Chapter 3

Chapter 3

The Capital Asset Pricing

Model

3. 1 The Capital Market Line

This section will deal with a variation of the model of the previous section. In addition to the n risky assets, we will now suppose there is an additional asset with special properties . This asset will be risk free and as such will have a zero variance and a zero covariance with the remaining n risky assets. A good example of a risk free asset is Treasury bills or T-bills .

Let Xn+l denote the proportion of wealth invested in the risk free asset and let r denote its return. The expected return of this portfolio is

=

Its variance is

41

42 The Capital Asset Pricing Model Chap. 3

The covariance matrix for this n + 1 dimensional problem is

for which the last row and column contain all zeros corresponding to the risk free asset . This matrix is positive semidefinite, whereas E, the covariance matrix for the n risky assets, is assumed to be positive definite. The efficient portfolios are the optimal solutions for (2 .3) . For the case at hand, (2 .3) becomes

minimize : -t (tl , r) [ n+l ] + [ n+l ] 1 [ ;, subject to : l'x + Xn+l = 1

l [ n+l l } (3. 1 )

In partitioned matrix form, the optimality conditions for (3 . 1 ) are

(3.2)

and

l'x + Xn+l = 1 . (3.3) The second partition of (3 .2) gives the scalar multiplier u:

u = tr.

Using this, the first partition of (3 .2) becomes

Ex = t (l-l - rl ) . (3.4) Because E is positive definite, it is also nonsingular (see Exercise 1 . 7) . Thus, we can obtain the efficient risky assets as

(3 .5)

and the efficient risk free asset

(3 .6)

Sec. 3 . 1 The Capital Market Line 43

Using (3 .5) and (3 .6) we can calculate an efficient portfolio 's variance and expected return as follows:

tJ.L'"E-1 (JL - rl) + r - trl'"E- 1 (JL - rl) r + t [J.L'"E-1 (J.L - rl) - rl'"E- 1 (JL - rl) ] r + t [ (JL - rl) '"E- 1 (JL - rl) ] .

(3 .7)

(3 .8)

Note that since L, is positive definite, so too is L,-1 (see Exercise 1 . 7) and thus

(JL - rl)'"E-1 (JL - rl) > 0

provided JL =I= rl . With this restriction, it follows from (3 .7) and (3 .8) that a; and f.Lp are both strictly increasing functions of t .

Observe from (3 .6) that Xn+l (t) i s strictly decreasing i f and only if l'L,-1 (JL - rl) > 0; i .e . ,

l'L,-1 JL r <

l'L,- 1 [. (3 .9)

Because L,-1 is symmetric (Exercise 1 . 7) and the transpose of a scalar is just the scalar ,

and (3.9) becomes

(3 . 10)

From (2 .7) and (2 . 1 1 ) , the quantity on the right-hand side of this inequality is a0 , the expected return of the minimum variance portfolio . Thus we have shown that the allocation in the risk free asset will be a strictly decreasing function of t if and only if the risk free rate is strictly less than the expected return of the minimum variance portfolio . We will assume that r < a0 throughout the remainder of this section.


Because Xn+l (t) is a strictly decreasing function of t , as t is increased eventually Xn+l (t) is reduced to zero . This occurs when

(3 . 1 1 )

The corresponding portfolio of risky assets is

(3 .12)

Xm is called the market portfolio (hence the m subscript ) . It is that efficient portfolio for which there are zero holdings in the risk free asset . Using (3 .5) , (3 . 1 1 ) and (3 . 12 ) we can write the efficient risky assets as

t x(t) = -Xm tm

(3 .13)

Using (3 . 7) and (3 .8) we can obtain the equation of the efficient frontier for this model:

(3. 14)

In mean-standard deviation space, the efficient frontier is a line. It is called the Capital Market Line (CML) and is illustrated in Figure 3 . 1 . Investors move up and down the Capital Market Line according to their aversion to risk. According to (3 .5) and (3 .6) , for t = 0 all wealth is invested in the risk free asset and none is invested in the risky assets. As t increases from 0, the amount invested in the risk free asset is reduced whereas the holdings in the risky assets increase. All investors must lie on the CML. From (3 . 13) , the sole thing that distinguishes individual investors is the proportion of the market portfolio they hold.

