Chapter 4

Chapter 4

Sharpe Ratios and Implied Risk Free Returns

The typical situation studied in Chapter 3 is illustrated in Figure 4 . 1 . The key quantities are the implied risk free return rm , the tangency point of the CML with the efficient frontier defined by J-lm and the slope, R, of the CML. It is apparent from Figure 4 . 1 that if any single one of r m, J-lm or R is changed then the other two will change accordingly. It is the purpose of this chapter to determine precisely what these changes will be.

In this chapter, we will look .at the following problems.

1 . Given the expected return on the market portfolio J-lm , what is the implied risk free return r m?

2 . Given the implied risk free return rm , what is the implied market portfolio Xm or equivalently, what is J-lm or tm from which all other relevant quantities can be deduced?

3. Given the slope R of the CML, what is the implied market portfolio Xm or equivalently, what is J-lm or tm from which all other relevant quantities can be deduced?

59

60 Sharpe Ratios and Implied Risk Free Returns

Jlp Implied CML

Implied risk free return a,

r m

1 0 112

0

Chap. 4

Figure 4.1 Risk Free Return and CML Implied by a Market Portfolio.

This analysis is important because if an analyst hypothesizes that one of J.lm , rm or R is changed, the effect on the remaining two parameters will immediately be known.

In Section 4 . 1 , we will perform this analysis for the basic Markowitz model with just a budget constraint . In Section 4 .2 , we will provide a similar analysis from a different point of view: one which will allow generalization of these results to problems which have general linear inequality constraints.

4 . 1 Direct Derivation

In this section we will look at problems closely related to the CML of the previous chapter but from an opposing point of view. We will

Sec. 4. 1 Direct Derivation 61

continue analyzing the model problem

minimize :

subject to :

-t(Jl , r) [ n+l ] + [ n+l r [ ; l'x + Xn+l = 1 , Xn+l >

: ] [ n+l l } (4 . 1 )

Note that we have made explicit the nonnegativity constraint on the risk free asset .

Consider the efficient frontier for the risky assets . Let a0 , at , {30 and {32 be the parameters which define this efficient frontier . Suppose we pick a point on it with coordinates (am , J.Lm) satisfying f..Lm > a0 . Next we require that point to correspond to the market portfolio . The corresponding CML must be tangent to the efficient frontier at this point . We can determine this line as follows. The efficient frontier for the risky assets (2 . 18) can be rewritten as

from which

( 2 !

f..Lp = a0 + (a1 aP - (30 ) ) 2 ,

df..Lp a1am dap (a1 (a - (30) ) !

The tangent line to the efficient frontier at (am , J.Lm) is thus

a1am ( ) f..Lp = f..Lm + ( ( 2

r-1 ) ) ! ap - am . al am - fJO 2

We have thus shown the following result :

(4 .2)

(4 .3)

Lemma 4.1 Let (am , J.Lm) be any point on the efficient frontier with f..Lm > a0 . Then the equation of the line tangent to the efficient frontier at that point is

62 Sharpe Ratios and Implied Risk Free Returns Chap. 4

We next use Lemma 4 . 1 to prove the key results of this section.

Theorem 4.1 (a) Let the expected return on the market portfolio be J.lm with J.lm > Go . Then the implied risk free return is

rm = Go - Gt/3o 'th t _ 1-lm - Go Wl m - .

1-lm - Go G1

(b) Let the risk free return be r m with r m < Go . Then the expected return of the implied market portfolio is given by

Gd3o f3o 1-lm = Go + with tm = Go - rm Go - rm

(c) Let R be the given slope of the CML with R > ..,f!J;,. Then the implied risk free return r m is given by

Proof: (a) rm is the intercept of the line specified in Lemma 4 . 1 , with ap = 0 . Thus

But (am , J.Lm) lies on the efficient frontier (4 .2 ) ; i .e . ,

Substitution of this into ( 4 .4) gives the intercept

Go +

= Go -

( 2 ) ) ! G1a! (GI am - f3o 2 - ( ( 2 R ) ) !

