university of ziirich

MODERN PORTFOLIO 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). Dif- ficulties 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).

ASTIN BULLETIN Vol. 18, No. 2

128 MOLLER

2. T H E M A R K O W I T Z A P P R O A C H

Whereas in ac tuar ia l science the law of large numbers plays a central role this is not the case in po r t fo l io theory . Due to the cor re la t ion between the re turns on financial assets, d ivers i f icat ion al lows in general only for a reduc t ion but not for an e l imina t ion o f the risk. MARKOWITZ (1952) was the first who took the covar iances between the rates o f re turn into account .

2.1. The M o d e l

There are N assets h = 1, . . . , N (e.g. c o m m o n stocks) . An investment o f one unit o f money (e.g. 1 ($)) in asset h leads to a s tochast ic re turn Rh. The first and second momen t s o f Rt . . . . , RN are assumed to exist.

The vector o f expected values and the covar iance matr ix are deno ted 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 po r t fo l io is given by a vector x e R N, with N Eh= ~ Xh = 1. Xh denotes the weight o f asset h = I . . . . . N. z Hence the overal l re turn on a po r t fo l io x is given by the r a n d o m var iable

N

R (x) := ~ xI, Rh h = l

and one ob ta ins immedia t e ly

E [ R ( x ) ] = # ' x , V a r [ R ( x ) ] = x ' F x .

DEFINITION. A por t fo l io x* is called (mean-var iance) efficient if there exists no po r t fo l io x with

E [ R ( x ) ] /> E [ R ( x * ) ] , V a r [ R ( x ) ] < V a r [ R ( x * ) ] .

The M a r k o w i t z a p p r o a c h is a me thod to calcula te mean-va r i ance efficient por t - folios. Hence, the M a r k o w i t z a p p r o a c h is based on mean-var iance analysis , where the var iance o f the overal l rate o f re turn is taken as a risk measure and the expected value measures prof i tabi l i ty . In con t ras t to expected uti l i ty maximiza t ion , mean-va r i ance analysis takes into account only the first two momen t s and there is no clear theoret ica l founda t ion . The special a s sumpt ions under which mean-var iance analysis is consis tent with expected uti l i ty maximiza t ion are discussed in Section 5.

The M a r k o w i t z a p p r o a c h can be formal ized as fol lows:

min l ]2x ' Vx x E R ,~'

2. x~, < 0 corresponds to a short position with respect to asset h.

MODERN PORTFOLIO THEORY

subject to

e~x = 1 3 (M)

p. 'x/> r,

129

where r is the min imum level for the expected value o f the overall rate o f return. Here, the min imum level r is assumed to be exogenously given. In applications, individual investors choose r in accordance with their risk aversion.

( M ) is a quadrat ic convex opt imizat ion 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 condit ions 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

or if (2) is taken into account

(5)

with

X ~" }kl V-le + ~2 V- IV.

X = VX I + (1 - v ) x 2

V _ t x 1= v-l____e_ x 2 = {eT.~=.i-p. i f e v - ¿ ~ # o

e ' V-~e ' ~ x t + V-~p. else

By varying the min imum level o f return r, all mean-variance efficient portfol ios 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 ) x l + [1 - - v(r)] x 2.

Thus, any efficient portfol io is a combina t ion o f two fixed reference portfol ios x I and x 2. INGERSOLL (1987, p. 86) shows that x I is the global min imum variance portfol io with an expected value o f return

= g,' V- le (7) rmin e ' V-I-'"---~

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> rmin , (3) and (6) lead to

(8) oZ(r):= Var lR[x( r ) ] } = vZx~'Vx ~ + 2 v ( l - v)xl'VxZ + (1 - v)2x2"Vx z

(9) 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. Avai lab i l i t y 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

i = ( X o , x ) ~ R N+l,

with #o = R0, #h = E ( R h ) , h = 1 . . . . . N

Vm = Cov(Ri,, RI), h, I = 1 . . . . . N

N

with ~ Xh = 1 h=0

N

R ( i ) : = ~ xhR, . h=0

Mean-variance efficient portfolios now result f rom the optimization problem

rain l /2x 'Vx

subject to ~ ' i = 1 4

( M ' ) ~ ' i ~> r.

rmin

e•fficienJtlfrontier iA t t

x inefficient branch \

FIGURE I .

A = ( ( . ~ l ' v x _ l ) 1 /2 ,~ 'x_ 1]

B = ( (x_2"Vx_2) l t2 ,~ 'x_ 2)

o ( r )

4. ~' = ( I , I . . . . . I ) ~ R ~+t .

