1

Portfolio Efficiency: Traditional Mean-Variance Analysis

versus Linear Programming

Steve Eli Ahiabu University of Toronto†

Spring 2003

† Please send comments to Stephen.ahiabu@utoronto.ca I thank Prof. Adonis Yatchew for his comments and suggestions in this project. All remaining shortcomings remain entirely mine.

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Abstract

So strong is the influence of Markovitz [1952] on modern finance that portfolio selection tasks and efficiency tests are dominated by one definition of mean-variance efficiency. Not much regard is paid to the fact that standard mean-variance utility functions satisfy the necessary and sufficient conditions of expected utility theory if and only if return distributions are elliptical. In this paper, I explore efficiency implications of portfolios using the mainstream mean-variance methodology and compare my results to a new test approach which assumes only nonsatiation and concavity of the utility functions; similar to Arrow [1971]. The results are interesting and suggest that better diversification is required of the otherwise popular portfolio, the value-weighted S&P 500.

Keywords: MV-efficiency, HL-efficiency, Spanning, Intersection

I Introduction

The concepts of Spanning and Intersection are major bedrocks of modern finance

in general and portfolio selection theory in particular. Given market completeness, the

market portfolio frontier is said to span all asset returns in the mean-variance sense. A

portfolio A with Nn + assets is said to span a narrower portfolio B of n assets if the

efficient frontiers of the two portfolios coincide. In fact, an investor holding the efficient

allocation B does not benefit, mean-variance wise, by adding any of the extra N assets

in the broader portfolio A . Thus there is no benefit from further expansion or contraction

of the current portfolio. Spanning occurs if no investor benefits from any diversification

moves irrespective of their degree of risk aversion.

If however the efficient frontiers intersect, then there is exactly one utility

function for which an investor does benefit. Only investors with that utility function will

be optimizing if they fail to add assets from the broader portfolio. Alternatively, there

exists one coefficient of risk aversion for which diversification benefits does not occur.

Intersection thus tests whether or not we can fine any rational agent with any conceivable

degree of risk aversion, however absurd as often suggested in the literature on the equity

premium puzzle, who might not benefit from diversification. Spanning thus implies

intersection but the reverse is not the case. Assets that are spanned can be ignored for the

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purpose of portfolio selection. For diversification purposes therefore, it is a standard task

for an investor to test for potential benefits from extending portfolio content to a broader

spectrum either for higher return, to economize on risk or in general, to achieve some

preferred return distribution.

Mean-variance analysis is a popular tool for analyzing portfolio efficiency. The

procedure typically involves maximizing a specified mean-variance utility function

subject some constraints including feasibility.1 Since Markovitz [1952, 1959] and Tobin

[1958] , mean-variance analysis has been dominated by one definition of efficiency.

According to these authors, a portfolio is efficient if there exists no other portfolio within

the feasible set that is characterized either by a higher expected return with no worse

volatility or by a lower volatility plus no worse mean return. In this paper, I refer to this

approach which is the most popular as the standard or mainstream mean-variance theory.

It is important however to stress that standard mean-variance analysis is only a

subset to the broader framework of Stochastic Dominance and is not necessarily

consistent with expected utility theory (see Arrow [1952]). It is only consistent if the

return distribution is elliptical, which is hardly the case with finance data. In particular,

an investor may prefer an asset with lower mean return plus higher volatility if that asset

has more preferred distributional properties including skewness, kurtosis and persistence

compared to the benchmark. That is, standard Markovitz-type definition of mean-

variance efficiency is a special case which is particularly not ideal given the nature of

financial data.

Hanoch and Levy [1970] provide an alternative definition of mean-variance

efficiency which is more consistent and free of distributional assumptions. In their

framework, a portfolio is efficient if there exists a monotone and concave (utility)

function that rationalizes that portfolio. Post [2001] uses this definition in his proposed

linear programming test for spanning and intersection using second order stochastic

dominance. Thus, by using second order stochastic dominance, his test is mindful only of

two of most important properties of expected utility theory, non-satiation and risk

1 Other approached may include minimizing variance subject to a minimum desirable return.

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aversion. Further, by using linear programming, the test avoids the computational burden

of quadratic solutions which characterized mainstream mean-variance analysis and has

the additional flexibility to incorporate logical extensions such as transaction costs.

In this paper, I evaluate portfolio efficiency and mean-variance spanning and

intersection via the restrictive standard approach initially using the simple Sharpe Ratios2

and next with the popular full characterization. I then repeat the task using the Hanoch-

Levy definition of efficiency and the compare the results of both branches. The rest of the

paper is organized as follows. The next section reviews the literature on portfolio

efficiency and mean-variance spanning and intersection. Section III presents the tests

considered in this paper as well as methods and algorithm employed. In section IV, I

present the data and the main results. I also discuss briefly the implications. Section V

concludes.

