Investments

Lecture 4: Portfolio Theory, CAPM and APT

Instructor: Chyi-Mei ChenCourse website: http://www.fin.ntu.edu.tw/∼cchen/

1. In this note we shall go over briefly the distribution-based and preference-based fund separation theorems, and two linear pricing models in staticeconomies, namely the CAPM (Capital Asset Pricing Model) and theAPT (Arbitrage Pricing Theory).

2. A developing economy typically has highly incomplete financial mar-kets. Market incompleteness, however, will not cause much loss inwelfare if, given a large number of (and hence a nearly complete set of)traded assets, when we vary investors’ initial wealth and/or preferences,investors’ optimal portfolios are always portfolios of a small number offixed portfolios, for when that happens, most financial markets can beclosed down without affecting trading efficiency, as long as those fixedportfolios are still available for trading. We shall focus on the situationwhere investors’ optimal portfolios are always portfolios of two fixedportfolios, and we refer to this situation by saying that two-fund sep-aration holds. If one of the separating portfolios is riskless, then weshall refer to that riskless portfolio as money, and we say that mone-tary separation holds. When two-fund separation holds, a developingeconomy can still attain full trading efficiency if two properly chosenfinancial assets are available for trading.

3. We shall start with the definition and implications of distribution-basedtwo-fund separation, which lead to the well-known Capital Asset Pric-ing Models (the CAPMs), and then we shall compare the CAPMs to theAPT. Finally we review the conditions on the primitives of the econ-omy that ensure that two-fund separation holds in equilibrium. In thatregard, financial economists have found restrictions on return distribu-tions alone (Ross 1978; Litzenberger and Ramaswamy 1979; Chamber-lain 1983; Owen and Rabinovitch 1983), and restrictions on preferences

1

alone (Cass and Stiglitz 1970), that imply two-fund separation.1 Theformer and the latter restrictions lead to respectively the distribution-based and the preference-based separation theorems.

4. Consider a two-period economy with perfect financial markets, whereI price-taking investors can trade N risky assets and perhaps a risklessasset also. When the riskless asset exists, we shall refer to it as asset 0.Let qj be the net supply of asset j, and pj be the equilibrium price ofasset j, for all j = 0, 1, · · · , N . Assets generate consumption (or cashflows) at date 1. An investor i is endowed with some traded securitieswhose date-0 value is Wi > 0, and he consumes only at date 1. ThusWi is also the date-0 value of investor i’s equilibrium asset holdings.Define

Wm ≡I∑

i=1

Wi,

which denotes the aggregate wealth at date 0.

Definition 1 In the absence of the riskless asset, a portfolio is a vectorwN×1 with

w′1 = 1,

where ′ stands for matrix transpose, and 1N×1 is the vector of whichall elements are equal to one. Also, a vector wN×1 with w

′1 = 0 isreferred to as an arbitrage portfolio. In the presence of a riskless assetwith rate of return rf , a portfolio is any vector

x(N+1)×1 =

[1−w′1

w

],

1Suppose that markets are complete, and consider how an investor’s favorite portfolio ofall traded assets may vary with his initial wealth. If no matter how his initial wealth varies,his favorite portfolio can always be represented as a small number of fixed portfolios, thenwe say that his consumption-and-investment behavior exhibits fund separation. Here theterm separation stems from the fact that, in solving for his utility-maximizing consumptionand investment decisions, the investor can first find his favorite portfolio and then decidehow to allocate his initial wealth to his favorite portfolio and to current consumption.That is, the investor’s optimal portfolio and his optimal consumption problems can besolved separately. See for example section 2.2 of Rubinstein, M., 1974, An AggregationTheorem for Securities Markets, Journal of Financial Economics, 1, 225-244.

2

where the first element of x is the portfolio weight for the riskless asset(asset 0). Correspondingly, an arbitrage portfolio is a vector

x(N+1)×1 =

[−w′1w

].

Thus a portfolio is simply a way to allocate an investor’s initial wealthon the traded assets, and an arbitrage portfolio is a way to take posi-tions in multiple assets without costing an investor anything at date 0.Note that for all j = 1, 2, · · · , N , asset j can also be represented as aportfolio, which has only one non-zero element appearing at the j-thplace.

For all j = 1, 2, · · · , N , let r̃j be the rate of return on risky asset j, andlet r̃N×1 be the column vector of which the j-th element is r̃j. DefineeN×1 as the column vector of which the j-th element is E[r̃j]. Definealso VN×N as the square matrix of which the (k, j)-th element (thatappears on the k-th row and the j-th column) is cov(r̃k, r̃j). We call Vthe covariance matrix for the random vector r̃. Let r̃w be the (random)rate of return on the portfolio w; that is, if you spend 1 dollar to holdthe portfolio w at time 0, then you will get 1 + r̃w dollars at time 1.

Theorem 1 The following statements are true.(i) V = E[(r̃− e)(r̃− e)′].(ii) V is positive semi-definite. V is positive definite if and only if itis impossible to form a riskless portfolio of the N risky assets. In caseN = 2, such a riskless portfolio can be constructed if and only if r̃1 andr̃2 are perfectly correlated.(iii) The expected value of r̃w is w

′e. The variance of r̃w is w′Vw.

The covariance of r̃w1 and r̃w2 is w′1Vw2.

Proof. Consider part (i). Note that the (k, j)-th element of the matrixE[(r̃− e)(r̃− e)′] is

E[(r̃k − E[r̃k])(r̃j − E[r̃j])],

which is exactly the definition of cov(r̃k, r̃j).

3

Next, consider part (ii). Pick any xn×1, and observe that

x′Vx = x′E[(r̃− e)(r̃− e)′]x

= E[x′(r̃− e)(r̃− e)′x]

= E[(x′r̃− x′e))((r̃′x− e′x)]

= E[(x′r̃− E[x′r̃])2] = var[x′r̃] ≥ 0.

Thus by definition, V is positive semi-definite. Note also that an equiv-alence condition for V not to be positive definite is the existence ofxn×1 ̸= 0 such that

var[x′r̃] = x′Vx = 0,

so that x is either an arbitrage portfolio generating a sure return, orxx′1

is a riskless portfolio.

Now, if N = 2, we have

V2×2 =

[var[r̃1] cov(r̃1, r̃2)

cov(r̃1, r̃2) var[r̃2]

],

and if V is singular, then its determinant must equal zero, and so

cov(r̃1, r̃2)2

var[r̃1]var[r̃2]= 1,

and so the coefficient of correlation for (r̃1, r̃2) equals either 1 or −1.Finally, consider part (iii). By definition of rate of return, we have

r̃w =1 +w′r̃

1− 1 = w′r̃ =

N∑j=1

wj r̃j.

Hence we have

E[r̃w] = E[N∑j=1

wj r̃j] =N∑j=1

wjE[r̃j] = w′e.

Similarly, mimicking the proof for part (ii), one can easily show that

var[r̃w] = w′Vw.

4

Finally, note that

w′1Vw2 = E[(w′1r̃−w′1e)(w′2r̃−w′2e)′]

= E[(w′1r̃−w′1e)(w′2r̃−w′2e)] = cov(w′1r̃,w′2r̃).

This finishes the proof. ∥

5. Now we define the distributional two-fund separation.

Definition 2 The equilibrium rates of return r̃ exhibit two-fund sepa-ration if and only if there exist two portfolios w1 and w2 such that forany feasbile portfolio w, there exists a constant λ(w) ∈ ℜ such that

λ(w)w′1r̃+ (1− λ(w))w′2r̃ ≥SSD w′r̃.

That is, for each portfolio w, we can find a portfolio

λ(w)w1 + (1− λ(w))w2,

which is composed of only the two portfolios w1 and w2, such that allrisk-averse investors prefer λ(w)w1 + (1− λ(w))w2 to w.

Here recall the definition and the main theorem of second-order stochas-tic dominance:

Theorem 2 A random terminal wealth x̃ stochastically dominates an-other random terminal wealth ỹ if and only if one of the following threeequivalence conditions holds:(i) E[u(x̃)] ≥ E[u(ỹ)] for all concave u : ℜ → ℜ;(ii) E[x̃] = E[ỹ] and for all z ∈ ℜ,

∫ z−∞[Fx(t)− Fy(t)]dt ≤ 0, where Fj

is the distribution function of j̃, for j = x, y;(iii) There exists a random variable ẽ such that for each and every re-alization x of x̃, E[ẽ|x] = 0, and moreover, Fy is also the distributionfunction of x̃+ ẽ.

The following proposition follows from the preceding theorem immedi-ately.

5

Proposition 1 If x̃ ≥SSD ỹ, then E[x̃] = E[ỹ] and var[x̃] ≤var[ỹ].

Proof. By the preceding theorem, we know that for some ẽ withE[ẽ|x̃] = 0 the two random variables ỹ and x̃+ ẽ have the same distri-bution function, and hence they have the same variance and expectedvalue also. Now, observe that

E[ỹ] = E[x̃] + E[ẽ] = E[x̃] + E[E[ẽ|x̃]] = E[x̃],

where the second equality follows from the law of iterated expectations.Hence E[ẽ] = 0. Now, observe also that

cov(ẽ, x̃) = E[ẽx̃]− E[x̃]E[ẽ]

= E[ẽx̃] = E[E[ẽx̃|x̃]] = E[x̃E[ẽ|x̃]]

= E[x̃ · 0] = 0,

so that

var[ỹ] = var[x̃+ ẽ] = var[x̃] + var[ẽ] + 2cov(ẽ, x̃)

= var[x̃] + var[ẽ] ≥ var[x̃],

where the inequality follows from var[ẽ] ≥ 0. ∥

The definition of two-fund separation and the preceding propositiontogether imply that if two-fund separation holds in equilibrium, thenan investor’s equilibrium portfolio must have the minimum variance ofreturn in the class of portfolios promising the same expected rate ofreturn. This brings us to our next task, which is to characterize theset of portfolios each having the minimum return variance among theportfolios with the same expected rate of return.2 We shall refer to

2Alternatively, if each investor is endowed with a mean-variance utility functionU(E[W̃ ], var[W̃ ]), where U(·, ·) is increasing in its first argument and decreasing in itssecond argument, then every investor’s optimal portfolio must be a frontier portfolio (tobe defined below). Since U(·, ·) is decreasing in its second argument, the investor is said tobe variance averse. In general, an investor’s preference over feasible investment projectsmay violate the independence axiom if that preference can be represented by a mean-variance utility function. The following is an example. (Notation follows from Lecture

6

this set of portfolios the portfolio frontier, and its elements the frontierportfolios. Note that rational investors’ equilibrium choices must befrontier portfolios if two-fund separation holds in equilibrium. We shalldistinguish the case where no riskless asset is present from the casewhere a riskless asset is present.

6. First we consider the case where there does not exist a riskless asset.

Definition 3 A portfolio w is a frontier portfolio if it has the min-imum variance of return among the portfolios that promise the sameexpected rate of return. That is, w is a frontier portfolio if and only iffor all portfolios w1,

w′1e = w′e ⇒ w′1Vw1 ≥ w′Vw.

Since we have assumed that there does not exist a riskless asset, eachr̃j has a positive variance. We shall further assume that there does notexist a riskless portfolio (nor a riskless arbitrage portfolio). Recall fromTheorem 1 that the latter assumption implies immediately that V ispositive definite, and therefore non-singular.