G1 (a! - f3o) - Gta! (Gt (a - f3o) ) !

Gtf3o 1-lm - Go

G} am - f-'0 2

(4.4)

Sec. 4 . 1 Direct Derivation

Noting that /-lm = ao + a1 tm and solving for tm gives

which completes the proof of part (a) .

(b) Solving for 1-lm in part (a) gives

/-lm ' ao - rm [ f3o ] a1 , ao - rm

63

(4 .5)

which verifies the desired result for 1-lm Recall that J.lm = a0 + a1tm . Comparing this formula with ( 4 .5 ) it follows that

f3o ' ao - rm

as required.

(c) From Lemma 4 . 1 , the slope of the CML is

( a1 (a;, - f3o) ) .

Equating this to R, squaring and solving for a;, gives

But a;, = {30 + t;,/32 Equating the two expressions for a;, and using the fact that /32 = a1 gives

/-lm


We now use this expression for J.Lm in the expression for the implied risk free return in part (a) . This gives

a1f3o

and this completes the proof of the theorem. 0

In Theorem 4 . 1 (a) recall from (2 . 12) , (2 . 13) and (2 . 17) that a1 > 0 and {30 > 0. It then follows that r m < a0 and that r m tends to ao as J.Lm tends to infinity. See Figure 4 . 1 .

We illustrate Theorem 4 . 1 in the following example.

Example 4.1 For the given data, answer the following questions.

(a) Find the risk free return rm , implied by a market expected return of J.Lm = 1 .4 . Also find the associated tm , J.Lm and 0";. . What market expected return is implied by r m?

(b) Suppose the risk free return is changed from r m = 1 .050 to r m = 1 .055. What is the change in the expected return on the market portfolio?

(c) If the risk free return is rm = 1 .05, what is the market portfolio?

The data are 1 . 1 0 .0100 0 .0007 0 0 0. 0006 1 . 15 0 .0007 0 .0500 0 .0001 0 0

J.L 1 . 2 ' = 0 0 .0001 0 .0700 0 0 1 .27 0 0 0 0 .0800 0 1 .3 0 .0006 0 0 0 0 .0900

The efficient set coefficients could be found by hand calculation. How-ever , this would require inverting the (5 , 5 ) covariance matrix which is

Sec. 4 . 1 Direct Derivation 65

quite tedious by hand. Rather, we use the computer program "EFMVcoeff.m" (Figure 2 .6) to do the calculations. Running the program with the given data produces

0 .6373 -4.3919 0 . 1 209 0 .2057

ho = 0.0927 h1 = 0 .8181 0 .0812 1 . 59 1 1 0 .0680 1 . 7770

and

o:0 = 1 . 1427, o:1 = 0 .7180, /30 = 0 .0065 , /32 = 0 .7180 .

(a) We can use Theorem 4 . 1 (a) to calculate

Tm = O:o -

We also calculate

1 . 1427 _ 0 .7180 X 0 .0065

1 .4 - 1 . 1427

f.lm - O:o 1 . 4 - 1 . 1427 tm = 0:1 =

0 .7180 = 0 .3584.

1 . 1246.

Furthermore, with this value of r m we can use Theorem 4 . 1 (b) to calculate

f3oo:1 f.lm = O:o + = 1 .4 , O:o - Tm as expected . Finally, we calculate O"; according to

O" = f3o + t/32 = 0 .0987.

(b) We use Theorem 4. 1 (b) with rm = 1 .050 to calculate

f.lm = O:o + o:1/3o

O:o - Tm = 1 . 1930.

Similarly, for rm = 1 . 055 , f.lm = 1 . 1959 so the change in the expected return on the market portfolio implied by the change in the risk free return is 0 .0029 .


