<< ÒÚ. 3(‚ÒÂ„Ó 17)—Œƒ≈–∆¿Õ»≈ >>
The contingent claims price vector pc points in to the positive orthant. We saw in sec-
tion 3.3 that m(s) = u0 [c(s)]/u0 (c). Now, marginal utility should always be positive (people
always want more), so the marginal rate of substitution and discount factor are always non-
negative, m > 0 and pc > 0. Don‚Ä™t forget, m and pc are vectors, or random variables. Thus,
m > 0 means the realization of m is positive in every state of nature, or, equivalently every
element of the vector m is positive.
The set of payoffs with any given price lie on a (hyper)plane perpendicular to the contin-

60
SECTION 3.5 STATE DIAGRAM AND PRICE FUNCTION

State 2
Payoff

Price = 2
Riskfree rate
pc
Price = 1 (returns)

State 1 contingent claim
State 1 Payoff
Price = 0 (excess returns)

Figure 7. Contingent claims prices (pc) and payoffs.

gent claim price vector. We reasoned above that the price of the payoff x must be given by its
contingent claim value (3.48),
X
(50)
p(x) = pc(s)x(s).
s

Interpreting pc and x as vectors, this means that the price is given by the inner product of the
contingent claim price and the payoff.
If two vectors are orthogonal ‚Ä“ if they point out from the origin at right angles to each
other ‚Ä“ then their inner product is zero. Therefore, the set of all zero price payoffs must lie
on a plane orthogonal to the contingent claims price vector, as shown in Ô¬Ågure 7.
More generally, the inner product of two vectors x and pc equals the product of the mag-
nitude of the projection of x onto pc times the magnitude of pc. Using a dot to denote inner
product,
X
p(x) = pc(s)x(s) = pc ¬· x = |pc| √— |proj(x|pc)| = |pc| √— |x| √— cos(Œ∏)
s

where |x| means the length of the vector x and Œ∏ is the angle between the vectors pc and
x. Since all payoffs on planes (such as the price planes in Ô¬Ågure 7) that are perpendicular
to pc have the same projection onto pc, they must have the same price. (Only the price = 0
plane is, strictly speaking, orthogonal to pc. Lacking a better term, I‚Ä™ve called the nonzero

61
CHAPTER 3 CONTINGENT CLAIMS MARKETS

price planes ‚Äúperpendicular‚Äù to pc.) When vectors are Ô¬Ånite-dimensional, the prime notation
is commonly used for inner products, pc0 x. This notation does not extend well to inÔ¬Ånite-
dimensional spaces. The notation hpc|xi is also often used for inner products.
Planes of constant price move out linearly, and the origin x = 0 must have a price of
zero. If payoff y = 2x, then its price is twice the price of x,
X X
p(y) = pc(s)y(s) = pc(s)2x(s) = 2 p(x).
s s

Similarly, a payoff of zero must have a price of zero.
We can think of p(x) as a pricing function, a map from the state space or payoff space
in which x lies (RS in this case) to the real line. We have just deduced from the deÔ¬Ånition
(3.50) that p(x) is a linear function, i.e. that

p(ax + by) = ap(x) + bp(y).

The constant price lines in Figure 7 are of course exactly what one expects from a linear
function from RS to R. (One might draw the price on the z axis coming out of the page.
Then the price function would be a plane going through the origin and sloping up with iso-
price lines as given in Figure 7.)
Figure 7 also includes the payoffs to a contingent claim to the Ô¬Årst state. This payoff is
one in the Ô¬Årst state and zero in other states and thus located on the axis. The plane of price
= 1 payoffs is the plane of asset returns; the plane of price = 0 payoffs is the plane of excess
returns. A riskfree unit payoff (the payoff to a risk-free pure discount bond) lies on the (1, 1)
point in Figure 7; the riskfree return lies on the intersection of the 45o line (same payoff in
both states) and the price = 1 plane (the set of all returns).

Geometry with m in place of pc.
The geometric interpretation of Figure 7 goes through with the discount factor m in the
place of pc. We can deÔ¬Åne an inner product between the random variables x and y by

x ¬· y ‚â° E(xy),

and retain all the mathematical properties of an inner product. For this reason, random vari-
ables for which E(xy) = 0 are often called ‚Äúorthogonal.‚Äù
This language may be familiar from linear regressions. When we run a regression of y on
x,

y = b0 x + Œµ

we Ô¬Ånd the linear combination of x that is ‚Äúclosest‚Äù to y, by minimizing the variance or ‚Äúsize‚Äù
of the residual Œµ. We do this by forcing the residual to be ‚Äúorthogonal‚Äù to the right hand vari-
able E(xŒµ) = 0. The projection of y on x is deÔ¬Åned as the Ô¬Åtted value, proj(y|x) =

62
SECTION 3.5 STATE DIAGRAM AND PRICE FUNCTION

b0 x =E(xx0 )‚à’1 E(yx0 )x. This ideal is often illustrated by a residual vector Œµ that is perpen-
dicular to a plane deÔ¬Åned by the right hand variables x. Thus, when the inner product is
deÔ¬Åned by a second moment, the operation ‚Äúproject y onto x‚Äù is a regression. (If x does not
include a constant, you don‚Ä™t add one.)
The geometric interpretation of Figure 7 also is valid if we generalize the setup to an
inÔ¬Ånite-dimensional state space, i.e. if we think of continuously-valued random variables.
Instead of vectors, which are functions from RS to R, random variables are (measurable)
functions from ‚„¦ to R. Nonetheless, we can still think of them as vectors. The equivalent
of Rs is now a Hilbert space L2 , which denotes spaces generated by linear combinations
of square integrable functions from ‚„¦ to the real line, or the space of random variables with
Ô¬Ånite second moments. We can still deÔ¬Åne an ‚Äúinner product‚Äù between two such elements
by x ¬· y = E(xy), and p(x) = E(mx) can still be interpreted as ‚Äúm is perpendicular to
(hyper)planes of constant price.‚Äù Proving theorems in this context is a bit harder. You can‚Ä™t
just say things like ‚Äúwe can take a line perpendicular to any plane,‚Äù such things have to be
proved. Sometimes, Ô¬Ånite-dimensional thinking can lead you to errors, so it‚Ä™s important to
prove things the right way, keeping the Ô¬Ånite dimensional pictures in mind for interpretation.
Hansen and Richard (1987) is a very good reference for the Hilbert space machinery.

63
Chapter 4. The discount factor
Now we look more closely at the discount factor. Rather than derive a speciÔ¬Åc discount factor
as with the consumption-based model in the last chapter, I work backwards. A discount factor
is just some random variable that generates prices from payoffs, p = E(mx). What does this
expression mean? Can one always Ô¬Ånd such a discount factor? Can we use this convenient
representation without implicitly assuming all the structure of the investors, utility functions,
complete markets, and so forth?
The chapter focuses on two famous theorems. The law of one price states that if two
portfolios have the same payoffs (in every state of nature), then they must have the same
price. A violation of this law would give rise to an immediate kind of arbitrage proÔ¬Åt, as you
could sell the expensive version and buy the cheap version of the same portfolio. The Ô¬Årst
theorem states that there is a discount factor that prices all the payoffs by p = E(mx) if and
only if this law of one price holds.
In Ô¬Ånance, we reserve the term absence of arbitrage for a stronger idea, that if payoff A
is always at least as good as payoff B, and sometimes A is better, then the price of A must
be greater than the price of B. The second theorem is that there is a positive discount factor
that prices all the payoffs by p = E(mx) if and only if there are no arbitrage opportunities,
so deÔ¬Åned.
These theorems are useful to show that we can use stochastic discount factors without
implicitly assuming anything about utility functions, aggregation, complete markets and so
on. All we need to know about investors in order to represent prices and payoffs via a discount
factor is that they won‚Ä™t leave law of one price violations or arbitrage opportunities on the
table. These theorems can be used to describe aspects of a payoff space (such as law of one
price, absence of arbitrage) by restrictions on the discount factor (such as it exists and it is
positive). Chapter 18 shows how it can be more convenient to impose and check restrictions
on a single discount factor than it is to check the corresponding restrictions on all possible
portfolios. Chapter 7 discusses these and other implications of the existence theorems.
The theorems are credited to Ross (1978), and Harrison and Kreps (1979). My presenta-
tion follows Hansen and Richard (1987).

