Two fund separation and linear valuation

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Two fund separation and linear valuation

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Title: Two fund separation and linear valuation

1
Chapter 4

Two fund separation and linear valuation

2
Main objective

we give distributional conditions on the rates of
return on assets so that individuals will
optimally choose to hold portfolios on the
portfolio frontier.
Discuss the Capital Asset Pricing Model.(CAPM)
Also discuss the Arbitrage Pricing Theory(APT).

3
4.1 section

Two (mutual) fund separation phenomenon
Define given any feasible portfolio, there
exists a portfolio of two mutual funds such that
individuals prefer at least as much as the
original portfolio. The phenomenon is termed two
(mutual) fund separation.
E.g. if individuals prefer frontier portfolios,
they can simply hold a linear combination of two
frontier portfolios or mutual funds.

This chapter assume that
1. all assets are risky.
2. asset returns have finite second moments.
3. no two asset returns are perfectly
correlated, which imply that the
variance-covariance matrix of asset returns
exists and is positive definite.
Then portfolio frontier exists, and every
frontier portfolio is uniquely determined in that
there is a unique set of portfolio weights
associated with each frontier portfolio.

5
4.2 section

The definition of two fund separation
a vector of asset rate of returns is
said to exhibit two fund separation if there
exist two mutual funds and such that for
any portfolio q there exists a scalar such
that
for all concave u(.)

Now termed that the vector of asset rates of
return, , exhibits two fund separation.
We first claim that the separating mutual funds
and must be frontier portfolios.

To see this, for any portfolio q,
from the definition of two fund separation,
This is equivalent to

By second degree stochastic dominance.
Suppose is not a frontier portfolio. The
there must exist a portfolio that has a
variance strictly smaller than that of any
portfolio formed by and .
This contradicts the hypothesis that and
are separating portfolios.
Hence and must be on the portfolio
frontier.

9
4.3 section

If there exists two fund separation, any two
distinct frontier portfolios can be separating
funds.
In particular, we can pick any frontier
portfolio, and its zero covariance
portfolio, zc(p), to be the separating portfolios.

By Sections 3.16 and 3.17, for any portfolio q,
Where
Note that is
the rate of return of the dominating portfolio.

11
4.4 section

We show that
two fund separation
First proof the necessity of (4.3.3)
proof. As is the return on the dominating
portfolio, a0 is a solution of the following
program

A necessary condition for a0 to be a solution is
Suppose does not hold for
some q. Denote by and
the cumulative distribution function of
by F(.).

As 0 there must exist a real number z
such that
By hypothesis that (4.3.3) does not hold for some
q. we have (4.4.2) can not be zero for all z.
if the equality of (4.4.2) does not hold for all
z, that is,

That is,
Then
Which contradicts the fact that

Now consider a concave utility function that is
piecewise linear
Where gt0

Then
Which contradicts (4.4.1). Thus (4.3.3) is a
necessary condition for two fund separation

17
4.5 section

Now show that the sufficient for two fund
separation. That is, the separating portfolios
are p and its zero covariance portfolio. If q is
a portfolio, then the dominating portfolio is
To see this, let U(.) be any concave utility
function.
Thus (4.3.3) is the sufficient for two fund
separation

18
4.6 section

We will say that a vector of asset returns
exhibits one fund separation if there exists a
(feasible) portfolio such that every risk
averse individual prefers to any other
feasible portfolio.
That is, when one fund separation, there must
exist a mutual fund such that for any
portfolio q, for all concave u(.)

By second degree stochastic dominance, have
Therefore, write
Thus, One fund separation

Note that
1. one fund separation can be viewed as a
degenerate case of two fund separation.
2. When all assets have the same expected rate of
return, the portfolio frontier degenerates to a
single point, the minimum variance portfolio.
3.The portfolio frontier is thus trivially
generated by a single mutual fund.
4. A necessary and sufficient condition for one
fund separation is given by (4.3.3) with

21
4.7 section

We shall give concrete examples of distributions
of rates of return that exhibit two fund and one
fund separation, respectively.
1. If rates of return are multivariate normally
distributed and have nonidentical expectations,
two fund separation obtains.

By
As any linear combination of normal random
variables is itself normal,
It follows that , , and are
normal random variables.
Recall section 3.17
Have

Consequently,
We have thus shown that two fund separation
obtains.
2. When returns have identical expectations,
multivariate normality implies one fund
separation.

