<< ŮÚū. 4(‚ŮŚ„Ó 17)—őńŇ–∆ņÕ»Ň >>
prices on X via an inner product. Re‚ą— is orthogonal to planes of constant mean in Re as x‚ą— is
orthogonal to planes of constant price. Algebraically, in analogy to p(x) = E(x‚ą— x) we have

E(Re ) = E(Re‚ą— Re ) ‚ąÄRe ‚ąą Re . (70)

This fact can serve as an alternative deÔ¬Āning property of Re‚ą— .
5) Re‚ą— and R‚ą— are orthogonal,

E(R‚ą— Re‚ą— ) = 0.

More generally, R‚ą— is orthogonal to any excess return.
6) The mean variance frontier is given by

Rmv = R‚ą— + wRe‚ą— .
¬£ ¬¤
To prove this, E(R2 ) = E (R‚ą— + wRe‚ą— + n)2 = E(R‚ą—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) R‚ą— is the minimum second moment return. Graphically, R‚ą— 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) Re‚ą— has the same Ô¬Ārst and second moment,

E(Re‚ą— ) = E(Re‚ą—2 ).

Just apply fact (5.70) to Re‚ą— itself. Therefore

var(Re‚ą— ) = E(Re‚ą—2 ) ‚ą’ E(Re‚ą— )2 = E(Re‚ą— ) [1 ‚ą’ E(Re‚ą— )] .

9) If there is a riskfree rate, then Re‚ą— can also be deÔ¬Āned as the residual in the projection
of 1 on R‚ą— :
E(R‚ą— ) ‚ą— 1
Re‚ą— = 1 ‚ą’ proj(1|R‚ą— ) = 1 ‚ą’ R = 1 ‚ą’ f R‚ą— (71)
E(R‚ą—2 ) R

You‚Ä™d never have thought of this without looking at Figure 14! Since R‚ą— and Re are orthog-
onal and together span X, 1 = proj(1|Re ) + proj(1|R‚ą— ). You can also verify this statement
analytically. Check that Re‚ą— so deÔ¬Āned is an excess return in X ‚Ä“ its price is zero‚Ä“, and
E(Re‚ą— Re ) = E(Re ); E(R‚ą— Re‚ą— ) = 0.

90
SECTION 5.5 A COMPILATION OF PROPERTIES OF R‚ą— , Re‚ą— AND x‚ą—

As a result, Rf has the decomposition

Rf = R‚ą— + Rf Re‚ą— . (72)

Since Rf > 1 typically, this means that R‚ą— +Re‚ą— 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 = R‚ą— + Re‚ą— . 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 Re‚ą— that you add to R‚ą— 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) |R‚ą— )
= proj(1|Re ) + proj(1|R‚ą— )

to deduce an analogue to equation (5.71),

E(R‚ą— ) ‚ą—
e‚ą— ‚ą—
(73)
R = proj(1|X) ‚ą’ proj(1|R ) = proj(1|X) ‚ą’ R
E(R‚ą—2 )

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

E(R‚ą—2 )
1
f
(74)
R= = .
E(x‚ą— ) E(R‚ą— )

If not, this gives a ‚Äúzero beta rate‚ÄĚ interpretation of the right hand expression.
11) Since we have a formula x‚ą— = p0 E(xx0 )‚ą’1 x for constructing x‚ą— from basis assets
(see section 4.1), we can construct R‚ą— in this case from

x‚ą— p0 E(xx0 )‚ą’1 x
‚ą—
R= =0 .
p(x‚ą— ) p E(xx0 )‚ą’1 p

(p(x‚ą— ) = E(x‚ą— x‚ą— ) leading to the denominator.)
12) We can construct Re‚ą— from a set of basis assets as well. Following the deÔ¬Ānition to
project one on the space of excess returns,

Re‚ą— = E(Re )0 E(Re Re0 )‚ą’1 Re

where Re is the basis set of excess returns. (You can always use Re = R‚ą’R‚ą— if you want).
This construction obviously mirrors the way we constructed x‚ą— in section 4.1, and you can
see the similarity in the result, with E in place of p, since Re‚ą— 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 )‚ą’1 x
1‚ą—
Re‚ą— = 1 ‚ą’ (75)
R =1‚ą’ f 0 .
Rf R p E(xx0 )‚ą’1 p

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

proj(1|X) = E(x)0 E(xx0 )‚ą’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 )|
ŌÉ(m)
‚Č• .
ŌÉ(Re )
E(m)

and hence
E(Re )
ŌÉ(m)
min = max e
{all excess returns R in X} ŌÉ(Re )
{all m that price x‚ąąX} E(m)

The discount factors on the frontier can be characterized analogously to the mean-variance
frontier of asset returns,

m = x‚ą— + we‚ą—
0
e‚ą— ‚Č° 1 ‚ą’ proj(1|X) = proj(1|E) = 1 ‚ą’ E(x) E(xx0 )‚ą’1 x
E = {m ‚ą’ x‚ą— } .

