Markowitz Principles for Multi-Period Portfolio Selection Problems - - PowerPoint PPT Presentation

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Markowitz Principles for Multi-Period Portfolio Selection Problems with Moments of any Order and Constraints

Thamayanthi Chellathurai Enterprise Risk & Portfolio Management Bank of Montreal, Toronto

Disclaimer: The opinion expressed in this talk is that of the author only, and not that of the Bank of Montreal

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Outline

1. Single-Period Portfolio Selection 2. Limitations 3. Multi-Period Portfolio Selection 4. Multi-Period Portfolio Selection with Non- Negative Wealth Constraints 5. Numerical Results 6. Conclusion

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Markowitz Single-Period Model

= t T t = t

horizon Planning rate Interest T t at asset i

• f

unit

• ne
• f

Price t at asset i

• f

unit

• ne
• f

Price

th , th ,

→ → = → = → T r S S

T i i

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Markowitz Single-Period Model

Risk-free Asset Risky Assets ( N )

,

S

rT

e S

,

, i

S

T i

S ,

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t t =

T t=

,

U

,

S

,

U

T

S

, , 1

U

, 1

U

T

S ,

1 , 1

S

, N

U

, N

U

, N

S

T N

S

,

) ( W ) (T W

t at transacted asset i

• f

Shares

• f

Number ... ) ( ... ) (

th , , , , 1 , 1 , , , , , 1 , 1 , ,

= → + + + = + + + =

i T N N T T N N

U S U S U S U T W S U S U S U W

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? ), ( Given : Problem ) ( ) (

, , , , , 1 , , ,

=           − + =

∑

= i i T T i N i i T

U W S S S S U S W S T W

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                            − − − =                     =

, , , , , , 2 , , 2 , , 1 , , 1 , . , 2 , 1

. . . , . . S S S S S S S S S S S S Q U U U A

N T T N T T T T N

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[ ] [ ]

{ }

matrix Covariance ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) (

, , , , ,

→ − − = =           + =           + =

T T T T T T T

Q E Q Q E Q E C A C A S T W Var A Q E S W S T W E A Q S W S T W

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Markowitz Problem

T G T T T T

S T W A Q E S W S T W A C A S T W Var

, , , ,

) ( ) ( ) ( ) ( E subject to 2 1 ) ( 2 1 minimizes A that Find = + =           =          

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) ( ) ( ) ( ) ( ) ( ) ( Frontier Efficient ) ( ) ( 1 ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) (

1 , , , , 1 , , , , , , , , 1 1 , ,

Q E C Q E S W S T W Std S W S T W E Q E C Q E S W S T W S W S T W Std S W S T W S W S T W E Q E C Q E C Q E S W S T W A

T T T T T G T T G T T T G

• pt

− − − −

          − =           −           − =           −           − =           −       − =

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• Short-sale restrictions
• Bounds on asset holdings
• Transaction costs
• Sensitivity to E(Q) and C

( Perold 1984, Best & Grauer 1991, Best & Hlouskova 2008, Roman & Mitra 2009, Best 2010 )

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Issues

• 1. Single Period
• Now or Never
• Volatility is Risk NOT an

Opportunity

(Fernholz & Shay 1982, Luenberger 1998)

• 2. Moments
• First and Second only

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Multi-Period Markowitz Principles

( Proc. Roy. Soc., 2002 )

saction after tran t t at held asset i

• f
• f

shares

• f

Number t t at asset i

• f

unit

• ne
• f

Price

j th , j th ,

= → = →

j i j i

U S

t

1 2 M+1 M

……

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Multi-Period Markowitz Principles