4.1 Law of one price and existence of a discount factor

DeÔ¬Ånition of law of one price; price is a linear function.
p = E(mx) implies law of one price.
The law of one price implies that a discount factor exists: There exists a unique x‚à— in X
such that p = E(x‚à— x) for all x ‚àà X = space of all available payoffs.

64
SECTION 4.1 LAW OF ONE PRICE AND EXISTENCE OF A DISCOUNT FACTOR

Furthermore, for any valid discount factor m,

x‚à— = proj(m | X).

So far we have derived the basic pricing relation p = E(mx) from environments with a
lot of structure: either the consumption-based model or complete markets.
Suppose we observe a set of prices p and payoffs x, and that markets ‚Ä” either the mar-
kets faced by investors or the markets under study in a particular application ‚Ä” are in-
complete, meaning they do not span the entire set of contingencies. In what minimal set
of circumstances does some discount factor exists which represents the observed prices by
p = E(mx)? This section and the following answer this important question. This treatment
is a simpliÔ¬Åed version of Hansen and Richard (1987), which contains rigorous proofs and
some technical assumptions.

Payoff space

The payoff space X is the set (or a subset) of all the payoffs that investors can purchase,
or it is a subset of the tradeable payoffs that is used in a particular study. For example, if there
are complete contingent claims to S states of nature, then X = RS . But the whole point is
that markets are (as in real life) incomplete, so we will generally think of X as a proper subset
of complete markets RS .
The payoff space will include some set of primitive assets, but investors can also form
new payoffs by forming portfolios. I assume that investors can form any portfolio of traded
assets:
A1: (Portfolio formation) x1 , x2 ‚àà X ‚á’ ax1 + bx2 ‚àà X for any real a, b.
Of course, X = RS for complete markets satisÔ¬Åes the portfolio formation assumption. If
there is a single basic payoff x, then the payoff space must be at least the ray from the origin
through x. If there are two basic payoffs in R3 , then the payoff space X must include the
plane deÔ¬Åned by these two payoffs and the origin. Figure 8 illustrates these possibilities.
The payoff space is not the space of returns. The return space is a subset of the payoff
space; if a return R is in the payoff space, then you can pay a price \$2 to get a payoff 2R, so
the payoff 2R with price 2 is also in the payoff space. Also, ‚à’R is in the payoff space.
Free portfolio formation is in fact an important and restrictive simplifying assumption. It
rules out short sales constraints, bid/ask spreads, leverage limitations and so on. The theory
can be modiÔ¬Åed to incorporate these frictions, but it is a substantial modiÔ¬Åcation.
If investors can form portfolios of a vector of basic payoffs x (say, the returns on the
NYSE stocks), then the payoff space consists of all portfolios or linear combinations of these
original payoffs X = {c0 x} where c is a vector of portfolio weights. We also can allow truly
inÔ¬Ånite-dimensional payoff spaces. For example, investors might be able to trade nonlinear

65
CHAPTER 4 THE DISCOUNT FACTOR

State 3 (into page)
State 2 State 2

x2
X x1
x

State 1 State 1
Single Payoff in R2 Two Payoffs in R3

Figure 8. Payoff spaces X generated by one (left) and two (right) basis payoffs.

functions of a basis payoff x, such as call options on x with strike price K, which have payoff
max [x(s) ‚à’ K, 0] .

The law of one price.

A2: (Law of one price, linearity) p(ax1 + bx2 ) = ap(x1 ) + bp(x2 )
It doesn‚Ä™t matter how one forms the payoff x. The price of a burger, shake and fries must
be the same as the price of a happy meal. Graphically, if the iso-price curves were not planes,
then one could buy two payoffs on the same iso-price curve, form a portfolio whose payoff
is on the straight line connecting the two original payoffs, and sell the portfolio for a higher
price than it cost to assemble it.
The law of one price basically says that investors can‚Ä™t make instantaneous proÔ¬Åts by
repackaging portfolios. If investors can sell securities, this is a very weak characterization
of preferences. It says there is at least one investor for whom marketing doesn‚Ä™t matter, who
values a package by its contents. The law is meant to describe a market that has already
reached equilibrium. If there are any violations of the law of one price, traders will quickly
eliminate them so they can‚Ä™t survive in equilibrium.
A1 and A2 also mean that the 0 payoff must be available, and must have price 0.

The Theorem

The existence of a discount factor implies the law of one price. This is obvious to the
point of triviality: if x = y + z then E(mx) = E[m(y + z)]. The hard, and interesting part
of the theorem reverses this logic. We show that the law of one price implies the existence of
a discount factor.

66
SECTION 4.1 LAW OF ONE PRICE AND EXISTENCE OF A DISCOUNT FACTOR

Theorem: Given free portfolio formation A1, and the law of one price A2, there
exists a unique payoff x‚à— ‚àà X such that p(x) = E(x‚à— x) for all x ‚àà X.

x‚à— is a discount factor. A1 and A2 imply that the price function on X looks like Figure
7: parallel hyperplanes marching out from the origin. The only difference is that X may be a
subspace of the original state space, as shown in Figure 8. The essence of the proof, then, is
that any linear function on a space X can be represented by inner products with a vector that
lies in X.
Proof 1: (Geometric.) We have established that the price is a linear function as shown
in Figure 9. (Figure 9 can be interpreted as the plane X of a larger dimensional space as in
the right hand panel of Figure 8, laid Ô¬‚at on the page for clarity.) Now we can draw a line
from the origin perpendicular to the price planes. Choose a vector x‚à— on this line. Since the
line is orthogonal to the price zero plane we have 0 = p(x) = E(x‚à— x) for price zero payoffs
x immediately. The inner product between any payoff x on the price = 1 plane and x‚à— is
|proj(x|x‚à— )| √— |x‚à— | Thus, every payoff on the price = 1 plane has the same inner product
with x‚à— . All we have to do is pick x‚à— to have the right length, and we obtain p(x) = 1 =
E(x‚à— x) for every x on the price = 1 plane. Then, of course we have p(x) = E(x‚à— x) for
payoffs x on the other planes as well. Thus, the linear pricing function implied by the Law
¬¤
of One Price can be represented by inner products with x‚à— .

Price = 2

x* Price = 1 (returns)

Price = 0 (excess returns)

Figure 9. Existence of a discount factor x‚à— .

The basic mathematical point is just that any linear function can be represented by an

67
CHAPTER 4 THE DISCOUNT FACTOR

inner product. The the Riesz representation theorem extends the proof to inÔ¬Ånite-dimensional
payoff spaces. See Hansen and Richard (1987).
Proof 2: (Algebraic.) We can prove the theorem by construction when the payoff space
X is generated by portfolios of a N basis payoffs (for example, N stocks). This is a common
situation, so the formulas are also useful in practice. Organize the basis payoffs into a vector
¬£ ¬¤0
x = x1 x2 ... xN and similarly their prices p. The payoff space is then X = {c0 x}.
We want a discount factor that is in the payoff space, as the theorem requires. Thus, it must be
of the form x‚à— = c0 x. Construct c so that x‚à— prices the basis assets. We want p =E(x‚à— x) =
E(xx0 c). Thus we need c = E(xx0 )‚à’1 p. If E(xx0 ) is nonsingular, this c exists and is unique.
A2 implies that E(xx0 ) is nonsingular (after pruning redundant rows of x). Thus,

x‚à— = p0 E(xx0 )‚à’1 x (51)

is our discount factor. It is a linear combination of x so it is in X. It prices the basis assets
x by construction. It prices every x ‚àà X : E[x‚à— (x0 c)] = E[p0 E(xx0 )‚à’1 xx0 c] = p0 c. By
0
linearity, p(c0 x) = c p.

