马克维茨投资组合选择

马克维茨投资组合选择

Portfolio Selection

Harry Markowitz

The Journal of Finance

, Vol. 7, No. 1. (Mar., 1952),

pp. 77-91.

Stable URL:

/sici?sici=0022-1082%28195203

The Journal of Finance

is currently published by

American Finance Association.

Your

use

of

the

JSTOR

archive

indicates

your

acceptance

of JSTOR's Terms and Conditions of Use, available at

/about/

. JSTOR's Terms

and Conditions of Use provides, in part, that unless

you have obtained

prior

permission,

you

may

not

download

an

entire

issue

of a journal or multiple copies of articles, and you

may use content in

the JSTOR archive only for your personal,

non- commercial use.

Please

contact

the

publisher

regarding

any

further

use

of this work. Publisher contact information may be

obtained at

/journals/

.

Each copy of any part of a JSTOR transmission must

contain the same copyright notice that appears on the

screen or printed

page of such transmission.

The JSTOR Archive is a trusted digital repository

providing for long-term preservation and access to

leading academic

journals

and

scholarly

literature

from

around

the

world.

The Archive is supported by libraries, scholarly

societies, publishers,

and foundations. It is an initiative of JSTOR, a

not-for-profit

organization

with

a

mission

to

help

the

scholarly community take

advantage of advances in technology. For more

information regarding JSTOR, please contact

support@.

Sun Oct 21 07:53:25 2007

PORTFOLIO SELECTION*

HARRYMARKOWITZ

The Rand Corporation

THEPROCESS

OF

SELECTING

a

portfolio

may

be

divided

into

two stages.

The

first

stage

starts

with

observation

and

experience

and ends with

beliefs about the future performances of available

securities. The

second stage starts with the relevant beliefs about

future performances

and ends with the choice of portfolio. This paper is

concerned with the

second stage. We first consider the rule that the

investor does (or should)

maximize

discounted

expected,

or

anticipated,

returns.

This rule is rejected

both as a hypothesis to explain, and as a maximum to

guide investment

behavior. We next consider the rule that the investor

does (or

should)

consider

expected

return

a

desirable

thing

and

variance of return

an

undesirable

thing.

This

rule

has

many

sound

points,

both as a

maxim for,

and

hypothesis about,

investment behavior.

We illustrate

geometrically

relations

between

beliefs

and

choice of

portfolio according

to the

One

type

of

rule

concerning

choice

of

portfolio

is

that

the investor

does (or should) maximize the discounted (or

capitalized) value of

future returns.l Since the future is not known with

certainty, it must

be

discount. Variations

of

this

type

of

rule

can

be

suggested.

Following

Hicks,

we could let

returns

include

an

allowance

for

risk.2

Or, we could let

the rate at which we capitalize the returns from

particular securities

vary with risk.

The hypothesis (or maxim) that the investor does (or

should)

maximize discounted return must be rejected. If we

ignore market imperfections

the foregoing rule never implies that there is a

diversified

portfolio which is preferable to all non-diversified

portfolios. Diversification

is

both

observed

and

sensible;

a

rule

of

behavior

which

does

not imply the superiority of diversification must be

rejected both as a

hypothesis and as a maxim.

* This paper is based on work done by the author while

at the Cowles Commission for

Research

in

Economics

and

with

the

financial

assistance

of the Social Science Research

Council. I t will be reprinted as Cowles Commission

Paper, New Series, No. 60.

1. See, for example, J.B. Williams,

The Theory of

Investment Value

(Cambridge, Mass.:

Harvard University Press, 1938), pp. 55-75.

2. J. R. Hicks,

V a l ~ eand Capital

(New York: Oxford

University Press, 1939), p. 126.

Hicks applies the rule to a firm rather than a

portfolio.

78

The

Journal

of Finance

The foregoing rule fails to imply diversification no

matter how the

anticipated returns are formed; whether the same or

different discount

rates

are

used

for

different

securities;

no

matter

how

these discount

rates

are

decided

upon

or

how

they

vary

over

time.3

The

hypothesis

implies that the investor places all his funds in the

security with the

greatest discounted value. If two or more securities

have the same value,

then

any

of

these

or

any

combination

of

these

is

as

good

as any

other.

We can see this analytically: suppose there are N

securities; let

rit

be

the anticipated return (however decided upon) at time

t

per dollar invested

in security

i;

let djt be the rate at which the return

on the

ilk

security at time

t

is discounted back to the present;

let

Xi

be the relative

amount

invested

in

security

i

.

