MODERN P O R T F O L I O THEORY: SOME MAIN RESULTS
BY HEINZ H. MOLLER
University of Ziirich
ABSTRACT
This article summarizes some main results in modern portfolio theory. First, the
Markowitz approach is presented. Then the capital asset pricing model is derived
and its empirical testability is discussed. Afterwards Neumann-Morgenstern
utility theory is applied to the portfolio problem. Finally, it is shown how optimal
risk allocation in an economy may lead to portfolio insurance.
KEYWORDS
Modern portfolio theory; Markowitz approach; capital asset pricing model;
Neumann-Morgenstern utilities; portfolio insurance.
1. INTRODUCTION
Starting with MARKOWITZ' (1952) pioneering work, modern portfolio theory has
developed to a highly sophisticated field of research. In addition it became more
and more obvious that for a large class of insurance problems a separate analysis
of actuarial and financial risks is inappropriate. Of course modern portfolio
theory is typically applied to common stocks. However, it can also be applied to
bonds if there are risks with respect to default, exchange rates, inflation, etc.
These facts, the increasing importance of new financial instruments, and the
availability of computer capacities explain the growing interest of actuaries in
modern portfolio theory.
In this paper some main results of modern portfolio theory are presented.
However, some important aspects such as arbitrage pricing theory, multiperiod
models, etc. are not treated here. ~ The paper is organized as follows: Section 2
deals with the Markowitz approach. In Section 3 the Capital Asset Pricing Model
(CAPM) is derived (SHARPE, 1963, 1964; LINTNER, 1965; BLACK, 1972). Difficulties with respect to the testability of CAPM are discussed in Section 4
(ROLL, 1977). In Section 5 von Neumann-Morgenstern utility theory is applied
to the portfolio problem and a generalized version of the CAPM-relationship is
presented (CASS and STIGLITZ, 1970; MERTON, 1982). Finally, in Section 6 it is
shown how the optimal risk allocation in an economy may lead to portfolio
insurance (BORCH, 1960; LELAND, 1980).
I.
For a rigorous and comprehensive representation of modern portfolio theory see INGERSOLL
(1987), An intuitive introduction is given by COPELAND and WESTON (1988) or HARRINGTON
(1987).
A S T I N B U L L E T I N Vol. 18, No. 2
128
MOLLER
2.
THE
MARKOWITZ
APPROACH
W h e r e a s in a c t u a r i a l science the law o f large n u m b e r s plays a central role this is
not the case in p o r t f o l i o t h e o r y . Due to the c o r r e l a t i o n between the returns on
financial assets, diversification allows in general o n l y for a r e d u c t i o n but not for
an e l i m i n a t i o n o f the risk. MARKOWITZ (1952) was the first who t o o k the
c o v a r i a n c e s between the rates o f return into a c c o u n t .
2.1. T h e M o d e l
T h e r e are N assets h = 1, ..., N (e.g. c o m m o n stocks). A n investment o f one unit
o f m o n e y (e.g. 1 ($)) in asset h leads to a stochastic return Rh. The first and second
m o m e n t s o f Rt . . . . , RN are a s s u m e d to exist.
T h e vector o f expected values a n d the c o v a r i a n c e m a t r i x are d e n o t e d by
/z ~ R N,
with
#h = E ( R h )
h = I ..... N
and
VE R N2, with V~,t = Cov(Rh, Rt)
h, 1 = 1. . . . . N.
A p o r t f o l i o is given by a vector x e R N, with Eh=
N ~ Xh = 1. Xh denotes the weight
o f asset h = I . . . . . N. z Hence the overall return on a p o r t f o l i o x is given by the
r a n d o m variable
N
R (x) := ~
xI, Rh
h=l
and one o b t a i n s i m m e d i a t e l y
E[R(x)] = #'x, Var[R(x)] = x 'Fx.
DEFINITION. A p o r t f o l i o x* is called ( m e a n - v a r i a n c e ) efficient if there exists no
p o r t f o l i o x with
E [ R ( x ) ] /> E [ R ( x * ) ] , V a r [ R ( x ) ] < V a r [ R ( x * ) ] .