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 )|
ŌÉ(m)
‚Č• .
ŌÉ(Re )
E(m)

Quickly,

0 = E(mRe ) = E(m)E(Re ) + ŌĀm,Re ŌÉ(m)ŌÉ(Re ),

92
SECTION 5.6 MEAN-VARIANCE FRONTIERS FOR m: THE HANSEN-JAGANNATHAN BOUNDS

and |ŌĀ| ‚Č¤ 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 speciÔ¬Āc 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), ŌÉ(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 we‚Ä™re after. For any hypothetical risk-free rate, Ô¬Ānd the highest Sharpe
ratio. That is, of course the tangency portfolio. Then the slope to the tangency portfolio gives
the ratio ŌÉ(m)/E(m). Figure 16 illustrates.

E(R)

ŌÉ(m) =
ő•(Re)/ŌÉ(Re)

1/E(m) E(Re)/ŌÉ(Re)

ŌÉ(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 )
ŌÉ(m)
(76)
min = max e .
{all excess returns R in X} ŌÉ(Re )
{all m that price x‚ąąX} 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 ‚Ä“ that satisfy p = E(mx) :

m = E(m) + [p ‚ą’ E(m)E(x)] ő£‚ą’1 [x ‚ą’ E(x)] + őµ (77)

where ő£ ‚Č° cov(x, x0 ) and E(őµ) = 0, E(őµx) = 0. You can think of this as a regression or
projection of any discount factor on the space of payoffs, plus an error. Since ŌÉ2 (őµ) > 0,
this representation leads immediately to an explicit expression for the Hansen-Jagannathan
bound,
0
ŌÉ 2 (m) ‚Č• (p ‚ą’ E(m)E(x)) ő£‚ą’1 (p ‚ą’ E(m)E(x)) . (78)

As all asset returns must lie in a cup-shaped region in {E(R), ŌÉ(R)} space, all discount
factors must lie in a parabolic region in E(m), ŌÉ 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 Ô¬Ānd 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 x‚ą— . Any m must
be of the form

m = x‚ą— + we‚ą— + n.

Here, I have just broken up the residual őµ in the familiar representation m = x‚ą— + őµ into two
components. e‚ą— is deÔ¬Āned as the residual from the projection of 1 onto X or, equivalently the
projection of 1 on the space E of ‚Äúexcess m‚Ä™s,‚ÄĚ random variables of the form m ‚ą’ x‚ą— .

e‚ą— ‚Č° 1 ‚ą’ proj(1|X) = proj(1|E).

e‚ą— generates means of m just as Re‚ą— did for returns:

E(m ‚ą’ x‚ą— ) = E[1 √— (m ‚ą’ x‚ą— )] = E[proj(1|E)(m ‚ą’ x‚ą— )].

Finally n, deÔ¬Āned as the leftovers, has mean zero since it‚Ä™s orthogonal to 1 and is orthogonal
to X.
As with returns, then, the mean-variance frontier of m0 s is given by

m‚ą— = x‚ą— + we‚ą— . (79)

If the unit payoff is in the payoff space, then we know E(m), and the frontier and bound
are just m = x‚ą— , ŌÉ 2 (m) ‚Č• ŌÉ2 (x‚ą— ). 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 = x‚ą— + 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 Ô¬Ānite-dimensional cases discussed above. It‚Ä™s more general, since it can be used
in inÔ¬Ānite-dimensional payoff spaces as well. Along with the corresponding return formula
Rmv = R‚ą— + wRe‚ą— , 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 Ô¬Ānd equations for its components. We
Ô¬Ānd x‚ą— as before, it is the portfolio c0 x in X that prices x:

x‚ą— = p0 E(xx0 )‚ą’1 x.

Similarly, let‚Ä™s Ô¬Ānd e‚ą— . The projection of 1 on X is

proj(1|X) = E(x)0 E(xx0 )‚ą’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,

e‚ą— = 1 ‚ą’ E(x)0 E(xx0 )‚ą’1 x.

