ńņš. 4 |

orthogonal to planes of constant price. Algebraically, in analogy to p(x) = E(xā— x) we have

E(Re ) = E(Reā— Re ) āRe ā Re . (70)

This fact can serve as an alternative deļ¬ning property of Reā— .

5) Reā— and Rā— are orthogonal,

E(Rā— Reā— ) = 0.

More generally, Rā— is orthogonal to any excess return.

6) The mean variance frontier is given by

Rmv = Rā— + wReā— .

Ā£ Ā¤

To prove this, E(R2 ) = E (Rā— + wReā— + n)2 = E(Rā—2 ) + w2 E(Re2 ) + E(n2 ), and

E(n) = 0, so set n to zero. The conditional mean-variance frontier allows w in the con-

ditioning information set. The unconditional mean variance frontier requires w to equal a

constant.

7) Rā— is the minimum second moment return. Graphically, Rā— is the return closest to

the origin. To see this, using the decomposition in #6, and set w2 and n to zero to minimize

second moment.

8) Reā— has the same ļ¬rst and second moment,

E(Reā— ) = E(Reā—2 ).

Just apply fact (5.70) to Reā— itself. Therefore

var(Reā— ) = E(Reā—2 ) ā’ E(Reā— )2 = E(Reā— ) [1 ā’ E(Reā— )] .

9) If there is a riskfree rate, then Reā— can also be deļ¬ned as the residual in the projection

of 1 on Rā— :

E(Rā— ) ā— 1

Reā— = 1 ā’ proj(1|Rā— ) = 1 ā’ R = 1 ā’ f Rā— (71)

E(Rā—2 ) R

Youā™d never have thought of this without looking at Figure 14! Since Rā— and Re are orthog-

onal and together span X, 1 = proj(1|Re ) + proj(1|Rā— ). You can also verify this statement

analytically. Check that Reā— so deļ¬ned is an excess return in X ā“ its price is zeroā“, and

E(Reā— Re ) = E(Re ); E(Rā— Reā— ) = 0.

90

SECTION 5.5 A COMPILATION OF PROPERTIES OF Rā— , Reā— AND xā—

As a result, Rf has the decomposition

Rf = Rā— + Rf Reā— . (72)

Since Rf > 1 typically, this means that Rā— +Reā— is located on the lower portion of the mean-

variance frontier in mean-variance space, just a bit to the right of Rf . If the risk free rate were

one, then the unit vector would lie in the return space, and Rf = Rā— + Reā— . Typically, the

space of returns is a little bit above the unit vector. As you stretch the unit vector by the

amount Rf to arrive at the return Rf , so you stretch the amount Reā— that you add to Rā— to

get to Rf .

If there is no riskfree rate, then we can use

proj(1|X) = proj(proj (1|X) |Re ) + proj(proj (1|X) |Rā— )

= proj(1|Re ) + proj(1|Rā— )

to deduce an analogue to equation (5.71),

E(Rā— ) ā—

eā— ā—

(73)

R = proj(1|X) ā’ proj(1|R ) = proj(1|X) ā’ R

E(Rā—2 )

10) If a riskfree rate is traded, we can construct Rf from Rā— via

E(Rā—2 )

1

f

(74)

R= = .

E(xā— ) E(Rā— )

If not, this gives a āzero beta rateā interpretation of the right hand expression.

11) Since we have a formula xā— = p0 E(xx0 )ā’1 x for constructing xā— from basis assets

(see section 4.1), we can construct Rā— in this case from

xā— p0 E(xx0 )ā’1 x

ā—

R= =0 .

p(xā— ) p E(xx0 )ā’1 p

(p(xā— ) = E(xā— xā— ) leading to the denominator.)

12) We can construct Reā— from a set of basis assets as well. Following the deļ¬nition to

project one on the space of excess returns,

Reā— = E(Re )0 E(Re Re0 )ā’1 Re

where Re is the basis set of excess returns. (You can always use Re = Rā’Rā— if you want).

This construction obviously mirrors the way we constructed xā— in section 4.1, and you can

see the similarity in the result, with E in place of p, since Reā— represents means rather than

prices. .

91

CHAPTER 5 MEAN-VARIANCE FRONTIER AND BETA REPRESENTATIONS

If there is a riskfree rate, we can also use (5.71),

1 p0 E(xx0 )ā’1 x

1ā—

Reā— = 1 ā’ (75)

R =1ā’ f 0 .

Rf R p E(xx0 )ā’1 p

If there is no riskfree rate, we can use (5.73) to construct Reā— . The central ingredient is

proj(1|X) = E(x)0 E(xx0 )ā’1 x.

5.6 Mean-variance frontiers for m: the Hansen-Jagannathan

bounds

The mean-variance frontier of all discount factors that price a given set of assets is related

to the mean-variance frontier of asset returns by

|E(Re )|

Ļ(m)

ā„ .

Ļ(Re )

E(m)

and hence

E(Re )

Ļ(m)

min = max e

{all excess returns R in X} Ļ(Re )

{all m that price xāX} E(m)

The discount factors on the frontier can be characterized analogously to the mean-variance

frontier of asset returns,

m = xā— + weā—

0

eā— ā” 1 ā’ proj(1|X) = proj(1|E) = 1 ā’ E(x) E(xx0 )ā’1 x

E = {m ā’ xā— } .

We derived in Chapter 1 a relation between the Sharpe ratio of an excess return and the

volatility of discount factors necessary to price that return,

|E(Re )|

Ļ(m)

ā„ .

Ļ(Re )

E(m)

Quickly,

0 = E(mRe ) = E(m)E(Re ) + Ļm,Re Ļ(m)Ļ(Re ),

92

SECTION 5.6 MEAN-VARIANCE FRONTIERS FOR m: THE HANSEN-JAGANNATHAN BOUNDS

and |Ļ| ā¤ 1. If we had a riskfree rate, then we know in addition

E(m) = 1/Rf .

Hansen and Jagannathan (1991) had the brilliant insight to read this equation as a restriction

on the set of discount factors that can price a given set of returns, as well as a restriction on

the set of returns we will see given a speciļ¬c discount factor. This calculation teaches us

that we need very volatile discount factors with a mean near one to understand stock returns.

This and more general related calculations turn out to be a central tool in understanding and

surmounting the equity premium puzzle, surveyed in Chapter 21.

We would like to derive a bound that uses a large number of assets, and that is valid if

there is no riskfree rate. What is the set of {E(m), Ļ(m)} consistent with a given set of asset

prices and payoffs? What is the mean-variance frontier for discount factors?

Obviously, the higher the Sharpe ratio, the tighter the bound. This suggests a way to

construct the frontier weā™re after. For any hypothetical risk-free rate, ļ¬nd the highest Sharpe

ratio. That is, of course the tangency portfolio. Then the slope to the tangency portfolio gives

the ratio Ļ(m)/E(m). Figure 16 illustrates.

E(R)

Ļ(m) =

Ī•(Re)/Ļ(Re)

1/E(m) E(Re)/Ļ(Re)

Ļ(R) E(m)

Figure 16. Graphical construction of the Hansen-Jagannathan bound.

