unit-3_pseudo_efficiency_complete.pdf

Portfolio Optimization

Unit 3: Classical Mean Variance Model Revisited:

Pseudo E!ciency

Duan LI & Xiangyu Cui

India Institute of Technology Kharagpur

May 26 - 30, 2014

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Outline

Literature and Introduction

Dual Realizations of Mean-Variance Pair

Pseudo EfficiencyType 1 and Best Investment PerformanceType 2 and Optimal Management of Initial Endowment

Conclusions

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Literature and introduction! Markowitz(1952) introduced seminal work on mean-variance

portfolio selection which laid the foundation for modern financialanalysis and led a remarkable development of a return-risk portfolioselection framework.

! Tobin(1958) revealed the famous mutual fund theorem that theoptimal portfolio of a mean-variance optimizer is a combination of ariskless asset and a risky fund.

! Sharp(1964), Lintner(1965) and Mossin(1966) introduced thecapital asset pricing model, independently, using di!erentapproaches.

! Black(1972) discussed the equilibrium of a capital market allowingshort selling and without riskless asset or with restricted borrowing.

! Merton(1972) derived the analytic expression of e"cient solutionsfor the unconstrained mean-variance portfolio selection.

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Mean-Variance Model with All Risky AssetsConsider the following classical mean-variance portfolio selection problemin a market of n risky assets with a total random return vector,r = (r1, r2, . . . , rn)!, of which the first and second order moments areknown as e = E (r), V = Cov(r):

(MV ) minx

12x !Vx

s.t. x !e = µ,

x !1 = x0, (1)

where x0 is an initial wealth level which is assumed to be positive and µis the pre-given expected wealth level.If we normalize x0 to 1, (MV ) becomes the mean-variance modeloriginally studied by Markowitz(1952), except that we allow short sellinghere in (MV ).

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Mean-Variance Model with All Risky Assets(Cont’d)

It can be verified that the optimal portfolio of (MV ) is given by

x(x0;µ) =x0

D(BV"11 ! AV"1e) +

µ

D(CV"1e ! AV"11), (2)

where

A = 1!V"1e = e!V"11,B = e!V"1e > 0,

C = 1!V"11 > 0,

D = BC ! A2 > 0.

The positiveness of D can be seen from the positiveness of(Ae ! B1)!V"1(Ae ! B1) = BD (Merton(1972)).A fact that has not been fully recognized in the literature is thatparameter A can be positive, negative or zero.

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The minimum variance set of problem (MV ) can be expressed as

!2 =C

D

!µ! A

Cx0

"2

+x20C. (3)

As the mean-variance pair of the minimum variance portfolio (MVP) is

given as (A

Cx0,

x20C), the upper branch of the minimum variance set,

{(µ,!) | !2 =C

D

!µ! A

Cx0

"2

+x20C, µ " A

Cx0},

constitutes the so-called mean-variance e"cient frontier.

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We denote all portfolio policies corresponding to the mean-variance pairson the minimum variance set of problem (MV ) boundary policies, whichcould be either e"cient or ine"cient.More specifically, the set of e"cient boundary policies, denoted by X e ,and the set of ine"cient boundary policies, denoted by X ie , can beexpressed explicitly as follows:

X e = {x(x0;µ) | x(x0;µ) is given in (2), µ " A

Cx0},

X ie = {x(x0;µ) | x(x0;µ) is given in (2), µ <A

Cx0}.

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Dual Realizations of Mean-Variance PairWe are interested in solving an inverse problem to find out which initialwealth level enables us to achieve a given mean-variance pair, (µ,!), inthe mean-variance space by adopting a boundary portfolio policy.Solving x0 from (3) yields the following two solutions when condition| µ |#

$B! holds:

x+0 =Aµ+

#D(B!2 ! µ2)

B, (4)

x"0 =Aµ!

#D(B!2 ! µ2)

B. (5)

Clearly, x+0 and x"0 represent two initial endowment levels which canachieve the given pair of (µ,!), where x+0 " x"0 holds whenever B!2 "µ2 and they are equal only when B!2 = µ2. Note that any pair (µ,!)that does not satisfy | µ |#

$B! is not achievable.

