Capital M arkets and Portfolio
Theory
Roland P ortait
From the class notes taken by Pen g C he n g
No vembre 2000
2
Table of Contents
Table of Contents
PART I Stand ard (One Period) P o rtfo lio The o ry 1
1 Portfolio Choices 2
1.A Framework and notations 2
1.A.i No Risk-free Asset
2
1.A.ii With Risk-free A sset
4
1.B Efficient portfolio in absence of a risk-free asset 6
1.B.i Efficiency criteria
6
1.B.ii Efficient portfolio and risk averse investors
8
1.B.iii Efficient set
9
1.B.iv Two funds separation (Black)
10
1.C E fficient portfolio with a risk-free asset 11
1.D HARA preferences and Cass-Stiglitz 2 fund separation 14
1.D.i HARA (Hyperbolic Absolute Risk Aversion)
14
1.D.ii Cass and Stiglitz separation
15
2 Ca pit al Ma rket Equ ilib rium 17

3.C.iii Continuous Time
30
i
Table of Contents
4 Ao A, Attainability and Com pleteness 32
4.A DeÞnitions 32
4.B Propositions on AoA and Completeness 35
4.B.i Correspondance between Q and Π :MainResults
35
4.B.ii Extensions
38
5 Alternative SpeciÞcations of Asset Prices 39
5.A Ito Process 39
5.B Diffusions 40
5.C Diffusion state variables 41
5.D Theory in the Ito-Diffusion Case 41
5.D.i Framew ork
41
5.D.ii Martingales
42
5.D.iii Redundancy and Completeness
42
5.D.iv Criteria for Recognizing a Complete Mark et
44
PART III State Variables Models: the PDE Approach 45
6Framework 46
7 Discoun ting Under Uncertainty 48
7.A Ito’s lemma and the Dynkin Operator 48
7.B The Feynman-Kac Theorem 48
8 The PDE Approach 50

,H) 65
12.A.ii Characterization and Composition of (h
,H) 65
12.A.iii The Numeraire P or tfolio and Radon-Nikodym Derivatives
69
12.BFirst Applications 69
12.B.i CAPM
70
12.B.ii Valuation
70
PART V Contin uous Time Portfolio Optimization 72
13 Dynamic Consump tion and P ortfolio Choices (The Merton
Model) 73
13.AFramework 73
13.A.i The Capital Market
73
13.A.ii The Investors (Consumers)’ Problem
74
13.BThe Solution 74
13.B.i Sketch of the Method
74
13.B.ii Optimal portfolios and L +2 funds separation
77
13.B.iii Intertemporal CAPM
78
14 THE ”EQUIVALENT” STATIC PR OBLEM (Cox-Huang,
Karatzas approach) 80
14.ATransforming the dynamic into a static problem 80
14.A.i The pure portfolio problem
80

Standard (One Period )
Portfolio Theory
Chapter 1 Portfolio Choices
Chapter 1
Po r tfolio Ch oice s
1.A Framework and notations
In all the following we co nsider a single period or time i nter val (0 1),hencetwo
instants t =0and t =1
Consider an asset whose price is S(t) (no dividends or dividends reinvested).
The r et urn of this asset between two points in time (t =0, 1) is:
R =
S (1) − S (0)
S (0)
We now consider t he case o f a portfolio. and di stinguish the case where a
riskless asset does not ex ist from t he case where a risk free asset is traded.
1.A.i N o R isk-free Asset
There are N tradable risky assets noted i =1, , N :
• The price of asset i is S
i
(t),t=0, 1.
• The return o f asset i is
R
i
=
S
i
(1) − S
i
(0)
S

(0) = x
i
=
n
i
S (0)
X (0)
(Note that x
i
(1) 6= x
i
). Besides the weigh ts sum up to one:
x
0
· 1=1
where x=(x
1
,x
2
, , x
N
)
0
and 1 is the unit vect o r.
• Th e retur n of the portfolio is the weigh te d av e r age of the returns of its
components:
R
X
= x
0

