ńņš. 3 |

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. Donā™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 ā“ if they point out from the origin at right angles to each

other ā“ 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 ļ¬gure 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 Ā· x = |pc| Ć— |proj(x|pc)| = |pc| Ć— |x| Ć— cos(Īø)

s

where |x| means the length of the vector x and Īø is the angle between the vectors pc and

x. Since all payoffs on planes (such as the price planes in ļ¬gure 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, Iā™ve called the nonzero

61

CHAPTER 3 CONTINGENT CLAIMS MARKETS

price planes āperpendicularā to pc.) When vectors are ļ¬nite-dimensional, the prime notation

is commonly used for inner products, pc0 x. This notation does not extend well to inļ¬nite-

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 deļ¬nition

(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 ļ¬rst state. This payoff is

one in the ļ¬rst 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 deļ¬ne an inner product between the random variables x and y by

x Ā· y ā” 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 āorthogonal.ā

This language may be familiar from linear regressions. When we run a regression of y on

x,

y = b0 x + Īµ

we ļ¬nd the linear combination of x that is āclosestā to y, by minimizing the variance or āsizeā

of the residual Īµ. We do this by forcing the residual to be āorthogonalā to the right hand vari-

able E(xĪµ) = 0. The projection of y on x is deļ¬ned as the ļ¬tted value, proj(y|x) =

62

SECTION 3.5 STATE DIAGRAM AND PRICE FUNCTION

b0 x =E(xx0 )ā’1 E(yx0 )x. This ideal is often illustrated by a residual vector Īµ that is perpen-

dicular to a plane deļ¬ned by the right hand variables x. Thus, when the inner product is

deļ¬ned by a second moment, the operation āproject y onto xā is a regression. (If x does not

include a constant, you donā™t add one.)

The geometric interpretation of Figure 7 also is valid if we generalize the setup to an

inļ¬nite-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 ā„¦ 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 ā„¦ to the real line, or the space of random variables with

ļ¬nite second moments. We can still deļ¬ne an āinner productā between two such elements

by x Ā· y = E(xy), and p(x) = E(mx) can still be interpreted as ām is perpendicular to

(hyper)planes of constant price.ā Proving theorems in this context is a bit harder. You canā™t

just say things like āwe can take a line perpendicular to any plane,ā such things have to be

proved. Sometimes, ļ¬nite-dimensional thinking can lead you to errors, so itā™s important to

prove things the right way, keeping the ļ¬nite 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 speciļ¬c 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 ļ¬nd 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 proļ¬t, as you

could sell the expensive version and buy the cheap version of the same portfolio. The ļ¬rst

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 ļ¬nance, 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 deļ¬ned.

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 wonā™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

Deļ¬nition 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 xā— in X

such that p = E(xā— x) for all x ā 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,

xā— = 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 ā” either the mar-

kets faced by investors or the markets under study in a particular application ā” 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 simpliļ¬ed 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 ā X ā’ ax1 + bx2 ā X for any real a, b.

Of course, X = RS for complete markets satisļ¬es 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 deļ¬ned 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, ā’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 modiļ¬ed to incorporate these frictions, but it is a substantial modiļ¬cation.

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

inļ¬nite-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) ā’ K, 0] .

The law of one price.

A2: (Law of one price, linearity) p(ax1 + bx2 ) = ap(x1 ) + bp(x2 )

It doesnā™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 canā™t make instantaneous proļ¬ts 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 doesnā™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 canā™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 xā— ā X such that p(x) = E(xā— x) for all x ā X.

xā— 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 ļ¬‚at on the page for clarity.) Now we can draw a line

from the origin perpendicular to the price planes. Choose a vector xā— on this line. Since the

line is orthogonal to the price zero plane we have 0 = p(x) = E(xā— x) for price zero payoffs

x immediately. The inner product between any payoff x on the price = 1 plane and xā— is

|proj(x|xā— )| Ć— |xā— | Thus, every payoff on the price = 1 plane has the same inner product

with xā— . All we have to do is pick xā— to have the right length, and we obtain p(x) = 1 =

E(xā— x) for every x on the price = 1 plane. Then, of course we have p(x) = E(xā— x) for

payoffs x on the other planes as well. Thus, the linear pricing function implied by the Law

Ā¤

of One Price can be represented by inner products with xā— .

