(L10)Markowitz_ns

Primbs/Investment Science 1

Topic #10Markowitz Portfolio Theory

Reading: Luenberger Chapter 6, Sections 6 - 10


Markowitz Portfolio Theory

The Markowitz Model

The Two Fund Theorem

Inclusion of a Risk Free Asset

The One Fund Theorem

Markowitz’s Message

Solving the Optimization


Picture of Markowitz

r

For a given mean return, you would like to minimize your risk or the variance.

x

Minimum variance point for a given mean return


The Markowitz Model

Markowitz formulated the problem of being on the efficient frontier as an optimization.

Assume there are n risky assets with

Mean returns:nrrr ,,, 21

Covariances: ij for i,j=1,...,n


Markowitz Optimization

Minimize:

n

jiijjiww

1,2

1 = 1/2 (Variance)

p

n

iii rrw

1

Subject to: = Mean Return

11

n

iiw = Weights sum to 1.

Note: (1) We are allowing short selling! (2) We assume all assets are risky!



The Markowitz Model







Solving a General Optimization),( uxf

11 ),( cuxg Min:

s.t.:

Step 1: Convert all constraints to zero on the right hand side.

22 ),( cuxg

),( uxfMin:

s.t.: 0),( 11 cuxg

0),( 22 cuxg



11 ),( cuxg Min:

s.t.:

Step 2: Associate a Lagrange multiplier with each constraint.

),( uxfMin:

s.t.:

22 ),( cuxg

0),( 11 cuxg

0),( 22 cuxg1

2



11 ),( cuxg Min:

s.t.:

Step 3: Form the Lagrangian by subtracting from the objective each constraint multiplied by its Lagrange multiplier.

),( uxfMin:

s.t.:

22 ),( cuxg

0),( 11 cuxg

0),( 22 cuxg1

2

)),(()),((),(),,,( 22211121 cuxgcuxguxfuxL



11 ),( cuxg Min:

s.t.:

Step 4: Compute the partial derivatives of the Lagrangian with respect to all its variables and set equal to zero.

22 ),( cuxg

022

11

x

g

x

g

x

f

x

L

022

11

u

g

u

g

u

f

u

L

0),( 111

cuxgL

0),( 222

cuxgL

)),(()),((),(),,,( 22211121 cuxgcuxguxfuxL



11 ),( cuxg Min:

s.t.:

Step 5: Solve these equations (for x, u, ) to find the optimal solution.

22 ),( cuxg

022

11

x

g

x

g

x

f

x

L

022

11

u

g

u

g

u

f

u

L

0),( 111

cuxgL

0),( 222

cuxgL



Minimize:

n

jiijjiww

1,2

1 = 1/2 (Variance)

p

n

iii rrw

1


11

n





Minimize:

n

jiijjiww

1,2

1

01

p

n

iii rrwSubject to:

011

n

iiw

Step 1



Minimize:

n

jiijjiww

1,2

1

01

p

n

iii rrwSubject to:

011

n

iiw

Step 2



Minimize:

n

jiijjiww

1,2

1

01

p

n

iii rrwSubject to:

011

n

iiw

Step 3

Form the Lagrangian:

n

ii

n

ipii

n

jiijji wrrwwwwL

111,

12

1),,(


Markowitz OptimizationStep 4

Form the Lagrangian:

n

ii

n

ipii

n

jiijji wrrwwwwL

111,

12

1),,(

Differentiate with respect to ,,iw

01

i

n

jjij rw for i=1,...,nwi

p

n

iii rrw

1

11

n

iiw

These equations characterize efficient funds.


Markowitz OptimizationStep 5

Differentiate with respect to ,,iw

01

i

n

jjij rw for i=1,...,nwi

p

n

iii rrw

1

11

n

iiw

These equations characterize efficient funds.

(n+2 equations, n+2 unknowns)Solve to obtain optimal weights.



The Markowitz Model







The Two-Fund Theorem

Theorem: Investors seeking minimum variance portfolios need only invest in combinations of two minimum variance funds.

Let’s make the following assumptions:

(1) Short selling is allowed.(2) All assets are risky.(3) All investors have the same estimates of

means, variances, and covariances.


Importance of the Two Fund Theorem

Only two efficient funds need to exist, and everyone can invest in them!

r x

xFund 1

Fund 2x

Another EfficientFund

It is just a portfolio of Fund 1 and Fund 2.


The Two-Fund TheoremTheorem: Investors seeking efficient portfolios need only

invest in combinations of two efficient funds.