MODERN PORTFOLIO THEORY 131

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, Xl 2 . . . . . 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 ,

1)

Remarks

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 im- posed, 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.

Ro

B ' ~ e r

%

% %

FIGURE 2.

A' = ( 0 , R o )

B ' = Q,.x.2" Vx2) I/2, ~" ~ "2 )

o ( r )

132 MULLER

3. THE CAPITAL ASSET PRICING MODEL (CAPM)

3.1. The Sharpe-Lintner Model

In the Sha rpe -L i n t ne 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 moments o f returns on

assets h = 0 . . . . . N, i.e.

~i=~, vi= V i= l ..... m,

3) Investor i seeks a mean-variance efficient portfol io ~i with E[R(£i)] = ri, where r~ > Ro i = 1, ..., m.

According to formula (6 ' ) (Section 2.2) the portfol io chosen by investor i is o f the form

(6 ' ) xi = u ( r / ) x I + [1 - v(r/)]x 2 i= 1 ..... m.

Total demand results in a por t fol io

1 ~ Wi~i, with W:= ~ Wi. (10) i M : = 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 iER x÷~

subject to

~ ' i = 1

(M") ~ ' i 1> rM

r M

ri ~

R o

FIGURE 3.

G ( r )


with

rM := " ~ Wiri . i=l

Hence, iff t n M = (Xao, xa ) satisfies the K u h n - T u c k e r conditions

(11) - Xi - X2Ro = 0

(12) V:,~ t - X l e - X2/J. = 0

(13) ~ ' i a m = 1

(14) ~ ' i ~ = rM

(15) X2 >/0.

Formulae (11) and (12) lead to

(16) Vx~ = h2 (p. - Roe).

From (11), (12) and #o = Ro one obtains

(17) x;7' v x Y = × ,~ ' iY + x2~'~,~'

or according to (11), (13) and (14)

(18) xfl ' Vxff = XZ(rM -- Ro).

Formulae (16) and (18) imply

Vxff j, - Roe (19) xa M' Vxa ~---~ = rM -- Ro"

On the other hand, total supply is given by the market por t fol io ~^ ! where all assets are held in propor t ion to their market values. In equilibrium total demand must be equal to total supply, i.e.

(20) iff t = ira.

Combining (19) and (20) leads to the equilibrium condit ion

VxM _tl-Roe (21) x M ' V x M rM-- Ro

or by taking into account

#1, = E ( R h ) , h = 1 . . . . . N N

r M = E ( R M ) , where R M = ~] x i~Rh h=O

xM'VxM = V a r ( R g )

X M' V = (Cov(Ri , R M ) . . . . . CoV(RN, RM ))

one obtains the CAPM-re la t ionship

(22) E ( R h ) - Ro - Cov(Rt , , R M ) [ E ( R M ) - Ro], h = ! . . . . . N Var(RM)

134 MOLLER

COMMENTS

1) /3h := Cov(Rh, RM)[Var(RM) is called the beta-coefficient of asset h. E(Rh) -Ro is the risk premium on asset h E(RM)- 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 undiversi- fiable. The error term Ch satisfies Cov(ch ,RM)=0 and corresponds to the unsystematic risk. In particular, the following decomposition is possible

Var(Rh) =/3h 2 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 beta- coefficients may occur as well.3

5) According to the CAPM relationship each asset h can be represented by a point (/3h, E(Rh)) which lies on the theoretical market line

E(Rh) = Ro + [E(RM)- 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).

E (R h )

Ro

theoretical market line

~,,.~-'~"~ 7 - ' ' empir ical market line

FIGURE 4.

5. In such a case Rh and RM are negatively correlated and E(Rh) < Ro must hold.


3.2. The Black 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 CAPM-re la t ionsh ip can be derived.

It is assumed that the f ramework of Section 2.1 holds and that there are m investors i = 1 . . . . . m with

initial wealth Wi,

first and second moments o f returns given by i = t z , V i = V

and seeking for mean-variance efficient portfol ios x~ with E[R(x i ) ] = r~, where ri i> rmin, i = 1 . . . . . m.

Under these condit ions it can easily be shown that in equilibrium the market por t fol io x M must be effÉcient and f rom the corresponding optimali ty condit ions one can derive the CAPM-re la t ionship

(23) E ( R h ) - E [ R ( x ° ) ] - Cov(Rh, RM) i E ( R M ) _ E[R(xO]] Var(RM)

h = l . . . . . N.

x ° is the so-called zero-beta portfol io 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.

rmin

~ ' x_ 0

M M

\ x_O .-~ . - - - - -

FIGURE 5.

x 2

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 , t t '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 Sharpe-Lin tner approach. In the next section we shall see that it is needed in order to discuss the empirical testability of the Sharpe-Lin tner model.