Theorem 3 Suppose that e is not proportional to 1. Then for allµ ∈ ℜ, there exists a unique portfolio w∗(µ) that solves the followingminimization problem:

minw

1

2w′Vw,

subject tow′1 = 1, w′e = µ.

Moreover,w∗(µ) = g + hµ,

2.) Suppose that Z = {1,−1} and the three lotteries p, q, r are defined as p(1) = 13 ,q(1) = 14 , r(1) =

15 , p(−1) =

23 , q(−1) =

34 , r(−1) =

45 . Recall that P contains all prob-

ability distributions on Z. Suppose that an investor’s preference on P can be defined bythe mean-variance utility function U = E[W̃ ] − var[W̃ ]. Then it can be verified that theinvestor prefers p to q, and yet she also prefers 12q +

12r to

12p+

12r.

7

where

g =bV−1e− aV−11

b2 − ac, h =

−cV−1e+ bV−11b2 − ac

,

anda ≡ e′V−1e, b ≡ 1′V−1e = e′V−11, c ≡ 1′V−11.

Proof. Define the Lagrangian

L(w, t, s) ≡ 12w′Vw − t(w′1− 1)− s(w′e− µ).

Since the Hessian of the objective function is

D2[1

2w′Vw] = V,

which is positive definite, we conclude that 12w′Vw is a convex function

of w, so that according to the Lagrange Theorem the optimal w mustsatisfy the following first-order condition:

D[L] = 0(N+2)×1 ⇔

∂L∂w1

∂L∂w2...∂L∂wN

∂L∂t

∂L∂s

= 0(N+2)×1.

Thus the optimal solution must satisfy

Vw = t1+ se, w′1 = 1, w′e = µ.

From here, we obtainw = V−1[t1+ se]

⇒ 1 = 1′w = t1′V−11+ s1′V−1e, µ = e′w = te′V−11+ se′V−1e,

8

which allows us to solve for t and s. Replacing the explicit solutionsfor t and s into

w = V−1[t1+ se],

we obtain the optimal solution

w∗(µ) = g + hµ,

where

g =bV−1e− aV−11

b2 − ac, h =

−cV−1e+ bV−11b2 − ac

,

anda ≡ e′V−1e, b ≡ 1′V−1e = e′V−11, c ≡ 1′V−11.

Observe that h = 0 if and only if

e =b

c1,

which has been ruled out by the assumption that e is not proportionalto 1. Thus each µ ∈ ℜ corresponds to a distinct w∗(µ).3

7. The preceding theorem shows that, since w∗(µ) is linear in µ, it canbe spanned by two points w∗(µ1) and w

∗(µ2), for any µ1 ̸= µ2. Verifythat, indeed,

w∗(µ) =µ− µ2µ1 − µ2

w∗(µ1) +µ1 − µµ1 − µ2

w∗(µ2).

In particular, letting µ1 = 1 and µ2 = 0, we can conclude that anyw∗(µ) can be spanned by g + h = w∗(100%) and g = w∗(0%). Inter-preted this way, a frontier portfolio can be constructed by holding theportfolio g = w∗(0%) and then adding an arbitrage portfolio hµ to it.Recall that an arbitrage portfolio is a portfolio that costs nothing; thesum of its portfolio weights is zero.

Proposition 2 The following minimization problem has a unique solu-tion, referred to as the minimum variance portfolio for obvious reasons:

minw

1

2w′Vw,

3If instead E[r̃j ] =bc for all j = 1, 2, · · · , N , then w

∗(µ) exists if and only if µ = bc .

9

subject tow′1 = 1.

The solution is denoted by wmvp, and in fact, wmvp = w∗( b

c).

Proof. Again the solution can be easily obtained by applying theLagrange theorem. Define the Lagrangian

L(w, t) ≡ 12w′Vw − t(w′1− 1).

The optimal solution must satisfy

∂L∂w1

∂L∂w2...∂L∂wN

∂L∂t

= 0(N+1)×1.

Thus the optimal solution must satisfy

Vw = t1, w′1 = 1.

From here, we obtain

w = V−1[t1] ⇒ 1 = 1′w = t1′V−11 ⇒ t = [1′V−11]−1

⇒ wmvp =V−11

1′V−11.

On the other hand, observe that

w∗(b

c) = g + h

b

c=

1

b2 − ac[bV−1e− aV−11− bV−1e+ b

2

cV−11]

=1

c(b2 − ac)[b2V−11− acV−11] = 1

cV−11

=V−11

1′V−11.

Hence wmvp = w∗( b

c). ∥

10

8. Now, defineσ2(µ) ≡ w∗(µ)′Vw∗(µ),

andσ(µ) ≡

√w∗(µ)′Vw∗(µ).

Note that σ2(µ) is the variance of the rate of return on the frontierportfolio with expected rate of return equal to µ. The graph of thefunction σ2(·) on the (µ, σ2)-space is a parabola:

σ2(µ) = h′Vhµ2 + 2g′Vhµ+ g′Vg,

where, by the fact that V is positive definite, the coefficient of µ2 isstrictly positive! Since wmvp ̸= 0, we know that for all µ ∈ ℜ,

σ2(µ) ≥ σ2(bc) > 0.

On the other hand, one can show that

[σ(µ)√

1c

]2 − [µ− b

c√dc2

]2 = 1,

where4

d = ac− b2,4We can show that d > 0. Verify that

0 < (be− a1)′V−1(be− a1) = a(ac− b2).

There is another way to see the sign of d. Note that V−1 is symmetric:

[V−1]′ = [V′]−1 = [V]−1 = V−1.

Note also that V−1 is positive definite: for each y ∈ ℜN , there exists an x ∈ ℜN suchthat y = Vx; this actually defines a one-to-one correspondence from ℜN to itself. Hence

∀y ̸= 0 ⇒ x = V−1y ̸= 0, 0 < x′Vx = x′(VV−1)Vx = x′V′V−1Vx = (Vx)′V−1(Vx) = y′V−1y,

showing that V−1 is positive definite. Hence V−1 is a legitimate covariance matrix for

some N ×1 random vector z̃. Recognizing this fact, we can now interpret b2

ac as the squareof the coefficient of correlation for the random variables 1′z̃ and e′z̃, which is less than 1.

11

so that the graph of the function σ(·) on the (µ, σ)-space is the rightpiece of a hyperbola.5

If we assume that investors care about only the first two moments ofW̃ , with their welfare increasing and decreasing in the first and in thesecond moments of W̃ respectively, then each and every investor willend up holding a frontier portfolio with an expected rate of returngreater than b

c!6 For this reason, we shall refer to a frontier portfolio

w∗(µ) as a mean-variance efficient (or simply efficient) portfolio if µ >bc. We shall also refer to the set of mean-variance efficient portfolios

the efficient frontier. The above conclusion is that, in the absence of ariskless asset, the efficient frontier is the upper half of the right piece ofa hyperbola in the (µ, σ)-space, composed of those frontier portfolioswith µ > b

c.

9. To prepare for our next main result, we give a few propositions.

Proposition 3 For each µ ̸= bc, there exists a unique µ′(µ) ∈ ℜ

such that the covariance of rates of return on respectively w∗(µ) andw∗(µ′(µ)) is zero. Moreover, for all µ ̸= b

c,

µ′(µ) =b

c−

ac−b2c2

µ− bc

,

so that

(µ− bc)(µ′(µ)− b

c) < 0.

Proof. One can obtain µ′(µ) by directly solving

(w∗(µ))′Vw∗(µ′(µ)) = 0,

5Here by convention, the horizontal axis measures σ.6This will be true if each investor is endowed with some mean-variance utility function.

Note that if an investor is endowed with a quadratic VNM utility function U(W ) =W − ρ2W

2, where the constant ρ > 0, then the investor will optimally hold a frontier

portfolio, since E[U(W̃ )] = E[W̃ ] − ρ2 (E[W̃ ])2 − ρ2var[W̃ ] is decreasing in var[W̃ ] given

E[W̃ ]. However, note that E[U(W̃ )] is not increasing in E[W̃ ], and hence we cannot besure if the investor will hold a mean-variance efficient portfolio.

12

using the formula

w∗(µ) = g + hµ, w∗(µ′(µ)) = g + hµ′(µ).

Indeed, we have, using d = ac− b2 > 0,

g′Vg =1

d2[a2c− ab2], h′Vh = 1

d2[ac2 − cb2],

and

g′Vh =1

d2[b3 − abc].

Thus

(w∗(µ))′Vw∗(µ′(µ)) = 0 ⇔ 1d2

[a2c−ab2+µµ′(ac2−cb2)+(µ+µ′)(b3−abc)] = 0

⇔ µ′ = ab2 − a2c+ µ(abc− b3)

µ(ac2 − cb2)− abc+ b3=

µbd− adµcd− bd

=µb− aµc− b

=µb− b2

c− d

c

µc− b

=b

c−

dc2

µ− bc

.

Now the last assertion becomes transparent. ∥

Proposition 4 The set of mean-variance efficient portfolios is a con-vex set; that is, a convex combination7 of a finite number of mean-variance efficient portfolios is a mean-variance efficient portfolio.

Proof. Note that a convex combination of a finite number of frontierportfolios is a frontier portfolio. Indeed, suppose that

0 ≤ α1, α2, · · · , αm ≤ 1,m∑k=1

αk = 1, µ1, µ2, · · · , µm >b

c.

7Recall that if X is a real vector space, then for any x1, x2, · · · , xm ∈ X andα1, α2, · · · , αm ∈ [0, 1] with

∑mk=1 αk = 1,

∑mk=1 αkxk is called a convex combination

of the m vectors x1, x2, · · · , xm ∈ X. A subset A ⊂ X is a convex set if for all x1, x2 ∈ Aand for all λ ∈ [0, 1], the convex combination λx1 + (1− λ)x2 ∈ A.

13

then we have

m∑k=1

αkw∗(µk) =

m∑k=1

αk[g + hµk] = g + h[m∑k=1

αkµk] = w∗(

m∑k=1

αkµk),

so that∑m

k=1 αkw∗(µk) is a frontier portfolio. Moreover, since

m∑k=1

αkµk >m∑k=1

αkb

c=

b

c,

∑mk=1 αkw

∗(µk) is actually a (mean-variance) efficient portfolio. ∥

Let qij be the number of shares of asset j held by investor i in equilib-rium, and let qi be the N × 1 vector of which the j-th element is qij.Let q be the N×1 vector of which the j-th element is qj, and PN×N bethe diagonal matrix of which the (j, j)-th element is pj. Since marketsclear in equilibrium, we know that for all j = 1, 2, · · · , N ,

I∑i=1

qi = q.

Note also that for all i = 1, 2, · · · , I,

Wi = 1′Pqi.

Since∑I

i=1 qi = q and since∑I

i=1Wi = Wm, we have

Wm = 1′Pq.

Definition 4 The market portfolio is defined as wm, of which the j-thelement is

wmj ≡pjqjWm

, ∀j = 1, 2, · · · , N.

That is, the market portfolio is simply the market-value-weighted port-folio. Equivalently, we can write

wm =Pq

Wm=

Pq

1′Pq.

14

Proposition 5 The following equation holds:

wm =I∑

i=1

WiWm

wi,

where wi is investor i’s equilibrium portfolio; that is,

wi =PqiWi

=Pqi1′Pqi

.

