% 34.25 σ13= −2.65
The input data in Table 5.5 are obtained with monthly rates of return from Excelon, International Holdings (IH), and Citi-Guys during a representative sample period.
It is possible for an extremely careful draftsman (with a very sharp pencil) to find the set of efficient portfolios made from three securities with a three-dimensional drawing. However, it is difficult to create a three-dimensional drawing with as much precise detail as we enjoyed when we analyzed two assets in two dimensions in the preceding section.
To solve a three-asset portfolio problem in two-dimensional space, the following seven steps will be performed:
1. Convert the formulas for the portfolio’s expected return (the isomeans) and the portfolio’s risk (the isovariances) from three to two variable equations.
2. Find the MVP.
3. Graph the isomean lines.
4. Graph the isovariance ellipses.
5. Delineate the efficient set (that is, draw the critical line).
6. Calculate the expected return,E(rp), and risk,σp2, for the efficient portfolios.
7. Graph the efficient frontier in [σ,E(r)] space.
5.5.1 Solving for One Variable Implicitly
In order to analyze the efficient frontier graphically, it is necessary to convert the three dimensional problem associated with a three-security portfolio to a two-dimensional problem that is easier to graph. This conversion is accomplished by transforming the weight of the third security into an implicit solution from the other two securities.
That is,
w3=1−w1−w2 (5.4)
First, the conversion of E(rp) will be considered. This conversion is done by substituting equation (5.4) into equation (2.12) to obtain equation (5.5):
E(rp)= 3
i=1
wiE ri
=w1E(r1)+w2E(r2)+w3E(r3) (2.12)
=w1E(r1)+w2E(r2)+
1−w1−w2
E(r3) by substituting forw3
=
E(r1)−E(r3)
w1+
E(r2)−E(r3)
w2+E(r3) (5.5)
Equation (5.5) is a linear equation in two variables (w1 and w2). Substituting the three stocks’ numerical values in Table 5.5 for E(ri) in equation (5.5) yields equation (5.6).
E(rp)=(2.07−1.01)w1+(0.21−1.01)w2+1.01
=1.06w1−0.80w2+1.01 (5.6)
Equation (5.6) gives the expected return of the three-security portfolio in terms ofw1andw2explicitly andw3implicitly. It is a linear function in two variables,w1 andw2, and can be readily graphed in two dimensions.
The three-security portfolio variance formula is similarly converted to two vari- ables by substituting equation (5.4) into equation (2.18a) to obtain equation (5.7).
σp2= 3
i=1
3 j=1
wiwjσij
=w21σ11+w22σ22+w23σ33+2w1w2σ12+2w1w3σ13+2w2w3σ23
=w21σ11+w22σ22+
1−w1−w22
σ33+2w1w2σ12+2w1
1−w1−w2 σ13
+2w2
1−w1−w2 σ23
=w21σ11+w22σ22+
1+w21+w22−2w1−2w2+2w1w2
σ33+2w1w2σ12 +
2w1−2w21−2w1w2 σ13+
2w2−2w1w2−2w22 σ23
=w21σ11+w22σ22+σ33+w21σ33+w22σ33−2w1σ33−2w2σ33+2w1w2σ33
+2w1w2σ12+2w1σ13−2w21σ13−2w1w2σ13
+2w2σ23−2w1w2σ23−2w22σ23
=
σ11+σ33−2σ13 w21+
2σ33+2σ12−2σ13−2σ23 w1w2 +
σ22+σ33−2σ23
w22+
−2σ33+2σ13
w1 +
−2σ33+2σ23
w2+σ33 (5.7)
Equation (5.7) is a second-degree equation in two variables,w1andw2. Recall that such equations have the following general quadratic form.3
Ax2+Bxy+Cy2+Dx+Ey+F=0 (5.8) Inserting the variances and covariances from Table 5.5 into equation (5.8) yields equation (5.9).
σp2=[48.20+34.25−2(−2.65)]w21
+[2(34.25)+2(7.82)−2(−2.65)−2(0.99)]w1w2
+[16.34+34.25−2(0.99)]w22+[−2(34.25)+2(−2.65)]w1 +[−2(34.25)+2(0.99)]w2+34.25
=87.75w21+87.46w1w2+48.61w22−73.80w1−66.52w2+34.25 (5.9)
To find the weights for three securities that minimize the portfolio variance in equation (5.7), we try all possible combinations of w1 and w2. Table 5.6 provides variances for various sets of the weights. By changingw1 andw2 with an increment of 0.1, we compute the portfolio variance with equation (5.9) that equals equation (5.7) with substituted values for the relevant variances and covariances.
With w1 =0.1, w2 =0.60, and w3 =0.3, we can find the MVP is σp2 =10.58.