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Reachable RegionThe reachable region in the mean-variance space is defined as

{(µ,!) | | µ | #$B!}.

! All interior points within the reachable region in the mean-variancespace can be realized by adopting one of the two boundary portfoliopolicies associated, respectively, with two di!erent initial wealthlevels.

! Any boundary point of the reachable region in the mean-variancespace is generated by a single boundary portfolio policy associatedwith one specific initial wealth level.

! The minimum variance sets associated with di!erent initial wealthlevels of x0 form a family of hyperbolas. i) When A is positive,decreasing the level of x0 moves the hyperbola downwards; and ii)When A is negative, decreasing the level of x0 moves the hyperbolaupwards.

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Reachable Region (Cont’d)

0

µ

σ

Part A

Part B

Part C

Figure : The reachable region and its partition

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Reachable Region (Cont’d)

! The set of all mean-variance pairs associated with the minimumvariance portfolio policies corresponding to di!erent initial wealthlevels is given by

{(µ,!) | µ = ± |A|$C!},

where {(µ,!) | µ = |A|!#C} represents all minimum variance points

with positive expected returns and {(µ,!) | µ = ! |A|!#C} represents

all minimum variance points with negative expected returns.

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Partition of Reachable Region! Part A:

{(µ,!) | | A |$C! # µ <

$B!}.

Every point in Part A is achieved by two e"cient boundary portfoliopolicies corresponding to two di!erent initial wealth levels.

! Part B:{(µ,!) | ! | A |$

C! # µ <

| A |$C!},

Any point in Part B is achieved by one e"cient boundary portfoliopolicy and one ine"cient boundary portfolio policy corresponding totwo di!erent initial wealth levels. When A = 0, part B vanishes.

! Part C:{(µ,!) | !

$B! < µ < ! | A |$

C! }.

Every point in Part C is achieved by two ine"cient boundaryportfolio policies.

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Pseudo E!ciencyType 1 and Best Investment Performance

Outline




Conclusions

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Pseudo Efficiency (Type 1)If the investment performance is measured by a mean-variance pair, thecommon assumption that the higher endowment the better does not hold.

DefinitionIf an e!cient mean-variance pair of problem (MV ) associated with initialendowment x0 can be also generated or even dominated by anothermean-variance pair generated by another boundary portfolio policyassociated with initial wealth x̂0 which is strictly less than x0, i.e.,

(µ,![C

D

!µ! A

Cx0

"2

+x20C]) % (µ̂,![

C

D

!µ̂! A

Cx̂0

"2

+x̂20C]), (6)

x̂0 < x0, (7)

then, the given mean-variance pair associated with endowment x0 istermed pseudo e!cient (type 1) and the corresponding e!cient boundaryportfolio policy x(x0;µ) is called pseudo e!cient policy (type 1).

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Pseudo Efficiency (Type 1) (Cont’d)

! In other words, if a mean-variance e"cient solution of (MV ) withrespect to a given initial wealth level x0 becomes ine"cient in anexpanded three dimensional objective space:{min (initial wealth level), max (expected future wealth),min (variance of the future wealth)}, it is pseudo e"cient.

! One important recognition from our earlier discussion on dualrealization is that, for any given mean-variance pair

(µ,$

CD

%µ! A

C x0&2

+ x20C ) in the interior of the reachable region,

there exists another initial wealth level x̂0 such that

x0 + x̂0 =2AµB

(8)

and (6) becomes an equality.

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Pseudo Efficiency (Type 1) (Cont’d)PropositionWhen x0 > 0 and A > 0, all mean-variance pairs within

{(µ,

'C

D

!µ! A

Cx0

"2

+x20C) | A

Cx0 # µ # B

Ax0}

are pseudo e!cient (type 1).When x0 > 0 and A < 0, all e!cient mean-variance pairs of (MV ),

{(µ,

'C

D

!µ! A

Cx0

"2

+x20C) | A

Cx0 # µ}

are pseudo e!cient (type 1).

RemarkSet {µ | A

C x0 # µ # BA x0} is non-empty as D = BC ! A2 > 0.

