770
JAN
MOSSIN
perceptions
of these
probability
distributions.3
The
yield
on a whole
portfolio is, of
course,
also a random
variable. The
portfolio
analyses
mentioned earlier
assume
that, in his
choice
among
all
the
possible
portfolios,
the
individual is
satisfied
to
be
guided
by its expected
yield
and its variance
only.
This
assumption
will
also
be
made in
the present
paper.4
It
is
important to make
precise
the
description
of a
portfolio
in
these terms.
It
is
obvious
(although the
point is rarely
made explicit) that the
holdings
of
the
various
assets must
be measured
in some kind of units. The
Markowitz
analysis,
for
exam-
ple, starts
by picturing
the investment
alternatives
open
to the
individual
as
a
point set
in a
mean-variance plane, each
point
representing
a
specific
investment
opportunity. The
question is: to what do this
expected
yield
and
variance
of
yield
refer? For
such a
diagram to make
sense, they
must
necessarily
refer
to
some
unit
common
to
all
assets.
An
example
of such
a
unit would
be
one dollar's
worth
of
investment
in each asset.
Such a
choice of units
would
evidently be
of
little
use
for
our
purposes, since we
shall consider the
prices
of
assets
as
variables to
be deter-
mined in
the market.
Consequently, we must
select
some
arbitrary "physical"
unit
of
measurement and
define expected
yield and variance of
yield
relative to this
unit. If, for
example,
we select one share
as our
unit for
measuring holdings
of
Standard
Oil stock
and
say
that the
expected yield
is
,u
and
the variance
a2, this
means
expected
yield
and
variance
of
yield per
share;
if
instead we
had
chosen a
hundred
shares as our
unit,
the relevant
expected yield
and variance of
yield would
have been
100 4e, and
10,000
a2. respectively.
We shall
find
it
convenient to give
an
interpretation
of
the
concept of "yield"
by
assuming discrete
market
dates
with
intervals
of one time unit. The
yield
to be
considered on
any asset
on a
given market
date
may
then
be
thought
of as the
value
per unit that the asset
will
have at
the next
market
date
(including possible
accrued
dividends,
interest,
or other
emoluments).
The terms
"yield"
and
"future
value"
may then be
used more or less
interchangeably.
We
shall, in general,
admit
stochastic
dependence
among yields
of
different
assets.
But
the
specification of the stochastic
properties poses
the
problem
of
identification of
"different" assets. It
will be
necessary to make
the
convention
that
two
units
of assets
are of the same
kind
only
if
their
yields
will be identical.
3
This
assumption is not crucial
for the analysis,
but simplifies it a
good deal. It also
seems
doubtful whether
the introduction of
subjective
probabilities would really be
useful
for
deriving
propositions
about market behavior.
In any case, it
may be argued, as
Borch [3, p. 439]
does:
"Whether two
rational persons on
the basis of the same
information can
arrive at different
evalua-
tions of the
probability of a specific
event, is a question
of semantics. That
they may act
differently
on the same
information is well
known, but this can
usually be explained
assuming that the
two
persons attach
different utilities to
the event."
4
Acceptance of
the von
Neumann-Morgenstern
axioms (leading to
their theorem
on
measur-
able
utility),
together with this
assumption, implies a
quadratic utility
function for yield (see
[4]).
But such a
specification is not strictly
necessary for the
analysis to follow,
and so, by the
principle
of Occam's razor,
has not been
introduced.