Proof: Let ),,,( 11

211

1nwwww 111 ,, Pr

be efficient funds. Hence they satisfy the equations for anefficient fund on a previous slide.

),,,( 222

21

2nwwww 222 ,, Prand

022

1

2

i

n

jjij rw

2

1

2P

n

iii rrw

11

2

n

iiw

2:011

1

1

i

n

jjij rw

1

1

1P

n

iii rrw

11

1

n

iiw

1:



011

1

1

i

n

jjij rw

1

1

1P

n

iii rrw

11

1

n

iiw

022

1

2

i

n

jjij rw

2

1

2P

n

iii rrw

11

2

n

iiw

1: 2:

213 )1( www 213 )1(

213 )1(

Solve 321 )1( PPP rrr for and set:Now consider an efficient fund with mean return 3

Pr

Are these the optimal weights and Lagrange multipliers corresponding to ?3

Pr

033

1

3

i

n

jjij rw

3

1

3P

n

iii rrw

11

3

n

iiw

3:

Yes!...We need to show



011

1

1

i

n

jjij rw

1

1

1P

n

iii rrw

11

1

n

iiw

022

1

2

i

n

jjij rw

2

1

2P

n

iii rrw

11

2

n

iiw

1: 2:

033

1

3

i

n

jjij rw

3

1

3P

n

iii rrw

11

3

n

iiw

3:

213 )1( www 213 )1( 213 )1(

))1(())1(())1(( 21212

1

1

ij

n

jjij rww

22

1

211

1

1 )1( i

n

jjiji

n

jjij rwrw 0

33

1

3

i

n

jjij rw



011

1

1

i

n

jjij rw

1

1

1P

n

iii rrw

11

1

n

iiw

022

1

2

i

n

jjij rw

2

1

2P

n

iii rrw

11

2

n

iiw

1: 2:

033

1

3

i

n

jjij rw

3

1

3P

n

iii rrw

11

3

n

iiw

3:

213 )1( www 213 )1( 213 )1(

n

iiii rww

1

21 ))1((

n

iii

n

iii rwrw

1

2

1

1 )1( 321 )1( PPP rrr

n

iii rw

1

3



011

1

1

i

n

jjij rw

1

1

1P

n

iii rrw

11

1

n

iiw

022

1

2

i

n

jjij rw

2

1

2P

n

iii rrw

11

2

n

iiw

1: 2:

033

1

3

i

n

jjij rw

3

1

3P

n

iii rrw

11

3

n

iiw

3:

213 )1( www 213 )1( 213 )1(

n

iiw

1

3

1)1(

n

ii

n

ii ww

1

2

1

1 )1(

n

iii ww

1

21 )1(

efficient! is ),,( 333 w



Asset Return Std. Dev. Covariance Matrix 1 2 3 Lambda (sum to one)Mu (mean return)Min Variance1 10% 0.1581139 1 0.025 0 0 1 10% 02 25% 0.244949 2 0 0.06 0 1 25% 03 20% 0.1581139 3 0 0 0.025 1 20% 0

Constraint (weights sum to 1)1 1 1 0 0 1Constraint (mean return)10% 25% 20% 0 0 0.25

SolutionsMin Variance Mean Return = 25%

w1 0.413793103 -0.244898w2 0.172413793 0.5102041w3 0.413793103 0.7346939Lagrange -0.01034483 0.0306122Mu -0.244898Mean Return0.167241379 0.25Variance 0.010344828 0.0306122Std. Dev. 0.101709526 0.1749636

Efficient Frontier 1 and 2Weight on Min VarianceWeight on mean returnw1 w2 w3 Mean ReturnVariance Std. Dev. Mean ReturnVariance

-1 2 -0.903589 0.84799437 1.055595 0.33276 0.091414 0.302348 0.4 0.265-0.9 1.9 -0.83772 0.81421534 1.023505 0.32448 0.08351 0.288981 0.385 0.23685-0.8 1.8 -0.771851 0.78043631 0.991414 0.31621 0.076011 0.275701 0.37 0.2104-0.7 1.7 -0.705982 0.74665728 0.959324 0.30793 0.068918 0.262522 0.355 0.18565-0.6 1.6 -0.640113 0.71287825 0.927234 0.29966 0.062229 0.249458 0.34 0.1626-0.5 1.5 -0.574243 0.67909923 0.895144 0.29138 0.055947 0.23653 0.325 0.14125-0.4 1.4 -0.508374 0.6453202 0.863054 0.2831 0.050069 0.223761 0.31 0.1216-0.3 1.3 -0.442505 0.61154117 0.830964 0.27483 0.044597 0.211179 0.295 0.10365-0.2 1.2 -0.376636 0.57776214 0.798874 0.26655 0.03953 0.198821 0.28 0.0874-0.1 1.1 -0.310767 0.54398311 0.766784 0.25828 0.034868 0.186731 0.265 0.07285

0

0.05

0.1

0.15

0.2

0.25

0.3

0 0.05 0.1 0.15 0.2 0.25 0.3

Standard Deviation

Mea

n R

etu

rn

EfficientFrontier1 and 2

1 and 3

2 and 3

Asset 1

Asset 2

Asset 3



The Markowitz Model







Inclusion of a Risk-Free Asset

We have assumed that all the assets are risky.