TABLE 5.6 Three-Security Portfolio Variances for Various Sets of Weights
w1 w2 w3
(=1−w1−w2)
σp2
0.0 1.0 0.0 16.34
0.0 0.9 0.1 13.76
0.0 0.8 0.2 12.14
0.0 0.7 0.3 11.50
0.0 0.6 0.4 11.84
0.0 0.5 0.5 13.14
0.0 0.4 0.6 15.42
0.0 0.3 0.7 18.67
0.0 0.2 0.8 22.89
0.0 0.1 0.9 28.08
0.0 0.0 1.0 34.25
... ... ... ...
0.1 1.0 −0.1 18.58
0.1 0.9 0.0 15.13
0.1 0.8 0.1 12.64
0.1 0.7 0.2 11.12
0.1 0.6 0.3 10.58←MVP
0.1 0.5 0.4 11.01
0.1 0.4 0.5 12.42
0.1 0.3 0.6 14.79
0.1 0.2 0.7 18.14
0.1 0.1 0.8 22.46
0.1 0.0 0.9 27.75
... ... ... ...
... ... ... ...
1.0 1.0 −1.0 117.75
1.0 0.9 −0.9 106.42
1.0 0.8 −0.8 96.06
1.0 0.7 −0.7 86.68
1.0 0.6 −0.6 78.26
1.0 0.5 −0.5 70.82
1.0 0.4 −0.4 64.35
1.0 0.3 −0.3 58.86
1.0 0.2 −0.2 54.33
1.0 0.1 −0.1 50.78
1.0 0.0 0.0 48.20
Of course, these answers are not accurate, because we consider only an increment of 0.1. If we consider a finer increment, such as 0.001, more accurate answers can be obtained. In this case, however, many more combinations of the weights will be considered.4 In fact, the accurate answers for the weights arew1=0.1442, w2=0.5545, andw3 =0.3013, and the minimum variance isσmvp2 =10.49.
5.5.2 Isomean Lines
The portfolio analysis graphing may begin after the formulas for the variance and expected return for the portfolio are reduced to two variables and the MVP weights are discovered. The isomean lines are graphed first in this example; but, just as easily, we could have begun with the isovariance ellipses.
After arbitrarily selecting a few values ofE(rp) in the neighborhood of theE(ri)’s of the securities in the portfolio, the isomean lines may be determined. By selecting four arbitrary values (0.4, 0.8, 1.2, and 1.6 percent per month) ofE(rp) and using equation (5.6), the formulas for four isomean lines are derived:
E(rp)=0.4%=1.06w1−0.80w2+1.01 E(rp)=0.8%=1.06w1−0.80w2+1.01 E(rp)=1.2%=1.06w1−0.80w2+1.01 E(rp)=1.6%=1.06w1−0.80w2+1.01
The easiest way to graph these four linear equations in a Cartesian plane, as shown in Figures 5.3 and 5.4, is to set one weight equal to zero and then solve the equation for the other weight. Because the isomean lines intersect thew1 axis when w2 is zero, and vice versa, this process will yield points on the two axes.
Connecting these points with a line yields the isomean lines. For example, the 0.5 percent isomean line must havew1= −0.5755 whenw2is set equal to zero:
0.4=1.06w1−0.80(0.0)+1.01⇒w1= −0.5755 When the 0.5 percent isomean hasw1=0.0 andw2=0.7625
0.4=1.06(0.0)−0.80w2+1.01⇒w2=0.7625 The points in Table 5.7 are derived similarly.
TABLE 5.7 Isomeans Isomean w1-axis intercept
givenw2=0
w2-axis intercept givenw1=0
0.4 −0.5755 0.7625
0.8 −0.1981 0.2625
1.2 0.1792 −0.2375
1.6 0.5566 −0.7375
Isomean lines 1.0
0.5
–0.5
–1.0 –1.0
E(rp) = 0.4 E(rp) = 0.8
E(rp) = 1.2 E(rp) = 1.6
–0.5 0.5 1.0
w2
w1
FIGURE 5.6 Four Isomean Lines for a Three-security Portfolio
Plotting the four isomeans from Table 5.7 yields the four parallel straight lines in Figure 5.6.
There are an infinite number of isomean lines, but only a few have been graphed.
The primary characteristic of the isomean lines is that they are all parallel to each other. Knowledge of this characteristic provides a good check when graphing the isomean lines. Because the budget constraint (w1+w2+w3 =1) is already contained in the isomean lines in Figure 5.6, the expected return implied in any isomean line is attainable without violating the budget constraint.
5.5.3 Isovariance Ellipses
The next step of the graphical analysis is graphing the isovariances. Isovariances are ellipses with a common center, orientation, and egg shape.
Graphing the isovariances should ideally be preceded by finding the MVP. The MVP is the center point for all the isovariance ellipses, it represents the portfolio with the least (but not necessarily zero) variance. It is impossible to graph isovariance ellipses for variances less than the variance of the MVP.