Proof. Note that

I∑i=1

WiWm

wi =I∑

i=1

WiWm

PqiWi

=I∑

i=1

PqiWm

=Pq

Wm= wm.∥

The preceding proposition shows that the market portfolio is a con-vex combination of individual investors’ equilibrium portfolios. Thusthe market portfolio will be mean-variance efficient if each individualinvestor chooses to hold a mean-variance efficient portfolio in equilib-rium.

Proposition 6 Fix any µ ̸= bc, and the associated µ′(µ). Then, for

any feasible portfolio w, we have

w′e = µ′(µ) +w′Vw∗(µ)

[w∗(µ)]′Vw∗(µ)[µ− µ′(µ)].

Proof. It is tedious but rather straightforward to prove this assertion.Simply use the expressions

µ′(µ) =b

c−

dc2

µ− bc

,

andw∗(µ) = g + hµ.∥

10. Now we introduce Fischer Black’s zero-beta CAPM.

15

Theorem 4 Suppose that in equilibrium of the date-0 financial mar-kets, two-fund separation holds and all investors hold mean-varianceefficient portfolios. Then in the absence of the riskless asset, for eachportfolio w,

w′e = µ′(µm) +w′Vwmw′mVwm

[µm − µ′(µm)],

whereµm = w

′me = E[r̃m]

denotes the expected rate of return on the market portfolio.

Proof. This theorem follows from the preceding propositions directly.Since the market portfolio is a convex combination of individual in-vestors’ equilibrium portfolios, it is mean-variance efficient, and hencethe assertion follows from Proposition 6. ∥

11. Now we move on to the case where the riskless asset exists. We shallassume that the riskless asset is in zero net supply, whereas all the Nrisky assets are in strictly positive net supply.

Again, let the (N + 1)-vector qi be investor i’s equilibrium holdingsof the N traded assets (where asset 0 is the riskless asset). Let the(N + 1)-vector q contain the net supplies of the N + 1 traded assets.Let P(N+1)×(N+1) be the diagonal matrix whose (j, j)-th element is theprice of asset j, pj. Let investor i’s equilibrium portfolio be

xi =

[1−w′i1

wi

],

where the N × 1-vector wi contains portfolio weights that investor iassigns to the N risky assets. Let xm be the market portfolio. Let Wiand Wm be respectively investor i’s initial wealth and the aggregatewealth at date 0. Then we have

Wi = 1′Pqi, xi =

PqiWi

, Wm = 1′Pq.

16

Note that

xm =Pq

Wm=

I∑i=1

PqiWm

=I∑

i=1

PqiWi

WiWm

=I∑

i=1

WiWm

xi,

where the second equality is the markets clearing condition. Hence wehave shown that the market portfolio is once again a convex combina-tion of the individual investors’ equilibrium portfolios.

12. Again, we shall start with the formulae for the first two moments ofportfolio returns.

Proposition 7 The expected value and variance of the rate of returnon portfolio

x =

[1−w′1

w

]are respectively

w′e+ (1−w′1)rfand

w′Vw.

The covariance of the rates of return on respectively portfolio

x1 =

[1−w′11

w1

]

and portfolio

x2 =

[1−w′21

w2

]is

w′1Vw2.

Proof. The expression for the expected rate of return is obvious. Thecovariance of the rates of return on respectively portfolio

x1 =

[1−w′11

w1

]

17

and portfolio

x2 =

[1−w′21

w2

]is

x′1

[0 00 V

]x2 =

[1−w′11 w′1

] [ 0 00 V

] [1−w′21

w2

]= w′1Vw2.∥

Now we can give a characterization of the portfolio frontier.

Proposition 8 Suppose that there exists j ∈ {1, 2, · · · , N} such thatE[r̃j] ̸= rf . Then for each µ ∈ ℜ, there exists an N × 1 vector w∗(µ)such that

w′e+(1−w′1)rf = µ = w∗(µ)′e+(1−w∗(µ)′1)rf ⇒ w′Vw ≥ w∗(µ)′Vw∗(µ).

That is,

x∗(µ) =

[1−w∗(µ)′1

w∗(µ)

]is the frontier portfolio with expected rate of return µ. Moreover, givenµ,

w∗(µ) =(µ− rf )V−1(e− rf1)(e− rf1)′V−1(e− rf1)

,

and

w∗(µ)′Vw∗(µ) =(µ− rf )2

(e− rf1)′V−1(e− rf1).

Proof. We must solve the following minimization

minw

1

2w′Vw,

subject tow′e+ (1−w′1)rf = µ.

Now the asserted formulae can be obtained by applying the LagrangeTheorem. More precisely, define the Lagrangian

L(w, t) ≡ 12w′Vw − t[w′e+ (1−w′1)rf − µ],

18

where one can easily verify that the functions f(w) = 12w′Vw and

g(w) = w′e+(1−w′1)rf−µ are respectively convex and affine functionsof w. Thus we can obtain the optimal solution by setting the gradientof L to the (N + 1) × 1 zero vector. That is, the optimal w∗ mustsatisfy

Vw∗ = t[e− rf1] ⇒ w∗ = tV−1[e− rf1],and

[w∗]′e+ (1− [w∗]′1)rf = µ ⇔ [e− rf1]′w∗ = µ− rf .It follows that

µ− rf = [e− rf1]′w∗ = t[e− rf1]′V−1[e− rf1],

and hence

t =µ− rf

[e− rf1]′V−1[e− rf1].

It follows that

w∗(µ) =(µ− rf )V−1(e− rf1)(e− rf1)′V−1(e− rf1)

,

and

w∗(µ)′Vw∗(µ) =(µ− rf )2

(e− rf1)′V−1(e− rf1).

Note that the supposition that there exists j ∈ {1, 2, · · · , N} such thatE[r̃j] ̸= rf implies that

(e− rf1) ̸= 0,so that by the fact that V−1 is positive definite (which we showed in apreceding footnote),

H ≡ (e− rf1)′V−1(e− rf1) > 0.∥

Corollary 1 Recall that in the preceding proposition,

x∗(µ) =

[1−w∗(µ)′1

w∗(µ)

], w∗(µ) =

(µ− rf )V−1(e− rf1)H

,

andH ≡ (e− rf1)′V−1(e− rf1) > 0.

Then, the following assertions are true.

19

• w∗(µ)′1 = 0 for all µ ∈ ℜ if and only if rf = bc .• If rf ̸= bc , then there exists a unique µ

∗ ∈ ℜ such that w∗(µ∗)′1 =1, where µ∗ > rf if and only if rf <

bc.

Proof. It is easy to see that

w∗(µ)′1 = 0, ∀µ ∈ ℜ,⇔ (µ− rf )1′V−1(e− rf1) = 0, ∀µ ∈ ℜ,

⇔ 1′V−1(e− rf1) = 0,⇔ b = 1′V−1e = rf1′V−11 = rfc,

⇔ rf =b

c.

Similarly, we have, if rf ̸= bc ,

w∗(µ∗)′1 = 1 ⇒ µ∗ = H1′V−1(e− rf1)

+ rf ,

⇒ µ∗ = Hb− rfc

+ rf ,

so that µ∗ > rf if and only if rf <bc. ∥

13. The preceding proposition shows that, in the presence of a risklessasset, the portfolio frontier is the union of two half lines on the µ − σspace (where the vertical and the horizontal axes measure respectivelythe expected value and the standard deviation of the rate of return ona portfolio):

σ(µ) =

µ−rf√

H, if µ ≥ rf ;

−µ−rf√H, if µ < rf .

Apparently, the minimum variance portfolio in the current case is

xmvp =

[10

];

that is, xmvp is the riskless asset. Thus the half line

σ(µ) =µ− rf√

H, ∀µ ≥ rf

contains all the mean-variance efficient portfolios, and is termed thecapital market line (CML).

20

14. Since we have assumed that the riskless asset is in zero net supplyand all risky assets in strictly positive supply, we have the followingtheorem.8

Theorem 5 Suppose that the riskless asset exists and two-fund sepa-ration holds in equilibrium with every investor holding a mean-varianceefficient portfolio.

• In equilibrium, rf < bc .• There exists a unique µ∗ ≥ rf such that

w∗(µ∗)′1 = 1;

that is, x∗(µ∗) contains only risky assets.

• Moreover, for all µ ≥ rf , there exists a(µ) ≥ 0 such that

x∗(µ) = a(µ)x∗(µ∗) + [1− a(µ)]xmvp.

• In fact, x∗(µ∗) = xm.

Proof. First observe that if instead bc= rf , then w

∗(µ)′1 = 0 for allµ ∈ ℜ, and so every investor would be taking a long position in theriskless asset in equilibrium (because Wi > 0 for all i = 1, 2, · · · , I),which implies that the market for the riskless asset cannot clear, acontradiction! Hence we know that b

c̸= rf . We shall prove below that

rf <bc.

Next, recall from the preceding Corollary that the equation

1 = w∗(µ)′1 =(µ− rf )1′V−1(e− rf1)(e− rf1)′V−1(e− rf1)

has a unique solution µ∗ if and only if

c(b

c− rf ) = 1′V−1(e− rf1) ̸= 0,

8Most of the preceding propositions were developed by Merton (1972).

21

and in that case we obtain

µ∗ = rf +H

b− crf̸= rf ,

and

w∗(µ∗) =V−1(e− rf1)1′V−1(e− rf1)

,

and moreover, rf <bcif and only if rf < µ

∗.

Now we show that the riskless asset and x∗(µ∗) span the entire portfoliofrontier. Note that, for all µ ∈ ℜ,

x∗(µ) =

[1−w∗(µ)′1

w∗(µ)

]

= a(µ)

[0

w∗(µ∗)

]+ [1− a(µ)]

[10

],

where

a(µ) = w∗(µ)′1 =(µ− rf )1′V−1(e− rf1)(e− rf1)′V−1(e− rf1)

.

In fact, to see that the last equality holds, simply note that

a(µ)w∗(µ∗) = w∗(µ)′1w∗(µ∗)

=(µ− rf )1′V−1(e− rf1)(e− rf1)′V−1(e− rf1)

× V−1(e− rf1)

1′V−1(e− rf1)

=(µ− rf )V−1(e− rf1)(e− rf1)′V−1(e− rf1)

= w∗(µ).

Thus we conclude that every frontier portfolio is a portfolio of theriskless asset and x∗(µ∗).

Now we show that x∗(µ∗) is nothing but the market portfolio xm. Byassumption, all the I investors hold mean-variance efficient portfoliosin equilibrium, and hence there must exist a1, a2, · · · , aI ∈ ℜ, such thatfor all i = 1, 2, · · · , I,

xi =

[1− ai

aiw∗(µ∗)

].

22

Since the riskless asset is in zero net supply, we have

0 = p0q0 =I∑

i=1

(1− ai)Wi ⇒I∑

i=1

aiWi = Wm.

It follows that

xm =I∑

i=1

WiWm

xi =I∑

i=1

WiWm

[1− ai

aiw∗(µ∗)

]=

[0

w∗(µ∗)

]= x∗(µ∗);

that is, x∗(µ∗) is exactly the market portfolio.

Finally, we show that bc> rf . We have shown above that

bc̸= rf . Now

suppose instead that bc< rf . We show above that this implies that

µ∗ < rf , which implies that the CML (the set of mean-variance efficientportfolios) contains portfolios that involve selling xm short and puttingall the money in the riskless asset. Since by assumption all investorshold portfolios on the CML, every investor is taking a long position inthe riskless asset, which implies that the market for the riskless assetcannot clear, a contradiction. Thus it must be that b

c> rf (and hence

µ∗ > rf ). ∥

15. Because of the preceding theorem, from now on we re-write µ∗ by µmor E[r̃m], where r̃m denotes the rate of return on the market portfolio.Now we are ready to introduce the (traditional) Sharpe-Lintner-MossinCAPM.