II Literature Review

The literature on Spanning and Intersection is vast and the intention here is not to

give a complete overview but merely to briefly recap the main branches and to indicate to

an interested reader as to where to look.

Applications of standard mean-variance analysis (MVA) abound. DeSantis

[1995] and Cumby and Glen [1990] employ MVA to question whether US-investors can

benefit from international diversification. Taking the viewpoint of a US investor who

initially only invests in the US, these authors study the question whether they can

enhance the mean-variance characteristics of their portfolio by also investing in other

developed markets. DeSantis [1994], Bekaert and Urias [1996], Errunza, Hogan and

Hung [1998], and DeRoon, Nijman and Werker [2001] investigate mean-variance

portfolio advantages to the US investor who holds assets in the developed markets in the

US, Japan and Europe by investing in emerging markets. Glen and Jorion [1993] take the

argument further by investigating whether mean-variance optimizing investors with well-

diversified international portfolios should add currency futures to their portfolios. That is,

2 Sharpe [1971]

5

should they hedge the currency risk that arises from positions taken in cross border stocks

and bonds?

Some authors have explored spanning and intersection in joint formulations

involving mean-variance frontiers and volatility bounds in what has come to be termed

“duality” tests. Ferson, Foerster, and Keim [1993], DeSantis [1994], Ferson [1995] and

Bekaert and Urias [1996] demonstrate that the hypothesis of mean-variance spanning and

intersection can be reformulated in terms of the volatility bounds similar to those by

Hansen and Jagannathan [1991]. In their framework, the question is whether the set of

additional assets contain information about the volatility of the stochastic discount factor

or pricing kernel that is not already present in the current portfolio. A mean-variance

improvement in this case occurs if the diversification into emerging markets for instance

provides tighter volatility bounds on the stochastic discount factor than returns from the

developed markets only. Bansal and Lehmann (1997) provide a bound on the mean of the

logarithm of the pricing kernel, using growth optimal portfolios. Balduzzi and Kallal

(1997) show how additional knowledge about risk premia may lead to sharper bounds on

the volatility of the discount factor and Balduzzi and Robotti (2000) use the minimum

variance discount factor to estimate risk premia associated with economic risk variables.

There is literature that uses conditioning information. Finance return data are

hardly independently and identically distributed (i.i.d.). Cochrane [1997] and Bekaert and

Urias [1996] develop models that allow the incorporation of conditional information in

their tests. Though their procedures are intuitive and involve only a rescaling of returns, a

disadvantage of this method is that the dimension of the estimation and testing problem

increases quickly. Harvey [1989], Campbell and Viceira [1998] and DeRoon, Nijman

and Werker [1998] show how the problem can be largely circumvented by assuming that

variances and covariances are homoskedastic, while expected returns are allowed to vary

over time, although this assumption is largely in conflict with the empirical evidence

regarding time-varying second moments. Appealing to this simplifying assumption

however, the authors show that the conditioning variables can easily be accounted for by

using them as additional regressors. The restrictions for the intersection and spanning

hypotheses then become similar to the restrictions in the standard case with i.i.d.

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variables. This way of incorporating conditional variables also has the additional

advantage that the regression estimates indicate the economic circumstances, i.e., for

what values of the conditioning variables, intersection and spanning can or can not be

rejected as demonstrated in Shanken [1990] and Ferson and Schadt [1996].

Markovitz [1952, 1959] and Tobin [1958] present quantitative approaches to

portfolio analysis. Their prescription remains dominant in practice to date. Markovitz

propose choosing the portfolio that minimizes variance subject to a restriction on the

mean return. These methodologies involve large quadratic programming solution rules.

To simplify the problem, Yamakazi and Konno [1991] present linear methods involving

mean absolute error analysis (MADA) while Young [1998] formalize a maximum (and

minimum) return approach requiring linear programming. The latter also establishes the

exact relationship between his minimax approach and Markovitz’s.

Most mean-variance analysis methodologies (methods above) consider when the

investment possibility set is given. Kandel and Stambaugh [1987] and De Roon, Nijman

and Werber [2001] propose mean-variance spanning and intersection tests in the case

where IPOs exist in the market.

Though dominated by standard mean-variance analysis, it is important to stress

that standard MVA is only a subset of the broader subject of Stochastic Dominance (SD).