Again, you can construct time-series of x‚ą— and e‚ą— from these deÔ¬Ānitions.
Finally, we now can construct the variance-minimizing discount factors
¬£ ¬¤
m‚ą— = x‚ą— + we‚ą— = p0 E(xx0 )‚ą’1 x + w 1 ‚ą’ E(x)0 E(xx0 )‚ą’1 x

or
0
m‚ą— = w + [p ‚ą’ wE(x)] E(xx0 )‚ą’1 x (80)

As w varies, we trace out discount factors m‚ą— on the frontier with varying means and vari-
ances. It‚Ä™s easiest to Ô¬Ānd mean and second moment:

E(m‚ą— ) = w + [p ‚ą’ wE(x)]0 E(xx0 )‚ą’1 E(x)

0
E(m‚ą—2 ) = [p ‚ą’ wE(x)] E(xx0 )‚ą’1 [p ‚ą’ wE(x)] ;

variance follows from ŌÉ 2 (m) = E(m2 ) ‚ą’ 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 efÔ¬Āciency 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 ŌÉ2 (m) s.t. p = E(mx), m > 0.
This is the ‚ÄúHansen-Jagannathan bound with positivity,‚ÄĚ 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 ŌĀ = 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 Re‚ą— lies at right angles to planes (in Re ) of constant mean return, as shown in
Ô¬Āgure 14.
2. Should we typically draw x‚ą— above, below or on the plane of returns? Must x‚ą— always lie
in this position?
3. Can you construct Re‚ą— from knowledge of m, x‚ą— , or R‚ą— ?
4. What happens to R‚ą— , Re‚ą— 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. x‚ą— = proj(m|X). Is R‚ą— = proj(m|R)?
6. We showed that all m are of the form x‚ą— + őµ. What about R‚ą’1 R?
7. Show that if there is a risk-free rate‚Ä”if the unit payoff is in the payoff space X‚Ä”then
Re‚ą— = (Rf ‚ą’ R‚ą— )/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 Ô¬Ārst 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 ‚Ä“ the thing you put on the right hand side in the regressions that
give betas ‚Ä“ and a return on the mean-variance frontier all carry the same information, and
given any one of them, you can Ô¬Ānd the others. More speciÔ¬Ācally,

1. p = E(mx) ‚á’ ő≤: Given m such that p = E(mx), m, x‚ą— , R‚ą— , or R‚ą— + wRe‚ą— all can
serve as reference variables for betas.
2. p = E(mx) ‚á’ mean-variance frontier. You can construct R‚ą— from x‚ą— = proj(m|X),
R‚ą— = x‚ą— /E(x‚ą—2 ), and then R‚ą— , R‚ą— + wRe‚ą— are on the mean-variance frontier.
3. Mean-variance frontier ‚á’ p = E(mx). If Rmv is on the mean-variance frontier, then
m = a + bRmv linear in that return is a discount factor; it satisÔ¬Āes p = E(mx).
4. ő≤ ‚á’ p = E(mx). If we have an expected return/beta model with factors f , then
m = b0 f linear in the factors satisÔ¬Āes 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, x‚ą— , and R‚ą— can all be the single factor in a single beta representation.

98
SECTION 6.1 FROM DISCOUNT FACTORS TO BETA REPRESENTATIONS

LOOP √ě m exists

p = E(mx)
f = m, x*, R*

m = b‚Ä™f m = a + bRmv
R* is on m.v.f.

E(Ri) = ő± + ő≤i‚Ä™ő»

f = Rmv

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

E(RR‚Ä™) nonsingular √ě Rmv exists

Figure 18. Relation between three views of asset pricing.

6.1.1 Beta representation using m

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

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

Thus,
cov(m, Ri )
1
i
E(R ) = ‚ą’ .
E(m) E(m)
Multiply and divide by var(m), deÔ¬Āne ő± ‚Č° 1/E(m) to get
¬µ ¬∂¬µ ¬∂
cov(m, Ri ) var(m)
i
E(R ) = ő± + ‚ą’ = ő± + ő≤ i,m ő»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 )‚ą’ő≥ . Furthermore,
the slope of this cross-sectional relationship ő»m is not a free parameter, though it is usually
treated as such in empirical evaluation of factor pricing models. ő»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 )‚ą’ő≥ .
The factor risk premium ő»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 ő»m < 0.

6.1.2 ő≤ representation using x‚ą— and R‚ą—

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-
Ô¬Āculties of real data. We have already seen the idea of ‚Äúfactor mimicking portfolios‚ÄĚ formed
by projection: project m on to X, and the resulting payoff x‚ą— also serves as a discount fac-
tor. Unsurprisingly, x‚ą— can also serve as the factor in an expected return-beta representa-
tion. It‚Ä™s even more useful if the reference payoff is a return. Unsurprisingly, the return
R‚ą— = x‚ą— /E(x‚ą—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 x‚ą— =
proj(m|X) or R‚ą— ‚Č° x‚ą— /E(x‚ą—2 ) as factors, e.g. E(Ri ) = ő± + ő≤ i,x‚ą— ő»x‚ą— and
E(Ri ) = ő± + ő≤ i,R‚ą— [E(R‚ą— ) ‚ą’ ő±].
Proof: Recall that p = E(mx) implies p = E [proj(m | X) x], or p = E(x‚ą— x).
Then

1 = E(mRi ) = E(x‚ą— Ri ) = E(x‚ą— )E(Ri ) + cov(x‚ą— , Ri ).