Price = 2

x* Price = 1 (returns)

Price = 0 (excess returns)

Figure 9. Existence of a discount factor xā— .

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 inļ¬nite-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

Ā£ Ā¤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 xā— = c0 x. Construct c so that xā— prices the basis assets. We want p =E(xā— x) =

E(xx0 c). Thus we need c = E(xx0 )ā’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,

xā— = p0 E(xx0 )ā’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 ā X : E[xā— (x0 c)] = E[p0 E(xx0 )ā’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 xā— in X. There may be many other discount factors

m not in X. In fact, unless markets are complete, there are an inļ¬nite number of random

variables that satisfy p = E(mx). If p = E(mx) then p = E [(m + Īµ)x] for any Īµ orthogonal

to x, E(Īµx) = 0.

Not only does this construction generate some additional discount factors, it generates

all of them: Any discount factor m (any random variable that satisļ¬es p = E(mx)) can be

represented as m = xā— +Īµ with E(Īµx) = 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 xā— is the only possible discount factor.

Reversing the argument, xā— 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) + Īµ)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* + Īµ 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 ācompletenessā 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 deļ¬nition 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 deļ¬nition of

arbitrage:

Deļ¬nition (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 ā„ 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 canā™t get for free a portfolio that might pay off positively, but

will certainly never cost you anything. This deļ¬nition is different from the colloquial use of

the word āarbitrage.ā Most people use āarbitrageā to mean a violation of the law of one price

ā“ a riskless way of buying something cheap and selling it for a higher price. āArbitragesā here

might pay off, but then again they might not. The word āarbitrageā is also widely abused.

āRisk arbitrageā is a Wall Street oxymoron that means making speciļ¬c kinds of bets.

An equivalent statement is that if one payoff dominates another, then its price must be

higher ā“ if x ā„ y, then p(x) ā„ p(y) (Or, a bit more carefully but more long-windedly, if

x ā„ y almost surely and x > y with positive probability, then p(x) > p(y). You canā™t forget

that x and y are random variables.)

m > 0 ā’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) = Ī² 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 āpositiveā

means āpositive in every state of natureā or āin every possible realization.ā

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 ā„ 0 and there are some states where x > 0. Thus, in some states

Ā¤

mx > 0 and in other states mx = 0. Therefore E(mx) > 0.

No arbitrage ā’ 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 ļ¬gure 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 ā’ 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, xā— . If it isnā™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 xā— such that p =

E(xā— x), and in a complete market this is the unique discount factor. Suppose that

xā— ā¤ 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:xā— (s)<0 Ļ(s)xā— (s) is

71

CHAPTER 4 THE DISCOUNT FACTOR

Ā¤

negative, negating the assumption of no-arbitrage.

The tough part comes if markets are incomplete. There are now many mā™s that price

assets. Any m of the form m = xā— + Ā², with E(Ā²x) = 0 will do. We want to show that at

least one of these is positive. But that one may not be xā— . Since the discount factors other

than xā— are not in the payoff space X, we canā™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 donā™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. Dufļ¬e 1992 has a more formal textbook treatment.) The basic

idea is another āto every plane there is a perpendicular lineā 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) ā x ā X.

Proof : Join (ā’p(x), x) together to form vectors in RS+1 . Call M the set of all

(ā’p(x), x) pairs,

M = {(ā’p(x), x); x ā X}

M is still a linear space: m1 ā M, m2 ā M ā’ am1 + bm2 ā M. No-arbitrage

means that elements of M canā™t have all positive elements. If x is positive, ā’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 ā’ R such that

+

F (ā’p, x) = 0 for (ā’p, x) ā M, and F (ā’p, x) > 0 for (ā’p, x) ā 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 (ā’p, x) = (1, m) Ā· (ā’p, x) = ā’p + m Ā· x or

ā’p + E(mx) using the second moment inner product. Finally, since F (ā’p, x) is

positive for (ā’p, x) > 0, m must be positive.

Ā¤

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 (ā’p, x) = ā’p + m Ā· 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

xā— perpendicular to X also prices assets. Again, p = E[(m + Īµ)x] if E(Īµx) = 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 ā“ 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 xā— in the payoff space is still perfectly valid ā” p(x) = E(xā— x) ā” 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, xā— 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 deļ¬ned 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) deļ¬ned 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 mā™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 xā— in continuous time

In terms of the covariance matrix of payoffs,

xā— = E(xā— ) + [pā’E(xā— )E(x)]0 Ī£ā’1 (xā’E(x)).