What the theorem does and does not say

The theorem says there is a unique x‚à— in X. There may be many other discount factors
m not in X. In fact, unless markets are complete, there are an inÔ¬Ånite number of random
variables that satisfy p = E(mx). If p = E(mx) then p = E [(m + Œµ)x] for any Œµ orthogonal
to x, E(Œµx) = 0.
Not only does this construction generate some additional discount factors, it generates
all of them: Any discount factor m (any random variable that satisÔ¬Åes p = E(mx)) can be
represented as m = x‚à— +Œµ with E(Œµx) = 0. Figure 10 gives an example of a one-dimensional
X in a two-dimensional state space, in which case there is a whole line of possible discount
factors m. If markets are complete, there is nowhere to go orthogonal to the payoff space X,
so x‚à— is the only possible discount factor.
Reversing the argument, x‚à— is the projection of any stochastic discount factor m on the
space X of payoffs. This is a very important fact: the pricing implications of any discount
factor m for a set of payoffs X are the same as those of the projection of m on X. This
discount factor is known as the mimicking portfolio for m. Algebraically,

p = E(mx) = E [(proj(m|X) + Œµ)x] = E [proj(m|X) x]

Let me repeat and emphasize the logic. Above, we started with investors or a contingent
claim market, and derived a discount factor. p = E(mx) implies the linearity of the pricing
function and hence the law of one price, a pretty obvious statement in those contexts. Here
we work backwards. Markets are incomplete in that contingent claims to lots of states of
nature are not available. We found that the law of one price implies a linear pricing function,
and a linear pricing function implies that there exists at least one and usually many discount
factors.

68
SECTION 4.2 NO-ARBITRAGE AND POSITIVE DISCOUNT FACTORS

Payoff space X

x*

m = x* + Œµ space of discount factors

Figure 10. Many discount facotors m can price a given set of assets in incomplete markets.

We do allow arbitrary portfolio formation, and that sort of ‚Äúcompleteness‚Äù is important
to the result. If investors cannot form a portfolio ax + by, they cannot force the price of this
portfolio to equal the price of its constituents. The law of one price is not innocuous; it is an
assumption about preferences albeit a weak one. The point of the theorem is that this is just
enough information about preferences to deduce the existence of a discount factor.

4.2 No-Arbitrage and positive discount factors

The deÔ¬Ånition of arbitrage: positive payoff implies positive price.
There is a strictly positive discount factor m such that p = E(mx) if and only if there are
no arbitrage opportunities.

No arbitrage is another, slightly stronger, implication of marginal utility, that can be re-
versed to show that there is a positive discount factor. We need to start with the deÔ¬Ånition of
arbitrage:
DeÔ¬Ånition (Absence of arbitrage): A payoff space X and pricing function p(x) leave
no arbitrage opportunities if every payoff x that is always non-negative, x ‚â• 0
(almost surely), and positive, x > 0, with some positive probability, has positive

69
CHAPTER 4 THE DISCOUNT FACTOR

price, p(x) > 0.

No-arbitrage says that you can‚Ä™t get for free a portfolio that might pay off positively, but
will certainly never cost you anything. This deÔ¬Ånition is different from the colloquial use of
the word ‚Äúarbitrage.‚Äù Most people use ‚Äúarbitrage‚Äù to mean a violation of the law of one price
‚Ä“ a riskless way of buying something cheap and selling it for a higher price. ‚ÄúArbitrages‚Äù here
might pay off, but then again they might not. The word ‚Äúarbitrage‚Äù is also widely abused.
‚ÄúRisk arbitrage‚Äù is a Wall Street oxymoron that means making speciÔ¬Åc kinds of bets.
An equivalent statement is that if one payoff dominates another, then its price must be
higher ‚Ä“ if x ‚â• y, then p(x) ‚â• p(y) (Or, a bit more carefully but more long-windedly, if
x ‚â• y almost surely and x > y with positive probability, then p(x) > p(y). You can‚Ä™t forget
that x and y are random variables.)

m > 0 ‚á’No-arbitrage

The absence of arbitrage opportunities is clearly a consequence of a positive discount
factor, and a positive discount factor naturally results from any sort of utility maximization.
Recall,
u0 [c(s)]
m(s) = Œ≤ 0 > 0.
u (c)
It is a sensible characterization of preferences that marginal utility is always positive. Few
people are so satiated that they will throw away money. Therefore, the marginal rate of
substitution is positive. The marginal rate of substitution is a random variable, so ‚Äúpositive‚Äù
means ‚Äúpositive in every state of nature‚Äù or ‚Äúin every possible realization.‚Äù
Now, if contingent claims prices are all positive, a bundle of positive amounts of con-
tingent claims must also have a positive price, even in incomplete markets. A little more
formally,
Theorem: p = E(mx) and m(s) > 0 imply no-arbitrage.

Proof: m > 0; x ‚â• 0 and there are some states where x > 0. Thus, in some states
¬¤
mx > 0 and in other states mx = 0. Therefore E(mx) > 0.

No arbitrage ‚á’ m > 0

Now we turn the observation around, which is again the hard and interesting part. As
the law of one price property guaranteed the existence of a discount factor m, no-arbitrage
guarantees the existence of a positive m.
The basic idea is pretty simple. No-arbitrage means that the prices of any payoff in the
positive orthant (except zero, but including the axes) must be strictly positive. The price =

70
SECTION 4.2 NO-ARBITRAGE AND POSITIVE DISCOUNT FACTORS

0 plane divides the region of positive prices from the region of negative prices. Thus, if the
region of negative prices is not to intersect the positive orthant, the iso-price lines must march
up and to the right, and the discount factor m, must point up and to the right. This is how
we have graphed it all along, most recently in Ô¬Ågure 9. Figure 11 illustrates the case that is
ruled out: a whole region of negative price payoffs lies in the positive orthant. For example,
the payoff x is strictly positive, but has a negative price. As a result, the (unique, since this
market is complete) discount factor m is negative in the y-axis state.

p = -1

p=0
x

p = +1

x*, m

Figure 11. Counter-example for no-arbitrage ‚á’ m > 0 theorem. The payoff x is positive,
but has negative price. The discount factor is not strictly positive

The theorem is easy to prove in complete markets. There is only one m, x‚à— . If it isn‚Ä™t
positive in some state, then the contingent claim in that state has a positive payoff and a
negative price, which violates no arbitrage. More formally,
Theorem: In complete markets, no-arbitrage implies that there exists a unique m >
0 such that p = E(mx).

Proof: No-arbitrage implies the law of one price, so there is an x‚à— such that p =
E(x‚à— x), and in a complete market this is the unique discount factor. Suppose that
x‚à— ‚â¤ 0 for some states. Then, form a payoff x that is 1 P those states, and zero
in
elsewhere. This payoff is strictly positive, but its price, s:x‚à— (s)<0 œÄ(s)x‚à— (s) is

71
CHAPTER 4 THE DISCOUNT FACTOR

¬¤
negative, negating the assumption of no-arbitrage.