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Markowitz’s Example

For this market of three risky assets with expected return vector

e = (1.162, 1.246, 1.228)!

and covariance,

V =

(

)0.0146 0.0187 0.01450.0187 0.0854 0.01040.0145 0.0104 0.0289

*

+ ,

the corresponding parameters are A = 8.0602, B = 9.3568, C = 6.9846,and D = 0.3872.

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Markowitz’s Example (Continued)! We consider an instance with investor’s initial wealth level equal to

1 and the pre-given expected return equal to µ = 1.160, which isgreater than A

C x0 = 1.154 and less than BA x0 = 1.1609.

! Furthermore, the optimal e"cient portfolio is specified by

x(x0 = 1;µ = 1.16) = (1.1075,!0.0471, 0.0296)!

and the corresponding e"cient mean-variance pair is given by(1.160, 0.0719).

! From (8) and Proposition 3.2, it can be verified that the followingboundary policy associated with a less initial wealth x̂0 = 0.9985,

x(x̂0 = 0.9985, µ = 1.16) = (0.9914,!0.0404, 0.0475)!

yields the same mean-variance pair of (1.160, 0.0719).! Thus, x(x0 = 1;µ = 1.16) is pseudo e"cient.

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Best Investment Performance

We now consider the following revised formulation of problem (MV ) byallowing investors the flexibility not to invest all his money into themarket,

(MV1) minx

12x !Vx

s.t. x !e = µ,

x !1 # x0, (9)

where the initial wealth level, x0, and the pre-set expected wealth level,µ, are both assumed to be positive.

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Best Investment Performance (Cont’d)PropositionThe optimal portfolio policy of (MV1) is given as,

x$(x̂0;µ) =x̂0

D(BV"11 ! AV"1e) +

µ

D(CV"1e ! AV"11), (10)

where x̂0 is the optimal investment amount invested into the market,

x̂0 =

,x0, A > 0 and µ > B

A x0,ABµ, otherwise. (11)

Furthermore, the mean-variance e!cient frontier of (MV1) is given by

µ =

- $DC !

2 ! DC2 x2

0 + AC x0, A > 0 and µ > B

A x0,$B!, otherwise,

(12)

which is a continuous function.19 / 33

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Best Investment Performance (Cont’d)Hint of the proof: Solving problem (MV1) is equivalent to solving atwo-phase optimization problem: Finding first the optimal initialinvestment level x̂0 from the following formulation,

minx̂0%x0

C

D

!µ! A

Cx̂0

"2

+x̂20C, (13)

and applying then the e"cient mean-variance policy x$(x̂0;µ).! When A < 0, the e"cient frontier of problem (MV1) is exactly the

upper boundary of the reachable region. In other words, all e"cientsolutions in the traditional mean-variance sense are essentiallypseudo e"cient (type 1).

! When A > 0, the e"cient frontier of problem (MV1) is acombination of the lower segment of the upper boundary of thereachable region and the upper segment of the e"cient frontier ofproblem (MV ) with initial wealth x0.

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Best Investment Performance (Cont’d)

0

µ

σ

MVP

Figure : The e"cient frontier of (MV1) with A > 0

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Pseudo E!ciencyType 2 and Optimal Management of Initial Endowment

Outline




Conclusions

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Optimal Management of Initial Endowment

Consider the following revised mean variance model for the optimalmanagement of the total endowment,

(MV2) maxx

x !e + (x0 ! x !1)

s.t.12x !Vx = !2,

x !1 # x0, (14)

where the initial wealth level, x0, is assumed to be positive and zerointerest is assumed to be applied to the money the investor places asideat time 0.

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Optimal Management of Initial Endowment(Cont’d)

PropositionThe optimal portfolio policy of (MV2) is given as,

x!(x̂0;!2) =

x̂0

D(BV"11!AV"1e)+

!"!2

CD! x̂2

0C 2D

+ACD

x̂0

#(CV"1e!AV"11),

(15)where

x̂0 =

$x0, if A > C and !2 > B

A x0,

(A! C)!%

CD+(A"C)2 , otherwise. (16)

Furthermore, the mean-variance e!cient frontier of (MV2) can be expressed as,

µ =

&'

(

%DC !

