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Unformatted text preview: Dellneating Efficient Portfolios In Chapter 4 we examined the return and risk characteristics of individaai securities and
began to study the attribates of combinations or portfolios of securities. In this chapter we
look at the risk and return characteristics of combinations of securities in more detail. We
start off with a reexamination of the attributes of combinations of two rislcy assets. In doing
so, we emphasize a geometric interpretation of asset combinations. it is a short step from
the analysis of the combination of two or more risky assets to the analysis of combinations
of all possible risky assets. After making this step, we can delineate that subset of portfow
lios that wilt be preferred by all investors who exhibit risk avoidance and who prefer more
return to less.l This set is usually called the efﬁcient set or efﬁcient frontier. Its shape wiil
differ according to the assumptions that are made with respect to the ability of the investor
to sell securities short as weli as his ability to lend and borrow funds.2 Alternative assump
tions about short sales and lending and borrowing are examined. COMBINATIONS OF TWO RISKY ASSETS REVISITED:
SHORT SALES NOT AiLOWED in Chapter 4 we began the analysis of combinations of risky assets, In this chapter we con~
tinue it. Previously, we treated the two assets as if they were individual assets, but nothing in the analysis so constrains them. In fact. when we talk about assets, we could equally Well be talking about portfolios of risky assets.
Recall from Chapter 4 that the expected return on a portfolio of two assets is given by §p3XA§A+XBE3 (5D
where X A is the fraction of the portfoiio heid in asset A
X3 is the fraction of the portfolio held in aSset B ‘In this chapter and most of those that follow, we assume that mean variance is the relevant space for portfolio
analysts. See Chapter 11 for an examination of other portfolio models. 2Short selling is deﬁned at a later point in this chapter. 68 4 1 :13 CHAWER 5 {DEUNEATENG EFFICFENT PCR’TFOLIOS 69 it}; is the expected return on the portfolio
EA is the expected return on the asset A
E3 is the expected return on the asset B In addition, since we require the investor to be fully invested, the fraction she invests in
A plus the fraction she invests in B must equal one, or XA+X3=1 We can rewrite this expression as XBEIMXA {52) Substituting Equation (5.2) into Equation (5.1), we can express the expected return on a
ponfoiio of two assets as E}: 3 XAR‘A +{1WXA}§3 Notice that the expected return on the portfolio is a simple—weighted average of the
expected returns on the individual secerities, and that the weights add to one. The same is
not necessarily true of the risk (standard deviation of the return} of the portfoiio. In
Chapter 4 the standard deviation of the return on the portfolio was shown to be equal to 0? = (X362, +X§o§ + QXAXBGABWZ Up is the standard deviation of the return on the portfolio
0‘,‘ is the variance of the return on security A
03 is the variance of the return on security B 0,43 is the covariance between the returns on security A and security B If we substitute Equation (5.2) into this expression, we obtain GP=[Xﬁ0i+(l—XA}20'%3+2XA{1—XA)GAB}U2 (5.3) ' Recaiiing that (FAB 3 (34303463 where pl, B is the correlation coefﬁcient between securi—
ties A and B, then Equation (5.3} becomes ”2
0,, :[Xgog +(lmXA)26§+2XA(1WXA)pABvoB] (5.4) The standard deviation of the portfolio is not, in general, a simpleweighted average of the
standard deviation of each security. Crossproduct terms are invoived and the weights do
not, 1n general, add to one. In order to learn more about this relationship, we now study
some speciﬁc cases involving different degrees of co—rnovernent between securities. We know that a correlation coefﬁcient has maximum value of +1 and minimum value of 1. A valae of +3 means that two securities will always move in perfect unison, while a value of —.1 means that their movements are exactly opposite to each other. We start with
an examination of these extreme cases; then we tam to an examination of some interme diate values for the correlation coefﬁcients. As an aid in interpreting results, we examine . a Speciﬁc example as well as general expressions for risk and return. For the example, we consider two stocks: a large manufacturer of automobiles (“Colonel Motors") and an electric 70 PART 2 Pourrouo mums utility company operating in a large eastern city (“Separated Edison”). Assume the stocks
have the following characteristics: Expected Return Colonel Motors (C) 14% 6%
Separated Edison (S) 8% 3% As you might suspect, the car manufacturer has a bigger expected return and a bigger
risk than the electric utility. Case i—Perfect Positive Correlation in = +1} Let the subscript C stand for Coiouei Motors and the subscript S stand for Separated
Edison. If the correlation coefﬁcient is +1, then the equation for the risk on the portfolio,
Equation (5.4), simpliﬁes to $2
0pm[XE~G%+(imx€)26§+2XC(iXc}0'c03]f (5.5) Note that the term in square brackets has the form}? + ZXY + Y2 and, thus. can be writ”
ten as [chrc + (1  Kicks]?