T h e M a r k o w i t z a p p r o a c h is a m e t h o d to calculate m e a n - v a r i a n c e efficient portfolios. Hence, the M a r k o w i t z a p p r o a c h is based on m e a n - v a r i a n c e analysis,
where the variance o f the overall rate o f return is taken as a risk m e a s u r e and
the expected value measures profitability. In c o n t r a s t to expected utility maxim i z a t i o n , m e a n - v a r i a n c e analysis takes into a c c o u n t only the first two m o m e n t s
a n d there is no clear theoretical f o u n d a t i o n . T h e special a s s u m p t i o n s under which
m e a n - v a r i a n c e analysis is consistent with expected utility m a x i m i z a t i o n are
discussed in Section 5.
T h e M a r k o w i t z a p p r o a c h can be f o r m a l i z e d as follows:
min l ] 2 x ' Vx
x E R ,~'
2. x~, < 0 corresponds to a short position with respect to asset h.
129
MODERN PORTFOLIO THEORY
subject to
e~x = 1 3
(M)
p.'x/> r,
where r is the m i n i m u m level for the expected value o f the overall rate o f return.
Here, the m i n i m u m level r is assumed to be exogenously given. In applications,
individual investors choose r in accordance with their risk aversion.
( M ) is a quadratic convex optimization problem. Under the assumption
A.1
1) V is positive definite,
2) e, # are linearly independent,
there exists a unique solution o f ( M ) and the K u h n - T u c k e r theorem can be
applied. The K u h n - T u c k e r conditions are given by
(1)
Vx - Xje - X2/.*= 0
(2)
e'x = 1
(3)
~'x/> r
(4)
X2 t> 0, ~z(tz'x - r) = 0.
Hence
X ~" }kl
V-le + ~2 V- IV.
or if (2) is taken into account
(5)
X =
VX I + (1 -
v)x 2
with
V_t
x 1= v-l____e_
e' V-~e'
x 2 = {eT.~=.i-p.
ife
~xt+
else
V-~p.
v-¿~#o
By varying the m i n i m u m level o f return r, all mean-variance efficient portfolios
are obtained, x ~ and x 2 do not depend on r. However. v depends on Xj resp. on
X2 and therefore ultimately on r. Hence, one gets
(6)
x(r)
= v(r)xl
+
[1 --
v(r)] x 2.
Thus, any efficient portfolio is a c o m b i n a t i o n o f two fixed reference portfolios
x I and x 2. INGERSOLL (1987, p. 86) shows that x I is the global m i n i m u m
variance portfolio with an expected value o f return
= g,' V - l e
(7)
rmin
V-I-'"---~
e'
Furthermore, it is easily seen that (3) holds with equality if and only if r 1> r m i n .
3. e ' = ( l , l ..... I)~R #.
130 )
MOLLER
Hence, for r t> r m i n , (3) and (6) lead to
(8)
(9)
oZ(r):= V a r l R [ x ( r ) ] } = vZx~'Vx ~ + 2 v ( l - v)xl'VxZ + (1
-
v)2x2"Vx z
r = v/.t'x I +(1 - v)/z'x 2.
From (8) and (9) one can derive that there is a hyperbolic relationship between
r and a(r) (Figure 1).
2.2. Availability o f a riskless asset
If, in addition to the risky asset h = 1..... N (common stocks, etc.), a riskless
asset h = 0 with a deterministic return Ro (e.g. a treasury bill) is available, the
model changes as follows:
= (Izo, lz) E R N+I,
V~ R N2,
with
with
#o = R0, #h = E ( R h ) ,
Vm = Cov(Ri,, RI),
h = 1. . . . . N
h, I = 1 . . . . . N
N
i = ( X o , x ) ~ R N+l,
with
~
Xh = 1
h=0
N
R(i):= ~
xhR,.
h=0
Mean-variance efficient portfolios now result from the optimization problem
rain
l/2x'Vx
subject to
~'i = 1 4
~ ' i ~> r.
(M')
rmin
e•ffi
c
i
e
nJt
l
f
ront
i
e
r
iA
A =
( ( . ~ l ' v x _ l ) 1 / 2 , ~ ' x _ 1]
B =
( ( x _ 2 " V x _ 2 ) l t 2 , ~ ' x _ 2)
t
t
inefficient branch
x
\
o(r)
FIGURE I.
4. ~' = ( I , I . . . . . I ) ~ R ~+t.