Just like xā— in discrete time,

Āµ Ā¶0

dĪā— D

= ā’rf dt ā’ Āµ + ā’ r Ī£ā’1 dz.

ā—

Ī p

prices assets by construction in continuous time.

Being able to compute xā— 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

xā— = E(xā— ) + [pā’E(xā— )E(x)]0 Ī£ā’1 (xā’E(x)) (52)

where

Ā” Ā¢

Ī£ ā” E [xā’E(x)] [xā’E(x)]0

denotes the covariance matrix of the x payoffs. (We could just substitute E(xx0 ) = Ī£ +

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,

xā— = E(xā— ) + (xā’E(x))0 b,

and then ļ¬nding b to ensure that xā— prices the assets x :

Ā£ Ā¤

p = E(xā— )E(x) + E (xā’Ex)x0 b

74

SECTION 4.3 AN ALTERNATIVE FORMULA, AND Xā— IN CONTINUOUS TIME

so

b = Ī£ā’1 [pā’E(xā— )E(x)] .

If a riskfree rate is traded, then we know E(xā— ) = 1/Rf . If a riskfree rate is not traded ā“

if 1 is not in X ā“ then this formula does not necessarily produce a discount factor xā— 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, xā— = p0 E(xx0 )ā’1 x gives xā— = 0. xā— = 0 is in fact the

only discount factor in X that prices all the assets, but in this case itā™s more interesting (and

avoids 1/0 difļ¬culties 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

xā— = ā’ f E(Re )0 Ī£ā’1 (Re ā’E(Re )); Ī£ ā” 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 Īt = Et (Īt+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 Īt+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 Īt for all time t.

For calculations it is useful to ļ¬nd explicit formulas for a discount factors. Suppose a set

of securities pays dividends

Dt dt

and their prices follow

dpt

= Āµt dt + Ļt dzt

pt

where p and z are N Ć— 1 vectors, Āµt and Ļ t may vary over time, Āµ(pt , t,other variables),

E (dzt dzt ) = I and the division on the left hand side is element-by element. (As usual, Iā™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,

Āµ Ā¶0

dĪā— D

f f

Ī£ā’1 Ļdz. (53)

= ā’r dt ā’ Āµ + ā’r

ā—

Ī p

where Ī£ ā” ĻĻ0 again is the covariance matrix of returns. You can easily check that this

equation solves

ĀµĀ¶ Āµā—Ā¶

dp D dĪ dp

f

(54)

Et + dt ā’ r dt = ā’Et

Īā— p

p p

and

Āµ Ā¶

dĪā—

= ā’rf dt,

Et

Īā—

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; Īā— plus orthogonal noise will also act as a

discount factor:

dĪā—

dĪ

= ā— + dw; E(dw) = 0; E(dzdw) = 0.

Ī Ī

You can see that (4.53) is exactly analogous to the discrete time formula (4.52). (If you donā™t

like answers popping out of hats like this, guess a solution of the form

dĪ

= ĀµĪ dt + ĻĪ dz.

Ī

Then ļ¬nd ĀµĪ and ĻĪ 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 deļ¬ne 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 inļ¬nite-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 ) = Ī± + Ī² i,a Ī»a + Ī² i,b Ī»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 Ī»a = E(f a ). If the test assets are also

excess returns, then the intercept should be zero, Ī± = 0.

Much empirical work in ļ¬nance is cast in terms of expected return - beta representations

of linear factor pricing models, of the form

E(Ri ) = Ī± + Ī² i,a Ī»a + Ī² i,b Ī»b + . . . , i = 1, 2, ...N. (55)

77

CHAPTER 5 MEAN-VARIANCE FRONTIER AND BETA REPRESENTATIONS

The Ī² terms are deļ¬ned as the coefļ¬cients in a multiple regression of returns on factors,

Ri = ai + Ī² i,a fta + Ī² i,b ftb + . . . + Īµi ; 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 āfactorsā 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 āgood timesā or ābad timesā

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 Ī± and Ī» ā“ common for all assets ā“ 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 ), Ī² i,a ) pairs

to line up on a straight line with slope Ī»a and intercept Ī±.