Solving for the expected return,
cov(x‚ą— , Ri ) cov(x‚ą— , Ri ) var(x‚ą— )
1 1
i
(81)
E(R ) = ‚ą’ = ‚ą’
E(x‚ą— ) E(x‚ą— ) E(x‚ą— ) var(x‚ą— ) E(x‚ą— )
which we can write as the desired single-beta model,

E(Ri ) = ő± + ő≤ i,x‚ą— ő»x‚ą— .

Notice that the zero-beta rate 1/E(x‚ą— ) appears when there is no riskfree rate.
To derive a single beta representation with R‚ą— , recall the deÔ¬Ānition,
x‚ą—
‚ą—
R=
E(x‚ą—2 )
Substituting R‚ą— for x‚ą— , equation (6.81) implies that we can in fact construct a return
R‚ą— from m that acts as the single factor in a beta model,
¬µ ¬∂¬µ ¬∂
E(R‚ą—2 ) cov(R‚ą— , Ri ) E(R‚ą—2 ) cov(R‚ą— , Ri ) var(R‚ą— )
E(Ri ) = ‚ą’ = + ‚ą’
E(R‚ą— ) E(R‚ą— ) E(R‚ą— ) var(R‚ą— ) E(R‚ą— )

100
SECTION 6.2 FROM MEAN-VARIANCE FRONTIER TO A DISCOUNT FACTOR AND BETA REPRESENTATION

or, deÔ¬Āning Greek letters in the obvious way,
E(Ri ) = ő± + ő≤ Ri ,R‚ą— ő»R‚ą— (82)
Since the factor R‚ą— is also a return, its expected excess return over the zero beta rate
gives the factor risk premium ő»R‚ą— . Applying equation (6.82) to R‚ą— itself,
var(R‚ą— )
E(R‚ą— ) = ő± ‚ą’ (83)
.
E(R‚ą— )
So we can write the beta model in an even more traditional form
E(Ri ) = ő± + ő≤ Ri ,R‚ą— [E(R‚ą— ) ‚ą’ ő±]. (84)
¬•

Recall that R‚ą— is the minimum second moment frontier, on the lower portion of the mean-
variance frontier. This is why R‚ą— has an unusual negative expected excess return or factor
risk premium, ő»R‚ą— = ‚ą’var(R‚ą— )/E(R‚ą— ) < 0. ő± is the zero-beta rate on R‚ą— shown in
Figure15.
Special cases
A footnote to these constructions is that E(m), E(x‚ą— ), or E(R‚ą— ) cannot be zero, or you
couldn‚Ä™t divide by them. This is a pathological case: E(m) = 0 implies a zero price for the
riskfree asset, and an inÔ¬Ānite riskfree rate. If a riskfree rate is traded, we can simply observe
that it is not inÔ¬Ānite 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
inÔ¬Ānite 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(x‚ą— ) or var(R‚ą— ) 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 ‚á’ m = a + bRmv ; E(Ri ) ‚ą’ ő± = ő≤ i [E(Rmv ) ‚ą’ ő±]

We have seen that p = E(mx) implies a single‚ą’ő≤ model with a mean-variance efÔ¬Ācient
reference return, namely R‚ą— . The converse is also true: for (almost) any return on the mean-
variance frontier, we can deÔ¬Āne 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 efÔ¬Ācient return. Also, expected returns mechanically follow a single‚ą’ő≤
representation using the mean-variance efÔ¬Ācient return as reference.
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 = x‚ą— is proportional to R‚ą— . The mean-variance frontier is
R‚ą— + wRe‚ą— . 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 Re‚ą— is generated by the
unit vector, we can get rid of the Re‚ą— component and get back to the discount factor x‚ą— 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 R‚ą— and x‚ą— do
not. If b = 0, though, just stretching up and down the 1 vector will not get us to x‚ą— . Thus, we
can only get a discount factor of the form a + bRmv if Rmv is on the frontier.
You may remember that x‚ą— is not the only discount factor ‚Ä“ all discount factors are of the
form m = x‚ą— + őµ with E(őµx) = 0. Perhaps a + bR gives one of these discount factors, when
R is not on the mean-variance frontier? This doesn‚Ä™t work, however; n is still in the payoff
space X while, by deÔ¬Ānition, őµ is orthogonal to this space.
If the mean-variance efÔ¬Ācient 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 can‚Ä™t get back to x‚ą— 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 ‚Äúconstant-mimicking
portfolio return.‚ÄĚ 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(R‚ą— + wRe‚ą— + 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 Ô¬Ānd a and b to

102
SECTION 6.2 FROM MEAN-VARIANCE FRONTIER TO A DISCOUNT FACTOR AND BETA REPRESENTATION

bRmv

ier
E - ŌÉ 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 R‚ą— and Re‚ą— .