(0)
=
N
X
i=1
x
i
·
S
i
(1)
S
i
(0)
=
N
X
i=1
x
i
· (1 + R
i
)
= 1 +
N
X
i=1
x
i
R

i
,R
j
),then:
var (R
X
)=var (x
0
R)
= x
0
Γx
=
N
X
i=1
N
X
j=1
x
i
x
j
σ
ij
1.A.ii With Risk-free A s set
We now have N +1 assets, with asset 0 being the risk-free asset, and the remaining
N assets being the risky assets.
4
Chapter 1 Portfolio Choices

x
i
Note that now
x
0
· 1 6=1
where x=(x
1
,x
2
, , x
N
)
0
denotes the weights in the N risky assets.
• The retur n of the portfolio is:
R
X
= x
0
r +
N
X
i=1
x
i
R
i
= r +
N

Γx ≥ 0).Insomecases
it is positive deÞnite (∀x 6=0, x
0
Γx > 0).
DeÞnit ion 1
Assets i = 1, 2, , N are redundant if there ex ist N scalars λ
1
, λ
2
, , λ
N
such
that
P
N
i=1
λ
i
R
i
= k,wherek is a constant. Then the portfolio λ
is risk-free.
Proposition 1
The N assets i = 1, 2, , N are not redundant iff Γ is positive deÞnite (i.e. non-
singular or invertible).
5
Chapter 1 Portfolio Choices
Proof
Assume that the assets are redundant, then there exist N scalars λ
1

i
= k
Q.E.D.
Remark 1
In the following s ections we will assume that the assets are non-re dundant (it is
always possible to “drop” redundant assets if any).
1.B Efficient portfolio in absence of a risk-free asset
1.B.i E fficiency criteria
DeÞnit ion 2
Portfolio (x
∗
,X
∗
) is efficient if ∀y
, σ
Y
< σ
X
∗
⇒ µ
Y
<µ
X
∗
and σ
Y
=
σ
X
∗

, λ
¶
= x
0
µ −
θ
2
x
0
Γx − λx
0
1
The Þrst order condition
¡
∂L
∂x
=0
¢
writes:
µ − θΓx
∗
−λ1 =0
or equivalently, for i =1, , N :
µ
i
= λ + θ
N
X
j=1
x

j
σ
ij
. We then have:
µ
i
= λ + θ
N
X
j=1
x
∗
j
cov (R
i
,R
j
)
= λ + θ · cov


R
i,
N
X
j =1
x
∗
j
R

X
i=1
N
X
j=1
x
i
x
j
σ
ij
=
N
X
i=1
x
i
· cov


R
i
,
N
X
j =1
x
j
R
j

), equivalent to (P ) :
(P
0
)max
µ
x
0
µ−
θ
2
x
0
Γx
¶
s.t. : x
0
1 = 1
( (P ) an d (P
0
) yield the same solutions since they have the same Lagrangian)
(P
0
) writes, equivalently:
Max E [R
X
] −
θ
2
var (R
X

x
0
Γx s.t. : x
0
1 =1
The Lagrangia n is then:
L (x,λ)=
1
2
x
0
Γx − λx
0
1
Call k
1
the s olution. The Þrst order condition giv es:
Γk
1
− λ1 =0
Togeth er with the constraint k
0
1
1 =1gives:
λ =
1
1
0
Γ
−1

0
) is:
µ − θΓx
∗
−λ1 =0
DeÞne risk tolerance
b
θ as the inverse of risk a version, i.e.
b
θ =
1
θ
Then x
∗
can be solved as:
x
∗
=
b
θΓ
−1
¡
µ − λ1
¢
To Þnd λ,usetheconstraint1
0
x
∗
=1,i.e.
1=1