The CML can be written in a perhaps more revealing way as follows. Recall from (3 .5) that part of the optimality conditions for (3 . 1 ) are

(3 .15)

Taking the inner product of both sides with (J..t - rl) gives

(3 .16)


At the market portfolio , the holdings in the risk free asset have been reduced to zero . Consequently,

(3 . 1 7)

Using (3 . 17) in (3 . 16 ) , taking square roots and rearranging gives

1 [JL: r r = [ (JL - rl) 'E- 1 (JL - rl) ] ! . (3 . 18) Now, taking the inner product of both sides of (3 . 15 ) with Exm gives

so that 1 am [tm (JLm - r)] 2 ,

or,

1 tl (3 . 19)

Substitution of (3 . 19) into (3 . 1 8) , rearranging and simplifying, gives

(3 .20)

Substituting the quantity in the left-hand side of (3 . 14) shows that the Capital Market Line can also be expressed as

(3 .21 )

Equation (3 .21 ) shows that the slope of the CML is

This is in agreement with direct observation in Figure 3 . 1 .

46

r

0

The Capital Asset Pricing Model

Capital Market Line

Figure 3 .1

f-- X m

Capital Market Line.

cr p

Chap. 3

Consider the following implication of this model. Suppose in an economy that everyone is mean-variance efficient and everyone "sees" the same covariance matrix L: and the same vector of expected returns JL. Let two typical investors be Mr. X and Ms. Y and suppose their t values are tx and ty , respectively. Note that tx and iy could be obtained from the amounts Mr. X and Ms. Y invest in the risk free asset by means of (3 .6) . Let Mr. X and Ms. Y's holdings in the risky assets be denoted by x and y , respectively. From (3 . 13) ,

and,

This implies that their holdings in risky assets are proportional; i .e . , x = (tx/ty )y . For example, suppose n = 4 and Mr. X's risky holdings are x = (0 . 1 , 0 . 1 1 , 0 . 12 , 0 . 13) ' . Suppose we knew Ms. Y's holdings in just the first risky asset were 0 .2 . But then her holdings in the remaining risky assets must be in the same proportion (namely 2) as Mr. X's. Thus we would know the remainder of her holdings: y = (0 .2 , 0 .22 , 0 .24, 0 .26)' .


We next turn to the question of what happens when t exceeds tm . By definition , if the Capital Market Line were to continue, holdings in the risk free asset would become negative. This would allow short selling the risk free asset , which we will not allow in this model . For all t tm , we shall impose the condition Xn+l (t) = 0 . Substituting Xn+l = 0 everywhere in (3 . 1 ) , the problem (3 . 1 ) reduces to (2 .3) , the n risky asset problem of the previous section for t tm Figure 3 .2 shows the Capital Market Line and the efficient frontier for the n risky asset problem superimposed, in mean-standard deviation space.

llp

r

0 112 0

Figure 3.2 Capital Market Line and Efficient Frontier for Risky Assets.

Implicit in the model problem (3 . 1 ) is the assumption that the risk free asset must be nonnegative. The key result for this problem is that the efficient frontier for it is composed of two pieces (see Figure 3.3) . The first piece is the Capital Market Line going from r to the market portfolio Xm The second piece is a part of the hyperbola (2 . 18) , the efficient frontier for the risky assets . The two pieces of the efficient frontier are shown as the solid line in Figure 3 .3 .

48

r

0

The Capital Asset Pricing Model Chap. 3

CML

13112 0

Efficient Frontier for combined problem

cr p

Figure 3.3 Two Pieces of the Efficient Frontier.

Figure 3 .3 is drawn in a special way, and not by accident . The mean and standard deviation at Xm on the CML is identical to that for the risky asset part of the frontier . This is always the case. Indeed, the two corresponding portfolios are both Xm . To see this, from (2 .8) and (3 . 1 1 ) we can calculate

Since the two portfolios are identical , their means are equal and so too are their standard deviations. Thus, the efficient frontier is continuous.


Not only is the efficient frontier continuous, it is also differentiable. One way to think of this is that in mean-standard deviation space, the CML is tangent to the efficient frontier for the risky assets. This is shown in Figure 3 .3 . From (3 . 14) , the slope of the CML is

( (f.L - rl) 'L:- 1 (f.L - rl) ) . (3 .22) To verify the differentiability assertion, we need only show that the slope of the efficient frontier for risky assets at Xm has this same value. A convenient way to do this is to use (2 . 14) , namely

( 2 ) 1 /Lp = ao + a1t and ap = f3o + 2f31t + f32t 2 Using the chain rule of calculus , /31 = 0 (2 . 15 ) and /32 = a1 (2 . 17) , we determine

aP t

When t = tm , (3 .7) with (3 . 12) imply 1

= [ (/L - rl) 'L:- 1 (/L - rl) ] 2 aP (l'L:- 1 (1-l - rl) )2 ' so that with (3 . 1 1 ) we have

df.Lp = ((f.L - rl) 'L:- 1 (1-l - rl) ) , dap

(3 .23)

(3 .24)

(3.25)

and this is identical to (3 .22) . Thus the slope of the CML is equal to the slope of the efficient frontier for risky assets at Xm and the total efficient frontier is indeed differentiable.