1 = E(mR‚ą— ) = aE(R‚ą— ) + bE(R‚ą—2 )
0 = E(mRe‚ą— ) = aE(Re‚ą— ) + bwE(Re‚ą—2 ) = (a + bw) E(Re‚ą— ).

Solving for a and b,
w
a=
wE(R‚ą— ) ‚ą’ E(R‚ą—2 )
1
b=‚ą’ .
‚ą— ) ‚ą’ E(R‚ą—2 )
wE(R
Thus, if it is to price R‚ą— and Re‚ą— , the discount factor must be

w ‚ą’ (R‚ą— + wRe‚ą— + n)
(86)
m= .
wE(R‚ą— ) ‚ą’ E(R‚ą—2 )

Now, let‚Ä™s see if m prices an arbitrary payoff xi . Any xi ‚ąą X can also be decom-

103
CHAPTER 6 RELATION BETWEEN DISCOUNT FACTORS, BETAS, AND MEAN-VARIANCE FRONTIERS

posed as

xi = y i R‚ą— + wi Re‚ą— + ni .

(See Figure 14 if this isn‚Ä™t obvious.) The price of xi is yi , since both Re‚ą— and ni are
zero-price (excess return) payoffs. Therefore, we want E(mxi ) = y i . Does it?
¬µ ¬∂
(w ‚ą’ R‚ą— ‚ą’ wRe‚ą— ‚ą’ n)(y i R‚ą— + wi Re‚ą— + ni )
E(mxi ) = E
wE(R‚ą— ) ‚ą’ E(R‚ą—2 )

Using the orthogonality of R‚ą— , Re‚ą— n; E(n) = 0 and E(Re‚ą—2 ) = E(Re‚ą— ) to sim-
plify the product,

wyi E(R‚ą— ) ‚ą’ y i E(R‚ą—2 ) ‚ą’ E(nni ) E(nni )
i i
E(mx ) = =y ‚ą’ .
wE(R‚ą— ) ‚ą’ E(R‚ą—2 ) wE(R‚ą— ) ‚ą’ E(R‚ą—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 ‚ąą X is to insist that n = 0.
Obviously, this construction can‚Ä™t work if the denominator of (6.86) is zero, i.e. if
w = E(R‚ą—2 )/E(R‚ą— ) = 1/E(x‚ą— ). If there is a riskfree rate, then Rf = 1/E(x‚ą— ),
so we are ruling out the case Rmv = R‚ą— + Rf Re‚ą— , which is the risk free rate. If
ňÜ
there is no riskfree rate, I interpret R = R‚ą— + E(R‚ą—2 )/E(R‚ą— )Re‚ą— as a ‚Äúconstant
mimicking portfolio return‚ÄĚ in section 5.3, and I give a graphical interpretation of
¬•
this special case in section 6.1

We can generalize the theorem somewhat. Nothing is special about returns; any payoff of
the form yR‚ą— +wRe‚ą— or yx‚ą— +wRe‚ą— 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 + ¬≤, E(¬≤x) = 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 ) = ő± + ő»0 ő≤ i ‚á” 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, x‚ą— or
R‚ą— as factors. Let‚Ä™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
Ô¬Ānd it convenient to normalize to E(m) = 1, or m = 1 + b0 [f ‚ą’ E (f )]. Then,
Theorem: Given the model

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

one can Ô¬Ānd ő» such that

E(Re ) = ő≤ 0 ő» (88)

where ő≤ are the multiple regression coefÔ¬Ācients of excess returns Re on the factors.
Conversely, given ő» in (6.88), we can Ô¬Ānd b such that (6.87) holds.
Proof: From (6.87)

0 = E(mRe ) = E(Re ) + b0 cov(f, Re )
E(Re ) = ‚ą’b0 cov(f, Re ).

From covariance to beta is quick,

E(Re ) = ‚ą’b0 var(f )var(f )‚ą’1 cov(f, Re ) = ő»0 ő≤

Thus, ő» and b are related by

ő» = ‚ą’var(f )b.

¬•

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 Ô¬Ānd ő± and ő» such that

E(Ri ) = ő± + ő»0 ő≤ i , (90)

where ő≤ i are the multiple regression coefÔ¬Ācients of Ri on f with a constant. Con-
versely, given ő± and ő» in a factor model of the form (6.90), one can Ô¬Ānd a, b such
that (6.89) holds.
Proof: We just have to construct the relation between (ő±, ő») 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) = ‚ą’ =‚ą’
E(m) E(m) a a
ő≤ i is the vector of the appropriate regression coefÔ¬Ācients,
¬° ¬Ę‚ą’1
ő≤ i ‚Č°E ff 0 E(fRi ),