−1
µ −
b
θλ1
0
Γ
−1
1 =
b
θθ
This solves for λ:
λ =
1
0
Γ
−1
µ−θ
1
0
Γ
−1
1
Then:
x
∗
=
b
θΓ
−1
¡

b
θΓ
−1
µ
µ −
1
0
Γ
−1
µ
1
0
Γ
−1
1
·1
¶
9
Chapter 1 Portfolio Choices
We recognize in the Þrst term the minimum variance portfolio (k
1
)andwecall
k
2
the second term :
k
1
=
Γ
−1

b
θk
2
Note that k
0
1
1 =1and x
∗0
1 =1,thereforek
0
2
1 =0.Anyefficient portfolio is thus
the sum of k
1
(the minimum variance portfolio) and k
2
wh ich is a zero w eight
(zero in vestment) portfolio. As it could be expected, an inv estor with a zero risk
tolerance will hold on ly k
1
; If he has a positive risk tolerance
b
θ h e will add a risk
taking the form
b
θk
2
in order to increase the expected return. The efficien t set can
now be c aracterized as:
ES =

of the constraint on variance).
In the (σ,R) space the efficien t frontier is an hyperbola.
1.B.iv Two funds separ ation (Black)
Theorem 2
Consider any two efficient portfolio x and y:
1. Any c on v ex com bination of x and y is efficien t, i.e.∀ u ∈ [0, 1] ,ux+(1− u) y ∈
ES
2. Any efficien t portfolio is a combination of x and y (not necessarily a convex
combination)
3. The whole parabola (efficie nt and ineffic ient fron tier) is generated by (all)
combinations of x and y
10
Chapter 1 Portfolio Choices
Proof
• Sinc e x∈ ES and y∈ ES, for some positive
b
θ
X
and
b
θ
Y
,wehave:
x = k
1
+
b
θ
X
k

b
θ
Z
k
2
With
b
θ
Z
> 0, we can conclude that z∈ ES.
• Let z∈ ES,thenz = k
1
+
b
θ
Z
k
2
for some
b
θ
Z
> 0. For any x∈ ES and
y∈ ES:
ux +(1−u) y = k
1
+
h
u
b

θ
Y
b
θ
X
−
b
θ
Y
Then the combination u
∗
x +(1−u
∗
) y = z
Q.E.D.
1.C Efficient portfolio with a risk-free asset
Consider Þgure 1 w here the upper branch of the hy perbola EFR represents, in the
(σ,E) space, the efficient portfolios in absen ce of a riskless asset. Assume now tha t
exists a risk free asset 0 yielding the certain return r. M stands for the tangency
point of the hyperbola EFR w ith a straigh t lin e drown from r representing asset 0.
Point M represen ts a portfolio com posed only of risky assets, called the tangent
portfolio.
11
Chapter 1 Portfolio Choices
• Efficient frontier in presence of a riskless asset
σ
E
r
M
X

M
Γ
−1
¡
µ−r1
¢
b
θ
M
=
1
1
0
Γ
−1
¡
µ−r1
¢
Proof
1, 2, 3, 4 are standard and easy to prove. Let us proove 5 and 6: x
∗
∈ ES solves:
max
1
r + x
∗0
¡
µ
−r1
¢

−r1
¢
The tangent portfolio is an efficient portfolio, therefor e, m
=
b
θ
M
Γ
−1
¡
µ
−r1
¢
.Also:m
0
1 = 1,
then:
b
θ
M
=
1
1
0
Γ
−1
¡
µ
−r1
¢

they would all hold combinations of 0 and M and the tangent p ortfolio M would necessarily
coincide with the market portfolio.
1.D HARA preferences and Cass-Stiglitz 2 fund separation
A rational agent (in the sense of Von Neumann-Morgenstern) should maximize
the expected utility of we a lth E [U (W )].
1.D.i H AR A (Hyperbolic Absolute Risk Aversion)
A utility function U (W ) belongs to HA R A c lass if it wr ites :
U (W )=
γ
1 − γ
·
b
θ +
W
γ
¸
1−γ
Some restrictions ar e imposed on the coefficien ts γ and
b
θ and the domain of
deÞnition.
The absolute risk tolerance (ART) and absolute risk aversion (ARA ) are:
ART =
1
ARA
= −
U
0
U
00