Ī² i,a is interpreted as the amount of exposure of asset i to factor a risks, and Ī»a is inter-

preted as the price of such risk-exposure. Read the beta pricing model to say: āfor each unit

of exposure Ī² to risk factor a, you must provide investors with an expected return premium

Ī»a .ā 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 (Ī±, Ī») and to test the model (5.55) is to run a

cross sectional regression of average returns on betas,

E(Ri ) = Ī± + Ī² i,a Ī»a + Ī² i,b Ī»b + . . . + Ī±i , i = 1, 2, ...N. (57)

Again, the Ī² i are the right hand variables, and the Ī± and Ī» are the intercept and slope coef-

ļ¬cients that we estimate in this cross-sectional regression. The errors Ī±i are pricing errors.

The model predicts Ī±i = 0, and they should be statistically insigniļ¬cant in a test. (I intention-

ally use the same symbol for the intercept, or mean of the pricing errors, and the individual

pricing errors Ī±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 coefļ¬cients 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-speciļ¬c or ļ¬rm-speciļ¬c characteristics,

such as the size of the ļ¬rm, 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 ālargeā 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 ālargeā

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 ā“ on betas rather than characteristics ā“ 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 = Ī±.

We can impose this condition rather than estimate Ī± in the cross-sectional regression (5.57).

If there is no risk-free rate, then Ī± must be estimated in the cross-sectional regression. Since

it is the expected return of a portfolio with zero betas on all factors, Ī± 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 justiļ¬ed, division of labor between models of interest rates and

models of equity risk premia.) Differencing (5.55) between any two returns Rei = Ri ā’ Rj

(Rj does not have to be risk free), we obtain

E(Rei ) = Ī² i,a Ī»a + Ī² i,b Ī»b + . . . , i = 1, 2, ...N. (58)

Here, Ī² ia represents the regression coefļ¬cient 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 Ī» coef-

ļ¬cients. 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 ) = Ī»a , and so forth. We can then

write the factor model as

E(Rei ) = Ī² i,a E(f a ) + Ī² 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 Ī± 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 ļ¬nd or deļ¬ne 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 āmean-

variance frontierā 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 ļ¬ll out a hyperbolic curve through the

two assets. The curve is sharper the less correlated are the two assets, because the portfolio

variance beneļ¬ts from increasing diversiļ¬cation. 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, Ļ} 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 inļ¬nite 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

Ļ(R)

Figure 13. Mean-variance frontier

5.2.1 Lagrangian approach to mean-variance frontier

The standard deļ¬nition 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,

Ā£ 0Ā¤

E ā” E(R), and denote by Ī£ the variance-covariance matrix Ī£ = E (R ā’ E)(R ā’ E) .

A portfolio is deļ¬ned 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 āchoose a portfolio to minimize vari-

ance for a given meanā is then

min{w} w0 Ī£w s.t. w0 E = Āµ; w0 1 = 1. (59)

Solution: Let

A = E 0 Ī£ā’1 E; B = E 0 Ī£ā’1 1; C = 10 Ī£ā’1 1.

81

CHAPTER 5 MEAN-VARIANCE FRONTIER AND BETA REPRESENTATIONS

Then, for a given mean portfolio return Āµ, the minimum variance portfolio has variance

CĀµ2 ā’ 2BĀµ + A

var (Rp ) = (60)

AC ā’ B 2

and is formed by portfolio weights

E (CĀµ ā’ B) + 1 (A ā’ BĀµ)

w = Ī£ā’1 .

(AC ā’ 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 ļ¬nd it by minimizing (5.60)

over Āµ, giving Āµmin var = B/C. The weights of the minimum variance portfolio are thus

w = Ī£ā’1 1/(10 Ī£ā’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 Āµ. Thus, if you take the portfolios

corresponding to any two distinct mean returns Āµ1 and Āµ2 , the weights on a third portfolio

with mean Āµ3 = Ī»Āµ1 + (1 ā’ Ī»)Āµ2 are given by w3 = Ī»w1 + (1 ā’ Ī»)w2 .