131
MODERN PORTFOLIO THEORY
Under the assumption
A'I
1) V is positive definite
2) f~, ~ are linearly independent
mean-variance efficient portfolios are still of the form
~(r) = v ( r ) i l + [1 - v(r)]x 2
(6')
but now one can show that
~1' = ( 1 , 0 . . . . . 0), ]~2' ~---(0, Xl2 . . . . . X 2 )
holds, i.e. every mean-variance efficient portfolio is a combination of the riskless
investment i ~with a reference portfolio i 2, consisting exclusively of risky assets.
Furthermore, the efficient frontier degenerates to a straight line (Figure 2).
2.3,
Remarks
1) It is important to note that the special structure of the set of mean-variance
efficient portfolios provides the basis for the Capital Asset Pricing Model
(see Section 3).
2) In practical applications additional constraints sometimes have to be imposed, such as exclusion of short sales, bounds on the weights of individual
assets, etc. With constraints of this type, the optimization problem is still
quadratic convex and powerful numerical methods are available. However,
in general the special structure of the efficient frontier is destroyed.
3) A detailed and careful analysis of the Markowitz approach can be found in
the appendix of ROLL'S (1977) article or in ]NGERSOLL'S (1987) chapter 4.
B
'
~
e
r
A' = ( 0 , R o )
Ro
B' = Q,.x.2"Vx2) I/2, ~" ~ "2)
%
%
%
o(r)
FIGURE 2.
132
MULLER
3. THE CAPITAL ASSET PRICING MODEL (CAPM)
3.1. The Sharpe-Lintner Model
In the S h a r p e - L i n t n e r Model (SHARPE, 1963, 1964; LINTNER, 1965), there is a
riskless asset h = 0 and N risky assets h = 1,..., N.
There are m investors i = 1 ..... m who are characterized as follows:
1) Wi > 0 is the initial wealth o f investor i.
2) Investors i = l , . . . , m agree on the first and second m o m e n t s o f returns on
assets h = 0 . . . . . N, i.e.
~i=~,
vi= V
i = l ..... m,
3) Investor i seeks a mean-variance efficient portfolio ~i with E[R(£i)] = ri,
where r~ > Ro i = 1, ..., m.
According to formula ( 6 ' ) (Section 2.2) the portfolio chosen by investor i is o f
the form
(6')
xi =
u(r/)x
I +
[1
v(r/)]x 2
-
i= 1..... m.
Total d e m a n d results in a portfolio
i M : = W1 i =~~ Wi~i,
(10)
with
W:= ~°,
~ Wi.
By means o f ( 6 ' ) one can show (See Fig. 3) that i M lies on the efficient frontier
and can be represented as the solution o f
min
l]2x' Vx
i E R x÷~
subject to
~'i = 1
~ ' i 1> rM
(M")
rM
ri
~
Ro
G(r)
FIGURE 3.
133
MODERN PORTFOLIO THEORY
with
rM := " ~
Wiri.
i=l
Hence, ifft = (Xao,
n xaM ) satisfies the K u h n - T u c k e r conditions
(11)
- Xi - X2Ro = 0
(12)
(13)
V:,~t - X l e -
X2/J. = 0
~ ' i am = 1
(14)
~'i~
(15)
= rM
X2 >/0.
Formulae (11) and (12) lead to
(16)
Vx~ = h2 (p. - Roe).
F r o m (11), (12) and #o = Ro one obtains
(17)
x;7' v x Y = × , ~ ' i Y + x2~'~,~'
or according to (11), (13) and (14)
(18)
xfl' Vxff = XZ(rM -- Ro).
Formulae (16) and (18) imply
Vxff
(19)
j, - Roe
xaM' Vxa
~---~ = rM -- Ro"
On the other hand, total supply is given by the market portfolio ~ ^ ! where all
assets are held in p r o p o r t i o n to their market values. In equilibrium total d e m a n d
must be equal to total supply, i.e.
(20)
iff t = ira.
C o m b i n i n g (19) and (20) leads to the equilibrium condition
_tl-Roe
VxM
(21)
xM'Vx M
rM-- Ro
or by taking into account
#1, = E ( R h ) ,
h = 1. . . . . N
N
rM=E(RM),
where
RM=
~] xi~Rh
h=O
xM'VxM = V a r ( R g )
X M' V = (Cov(Ri, R M ) . . . . . CoV(RN, RM ))
one obtains the C A P M - r e l a t i o n s h i p
(22)
E ( R h ) - Ro - C o v ( R t , , R M ) [ E ( R M ) - Ro],
Var(RM)
h = ! ..... N
134
MOLLER
COMMENTS
1) /3h := Cov(Rh, RM)[Var(RM) is called the beta-coefficient of asset h.