Since the standard deviation of the portfoiio is equal to the positive square root of this
expression, we know that
6;: = XCCTC +0—qu
While the expected return on the portfolio is
it} = XCR} + (1 ._ Xcm Thus with the correlation coefficient equal to +i, both risk and return of the portfolio
are simply iinear combinations of the risk and return of each security. In footnote 3 we
Show that the form of these two equations means that ail combinations of two securities
that are perfectly correlated wiil lie on a straight line in risk and return space3 We now
illustrate that this is true for the stocks in our example. For the two stocks under study 3Solving for XC in the expression for standard deviation yields Substituting this into the expression for expected return yields Jew “H‘s §C+[1~6P_°5]§S UC"GS 6c“‘5s
m w Tim? §~§
RP=[RS— C Sos]+(mcum§]op
“6‘05 5c”°s which is the equation of a straight line connecting security C and security S in expected return standard devia
tion space. {HAPTER 5 GEUNEATING EFFSCIENT PORTFOUOS 7 1 Table 5.? The Expected Return and Standard beviation of a Portfolio of Colonel Motors and
Separated Ed" E? 8.0 9.2 $0.4 11
0‘? 3.0 3.6 4.2 4.5 4.8 5.4 6.0 — we — 0 —cs ——
RP =—————UP 5 RC {hump SJRS
UC'GS GC‘GS Table 5.1 presents the return on a portfoiio for seiected values of X5 and Figure 5.1 pres
ents a graph of this relationship. Note that the relationshio is a straight line. ”i‘he equation
of the straight iine could easily be derived as follows. Utilizing the equation presented
above for o,» to solve for XC yields Up __ X0“? Substituting this expression for XC into the equation for Ru and rearranging yields“ §p=2+20p a In the case of perfectly correlated assets, the return and risk on the portfoiio of the two
assets is a weighted average of the return and risk on the individuai assets. There is no
reduction in risk from purchasing both assets. This can be seen by examining Figure 5.1 ﬁal 14.0 8.0 Figure 5.1 Relationship between expected return and standard deviation when p = +1. r‘Au alternative way to derive this equation is to substitute the appropriate values for the two ﬁrms into the some
lion derived in footnote 3. This yieids iip=8+6(6""3) "..1.., .. Y2 mar 2 Posrsouo Miss and noting that combinations of the two assets lie along a straight line connecting the two assets. Nothing has been gained by diversifying rather than purchasing the individual
assets. Case 2—Peri‘ect Negative Correfation [p a m i .0} We now examine the other extreme: two assets that move perfectly together but' in exactly
opposite directions. in this case the standard deviation of the portfolio 15 [from Equation
(5. 4) with p w Mi .0] 112
cpm [213503; +(—1 X5? a§w2xch images] (5.6) Once again the equation for standard deviation caa be simpliﬁed. The term in the brackets
is equivalent to either of the following tWo expressions: [Xco'CW(i" Xc)0‘s]2 GI" [uchrc + (1 — Xc)01]2 (5.7}
Thus 6,: is either m Xco‘c "(lXC)O'S 01'
O'pﬁWXCdc'i(1~Xc)ﬁs (5.3) Since we took the square root to obtain an eXpressiori for Up and since the squaie root of
a negative number is imaginary, either of the above equations hoicls only when its right—
hand side' IS positive. A further examination shows the right— hand side of one equation is
simply ~l times the other Thus each equation is valid only when the right hand side is
positive. Since one is always positive when the other is negative {except when both equa—
tions equal zero), there is a unique solution for the return and risk of any combination of
securities C and S These equations are ve1y similar to the ones we obtained when we had
a correlation of +1. Each also plots as a straight line when op is plotted against XC. Thus,
one would suSpect that an examination of the return on the portfolio of two assets as a
function of the standard deviation would yield two straight lines, one for each expression