Derivation: To derive the solution, introduce Lagrange multipliers 2Ī» and 2Ī“ on the con-

straints. The ļ¬rst order conditions to (5.59) are then

Ī£wā’Ī»E ā’ Ī“1 = 0

w = Ī£ā’1 (Ī»E + Ī“1). (61)

We ļ¬nd the Lagrange multipliers from the constraints,

E 0 w = E 0 Ī£ā’1 (Ī»E + Ī“1) = Āµ

10 w = 10 Ī£ā’1 (Ī»E + Ī“1) = 1

or

Ā· ĀøĀ· Āø Ā· Āø

E 0 Ī£ā’1 E E 0 Ī£ā’1 1 Ī» Āµ

=

10 Ī£ā’1 E 10 Ī£ā’1 1 Ī“ 1

Ā· ĀøĀ· Āø Ā· Āø

A B Ī» Āµ

=

B C Ī“ 1

82

SECTION 5.3 AN ORTHOGONAL CHARACTERIZATION OF THE MEAN-VARIANCE FRONTIER

Hence,

CĀµā’B

Ī»=

ACā’B 2

Aā’BĀµ

Ī“=

AC ā’ 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 = Rā— + wi Reā— + ni .

The mean-variance frontier is Rmv = Rā— + wReā—

Reā— is deļ¬ned as Reā— = proj(1|Re ). It represents mean excess returns, E(Re ) = E(Reā— Re )

āRe ā 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 Richardā™s approach is also valid when

we canā™t generate the payoff space by portfolios of a ļ¬nite 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 ļ¬rst describe any return by a three-way orthogonal decomposition.

The mean-variance frontier then pops out easily without any algebra.

5.3.1 Deļ¬nitions of Rā— , Reā—

I start by deļ¬ning two special returns. Rā— is the return corresponding to the payoff xā— that

can act as the discount factor. The price of xā— , is, like any other price, p(xā— ) = E(xā— xā— ).

Thus,

The deļ¬nition of Rā— is

xā— xā—

Rā— ā” (62)

=

p(xā— ) E(xā—2 )

83

CHAPTER 5 MEAN-VARIANCE FRONTIER AND BETA REPRESENTATIONS

The deļ¬nition of Reā— is

Reā— ā” proj(1 | Re ) (63)

Re ā” space of excess returns = {x ā X s.t. p(x) = 0}

Why Reā— ? We are heading towards a mean-variance frontier, so it is natural to seek a

special return that changes means. Reā— is an excess return that represents means on Re with

an inner product in the same way that xā— is a payoff in X that represents prices with an inner

product. As

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

so

E(Re ) = E(1 Ć— Re ) = E [proj(1 | Re ) Ć— Re ] = E(Reā— Re ).

If Rā— and Reā— 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 = Rā— + wi Reā— + ni

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

E(ni ) = 0.

The three components are orthogonal,

E(Rā— Reā— ) = E(Rā— ni ) = E(Reā— 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 = Rā— + wReā— (64)

for some real number w.

As you vary the number w, you sweep out the mean-variance frontier. E(Reā— ) 6= 0, so

adding more w changes the mean and variance of Rmv . You can interpret (5.64) as a ātwo-

84

SECTION 5.3 AN ORTHOGONAL CHARACTERIZATION OF THE MEAN-VARIANCE FRONTIER

fundā theorem for the mean-variance frontier. It expresses every frontier return as a portfolio

of Rā— and Reā— , with varying weights on the latter.

As usual, ļ¬rst Iā™ll argue why the theorems are sensible, then Iā™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 xā— vector is

perpendicular to planes of constant price; thus the Rā— vector lies perpendicular to the plane

of returns as shown. Go to Rā— .

Reā— 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 Rā— lies at

right angles to planes of constant price. Since Reā— is an excess return, it is orthogonal to Rā— .

Proceed an amount wi in the direction of Reā— , 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 = Rā— +wi Reā— +ni in a way that all three components are orthogonal.

Returns with n = 0, Rā— + wReā— , are the mean-variance frontier. Hereā™s why. Since

E(R2 ) = Ļ 2 (R) + E(R)2 , we can deļ¬ne 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 Rā— + wReā— line.

The graph also shows how Reā— 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 xā— or

Rā— vectors. I donā™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 xā— in X to repre-

sent prices in X by projecting m onto X, we ļ¬nd Reā— in Re by projecting of 1 onto Re . Yes,

a regression with one on the left hand side. Planes perpendicular to Reā— in Re are payoffs

with constant mean, just as planes perpendicular to xā— 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 deļ¬nitions, (5.62) and (5.63) we know that Reā— is an

85

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 Rā— and Reā— are orthogonal,

E(xā— Reā— )

ā— eā—

E(R R ) = = 0.