E ( R h ) - R o is the risk premium on asset h
E ( R M ) - Ro is the risk premium on the market portfolio.
2) Under stationarity and normality assumptions the beta-coefficients can be
estimated by applying the ordinary least square method to
Rh - Ro = oth +/3j~(RM -- Ro) + eh.
The term l~h(RM -- Ro) corresponds to the systematic risk which is undiversifiable. The error term Ch satisfies C o v ( c h , R M ) = 0 and corresponds to the
unsystematic risk. In particular, the following decomposition is possible
Var(Rh) =/3h2 Var(RM) + Var(eh).
3) For empirical beta estimation there is no general agreement with respect to the
measurement period and the interval choice. Monthly data over a five-year
period are widely used.
4) Typically the observed beta-coefficients are positive. However, negative betacoefficients may occur as well.3
5) According to the C A P M relationship each asset h can be represented by a
point (/3h, E(Rh)) which lies on the theoretical market line
E(Rh) = Ro + [ E ( R M ) - Ro]~h.
Often the empirical market line which is based on estimations for Bh, E(Rh),
E(RM) deviates substantially (see, e.g. BLACK, JENSEN and SCHOLES, 1972).
theoretical market line
E (R h )
~,,.~-'~"~ 7 - ' '
empirical market line
Ro
FIGURE 4.
5. In such a case Rh and RM are negatively correlated and E(Rh) < Ro must hold.
135
MODERN PORTFOLIO THEORY
3.2. T h e B l a c k M o d e l
To assume the existence of a riskless asset is somewhat questionable, especially
if one is interested in real returns. Fortunately, even without assuming the
existence o f a riskless asset a C A P M - r e l a t i o n s h i p can be derived.
It is assumed that the framework o f Section 2.1 holds and that there are m
investors i = 1 ..... m with
initial wealth Wi,
first and second m o m e n t s o f returns given by i = t z ,
Vi = V
and seeking for mean-variance efficient portfolios x~ with E [ R ( x i ) ] = r~, where
ri i> rmin, i = 1..... m.
Under these conditions it can easily be shown that in equilibrium the market
portfolio x M must be effÉcient and from the corresponding optimality conditions
one can derive the C A P M - r e l a t i o n s h i p
(23)
E ( R h ) - E [ R ( x ° ) ] - Cov(Rh, R M ) i E ( R M ) _ E [ R ( x O ] ]
Var(RM)
h = l ..... N.
x ° is the so-called zero-beta portfolio with the properties
a) CoV(RM, R ( x ° ) ) = 0.
b) There exists J, such that (5) holds, i.e. x ° = ux~+ (1 - v)x 2.
EXPLANATIONS
1) According to b) the point Z : = ((x°'Vx°)~/2, p ' x ° ) lies on the hyperbola
described in Section 2.1.
x2
M
M
rmin
~ ' x_0
.-~
\ . - - - - - x_O
FIGURE 5.
136
MOLLER
2) ROLL (1977) shows that:
Z lies on the inefficient branch of H.
The tangent in the point M = ((xM'Vx~t)I/2, tt'XM ) intersects the return
axis at (0,/~'x°).
COMMENTS. This model was developed by BLACK (1972). It is more than an
interesting alternative to the S h a r p e - L i n t n e r approach. In the next section we
shall see that it is needed in order to discuss the empirical testability of the
S h a r p e - L i n t n e r model.
4.
EMPIRICAL TESTABILITY OF THE S H A R P E - L I N T N E R MODEL
Before discussing testability we must recall that the S h a r p e - L i n t n e r model was
developed under restrictive assumptions, namely:
M.l
M.2
M.3
M.4
M.5
M.6
Evaluation of portfolios by mean-variance analysis,
Uniform planning horizon for all investors,
Homogeneous expectations,
Existence of a riskless asset,
Exclusion of transaction costs,
No restrictions on short sales.