The tough part comes if markets are incomplete. There are now many m‚Ä™s that price
assets. Any m of the form m = x‚à— + ¬≤, with E(¬≤x) = 0 will do. We want to show that at
least one of these is positive. But that one may not be x‚à— . Since the discount factors other
than x‚à— are not in the payoff space X, we can‚Ä™t use the construction of the last argument,
since that construction may yield a payoff that is not in X, and hence to which we don‚Ä™t
know how to assign a price. To handle this case, I adopt a different strategy of proof. (This
strategy is due to Ross 1978. DufÔ¬Åe 1992 has a more formal textbook treatment.) The basic
idea is another ‚Äúto every plane there is a perpendicular line‚Äù argument, but applied to a space
that includes prices and payoffs together. As you can see, the price = 0 plane is a separating
hyperplane between the positive orthant and the negative payoffs, and the proof builds on this
idea.
Theorem: No arbitrage implies the existence of a strictly positive discount factor,
m > 0, p = E(mx) ‚àÄ x ‚àà X.
Proof : Join (‚à’p(x), x) together to form vectors in RS+1 . Call M the set of all
(‚à’p(x), x) pairs,

M = {(‚à’p(x), x); x ‚àà X}

M is still a linear space: m1 ‚àà M, m2 ‚àà M ‚á’ am1 + bm2 ‚àà M. No-arbitrage
means that elements of M can‚Ä™t have all positive elements. If x is positive, ‚à’p(x)
must be negative. Thus, M is a hyperplane that only intersects the positive orthant
RS+1 at the point 0. We can then create a linear function F : RS+1 ‚á’ R such that
+
F (‚à’p, x) = 0 for (‚à’p, x) ‚àà M, and F (‚à’p, x) > 0 for (‚à’p, x) ‚àà RS+1 except
+
the origin. Since we can represent any linear function by a perpendicular vector,
there is a vector (1, m) such that F (‚à’p, x) = (1, m) ¬· (‚à’p, x) = ‚à’p + m ¬· x or
‚à’p + E(mx) using the second moment inner product. Finally, since F (‚à’p, x) is
positive for (‚à’p, x) > 0, m must be positive.
¬¤

In a larger space than RS+1 , as generated by continuously valued random variables, the
+
separating hyperplane theorem assures us that there is a linear function that separates the two
convex sets M and the equivalent of RS+1 , and the Riesz representation theorem tells us that
+
we can represent F as an inner product with some vector by F (‚à’p, x) = ‚à’p + m ¬· x.

What the theorem does and does not say

The theorem says that a discount factor m > 0 exists, but it does not say that m > 0 is
unique. The left hand panel of Figure 12 illustrates the situation. Any m on the line through
x‚à— perpendicular to X also prices assets. Again, p = E[(m + Œµ)x] if E(Œµx) = 0. All of
these discount factors that lie in the positive orthant are positive, and thus satisfy the theorem.
There are lots of them! In a complete market, m is unique, but not otherwise.

72
SECTION 4.2 NO-ARBITRAGE AND POSITIVE DISCOUNT FACTORS

The theorem says that a positive m exists, but it also does not say that every discount
factor m must be positive. The discount factors in the left hand panel of Figure 12 outside the
positive orthant are perfectly valid ‚Ä“ they satisfy p = E(mx), and the prices they generate
on X are arbitrage free, but they are not positive in every state of nature. In particular, the
discount factor x‚à— in the payoff space is still perfectly valid ‚Ä” p(x) = E(x‚à— x) ‚Ä” but it need
not be positive, again as illustrated in the left hand panel of Figure 12.

p=2
p=1

m
m>0

p=1
x* x*
p=2
X X

Figure 12. Existence of a discount factor and extensions. The left graph shows that the
positive discount factor is not unique, and that discount factors may also exist that are not
strictly positive. In particular, x‚à— need not be positive. The right hand graph shows that
each particular choice of m > 0 induces an arbitrage free extension of the prices on X to all
contingent claims.

This theorem shows that we can extend the pricing function deÔ¬Åned on X to all possible
payoffs RS , and not imply any arbitrage opportunities on that larger space of payoffs. It says
that there is a pricing function p(x) deÔ¬Åned over all of RS , that assigns the same (correct, or
observed) prices on X and that displays no arbitrage on all of RS . Graphically, it says that
we can draw parallel planes to represent prices on all of RS in such a way that the planes
intersect X in the right places, and the price planes march up and to the right so the positive
orthant always has positive prices. Any positive m generates such a no-arbitrage extension,
as illustrated in the right hand panel of Figure 12. In fact, there are many ways to do this.
Each different choice of m > 0 generates a different extension of the pricing function.
We can think of strictly positive discount factors as possible contingent claims prices.
We can think of the theorem as answering the question: is it possible that an observed and
incomplete set of prices and payoffs is generated by some complete markets, contingent claim
economy? The answer is, yes, if there is no arbitrage on the observed prices and payoffs. In
fact, since there are typically many positive m‚Ä™s consistent with a {X, p(x)}, there exist

73
CHAPTER 4 THE DISCOUNT FACTOR

many contingent claims economies consistent with our observations.
Finally, the absence of arbitrage is another very weak characterization of preferences. The
theorem tells us that this is enough to allow us to use the p = E(mx) formalism with m > 0.
As usual, this theorem and proof do not require that the state space is RS . State spaces
generated by continuous random variables work just as well.

4.3 An alternative formula, and x‚à— in continuous time

In terms of the covariance matrix of payoffs,

x‚à— = E(x‚à— ) + [p‚à’E(x‚à— )E(x)]0 Œ£‚à’1 (x‚à’E(x)).

Just like x‚à— in discrete time,
¬µ ¬∂0
dŒõ‚à— D
= ‚à’rf dt ‚à’ ¬µ + ‚à’ r Œ£‚à’1 dz.
‚à—
Œõ p
prices assets by construction in continuous time.

Being able to compute x‚à— is useful in many circumstances. This section gives an alterna-
tive formula in discrete time, and the continuous time counterpart.
A formula that uses covariance matrices
E(xx0 ) in our previous formula (4.51) is a second moment matrix. We typically summa-
rize data in terms of covariance matrices instead. Therefore, a convenient alternative formula
is

x‚à— = E(x‚à— ) + [p‚à’E(x‚à— )E(x)]0 Œ£‚à’1 (x‚à’E(x)) (52)

where
¬° ¬¢
Œ£ ‚â° E [x‚à’E(x)] [x‚à’E(x)]0

denotes the covariance matrix of the x payoffs. (We could just substitute E(xx0 ) = Œ£ +
E(x)E(x0 ), but the inverse of the sum is not very useful.) We can derive this formula by
postulating a discount factor that is a linear function of the shocks to the payoffs,

x‚à— = E(x‚à— ) + (x‚à’E(x))0 b,

and then Ô¬Ånding b to ensure that x‚à— prices the assets x :
¬£ ¬¤
p = E(x‚à— )E(x) + E (x‚à’Ex)x0 b

74
SECTION 4.3 AN ALTERNATIVE FORMULA, AND X‚à— IN CONTINUOUS TIME

so

b = Œ£‚à’1 [p‚à’E(x‚à— )E(x)] .

If a riskfree rate is traded, then we know E(x‚à— ) = 1/Rf . If a riskfree rate is not traded ‚Ä“
if 1 is not in X ‚Ä“ then this formula does not necessarily produce a discount factor x‚à— that is
in X. In many applications, however, all that matters is producing some discount factor, and
the arbitrariness of the risk-free or zero beta rate is not a problem.
This formula is particularly useful when the payoff space consists solely of excess returns
or price-zero payoffs. In that case, x‚à— = p0 E(xx0 )‚à’1 x gives x‚à— = 0. x‚à— = 0 is in fact the
only discount factor in X that prices all the assets, but in this case it‚Ä™s more interesting (and
avoids 1/0 difÔ¬Åculties when we want to transform to expected return/beta or other represen-
tations) to pick a discount factor not in X by picking a zero-beta rate or price of the riskfree
payoff. In the case of excess returns, for arbitrarily chosen Rf , then, (4.52) gives us
1 1
x‚à— = ‚à’ f E(Re )0 Œ£‚à’1 (Re ‚à’E(Re )); Œ£ ‚â° cov(Re )
Rf R

Continuous time
The law of one price implies the existence of a discount factor process, and absence of
arbitrage a positive discount factor process in continuous time as well as discrete time. At
one level, this statement requires no new mathematics. If we reinvest dividends for simplicity,
then a discount factor must satisfy

pt Œõt = Et (Œõt+s pt+s ) .