,

U

,

U

,

S

, 1

S

, N

S

1 ,

U

, 1

U

, N

U

1 , 1

U

, N

U

, 1

U

1 , − M

U

M

U ,

1 , N

U

1 , 1 − M

U

1 , − M N

U

M

U ,

1 M N

U ,

M

U ,

M

U ,

1

M N

U ,

1 ,

S

M

S ,

1 , + M

S

1 , 1

S

M

S ,

1 1 , 1 + M

S

1 , N

S

M N

S ,

1 , + M N

S

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t t =

,

U

,

S

, 1

U

, 1

S

, N

U

, N

S

) ( W

, , , 1 , 1 , ,

... ) (

N N S

U S U S U W + + + =

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1

t t =

,

U

1 ,

S

1 ,

U

, 1

U

1 , 1

U

1 , 1

S

, N

U

1 , N

U

1 , N

S

∑ ∑

= =

=

N i i i N i i i

S U S U

1 , 1 , 1 , ,

1 ,

S

1 , 1

S

1 , N

S

• Instant Transaction
• Frictionless

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T t t

M

= =

+1

M

U

,

1 , + M

S

M

U ,

1

) (T W

1 , 1 + M

S

M N

U

, 1 , + M N

S

∑ ∑ ∑ ∑

= + = = − =

= = = =

N i M i M i N i j i j i N i j i j i N i i i

S U T W M j S U S U S U W

1 , , , , , 1 , , ,

) ( , ... , 2 , 1 , ) ( Dynamics Wealth

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1 , − j

U

1 , 1 − j

U

1 , − j N

U

j

U

, j

U ,

1 j N

U

,

j

S

, j

S

, 1 j N

S

,

1 , , , −

− =

j j j

U U α

1 , 1 , 1 , 1 −

− =

j j j

U U α

1 , , , −

− =

j N j N j N

U U α

j

t t =

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Controls Stochastic M , ... 2, 1, j N, , ... 2, 1, i , Controls tic Determinis N , ... 2, 1, i , ) 2002 Soc., Roy. Proc. ( ) ( ) ( 1 M , ... 3, 2, k ) ( ) (

, , 1 1 1 , , , , , 1 , , , , , , , 1 , , , 1 , 1 , , 1 , 1

→ = = → =           − +           − + = + =           − + =

∑ ∑ ∑ ∑

− = = = = j i i k j N i j i j j i k k i N i i i k k i k k N i i i i

U S S S S U S S S S S t W S t W U S S S S S t W S t W α α

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Multi-Period Markowitz Principles

Processes ve Anticipati

• Non

Controls Stochastic M , ... 2, 1, j N, , ... 2, 1, i , Controls tic Determinis N , ... 2, 1, i , ) ( ) (

, , 1 1 , , , 1 , 1 , 1 , , , 1 , 1 , , 1 , 1

→ → = = → =           − +           − + =

∑ ∑ ∑

= = + + = + + + + j i i M j N i j i j j i M M i N i i i M M i M M

U S S S S U S S S S S t W S t W α α

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Parametrization of Stochastic Controls (Proc. Roy. Soc., 2008) determined be to knowns Un Parameters tic Determinis , , : Quadratic : Linear

, , , , 1 1 , , 1 , 1 , , , 1 , 1 , , , 1 , 1 , 1 , , , , , 1 , 1 , 1 , , , , , k l j i k j i j i N l l k k l j i j j k j j k j j l j j l N k k j i j j k j j k j i j i N k k j i j j k j j k j i j i

D B V D S S S S S S S S B S S S S V B S S S S V

∑∑ ∑ ∑

= = − − − − = − − = − −

          −           − +           − + =           − + = α α

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Multi-Period Markowitz Principles [ ][ ]

{ }

moments

• rder

higher and

• rder

Second moments

• rder

higher as well as Means ) ( ) ( ) ( ) ( ) ( ) ( ) ( E B , V , U Unknowns,

• f

vector tic Determinis terms quadratic & linear Prices, Asset

• f

vector Stochastic ) ( ) (

1 , 1 , 1 , 1 k j i, j i, i,0 , 1 , 1

→ → − − = =           + =           → → + =

+ + + + + +

C Q E Q E Q Q E Q E C A C A S t W Var A Q E S t W S t W A Q A Q S t W S t W

T T M M T M M T M M

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Markowitz Problem

) ( ) ( ) ( ) ( ) ( ) ( E ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( E subject to 2 1 ) ( 2 1 minimizes A that Find

1 , 1 , 1 , 1 , 1 1 1 , 1 , 1 1 , 1 , 1 , 1 1 , 1

Q E C Q E S W S t W Std S W S t W Q E C Q E C Q E S W S t W A S t W A Q E S W S t W A C A S t W Var

T M M M M T M G M

• pt

M G M T M M T M M − + + + + − − + + + + + + + +

          − =           −       − = = + =           =          

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With Non-Negative Wealth Constraints

∑ ∑ ∑ ∑

− = = = =

          − +           − + = + =           − + =

1 1 , 1 , , , , , 1 , , , , , , , 1 , , 1 , 1 , , 1 , 1

) ( ) ( 1 , ... , 3 , 2 ) ( ) (

k j j i N i j j i k k i i N i i k k i k k i N i i i

S S S S U S S S S S t W S t W M k U S S S S S t W S t W α

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With Non-Negative Wealth Constraints

es inequaliti Stochastic , ... , 2 , 1 , ) ( and ) ( ) ( E subject to ) ( Minimize