BLACK et al. (1972), BLUME and FRIEND (1973), FAMA and MACBETH (1973)
were a m o n g the first to test the S h a r p e - L i n t n e r model. The evidence provided
by their studies in favour of the CAPM-relationship is rather weak. 6 However,
as ROLL (1977) pointed out there is a serious problem with the empirical testability
of the S h a r p e - L i n t n e r model. Due to the fact that all types of bonds, real estate,
etc. should be contained in the market portfolio, this portfolio cannot be
reasonably approximated.
In order to analyse the consequences of this fact, the following points of Roll's
paper are particularly important:
P.l
Within the framework of the Black model the CAPM-relationship (23)
holds for any mean-variance efficient portfolio x q. 7 On the other hand, if
there is a linear relationship
(24)
E ( R h ) = al3h + b,
with
C°v(Rh'R[xq] )
Var(R[x q]) '
P.2
h = 1. . . . . N
the portfolio x q is mean-variance efficient (Ioc, cit., corollary 6,
pp. 165-166).
There is an interesting connection between the Black and the
6. In fact BLACK et al. 0972) and BLUME and FRIEND (1973) reject the Sharpe-Linmer model.
7. x ° is the corresponding zero-beta portfolio.
i 37
MODERN PORTFOLIO THEORY
S h a r p e - L i n t n e r models. T o see this, let h = 0 be the riskless and
h = ! ..... N the risky assets. Then the efficient frontier is a straight line L
(see Section 2.2). If attention is restricted to portfolios consisting only of
risky assets a constrained efficient frontier C results which is the upper
branch o f a hyperbola (see Section 2.1). Obviously L must be tangential to
C (Figure 6).
From now on, it is assumed that the riskless asset h = 0 is in zero net supply, i.e.
XoM = 0. s Then, in equilibrium the market portfolio x M must correspond to the
tangency point T (Figure 6). Hence, in equilibrium x M satisfies the CAPMrelationship for the Black model
(23)
E(Rh)-E[R(x°)]
=
CoV(Rh'R[xM]) {E(R[xM])-E(R[x°])]
V a r ( R [ x M])
However, due to explanation 2 (Section 3.2)
E(R [x °]
(25)
) = R0
holds and one obtains the C A P M - r e l a t i o n s h i p for the S h a r p e - L i n t n e r model
(22)
P.3
{ E ( R [ x M ] ) - R o } , h= 1
N.
E(Rh)-Ro- C°v(RI"R[xM])
V a r ( R [ x M])
.....
Suppose that the S h a r p e - L i n t n e r model is true, i.e. the market portfolio
x M satisfies the C A P M - r e l a t i o n s h i p (22). If the market proxy x p is different
f r o m x M two possibilities arise.
1) x p = x " ' , where x"" does not belong to the constrained efficient frontier
C (Fig. 7). Due to P. 1 the relationship (24) is no longer satisfied and the
application of statistical methods leads in general to a rejection of the
S h a r p e - L i n t n e r model.
2) x p = x " , where x m belongs to the constrained efficient frontier C (Figure
( 0 , R o)
.b
x_0
%
o(r)
FIGURE 6.
8. This condition can be satisfied by choosing an appropriate normalization.
138
MOLLER
•
( 0 , E (R ~ o , n ]))
:~
~x_ M
-
• X ms
- _ _ " ~ ? x Om
( 0 , R o)
g(r)
FIGURE 7.
7). Then, according to P . I ,
E(Rh) - E(R [x °''' ] ) = Cov(Rh, R [x'"] ) [E(R [x'"]) - E(R Ix °''' ] )}
Var(R [ x " ] )
h = l ..... N
holds. 9 However, x'" ~ x M (Figure 7) leads to
E ( R [ x °'''] ) # Ro
and again the application of statistical methods leads in general to a
rejection of the S h a r p e - L i n t n e r model.
Roll's critique gave rise to consternation among researchers. In a recent paper
Shanken (1987) presented a method to test the joint hypothesis that the
Sharpe-Lintner
model is true and that the correlation coefficient
p(R [xM], R [ x ' ] ) between the returns of the market portfolio x M and the proxy
x"' exceeds a given limit tS. For the equal-weighted CRSP index (an American
stock index developed by the Center of Research in Security Prices at the University of Chicago) and a limit for the correlation coefficient of,6 ~ 0.7 he had to
reject the joint hypothesis at the 95% confidence level. With multivariate proxies
consisting of a stock and a bond index Shanken reached similar conclusions.