Example 3.1 Find the equation of the CML using (3 . 14) , tm and the market portfolio Xm for the problem of Example 2 . 1 and a risk free return of r = 1 .02. Verify the equation of the CML by recalculating it using (3 . 2 1 ) .


The data for Example 2 . 1 has n = 3 , J.l = ( 1 . 1 , 1 . 2 , 1 .3 ) ' and L: = diag ( l0-2 ' 5 X w-2 ' 7 X w-2 ) . The required quantities all depend on L;-1 (J.L - rl) so we calculate this first :

L:-1 (J.L - rl) = 100 [ i ] [ : ] = [ 3 . ] (3 .26) 0 0 0.28 4

Thus

(!' - r l ) 'E-1 (I' - rl) [ 0 .08 0 . 1 8 0. 28 ] [ 3 .: ] 2 .4080 and since v'2 .4080 = 1 .5518 , it follows from (3 . 14) that the equation of the CML is

/-lp = 1 .02 + 1 . 5518 fJp .

From (3 . 1 1 ) , we have 1

l'L:- 1 (J.L - rl)

From (3 .26) ,

so that

l'E-1 (1' - rl) [ 1 I I ] [ 3 .: l 15 .6 , tm = 0 .064 1 .

(3.27)

Finally, we have from (3 . 1 2) and (3 .26) that the market portfolio is Xm tmL:-1 (J.L - rl)

0 .0641 [ 3:6 ] [ : ] 0 .2564

Sec. 3 .2 The Security Market Line 51

We next calculate the equation of the CML using (3 . 2 1 ) as follows. Using Xm above, we calculate

J-tm = J-t1Xm = 1 . 1 744 , IT! = x:nL:xm = 0 .0099 ,

and

O"m = 0.0995, 1-tm - r 1 . 1 744 - 1 .02

0 .0995

From (3 .2 1 ) , the equation of the CML is

/-tp = 1 .02 + 1 .5518 O"p ,

in agreement with (3 .27) .

3 . 2 The Security Market Line

1 .5518 .

In this section we continue using the model problem Section 3 . 1 and repeat that model problem here for convenience:

minimize : -t(J-t' , r) [ n+l ] + [ n+l ] 1 [ ;, subject to : l'x + Xn+l = 1

l [ :.+, l } (3 .28)

The optimality conditions for (3 .28) were formulated in Section 3 . 1 as (3 .4) and (3 .3) . We repeat these here.

L:x t (J-t - rl ) , (3 .29)

l'x + Xn+l = 1 . (3 .30)

We now let t = tm . It follows that x = Xm , Xn+l = 0 and l'xm = 1 . Equation (3 .29) now becomes


or,

Taking the inner product of both sides of (3 .31 ) with Xm gives

But JlXm = J.tm , l'xm = 1 and x:Exm = o-! , so (3 .32) becomes

Upon rearrangement , this becomes

J.tm - r 1 =

Substituting (3 .33) into (3 .31 ) gives [J.tm - r ] J.t = rl + o- :E

xm .

(3 .31 )

(3 .32)

(3 .33)

(3 .34)

Now let ri denote the random variable whose value is the return from asset i , and E(ri ) = J.ti , i = 1 , 2, . . . , n. That is , J.ti is the expected value of ri . The i-th component of (3 .34) can be written

But

[o-il (xmh + D"i2 (xmh + + D"in (Xm)n] is just the covariance of ri with the market portfolio Xm which has return

and satisfies

Sec . 3 .2 The Security Market Line

Thus, (3 .35) can be rewritten as [/1m - r ] f.Li = r + a covh , rm) i = 1 , 2 , . . . , n. 53

(3 .36)

Equation (3 .36) is called the Security Market Line (SML) and shows that the expected returns of all assets must plot on a line with slope

and 11 intercept r, in (cov (ri , rm) , E(ri ) ) space.

The market portfolio also lies on the SML. See Exercise 3 .6 .