Theorem 6 Suppose that the riskless asset exists and two-fund sepa-ration holds with every investor holding a mean-variance efficient port-folio. Suppose that the riskless asset is in zero net supply and all riskyassets in strictly positive supply. Then in equilibrium, for any feasibleportfolio

xk =

[1−w′k1

wk

],

if we denote its rate of return by r̃k ≡ w′kr̃ + (1 − w′k1)rf , then thefollowing CAPM equation holds:

E[r̃k] = rf + βk(E[r̃m]− rf ),

23

where

βk ≡cov(r̃k, r̃m)

var(r̃m).

Proof. Note that

cov(r̃k, r̃m) = w′kVwm = w

′kVw

∗(µ∗) =w′kVV

−1(e− rf1)1′V−1(e− rf1)

=w′k(e− rf1)

1′V−1(e− rf1)=

w′ke+ rf [1−w′k1]− rf1′V−1(e− rf1)

=E[r̃k]− rf

1′V−1(e− rf1),

where the denominator is equal to c( bc− rf ) > 0. Since the portfolio

xk was chosen arbitrarily, the above equation holds for the marketportfolio as well. Hence we have

var(r̃m) =E[r̃m]− rf

1′V−1(e− rf1).

Dividing cov(r̃k, r̃m) by var(r̃m), we obtain

βk =E[r̃k]− rfE[r̃m]− rf

,

and hence the assertion follows. ∥

16. The CAPM equation can be graphically demonstrated in the β − µspace, which is a line referred to as the security market line (SML). Itis seen from the SML that the expected rate of return on a portfoliodepends only on the beta (with respect to the market portfolio) of thatportfolio. A risky asset with a negative beta can have an expectedrate of return lower than rf , for example. The idea is that a rationalinvestor realizes that he will ultimately take a long position in themarket portfolio besides lending and borrowing at the riskless rate rf .From this perspective, every portfolio xk is just one of the ingredientassets making up the portfolio that he will be holding in equilibrium.The part of variability in the return of xk that is un-correlated with r̃mis not his concern; the risk contribution made by an ingredient portfolio

24

xk is captured by the covariance of r̃k and r̃m. More precisely, let thej-th element of wm be wjm (which equals

pjqjWm

), and observe that

var(r̃m) =N∑j=1

wjmcov(r̃j, r̃m),

and hence cov(r̃j, r̃m) is the risk contribution of asset j to the in-vestor’s equilibrium portfolio. Dividing both sides of the last equationby var(r̃m), we get

100% =N∑j=1

wjmβj,

so that the share of the risk contributed by asset j to the market port-folio is equal to βj times the portfolio weight wjm of asset j in themarket portfolio.

17. Recall that for any two random variables x̃, ỹ with finite, strictly pos-itive variances, we can find real numbers a, b and a random variable ẽsuch that

ỹ = a+ bx̃+ ẽ,

with cov[x̃, ẽ] = 0 = E[ẽ].9 In fact, one can verify that

b =cov[x̃, ỹ]

var[x̃].

Now, for any asset or portfolio j, by the above theorem we can write

r̃j = a+ br̃m + ẽ,

where we have b = βj. It follows that

var[r̃j] = β2j var[r̃m] + var[ẽ].

For obvious reasons, the left-hand side is referred to as the total risk ofasset or portfolio j, and the first term on the right-hand side the sys-tematic, or non-diversifiable, or non-idiosyncratic risk. The last termon the right-hand side is then referred to as the diversifiable risk, or

9This is the theorem of mean-variance projection.

25

idiosyncratic risk. The thrust of the CAPM is that only the non-diversifiable risk is priced; the diversifiable risk is irrelevant becauserational investors will hold a mean-variance efficient portfolio. More-over, since var[r̃m] is common to any two assets or portfolios j and k,in comparing the systematic risks we can focus on the comparison ofβj to βk.

18. Prior to Markowitz (1952, 1959), security analysis focused on pickingundervalued securities, and a portfolio was considered nothing but anaccumulation of the optimally picked securities. Markowitz was thefirst person to point out that merely accumulating the predicted win-ners is a poor portfolio selection procedure, for it ignores the effect ofportfolio diversification on risk reduction. He assumed that investors’preferences are increasing in the expected value and decreasing in thestandard deviation of portfolio returns, and he defined the efficientfrontier. His analysis provides a formal definition of diversification,and gives a measure (the beta) for the risk contribution of an ingredi-ent security. He also developed rules for the construction of an efficientportfolio. Markowitz’s portfolio theory implies that a firm should eval-uate investment projects in the same way that investors evaluate secu-rities. His normative analysis was applied by Treynor (1961), Sharpe(1964), Lintner (1965), and Mossin (1966) (who and Treynor are thesame person) to create a positive pricing theory of capital assets, whichis the CAPM that we have developed above. The SML is the most im-portant prediction of the CAPM, and it gives an explicit formula tocompute the cross-sectional risk-return trade-off.

19. Now we give a series of examples.

Example 1 Suppose that the riskless asset is present. Recall that forany two portfolios i and j with non-zero risk premia, we have

E[r̃i − rf ]E[r̃j − rf ]

=βiβj

.

Show that for any two portfolios i and j lying on the CML with non-zero

26

risk premia,10

E[r̃i − rf ]E[r̃j − rf ]

=σiσj

=βiβj

.

Example 2 Suppose that in a two-period perfect markets economy thereare 3 traded assets, labeled 1,2 and 3, where asset 1 is in zero net supply.Suppose that the following equilibrium data are valid.

e =

0.10.150.2

, V = 0 0 00 0.01 −0.010 −0.01 0.04

.(i) Suppose that the Sharpe-Lintner CAPM holds. Find the portfolioweights (on the 3 traded assets) for the market portfolio.(ii) Consider an efficient portfolio with an expected rate of return 0.12.Mr. A has $700; 000. How much money should he borrow or lend (atthe riskfree rate) if he intends to hold this portfolio?11

10Hint: Assume that for some λi, λj > 0,

r̃i = λir̃m + (1− λi)rf , r̃j = λj r̃m + (1− λj)rf .

Now, compute σi, σj , βi, and βj . In particular, verify that

σi = λiσm, σj = λjσm,

and thatβiβj

=cov[(1− λi)rf + λir̃m, r̃m]cov[(1− λj)rf + λj r̃m, r̃m]

.

11Hint: Deduce that rf = E[r̃1]. Now, suppose that r̃m = wr̃2 + (1− w)r̃3. Apply theSML to assets 2 and 3 to get

E[r̃2]− rfE[r̃3]− rf

=0.01w − 0.01(1− w)−0.01w + 0.04(1− w)

,

and hence show that E[r̃m] =16 . Now suppose that the efficient portfolio in part (ii) has

an expected rate of return equal to λrf + (1 − λ)E[r̃m] = 0.12. Obtain λ and show thatMr. A should lend $490, 000.

27

Example 3 Assume that the Sharpe-Lintner CAPM holds and you aregiven the following equilibrium data about the rates of return on assets1 and 2:

r̃1/r̃2 0.15 0.250.09 1

20

0.15 0 12

Suppose that E[r̃m] = 16% and that asset 1 is an efficient portfolio.(i) Compute cov(r̃2, r̃m).(ii) Is asset 2 also an efficient portfolio?12

Example 4 Suppose that the Sharpe-Lintner CAPM holds. Consideran asset whose date-1 random payoff is x̃. What is its date-0 price Px?Note that the asset’s rate of return

r̃x ≡x̃

Px− 1

must satisfy

E(r̃x) = rf +cov(r̃x, r̃m)

var(r̃m)[E(r̃m)− rf ],

or

E(x̃)

Px= (1 + rf ) +

cov(x̃, r̃m)

Pxvar(r̃m)[E(r̃m)− rf ] ≡ (1 + rf ) +

cov(x̃, r̃m)

Pxλ.

12Deduce that the coefficient of correlation between r̃1 and r̃2 is ρ1,2 = 1. Thus showthat to rule out arbitrage opportunities,

rf =σ2µ1 − σ1µ2

σ2 − σ1,

where σj =√var[r̃j ] and µj = E[r̃j ]. Now apply the CML to asset 1, and obtain σm =√

var[r̃m] = 0.04. Finally, apply the SML to asset 2 and get β2, which together with σmallows you to get cov(r̃2, r̃m). It is straightforward to verify whether asset 2 is lying onthe CML also. Indeed, verify that rf = 0, and that

σ1σ2

= 35 =β1β2

(cf. Example 1).

28

Multiply both sides by Px, we have

E(x̃) = Px(1 + rf ) + λcov(x̃, r̃m),

or,

Px =E(x̃)− λcov(x̃, r̃m)

1 + rf,

where the numerator of the last expression is called the certainty equiv-alent of this asset (or in short-hand notation, CEx), and λcov(x, rm) iscalled the risk premium in payoffs for this asset. We conclude that thefollowing two ways of computing the price of an asset are both valid:

Px =E(x̃)

1 + E(r̃x)=

CEx1 + rf

.

Now, we give an application of this CE formula.

Mr. X is the CEO of a large company, considering taking one of thefollowing two mutually exclusive investment projects, A and B. Thefeatures of these two projects can be summarized as follows.

• Both incur a date-0 cash outflow of $1, 000;• Both generate a sure date-1 cash revenue equal to $1, 500;• The two projects differ in their date-1 cash expenses (denoted byCA and CB respectively). There are three equally likely date-1states, referred to as boom, average, and recession. The followingtable summarizes the date-1 cash expenses of the two projects andthe realized rate of return on the market portfolio in each of thethree date-1 states.

states prob. CA CB rmboom 1

3500 600 20%

average 13

400 400 10%recession 1

3300 200 0%

Which project between A and B has a higher variance of date-1 cashflow? Which project has a higher NPV (net present value) at date 0?

29

Solution. It can be verified that both projects have the same expecteddate-1 cash flow, but project B’s date-1 cash flow has a higher variance.Nonetheless, the date-0 firm value is maximized when project B ischosen over project A, under the assumption that the CAPM holds atdate 0. The latter assumption says that the firm’s investors all takelong positions in the market portfolio and put the rest of their wealthin the riskless asset at date 0, so that they care about only the riskcontribution from the firm’s equity to their optimal portfolio. Thatis, the investors in valuing the firm care about only the correlation ofthe firm’s date-1 cash earnings and the random rate of return on themarket portfolio, not the variance of the firm’s date-1 cash earnings.

Now let us compute the date-0 present value PVj given that project jis taken at date 0. Assuming that −100% < rf < 10% = E[rm],

∀j = A,B, PVj =1500− 400− λcov(−Cj, rm)

1 + rf.

Since λ, 1 + rf are both positive by assumption, and since

cov(−CB, rm) < cov(−CA, rm),

we have

PVB > PVA ⇒ NPVB = PVB − 1000 > PVA − 1000 = NPVA,

implying that, according to the NPV-maximization criterion, projectB should be taken at date 0.

Example 5 Consider the following probability matrix:

ri/rM 0.1 0.30.08 1

414

0.12 14

14

Suppose the CAPM holds. What is rf? What is σi? Discuss.13

13Hint: Show that βi = 0.