As we sweep through values of E(m), the slope to the tangency becomes lower, and

the Hansen-Jagannathan bound declines. At the mean return corresponding to the minimum

variance point, the HJ bound attains its minimum. Continuing, the Sharpe ratio rises again

and so does the bound. If there were a riskfree rate, then we know E(m), the return frontier

is a V shape, and the HJ bound is purely a bound on variance.

This discussion implies a beautiful duality between discount factor volatility and Sharpe

ratios.

E(Re )

Ļ(m)

(76)

min = max e .

{all excess returns R in X} Ļ(Re )

{all m that price xāX} E(m)

93

CHAPTER 5 MEAN-VARIANCE FRONTIER AND BETA REPRESENTATIONS

We need formulas for an explicit calculation. In equation (), we found a representation

for the set of discount factors that price a given set of asset returns ā“ that satisfy p = E(mx) :

m = E(m) + [p ā’ E(m)E(x)] Ī£ā’1 [x ā’ E(x)] + Īµ (77)

where Ī£ ā” cov(x, x0 ) and E(Īµ) = 0, E(Īµx) = 0. You can think of this as a regression or

projection of any discount factor on the space of payoffs, plus an error. Since Ļ2 (Īµ) > 0,

this representation leads immediately to an explicit expression for the Hansen-Jagannathan

bound,

0

Ļ 2 (m) ā„ (p ā’ E(m)E(x)) Ī£ā’1 (p ā’ E(m)E(x)) . (78)

As all asset returns must lie in a cup-shaped region in {E(R), Ļ(R)} space, all discount

Ā© ĀŖ

factors must lie in a parabolic region in E(m), Ļ 2 (m) space, as illustrated in the right

hand panel of Figure 16.

We would like an expression for the discount factors on the bound, as we wanted an

expression for the returns on the mean variance frontier instead of just a formula for the

means and variances. As we found a three way decomposition of all returns, two of which

generated the mean-variance frontier of returns, so we can ļ¬nd a three way decomposition of

discount factors, two of which generate the mean-variance frontier of discount factors (5.78).

I illustrate the construction in Figure 17.

Any m must line in the plane marked M, perpendicular to X through xā— . Any m must

be of the form

m = xā— + weā— + n.

Here, I have just broken up the residual Īµ in the familiar representation m = xā— + Īµ into two

components. eā— is deļ¬ned as the residual from the projection of 1 onto X or, equivalently the

projection of 1 on the space E of āexcess mā™s,ā random variables of the form m ā’ xā— .

eā— ā” 1 ā’ proj(1|X) = proj(1|E).

eā— generates means of m just as Reā— did for returns:

E(m ā’ xā— ) = E[1 Ć— (m ā’ xā— )] = E[proj(1|E)(m ā’ xā— )].

Finally n, deļ¬ned as the leftovers, has mean zero since itā™s orthogonal to 1 and is orthogonal

to X.

As with returns, then, the mean-variance frontier of m0 s is given by

mā— = xā— + weā— . (79)

If the unit payoff is in the payoff space, then we know E(m), and the frontier and bound

are just m = xā— , Ļ 2 (m) ā„ Ļ2 (xā— ). This is exactly like the case of risk-neutrality for return

mean-variance frontiers.

94

SECTION 5.6 MEAN-VARIANCE FRONTIERS FOR m: THE HANSEN-JAGANNATHAN BOUNDS

X = payoff space

M = space of discount factors

x*+we* n

x*

m = x*+we*+n

1

proj(1| X)

e*

0

E(.)=1

E(.)=0

E = space of m-x*

Figure 17. Decomposition of any discount factor m = xā— + we + n.

95

CHAPTER 5 MEAN-VARIANCE FRONTIER AND BETA REPRESENTATIONS

The construction (5.79) can be used to derive the formula (5.78) for the Hansen-Jagannathan

bound for the ļ¬nite-dimensional cases discussed above. Itā™s more general, since it can be used

in inļ¬nite-dimensional payoff spaces as well. Along with the corresponding return formula

Rmv = Rā— + wReā— , we see in Chapter 8 that it extends more easily to the calculation of

conditional vs. unconditional mean-variance frontiers (Gallant, Hansen and Tauchen 1995).

It will make construction (5.79) come alive if we ļ¬nd equations for its components. We

ļ¬nd xā— as before, it is the portfolio c0 x in X that prices x:

xā— = p0 E(xx0 )ā’1 x.

Similarly, letā™s ļ¬nd eā— . The projection of 1 on X is

proj(1|X) = E(x)0 E(xx0 )ā’1 x.

(After a while you get used to the idea of running regressions with 1 on the left hand side and

random variables on the right hand side!) Thus,

eā— = 1 ā’ E(x)0 E(xx0 )ā’1 x.

Again, you can construct time-series of xā— and eā— from these deļ¬nitions.

Finally, we now can construct the variance-minimizing discount factors

Ā£ Ā¤

mā— = xā— + weā— = p0 E(xx0 )ā’1 x + w 1 ā’ E(x)0 E(xx0 )ā’1 x

or

0

mā— = w + [p ā’ wE(x)] E(xx0 )ā’1 x (80)

As w varies, we trace out discount factors mā— on the frontier with varying means and vari-

ances. Itā™s easiest to ļ¬nd mean and second moment:

E(mā— ) = w + [p ā’ wE(x)]0 E(xx0 )ā’1 E(x)

0

E(mā—2 ) = [p ā’ wE(x)] E(xx0 )ā’1 [p ā’ wE(x)] ;

variance follows from Ļ 2 (m) = E(m2 ) ā’ E(m)2 . With a little algebra one can also show

that these formulas are equivalent to equation (5.78).

As you can see, Hansen-Jagannathan frontiers are equivalent to mean-variance frontiers.

For example, an obvious exercise is to see how much the addition of assets raises the Hansen-

Jagannathan bound. This is exactly the same as asking how much those assets expand

the mean-variance frontier. It was, in fact, this link between Hansen-Jagannathan bounds

and mean-variance frontiers rather than the logic I described that inspired Knez and Chen

(1996) and DeSantis (1994) to test for mean-variance efļ¬ciency using, essentially, Hansen-

Jagannathan bounds.

96

SECTION 5.7 PROBLEMS

Hansen-Jagannathan bounds have the potential to do more than mean-variance frontiers.

Hansen and Jagannathan show how to solve the problem

min Ļ2 (m) s.t. p = E(mx), m > 0.

This is the āHansen-Jagannathan bound with positivity,ā and is strictly tighter than the Hansen-

Jagannathan bound. It allows you to impose no-arbitrage conditions. In stock applications,

this extra bound ended up not being that informative. However, in the option application of

this idea of Chapter (18), positivity is really important. That chapter shows how to solve for

a bound with positivity.

Hansen, Heaton and Luttmer (1995) develop a distribution theory for the bounds. Luttmer

(1996) develops bounds with market frictions such as short-sales constraints and bid-ask

spreads, to account for ludicrously high apparent Sharpe ratios and bounds in short term

bond data. Cochrane and Hansen (1992) survey a variety of bounds, including bounds that

incorporate information that discount factors are poorly correlated with stock returns (the

HJ bounds use the extreme Ļ = 1), bounds on conditional moments that illustrate how many

models imply excessive interest rate variation, bounds with short-sales constraints and market

frictions, etc.