(c) From Theorem 4 . 1 (b) with rm = 1 .05 we calculate tm = 0.0701 . The implied market portfolio i s now Xm = h0 + tmh1 , that is,

0 .3294 0 . 1353

Xm 0 . 1500 0 . 1928 0 . 1925

4 . 2 Optimization Derivation 0

We next analyze the problems addressed in Section 4 . 1 from a different point of view. Suppose rm is already specified and we would like to find the point on the efficient frontier corresponding to the implied market portfolio. In Figure 4 .2 , we assume rm is known and we would like to determine (O'm , flm) One way to solve this is as follows . Let (O'p , flp) represent an arbitrary portfolio and consider the slope of the

Jlp

r m

0

Figure 4.2 Maximizing the Sharpe Ratio.

Sec. 4 .2 Optimization Derivation 67

line joining (0, rm) and (ap , /-lp) The slope of this line is (f.-lp - rm)/ap and is called the Sharpe ratio. In Figure 4 .2 we can see the Sharpe ratio increases as (ap , /-lp) approaches (am , f.-lm) The market portfolio corresponds to where the Sharpe ratio is maximized. We thus need to solve the problem

rna I llx { f.-l1X - rm

X 1 (x1x) 2

(4 .6)

The problem ( 4.6) appears somewhat imposing, as the function to be maximized is the ratio of a linear function to the square root of a quadratic function. Furthermore, it appears from Figure 4 .2 that in (4.6) we should be requiring that f.-l1X > rm and we do this implicitly rather than making it an explicit constraint . The problem ( 4 .6) can be analyzed using Theorem 1 . 5 as follows.

From Theorem 1 . 6 (b) , the gradient of the objective function for ( 4.6) is

(x1x) f.-l - (f.-l1X - rm) (x1xtx x1x

(4 .7)

After rearranging, Theorem 1 . 5 states that the optimality conditions for ( 4.6) are

[ X1X ] _ [ (x1x)312 ] 1 _ 1

f.-l - X - 1 U l , l X - 1 , f.-l X - rm f.-l X - rm (4 .8)

where u is the multiplier for the budget constraint . Note that the quantities within square brackets are scalars, whereas f.-l, x and l are vectors .

Now, recall the optimization problem (2 .3) and its optimality conditions (2 .4) . We repeat these for convenience:

"n{ tu' + 1 1x I l1x = 1 } m1 - t-" x 2x (4 .9)


t-t - Ex = ul , and, l'x = 1 . (4. 10)

Comparing (4 .8) with (4 . 10) shows that the optimality conditions , and thus the optimal solutions for ( 4 .9) and ( 4 .6) will be identical provided

t

or equivalently

x'Ex 1-1X - rm '

x'Ex -t'x - t

We have just shown the following:

(4 . 1 1 )

(4 . 12)

Theorem 4.2 (a) For fixed rm, let x0 be optimal for (4 . 6} . Then xo is also optimal for (4 . 9} with t = xExo/ (-t'xo - rm)

(b) For fixed t = t 1 , let x1 = x(tt ) be optimal for (4 . 9} . Then x1 is optimal for (4 . 6} with rm = 1-1Xl - x Ext /t1 .

Theorems 4 . 1 (b) and 4 .2 (a) are closely related . Indeed, Theorem 4 .2 (a) implies the results of Theorem 4 . 1 (b) as we now show.

Let rm be given and let x0 be optimal for (4 .6) . Then Theorem 4 .2 (a) asserts that x0 i s optimal for (2 .3) with

(4. 13)

Because x0 is on the efficient frontier for (2 .3) we can write xEx0 = /30 + f32t2 and -t'x0 = a0 + a1 t . Substitution of these two expressions in t given by Theorem 4 .2 (a) gives

t = tm = f3o + f32t';, ao + a1 tm - rm

Sec. 4 .2 Optimization Derivation

Cross multiplying gives

Simplifying and solving for tm now gives

f3o ' ao - rm

which is the identical result obtained in Theorem 4. 1 (b) .

69

We can perform a similar analysis beginning with Theorem 4 .2 (b) . Set tm = t 1 and Xm = X1 in Theorem 4 .2 (b) . Because Xm is optimal for (2 .3 ) , J-lm = ao + tmal and 0";, = f3o + t;.f32 Then Theorem 4 .2 (b) implies

Multiplying through by tm , using a1 = /32 and simplifying gives

tm = [ f3o ] and J-lm = ao + [ f3o ] a1 , ao - rm ao - rm in agreement with Theorem 4 . 1 (b) .