30

Example 6 Consider the following probability matrix:

ri/rM 0.1 0.30.0 1

214

0.4 0 14

Does the Sharpe-Lintner CAPM hold?14

Solution. First we compute the first and second moments of (ri, rM).We have

E[ri] =3

4× 0 + 1

4× 0.4 = 0.1,

E[rM ] =1

2× 0.1 + 1

2× 0.3 = 0.2,

var[ri] =3

4× (0− 0.1)2 + 1

4× (0.4− 0.1)2 = 0.03,

var[rM ] =1

2× (0.1− 0.2)2 + 1

2× (0.3− 0.2)2 = 0.01,

cov(ri, rM) =1

2× (0.1− 0.2)× (0− 0.1) + 1

4× (0.3− 0.2)× (0− 0.1)

+0× (0.1− 0.2)× (0.4− 0.1) + 14× (0.3− 0.2)× (0.4− 0.1) = 0.01.

Thus we have

βi ≡cov(ri, rM)

var[rM ]=

0.01

0.01= 1.

If the Sharpe-Lintner CAPM holds, then we must have

0.1 = E[ri] = rf + βi(E[rM ]− rf ) = rf + (E[rM ]− rf ) = E[rM ] = 0.2,

which is a contradiction. Hence we conclude that the CAPM does nothold.

14Hint: Show that βi = 1. Now apply the SML to asset i and compare E[r̃i] andE[r̃m].

31

Example 7 Suppose that the traditional CAPM holds period by pe-riod. Consider an asset that promises to pay you the following per-unitrandom payoff at date 2:

M̃2

M̃1,

where Mt is the date-t value of the market portfolio of risky assets.Suppose further that the riskless rate from date 0 to date 1, r10, andthat from date 1 to date 2, r21, are respectively 5% and 7%. Determinethe date-0 and date-1 prices for the asset.15

Example 8 Suppose that in a two-period perfect-markets economy,two risky assets (assets 1 and 2) together with a riskless asset (asset 0)are traded at date 0, which generate cash flows at date 1. Suppose thatfor assets 1 and 2, we have the following data:

e =

[E[r̃1]E[r̃2]

]=

[0.240.08

], V =

[cov(r̃1, r̃1) cov(r̃1, r̃2)cov(r̃1, r̃2) cov(r̃2, r̃2)

]=

[0.04 −0.01−0.01 0.01

].

(i) Assume that asset 0 is in zero net supply and asset 1 is in strictlypositive supply. Suppose that (w, 1−w) is a portfolio of assets 1 and 2(where w ∈ ℜ), and it has zero covariance with asset 1. Find w.(ii) Continue to assume that asset 0 is in zero net supply and asset 1is in strictly positive supply. Suppose that asset 2 is also in zero netsupply. Suppose that the traditional CAPM holds in equilibrium. Whatis rf?

15Hint: Use the certainty equivalent formula to show that at date 1, the price of thatasset is

P1 =E[ M̃2

M̃1]− λcov[ M̃2

M̃1, r̃m]

1 + r21,

where

λ =E[r̃m]− r21var[r̃m]

, r̃m =M̃2

M̃1− 1.

Hence conclude that the date-1 price of that asset is non-random from investors’ perspec-tive at date 0. Conclude that at date 0, carrying that asset till date 1 and then selling it isa riskless trading strategy. Now you can compute the date-0 price of that asset accordingly:P0 =

P11+r10

= 2021 .

32

(iii) Now, ignore parts (i) and (ii). Assume instead that the risklessasset is in zero net supply and the risky assets are both in strictly posi-tive supply. Moreover, in addition to the e and V given above, you aretold that rf = 0.1. Suppose further that the traditional CAPM holds.Find E[rm], the expected rate of return on the market portfolio.

Solution. For part (i), we solve

cov(wr̃1 + (1− w)r̃2, r̃1) = 0,⇒ w =1

5.

For part (ii), we know that asset 1 is the market portfolio, and hencethe risky portfolio obtained in part (i) must have a zero beta, implyingthat its expected rate of return equals rf . Hence we have

rf =1

5× 0.24 + 4

5× 0.08 = 0.112.

Finally, for part (iii), letting (w, 1 − w) be the portfolio weights thatthe market portfolio assigns to assets 1 and 2, we have

−7 = E[r̃1]− rfE[r̃2]− rf

=cov(wr̃1 + (1− w)r̃2, r̃1)cov(wr̃1 + (1− w)r̃2, r̃2)

=5w − 11− 2w

,

yielding

w =2

3.

Hence we have

E[r̃m] =2

3× 0.24 + 1

3× 0.08 = 14

75.

Example 9 Suppose that in a two-period perfect-markets economy,two risky assets (assets 1 and 2) together with a riskless asset (as-set 0) are traded at date 0, which pay one-time cash flows at date 1.Suppose that for assets 1 and 2, we have the following data:

e =

[0.250.10

], V =

[0.04 zz 0.01

].

33

We shall assume that asset 0 is in zero net supply and assets 1 and 2are in non-negative supply.

Suppose that the Sharpe-Lintner CAPM holds in equilibrium. Supposethat the portfolio (1

5, 45), which consists of assets 1 and 2 only, has a zero

covariance with the market portfolio. Suppose that the market portfoliois either asset 1 alone or asset 2 alone. Find z.

Solution. Suppose that the market portfolio (generated by assets 1and 2 only) consists of a fraction w of the initial wealth allocated toasset 1. Define y = 100z. We have

0 = cov(wr̃1 + (1− w)r̃2,1

5r̃1 +

4

5r̃2)

=0.01

5(w × 4 + 4wy + (1− w)y + 4(1− w)× 1)

⇒ 4 + y + 3wy = 0.

Recall that w equals either zero or one. If w = 1, then y = −1; or else,y = −4. Note that if y = −4, so that z = −0.04, V is no longer apositive definite matrix: its determinant

(0.04)(0.01)− (−0.04)2 < 0!

Hence we conclude that w = 1, and hence

z = −0.01.

Example 10 Re-consider the three-asset economy described in Exam-ple 8, but assume instead that the riskless asset is in zero net supplyand the risky assets are both in strictly positive supply. Moreover, inaddition to the e and V given above, you are told that rf =

1361300

. Sup-pose that the traditional CAPM holds. Find E[rm] (the expected rate ofreturn on the market portfolio).

34

Solution Since the riskless asset is in zero net supply, we conclude thatthe market portfolio is a portfolio consisting of assets 1 and 2 only. Letthe market portfolio be (0, λ, 1−λ), where 0 is the portfolio weight forthe riskless asset, and λ is the portfolio weight for asset 1. Now wesolve for λ. Since the CAPM holds, we have from the SML

−112

=0.24− 136

1300

0.08− 1361300

=E[r1]− rfE[r2]− rf

=β1(E[rm]− rf )β2(E[rm]− rf )

=β1β2

=

cov(r1,λr1+(1−λ)r2)var(λr1+(1−λ)r2)cov(r2,λr1+(1−λ)r2)var(λr1+(1−λ)r2)

=cov(r1, λr1 + (1− λ)r2)cov(r2, λr1 + (1− λ)r2)

=λvar(r1) + (1− λ)cov(r1, r2)λcov(r1, r2) + (1− λ)var(r2)

=λ(0.04) + (1− λ)(−0.01)λ(−0.01) + (1− λ)(0.01)

⇒ λ = 34.

Thus the market portfolio (of traded assets 0,1,2) is

wm =

03414

.It follows that

E[rm] = 0× rf +3

4× E[r1] +

1

4× E[r2] = 0.2.

Example 11 Suppose that in the two-period economy, the markets forthe N risky assets are perfect. In the market for the riskless asset, how-ever, unlimited lending at the interest rate rf is allowed, but borrowingis completely prohibited. Draw the efficient frontier on the σ−µ space,assuming that rf > E[r̃mvp], where r̃mvp is the rate of return on theminimum variance portfolio composed of risky assets only. Do we stillhave 2-fund separation?

35

Solution. Yes, we do. Recall that two-fund separation holds when allthe efficient portfolios can be spanned by two fixed portfolios. In thecurrent case, although it takes more than 2 funds to span the portfoliofrontier, it only takes two funds to span the efficient frontier. See Figure1 in the attached pdf file frontier.pdf.

Example 12 Suppose that in the two-period economy, the markets forthe N risky assets are perfect. In the market for the riskless asset,however, the lending rate rL differs from the borrowing rate rB. As-sume that rB > E[r̃mvp] > rL, where r̃mvp is the rate of return on theminimum variance portfolio composed of risky assets only. Draw theefficient frontier on the σ − µ space.(i) Do we still have 2-fund separation? If not, and if we have k-fundseparation, what is the smallest k?(ii) How many distinct portfolios do we need to span the entire portfoliofrontier in this case?

Solution. For part (i), the answer is no. Now we need 3 funds to spanthe efficient frontier. On the other hand, for part (ii), it takes either3 or 4 funds to span the portfolio frontier. See Figures 2 and 3 in theattached pdf file frontier.pdf.

Example 13 Consider two risky assets with rates of return r̃1 and r̃2,and denote the expected value and standard deviation of r̃j by µj andσj. Suppose that

µ1 > µ2, σ1 > σ2.

A portfolio of these two assets can be conveniently denoted by (w, 1−w),where w is the portfolio weight assigned to (the percentage of the initialwealth spent on) asset 1.(i) Find the portfolio for the two assets with the smallest variance ofrate of return. From now on, we refer to this portfolio the minimumvariance portfolio of assets 1 and 2, or simply the mvp.16 Can themvp turn out to be asset 1 alone? If it can, when does this happen?Can it be asset 2 alone? If it can, when does this happen?

16Here you must detail the first-order and second-order conditions.

36

(ii) Now, suppose that the two risky assets are the only two tradedassets at date 0. Suppose also that every investor is endowed with amean-variance utility function (as defined in Problem 2 of Homework1). That is, every investor’s welfare is increasing in the expected valueand decreasing in the variance of the rate of return on the portfolio thathe chooses to hold at date 0. Suppose furthermore that

µ1 < µ2, σ1 > σ2.

Can there be an investor with a mean-variance utility function that iswilling to take a long position in asset 1 at date 0? Does your answerdepend on whether the two assets are in positive supply?

Solution. For part (i), we seek to

minw∈R

f(w) ≡ 12var(wr̃1 + (1− w)r̃2),

and it can be easily verified that, with ρ being the coefficient of corre-lation between r1 and r2,

f ′(w) = w(σ21 − 2ρσ1σ2 + σ22) + ρσ1σ2 − σ22,

and

f ′′(w) = σ21 − 2ρσ1σ2 + σ22 ≥ σ21 − 2 · 1 · σ1σ2 + σ22 = (σ1 − σ2)2 > 0,

so that f(·) is strictly convex. Thus the first-order condition is neces-sary and sufficient:

f ′(w∗) = 0 ⇒ w∗ = σ22 − ρσ1σ2

σ21 − 2ρσ1σ2 + σ22.

Apparently, asset 1 cannot be the mvp. For asset 2 to be the mvp,we need w∗ = 0 or cov(r̃1, r̃2) = σ

22.

17 This finishes part (i).

Before we examine part (ii), let us determine whether asset 1 and asset2 are respectively mean-variance efficient.