The advantages of using this standard form include its tractability, ease of testability (see

Huberman and Kandel [1987] and De Roon et al. [2001]) as well as flexibility to allow

for logical extensions such as transactions cost and short selling constraints. As pointed

by Bigoelow [1993], this standard definition of MVA is not necessarily consistent with

expected utility theory; it is consistent only if return distribution is elliptical. Hanoch and

Levy [1970] make this claim much more intuitive when in their characterization of

results when different classes of mean-variance utility functions are assumed. They

suggest that the stronger the restrictions assumed on admissible utility functions, the

closer one gets to individual complete preference ordering. Therefore, the number of

items in the efficient set is reduced as the condition for dominance becomes more

specialized as in the standard mean-variance utility case.

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Meyer [1979] presents necessary and sufficient conditions for testing whether or

not a given portfolio is efficient in the Hanoch-Levy sense. Similar to the case of standard

MVA, the implementation of such tests involve quadratic programming which can be

computationally tasking. Post [2001] demonstrates a way to derive necessary and

sufficient conditions that require only linear programming hence reducing the

computational burden enormously. His test also offers increased flexibility including the

opportunity to consider transaction costs and short selling constraints. In the next section,

I recap standard mean-variance analysis starting with the Sharpe ratio and then a full

characterization. Portfolio efficiency tests are highlighted for both standard MVA and for

the Hanock-Levy type utility due Post [2001].

III Methods and Procedures

The Sharpe ratio is perhaps the crudest and quick source of the standard mean-

variance criterion. A portfolio is mean-variance efficient if no alternative feasible

portfolio yields at least the same mean return with a lower variance or a higher mean

return with at worst the same variance.3 The ratio basically is4

( )( )

= tt rr

SR φωσωµ

where ( )rωµ and ( )rωσ are the mean return and standard deviation of the portfo lio ω .

tφ is simply data set available up to date t .

It is easy to spot potential flaws with the Sharpe ratio. For instance, this criterion

automatically ranks risk free assets as the most efficient – with a ratio of positive infinity

– regardless of the distribution of returns on risky assets. This is so because no other asset

has zero variance except the risk free asset.

3 An alternative measure of portfolio performance similar to the Sharpe Ratio is Jensen’s alpha following Jensen [1968] . 4 Other measurements of Sharpe Ratio use mean-to-variance ratio. This is simply a normalization and yields same portfolio ranking as the above formulation.

8

A more rigorous mean variance analysis follows. An investor is assumed to be

faced with the one period problem of maximizing the indirect utility of future wealth

( )[ ]1max +tt WuEω

subject to the constraints 11 ++ ′= ttt rWW ω and 1=′ niω (see Ingersoll

[1987]). ω is the vector of weights assigned to each of the n assets within the portfolio

and ni is an 1×n unit vector. 1+tr is an 1×n vector of next period returns. Utility u is

assumed to satisfy the usual properties. The agent’s problem thus can be rewritten as:

( )[ ] ( )nttt irWuE ωηωω

′−+′ + 1max 1

where η is a Lagrange multiplier. The first order condition of the above implies that

[ ] nttt irmE =++ 11

where 1+tm is the stochastic discount factor or pricing kernel which is assumed to exist if

the law of one price holds.5 1+tm is given as ( ) −++ ′′= ηω 11 tttt rWuWm .

As suggested by the previous section, and as shown by Ferson, Foerster, and

Keim [1993], DeSantis [1994], Ferson [1995] and Bekaert and Urias [1996], the concept

of mean-variance spanning and intersection has a dual interpretation in terms of volatility

bounds. In this regard, mean-variance spanning means that the volatility bound derived

from the returns 1,1 +tr is the same as the bound derived from ( )1,21,1 ; ++ tt rr . Therefore, the

minimum variance stochastic discount factors for 1,1 +tr , ( ) 11 +trm , are also the minimum

variance stochastic discount factors for ( )1,21,1 ; ++ tt rr , and the asset returns 1,2 +tr do not

provide information about the necessary volatility of stochastic discount factors that is

not already present in 1,1 +tr .

Using the definition for covariance ( ) ( ) ( ) ( )tttttttttt yExEyxEyx −=cov , we can

rewrite the above FOC as:

[ ] ( )[ ] ( )[ ] [ ]111

11

11 cov ++−

+−

++ −= tttttntttt rmmEimErE

5 Substituting the stronger assumption of “no arbitrage” for the “law of one price”, one can show that

01 >+tm . The same result is arrived at when one interprets the kernel function as the intertemporal rate

of substitution.

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The optimal portfolio weights ω can be found from the above if the utility function and

Lagrange multiplier η are known.

The problem becomes more tractable if one restricts the objective to mean-

variance optimization (that is assuming a mean-variance utility function). Assume that a

fund initially has n assets in its portfolio. A portfolio ω is mean-variance efficient if it is

chosen to optimization

ωωγ

µωω

∑′−′=2

max u (1)

subject to the constraint 1=′ niω where µ is an ( )1×n vector of gross returns. γ is the

coefficient of risk aversion and ∑ the ( )nn× variance covariance matrix of returns.

