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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
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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.
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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).
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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.
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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.
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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
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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.
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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,
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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 .
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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). ∥
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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.
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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.
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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.
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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.
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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.
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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
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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.
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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.
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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 .
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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.∥
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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.
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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.
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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
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= 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.
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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.
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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
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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