Chapter 21 discusses what the results of Hansen Jagannathan bound calculations and what

they mean for discount factors that can price stock and bond return data.

5.7 Problems

1. Prove that Reā— lies at right angles to planes (in Re ) of constant mean return, as shown in

ļ¬gure 14.

2. Should we typically draw xā— above, below or on the plane of returns? Must xā— always lie

in this position?

3. Can you construct Reā— from knowledge of m, xā— , or Rā— ?

4. What happens to Rā— , Reā— and the mean-variance frontier if investors are risk neutral?

(a) If a riskfree rate is traded.

(b) If no riskfree rate is traded?

(Hint: make a drawing or think about the case that payoffs are generated by an N

dimensional vector of basis assets x)

5. xā— = proj(m|X). Is Rā— = proj(m|R)?

6. We showed that all m are of the form xā— + Īµ. What about Rā’1 R?

7. Show that if there is a risk-free rateā”if the unit payoff is in the payoff space Xā”then

Reā— = (Rf ā’ Rā— )/Rf .

97

Chapter 6. Relation between discount

factors, betas, and mean-variance frontiers

In this chapter, I draw the connection between discount factors, mean-variance frontiers, and

beta representations. In the ļ¬rst chapter, I showed how mean-variance and a beta represen-

tation follow from p = E(mx) and (in the mean-variance case) complete markets. Here, I

discuss the connections in both directions and in incomplete markets, drawing on the repre-

sentations studied in the last chapter.

The central theme of the chapter is that all three representations are equivalent. Figure

18 summarizes the ways one can go from one representation to another. A discount factor, a

reference variable for betas ā“ the thing you put on the right hand side in the regressions that

give betas ā“ and a return on the mean-variance frontier all carry the same information, and

given any one of them, you can ļ¬nd the others. More speciļ¬cally,

1. p = E(mx) ā’ Ī²: Given m such that p = E(mx), m, xā— , Rā— , or Rā— + wReā— all can

serve as reference variables for betas.

2. p = E(mx) ā’ mean-variance frontier. You can construct Rā— from xā— = proj(m|X),

Rā— = xā— /E(xā—2 ), and then Rā— , Rā— + wReā— are on the mean-variance frontier.

3. Mean-variance frontier ā’ p = E(mx). If Rmv is on the mean-variance frontier, then

m = a + bRmv linear in that return is a discount factor; it satisļ¬es p = E(mx).

4. Ī² ā’ p = E(mx). If we have an expected return/beta model with factors f , then

m = b0 f linear in the factors satisļ¬es p = E(mx) (and vice-versa).

5. If a return is on the mean-variance frontier, then there is an expected return/beta model

with that return as reference variable.

The following subsections discuss the mechanics of going from one representation to the

other in detail. The last section of the chapter collects some special cases when there is

no riskfree rate. The next chapter discusses some of the implications of these equivalence

theorems, and why they are important.

Roll (197x) pointed out the connection between mean-variance frontiers and beta pricing.

Ross (1978) and Dybvig and Ingersoll (1982) pointed out the connection between linear

discount factors and beta pricing. Hansen and Richard (1987) pointed out the connection

between a discount factor and the mean-variance frontier.

6.1 From discount factors to beta representations

m, xā— , and Rā— can all be the single factor in a single beta representation.

98

SECTION 6.1 FROM DISCOUNT FACTORS TO BETA REPRESENTATIONS

LOOP Ć m exists

p = E(mx)

f = m, x*, R*

m = bā™f m = a + bRmv

R* is on m.v.f.

E(Ri) = Ī± + Ī²iā™Ī»

f = Rmv

proj(f|R) on m.v.f. Rmv on m.v.f.

E(RRā™) nonsingular Ć Rmv exists

Figure 18. Relation between three views of asset pricing.

6.1.1 Beta representation using m

p = E(mx) implies E(Ri ) = Ī± + Ī² i,m Ī»m . Start with

1 = E(mRi ) = E(m)E(Ri ) + cov(m, Ri ).

Thus,

cov(m, Ri )

1

i

E(R ) = ā’ .

E(m) E(m)

Multiply and divide by var(m), deļ¬ne Ī± ā” 1/E(m) to get

Āµ Ā¶Āµ Ā¶

cov(m, Ri ) var(m)

i

E(R ) = Ī± + ā’ = Ī± + Ī² i,m Ī»m .

var(m) E(m)

As advertised, we have a single beta representation.

For example, we can equivalently state the consumption-based model as: mean asset

returns should be linear in the regression betas of asset returns on (ct+1 /ct )ā’Ī³ . Furthermore,

the slope of this cross-sectional relationship Ī»m is not a free parameter, though it is usually

treated as such in empirical evaluation of factor pricing models. Ī»m should equal the ratio of

99

CHAPTER 6 RELATION BETWEEN DISCOUNT FACTORS, BETAS, AND MEAN-VARIANCE FRONTIERS

variance to mean of (ct+1 /ct )ā’Ī³ .

The factor risk premium Ī»m for marginal utility growth is negative. Positive expected

returns are associated with positive correlation with consumption growth, and hence negative

correlation with marginal utility growth and m. Thus, we expect Ī»m < 0.

6.1.2 Ī² representation using xā— and Rā—

It is often useful to express a pricing model in a way that the factor is a payoff rather than a

real factor such as consumption growth. In applications, we can then avoid measurement dif-

ļ¬culties of real data. We have already seen the idea of āfactor mimicking portfoliosā formed

by projection: project m on to X, and the resulting payoff xā— also serves as a discount fac-

tor. Unsurprisingly, xā— can also serve as the factor in an expected return-beta representa-

tion. Itā™s even more useful if the reference payoff is a return. Unsurprisingly, the return

Rā— = xā— /E(xā—2 ) can also serve as the factor in a beta pricing model. When the factor is also

a return, the model is particularly simple, since the factor risk premium is also the expected

excess return.

Theorem. 1 = E(mRi ) implies an expected return - beta model with xā— =

proj(m|X) or Rā— ā” xā— /E(xā—2 ) as factors, e.g. E(Ri ) = Ī± + Ī² i,xā— Ī»xā— and

E(Ri ) = Ī± + Ī² i,Rā— [E(Rā— ) ā’ Ī±].

Proof: Recall that p = E(mx) implies p = E [proj(m | X) x], or p = E(xā— x).

Then

1 = E(mRi ) = E(xā— Ri ) = E(xā— )E(Ri ) + cov(xā— , Ri ).

Solving for the expected return,

cov(xā— , Ri ) cov(xā— , Ri ) var(xā— )

1 1

i

(81)

E(R ) = ā’ = ā’

E(xā— ) E(xā— ) E(xā— ) var(xā— ) E(xā— )

which we can write as the desired single-beta model,

E(Ri ) = Ī± + Ī² i,xā— Ī»xā— .

Notice that the zero-beta rate 1/E(xā— ) appears when there is no riskfree rate.