5. A P P L I C A T I O N OF VON N E U M A N N - M O R G E N S T E R N UTILITY T H E O R Y
Mean-variance analysis is rather unsatisfactory on theoretical grounds. Fortunately, NM utility theory can be applied to the portfolio problem.
Throughout this section it is assumed that there is a riskless asset h = 0 with
a deterministic return Ro and N risky assets h = 1.... , N with stochastic returns
Rh. The overall return R ( i ) of a portfolio ii = (x0, xl ..... XN) is evaluated by a
NM utility function u : R ~ R.
9. x °'''
is the zero-beta portfolio which corresponds to x'".
MODERN PORTFOLIO THEORY
139
5.1. Efficient Portfolios
DEFINITION I. A p o r t f o l i o ~* is called o p t i m a l relative to the N M utility
u : R ~ R if it is a s o l u t i o n o f the o p t i m i z a t i o n p r o b l e m
max
E l u [ R ( i ) ] },10,tl
i E R,',+~
subject to
N
(0)
~
x,, = 1.
h=O
REMARK. T h e o p t i m i z a t i o n p r o b l e m (0) is equivalent to
(0')
max
(x,
.....
XN) ~ R '¢
E u Ro+ ~
h =
xh(Rh-Ro)
.
I
ASSUMPTIONS
B.I
B.2
T h e N M utility u : R ~ R is increasing, strictly c o n c a v e a n d c o n t i n u o u s l y
differentiable.
The r a n d o m variables Rh, h = 1. . . . . N are b o u n d e d .
PROPOSITION. U n d e r B.I a n d B.2 the p o r t f o l i o i * is o p t i m a l relative to u if
a n d only if
(26)
E l u ' [R(~*)] ( R h - Ro)} = 0, h = 1. . . . . N.
PROOF.
1) U n d e r B. 1 a n d B.2 the o b j e c t i v e function o f ( 0 ' ) is well defined a n d c o n c a v e
in (xl . . . . . XN).
2) Due to B.I a n d B.2 L e b e s g u e ' s t h e o r e m allows to reverse the o r d e r o f diff e r e n t i a t i o n a n d integration. ~z
DEFINITION 2. A p o r t f o l i o i * is called efficient if there exists a N M utility
v : R ~ R satisfying B.I such t h a t i * is o p t i m a l relative to v.
5.2. Mutual Fund Theorems
In Section 2 it was shown that the set o f m e a n - v a r i a n c e efficient p o r t f o l i o s can
always be s p a n n e d by two reference p o r t f o l i o s . In the f r a m e w o r k o f N M utility
t h e o r y this is no longer the case. 13 H o w e v e r , if there are restrictions on the class
I0. R ( i ) : = ~-~h
,N= 0 xhRh.
I 1. If Wo is the initial wealth, then o f course one has to evaluate the fmal wealth WoR(i). However,
the choice o f an appropriate scaling allows always for the normalization Wo = 1.
12. This problem is often overlooked in the literature.
13. Since NM utility theory takes third and higher moments of the overall return on a portfolio into
account one can show that the set of efficient portfolios becomes larger than under mean-variance
analysis.
140
MOLLER
of NM utilities and/or the class of returns distributions, the set of efficient
portfolios can still be spanned by a few references portfolios.
RESTRICTIONS ON RETURNS DISTRIBUTIONS. The following result is often
presented as a theoretical basis for mean-variance analysis in the case of
multivariate normal distribution of returns.
THEOREM. Suppose
that R = (Rt, ..., RN) has a multivariate normal
distribution
with a density f ( r ) = (27r)-N/2(det
V)-t/2
expl - l / 2 ( r - # ) ' V - l ( r - / z ) J , where V is a regular N x N covariance matrix.
Then, the set o f efficient portfolios is spanned by two reference portfolios
io), ~(z~, where
probability
~(t) = (1,0 ..... O) ~ R N÷ ~ is the riskless investment
~t2) is a fixed mean-variance efficient portfolio.
PROOF. See
pp. 272-273).