17In general, one can show that with N risky assets and one riskless asset, if r̃mvp standsfor the random rate of return on the mvp generated by N risky assets and r̃p is the rateof return on any portfolio p, then we have cov(r̃p, r̃mvp) =var(r̃mvp).

37

Note that (1) both assets are inefficient if µ1 < w∗µ1 + (1−w∗)µ2; (2)

both are efficient if µ2 > w∗µ1 + (1−w∗)µ2; and (3) asset 1 is efficient

while asset 2 inefficient if µ2 < w∗µ1 + (1 − w∗)µ2 < µ1. In case (1),

since µ1 > µ2, it must be that w∗ > 1. Similarly, w∗ < 0 in case (2)

and w∗ ∈ (0, 1) in case (3). We claim that w∗ < 1. To see this, notethat the sign of w∗ − 1 is the sign of ρ − σ1

σ2, which is negative. Thus

asset 1 is efficient. For w∗ > 0, it must be that ρ < σ2σ1. Thus asset 2 is

efficient if and only if ρ > σ2σ1.

Now consider part (ii). Apparently, if µ1 < µ2 and σ1 > σ2, then asset1 is dominated by asset 2 as a single asset. However, investors are notrequired to choose one single asset. In fact, they are allowed to chooseany portfolio generated by the two assets. It may still happen thatsome efficient portfolio is a convex combination of asset 1 and asset 2,and that happens if and only if w∗ > 0, or simply ρ < σ2

σ1, according

to our preceding discussion. There are two cases to consider: eitherasset 2 is efficient or it is inefficient. If asset 2 is efficient, then asset 1must be inefficient, and this case is consistent with both assets being inpositive net supply. If instead asset 2 is inefficient then asset 1 cannotbe in positive net supply: every mean-variance rational investor willchoose to hold an efficient portfolio in equilibrium, and so everyone isselling asset 1 short in equilibrium.

Example 14 Suppose that in a two-period economy there are 3 tradedassets, labeled 1,2 and 3. Suppose that the following data are valid andshort sale is completely prohibited (that is, the portfolio weights are allrequired to be non-negative).

e =

0.10.20.4

, V = 0.01 −0.02 −0.03−0.02 0.04 0.06−0.03 0.06 0.16

.Find the portfolio frontier (i.e. the frontier portfolio w∗(µ) for each tar-get expected rate of return µ ∈ ℜ). (Hint: Set up a minimization pro-gram with inequality and equality constraints, and apply Kuhn-Tuckertheorem.)

Solution. One can verify that V is positive semi-definite, but notpositive definite. Comparing the first and second rows of V reveals

38

that, essentially, for some constant α,

r̃2 = α− 2r̃1.

Comparing the first two elements of e reveals that

α = 0.4.

Now, defineν = 10µ, ∀µ ∈ [0.1, 0.4].

Given µ ∈ [0.1, 0.4], or ν ∈ [1, 4], we seek to

minw

100w′Vw

subject to

w =

x

1− x− y

y

,

x ≥ 0, y ≥ 0, x+ y ≤ 1,

and0.4(1− x− y) + (3x+ 2y − 2)e1 + ye3 = µ =

ν

10,

where e1 = 0.1 and e3 = 0.4 denote respectively E[r̃1] and E[r̃3].

It follows that

x = 2y − ν + 2 ≥ 0 ⇔ y ≥ ν2− 1,

and

1− x− y ≥ 0 ⇔ y ≤ ν − 13

.

Thus our minimization problem can be restated as, given ν ∈ [1, 4],

miny

f(y) = (8y − 3ν + 4)2 + 16y2 − 6y(8y − 3ν + 4)

39

subject to

max(0,ν

2− 1) ≤ y ≤ ν − 1

3.

The unconstrained minimum of the convex function f(·) appears at

y′ ≡ 15ν − 2032

.

Note thaty′ ≥ ν

2− 1 ⇔ ν ≤ 12;

y′ ≥ 0 ⇔ ν ≥ 43;

y′ ≤ ν − 13

⇔ ν ≤ 2813

.

Let us now take cases.

Case 1. Suppose that 1 ≤ ν ≤ 2, so that max(0, ν2− 1) = 0.

In this case, using the aforementioned properties of y′ and the fact thatf(·) is strictly convex, we have

y∗(ν) =

0, if 1 ≤ ν ≤ 4

3;

y′, if 43≤ ν ≤ 2.

Case 2. Suppose that 2 ≤ ν ≤ 4, so that max(0, ν2− 1) = ν

2− 1.

In this case, using the aforementioned properties of y′ and the fact thatf(·) is strictly convex, we have

y∗(ν) =

y′, if 2 ≤ ν ≤ 28

13;

ν−13, if 28

13≤ ν ≤ 4.

To sum up, the frontier portfolios can be characterized as follows.

40

w∗(µ) =

2− 10µ

10µ− 1

0

, if 0.1 ≤ µ ≤215;

−10µ+1216

14−65µ16

75µ−1016

, if215

≤ µ ≤ 1465;

4−10µ3

0

10µ−13

, if1465

≤ µ ≤ 0.4.

Example 15 Consider a two-period perfect markets economy wherethere are only two traded risky assets at date 0. Let the rates of returnon the two assets be r̃1 and r̃2, with their means, variances, and co-efficient of correlation being µ1, µ2, σ

21, σ

22, and ρ. You are told that

the mvp would remain the same if ρ were replaced by any otherρ′ ∈ (−1, 1). Suppose that E[r̃mvp] = 10% and µ1 = 5%. Supppsethat Mr. A cares about only the expected value and variance of the rateof return on his portfolio as we assumed in Portfolio Theory, and he hasan initial wealth $800, 000. If Mr. A wants to enjoy an expected rate ofreturn 13%, then how much money should he spend on asset 2? (Hint:Try to get information about σ1 and σ2 from the statement the mvpwould remain the same if ρ were replaced by any other ρ′ ∈ (−1, 1).Then use the fact that E[r̃mvp] = 10% and µ1 = 5% to deduce µ2.)

Solution. From Example 13, we can obtain the portfolio weight onasset 1 of the mvp, which is

σ22 − ρσ1σ2σ21 − 2ρσ1σ2 + σ22

,

41

and you are told that this portfolio weight remains the same when ρ isreplaced by ρ′, for any ρ, ρ′ ∈ (−1, 1). Hence we obtain

σ1σ2(σ21 − σ22)(ρ− ρ′) = 0,⇒ σ1 = σ2,

which in turn implies that the mvp is the equally weighted portfolioof assets 1 and 2. From here, since E[r̃mvp] = 10% and µ1 = 5%, weconclude that µ2 = 15%. If the portfolio (w, 1−w) yields an expectedrate of return equal to 13%, then it must be that w = 1

5, and hence

Mr. A should spend $800, 000× 45= $640, 000 on asset 2.

Example 16 Suppose that the Sharpe-Lintner CAPM holds in a two-period perfect markets economy where rf = 0. Suppose that it is equallylikely that r̃m = 10% and 14%.(i) Recall the price of market risk defined by

λ =E[r̃m]− rfvar(r̃m)

.

Compute λ.(ii) Suppose that in equilibrium the following data about firm A arevalid. Let x̃ be the date-1 total cash earnings of firm A, and let Dbe the face value of firm A’s debt that will be due at date 1. Sup-pose that conditional on r̃m = 10%, x̃ is equally likely to take any valuej ∈ {1, 2, · · · , 1000}, and conditional on r̃m = 14%, x̃ is equally likely totake any value k ∈ {1, 2, · · · , 100}. Suppose that 100 < D < 1000 andD is a positive integer. Assume that firm A is a corporation protectedby limited liability. Find the date-0 debt value for firm A as a functionof D. (Hint: Use the certainty equivalent formula developed in Exam-ple 4. Now apply the law of iterated expectations to E[min(x̃, D)] andcov(min(x̃, D), r̃m) by first conditioning these expectations on a realiza-tion of r̃m, and then taking the un-conditional expectations.)

Solution. It is easy to show that λ = 300, which is part (i).

For part (ii), let P be the bond price, and from the certainty equivalentformula mentioned in the hint we can get (since rf = 0)

P = E[ỹ]− λcov(ỹ, r̃m),

42

where ỹ is the date-1 payoff generated by one unit of the corporatebond. What is ỹ? Since the firm is protected by limited liability,ỹ = min(x̃, D).

By the law of iterated expectations, we have

P = E[ỹ]− λcov(ỹ, r̃m) = E[ỹ]− λE[ỹ(r̃m − µm)]

= E[ỹ(1− λ(r̃m − µm))]= E[E[ỹ(1− λ(r̃m − µm))|r̃m]]= E[E[ỹ|r̃m](1− λ(r̃m − µm))]

= prob.(r̃m = 10%){E[min(x̃, D)|r̃m = 10%](1− λ(10%− 12%)}+prob.(r̃m = 14%){E[min(x̃, D)|r̃m = 14%](1− λ(14%− 12%)}

=1

2× {

D(D+1)2

+ (1, 000−D)D1, 000

× 7}

+1

2×

100(100+1)2

100× (−5)

=1

2{− 7D

2

2, 000+

14, 007D

2, 000}+ 1

2{−505

2}

=−7D2 + 14, 007D − 505, 000

4, 000.

This finishes part (ii).

Example 17 Consider a two-period perfect markets economy wherethere are only two traded risky assets at date 0. Let the rates of returnon the two assets be r̃1 and r̃2, with their means, variances, and coeffi-cient of correlation being µ1, µ2, σ

21, σ

22, and ρ. Assume that (r̃1, r̃2) are

bivariate normal (meaning that any linear combination of r̃1 and r̃2 isagain a normal random variable).18 Suppose that there are only two

18Two random variables x̃, ỹ are bivariate normal if their joint density function

f(x, y) =1

2πσxσy√1− ρ2

e−

(x−µx)2

σ2x

+(y−µy)2

σ2y

−2ρ(x−µx)(y−µy)

σxσy

2(1−ρ2) , ∀x, y ∈ ℜ,

where µx, µy, σx, σy, ρ are the means, the standard deviations, and the coefficient of cor-relation of x̃ and ỹ.

43

investors at date 1, both seeking to maximize E[−e−W̃ ], where W̃ is aninvestor’s date-1 random wealth. For i = 1, 2, investor i is endowedwith one unit of asset i (which is the total supply of asset i) and noth-ing else. Let the date-0 equilibrium prices of the two assets be 1 andp respectively (so that asset 1 is taken as numeraire). Find the date-0market portfolio as a function of µ1, µ2, σ

21, σ

22, and ρ. (Hint: Notice

that investor i seeks to maximize E[W̃ ]− 12var(W̃ ) when choosing their

demands for the two assets, di1(p) and di2(p). Find the demands for the

two assets for each investor i, and then solve for the equilibrium pricep for asset 2 by imposing the markets clearing condition. Finally, recallthe definition of the market portfolio.)

Solution. By the hint, investor i given his initial wealth W i seeks to

maxdi1,d

i2∈ℜ

E[di1(1 + r̃1) + di2p(1 + r̃2)]−

1

2var[di1(1 + r̃1) + d

i2p(1 + r̃2)],

subject todi1 + d

i2p = W

i,

where W 1 = 1 and W 2 = p. Replacing di1 = Wi−di2p into the objective

function, and calling the latter L(di2p), we have

L(x) = E[(W i−x)(1+r̃1)+x(1+r̃2)]−1

2var[(W i−x)(1+r̃1)+x(1+r̃2)],

where L(·) can be easily verified to be concave. Rewrite L as

L(x) = E[W i(1 + r̃1) + x(r̃2 − r̃1)]−1

2var[W i(1 + r̃1) + x(r̃2 − r̃1)]

= W i(1+µ1)+x(µ2−µ1)−1

2[(W i)2σ21+x

2(σ21+σ22−2ρσ1σ2)+2W ix(ρσ1σ2−σ21)].