To derive a single beta representation with Rā— , recall the deļ¬nition,

xā—

ā—

R=

E(xā—2 )

Substituting Rā— for xā— , equation (6.81) implies that we can in fact construct a return

Rā— from m that acts as the single factor in a beta model,

Āµ Ā¶Āµ Ā¶

E(Rā—2 ) cov(Rā— , Ri ) E(Rā—2 ) cov(Rā— , Ri ) var(Rā— )

E(Ri ) = ā’ = + ā’

E(Rā— ) E(Rā— ) E(Rā— ) var(Rā— ) E(Rā— )

100

SECTION 6.2 FROM MEAN-VARIANCE FRONTIER TO A DISCOUNT FACTOR AND BETA REPRESENTATION

or, deļ¬ning Greek letters in the obvious way,

E(Ri ) = Ī± + Ī² Ri ,Rā— Ī»Rā— (82)

Since the factor Rā— is also a return, its expected excess return over the zero beta rate

gives the factor risk premium Ī»Rā— . Applying equation (6.82) to Rā— itself,

var(Rā— )

E(Rā— ) = Ī± ā’ (83)

.

E(Rā— )

So we can write the beta model in an even more traditional form

E(Ri ) = Ī± + Ī² Ri ,Rā— [E(Rā— ) ā’ Ī±]. (84)

Ā„

Recall that Rā— is the minimum second moment frontier, on the lower portion of the mean-

variance frontier. This is why Rā— has an unusual negative expected excess return or factor

risk premium, Ī»Rā— = ā’var(Rā— )/E(Rā— ) < 0. Ī± is the zero-beta rate on Rā— shown in

Figure15.

Special cases

A footnote to these constructions is that E(m), E(xā— ), or E(Rā— ) cannot be zero, or you

couldnā™t divide by them. This is a pathological case: E(m) = 0 implies a zero price for the

riskfree asset, and an inļ¬nite riskfree rate. If a riskfree rate is traded, we can simply observe

that it is not inļ¬nite and verify the fact. Also, in a complete market, E(m) cannot be zero

since, by absence of arbitrage, m > 0. We will see similar special cases in the remaining

theorems: the manipulations only work for discount factor choices that do not imply zero or

inļ¬nite riskfree rates. I discuss the issue in section 6.6

The manipulation from expected return-covariance to expected return-beta breaks down

if var(m), var(xā— ) or var(Rā— ) are zero. This is the case of pure risk neutrality. In this case,

the covariances go to zero faster than the variances, so all betas go to zero and all expected

returns become the same as the risk free rate.

6.2 From mean-variance frontier to a discount factor and beta

representation

Rmv is on mean-variance frontier ā’ m = a + bRmv ; E(Ri ) ā’ Ī± = Ī² i [E(Rmv ) ā’ Ī±]

We have seen that p = E(mx) implies a singleā’Ī² model with a mean-variance efļ¬cient

reference return, namely Rā— . The converse is also true: for (almost) any return on the mean-

variance frontier, we can deļ¬ne a discount factor m that prices assets as a linear function of

101

CHAPTER 6 RELATION BETWEEN DISCOUNT FACTORS, BETAS, AND MEAN-VARIANCE FRONTIERS

the mean-variance efļ¬cient return. Also, expected returns mechanically follow a singleā’Ī²

representation using the mean-variance efļ¬cient return as reference.

I start with the discount factor.

Theorem: There is a discount factor of the form m = a + bRmv if and only if Rmv

is on the mean-variance frontier, and Rmv is not the riskfree rate. (If there is no

riskfree rate, if Rmv is not the constant-mimicking portfolio return.)

Graphical argument

The basic idea is very simple, and Figure 19 shows the geometry for the complete markets

case. The discount factor m = xā— is proportional to Rā— . The mean-variance frontier is

Rā— + wReā— . Pick a vector Rmv on the mean-variance frontier as shown in Figure 19. Then

stretch it (bRmv ) and then subtract some of the 1 vector (a). Since Reā— is generated by the

unit vector, we can get rid of the Reā— component and get back to the discount factor xā— if we

pick the right a and b.

If the original return vector were not on the mean-variance frontier, then any linear com-

bination a + bRmv with b 6= 0 would point in some of the n direction, which Rā— and xā— do

not. If b = 0, though, just stretching up and down the 1 vector will not get us to xā— . Thus, we

can only get a discount factor of the form a + bRmv if Rmv is on the frontier.

You may remember that xā— is not the only discount factor ā“ all discount factors are of the

form m = xā— + Īµ with E(Īµx) = 0. Perhaps a + bR gives one of these discount factors, when

R is not on the mean-variance frontier? This doesnā™t work, however; n is still in the payoff

space X while, by deļ¬nition, Īµ is orthogonal to this space.

If the mean-variance efļ¬cient return Rmv that we start with happens to lie right on the

intersection of the stretched unit vector and the frontier, then stretching the Rmv vector and

adding some unit vector are the same thing, so we again canā™t get back to xā— by stretching and

adding some unit vector. The stretched unit payoff is the riskfree rate, so the theorem rules out

the riskfree rate. When there is no riskfree rate, we have to rule out the āconstant-mimicking

portfolio return.ā I treat this case in section 6.1.

Algebraic proof

Now, an algebraic proof that captures the same ideas.

Proof. For an arbitrary R, try the discount factor model

m = a + bR = a + b(Rā— + wReā— + n). (85)

We show that this discount factor prices an arbitrary payoff if and only if n = 0, and

except for the w choice that makes R the riskfree rate, or the constant-mimicking

portfolio return if there is no riskfree rate.

We can determine a and b by forcing m to price any two assets. I ļ¬nd a and b to

102

SECTION 6.2 FROM MEAN-VARIANCE FRONTIER TO A DISCOUNT FACTOR AND BETA REPRESENTATION

bRmv

ier

E - Ļ front Rf

Rmv

R*

1

x* = a + bRmv

Re*

0

Figure 19. There is a discount factor m = a + bRmv if and only if Rmv is on the

mean-variance frontier and not the risk free rate.

make the model price Rā— and Reā— .

1 = E(mRā— ) = aE(Rā— ) + bE(Rā—2 )

0 = E(mReā— ) = aE(Reā— ) + bwE(Reā—2 ) = (a + bw) E(Reā— ).

Solving for a and b,

w

a=

wE(Rā— ) ā’ E(Rā—2 )

1

b=ā’ .

ā— ) ā’ E(Rā—2 )

wE(R

Thus, if it is to price Rā— and Reā— , the discount factor must be

w ā’ (Rā— + wReā— + n)

(86)

m= .

wE(Rā— ) ā’ E(Rā—2 )

Now, letā™s see if m prices an arbitrary payoff xi . Any xi ā X can also be decom-

103

CHAPTER 6 RELATION BETWEEN DISCOUNT FACTORS, BETAS, AND MEAN-VARIANCE FRONTIERS

posed as

xi = y i Rā— + wi Reā— + ni .

(See Figure 14 if this isnā™t obvious.) The price of xi is yi , since both Reā— and ni are

zero-price (excess return) payoffs. Therefore, we want E(mxi ) = y i . Does it?

