PRINCIPLES BOX: DEFINITION OF DOMINANCE

A dominant asset has: (1) the lowest risk at its level of return, or (2) the highest expected return in its risk class.

3.2.1 Numerical Example

Assume an investor is trying to select one security from among five securities. These securities and their estimated return and risk statistics are listed in Table 3.1.

The five securities are compared in the two-dimensional Figure 3.5 with expected return on the vertical axis and risk on the horizontal axis. All investors that are averse to accepting risk would agree that GA dominates HT, because they both offer the same expected return but GA is less risky. And all investors that prefer higher returns over lower returns would agree that FT is dominated by AT, because they are both in the same risk class (that is,σ =4 percent) but AT offers a higher expected return. Thus, FT and HT can be eliminated from consideration: They would not make good individual investments.

At first glance it appears that the number of desirable choices has been narrowed from five to three. The truth is more complex. Portfolios of the three dominant securities can be combined to create an infinite number of choices that are located approximately along the liner_f Q (or to the right of it—depending on the correlation coefficients between the securities⁵). For example, a portfolio composed of 50 percent AT and 50 percent GA has an expected return of 7.5 percent and a standard deviation

TABLE 3.1 Investment Candidates

Name of Security Expected return E(r) (%)

Risk σ (%)

American Telephone (AT) 7 4

General Auto (GA) 8 5

Yellow Tractors (YT) 15 15

Fine Tires (FT) 3 4

Hot Tires (HT) 8 12

0 0.05 0.10 0.15 0.20s 0.05

0.10 0.15 0.20

E(r)

Q

0.04 = r_f

GA HT

AT

FT

YT

FIGURE 3.5 Investment Opportunities in Risk-return Space

of 4.5 percent, assuming AT and GA are perfectly positively correlated,ρAT,GA= +1.

The expected return from this portfolio is calculated as follows:

E r_p

= 2

i=1

w_iE r_i

=w_ATE R_AT

+w_GAE R_GA

(2.11)

=1/2(7%)+1/2(8%)=7.5 percent.

The standard deviation of returns from a two-asset portfolio is calculated with equation (2.14).

σp²=w²_ATσAT² +w²_GAσGA² +2w_ATw_GAσAT,GA (2.14)

= (1/2)²(4%)²+(1/2)²(5%)⁵+ 2(1/2) (1/2) (+1) (4%) (5%)

= 0.002025 σp=√

0.002025=0.045=4.5 percent.

Plotting this portfolio in Figure 3.5 produces a point halfway between AT and GA. If pointr_fdenotes investing in a risk-free asset at 4 percent return (for example, in a short-term FDIC-insured bank deposit), the points between pointsr_f and AT can be interpreted to be portfolios with varying proportions of bank deposit and AT shares. By borrowing at interest rater_f (that is, by using leverage) and investing in AT or GA, the investor can create points (investment opportunities) on the liner_f Q that lie to the right and higher than GA.

By eliminating two dominated securities, the choice has been limited to points AT, GA, YT, and the infinite number of diversified portfolios containing various

combinations of different securities, which we assume, to keep things simple, lie along the liner_f Q in Figure 3.5. Exactly which point alongr_f Q an investor selects depends on the investor’s personal preferences in the trade-off between risk and return.

3.2.2 Indifference Curves

Indifference curves can be used to represent investors’ preferred trade-off between risk,σ, and return,E(r). Indifference curves are drawn so an investor’s satisfaction is the same throughout their length; they are utility isoquants. If we assume that investors dislike (get disutility from) risk and like (receive positive utility from) positive expected returns, the indifference curves that result will be positively sloped.

The slope depends on the investor’s particular preferences for a risk-return trade- off. The indifference map of a timid, risk-averse investor is shown in Figure 3.6.

Figure 3.7 depicts a more aggressive risk-averse investor who will accept more risk to get a higher return. Both the timid and aggressive investors dislike risk. However, the timid investor in Figure 3.6 dislikes risk more than the aggressive investor in Figure 3.7.

The higher-numbered utility isoquants represent higher levels of satisfaction (happiness) for the investor. These curves grow more vertical as they rise, reflecting a diminishing willingness to assume risk.

s E(r) U₄ U₃ U₂ U₁

FIGURE 3.6 A Timid

Risk-averter’s Indifference Map

s E(r)

U₁ U₂ U₃ U₄

FIGURE 3.7 A Slightly More Aggressive Risk-averter’s Indifference Map

s E(r)

0.20

0.15

0.1

0.05

0.05 0.10 0.15

P

0 0.20

S Q

M

U₁ U₂ U₃ U₄

r_f

FIGURE 3.8 A Risk-return Preference Ordering

Combining the liner_f Q from Figure 3.5 with the indifference map from either Figure 3.6 or 3.7 will make this choice analysis determinate, as shown in Figure 3.8.

The investor will seek the highest indifference curve tangent to the dominant opportunity locus,r_f Q, and thereby maximize his or her utility at point M.

Asset P in Figure 3.8 is a dominated asset, it will suffer from lack of demand, and its price will fall.⁶ The rate of return is the ratio of the investor’s income (from cash dividends plus capital gains or losses) divided by the purchase price. After a price fall, the denominator of the ratio will be reduced enough to increase the value of the rate of return. This means that the equilibrium return on P will move towardr_f Q after the temporary capital losses cease. Points above the liner_f Q (like S) represent undervalued assets whose prices will be bid up. The resulting higher equilibrium price (that is, higher denominator in rate-of-return ratio) will lower the expected rate of return on this previously undervalued asset, and it will be relocated on the liner_f Q. An equilibrium rate of return is a rate of return on the rayr_f Q.

Equilibrium returns have no tendency to change.

3.3 RISK-RETURN SPACE

Consider plotting theE(r) andσpairs denoted [σ,E(r)] that represent the individual investments in a two-dimensional graph in risk-return space. Next, connect the dots representing the individual investments with lines representing possible portfolios made from various combinations of the individual assets. Plotting such a figure would generate a set of investment opportunities that might take on the escalloped quarter-moon shape in Figure 3.9A. Within this opportunity set are the individual securities and the diversified portfolios made up from those securities. Every point on the edge and within the escalloped quarter-moon shape in Figure 3.9A is a feasible investment opportunity. The left side of the opportunity set between points E and F is called theefficient frontierof the opportunity set. This efficient frontier is comprised of an infinite number of efficient portfolios.

E(r)

E

F

s

FIGURE 3.9A Investment Opportunities Without Borrowing and Lending

CML F E(r)

E

M

Q

s r_f

FIGURE 3.9B Investment Opportunities With Borrowing and Lending