(4 . 14)

(4 . 1 5 )

There are other somewhat simpler methods to derive the results of (4 . 15 ) (see Exercise 4 .4) . Using Theorem 4 .2 (b) for the analysis has the advantage of being easy to generalize for practical portfolio optimization as we shall see in Section 9 . 1 .

We next consider a variation of the previous problem. We assume that the value of the maximum Sharpe ratio, denoted by R, is known. R is identical to the slope of a CML. The questions to be answered are to what market portfolio does R correspond, and what is the associated implied risk free return r m? The situation is illustrated in Figure 4 .3 .

70 Sharpe Ratios and Implied Risk Free Returns

r

f- Jl - Rcr = r p p

Chap. 4

Figure 4.3 Maximize Implied Risk Free Return for Given Sharpe Ratio.

The line /-lp - Rap = r has slope R and is parallel to the specified CML. As r is increased (or equivalently /-lp - Rap is increased) , the line will eventually be identical with the CML. The optimization problem to be solved is thus

max{t-t'x - R(x'x) I l'x = 1 } . (4. 16)

This can be rewritten as an equivalent minimization problem

min{ - t-t'x + R(x'x) I l'x = 1 } . (4. 17)

The problem (4 . 1 7) is of the same algebraic form as ( 1 . 1 1 ) so that we can apply the optimality conditions of Theorem 1 . 5 to it . The dual feasibility part of the optimality conditions for ( 4 . 17) states

Sec. 4 .2 Optimization Derivation 71

(see Exercise 4 .5)

J.l - R x 1 = ul , (x'x) 2 (4. 18)

where u is the multiplier for the budget constraint . This can be rewrit-ten as

[ (x'x) l _ _ [ (x'x) l R J.l X - R u l . (4 . 19)

Comparing (4 . 19) with (4 . 10) shows that the optimality conditions, and thus the optimal solutions for (4 .9) and (4. 19) will be identical provided

t

or equivalently

R (x'x) t

Analogous to Theorem 4 .2 , we have

(4.20)

Theorem 4.3 (a) For fixed Sharpe ratio R, let x0 be optimal for (4 . 1 7} . Then Xo is also optimal for (4 . 9} with t = (xoE;o ) . (b} For fixed t = t1 , let Xt = x( t t ) be optimal for ( 4 . 9) . Then x1 is

I 1 optimal for (4 . 1 7} with R = (x1;1 ) "2" .

Theorems 4 . 1 (c) and 4 .3(b) are closely related. Indeed, Theorem 4.3 (b) implies the results of Theorem 4 . 1 (c) as we now show. Let R be the given slope of the CML, assume

R > J1j; ( 4 .21 ) and let tm = t1 in the statement of Theorem 4.3 (b) . We can determine tm and r m as follows. Theorem 4.3 (b) asserts that

R = (xxm) l/2


Squaring and cross multiplying give

t'!. R2 = {30 + t'!.f32 .

Solving this last for tm gives

(4 .22)

In addition, we can determine the implied risk free return as follows. The CML is /-Lp = rm + Rap . The point (am , J.Lm) is on this line, so J.Lm = rm + Ram . Therefore,

Solving for r m in this last gives

ao + tm (al - R2 ) {3,1/2 ao - (R2 !32 ) 1/2 (R2 - !32)

ao - [f3o (R2 - fJ2WI2 . (4 .23)

Equations (4 .23) and (4.22) show that Theorem 4. l (c) is a special case of Theorem 4 .3 (b) .

There are other somewhat simpler methods to deduce the previous result (see Exercise 4 .6) . The advantage of using Theorem 4.3 (b) is that it generalizes in a straightforward manner to practical portfolio optimization problems, as we shall see in Sections 9 . 1 .