Āµ Ā¶

(w ā’ Rā— ā’ wReā— ā’ n)(y i Rā— + wi Reā— + ni )

E(mxi ) = E

wE(Rā— ) ā’ E(Rā—2 )

Using the orthogonality of Rā— , Reā— n; E(n) = 0 and E(Reā—2 ) = E(Reā— ) to sim-

plify the product,

wyi E(Rā— ) ā’ y i E(Rā—2 ) ā’ E(nni ) E(nni )

i i

E(mx ) = =y ā’ .

wE(Rā— ) ā’ E(Rā—2 ) wE(Rā— ) ā’ E(Rā—2 )

To get p(xi ) = y i = E(mxi ), we need E(nni ) = 0. The only way to guarantee

this condition for every payoff xi ā X is to insist that n = 0.

Obviously, this construction canā™t work if the denominator of (6.86) is zero, i.e. if

w = E(Rā—2 )/E(Rā— ) = 1/E(xā— ). If there is a riskfree rate, then Rf = 1/E(xā— ),

so we are ruling out the case Rmv = Rā— + Rf Reā— , which is the risk free rate. If

Ė

there is no riskfree rate, I interpret R = Rā— + E(Rā—2 )/E(Rā— )Reā— as a āconstant

mimicking portfolio returnā in section 5.3, and I give a graphical interpretation of

Ā„

this special case in section 6.1

We can generalize the theorem somewhat. Nothing is special about returns; any payoff of

the form yRā— +wReā— or yxā— +wReā— can be used to price assets; such payoffs have minimum

variance among all payoffs with given mean and price. Of course, we proved existence not

uniqueness: m = a + bRmv + Ā², E(Ā²x) = 0 also price assets as always.

To get from the mean-variance frontier to a beta pricing model, we can just chain this

theorem and the theorem of the last section together. There is a slight subtlety about special

cases when there is no riskfree rate, but since it is not important for the basic points I relegate

the direct connection and the special cases to section 6.2.

6.3 Factor models and discount factors

Beta-pricing models are equivalent to linear models for the discount factor m.

E(Ri ) = Ī± + Ī»0 Ī² i ā” m = a + b0 f

104

SECTION 6.3 FACTOR MODELS AND DISCOUNT FACTORS

We have shown that p = E(mx) implies a single beta representation using m, xā— or

Rā— as factors. Letā™s ask the converse question: suppose we have an expected return - beta

model such as CAPM, APT, ICAPM, etc. What discount factor model does this imply? I

show that an expected return - beta model is equivalent to a model for the discount factor that

is a linear function of the factors in the beta model. This is an important and central result.

It gives the connection between the discount factor formulation emphasized in this book and

the expected return/beta, factor model formulation common in empirical work.

You can write a linear factor model most compactly as m = b0 f , letting one of the factors

be a constant. However, since we want a connection to the beta representation based on

covariances rather than second moments, it is easiest to fold means of the factors in to the

constant, and write m = a + b0 f with E(f) = 0 and hence E(m) = a.

The connection is easiest to see in the special case that all the test assets are excess returns.

Then 0 = E(mRe ) does not identify the mean of m, and we can normalize a arbitrarily. I

ļ¬nd it convenient to normalize to E(m) = 1, or m = 1 + b0 [f ā’ E (f )]. Then,

Theorem: Given the model

m = 1 + b0 [f ā’ E (f )] ; 0 = E(mRe ) (87)

one can ļ¬nd Ī» such that

E(Re ) = Ī² 0 Ī» (88)

where Ī² are the multiple regression coefļ¬cients of excess returns Re on the factors.

Conversely, given Ī» in (6.88), we can ļ¬nd b such that (6.87) holds.

Proof: From (6.87)

0 = E(mRe ) = E(Re ) + b0 cov(f, Re )

E(Re ) = ā’b0 cov(f, Re ).

From covariance to beta is quick,

E(Re ) = ā’b0 var(f )var(f )ā’1 cov(f, Re ) = Ī»0 Ī²

Thus, Ī» and b are related by

Ī» = ā’var(f )b.

Ā„

When the test assets are returns, the same idea works just as well, but gets a little more

drowned in algebra since we have to keep track of the constant in m and the zero-beta rate in

the beta model.

105

CHAPTER 6 RELATION BETWEEN DISCOUNT FACTORS, BETAS, AND MEAN-VARIANCE FRONTIERS

Theorem: Given the model

m = a + b0 f, 1 = E(mRi ), (89)

one can ļ¬nd Ī± and Ī» such that

E(Ri ) = Ī± + Ī»0 Ī² i , (90)

where Ī² i are the multiple regression coefļ¬cients of Ri on f with a constant. Con-

versely, given Ī± and Ī» in a factor model of the form (6.90), one can ļ¬nd a, b such

that (6.89) holds.

Proof: We just have to construct the relation between (Ī±, Ī») and (a, b) and show

that it works. Start with m = a + b0 f , 1 = E(mR), and hence

1 E(Rf 0 )b

1 cov(m, R)

(91)

E(R) = ā’ =ā’

E(m) E(m) a a

Ī² i is the vector of the appropriate regression coefļ¬cients,

Ā” Ā¢ā’1

Ī² i ā”E ff 0 E(fRi ),

so to get Ī² in the formula, continue with

1 E(Rf 0 )E(f f 0 )ā’1 E(f f 0 )b 0

1 0 E(f f )b

E(R) = ā’ = ā’Ī²

a a a a

Now, deļ¬ne Ī± and Ī» to make it work,

1 1

(6.92)

Ī±ā” =

E (m) a

1

Ī» ā” ā’ E(ff 0 )b = ā’ Ī±E [mf ]

a

Using (6.92) we can just as easily go backwards from the expected return-beta rep-

resentation to m = a + b0 f .

As always, we have to worry about a special case of zero or inļ¬nite riskfree rates.

We rule out E(m) = E(a + b0 f ) = 0 to keep (6.91) from exploding, and we rule

Ā„

out Ī± = 0 and E(f f 0 ) singular to go from Ī±, Ī², Ī» in (6.92) back to m.

Given either model there is a model of the other form. They are not unique. We can add to

m any random variable orthogonal to returns, and we can add spurious risk factors with zero

Ī² and/or Ī» , leaving pricing implications unchanged. We can also express the multiple beta

model as a single beta model with m = a + b0 f as the single factor, or use its corresponding

Rā— .

Equation (6.92) shows that the factor risk premium Ī» can be interpreted as the price of the

factor; A test of Ī» 6= 0 is often called a test of whether the āfactor is priced.ā More precisely,

106

SECTION 6.3 FACTOR MODELS AND DISCOUNT FACTORS

Ī» captures the price E(mf ) of the (de-meaned) factors brought forward at the risk free rate.

Ė Ė Ė

If we start with underlying factors f such that the demeaned factors are f = f ā’E(f),

" #

h i Ė

Ė E(f)

Ėā’E(f) = ā’Ī±

Ė

Ī» ā” ā’Ī± p f p(f)ā’

Ī±

Ī» represents the price of the factors less their risk-neutral valuation, i.e. the factor risk pre-

mium. If the factors are not traded, Ī» is the modelā™s predicted price rather than a market

price. Low prices are high risk premia, resulting in the negative sign. If the factors are re-

turns with price one, then the factor risk premium is the expected return of the factor, less Ī±,

Ī» = E(f ) ā’ Ī±.