Hence the optimal solution x satisfies

L′(x) = 0 ⇒ µ2 − µ1 − x(σ21 + σ22 − 2ρσ1σ2)−W i(ρσ1σ2 − σ21) = 0,

or, equivalently,

di2p =µ2 − µ1 +W iσ21 −W iρσ1σ2

σ22 + σ21 − 2ρσ1σ2

,

44

so that using W 1 +W 2 = 1 + p and the market clearing condition

d12 + d22 = 1,

we obtain

p =2(µ2 − µ1) + (1 + p)[σ21 − ρσ1σ2]

σ22 + σ21 − 2ρσ1σ2

,

implying that

p =2(µ2 − µ1) + σ21 − ρσ1σ2

σ22 − ρσ1σ2.

Thus p is higher if µ2 is higher or if µ1 is lower. By the definition ofthe market portfolio, we have

wm =

1

1+p

p1+p

=

σ22−ρσ1σ22(µ2−µ1)+σ21+σ

22−2ρσ1σ2

2(µ2−µ1)+σ21−ρσ1σ22(µ2−µ1)+σ21+σ

22−2ρσ1σ2

.

20. The equilibrium CAPM equation has a counterpart derived from a no-arbitrage argument, which is the APT (arbitrage pricing theory) pricingequation.19 The arbitrage pricing theory was first developed by StephenA. Ross (1976), but our discussion below will follow Huberman(1982).

21. Consider a sequence {En;n ∈ Z+} of two-period frictionless economies,where in economy En, there are n traded assets, of which the randomrates of return r̃n are affine functions of k random factors (best thoughtof as aggregate economic variables) and of their idiosyncratic noises;more precisely,

r̃n = en +Bnd̃+ ũn, (1)

19We have assumed thus far that there exists a competitive equilibrium, and proved thatthe CAPM must hold in equilibrium provided that the imposed conditions are met. Seefor example Nielsen (1990) for a set of conditions that ensure the existence of competitiveequilibrium for an economy where each investor is endowed with a (possibly non-linear)mean-variance utility function. We have mentioned earlier that a mean-variance utilityfunction may not be equivalent to an expected utility function. While the CAPM mustobviously hold in the equilibrium of an economy where investors have mean-variance util-ity functions, we will show below that the CAPM need not be incompatible with theequilibrium of an economy where investors are expected utility maximizers.

45

where enn×1 = E[r̃n], d̃k×1 is a random vector containing the outcomes

of k aggregate economic variables, Bnn×k is a non-random matrix, andũnn×1 gives the idiosyncratic risks for the n traded assets in economy En.Note that as n grows, the number of rows in r̃n, en,Bn, and ũn grows,but the random vector d̃ remains unchanged. (Note that we have puta superscript n on r̃, e,B, and ũ to emphasize that these matrices aredata pertaining to the n-th economy En.)

22. The main result below shows that as n tends to infinity, in orderthat these return data do not admit arbitrage opportunities definedby Stephen A. Ross, for almost all traded assets, the risk premia areapproximately linear functions of B. This result should be contrastedwith the traditional CAPM, where in equilibrium the risk premium ofeach asset is a linear function of the asset’s beta with respect to the rateof return on the market portfolio. Thus the APT gives a multi-betaextension of the CAPM.20

23. It will be assumed from now on that (i) E[ũn] = 0n×1; and (ii) thecovariance matrix of ũn is

Vnn×n ≡

σ21 0 · · · 00 σ22 · · · 0...

......

...0 0 · · · σ2n

,

and moreover, although as n grows, there will be more and more maindiagonal elements in Vn, these main diagonal elements σ2i are alwaysbounded above by a finite positive number T > 0. Recall from section17 that by performing a mean-variance projection, we can always writedown equation (1) as long as the random variables contained in r̃n

and d̃ all have finite variances. However, mean-variance projection

20However, the no-arbitrage argument employed in the derivation of the APT is differentfrom the equilibrium approach adopted in the development of the CAPM. For a genuinemulti-beta extension of the traditional CAPM in a continuous-time setting, see RobertMerton, 1973, An Intertemporal Capital Asset Pricing Model, Econometrica, and forthe case where Merton’s multi-beta CAPM reduces to a single-beta CAPM, see DouglasBreeden, 1979, An Intertemporal Asset Pricing Model with Stochastic Consumption andInvestment Opportunities, Journal of Financial Economics.

46

per se does not guarantee that the random variables contained in theprojection leftover ũn are pairwise uncorrelated! Thus that Vn is adiagonal matrix is an important and restrictive assumption made bythe APT theorists, which plays a crucial role in leading to the mainprediction of the theory.

24. Before we proceed further, let us consider a special case where there isa traded riskless asset, and where for all n ∈ Z+, ũn = 0n×1. In thiscase, for all n, we have

r̃n = en +Bnd̃.

Assume furthermore that the k random variables contained in d̃ arethe rates of return on k traded portfolios tracking perfectly k macroe-conomic risks. Then, we claim that, in order to rule out arbitrageopportunities, we must have

r̃n = rf1n×1 +Bn[d̃− rf1],

so that

E[r̃n] = rf1n×1 +Bn(E[d̃]− rf1),

which is a multi-beta version of the CAPM.

To see what happens, fix any j ∈ {1, 2, · · · , n} and consider the follow-ing portfolio strategy: spending (1 − ∑kh=1 βjh) on the riskless asset,and βjh on the portfolio with rate of return d̃h, for all h = 1, 2, · · · , k.(Note that βjh is the (j, h)-element of B

n.) This portfolio strategy costs1 dollar at date 0, and it generates

(1−k∑

h=1

βjh)rf +k∑

h=1

βjhd̃h

at date 1. If instead one spends the dollar on asset j at date 0, thenthe dollar will generate

ej +k∑

h=1

βjhd̃h

47

at date 1. Since the two strategies both cost a dollar at date 0, andsince they both generate risky cash flows

∑kh=1 βjhd̃h at date 1, we must

have

ej = (1−k∑

h=1

βjh)rf

in order to rule out arbitrage opportunities at date 0. Hence the claimis true.

In the following, we shall show that in an economy with a countablyinfinite number of traded assets where no arbitrage opportunities exist,and where for all n, the rates of return on the first n traded assets satisfy(1) with ũn having the diagonal covariance matrixVn, almost all tradedassets’ expected rates of return can be approximately represented by theabove multi-beta CAPM.

25. To obtain the main results of APT, we must first establish two simplelemmas, which we shall use to prove the two theorems below.Lemma APT-1 Given y,x1,x2, · · · ,xk ∈ ℜn, where k < n andXn×k ≡ [x1,x2, · · · ,xk] has rank k, there is a vector b ∈ ℜk and avector h ∈ ℜn such that

y = Xb+ h,

and thatX′h = 0k×1;

namely, h′xi = 0, for all i = 1, 2, · · · , k.Proof. Define

b ≡ (X′X)−1X′y,where the inverse exists because X has rank k. Now given b, define

h ≡ y −Xb,

and we have

X′h = X′(y −Xb) = X′y −X′Xb = X′y −X′y = 0k×1.

Lemma APT-2 For all z ∈ ℜn, we have

z′Vnz ≤ Tz′z.

48

Proof. Simply observe that

z′Vnz =n∑

i=1

σ2i z2i ≤ T

n∑i=1

z2i = Tz′z.

26. Now, given the return data in the n-economy En,

r̃n = en +Bnd̃+ ũn,

we apply Lemma APT-1 to write

en = ρn1+Bnqn + cn,

with en in place of y and [1,Bn] in place of X in Lemma APT-1. Thuscn corresponds to the vector h in Lemma APT-1. By Lemma APT-1, cn must be orthogonal to each and every column vector in [1,Bn].That is, we have

[cn]′1 = 0, [cn]′Bn = 0′1×k.

Definition APT-1 An arbitrage portfolio in economy En is any n-vector wn such that [wn]′1 = 0.Definition APT-2 An arbitrage opportunity in the sense of Ross isa sequence {wn;n ∈ Z+} of arbitrage portfolios in the correspondingsequence {En;n ∈ Z+} of two-period frictionless economies, such that21

21To understand the definition, consider a competitive securities markets economy inwhich every investor is endowed with a mean-variance utility function U(E[W̃ ], var[W̃ ]),which is strictly increasing in its first argument and strictly decreasing in its second argu-ment, and which satisfies

limE↑+∞

U(E, var) = +∞, ∀var ∈ ℜ+.

Now, if an arbitrage opportunity in the sense of Ross exists, then it is a feasible but notnecessarily optimal strategy for investor i to keep the initial wealthWi0 on one hand and tohold the arbitrage portfolio wn on the other hand, and this strategy will yield for investori the utility

U(E[W̃ ], var[W̃ ]) = U(Wi0[1 + (wn)′r̃n],W 2i0var[(w

n)′r̃n]),

which tends to +∞ as n tends to +∞, so that there does not exist an optimal tradingstrategy for investor i—the return data are incompatible to a competitive equilibrium.

49

limn→∞

E[(wn)′r̃n] = +∞,

limn→∞

var[(wn)′r̃n] = 0.

27. Theorem APT-1 Suppose that the sequence {En;n ∈ Z+} of two-period frictionless economies does not admit any arbitrage opportuni-ties in the sense of Ross. Then, there must exist some constant A > 0such that for all n ∈ Z+,

[cn]′[cn] = [en − ρn1−Bnqn]′[en − ρn1−Bnqn] ≤ A.

This implies that, given A, the following subset of ℜk+1,

Hn ≡ {

ρq

: [en − ρ1−Bnq]′[en − ρ1−Bnq] ≤ A},is non-empty.

Proof. Suppose not. Then, given A1 > 0, there must exist n1 ∈ Z+such that

[cn1 ]′[cn1 ] > A1,

and given A2 > [cn1 ]′[cn1 ] > A1, there must exist n2 > n1, n2 ∈ Z+

such that[cn2 ]′[cn2 ] > A2,

and so on, and so forth.

Then, the infinite sequence {[cn]′[cn]} must contain a subsequence{[cnl ]′[cnl ]} with

limnl→+∞

[cnl ]′[cnl ] = +∞.

Define for all nl ∈ Z+,

anl ≡ {[cnl ]′[cnl ]}−34 .

50

Observe that wnl ≡ anlcnl is an arbitrage portfolio in economy Enl ,according to Definition APT-1. We shall show that the sequence {wnl}is an arbitrage opportunity in the sense of Ross, as defined in DefinitionAPT-2, and hence we have a contradiction.

To this end, note that

limnl→∞

E[(wnl)′r̃nl ]

= limnl→∞

anlE[(cnl)′(enl +Bnld̃+ ũnl)]

= limnl→∞

anlE[(cnl)′(ρnl1+Bnlqnl + cnl +Bnld̃+ ũnl)]

= limnl→∞

anl [cnl ]′[cnl ]

= limnl→∞

{[cnl ]′[cnl ]}14 = +∞.