Sec. 4 .3 Free Problem Solutions 73

Implicit in ( 4 .23) is the assumption that R2 > /32 . To see why this is reasonable, observe that from ( 4 .3) the slope of the efficient frontier is

dMP a1p dP (al ( - f3o) )

I f3i

From (4 .24) it follows that

and

dMP > /3 d 2

' p

(4 .24)

We complete this section by stating a theorem which summarizes Theorems 4 .2 (b) and 4 .3 (b) .

Theorem 4.4 Let x( t 1 ) = x1 be a point on the efficient frontier for t = t1 . Then (M'Xl - rm) / (xxl ) is the maximum Sharpe ratio for 1 rm = M1Xl - xxdt1 , and, rm = M1X1 - R(xx1 ) 2 is the maximum risk free return with Sharpe ratio R = (x x1 ) /t1 .

4 . 3 Free Problem Solutions

Suppose we have solved

1 min{ - tM'x + 2x'x I l'x = 1 }

and obtained h0 , h1 , a0 , a1 = /32 and /30 . If we pick any t0 > 0 then we have a point on the efficient frontier and we can compute Mpo = a0+t0a1


and a = {30 + t{32 . But then ( 1 ) /-tpO is the optimal solution for the problem of maximizing the portfolio's expected return with the portfolio variance being held at a;a ; i .e . , (2 .2) with a; = a;0 Furthermore, (2) a is the minimum variance of the portfolio with the expected return being held at J-tp0 ; i .e . , (2 . 1 ) with /-tp = /-tpO

Table 4 . 1 shows these pairs for a variety of values of t for the problem of Example 2 . 1 . For example, for t = 0 . 10 , the minimum portfolio variance is 0 .0134043 when the expected return is required to be 1 . 19574. Conversely, the maximum expected return is 1 . 19574 when the variance is held at 0 .0134043 .

TABLE 4.1 Minimum Variance and Maximum Expected Returns For Example 2 .1 .

t Minimum For Specified Maximum For Specified Variance Return /-tp Expected Variance a;

Return 0 . 1000 0 .0134043 1 . 19574 1 . 19574 0 .0134043 0 . 1 1 1 1 0 .0148017 1 . 20236 1 .20236 0 .0148017 0 . 1 250 0 .0167553 1 .2 1064 1 .2 1064 0 .0167553 0 . 1429 0 .0196049 1 .22128 1 . 22128 0 .0196049 0 . 1 667 0 .0239953 1 .23546 1 .23546 0 .0239953 0 .2000 0.0312766 1 .25532 1 . 25532 0 .0312766 0 .2500 0 .0446809 1 .285 1 1 1 .285 1 1 0 .0446809 0 .3333 0.0736407 1 .33475 1 . 33475 0 .0736407 0 .5000 0 . 156383 1 .43404 1 .43404 0 . 156383 1 . 0000 0 .603191 1 . 73191 1 . 73191 0 .603191

Theorem 4.4 also says we have solved two other problems at t0 These are; (3)

is the maximum Sharpe ratio for rm = /-tpO - ajt0 , and, (4) rm = /-tpO - R(a) 112 is the maximum risk free return with Sharpe ratio R = (a) 1/2 /to .

Sec. 4 .3 Computer Programs 75

Table 4 .2 shows these pairs for a variety of values of t for the problem of Example 2 . 1 . For example, for t = 0. 10 , the maximum Sharpe ratio is 1 . 1 5777 for a fixed risk free return of 1 .0617 . Conversely, the maximum risk free return is 1 .06 17 for a specified Sharpe ratio of 1 . 15777.