2 ! DC2 x2

0 + AC x0, A > C and !2 > D+(A"C)2

(A"C)2C x20 ,

x0 +%

D+(A"C)2

C !, otherwise.(17)

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Efficient Frontiers of (MV ), (MV1) and (MV2)

µ

x0

0

MVP

σ

Figure : E"cient frontiers of (MV ), (MV1) and (MV2) when A > C

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Efficient Frontiers of (MV ), (MV1) and (MV2)

MVPσ

0

µ

x0

(a) A " C and A > 0

0

MVP

σ

µ

x0

(b) A < 0

Figure : E"cient frontiers of (MV ), (MV1) and (MV2)

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Pseudo Efficiency (Type 2)

DefinitionIf an e!cient mean-variance pair of problem (MV ) associated with initialendowment x0, (µ, C

D

%µ! A

C x0&2

+ x20C ), is not pseudo e!cient (type 1),

but is dominated by the e!cient mean-variance pair of problem (MV2),i.e.,

(µ,![C

D

!µ! A

Cx0

"2

+x20C]) % ((x$)!e + (x0 ! (x$)!1),!!2), (18)

where x$ is the optimal policy of (MV2) given in (15), then the givenmean-variance pair associated with endowment x0 is called pseudoe!cient (type 2) and the corresponding e!cient boundary portfoliopolicy x(x0;µ) is called pseudo e!cient policy (type 2).

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Pseudo Efficiency (Type 2) (Cont’d)

Propositioni) There is no type-2 pseudo e!cient mean-variance pair when A < 0.ii) When x0 > 0 and 0 < A # C, all mean-variance pairs within

{(µ,

'C

D

!µ! A

Cx0

"2

+x20C) | B

Ax0 # µ}

are pseudo e!cient (type 2).iii) When x0 > 0 and A > C, all mean-variance pairs within

{(µ,

'C

D

!µ! A

Cx0

"2

+x20C) | B

Ax0 # µ # (1 +

D + (A! C )2

(A! C )C)x0}

are pseudo e!cient (type 2).

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Markowitz’s ExampleApplying the two revised mean variance formulations, (MV1) and (MV2),to the Markowitz’s example yields the e"cient frontier of (MV1) as

µ =

, $0.0554!2 ! 0.0079 + 1.1540, µ > 1.1609,

3.0589!, 0 # µ # 1.1609,

and the e"cient frontier of (MV2) as

µ =

, $0.0554!2 ! 0.0079 + 1.1540, µ > 1.2055,

1 + 0.4702!, 0 # µ # 1.2055.

It can be seen that the mean variance pairs in{(µ, 18.0388(µ! 1.1540)2) + 0.1432) | 1.1540 # µ # 1.1609} are pseudoe"cient (type 1) and the mean variance pairs in{(µ, 18.0388(µ! 1.1540)2) + 0.1432) | 1.1609 # µ # 1.2055} are pseudoe"cient (type 2).

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Markowitz’s Example (Cont’d)

0.26 0.28 0.3 0.32 0.34 0.36 0.38 0.4 0.42 0.44 0.461.13

1.14

1.15

1.16

1.17

1.18

1.19

1.2

1.21

1.22

σ

µ

MVP

Figure : E"cient frontiers of (MV ), (MV1) and (MV2) for Markowitz’smarket setting

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Conclusions

Conclusion

! Since Markowitz published his seminal work on mean-varianceportfolio selection in 1952, almost all literature in the past halfcentury adhere their investigation to a budget constraint assumptionon this classical investment issue. In the mean-variance world for amarket of all risky assets, however, the common belief ofmonotonicity does not hold, i.e., not the larger amount you invest,the larger expected future amount you can expect for a given risk(variance) level.

! We examine this classical problem from an expanded three-objectiveframework: Maximizing the expected terminal wealth, minimizingthe risk (variance) of the terminal wealth and minimizing the initialinvestment level.

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Conclusions

! The concept of pseudo e!ciency is introduced to remove theset of portfolio policies which are e!cient in the originalmean-variance space, and are, however, ine!cient in this newlyintroduced three-dimensional space.

! By relaxing the binding budget spending constraint ininvestment, we derive an optimal scheme in managing initialendowment which dominates the traditional mean-variancee!cient frontier.

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unit-3_pseudo_efficiency_complete.pdf

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