Now assume there exists a risk-free asset withreturn rf.

r

rf


Inclusion of a Risk-Free AssetWhat happens when we combine the risk free assetwith a risky portfolio

risk free: )0,( fr

risky asset: ),( 2r

Let’s form a portfolio consisting of of the risk free asset and of the risk asset:

mean: rrf )1(

variance: 22)1(

standard deviation: )1(



For this portfolio we have

(mean, standard deviation)= ))1(,)1(( rrf

As we vary , this maps out a straight line

r

rf

x),( 2r


Expanded Feasible Region

r

rf

x Tangent to the feasible region of risky funds.F



The Markowitz Model







The One-Fund Theorem

Theorem: There is a single fund F of risky assets such that any efficient portfolio can be constructed as a combination of the fund F and the risk-free asset.

Let’s make the following assumptions:

(1) Short selling is allowed.(2) There is a risk free asset.(3) All investors have the same estimates of

means, variances, and covariances.




r

rf

x Tangent to the feasible region of risky portfolios.F


Computation of the One-Fund

r

rf


The one-fund is the fund of risky assets that results in the maximum slope with the risk-free rate.

Maximize this slope

Fund

fFund rrslope


Computation of the One-FundThe one-fund is the fund of risky assets that results in the maximum slope with the risk-free rate.

Maximize this slope

Fund

fFund rrslope

2/1

1,

1

)(max

n

jiijji

f

n

iii

w

ww

rrw

i

Take derivative wrt. wk for k=1...n and set equal to zero:

0

)()(

1,

1

2/1

1,1

2/1

1,

n

jiijji

n

jkjj

n

jiijjif

n

iiifk

n

jiijji

ww

wwwrrwrrww



0

)()(

1,

1

2/1

1,1

2/1

1,

n

jiijji

n

jkjj

n

jiijjif

n

iiifk

n

jiijji

ww

wwwrrwrrww

0

)(

)(

1,

11

n

jiijji

n

jkjjf

n

iii

fk

ww

wrrw

rr

n

jiijji

f

n

iii

ww

rrw

1,

1

)(

Let 0)(1

n

jkjjfk wrr

for k=1,...,n



0)(1

n

jkjjfk wrr

Solve for j and normalize

n

ii

jj

v

vw

1

for k=1,...,n

)(1

fk

n

jkjj rrw

for k=1,...,n

jj vw Let:

These are the weights of the One-Fund!

)(1

fk

n

jkjj rrv

for k=1,...,n



The Markowitz Model







The Message of Markowitz

The Two-Fund and One-Fund theorems are important consequences of Markowitz.

Beyond these, Markowitz says that we can form optimal portfolios which take advantage of the correlations between assets.

A serious difficulty with this theory is that among n assets, there are n(n-1)/2 covariances. This is a lot! Just consider trying to compute this for the market, which has thousands of assets!

Assets are valuable as members of portfolios!!!


Problems


The two-fund theorem:

Consider a market with 3 risky assets.

Two efficient funds areA – (0.25,0.25,0.50) with expected return 10%B – (0.10,0.70,0.20) with expected return 20%

What are the weights on an efficient portfolio with mean return a) 15% b) 30%


The one-fund theorem:

Consider a market with risky assets and a risk free asset.

The risk free return is 5%The one-fund has mean 10% and standard deviation 15%

What is the standard deviation on an efficient portfolio with mean return

a) 30%

b) 10%


There are two assets in the market with means and covariances given above. Write the optimization problem for a portfolio with minimum variance subject to a mean return constraint of 18%. Write the necessary conditions for the solution. Solve them.

If the risk free rate is 5%, compute the one-fund.

1.01 r

2.02 r

3.011

4.022 01.012

Optimizations:


Appendix 1: Markowitz Theory using

Linear Algebra


Markowitz Portfolio TheoryMarkowitz Portfolio Theory(The Structure of Optimal(The Structure of Optimal Portfolios) Portfolios)

The Markowitz Model





Constrained Optimization

Derivatives with Lin. Alg.