TABLE 4.2 Maximum Sharpe Ratios and Maximum Risk Free Returns For Example 2. 1 .

t Maximum For Risk Maximum For Specified Sharpe Free Return Risk Free Sharpe Ratio ratio Tm Return Tm R

0 . 1000 1 . 15777 1 .0617 1 .0617 1 . 15777 0. 1 1 1 1 1 .09496 1 .06915 1 .06915 1 .09496 0 . 1250 1 . 03554 1 .0766 1 .0766 1 . 03554 0. 1429 0 .980122 1 .08404 1 .08404 0 .980122 0. 1667 0. 929424 1 .09149 1 . 09 149 0 .929424 0 .2000 0 .88426 1 .09894 1 .09894 0 .88426 0. 2500 0 .845514 1 . 10638 1 . 10638 0 .845514 0 .3333 0 .814104 1 . 1 1383 1 . 1 1383 0 .814104 0 .5000 0 .790906 1 . 12128 1 . 12 128 0 . 790906 1 . 0000 0 . 776654 1 . 12872 1 . 12872 0 . 776654

Note that in Table 4.2 the two expressions for Sharpe ratios, namely (/-lpO - rm)/ (a-) 112 and R = (a-;0) 112 /to are numerically identical . Furthermore, the two expressions for the risk free return, namely rm = /-lpo - a-;0/to and rm = /-lpo - R(a-;0 ) 112 produce identical numerical results. The equality of these two pairs of expressions is true in general . See Exercise 4 .8 .

4.4 Computer Programs

Figure 4.4 displays the routine Example4pl .m which performs the calculations for Example 4 . 1 . The data is defined in lines 1-6 and line 8 invokes the routine EFMVcoeff (see Figure 2 .6) to compute the coeffi-

76 The Capital Asset Pricing Model Chap. 4

cients for the efficient frontier . The computations for parts (a) , (b) and (c) of Example 4 . 1 are then done in a straightforward manner.

1 % E x amp l e 4 p 1 . m 2 mu = [ 1 . 1 1 . 1 5 1 . 2 1 . 2 7 1 . 3 ] ' 3 S i gma = [ 1 . e-2 0 . 7 e-3 0 0 0 . 6e-3 4 0 . 7 e-3 5 . O e-2 1 . e-4 0 0 ;

0 1 . e- 4 7 . 0 e-2 0 0 6 0 0 0 8e-2 0 ;

0 . 6e-3 0 0 0 9 . e-2 ] 8 checkdat a ( S i gma 1 1 . e- 6 ) ; 9 [ alphaO 1 a lpha 1 1 bet a O 1 bet a2 1 h O 1 h 1 ]

10 11 %Examp l e 4 . 1 ( a ) 1 2 s t r i n g = ' Examp l e 4 . 1 ( a ) ' 13 mum = 1 . 4

EFMVcoe f f ( mu 1 S i gma ) ;

14 rm = a l pha O - ( a lpha 1 *bet a 0 ) / ( mum - a lpha O ) 15 tm = bet a O / ( a lpha O-rm ) 16 mum = a lpha O + tm * a l pha 1 11 s i gp2 = bet a O + tm* tm*bet a 2 1 8 19 % E x amp l e 4 . 1 ( b ) 2 0 s t r ing = ' Examp l e 4 . 1 ( b ) ' 21 rm = 1 . 0 5 0 0 0 22 t m = bet a O / ( alpha O - rm) 23 mum = a lp h a O + a lpha 1 * tm 24 s i gp 2 = bet a O + tm*tm*be t a 2 25 26 rm = 1 . 0 5 5 2 7 t m = b e t a O / ( a lp h a O - rm ) 28 mum = a lpha O + a lpha 1 * tm 29 s i gp2 = bet a O + tm*tm*be t a 2 30 31 % E xamp l e 4 . 1 { c ) 32 s t r i ng = ' Examp l e 4 . 1 ( c ) ' 33 rm = 1 . 0 5 34 t m = bet a O / ( a lphaO - rm) 35 port = h O + tm * h 1

Figure 4.4 Example4pl .m.