Optimal allocation requires the assigned weights to be generated as:

( )niηµγω −∑= −− 11 (2)

In the above, the Lagrange multiplier η can be interpreted as the zero-beta rate, i.e. the

return of the portfolio that is not correlated with the optimal portfolio.

The return vector on the entire market tR can be partitioned into ( )tt rr ,2,1 , ′′ , where

tr ,1′ is ( )1×n while tr ,2′ is ( )1×N . We regard tr ,1′ as the benchmark portfolio and tr ,2′ the

vector of test assets. If the benchmark portfolio is efficient mean-variance wise, then we

expect efficient portfolios to be of the form:

=

01ω

ω (3)

with 1ω being ( )1×n and ω being ( )[ ]1×+ Nn . From equation (2), in the general case

we have ( ) ωγηµ ∑=− ni , a partitioning of which implies

∑∑∑∑

=

−−

01

2221

1211

2

1 ωγ

ηµηµ

N

n

ii

(4)

The first equation in (4) implies ( )niηµγω −∑= −− 111

11 and substituting this into the

second line, we have:

( ) 11121

111212

−− ∑∑+∑∑−= nN iiηµ (5)

10

If there is only one value of ( )γη for which this condition holds, we say that there is

intersection. In this case, the two efficient frontiers intersect, and there can exist a rational

risk averse investor (i.e. an investor with a specific degree of risk aversion) who has no

benefit in terms of standard mean-variance tradeoff from including the extra N assets

into her portfolio. If this condition holds for every value of ( )γη , we say that there is

spanning. Thus, spanning implies:

011

11212 =∑∑− − µµ

011121 =−∑∑ −

Nn ii

Spanning means that the mean-variance frontier of the n assets completely coincides

with the mean-variance frontier of the Nn + assets and no investor, irrespective of

degree of risk aversion can benefit from further diversification. Intersection on the other

hand means that the two mean-variance frontiers have only one common point

(portfolio).

In the two asset case equation (5) becomes

121

212

1

212 1 µ

σσ

σσ

ηµ +

−= (6)

where subscripts 1 refer to the initial asset portfolio and 2 the asset for potential

addition. In (6), the ratio 21

21

σσ

coincidentally is the slope coefficient (beta) from

regressing the return of asset 1 on that of asset 2 . The hypothesis 02 =ω can thus be

tested by running the regression:

ttt urr ++= ,1,2 βα (7)

and testing the restriction ( )βηα −= 1 . Again, the Lagrange multiplier is

unobserved. Pre-multiplying equation (2) by ′2i , we have ( )2

12

121 iii ηµγω −∑′=′= −− .

Solving for relative risk aversion coefficient,

21

21

2 iii −− ∑′−∑′= ηµγ (8)

This implies that one can test for intersection (for a specific investor with a given degree

of risk aversion γ – hence η ) as well as test spanning (for all investors – irrespective of

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γ ) using the above methodology. In other words, testing intersection involves choosing

γ (hence η ) and testing whether the condition ( )βηα −= 1 holds. A test for

spanning implies this condition holds for all γ (hence η ). This requires the joint

hypothesis 1=β and 0=α .6

To test for intersection in this standard Markovitz-type utility framework, I

reparameterize and test 0=κ in the regression ( ) ttt urr +−+=− ηβκη ,1,2

for several values of γ (hence η ). The test of spanning involve the reparameterized

regression tttt urrr ++=− ,1,1,2 λα and the joint hypothesis 01 =−= βλ and

0=α .

Hanoch-Levy (HL) approach is critical of the above mainstream MVA due to

distributional assumptions which are inherent in the utility function specification. Since

utility functions are not observable, this provides rational for the use of general

assumptions such as nonsatiation and risk aversion. This notion is evident in their

alternative definition of portfolio efficiency. A portfolio Ω∈*ω is HL-efficient if it is

optimal relative to some function Uu ∈ where U is the set of all monotone concave

utility functions. This is a much stronger definition than the standard of mean-variance

definition of efficiency (see above). In contrast to the Sharpe ratio, the HL definition

typically classifies a riskless fund as inefficient, in consistency with Arrow [1971] since

stocks generally have higher mean return over the risk free rate. Thus, a HL-efficient

portfolio satisfies

( ) ( ) ( ) ( ) ( ) ( )[ ] 01

maxminmaxmin1

** =

−=− ∑∫∫ =Ω∈∈Ω∈∈

T

t ttUuUururu

TrdFrurdFru ωωωω

ωω

where ( )rF is the empirical return distribution function. The linear programming test by Post (2001) is designed using the above approach.