for Up. As we observe in a moment, this is, in fact, the case.5 The value of Up for Equation {5.7) or (5.8} is always smalier than the value of UP for the
case where p a +1 [Equation (5.5)] for all values of XC between 0 and 1. Thus the risk on
a portfolio of assets is always smaller when the correlation coefﬁcient is —1 than when it
is + 1. We can go one step farther. If two securities are perfectly negatively correlated (i.e.,
they move in exactly opposite directions), it should aiways be possible to ﬁnd some com
bination of these two securities that has zero risk. By setting either Equation {5.7) or {5.8)
equal to 0, we ﬁnd that a portfolio with X9 = G‘s/(0's + oc) wiil have zero risk. Since
or; > 0 and (r; + crc > 0'5, this implies that 0 < XC < 1 or that the zero risk portfolio will
always involve positive investment in both securities. 5This occurs for the same reason that the analysis for p — + i led to one straight line and the mathematics! proof
is analogous to that presented for the case of p w +1. CRAFTER 5 DEUNEATING EFFICJENT PORTFOLIOS 73 , . . _ 1
NOW iet us return to our example. Minimum Josh occurs when X0 : 3/(3 + 6) m 3.
Furthermore, for the case of perfect negative correlation, Ep=s+sxc .
0',» =6XC—3(1—Xc)' or
Up = *6Xc '9' 3(1— Xc) there are two equations relating op to KC. Only one is appropriate for any value of X9. The
appropriate equation to deﬁne Up for any vaiue of XC is that equation for which 0;» a 9. Note
that if of > 0 from one equation, then op < i} for the other. Table 52 presents the return on
the portfolio for selected values of XC and Figure 5.2 presents a graph of this relationship.6 Notice that a combination of the two secuiities exists that provides a portfoiio with zero
risk Employing the formula developed before for the composition of the zero1isk portfolio,
Xc shouid equal 3/(3 + 6) or 3. We can see this 18 correct from Figure 5. 2 or by substitub
ing 3 for Xc' 1n the equation for portfolio risk given previously. We have once again demon—
strated the most powerful result of diversification: the ability of combinations of securities
to reduce risk. In fact, it is not uncon'unon for combinations of two securities to have less
rislc than either of the assets in the combination. We have howiaexamined combinations of risky assets for perfect maritime and perfect
negative correlation. In Figure 5.3 we have plotted both of these relationships on the same
graph. From this graph we shoeid be able to see intuitively where portfolios of these two
stocks should lie if correlation coefﬁcients took on intermediate vaiues. From the expres—
sion for the standard deviation [Equation {5.4)}, we see that for any value for XC between
0 and l the lower the correlation, the lower the standard deviation of the portfolio. The
standard deviation reaches its lowest vaiue for p = 1 (curve SEC) and its highest value
for p = +1 (curve SAC). Therefore, these two curves should represent the limits within
which all portfoiios of these two securities must iie for intermediate values of the correlation
coefﬁcient. We would speculate that an intermediate correlation might produce a curve such as SOC in Figure 5.3. We demonstrate this'lbylreturning‘to'ourexample and con~~ '  ~  ~  Structing the relationship between risk and return for portfoiios 05 our two securities when
the correlation coefﬁcient is assumed to be 0 and +6.5. 'i’able 5.2 The Expected Return and Standard Deviation of a Portfoiio of Coionel Motors and
Separated Edison When 1) m 1 .. o (1.2 0.4 0.5 0.8 ' 1.0
RF 8.0 9.2 10.4 11.6 12.8 14.0 Up 3.0 1.2 0.6 2.4 4.2 6.0 s'i‘he equation for R» as a function of cry can be obtained by soiving the expression relating Up and Xe for Xe and
using this to eiiminate XC in the expression for RF. This yields Up+3
6+3 Epw8+s[ }w10+§~cp DI “3]a10w53159 apesmft‘ 74 FART 2 PORTFOUO ANALYSIS CHAPTER 5 DEUNEA‘ﬂNG EFFfClENT PORTFOLEOS 73 .3 Table 5.3 The Expected Return and Standard Deviation for a Portfolio of Colonel Motors and R — O . v.