so to get ő≤ in the formula, continue with

1 E(Rf 0 )E(f f 0 )‚ą’1 E(f f 0 )b 0
1 0 E(f f )b
E(R) = ‚ą’ = ‚ą’ő≤
a a a a
Now, deÔ¬Āne ő± and ő» to make it work,
1 1
(6.92)
ő±‚Č° =
E (m) a
1
ő» ‚Č° ‚ą’ E(ff 0 )b = ‚ą’ ő±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 inÔ¬Ānite riskfree rates.
We rule out E(m) = E(a + b0 f ) = 0 to keep (6.91) from exploding, and we rule
¬•
out ő± = 0 and E(f f 0 ) singular to go from ő±, ő≤, ő» 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
ő≤ and/or ő» , 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
R‚ą— .
Equation (6.92) shows that the factor risk premium ő» can be interpreted as the price of the
factor; A test of ő» 6= 0 is often called a test of whether the ‚Äúfactor is priced.‚ÄĚ More precisely,

106
SECTION 6.3 FACTOR MODELS AND DISCOUNT FACTORS

ő» captures the price E(mf ) of the (de-meaned) factors brought forward at the risk free rate.
ňú ňú ňú
If we start with underlying factors f such that the demeaned factors are f = f ‚ą’E(f),
" #
h i ňú
ňú E(f)
ňú‚ą’E(f) = ‚ą’ő±
ňú
ő» ‚Č° ‚ą’ő± p f p(f)‚ą’
ő±

ő» represents the price of the factors less their risk-neutral valuation, i.e. the factor risk pre-
mium. If the factors are not traded, ő» is the model‚Ä™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 ő±,
ő» = E(f ) ‚ą’ ő±.
Note that the ‚Äúfactors‚ÄĚ 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 speciÔ¬Āc 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 ‚ą— = proj(f|X)

factor-mimicking returns

proj(f|X)
f‚ą— =
p [proj(f |X)]

or factor-mimicking excess returns

f ‚ą— = 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 x‚ą— = proj(m|X) carries all of m0 s pricing implications on X; p(x) = E(mx) =
E(x‚ą— 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

DeÔ¬Āne the payoffs f ‚ą— by

f ‚ą— = proj(f|X)

Then, m = b0 f ‚ą— 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 ‚ą— 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 ) = ő±‚ą— + ő≤ ‚ą—0 ő»‚ą— , (94)

and Ô¬Ānd ő»‚ą— , ő±‚ą— using (6.92). The ő≤ ‚ą— are the regression coefÔ¬Ācients 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)
f‚ą— = .
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 don‚Ä™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 deÔ¬Āne

f ‚ą— = proj(f|Re )

then of course the excess returns f ‚ą— carry the same pricing implications as the factors f for a
set of excess returns; m = b0 f ‚ą— satisÔ¬Āes 0 = E(mRei ) and

E(Rei ) = ő≤ i,f ‚ą— ő» = ő≤ i,f ‚ą— E(f ‚ą— )

6.4 Discount factors and beta models to mean - variance frontier

From m, we can construct R‚ą— which is on the mean variance frontier
If a beta pricing model holds, then the return R‚ą— 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 R‚ą— 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.

It‚Ä™s easy to show that given m that we can Ô¬Ānd a return on the mean-variance frontier.
Given m construct x‚ą— = proj(m|X) and R‚ą— = x‚ą— /E(x‚ą—2 ). R‚ą— 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 , x‚ą— is linear in f ‚ą— = proj(f |X), so R‚ą— is linear in the factor
mimicking payoffs f ‚ą— or their returns f ‚ą— /p(f ‚ą— ).
Section 5.4 showed how we can span the mean-variance frontier with R‚ą— and a risk free
rate, if there is one, or the zero-beta, minimum variance, or constant-mimicking portfolio
ňÜ
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 efÔ¬Ācient return that is uncorre-
lated with another given mean-variance efÔ¬Ācient return. The minimum-variance return is just
that. The constant-mimicking portfolio return is the return on the payoff ‚Äúclosest‚ÄĚ to the unit
payoff. Each of these returns one has a representation in the standard form R‚ą— + wRe‚ą— 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 efÔ¬Ācient. Thus, I characterize each
one by Ô¬Ānding its weight w in a representation of the form R‚ą— + wRe‚ą— . 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 = R‚ą— + Rf Re‚ą— . (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 R‚ą—

The zero beta return for R‚ą— , denoted Rő± , is the mean-variance efÔ¬Ācient return uncorre-
lated with R‚ą— . Its expected return is the zero beta rate ő± = E(Ra ). This zero beta return has
representation

var(R‚ą— )
a ‚ą—
Re‚ą— ,
R =R + ‚ą— )E(Re‚ą— )
E(R
and the corresponding zero beta rate is

E(R‚ą—2 ) 1
ő± = E(Rő± ) = = .
E(R‚ą— ) E(x‚ą— )
The zero beta rate is found graphically in mean-standard deviation space by extending the
tangency at R‚ą— to the vertical axis. It is also the inverse of the price that x‚ą— and R‚ą— assign to
the unit payoff.