His test questions whether or not one can construct a monotone and concave utility

Uu ∈ that can rationalize the portfolio choice *ω . In other words, can we find support 6 An alternative test is spanning and intersection is Jensen’s alpha approach, due Jensen [1968].

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lines for a monotone and concave quadratic function that justifies the diversification

strategy evident in the current portfolio? If we can, then we can call such a function a

potential utility function for an optimizing agent with that portfolio. The above minimax

formulation has often appeared in the literature as measures of production efficiency7 as

well as in the finance literature (see Young [1998]). In the HL definition, the class of

monotone and concave quadratic utility functions is restricted to:

( ) 12,0: 3232

321 ≥∆+≤++=∈≡ αααααα rrruUuU

The first constraint 03 ≤α endures concavity. ∆ can be seen as the marginal

increase/decrease in return due to a portfolio reallocation and hence the second constraint

02 32 ≥∆+ αα restricts the function to be strictly increasing over the ent ire return

interval.

If the relevant portfolio is efficient, then such support lines must exist, and if such

support lines exist, then the portfolio must be efficient. There are two important issues to

note concerning the above statistic. First, the statistic does not represent a meaningful

performance measure that can be used to rank different portfolios with regards degree of

efficiency. Secondly, the support lines are used as an instrument for testing the efficiency

of the portfolio rather as an estimate of the utility function for which the given portfolio is

efficient. This is because once a portfolio is found to be efficient, there typically exists

multiple candidate utility functions that could equally justify that portfolio. The linear

programming formulation requires the test statistic

( )( )

( )( )

+∈∀≥+−+= ∑ =Θ∈

T

t titt NnirrrT 1 ,

*32

2,1,

* ......1021

:min θωωααθωξααθ

with ( ) 12:, 3232 ≥∆+ℜ×ℜ∈≡Θ − αααα . The problem hence has three choice

variables 32, ααθ and and 1++ Nn constraints. The portfolio *ω is efficient if and

only if ( ) 0* =ωξ .8

7 See Debreu (1951) and Farrell (1957). 8 See Post [2001] for proof.

13

In the above, the necessary condition follows direct from Kuhn-Turker conditions

of solving for ( )∑ =Ω∈

=T

t truT 1

* 1maxarg ωω

ω for Uu ∈ . Thus the condition

( )( ) Ω∈∀≥−′∑ =ωωωω 0

11

**T

t ttt rrruT

(9)

That is to say *ω is optimal for the concave monotone set of utility functions only if all

alternative portfolios Ω∈ω are enveloped by the tangent hyperplane defined by the

vector ( ) ( ) ( )( )**1

* ...,......... ωωω Trururu ′′≡′ . By construction (given our choice of 2α and

3α ), we know that ( ) ( )( )** , ωω ruru ′′′ is feasible i.e. ( ) ( )( ) Ω∈′′′ ** , ωω ruru . The

inequality (9) above implies that *ω is efficient only if ( ) 0* =ωξ , which is the

necessary condition for efficiency.

Sufficient conditions are established by using ( ) Ω∈32 ,αα for the optimal

portfolio and using some concave function ( ) 2*3

*2 WWWu αα += . If *ω is efficient i.e.

( ) 0* =ωξ , then

( ) ( ) ωωωωω ∑∑ =Ω∈=

′=′ T

t ttT

t rt rrurru1

*1

** max (10 )

Jensen’s inequality and u concave imply that ( ) ( ) ωωωω ∑∑ =Ω∈=

′≤′ T

t ttT

t t rruru1

*1

max for

all Ω∈ω . Given ( )Wu as above, we have ( ) ( ) ωωω ∑∑ ==′=

T

t ttT

t t rruru1

*1

. This

together with equation (10 ) gives ( ) ( )∑∑ ==′=

T

t rtT

t t rruru1

**1

* ωωω . Combining this

with the Jensen’s inequality gives ( ) ( ) ωωωω ∑∑ ==′≥′ T

t ttT

t rt rrurru1

*1

**

*0 ωω == when which simply implies the sufficient condition also being ( ) 0* =ωξ .

One observation about the test statistic is that there is always a feasible solution

for instance 01 32 == αα and , in which case ( )∑ =−=

T

t tti rrT 1

*,

1ωθ necessarily satisfies

the constraint. This refers to the case of a risk neutral agent with linear utility and who

seeks to maximize only expected return. A second observation is as follows. Consider an

initial portfolio of just one stock ( )1=n and set *ωω = . Thus tti rr ,1, = and hence

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0,* =− tit rr ω . Immedia tely, one gets the result 0=θ and the portfolio is efficient. In

other words, if there is only one asset in the investment opportunity set, then that single

asset makes an efficient portfolio.