E(xā—2 )

We deļ¬ne ni so that the decomposition adds up to Ri as claimed, and we deļ¬ne

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 deļ¬ne

ni ā” Ri ā’ Rā— ā’ wi Reā— .

ni is an excess return so already orthogonal to Rā— ,

E(Rā— ni ) = 0.

To show E(ni ) = 0 and ni orthogonal to Reā— , we exploit the fact that since ni is an

excess return,

E(ni ) = E(Reā— ni ).

Therefore, Reā— 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

donā™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(Rā— ) + wi E(Reā— )

Ļ2 (Ri ) = Ļ2 (Rā— + wi Reā— ) + Ļ2 (ni ).

Thus, for each desired value of the mean return, there is a unique wi . Returns with

Ā„

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, letā™s locate Rā— . Rā— is the minimum second moment return. One can see this fact

from the geometry of Figure 14: Rā— is the return closest to the origin, and thus the return

with the smallest ālengthā which is second moment. As with OLS regression, minimizing

the length of Rā— and creating an Rā— orthogonal to all excess returns is the same thing. One

can also verify this property algebraically. Since any return can be expressed as R = Rā— +

wReā— + n, E(R2 ) = E(Rā—2 ) + w2 E(Reā—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 Rā— 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 Rā— is on the lower, or āinefļ¬cientā segment of the mean-variance frontier. It is initially

surprising that this is the location of the most interesting return on the frontier! Rā— is not

the āmarket portfolioā or āwealth portfolio,ā which typically lie on the upper portion of the

frontier.

Adding more Reā— 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. Ī± is the āzero-beta rateā corresponding to Rā— . It is the expected return of any return

that is uncorrelated with Rā— . I demonstrate these properties in section 6.5.

Its value

3

E(Ri ) ā’ E(Rā— )

wi =

E(Reā— )

is not particularly enlightening.

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CHAPTER 5 MEAN-VARIANCE FRONTIER AND BETA REPRESENTATIONS

E(R)

Ri

R* + wiRe*

ni

R*

Ļ(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 Rā— and Reā— is most natural

in our setup. However, you can equivalently span the mean-variance frontier with any two

portfolios that are on the frontier ā“ any two distinct linear combinations of Rā— and Reā— . In

particular, take any return

RĪ± = Rā— + Ī³Reā— , Ī³ 6= 0. (65)

Using this return in place of Reā— ,

RĪ± ā’ Rā—

eā—

R =

Ī³

you can express the mean variance frontier in terms of Rā— and RĪ± :

Rā— + wReā— = Rā— + y (RĪ± ā’ Rā— ) (5.66)

= (1 ā’ y)Rā— + yRĪ±

where I have deļ¬ned a new weight y = w/Ī³.

88

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

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 Reā— to span the frontier. When there is a risk

free rate, it is on the frontier with representation

Rf = Rā— + Rf Reā—

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 āzero betaā return, which is uncorrelated

with Rā— , a āconstant-mimicking portfolioā return, which is the return on the traded payoff

Ė

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 Ī±.

Therefore, we can span the mean-variance frontier with Rā— and any of these risk-free rate

proxies.

5.5 A compilation of properties of Rā— , Reā— and xā—

The special returns Rā— , Reā— 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 donā™t look at the picture!

1)

1

E(Rā—2 ) = (67)

.

E(xā—2 )

To derive this fact, multiply both sides of (5.62) by Rā— , take expectations, and remember

Rā— is a return so 1 = E(xā— Rā— ).

2) We can reverse the deļ¬nition and recover xā— from Rā— via

Rā—

xā— = (68)

.

E(Rā—2 )

To derive this formula, start with the deļ¬nition Rā— = xā— /E(xā—2 ) and substitute from (5.67)

for E(xā—2 )

3) Rā— can be used to represent prices just like xā— . This is not surprising, since they both

point in the same direction, orthogonal to planes of constant price. Most obviously, from 5.68

E (Rā— x)

p(x) = E(xā— x) = āx ā X

E(Rā—2 )

For returns, we can nicely express this result as

E(Rā—2 ) = E(Rā— R) āR ā R. (69)

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CHAPTER 5 MEAN-VARIANCE FRONTIER AND BETA REPRESENTATIONS

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

4) Reā— represents means on Re via an inner product in the same way that xā— represents

ńņš. 3 |