The riskfree rate Rf is of course uncorrelated with R‚ą— . Risky returns uncorrelated with
R‚ą— 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 Rő± that is uncorrelated with R‚ą— we
have E(R‚ą— Rő± ) = E(R‚ą— )E(Rő± ), so

E(R‚ą—2 ) 1
ő±
ő± = E(R ) = = .
E(R‚ą— ) E(x‚ą— )
The Ô¬Ārst equality introduces a popular notation ő± for this rate. I call ő± the zero beta rate, and
Ra the zero beta return. There is no riskfree rate, so there is no security that just pays ő±.
As you can see from the formula, the zero-beta rate is the inverse of the price that R‚ą— and
x‚ą— assign to the unit payoff, which is another natural generalization of the riskfree rate. It is
called the zero beta rate because cov(R‚ą— , Rő± ) = 0 implies that the regression beta of Rő± on
R‚ą— is zero. More precisely, one might call it the zero beta rate on R‚ą— , since one can calculate
zero-beta rates for returns other than R‚ą— and they are not the same as the zero-beta rate for
R‚ą— In particular, the zero-beta rate on the ‚Äúmarket portfolio‚ÄĚ will generally be different from
the zero beta rate on R‚ą— .

110
SECTION 6.5 THREE RISKFREE RATE ANALOGUES

E(R)

ő±=
Rő±
E(R*2 )/ E(R* )
= 1/E(x*)
R*

ŌÉ(R)

Figure 20. Zero-beta rate ő± and zero-beta return Ra for R‚ą— .

111
CHAPTER 6 RELATION BETWEEN DISCOUNT FACTORS, BETAS, AND MEAN-VARIANCE FRONTIERS

I draw ő± 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 efÔ¬Ācient asset (a zero-beta asset) lies at the point so
constructed. To check this geometry, use similar triangles: p length ofp ‚ą— in Figure 20 is
The R
p
‚ą—2 ), and its vertical extent is E(R‚ą— ). Therefore, ő±/ E(R‚ą—2 ) = E(R‚ą—2 )/E(R‚ą— ),
E(R
or ő± = E(R‚ą—2 )/E(R‚ą— ). Since R‚ą— is on the lower portion of the mean-variance frontier, this
zero beta rate ő± is above the minimum variance return.
Note that in general ő± 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, x‚ą— 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 R‚ą— + wRe‚ą—
form. To do this, we want to Ô¬Ānd w such that
E(R‚ą—2 )
E(Ra ) = = E(R‚ą— ) + wE(Re‚ą— ).
‚ą—)
E(R
E(R‚ą—2 ) ‚ą’ E(R‚ą— )2 var(R‚ą— )
w= = .
E(R‚ą— )E(Re‚ą— ) E(R‚ą— )E(Re‚ą— )
Thus, the zero beta return is
var(R‚ą— )
a ‚ą—
Re‚ą— ,
R =R +
E(R‚ą— )E(Re‚ą— )

expression (6.103). Note that the weight is not E(Ra ) = E(R‚ą—2 )/E(R‚ą— ). 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(R‚ą— )
min. var. ‚ą—
Re‚ą— .
R =R +
1 ‚ą’ E(Re‚ą— )

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(R‚ą— )
Rmin. var. = R‚ą— + Re‚ą— . (96)
e‚ą— )
1 ‚ą’ E(R

112
SECTION 6.5 THREE RISKFREE RATE ANALOGUES

Taking expectations,
E(R‚ą— ) E(R‚ą— )
E(Rmin. var. ) = E(R‚ą— ) + E(Re‚ą— ) = .
1 ‚ą’ E(Re‚ą— ) 1 ‚ą’ E(Re‚ą— )
The minimum variance return retains the nice property of the risk free rate, that its weight on
Re‚ą— is the same as its mean,

Rmin. var. = R‚ą— + E(Rmin. var. )Re‚ą—

just as Rf = R‚ą— + Rf Re‚ą— . 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 R‚ą— + wRe‚ą— to minimize variance.

min var(R‚ą— + wRe‚ą— ) = E[(R‚ą— + wRe‚ą— )2 ] ‚ą’ E(R‚ą— + wRe‚ą— )2 =
w

= E(R‚ą—2 ) + w2 E(Re‚ą— ) ‚ą’ E(R‚ą— )2 ‚ą’ 2wE(R‚ą— )E(Re‚ą— ) ‚ą’ w2 E(Re‚ą— )2 .