To graph isovariances, it is necessary to solve equation (5.7) or (5.9) in terms of one of the variables (that is, weights), while treating the remaining variables as constants. Arbitrarily selectingw1 as the variable to be solved for, and treatingw2 as a constant, reduces equation (5.7) to a quadratic equation in one variable. The general form of a quadratic equation isax2+bx+c=0, where thexis a variable and the other symbols are any constant values. Solution of such second-order equations in one variable may be obtained with the well-known quadratic formula:
x=−b±√
b2−4ac 2a
Letx=w1in equations (5.7) and (5.9) and treatw2as a constant. Then, let a =all coefficients ofx2—that is, all coefficients ofw21 in equations (5.7) and
(5.9),
b =all coefficients of x—that is, all coefficients ofw1 in equations (5.7) and (5.9),
c=all values that are not coefficients ofx21orw1—that is, all constants, which includes thew2s and theσp2in equations (5.7) and (5.9).
For equation (5.7), set the entire expression equal to zero as follows:
0=
σ11+σ33−2σ13
w21+
2σ33+2σ12−2σ13−2σ23
w1w2 +
σ22+σ33−2σ23
w22+
−2σ33+2σ13
w1 +
−2σ33+2σ23
w2+σ33−σp2
Then the values ofa,b, andcare a=
σ11+σ33−2σ13
b=
2σ33+2σ12−2σ13−2σ23
w2+
−2σ33+2σ13
c=
σ22+σ33−2σ23
w22+
−2σ33+2σ23
w2+σ33−σp2
The value ofw1can be found by substituting these values ofa,b, andcinto the quadratic formula.
Following the procedure outlined above, equation (5.9) yields the following results:
σp2=87.75w21+87.46w1w2+48.61w22−73.80w1−66.52w2+34.25 0=87.75w21+87.46w1w2+48.61w22−73.80w1−66.52w2+34.25−σp2
Thus, a=87.75
b=87.46w2−73.80
c=48.61w22−66.52w2+34.25−σp2
Inserting these values ofa,b, andcinto the quadratic formula yields w1=−b±√
b2−4ac 2a
=
−
87.46w2−73.80
±
87.46w2−73.802
−4(87.75) 48.61w22−73.80w2+34.25−σp2
2(87.75) (5.10)
Equation (5.10) is the solution to equation (5.9) forw1 while treatingw2 as a constant. It is necessary to solve the formula to graph the isovariances. To obtain
points on the isovariance ellipse, some arbitrary values for σp2 andw2 are selected and equation (5.10) is then solved for two values ofw1. It is easiest to select a value forσp2 that is slightly larger than the variance of the MVP and select a value ofw2 at or near the MVP to get the first two values forw1. For example, forσp2=13.43 andw2=0.3, equation (5.10) yields the following two values forw1:
w1=
−[87.46(0.3)−73.80]±
[87.46(0.3)−73.80]2−4(87.75) 48.61(0.3)2−73.80(0.3)+34.25−15.29
2(87.75)
=0.388 and 0.154
The variance chosen in this example (σp2=13.43) is the variance of the portfolio whenE(rp)=1.20 percent. Of course, any value for σp2could have been chosen as long as it exceeded the variance of the MVP (that is, 10.49). In this way, we can find the value ofw1 for a given value ofw2 and the portfolio variance. Table 5.8 shows those values ofw1andw2. It is left as an exercise for the reader to verify the points on the isovariance ellipses listed in Table 5.8.
Beyond the given values ofw2, the solutions forw1 in equation (5.10) are not available, because the value of the equation inside the square root is negative.
The values ofw1andw2and the portfolio variances in Table 5.8 are graphed in w1, w2
space to obtain the isovariance ellipses in Figure 5.7.
In Figure 5.7,V1 is the isovariance ellipse with a variance ofσp2 =10.57. In a similar manner,V2,V3, andV4indicate the isovariance ellipses having the variances σp2 = 11.78, 13.43, and 20.36, respectively. The more outer ellipse indicates the greater variance (i.e.,V1<V2<V3<V4).
5.5.4 The Critical Line
The isomeans in Figure 5.6 and isovariances in Figure 5.7 are all graphed in the same
w1, w2
space in Figure 5.8. After the isomeans and isovariances are graphed, it is simple to determine the efficient set of portfolios. An efficient portfolio is defined as the portfolio with the maximum return for any given risk class. Because each isovariance traces out a risk class, the point where the highest-value isomean is just tangent to an isovariance is an efficient portfolio. OnV3isovariance ellipse, for example, pointOis an efficient portfolio. At this given level of the variance ofV3= 13.43, the portfolio at pointOearns the highest expected return of 1.2 percent. Note that pointOis the tangency point betweenV3and the isomean line ofE(rp)=1.2 percent. PointJis also on the same isovariance ellipseV3. However, the portfolio at pointJis dominated because it has a lower expected return than the portfolio at point O. The portfolio at pointO’ is also dominated (inefficient), because it has the same expected return as the portfolio at pointO, but has greater amount of variance of V3=13.43.