Note that the āfactorsā need not be returns (though they may be); they need not be orthog-

onal, and they need not be serially uncorrelated or conditionally or unconditionally mean-

zero. Such properties may occur as natural special cases, or as part of the economic deriva-

tion of speciļ¬c factor models, but they are not required for the existence of a factor pricing

representation. For example, if the riskfree rate is constant then Et (mt+1 ) is constant and at

least the sum b0f should be uncorrelated over time. But if the riskfree rate is not constant,

then Et (mt+1 ) = Et (b0f t+1 ) should vary over time.

Factor-mimicking portfolios

It is often convenient to use factor-mimicking payoffs

f ā— = proj(f|X)

factor-mimicking returns

proj(f|X)

fā— =

p [proj(f |X)]

or factor-mimicking excess returns

f ā— = proj(f|Re )

in place of true factors. These payoffs carry the same pricing information as the original

factors, and can serve as reference variables in expected return-beta representations

When the factors are not already returns or excess returns, it is convenient to express a beta

pricing model in terms of its factor mimicking portfolios rather than the factors themselves.

Recall that xā— = proj(m|X) carries all of m0 s pricing implications on X; p(x) = E(mx) =

E(xā— x). The factor-mimicking portfolios are just the same idea using the individual factors.

107

CHAPTER 6 RELATION BETWEEN DISCOUNT FACTORS, BETAS, AND MEAN-VARIANCE FRONTIERS

Deļ¬ne the payoffs f ā— by

f ā— = proj(f|X)

Then, m = b0 f ā— carries the same pricing implications on X as does m = b0 f :

p = E(mx) = E(b0 f x) = E [b0 (projf |X) x] = E [b0 f ā— x] . (93)

(I include the constant as one of the factors.)

The factor-mimicking portfolios also form a beta representation. Just go from (6.93) back

to an expected return- beta representation

E(Ri ) = Ī±ā— + Ī² ā—0 Ī»ā— , (94)

and ļ¬nd Ī»ā— , Ī±ā— using (6.92). The Ī² ā— are the regression coefļ¬cients of the returns Ri on the

factor-mimicking portfolios, not on the factors, as they should be.

It is more traditional to use the returns or excess returns on the factor-mimicking portfo-

lios rather than payoffs as I have done so far. To generate returns, divide the payoff by its

price,

proj(f |X)

fā— = .

p [proj(f|X)]

The resulting b will be scaled down by the price of the factor-mimicking payoff, and the

model is the same. Note you project on the space of payoffs, not of returns. Returns R are

not a space, since they donā™t contain zero.

If the test assets are all excess returns, you can even more easily project the factors on the

set of excess returns, which are a space since they do include zero. If we deļ¬ne

f ā— = proj(f|Re )

then of course the excess returns f ā— carry the same pricing implications as the factors f for a

set of excess returns; m = b0 f ā— satisļ¬es 0 = E(mRei ) and

E(Rei ) = Ī² i,f ā— Ī» = Ī² i,f ā— E(f ā— )

6.4 Discount factors and beta models to mean - variance frontier

From m, we can construct Rā— which is on the mean variance frontier

If a beta pricing model holds, then the return Rā— on the mean-variance frontier is a linear

combination of the factor-mimicking portfolio returns.

108

SECTION 6.5 THREE RISKFREE RATE ANALOGUES

Any frontier return is a combination of Rā— and one other return, a risk free rate or a risk

free rate proxy. Thus, any frontier return is a linear function of the factor-mimicking returns

plus a risk free rate proxy.

Itā™s easy to show that given m that we can ļ¬nd a return on the mean-variance frontier.

Given m construct xā— = proj(m|X) and Rā— = xā— /E(xā—2 ). Rā— is the minimum second

moment return, and hence on the mean-variance frontier.

Similarly, if you have a set of factors f for a beta model, then a linear combination of the

factor-mimicking portfolios is on the mean-variance frontier. A beta model is the same as

m = b0 f . Since m is linear in f , xā— is linear in f ā— = proj(f |X), so Rā— is linear in the factor

mimicking payoffs f ā— or their returns f ā— /p(f ā— ).

Section 5.4 showed how we can span the mean-variance frontier with Rā— and a risk free

rate, if there is one, or the zero-beta, minimum variance, or constant-mimicking portfolio

Ė

return R = proj(1|X)/p[proj(1|X)] if there is no risk free rate. The latter is particularly

nice in the case of a linear factor model, since we may consider the constant as a factor, so

the frontier is entirely generated by factor-mimicking portfolio returns.

6.5 Three riskfree rate analogues

I introduce three counterparts to the risk free rate that show up in asset pricing formulas

when there is no risk free rate. The three returns are the zero-beta return, the minimum-

variance return and the constant-mimicking portfolio return.

Three different generalizations of the riskfree rate are useful when a risk free rate or unit

payoff is not in the set of payoffs. These are the zero-beta return, the minimum-variance re-

turn and the constant-mimicking portfolio return. I introduce the returns in this section, and I

use them in the next section to state some special cases involving the mean-variance frontier.

Each of these returns maintains one property of the risk free rate in a market in which there

is no risk free rate. The zero-beta return is a mean-variance efļ¬cient return that is uncorre-

lated with another given mean-variance efļ¬cient return. The minimum-variance return is just

that. The constant-mimicking portfolio return is the return on the payoff āclosestā to the unit

payoff. Each of these returns one has a representation in the standard form Rā— + wReā— with

slightly different w. In addition, the expected returns of these risky assets are used in some

asset pricing representations. For example, the zero beta rate is often used to refer to the

expected value of the zero beta return.

Each of these riskfree rate analogues is mean-variance efļ¬cient. Thus, I characterize each

one by ļ¬nding its weight w in a representation of the form Rā— + wReā— . We derived such a

109

CHAPTER 6 RELATION BETWEEN DISCOUNT FACTORS, BETAS, AND MEAN-VARIANCE FRONTIERS

representation above for the riskfree rate as equation (5.72),

Rf = Rā— + Rf Reā— . (95)

In the last subsection, I show how each riskfree rate analogue reduces to the riskfree rate

when there is one.

6.5.1 Zero-beta return for Rā—

The zero beta return for Rā— , denoted RĪ± , is the mean-variance efļ¬cient return uncorre-

lated with Rā— . Its expected return is the zero beta rate Ī± = E(Ra ). This zero beta return has

representation

var(Rā— )

a ā—

Reā— ,

R =R + ā— )E(Reā— )

E(R

and the corresponding zero beta rate is

E(Rā—2 ) 1

Ī± = E(RĪ± ) = = .

E(Rā— ) E(xā— )

The zero beta rate is found graphically in mean-standard deviation space by extending the

tangency at Rā— to the vertical axis. It is also the inverse of the price that xā— and Rā— assign to

the unit payoff.