The Ô¬Ārst order condition is

0 = wE(Re‚ą— )[1 ‚ą’ E(Re‚ą— )] ‚ą’ E(R‚ą— )E(Re‚ą— )

E(R‚ą— )
w= .
1 ‚ą’ E(Re‚ą— )

6.5.3 Constant-mimicking portfolio return

The constant-mimicking portfolio return is deÔ¬Āned as the return on the projection of the
unit vector on the payoff space,
proj(1|X)
ňÜ
R= .
p [proj(1|X)]
It has the representation
‚ą—2
ňÜ = R‚ą— + E(R ) Re‚ą— .
R
E(R‚ą— )

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 deÔ¬Āne the rate of return on the mimicking portfolio for a

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

unit payoff,
proj(1|X)
ňÜ
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(R‚ą—2 ) e‚ą—
ňÜ
R = R‚ą— + ő±Re‚ą— = R‚ą— + (97)
R.
E(R‚ą— )
Note that the weight ő± equal to the zero beta rate creates the constant-mimicking return, not
E(R‚ą— ) ‚ą—
e‚ą—
(98)
R = proj(1|X) ‚ą’ R.
E(R‚ą—2 )
Take the price of both sides. Since the price of Re‚ą— is zero and the price of R‚ą— is one, we
establish
E(R‚ą— )
(99a)
p [proj(1|X)] = .
E(R‚ą—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 = R‚ą— + Rf Re‚ą— .

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 = R‚ą— + Rf Re‚ą— . (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(R‚ą—2 ) e‚ą—
ňÜ
constant-mimicking: R = R‚ą— + (101)
R
E(R‚ą— )

114
SECTION 6.6 MEAN-VARIANCE SPECIAL CASES WITH NO RISKFREE RATE

E(R‚ą— )
minimum-variance: Rmin. var. = R‚ą— + Re‚ą— (102)
e‚ą— )
1 ‚ą’ E(R

var(R‚ą— )
ő± ‚ą—
Re‚ą— .
zero-beta: R = R + (103)
‚ą— )E(Re‚ą— )
E(R
To establish that these are all the same when there is a riskfree rate, we need to show that
E(R‚ą—2 ) E(R‚ą— ) var(R‚ą— )
f
(104)
R= = =
E(R‚ą— ) 1 ‚ą’ E(Re‚ą— ) E(R‚ą— )E(Re‚ą— )
We derived the Ô¬Ārst equality above as equation (5.74). To derive the second equality, take
expectations of (6.95),

Rf = E(R‚ą— ) + Rf E(Re‚ą— ) (105)

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

E(R‚ą—2 )
= E(R‚ą— ) + Rf E(Re‚ą— ).
‚ą—)
E(R

Solving for Rf ,

E(R‚ą—2 ) ‚ą’ E(R‚ą— )2 var(R‚ą— )
Rf = = .
E(R‚ą— )E(Re‚ą— ) E(R‚ą— )E(Re‚ą— )

6.6 Mean-variance special cases with no riskfree rate

We can Ô¬Ānd a discount factor from any mean-variance efÔ¬Ācient return except the constant-
mimicking return.
We can Ô¬Ānd a beta representation from any mean-variance efÔ¬Ācient 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 inÔ¬Ānity. 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 ‚ąě > 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 ‚Äúwhen a riskfree rate is not traded, make sure you use
discount factors with 0 < E(m) < ‚ąě.‚ÄĚ However, it is potentially useful and it certainly is
traditional to specify the special return on the mean-variance frontier that leads to the inÔ¬Ānite
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 Ô¬Ānd a discount factor that is a linear function of
any mean-variance efÔ¬Ācient 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-
‚ą—2
variance efÔ¬Ācient return Rmv except one particular return, of the form R‚ą— + E(R ‚ą— )) Re‚ą— . This
E(R
return led to an inÔ¬Ānite 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 = x‚ą— + őµ,
E(őµx) = 0, the line through x‚ą— orthogonal to the payoff space X in the Ô¬Āgure. I draw the
unit payoff (the dot marked ‚Äú1‚ÄĚ 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 Ô¬Āgure. As the Ô¬Āgure
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,
ňÜ
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

116
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
ňÜ
mean-variance-efÔ¬Ācient return except the constant-mimicking return R.

of the residual in a projection (regression) on 1.
¬£ ¬¤ ¬£ ¬¤
var(x) = E (x ‚ą’ E(x))2 = E (x ‚ą’ proj(x|1))2 = ||x ‚ą’ 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 ‚á” discount factor and discount factor ‚á” single
beta representation, so at a superÔ¬Ācial 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 efÔ¬Ācient 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 ) = ő±Rmv + ő≤ i,Rmv [E(Rmv ) ‚ą’ ő±] ,

if and only if Rmv is mean-variance efÔ¬Ācient 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
ő±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 inÔ¬Ānite zero-beta rate.
Proof: The mean-variance frontier is Rmv = R‚ą— + wRe‚ą— . Any return is Ri =
R‚ą— + wi Re‚ą— + ni . Thus,

E(Ri ) = E(R‚ą— ) + wi E(Re‚ą— ) (106)

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