Quadratic Opt. w/ Lin. Alg.


Picture of Markowitz

r

For a given mean return, you would like to minimize your risk or the variance.

x

Minimum variance point for a given mean return


The Markowitz ModelMarkowitz formulated the problem of being on the minimum variance set as an optimization.

Assume there are n risky assets with

Mean returns: nrrr ,,, 21

Covariances: ij for i,j=1,...,n

Returns: nrrr ,,, 21

nr

r

r

r2

1

nr

r

r

r2

1

nnnn

n

n

21

22221

11211



Minimize:

n

jiijjiww

1,2

1 = 1/2 (Variance)

p

n

iii rrw

1


11

n




Markowitz OptimizationLinear Algebra

Minimize: wwT2

1= 1/2 (Variance)

pT rrw Subject to: = Mean Return

11Tw = Weights sum to 1.




The Markowitz Model










0

duu

fdx

x

fdf

0

duu

gdx

x

gdg

dxx

g

u

gdu

1

),( uxf

cuxg ),(Min:

s.t.:

At a minimum, variations in f(x,u) are equal to zero.

However, only variations that preserve the constraint are allowed.

Solve for du in terms of dx.



0

duu

fdx

x

fdf

dxx

g

u

gdu

1

At a minimum, variations in f(x,u) are equal to zero.

However, only variations that preserve the constraint are allowed.

Substitute into df dxx

g

u

g

u

f

x

fdf

1

We are at a minimum if: (and constraint is satisfied)

01

x

g

u

g

u

f

x

f



Let’s rewrite this condition

1

u

g

u

f 0

x

g

x

f and

0

x

g

x

f and0

u

g

u

f These equations along with the constraint are the optimality conditions

A convenient way to get to these equations is through the Lagrangian...

We are at a minimum if: (and constraint is satisfied)

01

x

g

u

g

u

f

x

f


The Lagrangian

0

x

g

x

f

x

L

0

u

g

u

f

u

L

0),(

cuxgL

OptimalityConditions

),( uxf

cuxg ),(Min:

s.t.:

)),((),(),,( cuxguxfuxL Define the Lagrangian

Setting the partial of the Lagrangian equal to zero gives the correct optimality conditions



The Markowitz Model









Derivatives using Linear Algebra

AAxx )(

TTx axa )(

TTTTx AxAxAxx )(

TTx AAx )(

AxAxx TTx 2)( if A is symmetric (i.e. A=AT).

Note: I use the convention that derivatives (i.e. gradients) are row vectors. This means the chain rule works from left to right:

x

yAxAyAxy TTTT

x

)( BAxABx TTT )( BAxABx TTTT

Bxy Let: then



The Markowitz Model









A Constrained Quadratic Optimization

AxxT21

cxbT

Solve to find optimum.

Min:

s.t.:

)(),( 21 cxbAxxxL TT Define the Lagrangian:

0),( TT bAxx

x

L

0),(

cbxxL T

Take partials:

0 bAx

cxbT

transpose

Assume A is symmetric



The Markowitz Model









Markowitz OptimizationLinear Algebra

Minimize: wwT2

1= 1/2 (Variance)


11Tw = Weights sum to 1.


Markowitz OptimizationMinimize: wwT

2

1

0 pT rrwSubject to:

011 Tw

Lagrange Mult.

)11()(2

1),,( T

pTT wrrwwwwL Lagrangian:

Optimality Conditions

01

rww

LT

0

pT

T

rwrL

011

wL T

T


The Structure of Optimality

Optimality Conditions

01 rw

0 pT rwr

011 wT

1

0

001

00

1

PT

T r

w

r

r

Rewrite as:


Solve for optimal w (and )by:

Given pr

1

0

001

00

1

PT

T r

w

r

r

Solving the Markowitz Problem

Rewrite as:

1

0

001

00

11

PT

T rr

rw


The Two Fund TheoremLet (w1,1,1) and (w2,2,2) be Markowitz solutions corresponding to and , respectively. Then the solution to the Markowitz problem for is: (w1,1,1)= (w1,1,1)+(1-)(w1,1,1) where solves:

1Pr

3Pr

2Pr

213 )1( PPP rrr

1

0

001

00

13

1

3

3

3

PT

T rr

rw

Proof:

1

0

)1(

1

0

001

00

121

1

PPT

T rrr

r

2

2

2

1

1

12

1

1

1

)1(

1

0

001

00

1

)1(

1

0

001

00

1

ww

rr

r

rr

r

PT

TP

T

T


Importance of Two Fund Theorem

Only two efficient funds need to exist, and everyone can invest in them!

r x

xFund 1

Fund 2x

Another EfficientFund

It is just a portfolio of Fund 1 and Fund 2.