To test the efficiency hypothesis, Post (2001) develops an alternative formulation

which he calls the “dual formulation” which is very similar that outlined above. The dual

statistic is given as:

( ) ( ) ( ) 0,:,max *** ≥≡Ω∈

ωωσωωµωψω

where the mean difference between the current portfolio *ω and a potential feasible

alternative is ( ) ( )∑ =−=

T

t tti rrT 1

*,

* 1, ωωωωµ . ( ) ( )( )∑ =

−−∆=T

t ttit rrrT 1

*,

** 1, ωωωωωσ is

the co-movement measure between both portfolios. Again, the derivation of this is quite

similar to that recapped above (see Post [200]). The portfolio *ω is HL-efficient if and

only if ( ) 0* =ωψ .

This paper adopts the dual formulation for the tests conducted in the next section.

The algorithm used is rather simply but not so fast for a large asset space and high

accuracy level. First, I decide the degree of precision that an investor may consider

important with regards portfolio weights. That is, an investor seeking fairly high accuracy

may choose weights to the decimal of say 5101 −× . Next, I guess a arbitrary positive value

for ψ . Then I take the current portfolio weight vector *ω and formulate all possible

alternative weight permutations to the required accuracy level bearing in mind the

investment constraints 11

=∑ +

=

Nn

i iω and ( )Nnii +∈∀≥ ,......10ω . Using each

hypothetical weight vector ω , I evaluate ( )*,ωωµ and ( )*,ωωσ . If ( ) 0, * ≥ωωσ and

( )*,ωωµ is less than the previously saved ψ , I replace ψ to equal the current ( )*,ωωµ

and also save the current weight vector. Then I try the next hypothetical weight vector.

My final *ψ is that remaining after the entire iteration is done.

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IV Data Description and Main Results

In the current paper, I use monthly return data on seven of the most widely

watched benchmark portfolios/indexes of stock market activity. The first is S&P 500

Table 1: Descriptive Statistics: monthly return data in percentages, July 1926 to Dec 2002.

PS & LB / MB / HB / LS / MS / HS /

Mean 0.9694 0.9137 0.9765 1.1974 1.0483 1.2680 1.4585

SD 5.6958 5.5060 5.9256 7.4732 7.9488 7.2302 8.4982

Minimum -28.7100 -28.1500 -27.7900 -35.3600 -32.3400 -30.8500 -33.7000

Maximum 41.6800 32.4700 51.6100 70.5600 64.3200 64.3900 82.0200

Skewness 0.4443 -0.1138 1.3966 1.6900 0.9777 1.4312 2.0744

Kurtosis 12.5277 8.1188 20.2693 21.2780 12.9412 18.1142 22.5401

Jarque-Bera 3482.505 997.3123 11646.53 13149.59 3904.517 9004.619 15187.38

Variance-Covariance Matrix PS & LB / MB / HB / LS / MS / HS /

PS & 32.4424

LB / 30.4441 30.3159

MB / 32.3927 29.1441 35.1124

HB / 38.6742 33.6901 41.3126 55.8484

LS / 38.3310 37.5513 38.7050 47.9873 63.1841

MS / 36.1469 33.6757 38.0098 48.4094 54.4517 52.2764

HS / 41.0795 33.6757 44.1789 58.3386 60.3318 59.1884 72.2200

Unconditional Correlation Matrix PS & LB / MB / HB / LS / MS / HS /

PS & 1.0

LB / 0.9708 1.0

MB / 0.9598 0.8933 1.0

HB / 0.9086 0.8188 0.9329 1.0

LS / 0.8466 0.8580 0.8217 0.8078 1.0

MS / 0.8777 0.8459 0.8872 0.8959 0.9474 1.0

HS / 0.8487 0.7883 0.8773 0.9186 0.8931 0.9633 1.0

Monthly return for S&P 500 were retrieved from CRSP Compuserve database. These returns are inclusive of dividends. Returns on Fama and French portfolios are from the web site of Kenneth French at http://mba.tuck.dartmouth.edu/pages/ faculty/ken.french/. Jarque-Bera [1980] provides an LM test for normality. In this case, I test the normality of returns. The result is distributed as Chi-Squared with 2 degrees of freedom. The test rejects the normality assumption in all 7 cases.

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value weighted cum-dividend return with a time range of July 1926 to December 2002

from CRSP database. The rest six are Fama and French book-to-market sorted and size-

sorted benchmark portfolio return data with the same time range.9 Summary descriptive

statistics are reported in table 1.