Calling pt+s = xt+s , this is precisely the discrete time p = E(mx) that we have studied all
along. Thus, the law of one price or absence of arbitrage are equivalent to the existence of a
or a positive Œõt+s . The same conditions at all horizons s are thus equivalent to the existence
of a discount factor process, or a positive discount factor process Œõt for all time t.
For calculations it is useful to Ô¬Ånd explicit formulas for a discount factors. Suppose a set
of securities pays dividends

Dt dt

and their prices follow
dpt
= ¬µt dt + œÉt dzt
pt
where p and z are N √— 1 vectors, ¬µt and œÉ t may vary over time, ¬µ(pt , t,other variables),
E (dzt dzt ) = I and the division on the left hand side is element-by element. (As usual, I‚Ä™ll
0

drop the t subscripts when not necessary for clarity, but everything can vary over time.)
We can form a discount factor that prices these assets from a linear combination of the

75
CHAPTER 4 THE DISCOUNT FACTOR

shocks that drive the original assets,
¬µ ¬∂0
dŒõ‚à— D
f f
Œ£‚à’1 œÉdz. (53)
= ‚à’r dt ‚à’ ¬µ + ‚à’r
‚à—
Œõ p
where Œ£ ‚â° œÉœÉ0 again is the covariance matrix of returns. You can easily check that this
equation solves
¬µ¬∂ ¬µ‚à—¬∂
dp D dŒõ dp
f
(54)
Et + dt ‚à’ r dt = ‚à’Et
Œõ‚à— p
p p
and
¬µ ¬∂
dŒõ‚à—
= ‚à’rf dt,
Et
Œõ‚à—
or you can show that this is the only diffusion driven by dz, dt with these properties. If there
f f
is a risk free rate rt (also potentially time-varying), then that rate determines rt . If there is
no risk free rate, (4.53) will price the risky assets for any arbitrary (or convenient) choice of
f
rt . As usual, this discount factor is not unique; Œõ‚à— plus orthogonal noise will also act as a
discount factor:
dŒõ‚à—
dŒõ
= ‚à— + dw; E(dw) = 0; E(dzdw) = 0.
Œõ Œõ
You can see that (4.53) is exactly analogous to the discrete time formula (4.52). (If you don‚Ä™t
like answers popping out of hats like this, guess a solution of the form
dŒõ
= ¬µŒõ dt + œÉŒõ dz.
Œõ
Then Ô¬Ånd ¬µŒõ and œÉŒõ to satisfy (4.54) for the riskfree and risky assets.)

4.4 Problems

1. Show that the law of one price loop implies that price is a linear function of payoff and
vice versa
2. Does the absence of arbitrage imply the law of one price? Does the law of one price
imply the absence of arbitrage? Answer directly using portfolio arguments, and indirectly
using the corresponding discount factors.
3. If the law of one price or absence of arbitrage hold in population, must they hold in a
sample drawn from that population?

76
Chapter 5. Mean-variance frontier and
beta representations
Much empirical work in asset pricing is couched in terms of expected return - beta represen-
tations and mean-variance frontiers. This chapter introduces expected return - beta represen-
tations and mean-variance frontiers.
I discuss here the beta representation, most commonly applied to factor pricing models.
Chapter 6 shows how an expected return/beta model is equivalent to a linear model for the
discount factor, i.e. m = b0 f where f are the right hand variables in the time-series regres-
sions that deÔ¬Åne betas. Chapter 9 then discusses the derivation of popular factor models such
as the CAPM, ICAPM and APT, i.e. under what assumptions the discount factor is a linear
function of other variables f such as the market return.
I summarize the classic Lagrangian approach to the mean-variance frontier. I then intro-
duce a powerful and useful representation of the mean-variance frontier due to Hansen and
Richard (1987). This representation uses the state-space geometry familiar from the existence
theorems. It is also useful because it is valid and useful in inÔ¬Ånite-dimensional payoff spaces,
which we shall soon encounter when we add conditioning information, dynamic trading or
options.

5.1 Expected return - Beta representations

The expected return-beta expression of a factor pricing model is

E(Ri ) = Œ± + Œ≤ i,a Œ»a + Œ≤ i,b Œ»b + . . .

The model is equivalent to a restriction that the intercept is the same for all assets in
time-series regressions.
When the factors are returns excess returns, then Œ»a = E(f a ). If the test assets are also
excess returns, then the intercept should be zero, Œ± = 0.

Much empirical work in Ô¬Ånance is cast in terms of expected return - beta representations
of linear factor pricing models, of the form

E(Ri ) = Œ± + Œ≤ i,a Œ»a + Œ≤ i,b Œ»b + . . . , i = 1, 2, ...N. (55)

77
CHAPTER 5 MEAN-VARIANCE FRONTIER AND BETA REPRESENTATIONS

The Œ≤ terms are deÔ¬Åned as the coefÔ¬Åcients in a multiple regression of returns on factors,

Ri = ai + Œ≤ i,a fta + Œ≤ i,b ftb + . . . + Œµi ; t = 1, 2, ...T. (56)
t t

This is often called a time-series regression, since one runs a regression across time for each
security i. The ‚Äúfactors‚Äù f are proxies for marginal utility growth. I discuss the stories used
to select factors at some length in chapter 9. For the moment keep in mind the canonical ex-
amples, f = consumption growth, or f = the return on the market portfolio (CAPM). Notice
that we run returns Rt on contemporaneous factors ftj . This regression is not about predict-
i

ing returns from variables seen ahead of time. Its objective is to measure contemporaneous
relations or risk exposure; whether returns are typically high in ‚Äúgood times‚Äù or ‚Äúbad times‚Äù
as measured by the factors.
The point of the beta model(5.55) is to explain the variation in average returns across
assets. I write i = 1, 2, ...N in (5.55) to emphasize this fact. The model says that assets with
higher betas should get higher average returns. Thus the betas in (5.55) are the explanatory (x)
variables, which vary asset by asset. The Œ± and Œ» ‚Ä“ common for all assets ‚Ä“ are the intercept
and slope in this cross-sectional relation. For example, equation (5.55) says that if we plot
expected returns versus betas in a one-factor model, we should expect all (E(Ri ), Œ≤ i,a ) pairs
to line up on a straight line with slope Œ»a and intercept Œ±.
Œ≤ i,a is interpreted as the amount of exposure of asset i to factor a risks, and Œ»a is inter-
preted as the price of such risk-exposure. Read the beta pricing model to say: ‚Äúfor each unit
of exposure Œ≤ to risk factor a, you must provide investors with an expected return premium
Œ»a .‚Äù Assets must give investors higher average returns (low prices) if they pay off well in
times that are already good, and pay off poorly in times that are already bad, as measured by
the factors.
One way to estimate the free parameters (Œ±, Œ») and to test the model (5.55) is to run a
cross sectional regression of average returns on betas,

E(Ri ) = Œ± + Œ≤ i,a Œ»a + Œ≤ i,b Œ»b + . . . + Œ±i , i = 1, 2, ...N. (57)

Again, the Œ≤ i are the right hand variables, and the Œ± and Œ» are the intercept and slope coef-
Ô¬Åcients that we estimate in this cross-sectional regression. The errors Œ±i are pricing errors.
The model predicts Œ±i = 0, and they should be statistically insigniÔ¬Åcant in a test. (I intention-
ally use the same symbol for the intercept, or mean of the pricing errors, and the individual
pricing errors Œ±i .) In the chapters on empirical technique, we will see test statistics based on
the sum of squared pricing errors.
The fact that the betas are regression coefÔ¬Åcients is crucially important. If the betas are
also free parameters then there is no content to the equation. More importantly (and this is
an easier mistake to make), the betas cannot be asset-speciÔ¬Åc or Ô¬Årm-speciÔ¬Åc characteristics,
such as the size of the Ô¬Årm, book to market ratio, or (to take an extreme example) the letter of
the alphabet of its ticker symbol. It is true that expected returns are associated with or corre-
lated with many such characteristics. Stocks of small companies or of companies with high

78
SECTION 5.1 EXPECTED RETURN - BETA REPRESENTATIONS

book/market ratios do have higher average returns. But this correlation must be explained by
some beta. The proper betas should drive out any characteristics in cross-sectional regres-
sions. If, for example, expected returns were truly related to size, one could buy many small
companies to form a large holding company. It would be a ‚Äúlarge‚Äù company, and hence pay
low average returns to the shareholders, while earning a large average return on its holdings.
The managers could enjoy the difference. What ruins this promising idea? . The ‚Äúlarge‚Äù
holding company will still behave like a portfolio of small stocks. Thus, only if asset returns
depend on how you behave, not who you are ‚Ä“ on betas rather than characteristics ‚Ä“ can a
market equilibrium survive such simple repackaging schemes.
Some common special cases
If there is a risk free rate, its betas in (5.55) are all zero,2 so the intercept is equal to the
risk free rate,

Rf = Œ±.