We

exclude

short

sales,

thus Xi

2

0

for all

i .

Then the discounted anticipated return of

the portfolio is

Ri = x

m

di,

T i t

is

the discounted

return

of

the

ith

security,

therefore

t-1

R = ZXiRi where Ri is independent of Xi. Since Xi

2

0

for all

i

and

ZXi

= 1, R is a weighted average of Ri with the Xi

as non-negative

weights.

To

maximize

R,

we

let

Xi

=

1

for

i

with

maximum

Ri.

If several Ra,,

a

= 1, .. . ,K are maximum then any

allocation with

maximizes R. In no case is a diversified portfolio

preferred to all nondiversified

poitfolios.

It

will

be

convenient

at

this

point

to

consider

a

static

model. Instead

of speaking of the time series of returns from the

ith

security

(ril,

ri2)

.

.

.

,rit,

.

.

.)

we

will

speak

of

flow

of returns

the

ith

security.

The

flow

of

returns

from

the

portfolio

as a whole is

3. The results depend on the assumption that the

anticipated returns and discount

rates are independent of the particular investor's

portfolio.

4. If short sales were allowed, an infinite amount of

money would be placed in the

security with highest

r.

Portfolio Selection

79

R

=

ZX,r,.

As

in

the

dynamic

case

if

the

investor

wished

to maximize

return

from

the

portfolio

he

would

place

all his funds in

that security with maximum anticipated returns.

There is a rule which implies both that the investor

should diversify

and that he should maximize expected return. The rule

states that the

investor

does

(or

should)

diversify

his

funds

among

all

those securities

which give maximum expected return. The law of large

numbers will

insure that the actual yield of the portfolio will be

almost the same as

the

expected

yield.5

This

rule

is

a

special

case

of

the

expected returnsvariance

of

returns

rule

(to

be

presented

below).

It

assumes

that

there

is

a

portfolio

which

gives

both

maximum

expected

return

and minimum

variance, and it commends this portfolio to the

investor.

This

presumption,

that

the

law

of

large

numbers

applies

to a portfolio

of securities, cannot be accepted. The returns from

securities are

too

intercorrelated.

Diversification

cannot

eliminate

all variance.

The portfolio with maximum expected return is not

necessarily the

one

with

minimum

variance.

There

is

a

rate

at

which

the

investor can

gain expected return by taking on variance, or reduce

variance by giving

up expected return.

We

saw

that

the

expected

returns

or

anticipated

returns

rule is inadequate.

Let us now consider the expected returns-variance of

returns

(E-V)

rule.

It

will

be

necessary

to

first

present

a

few

elementary

concepts and results of mathematical statistics. We

will then show

some implications of the E-V rule. After this we will

discuss its plausibility.

In our presentation we try to avoid complicated

mathematical statements

and proofs. As a consequence a price is paid in terms

of rigor and

generality.

The

chief

limitations

from

this

source

are

(1) we do not

derive

our

results

analytically

for

the

n-security

case;

instead, we

present them geometrically for the 3 and 4 security

cases; (2) we assume

static

probability

beliefs.

In a

general

presentation

we must recognize

that the probability distribution of yields of the

various securities is a

function

of

time.

The

writer

intends

to

present,

in

the

future, the general,

mathematical treatment which removes these

limitations.

We will need the following elementary concepts and

results of

mathematical statistics:

Let

Y

be

a

random

variable,

i.e.,

a

variable

whose

value

is decided by

chance. Suppose, for simplicity of exposition, that Y

can take on a

finite number of values yl, yz, . . . ,y,~L. et the

probability that Y =

5.

U'illiams,

op. cit.,

pp.

68, 69.

80 The Journal of Finance

yl, be pl; that Y = y2 be

pz

etc. The expected value

(or mean) of Y is

defined to be

The variance of

Y

is defined to be

V is the average squared deviation of Y from its

expected value. V is a

commonly

used

measure

of

dispersion.

Other

measures

of

dispersion,

closely related to V are the standard deviation, u

=

./V

and the coefficient

of variation, a/E.

Suppose we have a number of random variables:

R1, . . . ,R,. If R is

a weighted sum (linear combination) of the Ri

then

R

is

also

a random

variable.

(For

example R1,

may

be the number

which

turns

up

on

one

die;

R2,

that

of

another

die,

and

R the sum of

these numbers. In this case

n

= 2,

a1

= a2 = 1).