P __.A._. _..
I 0 0.2 0.4 0.6 0.3 1.0
R 8.0 9.2 10.4 .11.6 12.8 14.0
1° .
" 5,, 3.00 2.68 3.00 3.79 4.84 5.0
14.0 C :: ,,.,_,_ . _
to 0 For our exampie this yields
I in.
Z 2
a. =[(6)2 XE. +(3) (l—XC) ]
8.6 S : . up = [45% 91st + 9]1le Tabie 5.3 presents the returns and standard deviation on the portfolio of Colonel Motors
end Separated Edison for seiected veiues of X6. . ‘ ‘ .
A graphical presentation of the risic and return on these portfolios IS shown in Figure S4.
. There is one point on this figure that is worth speciai attention: the portfolio that'has mm:—
3‘6 50 5p mum risk. This portfolio can be found in general by looking at the equation for risk: Figure 5.2 Rciationship between expected return and standard deviation when p = ml. ”one 2 [XEG'E +(l chf 0'? +2XciIWXC)UCOSpCS] To ﬁnd the value of X5 that minimizes this equation, we take the derivativeof it with
respect to XC. set the derivative equal to zero, and solve for XC. The denvauve is 85!, [1)[2XCdé ' 20E +2Xco‘g + Zﬁco'spcs ~4XCO'CO'SPCS]
m :: w WWW—W
3X5 2 [thfé +(1—XC)ZO'§ +2XC(1~XC)OCGSPCS] Setting this equal to zero and solving for XC yields RP
C 14.0 Figure 5.3 Relationship between expected return and standard deviation for various correlaton 8 0
coefﬁcients. 5 Case 3~No Relationship between Returns on the Assets [p = O}
The expression for return on the portfolio remains unchanged; but noting that the covari—
ance term drops out, the expression for standard deviation becomes 3.0 5.0 G 0p = [XEOg +0 _ X6} 63] Figure 5.4 Relationship between expected return and standard devration when p = 0. 76 PART 2 PORTFOLiO MALYSIS 2
U "(5 U
XC— S C SPCS ""' W 5.9
G%+G§—ZGCGSPCS ( ) in the present case (pa; = 0), this reduces to
2
o
X C m "”41...