We can impose this condition rather than estimate Œ± in the cross-sectional regression (5.57).
If there is no risk-free rate, then Œ± must be estimated in the cross-sectional regression. Since
it is the expected return of a portfolio with zero betas on all factors, Œ± is called the (expected)
zero-beta rate in this circumstance.
We often examine factor pricing models using excess returns directly. (There is an im-
plicit, though not necessarily justiÔ¬Åed, division of labor between models of interest rates and
models of equity risk premia.) Differencing (5.55) between any two returns Rei = Ri ‚à’ Rj
(Rj does not have to be risk free), we obtain

E(Rei ) = Œ≤ i,a Œ»a + Œ≤ i,b Œ»b + . . . , i = 1, 2, ...N. (58)

Here, Œ≤ ia represents the regression coefÔ¬Åcient of the excess return Rei on the factors.
It is often the case that the factors are also returns or excess returns. For example, the
CAPM uses the return on the market portfolio as the single factor. In this case, the model
should apply to the factors as well, and this fact allows us to directly measure the Œ» coef-
Ô¬Åcients. Each factor has beta of one on itself and zero on all the other factors, of course.
Therefore, if the factors are excess returns, we have E(f a ) = Œ»a , and so forth. We can then
write the factor model as

E(Rei ) = Œ≤ i,a E(f a ) + Œ≤ i,b E(f b ) + . . . , i = 1, 2, ...N.

The cross-sectional beta pricing model (5.55)-(5.58) and the time-series regression def-
inition of the betas in (5.56) look very similar. It seems that one can take expectations of

The betas are zero because the risk free rate is known ahead of time. When we consider the effects of
2

conditioning information, i.e. that the interest rate could vary over time, we have to interpret the means and betas as
conditional moments. Thus, if you are worried about time-varying risk free rates, betas, and so forth, either assume
all variables are i.i.d. (and thus that the risk free rate is constant), or interpret all moments as conditional on time
t information.

79
CHAPTER 5 MEAN-VARIANCE FRONTIER AND BETA REPRESENTATIONS

the time-series regression (5.56) and arrive at the beta model (5.55), in which case the latter
would be vacuous since one can always run a regression of anything on anything. The differ-
ence is subtle but crucial: the time-series regressions (5.56) will in general have a different
intercept ai for each return i, while the intercept Œ± is the same for all assets in the beta pric-
ing equation (5.55). The beta pricing equation is a restriction on expected returns, and thus
imposes a restriction on intercepts in the time-series regression.
In the special case that the factors are themselves excess returns, the restriction is partic-
ularly simple: the time-series regression intercepts should all be zero. In this case, we can
avoid the cross-sectional regression entirely, since there are no free parameters left.

5.2 Mean-variance frontier: Intuition and Lagrangian
characterization

The mean-variance frontier of a given set of assets is the boundary of the set of means and
variances of the returns on all portfolios of the given assets. One can Ô¬Ånd or deÔ¬Åne this bound-
ary by minimizing return variance for a given mean return. Many asset pricing propositions
and test statistics have interpretations in terms of the mean-variance frontier.
Figure 13 displays a typical mean-variance frontier. As displayed in Figure 13, it is com-
mon to distinguish the mean-variance frontier of all risky assets, graphed as the hyperbolic
region, and the mean-variance frontier of all assets, i.e. including a risk free rate if there is
one, which is the larger wedge-shaped region. Some authors reserve the terminology ‚Äúmean-
variance frontier‚Äù for the upper portion, calling the whole thing the minimum variance fron-
tier. The risky asset frontier is a hyperbola, which means it lies between two asymptotes,
shown as dotted lines. The risk free rate is typically drawn below the intersection of the
asymptotes and the vertical axis, or the point of minimum variance on the risky frontier. If it
were above this point, investors with a mean-variance objective would try to short the risky
assets, which cannot represent an equilibrium.
In general, portfolios of two assets or portfolios Ô¬Åll out a hyperbolic curve through the
two assets. The curve is sharper the less correlated are the two assets, because the portfolio
variance beneÔ¬Åts from increasing diversiÔ¬Åcation. Portfolios of a risky asset and risk free rate
give rise to straight lines in mean-standard deviation space.
In Chapter 1, we derived a similar wedge-shaped region as the set of means and variances
of all assets that are priced by a given discount factor. This chapter is about incomplete
markets, so we think of a mean-variance frontier generated by a given set of assets, typically
less than complete.
When does the mean-variance frontier exist? I.e., when is the set of portfolio means and
variances less than the whole {E, œÉ} space? We basically have to rule out a special case: two
returns are perfectly correlated but yield different means. In that case one could short one,
long the other, and achieve inÔ¬Ånite expected returns with no risk. More formally, eliminate
purely redundant securities from consideration, then

80
SECTION 5.2 MEAN-VARIANCE FRONTIER: INTUITION AND LAGRANGIAN CHARACTERIZATION

Theorem: So long as the variance-covariance matrix of returns is non-singular, there
is a mean-variance frontier.

To prove this theorem, just follow the construction below. This theorem should sound
very familiar: Two perfectly correlated returns with different mean are a violation of the law
of one price. Thus, the law of one price implies that there is a mean variance frontier as well
as a discount factor.

E(R)
Mean-variance frontier

Risky asset frontier
Tangency portfolio
of risky assets
Original assets

Rf

œÉ(R)

Figure 13. Mean-variance frontier

5.2.1 Lagrangian approach to mean-variance frontier

The standard deÔ¬Ånition and the computation of the mean-variance frontier follows a brute
force approach.
Problem: Start with a vector of asset returns R. Denote by E the vector of mean returns,
¬£ 0¬¤
E ‚â° E(R), and denote by Œ£ the variance-covariance matrix Œ£ = E (R ‚à’ E)(R ‚à’ E) .
A portfolio is deÔ¬Åned by its weights w on the initial securities. The portfolio return is w0 R
where the weights sum to one, w0 1 =1. The problem ‚Äúchoose a portfolio to minimize vari-
ance for a given mean‚Äù is then

min{w} w0 Œ£w s.t. w0 E = ¬µ; w0 1 = 1. (59)

Solution: Let

A = E 0 Œ£‚à’1 E; B = E 0 Œ£‚à’1 1; C = 10 Œ£‚à’1 1.

81
CHAPTER 5 MEAN-VARIANCE FRONTIER AND BETA REPRESENTATIONS

Then, for a given mean portfolio return ¬µ, the minimum variance portfolio has variance
C¬µ2 ‚à’ 2B¬µ + A
var (Rp ) = (60)
AC ‚à’ B 2
and is formed by portfolio weights
E (C¬µ ‚à’ B) + 1 (A ‚à’ B¬µ)
w = Œ£‚à’1 .
(AC ‚à’ B 2 )

Equation (5.60) shows that the variance is a quadratic function of the mean. The square
root of a parabola is a hyperbola, which is why we draw hyperbolic regions in mean-standard
deviation space.
The minimum-variance portfolio is interesting in its own right. It appears as a special case
in many theorems and it appears in several test statistics. We can Ô¬Ånd it by minimizing (5.60)
over ¬µ, giving ¬µmin var = B/C. The weights of the minimum variance portfolio are thus
w = Œ£‚à’1 1/(10 Œ£‚à’1 1).