4. EMPIRICAL TESTABILITY OF THE SHARPE-LINTNER MODEL

Before discussing testability we must recall that the Sharpe-Lin tner model was developed under restrictive assumptions, namely:

M.l Evaluation of portfolios by mean-variance analysis, M.2 Uniform planning horizon for all investors, M.3 Homogeneous expectations, M.4 Existence of a riskless asset, M.5 Exclusion of transaction costs, M.6 No restrictions on short sales.

BLACK et al. (1972), BLUME and FRIEND (1973), FAMA and MACBETH (1973) were among the first to test the Sharpe-Lin tner model. The evidence provided by their studies in favour of the CAPM-relat ionship is rather weak. 6 However, as ROLL (1977) pointed out there is a serious problem with the empirical testability of the Sharpe-Lin tner 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-relat ionship (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

the portfolio x q is pp. 165-166).

C ° v ( R h ' R [ x q ] ) h = 1 . . . . . N Var(R[x q]) '

mean-variance efficient (Ioc, cit., corollary 6,

P.2 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.

MODERN PORTFOLIO THEORY i 37

S h a r p e - L i n t n e r models . To 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 a t tent ion is restricted to por t fol ios consisting only of risky assets a constra ined efficient front ier C results which is the upper branch o f a hyperbo la (see Section 2.1). Obvious ly L must be tangential to C (Figure 6).

F rom now on, it is assumed that the riskless asset h = 0 is in zero net supply, i.e. Xo M = 0. s Then, in equil ibrium the marke t por t fo l io x M must cor respond to the tangency point T (Figure 6). Hence, in equi l ibr ium x M satisfies the CAPM- relat ionship for the Black model

(23) E(Rh)-E[R(x°)] CoV(Rh'R[xM]) {E(R[xM])-E(R[x°])] = Var (R[x M])

However , due to explanat ion 2 (Section 3.2)

(25) E(R [x °] ) = R0

holds and one obtains the CAPM-re la t ionsh ip for the S h a r p e - L i n t n e r model

C°v(RI"R[xM]) {E(R[xM]) -Ro} , h= 1 N. (22) E(Rh)-Ro- V a r ( R [ x M]) . . . . .

P.3 Suppose that the S h a r p e - L i n t n e r model is true, i.e. the marke t por t fo l io x M satisfies the CAPM-re la t ionsh ip (22). If the marke t proxy x p is different f rom x M two possibilities arise.

1) x p = x " ' , where x"" does not belong to the constra ined efficient frontier C (Fig. 7). Due to P. 1 the relat ionship (24) is no longer satisfied and the appl icat ion 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 constra ined efficient front ier C (Figure

( 0 , R o) .b x_ 0

%

FIGURE 6.

8. This condition can be satisfied by choosing an appropriate normalization.

o(r)

138 MOLLER

( 0 , E (R ~ o , n ]))

( 0 , R o)

• ~ x _ M

: ~ - - _ _ " ~ ? x Om

• X ms

FIGURE 7.

g(r)

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 Sharpe-Lin tner 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-Lin tner 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 Univer- sity 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 THEORY

Mean-variance analysis is rather unsatisfactory on theoretical grounds. For- tunately, 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'".


5.1. Efficient Portfolios

DEFINITION I. A por t fo l io ~* is called op t ima l relat ive to the NM u : R ~ R if it is a so lu t ion o f the op t imiza t ion p rob lem

max E l u [ R ( i ) ] },10,tl i E R,',+~

subject to N

(0) ~ x,, = 1. h = O

util i ty

REMARK. The op t imiza t ion p rob lem (0) is equivalent to

( 0 ' ) max E u Ro+ ~ x h ( R h - R o ) . (x, . . . . . XN) ~ R '¢ h = I

ASSUMPTIONS

B.I The NM util i ty u : R ~ R is increasing, s tr ict ly concave and con t inuous ly di f ferent iable .

B.2 The r a n d o m variables Rh, h = 1 . . . . . N are bounded .

Under B.I and B.2 the po r t fo l io i* is op t imal relat ive to u if

E l u ' [R(~*)] ( R h - Ro)} = 0, h = 1 . . . . . N.

PROPOSITION. and only if

(26)

PROOF.

1) Under B. 1 and B.2 the objec t ive funct ion o f ( 0 ' ) is well defined and concave in (xl . . . . . XN).