2
b
θ
i.e. the quadra tic utility function.
3. Using a q u ad ratic utility function implies a mean-varian ce criteria; Indeed:
min var (R
X
) s.t. E [R
X
]=
b
E (and x
0
1 =1)
⇔ min E [R
2
X
] s.t. E [R
X
]=
b
E (and x
0
1 =1)
⇔ min E [X
2
(1)] s.t. E [X (1)] = X (0) ·
h
1+
b

agents).
—
Indifference to skewness (only the two Þrst moments of W matter), whereas most
investors actually like skewness.
1.D.ii Cass and Stiglitz separation
Cass and Stiglitz showed that all HA R A investors sharing the same exponential
15
Chapter 1 Portfolio Choices
parameter γ ca n build their optimal portfolios by mixing the two same funds.
When a risk free asset exists it can be chosen as one of the two funds. Since all
quadra tic (me an -varianc e ) inv estors exhib it the same γ(= −1) Tobin and Black
2 fund separation are par ticular cases of Cass and Stiglitz separation . Ca ss and
Stiglitz conditions on the u tility functions for s ep aration to hold for invest ors
sharing the sam e exponential parameter are sum marized in the follo wing table
Complete M arket Incom plete M arket
@r (under c omplete markets ∃ r) quadratic or CRRA
2
∃ r class wider th an HA RA HARA
2
in the particular case of CRRA one fund suffices (for a given
γ
the portfolio is the same for all W
16
Chapter 2 Capital M a rket Equilibrium
Chapter 2
Capital M arket E quilibrium
2.A CAPM
2.A.i T he Model
Consider again N risky assets (a risk free asset may exist or not). Th e market
value of asset i is V

]
= λ + θcov (R
M
,R
i
)
2. Co nversely, if there exist θ and λ suc h that, for i =1, , N : µ
i
=
λ + θcov (R
M
,R
i
),then(H) is true.
17
Chapter 2 Capital M a rket Equilibrium
Proof
The proof comes directly from Theorem 1.
Q.E.D.
Remark 8 θ can b e interpreted as the r isk aversion of the average (representative) investor.
Remark 9
CAPM holds for any portfolio (x
,X).
Indeed, call R
X
its return and consider the case where no risk free asset exists
(x
0
1 =1):
E [R

x
i
cov (R
M
,R
i
)
= λ + θcov
Ã
R
M
,
N
X
i=1
x
i
R
i
!
= λ + θcov (R
M
,R
X
)
Remark 10
The proof follows the same lines when the portfolio contains a risk fre e asset
with weight x
0
Remark 11 λ and θ are the same for all assets or portfolios

µ
M
−λ
σ
2
M
¸
cov (R
M
,R
i
)
18
Chapter 2 Capital M a rket Equilibrium
DeÞne:
β
i
=
cov (R
M
,R
i
)
σ
2
M
Then we may w rite the CAPM equation in the alternative form:
E [R
i
]=λ + β

N
] is the s et of all linear combinations of v
1
, v
2
, , v
N
, or line a r
subspace generated by v
1
, v
2
, , v
N
. Th e dim e ns ion of [vect (Γm)]
⊥
is thus N −1
and there are an inÞnity of 0-beta portfolios. Now, from the general CAPM, w e
would have: λ = µ
Z
;Thus:
Corollary 1
(0−beta CAPM) If M is efficient, for any zero beta portfolio or asset Z: E [R
i
]=
µ
Z
+ β
i
(µ

) are known), what is its p rice V (0) at time 0?
19