We can get to any point on the mean-variance frontier by starting with two returns on
the frontier and forming portfolios. The frontier is spanned by any two frontier returns.
To see this fact, notice that w is a linear function of ¬µ. Thus, if you take the portfolios
corresponding to any two distinct mean returns ¬µ1 and ¬µ2 , the weights on a third portfolio
with mean ¬µ3 = Œ»¬µ1 + (1 ‚à’ Œ»)¬µ2 are given by w3 = Œ»w1 + (1 ‚à’ Œ»)w2 .
Derivation: To derive the solution, introduce Lagrange multipliers 2Œ» and 2Œ¥ on the con-
straints. The Ô¬Årst order conditions to (5.59) are then
Œ£w‚à’Œ»E ‚à’ Œ¥1 = 0

w = Œ£‚à’1 (Œ»E + Œ¥1). (61)
We Ô¬Ånd the Lagrange multipliers from the constraints,

E 0 w = E 0 Œ£‚à’1 (Œ»E + Œ¥1) = ¬µ

10 w = 10 Œ£‚à’1 (Œ»E + Œ¥1) = 1
or
¬· ¬∏¬· ¬∏ ¬· ¬∏
E 0 Œ£‚à’1 E E 0 Œ£‚à’1 1 Œ» ¬µ
=
10 Œ£‚à’1 E 10 Œ£‚à’1 1 Œ¥ 1

¬· ¬∏¬· ¬∏ ¬· ¬∏
A B Œ» ¬µ
=
B C Œ¥ 1

82
SECTION 5.3 AN ORTHOGONAL CHARACTERIZATION OF THE MEAN-VARIANCE FRONTIER

Hence,

C¬µ‚à’B
Œ»=
AC‚à’B 2

A‚à’B¬µ
Œ¥=
AC ‚à’ B 2

Plugging in to (5.61), we get the portfolio weights and variance.

5.3 An orthogonal characterization of the mean-variance frontier

Every return can be expressed as Ri = R‚à— + wi Re‚à— + ni .
The mean-variance frontier is Rmv = R‚à— + wRe‚à—
Re‚à— is deÔ¬Åned as Re‚à— = proj(1|Re ). It represents mean excess returns, E(Re ) = E(Re‚à— Re )
‚àÄRe ‚àà Re

The Lagrangian approach to the mean-variance frontier is straightforward but cumber-
some. Our further manipulations will be easier if we follow an alternative approach due to
Hansen and Richard (1987). Technically, Hansen and Richard‚Ä™s approach is also valid when
we can‚Ä™t generate the payoff space by portfolios of a Ô¬Ånite set of basis payoffs c0 x. This hap-
pens, for example, when we think about conditioning information in Chapter 8. Also, it is the
natural geometric way to think about the mean-variance frontier given that we have started
to think of payoffs, discount factors and other random variables as vectors in the space of
payoffs. Rather than write portfolios as combinations of basis assets, and pose and solve a
minimization problem, we Ô¬Årst describe any return by a three-way orthogonal decomposition.
The mean-variance frontier then pops out easily without any algebra.

5.3.1 DeÔ¬Ånitions of R‚à— , Re‚à—

I start by deÔ¬Åning two special returns. R‚à— is the return corresponding to the payoff x‚à— that
can act as the discount factor. The price of x‚à— , is, like any other price, p(x‚à— ) = E(x‚à— x‚à— ).
Thus,
The deÔ¬Ånition of R‚à— is

x‚à— x‚à—
R‚à— ‚â° (62)
=
p(x‚à— ) E(x‚à—2 )

83
CHAPTER 5 MEAN-VARIANCE FRONTIER AND BETA REPRESENTATIONS

The deÔ¬Ånition of Re‚à— is

Re‚à— ‚â° proj(1 | Re ) (63)

Re ‚â° space of excess returns = {x ‚àà X s.t. p(x) = 0}

Why Re‚à— ? We are heading towards a mean-variance frontier, so it is natural to seek a
special return that changes means. Re‚à— is an excess return that represents means on Re with
an inner product in the same way that x‚à— is a payoff in X that represents prices with an inner
product. As

p(x) = E(mx) = E[proj(m|X)x] = E(x‚à— x),

so

E(Re ) = E(1 √— Re ) = E [proj(1 | Re ) √— Re ] = E(Re‚à— Re ).

If R‚à— and Re‚à— are still a bit mysterious at this point, they will make more sense as we use
them, and discover their many interesting properties.
Now we can state a beautiful orthogonal decomposition.
Theorem: Every return Ri can be expressed as

Ri = R‚à— + wi Re‚à— + ni

where wi is a number, and ni is an excess return with the property

E(ni ) = 0.

The three components are orthogonal,

E(R‚à— Re‚à— ) = E(R‚à— ni ) = E(Re‚à— ni ) = 0.

This theorem quickly implies the characterization of the mean variance frontier which we
are after:
Theorem: Rmv is on the mean-variance frontier if and only if

Rmv = R‚à— + wRe‚à— (64)

for some real number w.

As you vary the number w, you sweep out the mean-variance frontier. E(Re‚à— ) 6= 0, so
adding more w changes the mean and variance of Rmv . You can interpret (5.64) as a ‚Äútwo-

84
SECTION 5.3 AN ORTHOGONAL CHARACTERIZATION OF THE MEAN-VARIANCE FRONTIER

fund‚Äù theorem for the mean-variance frontier. It expresses every frontier return as a portfolio
of R‚à— and Re‚à— , with varying weights on the latter.
As usual, Ô¬Årst I‚Ä™ll argue why the theorems are sensible, then I‚Ä™ll offer a simple algebraic
proof. Hansen and Richard (1987) give a much more careful algebraic proof.

5.3.2 Graphical construction

Figure 14 illustrates the decomposition. Start at the origin (0). Recall that the x‚à— vector is
perpendicular to planes of constant price; thus the R‚à— vector lies perpendicular to the plane
of returns as shown. Go to R‚à— .
Re‚à— is the excess return that is closest to the vector 1; it lies at right angles to planes (in
Re ) of constant mean return, shown in the E = 1, E = 2 lines, just as the return R‚à— lies at
right angles to planes of constant price. Since Re‚à— is an excess return, it is orthogonal to R‚à— .
Proceed an amount wi in the direction of Re‚à— , getting as close to Ri as possible.
Now move, again in an orthogonal direction, by an amount ni to get to the return Ri . We
have thus expressed Ri = R‚à— +wi Re‚à— +ni in a way that all three components are orthogonal.
Returns with n = 0, R‚à— + wRe‚à— , are the mean-variance frontier. Here‚Ä™s why. Since
E(R2 ) = œÉ 2 (R) + E(R)2 , we can deÔ¬Åne the mean-variance frontier by minimizing second
moment for a given mean. The length of each vector in Figure 14 is its second moment, so
we want the shortest vector that is on the return plane for a given mean. The shortest vectors
in the return plane with given mean are on the R‚à— + wRe‚à— line.
The graph also shows how Re‚à— represents means in the space of excess returns. Ex-
pectation is the inner product with 1. Planes of constant expected value in Figure 14 are
perpendicular to the 1 vector, just as planes of constant price are perpendicular to the x‚à— or
R‚à— vectors. I don‚Ä™t show the full extent of the constant expected payoff planes for clarity; I
do show lines of constant expected excess return in Re , which are the intersection of constant
expected payoff planes with the Re plane. Therefore, just as we found an x‚à— in X to repre-
sent prices in X by projecting m onto X, we Ô¬Ånd Re‚à— in Re by projecting of 1 onto Re . Yes,
a regression with one on the left hand side. Planes perpendicular to Re‚à— in Re are payoffs
with constant mean, just as planes perpendicular to x‚à— in X are payoffs with the same price.