, 1 , 1 1 , 1 1 , 1

• =

≥ =                    

+ + + + + +

M k S t W S t W S t W S t W Var

k k M G M M M M M

Want:

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How do we model

, ) 3 ( ) ( Minimize ) 2 ( ) ( ) 1 ( by replaced is ) ( ) ( ) ( ) (

1 1 , 1 1 1 1 , 1 1 1 , 1 , 1 , 1 1 , 1 , , 1 , 1 , , 1 , 1 , , 1 , 1 , , 1 , 1

≥ ≥ = ≥           + −           − + = ≥           − + =

∑ ∑ ∑

= = = i i i N i i N i i i i N i i i

Var E S t W S S U S S S S S t W U S S S S S t W S t W η η φ φ η η φ

?

Stochastic Surplus Variable

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With Constraints

m j i m k j i j i i m m T m T m m T M M

B V U A Q M m A S t W A Q S t W S t W

, , , , , , 1 , 1

, Unknowns,

• f

Vector tic Determinis ) ( , , Unknowns,

• f

Vector tic Determinis terms quadratic and linear Prices, Asset

• f

Vectors , , , ... , 2 , 1 , ) ( ) ( ) ( η η η β ψ η β ψ φ → ≥ → → = + + = + =

+ +

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Objectives

{ }

ed unrestrict , (5) , ... , 2 , 1 , Minimize ) 4 ( , ... , 2 , 1 , ) ( ) 3 ( ) ( ) ( E ) 2 ( ) ( Minimize (1)

1 , 1 1 , 1 1 , 1

A M m Var M m E S t W S t W S t W Var

m m M G M M M M M

≥ = = = =                    

+ + + + + +

η φ φ

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Markowitz Problem { }

given ed unrestrict , ) 3 ( , ... , 2 , 1 , ) ( ) 2 ( ) ( ) ( E ) 1 ( subject to ) ( ) ( 2 1 ) F(A, Minimize

1 , 1 1 , 1 1 1 , 1 1

→ ≥ ≥ = = =                             + =

+ + + + = + + +

∑

i m M G M M M M i M M M i i

A M m E S t W S t W S t W Var Var θ η φ θ φ θ η

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Numerical Results

dt r S dS = Asset Free

• Risk

T = 5 Yr, N = 2 r = 5% 1 = 2 = 9% σ1 = σ2 = 40% dt dB dB E dB dt S dS dB dt S dS ρ σ µ σ µ = + = + = ) ( Assets Risky

2 1 2 2 2 2 2 1 1 1 1 1

S0(0) = 1 S1(0) = 10 S2(0) = 10 θi = 1, i = 1, 2, … , M+1

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50 100 150 200 250 300 350 100 110 120 130 140 150 160 170 Standard deviation of discounted terminal wealth Expected discounted terminal wealth Efficient frontier (ρ = 0.5) Without Constraints (M =6) With Constraints (M=6)

Dim(A) = 26 Dim(η) = 24

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50 100 150 200 250 300 350 400 100 110 120 130 140 150 160 170 Standard deviation of discounted terminal wealth Expected discounted terminal wealth Efficient frontier (ρ = 0.5) Without Constraints (M =11) With Constraints (M=11)

Dim(A) = 56 Dim(η) = 99

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50 100 150 200 250 100 110 120 130 140 150 160 170 Standard deviation of discounted terminal wealth Expected discounted terminal wealth Efficient frontier (ρ = − 0.5) Without Constraints (M =6) With Constraints (M=6)

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50 100 150 200 250 100 110 120 130 140 150 160 170 Standard deviation of discounted terminal wealth Expected discounted terminal wealth Efficient frontier (ρ = − 0.5) Without Constraints (M =11) With Constraints (M=11)

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50 100 150 200 250 300 100 110 120 130 140 150 160 170 Standard deviation of discounted terminal wealth Expected discounted terminal wealth Efficient frontier (ρ = 0.0) Without Constraints (M =6) With Constraints (M=6)

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50 100 150 200 250 300 100 110 120 130 140 150 160 170 Standard deviation of discounted terminal wealth Expected discounted terminal wealth Efficient frontier (ρ = 0.0) Without Constraints (M =11) With Constraints (M=11)

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Conclusion

• Moments of Any Order
• Only Moments – No Assumption on Dynamics
• Few Rebalancings = 90% Continuous-trading

( Proc.Roy.Soc., 2008 )

• Estimation of Moments ?
• Transaction Costs, Consumption ?

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Thank you