Example 3.2 For the problem of Example 3 . 1 , find the equation of the SML and numerically verify that f11 , 112 and 113 lie on it .

From the results of Example 3 . 1 , we have f.Lm = 1 . 1744 and am = 0 .0995 so that the slope of the SML is

J-lm - r 1 . 1744 - 1 . 02 a = (0 .0995) 2 = 15 .5999.

We can write the Security Market Line in vector format : [/1 - r ] p, = rl + : Xm , 1 .02 + 15 .5999Xm [ 1 .02 ]

1 .02 We can also write the SML in component form as

f.Li = r + 15 . 5999 x cov (ri , rm) ,

for i = 1 , 2 , . . . , n. It is straightforward to compute [ 0 .005 1 l Xm = 0 .01 1 5 .

0 .0179

(3 .37)


Inserting this last into the right-hand side of (3 .37) and simplifying gives in agreement with the left-hand side (namely J-t) and so the J-ti , i = 1 , 2 , 3 do indeed plot on the SML. See Figure 3 .4 . 0

ll;

Jl2 = 1 .2 Jl = 1. 174 m

r = 1 .02

(0,0.9)

0.0099

0.005 1 0.0 1 1 5 0.0 179

Figure 3.4 SML for Example 3 .2 .

3 . 3 Computer Programs

Cov(ri , r ,j

Figure 3 .5 shows the m-file Example3pl .m which is the main routine for plotting the CML for Example 3 . 1 . Lines 2-4 set up the data. Line 5 defines the lower , upper and increment on t for plotting the CML using (3 .7) and (3 .8) . Line 6 performs the validity check on the covariance matrix. Line 7 calls the function EFMVcoeff (Figure 2 .6) to obtain the coefficients of the efficient frontier for the risky assets. Lines 8 and 9 call CMLplot (Figure 3 .6) to plot the CML.

Sec. 3 .3 Computer Programs

1 %Examp l e 3 p 1 . m

2 mu = [ 1 . 1 1 . 2 1 . 3 ] '

a S i gma = [ 1 . e-2 0 0 ; 0 S . Oe-2 0 0 0 7 . 0 e-2 ] 4 r = 1 . 0 2 s t l ow=O . ; t h i gh= 1 . ; t i nc= 0 . 0 1 ;

6 che ckdat a ( S i gma , 1 . e- 6 ) ;

7 [ a 1pha O , a 1pha 1 , bet a O , bet a 2 , h O , h 1 ] = EFMVc oe f f ( mu , S i gma )

s CMLp 1 ot ( t 1 ow , t h i g h , t i n c , a lpha O , a lpha 1 , bet a O , . . . 9 b e t a 2 , mu , S i gma , r ) ;

Figure 3.5 Example3pl .m.

55

Figure 3 .6 shows the m-file CMLplot .m which plots the Capital Market Line. Its input parameters are the limits on t for plotting the CML, the coefficients of the efficient frontier and the problem data J.L, and r. Line 4 calls EFMSDplot (Efficient Frontier Mean-Standard Deviation plot) to plot the efficient frontier in meanstandard deviation space (see Figure 3 .8 ) . Line 5 keeps the current plot open for further additions. Line 8 calculates the slope of the CML, namely [ (J.L - rl) '- 1 (J.L - rl) ] 112 . Lines 9 and 10 compute tm and Xm , respectively. Lines 1 1-13 compute points on the CML and line 14 plots them. Lines 15 through 18 plot the extension of the CML using a dashed line. Lines 19 through 25 label some key points in the graph.

56 The Capital Asset Pricing Model

1 % CMLp l ot . m

2 funct i o n CMLp lot ( t l o w , t h i g h , t i n c , a lpha O , a lpha l , . . .

3 bet a O , bet a 2 , mu , S i gma , r )

Chap. 3

4 EFMSDp lot ( t l o w , t h i gh , t i n c , alphaO , a lpha 1 , bet a O , beta 2 ) ;

5 h o l d on 6 n = l e ngth ( mu ) ; e l l = o n e s ( 1 , n ) ' ;

1 coe f f = ( mu - r * e l l ) ' * S i gma A - 1 * ( mu - r * e l l )

8 coe f f s qrt = coe f f . A O . S

9 tm = 1 . / ( e l l ' * S i gma A - 1 * ( mu - r * e l l ) ) 10 xm = tm * S i gma A - 1 * ( mu - r * e l l )

u t = t l ow : t i n c . A 2 : tm ; 12 mup = r + t * c oe f f ;