5.3.3 Algebraic argument

Now, an algebraic proof of the decomposition and characterization of mean variance frontier.
The algebra just represents statements about what is at right angles to what with second
moments.
Proof: Straight from their deÔ¬Ånitions, (5.62) and (5.63) we know that Re‚à— is an

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CHAPTER 5 MEAN-VARIANCE FRONTIER AND BETA REPRESENTATIONS

R=space of returns (p=1)

R*+wiRe* ni
R*
Ri=R*+wiRe*+ni

1

Re*
0
E=1
E=0
Re = space of excess returns (p=0)

Figure 14. Orthogonal decomposition and mean-variance frontier.

excess return (price zero), and hence that R‚à— and Re‚à— are orthogonal,

E(x‚à— Re‚à— )
‚à— e‚à—
E(R R ) = = 0.
E(x‚à—2 )

We deÔ¬Åne ni so that the decomposition adds up to Ri as claimed, and we deÔ¬Åne
wi to make sure that ni is orthogonal to the other two components. Then we prove
that E(ni ) = 0. Pick any wi and then deÔ¬Åne

ni ‚â° Ri ‚à’ R‚à— ‚à’ wi Re‚à— .

ni is an excess return so already orthogonal to R‚à— ,

E(R‚à— ni ) = 0.

To show E(ni ) = 0 and ni orthogonal to Re‚à— , we exploit the fact that since ni is an
excess return,

E(ni ) = E(Re‚à— ni ).

Therefore, Re‚à— is orthogonal to ni if and only if we pick wi so that E(ni ) = 0. We

86
SECTION 5.4 SPANNING THE MEAN-VARIANCE FRONTIER

don‚Ä™t have to explicitly calculate wi for the proof.3
Once we have constructed the decomposition, the frontier drops out. Since E(ni ) =
0 and the three components are orthogonal,

E(Ri ) = E(R‚à— ) + wi E(Re‚à— )

œÉ2 (Ri ) = œÉ2 (R‚à— + wi Re‚à— ) + œÉ2 (ni ).

Thus, for each desired value of the mean return, there is a unique wi . Returns with
¬•
ni = 0 minimize variance for each mean.

5.3.4 Decomposition in mean-variance space

Figure 15 illustrates the decomposition in mean-variance space rather than in state-space.
First, let‚Ä™s locate R‚à— . R‚à— is the minimum second moment return. One can see this fact
from the geometry of Figure 14: R‚à— is the return closest to the origin, and thus the return
with the smallest ‚Äúlength‚Äù which is second moment. As with OLS regression, minimizing
the length of R‚à— and creating an R‚à— orthogonal to all excess returns is the same thing. One
can also verify this property algebraically. Since any return can be expressed as R = R‚à— +
wRe‚à— + n, E(R2 ) = E(R‚à—2 ) + w2 E(Re‚à—2 ) + E(n2 ). n = 0 and w = 0 thus give the
minimum second moment return.
In mean-standard deviation space, lines of constant second moment are circles. Thus,
the minimum second-moment return R‚à— is on the smallest circle that intersects the set of all
assets, which lie in the mean-variance frontier in the right hand panel of Figure 19. Notice
that R‚à— is on the lower, or ‚ÄúinefÔ¬Åcient‚Äù segment of the mean-variance frontier. It is initially
surprising that this is the location of the most interesting return on the frontier! R‚à— is not
the ‚Äúmarket portfolio‚Äù or ‚Äúwealth portfolio,‚Äù which typically lie on the upper portion of the
frontier.
Adding more Re‚à— moves one along the frontier. Adding n does not change mean but does
change variance, so it is an idiosyncratic return that just moves an asset off the frontier as
graphed. Œ± is the ‚Äúzero-beta rate‚Äù corresponding to R‚à— . It is the expected return of any return
that is uncorrelated with R‚à— . I demonstrate these properties in section 6.5.

Its value
3

E(Ri ) ‚à’ E(R‚à— )
wi =
E(Re‚à— )

is not particularly enlightening.

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CHAPTER 5 MEAN-VARIANCE FRONTIER AND BETA REPRESENTATIONS

E(R)
Ri
R* + wiRe*
ni

R*

œÉ(R)

Figure 15. Orthogonal decomposition of a return Ri in mean-standard deviation space.

5.4 Spanning the mean-variance frontier

The characterization of the mean-variance frontier in terms of R‚à— and Re‚à— is most natural
in our setup. However, you can equivalently span the mean-variance frontier with any two
portfolios that are on the frontier ‚Ä“ any two distinct linear combinations of R‚à— and Re‚à— . In
particular, take any return

RŒ± = R‚à— + Œ≥Re‚à— , Œ≥ 6= 0. (65)

Using this return in place of Re‚à— ,
RŒ± ‚à’ R‚à—
e‚à—
R =
Œ≥
you can express the mean variance frontier in terms of R‚à— and RŒ± :

R‚à— + wRe‚à— = R‚à— + y (RŒ± ‚à’ R‚à— ) (5.66)
= (1 ‚à’ y)R‚à— + yRŒ±

where I have deÔ¬Åned a new weight y = w/Œ≥.

88
SECTION 5.5 A COMPILATION OF PROPERTIES OF R‚à— , Re‚à— AND x‚à—

The most common alternative approach is to use a risk free rate or a risky rate that some-
how behaves like the risk free rate in place of Re‚à— to span the frontier. When there is a risk
free rate, it is on the frontier with representation

Rf = R‚à— + Rf Re‚à—

I derive this expression in equation (5.72) below. Therefore, we can use (5.66) with Ra = Rf .
When there is no risk free rate, several risky returns that retain some properties of the risk free
rate are often used. In section 5.3 below I present a ‚Äúzero beta‚Äù return, which is uncorrelated
with R‚à— , a ‚Äúconstant-mimicking portfolio‚Äù return, which is the return on the traded payoff
ÀÜ
closest to unity, R = proj(1|X)/p[proj(1|X)] and the minimum variance return. Each of
these returns is on the mean-variance frontier, with form 5.65, though different values of Œ±.
Therefore, we can span the mean-variance frontier with R‚à— and any of these risk-free rate
proxies.

5.5 A compilation of properties of R‚à— , Re‚à— and x‚à—

The special returns R‚à— , Re‚à— that generate the mean variance frontier have lots of interesting
and useful properties. Some I derived above, some I will derive and discuss below in more
detail, and some will be useful tricks later on. Most properties and derivations are extremely
obscure if you don‚Ä™t look at the picture!
1)
1
E(R‚à—2 ) = (67)
.
E(x‚à—2 )
To derive this fact, multiply both sides of (5.62) by R‚à— , take expectations, and remember
R‚à— is a return so 1 = E(x‚à— R‚à— ).
2) We can reverse the deÔ¬Ånition and recover x‚à— from R‚à— via
R‚à—
x‚à— = (68)
.
E(R‚à—2 )
To derive this formula, start with the deÔ¬Ånition R‚à— = x‚à— /E(x‚à—2 ) and substitute from (5.67)
for E(x‚à—2 )
3) R‚à— can be used to represent prices just like x‚à— . This is not surprising, since they both
point in the same direction, orthogonal to planes of constant price. Most obviously, from 5.68
E (R‚à— x)
p(x) = E(x‚à— x) = ‚àÄx ‚àà X
E(R‚à—2 )
For returns, we can nicely express this result as

E(R‚à—2 ) = E(R‚à— R) ‚àÄR ‚àà R. (69)

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CHAPTER 5 MEAN-VARIANCE FRONTIER AND BETA REPRESENTATIONS

This fact can also serve as an alternative deÔ¬Åning property of R‚à— .
4) Re‚à— represents means on Re via an inner product in the same way that x‚à— represents
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