The riskfree rate Rf is of course uncorrelated with Rā— . Risky returns uncorrelated with

Rā— earn the same average return as the risk free rate if there is one, so they might take the

place of Rf when the latter does not exist. For any return RĪ± that is uncorrelated with Rā— we

have E(Rā— RĪ± ) = E(Rā— )E(RĪ± ), so

E(Rā—2 ) 1

Ī±

Ī± = E(R ) = = .

E(Rā— ) E(xā— )

The ļ¬rst equality introduces a popular notation Ī± for this rate. I call Ī± the zero beta rate, and

Ra the zero beta return. There is no riskfree rate, so there is no security that just pays Ī±.

As you can see from the formula, the zero-beta rate is the inverse of the price that Rā— and

xā— assign to the unit payoff, which is another natural generalization of the riskfree rate. It is

called the zero beta rate because cov(Rā— , RĪ± ) = 0 implies that the regression beta of RĪ± on

Rā— is zero. More precisely, one might call it the zero beta rate on Rā— , since one can calculate

zero-beta rates for returns other than Rā— and they are not the same as the zero-beta rate for

Rā— In particular, the zero-beta rate on the āmarket portfolioā will generally be different from

the zero beta rate on Rā— .

110

SECTION 6.5 THREE RISKFREE RATE ANALOGUES

E(R)

Ī±=

RĪ±

E(R*2 )/ E(R* )

= 1/E(x*)

R*

Ļ(R)

Figure 20. Zero-beta rate Ī± and zero-beta return Ra for Rā— .

111

CHAPTER 6 RELATION BETWEEN DISCOUNT FACTORS, BETAS, AND MEAN-VARIANCE FRONTIERS

I draw Ī± in Figure 20 as the intersection of the tangency and the vertical axis. This

is a property of any return on the mean variance frontier: The expected return on an asset

uncorrelated with the mean-variance efļ¬cient asset (a zero-beta asset) lies at the point so

constructed. To check this geometry, use similar triangles: p length ofp ā— in Figure 20 is

The R

p

ā—2 ), and its vertical extent is E(Rā— ). Therefore, Ī±/ E(Rā—2 ) = E(Rā—2 )/E(Rā— ),

E(R

or Ī± = E(Rā—2 )/E(Rā— ). Since Rā— is on the lower portion of the mean-variance frontier, this

zero beta rate Ī± is above the minimum variance return.

Note that in general Ī± 6= 1/E(m). Projecting m on X preserves asset pricing implica-

tions on X but not for payoffs not in X. Thus if a risk free rate is not traded, xā— and m may

differ in their predictions for the riskfree rate as for other nontraded assets.

The zero beta return is the rate of return on the mean-variance frontier with mean equal to

the zero beta rate, as shown in Figure 20. We want to characterize this return in Rā— + wReā—

form. To do this, we want to ļ¬nd w such that

E(Rā—2 )

E(Ra ) = = E(Rā— ) + wE(Reā— ).

ā—)

E(R

Solving, the answer is

E(Rā—2 ) ā’ E(Rā— )2 var(Rā— )

w= = .

E(Rā— )E(Reā— ) E(Rā— )E(Reā— )

Thus, the zero beta return is

var(Rā— )

a ā—

Reā— ,

R =R +

E(Rā— )E(Reā— )

expression (6.103). Note that the weight is not E(Ra ) = E(Rā—2 )/E(Rā— ). When there is no

risk free rate, the weight and the mean return are different.

6.5.2 Minimum variance return

The minimum variance return has the representation

E(Rā— )

min. var. ā—

Reā— .

R =R +

1 ā’ E(Reā— )

The riskfree rate obviously is the minimum variance return when it exists. When there is

no risk free rate, the minimum variance return is

E(Rā— )

Rmin. var. = Rā— + Reā— . (96)

eā— )

1 ā’ E(R

112

SECTION 6.5 THREE RISKFREE RATE ANALOGUES

Taking expectations,

E(Rā— ) E(Rā— )

E(Rmin. var. ) = E(Rā— ) + E(Reā— ) = .

1 ā’ E(Reā— ) 1 ā’ E(Reā— )

The minimum variance return retains the nice property of the risk free rate, that its weight on

Reā— is the same as its mean,

Rmin. var. = Rā— + E(Rmin. var. )Reā—

just as Rf = Rā— + Rf Reā— . When there is no risk free rate, the zero-beta and minimum

variance returns are not the same. You can see this fact clearly in Figure 20.

We can derive expression (6.96) for the minimum variance return by brute force: choose

w in Rā— + wReā— to minimize variance.

min var(Rā— + wReā— ) = E[(Rā— + wReā— )2 ] ā’ E(Rā— + wReā— )2 =

w

= E(Rā—2 ) + w2 E(Reā— ) ā’ E(Rā— )2 ā’ 2wE(Rā— )E(Reā— ) ā’ w2 E(Reā— )2 .

The ļ¬rst order condition is

0 = wE(Reā— )[1 ā’ E(Reā— )] ā’ E(Rā— )E(Reā— )

E(Rā— )

w= .

1 ā’ E(Reā— )

6.5.3 Constant-mimicking portfolio return

The constant-mimicking portfolio return is deļ¬ned as the return on the projection of the

unit vector on the payoff space,

proj(1|X)

Ė

R= .

p [proj(1|X)]

It has the representation

ā—2

Ė = Rā— + E(R ) Reā— .

R

E(Rā— )

When there is a risk free rate, it is the rate of return on a unit payoff, Rf = 1/p(1). When

there is no risk free rate, we might deļ¬ne the rate of return on the mimicking portfolio for a

113

CHAPTER 6 RELATION BETWEEN DISCOUNT FACTORS, BETAS, AND MEAN-VARIANCE FRONTIERS

unit payoff,

proj(1|X)

Ė

R= .

p [proj(1|X)]

I call this object the constant-mimicking portfolio return.

The mean-variance representation of the constant-mimicking portfolio return is

E(Rā—2 ) eā—

Ė

R = Rā— + Ī±Reā— = Rā— + (97)

R.

E(Rā— )

Note that the weight Ī± equal to the zero beta rate creates the constant-mimicking return, not

the zero beta return. To show (6.97), start with property (5.73),

E(Rā— ) ā—

eā—

(98)

R = proj(1|X) ā’ R.

E(Rā—2 )

Take the price of both sides. Since the price of Reā— is zero and the price of Rā— is one, we

establish

E(Rā— )

(99a)

p [proj(1|X)] = .

E(Rā—2 )

Solving (6.98) for proj(1|X), dividing by (6.99a) we obtain the right hand side of (6.97).

6.5.4 Risk free rate

The risk free rate has the mean-variance representation

Rf = Rā— + Rf Reā— .

The zero-beta, minimum variance and constant-mimicking portfolio returns reduce to this

formula when there is a risk free rate.