1 3 s i gmap = t * coe f f s qrt ;

14 p l ot ( s i gmap , mup , ' k ' )

15 t = tm : t i n c . A 2 : 3 * tm; 16 mup = r + t * coe f f ;

1 1 s i gmap = t * coe f f s qrt ;

18 p l ot ( s i gmap , mup , ' k - . ' )

19 mup = r + tm * coe f f ; s i gmap = tm* coe f f s qrt ; 20 x l ab e l ( ' P o rt f o l i o St andard Deviat i o n \ s i gma_p ' ) 2 1 y l ab e l ( ' P o rt f o l i o Me an \mu_p ' ) 22 t ext ( s i gmap , mup , . . .

23 ' \ l e ft a r r ow { \ rm ma rket p o r fo l i o } , t = Lm ' ) 24 t ext ( s i gmap / 2 . , ( mup + r ) / 2 . , . . .

' \ l e f t a r r ow { \ rm Cap i t a l Market L i n e } ' ) 26 t ext ( 2 * s i gmap , r + 2 * t iD * C O e f f , . . .

27 ' \ l e ft a r row { \ rm Ext e n s i on o f C ap i t a l Ma rket L i n e } ' )

Figure 3.6 CMLplot .m.

Figure 3 . 7 shows EFMSDplot .m which plots the efficient frontier for the risky assets in mean-standard deviation space. Lines 4 through 6 compute points on the efficient frontier and its lower extension and plots them. Lines 9 through 13 construct the axes. Figure 3 .8 shows the final plot .

Sec. 3 .3 Computer Programs

1 % EFMSDp lot . m

2 funct ion EFMSDplot ( t l o w , t h i g h , t i n c , a lpha O , . . .

a l pha l , bet a 0 , b e t a 2 )

4 t = t l ow : t i nc : t h i g h ; 5 mup = a lphaO + t * a lpha l ; 6 mup low = alphaO - t * alpha l ; 1 st ddevp = ( bet a O + t . 2 * bet a 2 ) . o . s ; s p l ot ( s t ddevp , mup , ' k ' , s t ddevp , mup l o w , ' k-- ' ) 9 xmi n 0 . ;

10 xmax ( b e t a O + t h i gh . 2 * bet a 2 ) . o . S ;

11 ymi n a lpha O t h i gh * a lpha l ;

12 ymax a lpha O + t h i gh * a lpha l ;

13 axi s ( [ xmin xmax ymi n ymax ) )

Figure 3.7 EFMSDplot .m.

1 .6

1 .4 a.

::::l.. c: til 1 .2 Q) ::E .Q 0 0..

0.8

0.6

0 0 . 1

/ /

/ nsion of Capital Market Line

/ /

market porfolio, t = t m

pital Market Line '

0.2 0.3 0.4 0.5 Portfolio Standard Deviation cr

0.6

Figure 3.8 Capital Market Line: Exampe3p l .m.

57

0.7


3 . 4 Exercises

3 . 1 Each of the following problems is to be augmented with a risk free asset with return r = 1 .05. Determine the efficient portfolios and the two pieces of the efficient frontier .

(a) n = 2 , I-t = ( 1 . 1 , 1 .2 ) ' and r: = diag(l0-2 , w-1 ) ,

(b) n = 3 , J-l = ( 1 . 1 , 1 . 15 , 1 . 2) ' and r: = diag(l0-4 , w-3 , w-2) .

3 .2 Suppose the data for problem (2 . 1 1 ) (with no nonnegativity constraints) is augmented by a risk free asset with r = 1 . 1 . Determine all the efficient portfolios , Xm and draw the two pieces of the efficient frontier in (ap , /-lp) space.

3 .3 Suppose r: = diag(a1 , a2 , . . . , an) and ai > 0, i = 1 , 2 , . . . , n. Show that on the CML each investor i has holdings proportional to

J-t - r --- , i = 1 , 2 , . . . , n .

(Ji

Obtain an explicit formula for Xm

3 .4 Consider a universe of two assets . The first asset is risky and has an expected return and variance of 1 . 2 and w-2 , respectively. The second asset is risk free and has a return of 1 . 1 .

(a) Find the equations of the two pieces of the efficient frontier .

(b) Explain your results in part (a) in terms of any assumptions that may or may not have been made.

3 . 5 Find the SML for each of the problems of Exercise 3 . 1 . Also plot the SML as in Figure 3 .4 .

3 .6 Show that the market portfolio lies on the SML.

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Chapter 3