MERTON
(1982,
theorem
4.11,
p. 631)
or
ROSS (1978,
RESTRICTIONS ON THE CLASS OF NM UTILITIES. Let U be a class of NM utilities
u : R ~ R. The set of portfolios which are optimal relative to some u~ U is
denoted by xI,e(U). HAKANSSON (1969) and CASS and STIGLITZ (1970) were the
first to look for classes U such that 9 e ( U ) is spanned by two reference portfolios
i (~) and ~(2), where i ° ) = ( 1 , 0 ..... 0). Under regularity assumptions ~4 with
respect to returns distribution they show that the following classes U(c),
c ~ ( - oo, 0) t..J (0, ~ ] have this property:
(27a)
a) U ( c ) = I ulu(w)=(w-ff/k)l-c'w~>l-c
=
=
l-c
(27c)
c) U ( c ) = l u l u ( w ) = l n ( w -
(27d)
d) U ( c ) = l u l u ( w ) =
ff"kI v c ~ ( 0 , 1 ) U ( l , o o )
, w ~< ff'k
ff'k), W ~> ff/k},
-exp(-tSkw)},
v c ~ ( - oo,0)
c=l
c=oo
where Ig,rk can be interpreted as the subsistence level of wealth in a) and c) and
as the satiation level of wealth in b) (see INGERSOLL, 1987, pp. 146--147).
COMMENTS
1) These classes are also important in risk theory. According to Borch's
theorem there is linear risk sharing within each class.
2) The union of these classes represents the HARA-class (Hyperbolic Absolute
14. A full specification of these conditions is beyond the scope of this paper.
MODERN PORTFOLIO THEORY
141
Risk A v e r s i o n ) , which is c h a r a c t e r i z e d by
u"(w) = __1
> O.
u'(w)
a+bw
3) F o r c = - 1 the class o f q u a d r a t i c N M utilities results a n d m e a n - v a r i a n c e
analysis is o b t a i n e d as a special case.
5.3. A risk Measure f o r Individual Securities
In m e a n - v a r i a n c e analysis the beta-coefficients are used to m e a s u r e the risk o f an
individual security relative to the m a r k e t p o r t f o l i o . T h e N M utility f r a m e w o r k
allows for the following g e n e r a l i z a t i o n :
DEFINITION 3. Let i K be an o p t i m a l p o r t f o l i o relative to a N M utility
u : R --* R satisfying B . I . Then
(28)
bl~:=
Cov(u'[R(~r)],Rh)
Cov(u' [R(iX)],R(ig))
h=l
'
... N
'
is called the m e a s u r e o f risk o f asset h relative to p o r t f o l i o ~ K
REMARK. bhr coincides with the beta-coefficient o f asset h if ~g is the m a r k e t
p o r t f o l i o and if
u is q u a d r a t i c
or
u is twice differentiable a n d R = (R~ . . . . . R N ) has a m u l t i v a r i a t e n o r m a l
d i s t r i b u t i o n (see INGERSOLL, 1987, pp. 13--14).
PROPERTIES OF bhr.
P.l
If i k is an o p t i m a l p o r t f o l i o relative to a NM utility u satisfying B.l and
if s o m e regularity c o n d i t i o n s are satisfied then
E(Rh)-Ro=bhK{E[R(xr)]
(29)
-Rol,
h= l ..... N
holds.
P.2 U n d e r regularity c o n d i t i o n s , for all efficient p o r t f o l i o s ~K, ,~L
bhK = b~,, ¢~ b~- = b~,
b,f > b~, c~ bhL > b~, '
h , h ' E [ l ..... N}
holds.
F o r a p r o o f o f P . I and P.2 see INGERSOLL (1987, p. 134).
INTERPRETATION OF P.I AND P.2.
ad P . I If there are m investors with h o m o g e n e o u s
e x p e c t a t i o n s , whose
142
MOL'I.ER
preferences can be represented by NM utilities, u i, i = ! ..... rn, then in
general the market portfolio is not efficient (see INGERSOLL, 1987,
pp. 140--143). However, in the special case, where
u i ~ U ( c ) , i = 1 . . . . . rn, for some c ~ ( - oo, 0) A (0, oo], [see (27)],
there exists UME U ( c ) such that the market portfolio i M is optimal
relative to UM. Then, P. 1 leads to the following generalized version of
the CAPM-relationship
(30)
Cov(u,~.1[R(~M)], Rh)
I E [ R ( x M ) ] -- Rol,
E ( R h ) - Ro = C o v ( u ~ [ R ( i M ) ] , R ( ~ M ) )
h = l ..... N.
ad P.2 bff leads to a complete pre-ordering on the set of securities [ 1..... N I .