In the same vein, portfolios at pointsM,N, andPare efficient portfolios at the variance levels of 10.57, 11.78, and 20.36, respectively. The straight line starting from the MVP and connecting these points is the critical line. Thiscritical lineis the locus of points in (w1, w2) space representing the efficient set. In Figure 5.8, the set of efficient portfolios starts at the MVP and runs downward to the right through
TABLE 5.8 Isovariance Points in
w1, w2 Space
w2 Two values ofw1 Variance (σp2)
less n.a. n.a. n.a.
0.5 0.178 0.165 10.57
0.6 0.139 0.104 10.57
greater n.a. n.a. n.a.
less n.a. n.a. n.a.
0.4 0.307 0.135 11.78
0.5 0.289 0.054 11.78
0.6 0.240 0.003 11.78
0.7 0.163 −0.019 11.78
greater n.a. n.a. n.a.
less n.a. n.a. n.a.
0.3 0.388 0.154 13.43
0.4 0.383 0.059 13.43
0.5 0.352 −0.009 13.43
0.6 0.303 −0.066 13.43
0.7 0.236 −0.093 13.43
0.8 0.145 −0.101 13.43
greater n.a. n.a. n.a.
less n.a. n.a. n.a.
0.0 0.557 0.284 20.36
0.1 0.593 0.148 20.36
0.2 0.593 0.049 20.36
0.3 0.576 −0.033 20.36
0.4 0.546 −0.103 20.36
0.5 0.505 −0.163 20.36
0.6 0.456 −0.213 20.36
0.7 0.397 −0.254 20.36
0.8 0.329 −0.285 20.36
0.9 0.248 −0.304 20.36
1.0 0.150 −0.306 20.36
1.1 0.019 −0.275 20.36
greater n.a. n.a. n.a.
Note: n.a. indicates not available.
pointsM, N,O, andP. On the other hand, the dotted critical line starting at the MVP and running upward by connecting pointsL, K, J, and Iindicates inefficient (dominated, the least desirable) portfolios.
Once the critical line is graphed, the efficient frontier may be graphed with ease.
Reading weights (w1, w2) off the critical line at a few points (such as pointsM,N, O, andPin Figure 5.8), it is possible to calculate theE(r)andσ of portfolios that have the highest rate of return for their risk class. Table 5.9 shows these values.
The efficient frontier is found by plotting E(rp) and σp as done in Figure 5.9. The graphical method is only approximate because of problems in pencil drafting. This completes the graphical portfolio analysis.
w2
w1 MVP smvp2 = 10.49 V1: sp2= 10.57 V2: sp2= 11.78 V3: sp2= 13.43 V4: sp2= 20.36 V1
V2 V3
V4
FIGURE 5.7 Isovariance Ellipses
J I
V2 V4 V3 w2
w1 smvp2 = 10.49 V1: sp2= 10.57 V2: sp2= 11.78 V3: sp2= 13.43 V4: sp2= 20.36
P O
O′ MN
E(rp) = 0.4 E(rp) = 0.8
E(rp) = 1.2 E(rp) = 1.6
KL V1
FIGURE 5.8 Three-asset Graphical Solution
5.5.5 Inefficient Portfolios
Inexperienced portfolio analysts must take care not to draw the critical line in the wrong direction away from the MVP. Such a line would be the locus of points representing the set of most inefficient portfolios. The dotted line in Figure 5.8 is such a line; it is the locus of points representing the minimum return in each risk class; these portfolios are highly undesirable investments.
Earlier in this chapter the objective of portfolio analysis was said to be the deter- mination of the efficient set of portfolios. The efficient set is represented by the infinite number of portfolios whose weights lie along the critical line. The reader who under- stands this and the preceding chapters will recognize that portfolio analysis utilizes
TABLE 5.9 The Weights for the Portfolios on the Critical Line
Points w1 w2 w3 E(rp) σp2 σp
K −0.0051 0.7557 0.2494 0.40 11.78 3.43
M 0.1820 0.5036 0.3144 0.80 10.57 3.25
MVP 0.1445 0.5540 0.3015 0.72 10.49 3.24
O 0.3691 0.2515 0.3794 1.20 13.43 3.66
P 0.5562 −0.0006 0.4444 1.60 20.36 4.51
E (r)
0.8 1.2 1.6
0.4
3.5 4.0 4.5
M
K
I MVP
O
P
s
FIGURE 5.9 Three-asset Efficient Frontier
Markowitz diversification. In fact, it could be said that the objective of portfolio analysis is to maximize the benefits from Markowitz’s optimal diversification at each possible rate of return.