It will be important for us to know how the expected

value and

variance of the weighted sum (R) are related to the

probability distribution

of the R1, . . . ,R,. We state these relations below;

we refer

the reader to any standard text for proof.6

The expected value of a weighted sum is the weighted

sum of the

expected

values.

I.e.,

E(R) =

alE(R1) +aZE(R2) +

.

. .

+ a,E(R,)

The variance of a weighted sum is not as simple. To

express it we must

define

i.e.,

the

expected

value

of

[(the

deviation

of

R1

from

its mean) times

(the deviation of R2 from its mean)]. In general we

define the covariance

between Ri and R as

~ i

=

j

E

( [Ri

-E

(Ri) I [Ri

-E (Rj)

I

f

uij may be expressed in terms of the familiar

correlation coefficient

(pij). The covariance between Ri and Rj is equal to

[(their correlation)

times (the standard deviation of Ri) times (the

standard deviation of

Rj)l:

U i j = P i j U i U j

6. E.g.,J. V. Uspensky,

Introduction to mathematical

Probability

(New York: McGraw-

Hill, 1937), chapter 9, pp. 161-81.

Portfolio Selection

The variance of a weighted sum is

If we use the fact that the variance of Ri is uii then

Let Ri be the return on the

iN

security. Let pi be the

expected vaIue

of Ri; uij, be the covariance between Ri and Rj (thus

uii is the variance

of Ri). Let Xi be the percentage of the investor's

assets which are allocated

to the

ith

security. The yield (R) on the portfolio as

a whole is

The

Ri

(and

consequently

R)

are

considered

to

be

random

variables.'

The Xi are not random variables, but are fixed by the

investor. Since

the

Xi

are

percentages

we

have

ZXi

=

1.

In

our

analysis

we will exclude

negative values of the

Xi

(i.e., short sales);

therefore Xi > 0 for

all

i.

The

return

(R)

on

the

portfolio

as

a

whole

is

a

weighted

sum of random

variables

(where

the

investor

can

choose

the

weights).

From our

discussion of such weighted sums we see that the

expected return

E

from the portfolio as a whole is

and the variance is

7.

I.e.,

we

assume

that

the

investor

does

(and

should)

act as if he had probability beliefs

concerning these variables. I n general we ~voulde

xpect that the investor could tell us, for

any two events (A and B), whether he personally

considered A more likely than B, B more

likely

than

A,

or

both

equally

likely.

If

the

investor

were consistent in his opinions on such

matters, he would possess a system of probability

beliefs. We cannot expect the investor

to be consistent in every detail. We can, however,

expect his probability beliefs to be

roughly

consistent

on

important

matters

that

have

been

carefully considered. We should

also expect that he will base his actions upon these

probability beliefs- even though they

be in part subjective.

This

paper

does

not

consider

the

difficult

question

of

how investors do (or should) form

their probability beliefs.

82

The Journal of Finance

For fixed probability beliefs (pi, oij) the investor

has a choice of various

combinations of E and V depending on his choice of

portfolio

XI,

. . . ,

XN.

Suppose that the set of all obtainable

(E, V) combinations

were as in Figure E-V rule states that the

investor would

(or

should)

want

to

select

one

of

those

portfolios

which

give rise to the

(E, V) combinations indicated as efficient in the

figure; i.e., those with

minimum V for given E or more and maximum E for given

V or less.

There are techniques by which we can compute the set

of efficient

portfolios and efficient (E, V) combinations

associated with given pi

attainable

E, V combinations

and

oij.

We

will

not

present

these

techniques

here.

We

will, however,

illustrate geometrically the nature of the efficient

surfaces for cases

in which N (the number of available securities) is

small.

The calculation of efficient surfaces might possibly

be of practical

use. Perhaps there are ways, by combining statistical

techniques and

the

judgment

of

experts,

to

form

reasonable

probability

beliefs (pi,

aij).

We could use these beliefs to compute the

attainable efficient

combinations of (E, V). The investor, being informed

of what (E, V)

combinations were attainable, could state which he

desired. We could

then find the portfolio which gave this desired

combination.

Portfolio Selection

83

Two conditions-at least-must be satisfied before it

would be practical

to

use

efficient

surfaces

in

the

manner

described

above.

First, the

investor

must

desire

to

act

according

to

the

E-V

maxim.

Second, we

must

be

able

to

arrive

at

reasonable

pi

and

uij.

We

will

return to these

matters later.

Let us consider the case of three securities. In the