Let denote the return on the minimum
variance portfolio. For any feasible portfolio q,
we can always write
We know
Hence we know
Then yields
This establishes that one fund separation obtains.

25
4.8 section

In this and the next section, we will show that
when two fund separation obtains and markets for
risky assets are in equilibrium, the equilibrium
relation among asset returns is linear.
First give a definition of a market portfolio.

Let be individual is initial wealth , and
let be the proportion of the initial wealth
invested in the j-th security by individual i.
The total wealth in the economy is
Where I is the total number of individuals in the
economy

In equilibrium, the total wealth is equal
to the total value of securities.
For markets to clear, we must have
Dividing by , we have
That is, the market portfolio weights are a
convex combination of the portfolio weights for
individuals.

28
4.9 section

We claim that when two fund separation obtains
and in market equilibrium, the market portfolio
is a frontier portfolio.
Next from section 3.16 that if p is a frontier
(but not mvp) and q is any feasible portfolio, we
have

Hence, if the market portfolio is not the minimum
variance portfolio (mvp), we get
Where
Since any risky asset is itself a feasible
portfolio, also have
For all j1,2,, N . Therefore the equilibrium
relation among asset returns is linear.

30
4.10 section

Let us rewrite relation (4.9.3)
When (i.e. the market
portfolio is an efficient portfolio, zc(m) is
inefficient)
Note that the equilibrium expected rate of return
on a risky asset depends upon the covariance of
its rate of return with the rate of return on the
market portfolio.

Figure

security market line
32

When 0

security market line
33
4.11 section

In this section, we shall analyze a special case
of two fund separation and show that the market
portfolio is an efficient portfolio.
Thus a relation like (4.10.1) is valid in
equilibrium.
Let us assume that individuals utility functions
are increasing and strictly concave.
Also assume that the rates of return on assets
are multivariate normally distributed.
We show first that each individual will choose to
hold an efficient portfolio.

The expected utility of individual is choice is
Where denotes a standard normal random
variable.
Defining

Differentiating
Where
That is, individual i prefers a higher expected
rate of return, ceteris paribus.

Also get
Where
So a strictly risk averse individual prefers a
portfolio with a lower standard deviation,
ceteris paribus.

Now we show that individuals will choose to hold
efficient portfolios.
Totally differentiation
(indifference curves)
Seeing figure.

Figure
Thus an individual will choose the point of
tangency. Then his strictly positively sloped
indifference curves imply that he will choose an
efficient portfolio.

We know that all individuals hold efficient
portfolios. Therefore, the market portfolio is an
efficient portfolio.
Furthermore , we have for any feasible portfolio
This is known as the Zero-Beta Capital Asset
Pricing Model which was developed by Black(1972)
and Lintner(1969).

40
4.12 section

The above analyses, we have assumed that there is
no riskless asset.
These next three sections are devoted to the case
where there is a riskless asset.
Suppose first the expected rate of return on the
minimum variance portfolio exceeds the riskless
rate,
i.e.,

We know that the rate of return on any feasible
portfolio q can be expressed as
With
We claim that
two fund
separation.

We prove the necessity part first.
Suppose that two fund separation holds.
Thus two separating portfolios must be frontier
portfolios and can be chosen to be portfolio e
and the riskless asset.
Similar to section 4.4 then show that
For all portfolios q.

As is nonstochastic and we can always pick q
to be any risky asset j, it follows that
It is implied by two fund separation when a
riskless asset exists.

The sufficiency part follows easily from the
arguments of section 4.5.
Thus implies two fund separation
when a riskless asset exists.
Similar also show that when gtA/C
two fund separation
where

45
4.13 section

Assume that two fund separation holds.
When A/C and risky assets are in
strictly positive supply, the tangent portfolios
must be the market portfolios of risky assets in
equilibrium.
Hence, using(3.19.1), we know
For any portfolio q in the market equilibrium.
This is the traditional Capital Asset Pricing
Model (CAPM) independently derived by
Lintner(1965), Mossin(1965), and Sharoe(164).

When A/C, the story is a little different.
We claim that the riskless asset is in strictly
positive supply and the risky assets are in zero
net supply in equilibrium.
Note that the relation(3.19.1) holds independent
of the relationship between and A/C

47
4.14 section

In this section we will show that when investors
have strictly increasing and concave utility
functions, the risk premium of the market
portfolio must be strictly positive when the
risky assets are in strictly positive supply and
the presence of two fund separation.