Interestingly, the S&P 500 has one of the lowest mean returns among the seven

portfolios. In compensation, it does exhibit modest volatility. The index correlates highest

with F&F portfolio coded LB / and lowest with LS / . The portfolio LB / seems to be a

rather conservative portfolio exhibiting the lowest return, volatility, kurtosis and negative

skewness. HS / seems to be the most adventurous portfolio. Jarque-Bera [1980] test for

normality reports strong rejection of the null in all seven cases. A potential implication is

that standard mean-variance analysis Markovitz [1952] is flawed.

Next, I compare conditional Sharpe ratio for the above portfolios. Sharpe Ratios

reported in figure 1 do not show overwhelming performance advantage for or against any

portfolio since all the graphs seem neck-to-neck, perhaps in exception of F&F LS /

where S&P seems to do better. However, this is far from being conclusive since this pair

as well exhibit the lowest correlation which suggests a good avenue for risk curbing

diversification. Further, as highlighted earlier, the Sharpe ratio can be highly

uninformative and even erroneous. For instance, two risk free assets with different returns

will be ranked equally with a ratio of positive infinity because both have zero return

variance.

9 See http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_library.html

17

Figure 1: Conditional Sharpe Ratios, S&P verses F&F benchmark portfolios.

These ratios start from the 201st observation (from March 1943) up to the 918th (Dec 2002). The first 200 observations were used as previous information (conditioning information) to calculate the ratio reported at date 201 and so on. The solid lines are moving (conditional) Sharpe ratios for S&P 500 while the broken lines are moving Sharpe ratios for the F&F portfolio reported in each title.

18

Tests for intersection were carried out for varying degrees of relative risk aversion

using standard mean-variance utility maximization as outlined in the previous section.

The question is whether or not a given agent with specific degree of risk aversion γ

(hence η ) will be benefit from diversifying beyond the S&P benchmark and add of the

F&F portfolios. Table 2 shows t-statistics for ( )βηα −= 1:0H in using the

reparameterized regression ( ) ttt urr +−+=− ηβκη ,1,2 .

Table 2: t -statistic for different degrees of relative risk aversion.

γ LB / MB / HB / LS / MS / HS / 0.00 0.00 0.00 0.0001 0.00 0.0002 0.0003

0.25 0.0003 0.00 -0.0007 -0.0008 -0.0004 -0.0009

0.50 0.0005 0.00 -0.0016 -0.0017 -0.0009 -0.0021

0.75 0.0008 0.00 -0.0025 -0.0026 -0.0015 -0.0034

1.00 0.0011 0.00 -0.0033 -0.0034 -0.0021 -0.0046

1.25 0.0014 0.00 -0.0042 -0.0043 -0.0026 -0.0058

1.50 0.0017 0.0001 -0.0051 -0.0052 -0.0032 -0.007

1.75 0.002 0.0001 -0.0059 -0.006 -0.0037 -0.0082

2.00 0.0023 0.0001 -0.0068 -0.0069 -0.0043 -0.0094

2.50 0.0029 0.0001 -0.0085 -0.0086 -0.0054 -0.0118

3.00 0.0035 0.0001 -0.0103 -0.0103 -0.0065 -0.0142

4.00 0.0046 0.0001 -0.0137 -0.0138 -0.0087 -0.0191

5.00 0.0058 0.0002 -0.0172 -0.0173 -0.011 -0.0239

10.00 0.0116 0.0003 -0.0345 -0.0346 -0.0221 -0.0481

20.00 0.0232 0.0006 -0.0691 -0.0693 -0.0444 -0.0964

As explained in Section III, in all the above cases, the variable tr ,1 refers to S&P 500 returns while tr ,2

refers to the returns of the F&F portfolio noted at the top of each column. η is derived from γ and the

appropriate variance-covariance matrix using equation (8) above. That is,

nn

n

ii

i1

1

−

−

∑′−∑′

=γµ

η .

Table 2 above suggests a definite case of non-rejection of the hypothesis of

intersection for all degrees of risk aversion even as high as 20. For the range of relative

risk aversion considered, agents do not benefit from diversification. Agents will not be

acting inefficiently by failing to add any F&F portfolio to the benchmark S&P 500. As

one considers agents with higher risk aversion, the intersection the value of the statistic

rises slightly within reasonable ranges of γ yet the hypothesis is still no where close to

19

the rejection zone. The first row of table 2 refers to the risk neutral agent who is

interested only in expected return. One way to interpret these results is that from the

perspective of a risk neutral agent, the mean return of all seven assets are not statistically

different hence no diversification suggested on the first row. Comparing table 2 to table 1

suggests reasons why diversification benefits are not prevalent via the mean-variance

approach. Most of the F&F portfolios have slightly higher returns than S&P 500 return.

However, the former have higher volatility as well as strong correlation with the latter.