According to P.2, this pre-ordering does not depend on the underlying
efficient portfolio ~A'.
6. PORTFOLIO INSURANCE
Throughout this section, it is assumed that there is a riskless asset with a deterministic return Ro and a reference portfolio with the stochastic return RM.
Usually, the protection of the reference portfolio by a put option is called
portfolio insurance. Obviously, the hedged position consisting of the reference
portfolio and a put option has a return R , , which is a convex function of RM
(Figure 8).
On the other hand, LELAND (1980) shows that in the limit any twice continuously differentiable convex payoff function Y ( R M ) can be generated by
combining the reference portfolio, the riskless asset and put options. Therefore,
any investment policy with a strictly convex payoff function is called a general
portfolio insurance policy (Figure 9). LELAND (1980) shows how individual
investors should deviate from the market portfolio.
In the following we explain general portfolio insurance in a slightly different
framework.
RH
RM
FIGURE 8.
MODERN PORTFOLIO THEORY
143
Y (R M )
RM
FIGURE9.
ASSUMPTIONS
C.I
There are
rnt investors with N M utilities ui ~ U(c), i = 1. . . . . mt
investors with NM utilities uiE U(c'),
rn-mr
i = ml + 1. . . . . rn, where c, c' E (0, oo).
C.2
The initial wealth in the e c o n o m y consists o f Wo units o f the riskless asset
a n d WM units o f the reference portfolio.
According
to
C.2
total
final
wealth
is given
by
the
random
variable
WoRo + WMRM. t5 F r o m Borch's theorem o n e sees immediately that all Pareto
efficient allocations (Yt . . . . . Y.,) are of the f o r m
(31)
Yi =
a~ +
b ~ X l,
Yi=ai2+bi2X 2,
i = 1..... ml,
i=ml+l
. . . . . m,
with
g l + a 2 = WoRo,
(32)
b l X I + b 2 X 2 = WMRM,
where
III i
i= I
b l = ~ b),
i=l
i=mt+ I
02=
b~.
i=ml+
I
F u r t h e r m o r e , according to Section 5.2 the following aggregation is possible:
Investors i = 1. . . . . m~, (rnt + 1. . . . . m ) can be represented by a single investor
15. Ro and RM are considered here as exogenously given.
144
MULLER
with a N M utility ~ ( ~ ) . ~ a n d ,~ are c h a r a c t e r i z e d by
a'(w) = (w - ff,)-c
h' ( w ) = ( w - f i : ) - c '
A p p l y i n g B o r c h ' s t h e o r e m o n c e m o r e leads to
(a 1 + btX
(33)
I _ Ig/)-~"
(a 2 + b2X 2-
~V)_C,= k.
a.e.
w i t h h, > 0,
or by c o n s i d e r i n g (32)
(A l +blxl)
(34)
(A 2 + W M R M - -
-c
btXl)
-''' = k
a.e.
where
zl=al_l~
A2= a2-
"
~V.
Finally, one obtains
(35)
W M R M = b ~X 1 - A z + k l / , . ' ( A t + b l X l ) C / C '.
(35)
F r o m (35) o n e c o n c l u d e s :
(1) c = c ' :
X I is l i n e a r in R M
(2) c > c ' : X ~ is c o n c a v e in R M
(3) c < c ' : X 1 is c o n v e x in R M.
INTERPRETATION
1) F o r c = c ' l i n e a r risk s h a r i n g is P a r e t o - e f f i c i e n t a n d n o o p t i o n s are needed
in the e c o n o m y .
2) F o r c < c '
i n v e s t o r s i = l . . . . . m~, c h o o s e a g e n e r a l p o r t f o l i o i n s u r a n c e
policy. It is i m p o r t a n t to n o t e that c is n o t directly related to the a b s o l u t e
or relative risk a v e r s i o n . 16
ACKNOWLEDGEMENTS
T h e a u t h o r is i n d e b t e d to a n a n o n y m o u s referee for very h e l p f u l s u g g e s t i o n s .
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HEINZ MULLER
lnstitut f i i r Empirische Wirtschaftsforschung, Universitiit Ziirich, Kleinstrasse 15,
CH - 8008 Ziirich, Switzerland.