2) Due to B.I and B.2 Lebesgue ' s theorem al lows to reverse the o rde r o f dif- fe rent ia t ion and in tegra t ion . ~z

DEFINITION 2. A por t fo l io i* is called efficient if there exists a NM util i ty v : R ~ R sat isfying B.I such tha t i * is op t ima l relat ive to v.

5.2. Mutual Fund Theorems

In Sect ion 2 it was shown that the set o f mean-va r i ance efficient po r t fo l ios can a lways be spanned by two reference por t fo l ios . In the f r amework o f N M util i ty theory this is no longer the case. 13 However , if there are res t r ic t ions on the class

~-~h = 0 I0. R ( i ) : = ,N xhRh.

I 1. I f Wo is the init ial wealth, then o f course one has to evaluate the fmal wealth WoR(i ) . However, the choice o f an appropr iate scaling allows always for the normal izat ion 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 probability 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

~(t) = (1,0 . . . . . O) ~ R N÷ ~ is the riskless investment

~t2) is a f ixed mean-variance efficient portfolio.

PROOF. See MERTON (1982, theorem 4.11, p. 631) or ROSS (1978, pp. 272-273).

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 9e (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 ff"k I v c ~ ( 0 , 1 ) U ( l , o o )

= = , w ~< ff'k vc~ ( - oo,0) l - c

c) U ( c ) = l u l u ( w ) = l n ( w - ff'k), W ~> ff/k}, d) U ( c ) = l u l u ( w ) = - exp ( - t Skw)} , c = o o

(27c)

(27d)

c = l

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.


Risk Avers ion) , which is charac te r ized by

u"(w) = _ _ 1 > O. u ' ( w ) a + b w

3) For c = - 1 the class o f quad ra t i c NM utili t ies results and mean-var iance analysis is ob ta ined as a special case.

5.3. A risk Measure f o r Individual Securities

In mean-var iance analysis the beta-coefficients are used to measure the risk o f an individual securi ty relat ive to the marke t por t fo l io . The NM util i ty f r amework al lows for the fol lowing genera l iza t ion:

DEFINITION 3. Let i K be an op t ima l po r t fo l io relat ive to a NM util i ty u : R --* R sat isfying B. I . Then

(28) bl~:= C o v ( u ' [ R ( ~ r ) ] , R h ) h = l ... N C o v ( u ' [ R ( i X ) ] , R ( i g ) ) ' '

is called the measure o f risk o f asset h relat ive to por t fo l io ~K

REMARK. bh r coincides with the beta-coefficient o f asset h if ~g is the marke t por t fo l io and if

u is quadra t i c

or

u is twice di f ferent iable and R = (R~ . . . . . RN) has a mul t iva r ia te normal d i s t r ibu t ion (see INGERSOLL, 1987, pp. 13--14).

PROPERTIES OF bh r .

P . l If i k is an op t imal por t fo l io relat ive to a NM util i ty u sat isfying B.l and if some regular i ty condi t ions are satisfied then

(29) E ( R h ) - R o = b h K { E [ R ( x r ) ] - R o l , h = l . . . . . N

holds. P.2 Under regular i ty condi t ions , for all efficient por t fo l ios ~K, ,~L

bh K = b~,, ¢~ b~- = b~,

b,f > b~, c~ bhL > b~, ' h , h ' E [ l . . . . . N}

holds.

For a p r o o f of 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 homogeneous expec ta t ions , whose

142 MOL'I.ER

(30)

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-relat ionship

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 . . . . . NI . 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 con- tinuously 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

FIGURE 8.

R M

Y (R M )

MODERN PORTFOLIO THEORY

FIGURE 9.

RM

143

ASSUMPTIONS

C.I There are

rnt investors with NM utilities ui ~ U(c), i = 1 . . . . . mt

r n - m r investors with NM utilities uiE U(c'),

i = ml + 1 . . . . . rn, where c, c' E (0, oo).

C.2 The initial wealth in the economy consists of Wo units of the riskless asset and WM units of the reference portfolio.

According to C.2 total final wealth is given by the r andom variable WoRo + WMRM. t5 From Borch's theorem one sees immediately that all Pareto efficient al locations (Yt . . . . . Y.,) are of the form

Yi = a~ + b~X l, i = 1 . . . . . ml,

Yi=ai2+bi2X 2, i = m l + l . . . . . m, (31)

with

(32)

where

g l + a 2 = WoRo,

b l X I + b2X 2 = WMRM,

I I I i

i= I i=mt+ I

b l = ~ b), 02= b~. i = l i = m l + I

Fur thermore , 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 ut i l i ty ~ (~) . ~ a n d ,~ are charac te r ized by

a ' ( w ) = ( w - f f , ) - c

h' ( w ) = ( w - f i : ) - c '

A p p l y i n g Borch ' s t h e o r e m once m o r e leads to

( a 1 + b t X I _ Ig/)-~"

(33) ( a 2 + b 2 X 2 - ~V)_C,= k.

or by cons ide r i n g (32)

(34)

where

(A 2

( A l + b l x l ) - c

+ W M R M - - b t X l ) - ' ' ' = k

z l = a l _ l ~ "

A 2 = a 2 - ~V.