2 2
oc + os Continuing with the previous exampie, the value of XC that minimizes risk is 9 i
:mmOQG
91—36 5 This is the minimum risk portfolio that was shown in ﬁgure 5.4. XCw: Case 4mlntermediate Risk to == 0.5} The correlation between any two actual stocks is almost always greater than 0 and con—
siderably less than I. To show a more typical relationship between risk and return for two
stocks, we have chosen to examine the relationship when p m +0.5. The equation for the risk of portfolios composed of Colonel Motors and Separated
Edison when the correlation is 0.5 is a. = [(6)2 x3 +(3}2(1— m2 +2Xc(l .. avatar29]” 0',» = (273:3 + 9)”2
Table 5.4 presents the returns and risks on alternative portfolios of our two stocks when
the correlation between them is 0.5. This riskreturn relationship is plotted in Figure 5.5 along with the risk—return relation
ships for other intermediate vaiues of the correlation coefﬁcient. Notice that in this exam
ple if p = 0.5, then the minimum risk is obtained at a value och = 0 or where the investor
has placed 100% of his funds in Separated Edison. This point could have been derived ana—
lytically §rom Equation (5.9). Employing this equation yields 9 w13(05) X“ ﬁ 9 + 36 u 2(ts){o.s) = In this example (i.e., pa, = 0.5), there is no combination of the two securities that is less
risky than the least risky asset by itself. though combinations are still less risky than they
ware in the case of perfect positive correlation. The particular value of the correlation coef—
ﬁcient for which no combination of two securities is 16:33 risky than the least risky security
depends on the characteristics of the assets in Question. Speciﬁcally, for all assets there is Table 5.4 The Expected Return and Standard Deviation of a Portfolio of Coionel Motors and
Separated Edison When p = 9.5 Pic 0 0.2 0.4 0.6 as 1.0 is 3.0 9.2 10.4 11.6 12.8 14.0
a,» 3.00 3.17 3.65 4.33 5.13 6.00 cmPTER 5 DELINEATING EFFiCiENT PORTFOLIOS 77 37°: 14.0 8.0 Figure 5.5 Relationship between expected return and standard deviation of return for various
correlation coefﬁcients. some value of pISuch that the risk on the portfolio can no longer be made less than the risk
of the least risky asset in the portfolio.7 We have developed some insights into combinations of two securities or portfolios from
the anaiysis performed to this point. First, we have noted that the lower (cioscr to l.0)
the correlation coefﬁcient between assets, all other attributes held constant, the higher the
payoff from diversification. Second, we have seen that combinations of two assets can
never have more risk than that found on a straight iinc connecting the two assets in
expected return standard deviation space. Finaliy, we have produced a simple expression
for ﬁnding the minimum variance portfolio when two assets are combined in a portfolio.
We can use this to gain more insight into the shape of the curve along which all possible
combinations of assets must lie in expected return standard deviation space. This curve,
which is called the portfolio possibilities curve, is the subject of the next section. THE SHAPE OF THE PORTFOLIO POSSIBILITEES CURVE Reexamine the earlier ﬁgures in this chapter and note that the portion of the portfolio pos—
sibility curve that lies above the minimum variance portfolio is concave while that which
lies below the minimum variance portfolio is convert.3 This is not due to the peculiarities of
the examples we have chosen but rather is a general characteristic of all portfolio problems. 7The value of the correlation coefﬁcient where this occurs is easy to determine. Equation (5.9) is the expression
for the fraction of the portfolio to be held in KC to minimize risk. Assume X5 is the least risky asset. When XC
equals zero in Equation (5.9), that means that i00% of the funds are invested in the least risky asset (in, X3
equals 1) to obtain the least risky portfolio. Setting X9 equal to zero in Equation (5.9) and solving for p55 gives
9.35 = urging. So when sic5 is equal to osfoc, Xc will equal zero, and the least risky “combination" assets will
be 100% invested in the least risky asset alone. If p55 is greater than oslo'c, then the least risky combination
involves short soiling C. 3A concave curve is one where a straight line connecting any two points on the curve lies entirely under the curve.
Ifa curve is convex, a straight line connecting any two points lies totally above the curve. The only exception to
this is that a straight line is both convex and concave and so can be referred to as either. 78
mar 2 PORTFOLIO ANALYSIS CHAPTER 5 CELINE/MING EFFICIENT PORTFOLIOS 79 Figure 5.? Various possibie relationships between expected return and standard dcvranon of
return when the minimum variance portfolio is combined with portfolio S. ave Now that we understand the risk—return properties of combinations of two assets, we are
ammo ; in a position to study the attributes of combinations of all risky assets. ' {ﬂue shape depicted in 5.65 cannot be The Efficient Frontier with No Short Sales in theory we could plot...
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