Sec. 4 .3 Computer Programs

1 % S harpeRat i o s . m 2 mu = [ 1 . 1 1 . 2 1 . 3 ] ' 3 S i gma = [ 1 . e-2 0 0 ; 0 S . O e-2 0 ; 0 0 7 . 0 e-2 ] 4 checkdat a ( S i gma , 1 . e - 6 ) ;

77

5 [ a lpha O , a lpha 1 , bet a O , bet a 2 , h O , h 1 ] EFMVcoe f f ( mu , S i gma ) ; 6 for i = 1 : 1 0 7 t = 1 / ( 1 1 - i ) 8 x 1 = h O + t . * h 1 ; 9 mup = mu ' * X 1 ;

10 s i gp 2 = x 1 ' * S i gma * x 1 ; 1 1 s t r = ' Mi n imum v a r i a n c e % g f o r f i x e d e xpe c t e d r e t u r n o f 1 2 % g\n ' ; 13 fpr int f ( s t r , s i gp 2 , mup ) 14 s t r = ' Max imum expe c t e d return % g f o r f i x e d v a r i a n c e o f 1 s % g\n ' ; 16 fp r i n t f ( s t r , mup , s i gp2 ) 11 rm = mup - s i gp2 / t ; 18 S rat i o = ( mup - rm ) / s i gp2 ' ( 1 / 2 ) ; 19 fp r i nt f ( ' Maximum S h a rpe rat i o i s % g f o r rm % g\ n ' , 20 S r at i o , rm ) 21 S r at i o = s i gp2 ' ( 1 . / 2 . ) It ; 22 rm = mup - S r at i o * s i gp2 ' ( 1 / 2 ) ; 23 fp r i nt f ( ' Max r i s k free r e t u r n i s % g f o r S h a rpe rat i o 24 % g\n ' . . . 25 , rm, S rat i o ) 2 6 end

Figure 4.5 SharpeRatios.m.

Figure 4 .5 shows the routine SharpeRatios .m which computes the solution for four problems at each of ten points on the efficient frontier for the data of Example 2 . 1 . Lines 2 and 3 define the data and line 4 checks the covariance matrix. Line 5 uses the function EFMV coeff (see Figure 2.6) to calculate the coefficients of the efficient frontier . The "for" loop from line 6 to 20, constructs various values of t . For each such t, line 8 constructs the efficient portfolio for t and lines 9 and 10 compute the mean and variance of that portfolio . Then, as discussed in Section 4 .3 , the computed portfolio mean is the maximum portfolio mean with the variance being held at the computed level. Conversely, the computed variance is the minimum variance when the

78 The Capital Asset Pricing Model Chap. 4

portfolio expected return is held at the computed level. These values are summarized in Table 4 . 1 .

4 . 5 Exercises

4 . 1 For the data of Example 2 . 1 , determine the implied risk free return corresponding to (a) /-lp = 1 .2 , (b) /-lp = 1 .25 .

4 .2 What market portfolio would give an implied risk free return of rm = 0?

4 .3 Suppose rm = 1 . 1 is the risk free return for the data of Example 2 . 1 . Find the implied market portfolio .

4 .4 If we use the fact that the optimal solution must lie on the efficient frontier , then (4.6) can be formulated as

where the maximization is over all t 0. Show the optimal solution for this is identical to (4 . 15) .

4 .5 Verify (equation 4 . 18) .

4 .6 If we use the fact that the optimal solution must lie on the efficient frontier , then ( 4 . 1 7) can be formulated as

(4.25)

where the maximization is over all t 0 . Show that the optimal solution for this is identical to ( 4 .22) .

4. 7 Given the coefficients of the efficient frontier o:o = 1 . 13 , o:1 = 0. 72, f3o = 0 .007 and (32 = 0 . 72 .

(a) Find the risk free return rm implied by a market expected return of 1-lm = 1 .37. Also find the associated tm , 1-lm and a! . What market expected return is implied by r m?

Sec. 4.4 Exercises 79

(b) Suppose the expected return on the market changes from J.Lm = 1 .35 to J.Lm = 1 .36. What is the implied change in the risk free return?

4.8 Let J.Lpo and CJpo be as in Section 4 .3 .

(a) Show the Sharpe ratios (J.Lpo - rm)/ (CJ;0) 112 and R (CJ;0) 112 jt0 are identical, where rm = J.lpo - CJ;o/to .

(b) Show the risk free returns r m = J.Lpo - CJ;0/to and r m J.Lpa - R(CJ;0 ) 112 are identical , where R = (CJ;0 ) 112 jt0 .

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