F ina l ly , one o b t a i n s

(35) W M R M = b ~ X 1 - A z

F r o m (35) one conc ludes :

(1) c = c ' : X I is l inear in R M (2) c > c ' : X ~ is co n cav e in R M

(3) c < c ' : X 1 is convex in R M.

INTERPRETATION

+ k l / , . ' ( A t + b l X l ) C / C '.

a . e .

w i t h h, > 0,

a.e.

(35)

1) For c = c ' l inear risk s h a r i n g is Pare to-ef f ic ient and no o p t i o n s are needed in the e c o n o m y .

2) Fo r c < c ' inves tors i = l . . . . . m~, choose a genera l po r t fo l i o i n su rance pol icy. It is i m p o r t a n t to no t e tha t c is no t di rect ly related to the abso lu t e or relat ive risk avers ion . 16

ACKNOWLEDGEMENTS

The a u t h o r is i ndeb t ed to an a n o n y m o u s referee for very he lpful sugges t ions .

REFERENCES

BLACK, F. (1972). Capital market equilibrium with restricted borrowing, Journal o f Business 45, 444-454.

BLACK, F., JENSEN, M. C., and SCHOLES, M. (1972). The capital asset pricing model: some empirical tests, reprinted in M. C. Jensen, ed., Studies in the Theory o f Capital Market. Praeger, New York, 1972, 79-124.

16. c= -wu"(w)[u'(w) holds only if If', = 0 in (27a) and (27c).


BLUME, M. and FRIEND, I. (1973). A new look at the capital asset pricing model, Journal of Finance, March 1973, 19-34.

BORCH, K. 0960). The safety loading of insurance premiums, Skandinansk Akluarietidskrift 43, 163-184.

CASS, D. and STIGLtTZ, J. E. (1970). The structure of investor preferences and asset returns, and separability in portfolio allocation: a contribution to the pure theory of mutual funds, Journal of Economic Theory 2, 122-160.

COPELAND, T. E. and WESTON, J. F. (1988). Financial Theory and Corporate Policy. Addison- Wesley Publishing Company, third edition.

FAMA, E. F. and MACBETH, J. (1973). Risk, return and equilibrium: empirical test, Journal of Political Economy, May/June 1973, 607-635.

HAKANSSON, N. H. (1969). Risk disposition arid the separation property in portfolio selection, Jour- nal of Financial and Quantitative Analysis, 401-416.

HARRtNGTON, D. R. (1987). Modern Portfolio Theory, the Capital Asset Pricing Model and Arbitrage Pricing Theory: a user's guide. Prentice-Hall, Inc., second edition.

INGERSOt.L, J. E. (1987). Theory of Financial Decision Making, Rowman & Littlefield, New Jersey. LELAND, H. E. (1980). Who should buy portfolio insurance?, Journal of Finance 35, May 1980,

581-594. LINTNER, J. (1965). The valuation of risky assets and the selection of risky investments in stock port-

folios and capital budgets. Review of Economics and Statistics, February 1965, 13-37. MARKOWrrZ, H. (1952). Portfolio selection, Journal of Finance 7, 77-91. MER'rON, R. C. (1982). On the microeconomic theory of investment under uncertainty. In Handbook

of Mathematical Economics, Vol. II, North-Holland, Amsterdam. ROLL, R. (1977). A critique of the asset pricing theory's tests, Journal of Financial Economics,

March 1977, 129-176. ROSS, S. A. (1978). Mutual fund separation in financial theory: the separating distributions, Journal

of Economic Theory 17, 254-286. SHANKEN, J. (1987). Multivariate proxies and asset pricing relations: living with the Roll critique,

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1963, 277-293. SHARP•, W. F. (1964). Capital asset prices: a theory of market equilibrium under conditions of risk,

Journal of Finance, September 1964, 425-442.

HEINZ MULLER

lnstitut f i i r Empirische Wirtschaftsforschung, Universitiit Ziirich, Kleinstrasse 15,

CH-8008 Ziirich, Switzerland.

university of ziirich

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