We first claim that an investor will never choose
to hold a portfolio whose expected rate of return
is strictly lower than the riskless rate when his
utility function u(.) is strictly increasing and
concave.
Let be the random return of a portfolio chosen
by u(.) such that
Note that
This contradicts the hypothesis that the
individual chooses to hold the portfolio with a
random rate of return

when the risky assets are in strictly positive
supply and the presence of two fund separation.
When A/C , an individual puts all his wealth
into the riskless asset and holds a
self-financing portfolio.
this contradicts that the risky assets are in
strictly positive supply
Thus A/C

When gtA/C, then no investor holds a strictly
positive amount of the market portfolio.
This is inconsistent with market clearing.
Thus in equilibrium, it must be the case that
ltA/C and the risk premium of the market
portfolio is strictly positive.
The half line in the standard deviation-expected
rate of return space composed of efficient
frontier portfolios and the riskless asset is
termed the Capital Market Line.

We note that in the above proof when there exists
a riskless asset, an investor will never choose
to hold a portfolio having an expected rate of
return strictly less than the riskless rate we
only used the fact that an investor can always
invest all his money in the riskless asset.
Hence the conclusion holds even when
short-selling portfolio is not allowed.

When investors choose to hold efficient frontier
portfolios and the riskless asset is in strictly
positive supply, we have
For any feasible portfolio and

When the riskless borrowing is allowed at a rate
strictly higher than the riskless lending rate,
we have
For any feasible portfolio and
Where and denote the riskless lending rate
and borrowing rate, respectively.
These relations is termed the constrained
borrowing versions of the CAPM

54
4.15 section

In this and the next section, we will discuss two
simple situations where we can explicitly write
down how the risk premium of the market portfolio
is related to investors optimal portfolio
decisions.
First, suppose that there exists a riskless asset
with a rate of return and that the rates of
return of risky assets are multivariate normally
distributed.

Section 1.18 shows that
Where
Using the definition of a covariance,
Under normally distributed, have

Steins lemma
56

By steins lemma,
Defining the i-th investors global absolute risk
aversion
Dividing by summing across i.

We get
Where
Note that is the harmonic mean of
individuals global absolute risk aversion.
We can interpret to be the
aggregate relative risk aversion of the economy
in equilibrium.

Relation (4.15.4) implies that
Substituting (4.15.5) into (4.15.4) gives the
familiar CAPM.

(4.15.4)
(4.15.5)
59
4.16 section

For example
Now assume that utility functions are quadratic
Easy obtain CAPM

60
4.17 section

In the context of the CAPM, a risky assets beta
with respect to the market portfolio is a
sufficient statistic for its contribution to the
riskiness of an individuals portfolio.
Risky assets whose payoffs are positively
correlated with those of the market portfolio
have positive risk premiums.
In such event, the higher the asset beta, the
higher the risk premium.
The intuition of this relationship can be
understood as follows. Seeing p103.

61
4.18 section

Recall from previous analyses, two fund
separation implies that there exists a linear
relation among expected returns of assets.
In equilibrium, the coefficients of the linear
relation are identified to be related to the
betas of asset returns with respect to the
return on the market portfolio.
As saying that the rate of return on an risky
asset is generated by a one factor model plus a
random noise whose conditional expectation given
the factor is identically zero.
We will give mult-factors model in following
section

62
4.19 section

We consider a sequence of economies with
increasing numbers of assets.
In the n-th economy, there are n risky assets and
a riskless asset.
The rates of return on risky assets are generated
by a K-factor model
Where
And
Where are rates of return on portfolios
Using matrix notation, we can write as

63
4.20 section

We first note that if
then there exists an exact linear relation among
expected rates of return on assets in the n-th
economy, if one cannot create something out of
nothing.
This follows since the returns on risky assets
are completely spanned by the K factors
(portfolios) and the riskless asset.

Formally, consider a portfolio of the K factors
and the riskless asset, , with
Where is the proportion invested in the
riskless asset and is the proportion
invested in the k-th factor.

The rate of return on this portfolio is
Note that the factor components of the rate of
return on replicate those of asset j.

We show that must be equal to
We shall show that if this is not the case,
something can be created out of nothing.
Suppose that

We establish a portfolio by investing one dollar
in and shorting one dollars worth of
security j. this portfolio costs nothing. Its
rate of return, however, is
Which is riskless and strictly positive. Hence
something has been created out of nothing, or
there exists a free lunch.

Reversing the above arguments for the case
we thus have
From
Obtain
That is, there exists an exact linear relation
among expected asset reurns.