The test for spanning asks whether or not no agent will benefit from

diversification, thus irrespective of the degree of risk aversion. From the test of

intersection above, it is apparent that spanning (that no agent benefits) will most probably

not be rejected either given the rather low statistics reported. Table 3 below shows Wald

statistics for the joint hypothesis 1=β and 0=α in each of the six cases. The test is

conducted here using the reparameterized regression tttt urrr ++=− ,1,1,2 λα

with 001:0 ==−= αβλ andH .

Table 3: Wald Statistic irrespective of degrees of relative risk aversion.

LB / MB / HB / LS / MS / HS / Wald 0.0000 0.0000 0.0000 0.0002 0.0005 0.0012

Again the variable tr ,1 refers to S&P 500 returns while tr ,2 F&F portfolios. The statistic is

derived as ( ) ( ) ( ) ( ) ( )δδδ

δδδ

δδ

δ ˆˆˆˆ

ˆ2

hd

dhddld

ddh

h

′−

′′

distributed 2χ with degree of freedom equal

the number of restrictions under 0H .

Wald statistic with 2 degrees of freedom has 5% critical value of 5.99. According

to table 3, spanning cannot be rejected in any of the six cases. In words, agents,

irrespective of degree of risk aversion will not benefit from diversification beyond the

S&P benchmark if the other available assets are the Fama and French portfolios atop each

column.

As is obvious from the above tests, mainstream mean-variance analysis depends

strongly on distributional assumptions which are inaccurate as evident from the

20

descriptive statistics and the Jarque-Bera test statistics in table 1. HL-efficiency thus

offers an opportunity to test efficiency of portfolios relying on acceptable properties of

utility functions being nonsatiation and risk aversion; a test with fewer return distribution

assumptions.

Table 4: Dual formulation statistic *ψ comparing the value-weighted S&P 500 to each F&F portfolio.

LB / MB / HB / LS / MS / HS / *ψ 0.00 6.4900 209.2800 0.00 274.1000 448.9500

0* =ψ implies a stand-alone S&P 500 portfolio is efficient and there is no need to diversity if the only other asset available is the F&F Portfolio on each column. For instance in the 3rd column, if asset

MB/ is available, then a portfolio made solely of S&P 500 is sub-optimal. In contrast to the standard mean-variance tests, the dual test statistic above rejects

efficiency of a portfolio made solely of the value weighted S&P 500 when alternative

assets MB / , HB / , MS / and HS / are available for diversification purposes. It is

important to caution again that values reported in table 4 only compare the S&P to each

of the six Fama and French portfolios. The dual statistic ψ by design does not offer an

opportunity to rank these six Fama and French portfolios based only on information

contained in the table above.

V Conclusion

Sharpe [1971] has remarked that “if the essence of the portfolio analysis problem

can be adequately captured in a form suitable for linear programming methods, the

prospect for practical application would be greatly enhanced”. In the current paper, I have

demonstrated an example of the growing number of applications of linear programming

to effectively answer topical questions in finance which otherwise would have required a

large quadratic solution.

According to the standard mean-variance utility function approach similar to

Markovitz [1952], spanning and intersection analyzes the effect that the introduction of

additional assets has on the mean-variance frontier. If the mean-variance frontier of the

benchmark assets and the frontier of the benchmark plus the new assets have exactly one

point in common, then they intersect. This is what is termed “intersection”. This means

21

that an agent with that mean-variance utility function is optimizing my holding that

benchmark. For an agent with that utility function, there is no benefit in standard mean-

variance utility from adding the new assets. If the mean-variance frontier of the

benchmark assets plus the new assets coincides with the frontier of the benchmark assets

only, there is “spanning”. In this case no standard mean-variance utility optimizing

investor can benefit from adding the new assets to her initial portfolio of the benchmark

assets. The forgoing definition is only accurate if return distribution is elliptical. Hanoch

and Levy [1970] provide alternative definition for mean-variance efficiency much more

consistent with Arrow’s theorem even in a world of non-elliptical returns.

In this paper, I implemented tests of spanning and intersection for a portfolio

made solely of the value-weighted S&P 500 as a benchmark portfolio using both test

approaches: mainstream mean-variance utility approach using quadratic optimization and

a second test designed to satisfy the Hanoch-Levy definition of portfolio efficiency and

make use of linear programming tools. The objective is to test whether there exists

spanning and intersection of six Fama and French portfolios, considered here as assets

available for possible diversification.

While standard mean-variance tests are unable to reject the hypothesis of

efficiency of the benchmark, the Hanoch-Levy approach rejects in four of the six cases.

There are therefore mean-variance utility benefits from extending the asset holding

beyond the S&P benchmark. The difference in results is accounted for by distributional

assumptions inherent in mainstream mean-variance optimization theory.

22

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