On the other hand, we have the limit of the variances of the grossreturns on these arbitrage portfolios being

0 ≤ limnl→∞

var[(wnl)′r̃nl ] = limnl→∞

a2nl(cnl)′V(cnl)

≤ T limnl→∞

a2nl(cnl)′(cnl) = T lim

nl→∞{[cnl ]′[cnl ]}−

12

= 0 ⇒ limnl→∞

var[(wnl)′r̃nl ] = 0.

Thus we have indeed obtained an arbitrage opportunity defined byStephen Ross, which is a contracdition. Thus there must exist a con-stant A ≥ 0 such that for all n ∈ Z+,

[cn]′[cn] = [en − ρn1−Bnqn]′[en − ρn1−Bnqn] ≤ A.

This finishes the proof for the first assertion.

The second assertion is obvious, for Hn contains at least the vector ρn

qn

that satisfies

en = ρn1+Bnqn + cn.∥

51

28. Let R(n) be the rank of Bn. Note that R(n) ≤ R(n + 1) ≤ k. Thus,as n increases unboundedly, R(n) will converge to, say R(n); that is,R(n) = R(n) for all n ≥ n. Fix any n ≥ n, we can assume that all thecolumns in Bn are linear combinations of the first R(n) columns of Bn.This fact together with the preceding theorem then implies that givenA, for any n ≥ n, the set

Hn ≡ {

ρq

: [en − ρ1−Bnq]′[en − ρ1−Bnq] ≤ A,qR(n)+1 = qR(n)+2 = · · · = qk = 0}

is non-empty. Note that Hn+1 ⊂ Hn and for all n ≥ n, Hn is a compactset. This implies that ∩

n∈Z+Hn

is non-empty.22 Thus there must exist a constant A and some (k + 1)vector ρ

q

satisfying

∞∑i=1

[E[r̃i]− (ρ+ βi1q1 + βi2q2 + · · ·+ βikqk)]2 < A.

Define

ci ≡ E[r̃i]− (ρ+ βi1q1 + βi2q2 + · · ·+ βikqk).

Since the series∑∞

i=1 c2i converges, we must have

limi→∞

c2i = 0,

22Any decreasing sequence of non-empty closed sets in a compact space has a non-emptyintersection.

52

or equivalently, given any ϵ > 0, however small, there must exist N(ϵ) ∈Z+ such that

i > N(ϵ) ⇒ |ci| < ϵ;

that is, except for the first N(ϵ) assets, all other assets i > N(ϵ) musthave (recall that ei ≡ E[r̃i])

|E[r̃i]− (ρ+ βi1q1 + βi2q2 + · · ·+ βikqk)| < ϵ.

In plain words, almost all assets have their risk premia being approx-imately represented as affine functions of the k beta’s with respect tothe k aggregate economic variables.23

29. In the above it has been assumed that there does not exist a risklessasset. Now we introduce the riskless asset, and refer to it as asset 0,which is assumed to exist in each and every economy En (so that En has(n + 1) assets). In this case, an arbitrage portfolio can be representedas an (n+ 1)-vector [

−1′cncn

],

where unlike in the preceding sections, the n-vector cn is no longerrequired to have its elements sum up to one.

30. Theorem APT-2 Suppose that the sequence {En;n ∈ Z+} of two-period frictionless economies (with the riskless asset having rate of re-turn rf ) does not admit any arbitrage opportunities in the sense ofRoss. Let us redefine the notation. Let r̃n now denote the excess ratesof return vector. Its mean, en, becomes the risk premia vector. Notethat Vn remains to be the covariance matrix of r̃n under this new def-inition. Then, there must exist some constant A > 0 such that for all

23When k = 1, the APT equation

E[r̃i] ∼ ρ+ βi1q1,

which holds approximately for almost all assets, should be contrasted with Fischer Black’szero-β CAPM; see Theorem 4. By letting ρ = µ′(µm) and q1 = µm − µ′(µm), the APTequation becomes the zero-β CAPM equation.

53

n ∈ Z+,24[en −Bnqn]′[en −Bnqn] ≤ A.

Proof Applying Lemma APT-1, we can write

en = Bnqn + cn,

with en in place of y and Bn in place of X in Lemma APT-1. Thus cn

corresponds to the vector h in Lemma APT-1. By Lemma APT-1, cn

must be orthogonal to each and every column vector in Bn. That is,we have

[cn]′Bn = 0′1×k.

Now if the assertion fails to be true, then the infinite sequence {[cn]′[cn]}must contain a subsequence {[cnl ]′[cnl ]} with

limnl→∞

[cnl ]′[cnl ] = +∞.

In this case, define for all nl ∈ Z+,

anl ≡ {[cnl ]′[cnl ]}−34 .

Observe that

wnl ≡ anl

[−1′cnlcnl

]is an arbitrage portfolio in economy Enl , according to Definition APT-1.We shall show that the sequence {wnl} is an arbitrage opportunity inthe sense of Ross, as defined in Definition APT-2. This will establisha contradiction.

To this end, note that the limit of the risk premium on wnl

limnl→∞

E[anl(cnl)′r̃nl ]

= limnl→∞

anlE[(cnl)′(enl +Bnld̃+ ũnl)]

= limnl→∞

anlE[(cnl)′(Bnlqnl + cnl +Bnld̃+ ũnl)]

24Again, if k = 1, then the APT equation looks just like the Sharpe-Lintner CAPMequation if q1 = E[r̃m]− rf .

54

= limnl→∞

anl [cnl ]′[cnl ]

= limnl→∞

{[cnl ]′[cnl ]}14 = +∞.

On the other hand, we have the limit of the variances of the (gross)returns on these arbitrage portfolios being

limnl→∞

var[−anl1′cnl(1 + rf ) + anl(cnl)′(1 + rf1+ r̃nl)]

= limnl→∞

a2nl(cnl)′V(cnl)

≤ T limnl→∞

a2nl(cnl)′(cnl)

= T limnl→∞

{[cnl ]′[cnl ]}−12 = 0.∥

31. We have assumed in the preceding review of APT that there existan infinite number of traded assets whose rates of return are linearlyrelated to a fixed number (k) of macroeconomics variables representingthe systematic risks. Without specifying what the k macroeconomicvariables are, we now show that the APT holds trivially in an economywith a finite number (n) of risky traded assets. In fact, in the absenceof a riskless asset, if we let k = n and define

d̃ = r̃− e, Bn×n = In×n, ũ = 0n×1,

in economy En, then we have

e = ρ1+Bq,

whereρ = 0, q = e.

Note that although the APT holds exactly (rather than approximately)for economy En, it does not give any useful predictions regarding thecross-sectional relationships among the expected rates of return ontraded assets.

55

32. Now, reconsider economy En in the absence of a riskless asset. Supposethat we make the following stronger assumption

E[ũj|d̃] = 0, ∀j = 1, 2, · · · , n.

Suppose that there exists an investor with von Neumann-Morgensternutility function u such that u′′ < 0 < u′, and that for this investor theoptimal portfolio w∗ is such that

∑nj=1w

∗j ũj = 0; that is, w

∗ containsno unsystematic risk. We claim that the APT must hold exactly. Tosee this, note that w∗ must solve the following maximization problem:

maxw∈ℜn

E[u(W0(1 +w′r̃))]

subject tow′1 = 1.

Let γ be the Lagrange multiplier for the constraint. The first-orderconditions give

E[u′(W0(1 + [w∗]′r̃))r̃] = γ1.

Definef ≡ u′(W0(1 + [w∗]′r̃)) > 0,

and note that f depends on d̃ but not on ũ. The first-order conditionscan be re-written as

E[f · (e+Bd̃+ ũ)] = γ1.

It follows thateE[f ] = γ1+BE[f d̃] + E[f ũ]

= γ1+BE[f d̃],

sinceE[f ũ] = E[E[f ũn)|d̃]]

= E[fE[ũ|d̃]] = 0.

It follows that

e =γ

E[f ]1+BE[

f

E[f ]d̃],

which is an exact APT relationship.

56

33. Now we turn to the necessary and sufficient conditions for two-fundseparation. The following conditions are taken from Litzenberger andRamaswamy (1979); Ross (178) gives conditions for general k-fund sep-aration. We now state two theorems without proofs; see respectivelysections 4.4 and 4.12 of Huang and Litzenberger (1988) for the proofs.

Theorem 7 Suppose that the riskless asset does not exist and that e isnot proportional to 1. Fix µ1, µ2 ∈ ℜ, µ1 ̸= µ2. The equilibrium ratesof return r̃ exhibit two-fund separation if and only if for all portfoliosw,

[w′e− µ2µ1 − µ2

w∗(µ1) +µ1 −w′eµ1 − µ2

w∗(µ2)]′r̃

= E[w′r̃|[w′e− µ2

µ1 − µ2w∗(µ1) +

µ1 −w′eµ1 − µ2

w∗(µ2)]′r̃].

To understand this theorem, recall that when two-fund separationholds, given any µ1 ̸= µ2, w∗(µ1) and w∗(µ2) can be the two sepa-rating funds, and given any portfolio w, the portfolio

w∗(w) ≡ w′e− µ2

µ1 − µ2w∗(µ1) +

µ1 −w′eµ1 − µ2

w∗(µ2)

stochastically dominatesw in the second degree. Theorem 2 then showsthat for some random variable ϵ̃ with

E[ϵ̃|w∗(w)′r̃] = 0,

the two random variables w∗(w)′r̃ + ϵ̃ and w′r̃ have the same distri-bution function. Here, Theorem 7 gives the stronger result that thetwo random variables w∗(w)′r̃ + ϵ̃ and w′r̃ must be the same randomvariable! To obtain ϵ̃, here we can perform mean-variance projection25

25Given any two random variables x̃, ỹ both with finite positive variances, there existconstants a, b and random variable ũ such that ỹ = a+ bx̃+ ũ with

E[ũ] = cov(ũ, x̃) = 0.

We refer to bx̃ the projected value of ỹ on x̃, and a+ ũ the residual from that projection.

57

of w′r̃ on w∗(w)′r̃, and the residual from the projection is exactly theϵ̃ that we were looking for; that is, it satisfies

E[ϵ̃|w∗(w)′r̃] = 0.

Theorem 8 Suppose that the riskless asset exists and is in zero netsupply with equilibrium rate of return rf , that the N risky assets are instrictly positive supply, and that there exists j ∈ {1, 2, · · · , N} such thatE[r̃j] ̸= rf . The equilibrium rates of return rf and r̃ exhibit two-fundseparation if and only if for all portfolios w,

E[w′r̃+(1−w′1)rf−w′e−w′1rfE[r̃m]− rf

r̃m+E[r̃m]−w′e− (1−w′1)rf

E[r̃m]− rfrf |r̃m] = 0.

The idea of Theorem 8 is similar to that of Theorem 7. We simplyreplace w∗(µ1) and w

∗(µ2) in Theorem 7 by the market portfolio andthe riskless asset, for the latter can be the two separating funds in thepresence of the riskless asset.

34. A special class of distributions is of particular interest, because thosedistributions not only imply two-fund separation (and hence they sat-isfy respectively Theorems 7 and 8), but they also identify every ex-pected utility maximizer with a mean-variance utility maximizer.

Definition 5 A random vector x̃n×1 is elliptically distributed if its den-sity takes the form

f(x) = |Ω|−12 g[(x− e)′Ω−1(x− e);n],

where g(·) is some univariate function with parameter n, the n × npositive definite matrix Ω is called the dispersion matrix, and e is,ag