Again, we derived in equation (5.72) that the riskfree rate has the representation,

Rf = Rā— + Rf Reā— . (100)

Obviously, we should expect that the zero-beta return, minimum-variance return, and constant-

mimicking portfolio return reduce to the riskfree rate when there is one. These other rates

are

E(Rā—2 ) eā—

Ė

constant-mimicking: R = Rā— + (101)

R

E(Rā— )

114

SECTION 6.6 MEAN-VARIANCE SPECIAL CASES WITH NO RISKFREE RATE

E(Rā— )

minimum-variance: Rmin. var. = Rā— + Reā— (102)

eā— )

1 ā’ E(R

var(Rā— )

Ī± ā—

Reā— .

zero-beta: R = R + (103)

ā— )E(Reā— )

E(R

To establish that these are all the same when there is a riskfree rate, we need to show that

E(Rā—2 ) E(Rā— ) var(Rā— )

f

(104)

R= = =

E(Rā— ) 1 ā’ E(Reā— ) E(Rā— )E(Reā— )

We derived the ļ¬rst equality above as equation (5.74). To derive the second equality, take

expectations of (6.95),

Rf = E(Rā— ) + Rf E(Reā— ) (105)

and solve for Rf . To derive the third equality, use the ļ¬rst equality from (6.104) in (6.105),

E(Rā—2 )

= E(Rā— ) + Rf E(Reā— ).

ā—)

E(R

Solving for Rf ,

E(Rā—2 ) ā’ E(Rā— )2 var(Rā— )

Rf = = .

E(Rā— )E(Reā— ) E(Rā— )E(Reā— )

6.6 Mean-variance special cases with no riskfree rate

We can ļ¬nd a discount factor from any mean-variance efļ¬cient return except the constant-

mimicking return.

We can ļ¬nd a beta representation from any mean-variance efļ¬cient return except the

minimum-variance return.

I collect in this section the special cases for the equivalence theorems of this chapter. The

special cases all revolve around the problem that the expected discount factor, price of a unit

payoff or riskfree rate must not be zero or inļ¬nity. This is typically an issue of theoretical

rather than practical importance. In a complete, arbitrage free market, m > 0 so we know

E(m) > 0. If a riskfree rate is traded you can observe ā > E(m) = 1/Rf > 0. However,

in an incomplete market in which no riskfree rate is traded, there are many discount factors

with the same asset pricing implications, and you might have happened to choose one with

E(m) = 0 in your manipulations. By and large, this is easy to avoid: choose another of the

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CHAPTER 6 RELATION BETWEEN DISCOUNT FACTORS, BETAS, AND MEAN-VARIANCE FRONTIERS

many discount factors with the same pricing implications that does not have E(m) = 0. More

generally, when you choose a particular discount factor you are choosing an extension of the

current set of prices and payoffs; you are viewing the current prices and payoffs as a subset

of a particular contingent-claim economy. Make sure you pick a sensible one. Therefore, we

could simply state the special cases as āwhen a riskfree rate is not traded, make sure you use

discount factors with 0 < E(m) < ā.ā However, it is potentially useful and it certainly is

traditional to specify the special return on the mean-variance frontier that leads to the inļ¬nite

or zero implied riskfree rate, and to rule it out directly. This section works out what those

returns are and shows why they must be avoided.

6.6.1 The special case for mean variance frontier to discount factor

When there is no riskfree rate, we can ļ¬nd a discount factor that is a linear function of

any mean-variance efļ¬cient return except the constant-mimicking portfolio return.

In section 6.2, we saw that we can form a discount factor a + bRmv from any mean-

ā—2

variance efļ¬cient return Rmv except one particular return, of the form Rā— + E(R ā— )) Reā— . This

E(R

return led to an inļ¬nite m. We now recognize this return as the risk-free rate, when there is

one, or the constant-mimicking portfolio return, if there is no riskfree rate.

Figure 21 shows the geometry of this case. To use no more than three dimensions I had to

reduce the return and excess return spaces to lines. The payoff space X is the plane joining

the return and excess return sets as shown. The set of all discount factors is m = xā— + Īµ,

E(Īµx) = 0, the line through xā— orthogonal to the payoff space X in the ļ¬gure. I draw the

unit payoff (the dot marked ā1ā in Figure 21) closer to the viewer than the plane X, and I

draw a vector through the unit payoff coming out of the page.

Take any return on the mean-variance frontier, Rmv . (Since the return space only has two

dimensions, all returns are on the frontier.) For a given Rmv , the space a + bRmv is the plane

spanned by Rmv and the unit payoff. This plane lies sideways in the ļ¬gure. As the ļ¬gure

shows, there is a vector a + bRmv in this plane that lies on the line of discount factors.

Next, the special case. This construction would go awry if the plane spanning the unit

payoff and the return Rmv were parallel to the plane containing the discount factor. Thus,

Ė

the construction would not work for the return marked R in the Figure. This is a return

corresponding to a payoff that is the projection of the unit payoff on to X, so that the residual

will be orthogonal to X, as is the line of discount factors.

With Figure 21 in front of us, we can also see why the constant-mimicking portfolio return

is not the same thing as the minimum-variance return. Variance is the size or second moment

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SECTION 6.6 MEAN-VARIANCE SPECIAL CASES WITH NO RISKFREE RATE

^

Discount factors Constant-mimicking return R

Rmv R*

R

a+b Rmv x*

1

X

Re

0

Figure 21. One can construct a discount factor m = a + bRmv from any

Ė

mean-variance-efļ¬cient return except the constant-mimicking return R.

of the residual in a projection (regression) on 1.

Ā£ Ā¤ Ā£ Ā¤

var(x) = E (x ā’ E(x))2 = E (x ā’ proj(x|1))2 = ||x ā’ proj(x|1)||2

Thus, the minimum variance return is the return closest to extensions of the unit vector. It is

formed by projecting returns on the unit vector. The constant-mimicking portfolio return is

the return on the payoff closest to 1 It is formed by projecting the unit vector on the set of

payoffs.

6.6.2 The special case for mean-variance frontier to a beta model

We can use any return on the mean-variance frontier as the reference return for a single

beta representation, except the minimum-variance return.

We already know mean variance frontiers ā” discount factor and discount factor ā” single

beta representation, so at a superļ¬cial level we can string the two theorems together to go

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CHAPTER 6 RELATION BETWEEN DISCOUNT FACTORS, BETAS, AND MEAN-VARIANCE FRONTIERS

from a mean-variance efļ¬cient return to a beta representation. However it is more elegant to

go directly, and the special cases are also a bit simpler this way.

Theorem: There is a single beta representation with a return Rmv as factor,

E(Ri ) = Ī±Rmv + Ī² i,Rmv [E(Rmv ) ā’ Ī±] ,

if and only if Rmv is mean-variance efļ¬cient and not the minimum variance return.

This famous theorem is given by Roll (1976) and Hansen and Richard (1987). We rule

out minimum variance to rule out the special case E(m) = 0. Graphically, the zero-beta rate

is formed from the tangency to the mean-variance frontier as in Figure 20. I use the notation

Ī±Rmv to emphasize that we use the zero-beta rate corresponding to the particular mean-

variance return Rmv that we use as the reference return. If we used the minimum-variance

return, that would lead to an inļ¬nite zero-beta rate.

Proof: The mean-variance frontier is Rmv = Rā— + wReā— . Any return is Ri =

Rā— + wi Reā— + ni . Thus,

E(Ri ) = E(Rā— ) + wi E(Reā— ) (106)

ńņš. 4 |