69
4.21 section

When the are not zeros, the story is a
little complicated.
Formally, in economy n, a portfolio of the n
risky assets and riskless asset is an arbitrage
portfolio if it costs nothing.
An arbitrage opportunity (in the limit) is a
sequence of arbitrage portfolios whose expected
rates of return are bounded below away from zero,
while their variances converge to zero.
That is, it almost a free lunch.

We wish to show that if there is no arbitrage
opportunity, then a linear relation among
expected asset returns will hold approximately
for most of the assets in a large economy.
We first claim that
For most of the asset in large economies.

To see this, we fix , however small. Let
N(n) be the number of assets in the n-th economy
such that the absolute value of the difference
between the two sides of (4.21.1) is greater than
Without loss of generality, assume that

If we can show that there exists such that
for all n, we are done.
This follows since there will be at most assets
that satisfy(4.21.2) for arbitrarily large n and
can be arbitrarily small.
We shall proceed by contraposition.

Suppose that there does not exist a finite
such that for all n.
Then there must exist a subsequence of
denoted by,
such that when
We construct a sequence of arbitrage portfolios
as follows.
First, construct arbitrage portfolios
that have no factor risk in economy .

The rate of return for the j-th arbitrage
portfolio is
where

Next, form a portfolio of these arbitrage
portfolios with a constant weight, on
each.
The resulting portfolio is still an arbitrage
portfolio with an expected rate of return
And a variance

Since as , the variance
of the sequence of arbitrage portfolios converge
to zero, while their expected rates of return are
bounded below away from zero by .
Thus there exists an arbitrage opportunity, a
contradiction.
We hence conclude that there must exists a finite
such that for all n.

77
4.22 section

Now using the fact proven above that for a given
, however small, there exist at most
risky assets such that
We know
For all but at most risky assets in any
economy.
Thus, for economies with the number of assets
much larger than , a linear relation among
expected asset returns holds approximately for
most of the assets.
This relation is the Arbitrage Pricing Theory
(APT) originated by Ross (1976)

78
4.23 section

For the APT to make predictions in an economy
with a finite number of assets, would like to
bound the deviation from the APT linear relation
for any asset.
Using equilibrium rather than arbitrage
arguments, therefore called equilibrium APT
Suppose that there are N risky assets in the
economy, indexed by j1,2,N, and a riskless
asset.
These risky assets are in strictly positive
supply.
The rate of return on the riskless asset is

The rates of return on risky assets are generated
by a K-factor model
With
We also assume that the random variables
are independent and that are rates of
return on portfolios.

We assume that agents have utility functions that
are increasing, strictly concave, and absolute
risk aversion
From section 4.20 that, when
Or equivalently
Our purpose here is to bound the deviation of
from the right hand side of (4.23.3), in market
equilibrium, when is not identically zero.

81
4.24 section

Consider a portfolio of the K factors and the
riskless asset having a rate of return
An arbitrage portfolio is constructed by
investing one dollar in the above portfolio and
shorting one dollars worth of risky asset j.
This arbitrage portfolio has a rate of return
i.e.

Let be individual is initial wealth, let be
individual is time-1 random wealth.
Then is the unique solution to the
following problem
The first order necessary condition for
to be an optimum is

Using , the above relation can be
written as
Now we claim that, for all j such that

To see this, note that risky assets are in
strictly positive supply. Suppose risky asset j
with is held in a strictly positive
amount in equilibrium by individual i.
By the strict concavity of and the assumption
that is independent of all other random
variable,
We know that

Then (4.24.3)holds for j by the fact that the
above arguments can be applied to all j such that
. Thus we have that (4.24.3) holds for all
such js.
As a consequence of the above analysis, we can
also conclude that if , then asset j is
held in a strictly positive amount by all
individuals.

86
2.25 section

Now fix risky asset j with . Let
be the dollar amount invested in
the risky asset j by individual i.
Define
Taylors expansion
Where all the other random variables are
independent and is a random variable whose
value lies between and .

By the assumption that 1, we have
Therefore
We claim that

To see this, we first note that since
Where the second inequality follows from the
concavity of

Next, we observe that
by
Therefore

Now we obtain
Note also that, since , by the
conditional Jensens inequality, we have

Finally, by
We have
Or equivalently,

Let the total market value of asset j be denoted
by
In equilibrium there exists an i such that
, where I is the number of individuals in the
economy.
We thus have
Where we gives an explicit bound of the deviation
of from the APT relation.

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