Theorem: Investors seeking efficient portfolios need only invest in combinations of two efficient funds.



The Markowitz Model










We have assumed that all the assets are risky.

Now assume there exists a risk-free asset withreturn rf.

r

rf


Inclusion of a Risk-Free AssetWhat happens when we combine the risk free assetwith a risky portfolio

risk free: )0,( fr

risky asset: ),( 2r

Let’s form a portfolio consisting of of the risk-free asset and of the risky asset:

mean: rrf )1(

variance: 22)1(

standard deviation )1(



For this portfolio we have

(mean, standard deviation)= ))1(,)1(( rrf

As we vary , this maps out a straight line

r

rf

x),( 2r


Expanded Feasible Region

r

rf




The Markowitz Model











r

rf

x Tangent to the feasible region of risky portfolios.F


Derivation of the One-Fund Theorem

Minimize: wwT2

1= 1/2 (Variance)

pT

f rrwrw 0Subject to: = Mean Return

110 Tww = Weights sum to 1.

Let w be the weights on risky assets, and w0 the weight on the risk free asset.

110Tww



Minimize: wwT2

1

pfT

f rrrwr )1(Subject to:

))1((2

1),( pf

Tf

T rrrwrwwwL Lagrangian:

0)1(

f

T

rrww

L

0)1(

pT

ff

T

rwrrrL


Lagrange Mult.



0)1(

f

T

rrww

L

0)1(

pT

ff

T

rwrrrL


But these weights won’t sum to 1. So normalize to sum to 1.

))1(())1((1

1))1((

))1((1

1 11

11

rrrr

rrrr

w ff

Tff

T

This does not depend on . This is the one-fund of risky assets.Pr

)1(1 rrw f


Appendix 2: When the 1-Fund and 2-Fund

Theorems Hold


When the two fund theorem holds

Short selling allowed

No short selling allowed


Yes!

No!


No Short Selling

Minimize: wwT2

1= 1/2 (Variance)


11Tw = Weights sum to 1

0iw = No short selling


No Short Selling

i

iiT

pTT wwrrwwwwL )11()(

2

1),,(

With the inequality constraint, we must use the Kuhn-Tucker conditions.

01

ii

T

rww

L

pT rwr

11 wT

0iw

0i0iiw

This condition is not linear in ),( ii w


When the One-Fund Theorem Holds

Short Sell

No Short Sell

Short Sell No Short SellRisky Assets

Risk Free Asset:

Yes No

Yes No


Pictures: Short Selling of Risk Free Allowed

r

rf

x Tangent to the feasible region of risky portfolios.(Doesn’t matter if this is under short selling or noshort selling of risky assets!)

F

Efficient Frontier


When the One-Fund Theorem Holds

Short Sell

No Short Sell

Short Sell No Short SellRisky Assets

Risk Free Asset:

Yes No

Yes No


Pictures: Short Selling of Risk Free Not Allowed

r

rf

xF

Efficient Frontier

Tangent to the feasible region of risky portfolios.(Doesn’t matter if this is under short selling or noshort selling of risky assets!)


General Optimization

Minimize: wwT2

1= 1/2 (Variance)

pfT

f rrrwr )1(Subject to: = Mean Return

0iw = No Short Risky

011 wT = No Short Risk-free


First Order Optimality Conditions

01)1( frrww

L

0)1( pfT

f rrrwr

0iw

011 wT

0i

0

0iiw

0)11( wT


Allow Shorting of the Risk-Free

0iw

011 wT

0i

0

0iiw

0)11( wT

Remove the restriction on no shorting of the risk free asset.Keep restriction on shorting of risky assets.

01)1( frrww

L

0)1( pfT

f rrrwr


One-Fund Theorem Holds!

0iw 0i 0iiw

Note that if (w,rP-rf) is a solution, then so is a(w,rP-rf) for any .0a

Hence, a one-fund theorem holds in this case!

(Think about the picture and this should be clear!)

0)1( frrww

L

0)1( pfT

f rrrwr


Allow Shorting of Risky

0iw

011 wT

0i

0

0iiw

0)11( wT

Remove the restriction on no shorting of the risky assets.Keep the restriction on shorting of risk free asset.

01)1( frrww

L

0)1( pfT

f rrrwr


One-Fund Theorem Doesn’t Hold!

011 wT 0 0)11( wT

A one-fund theorem cannot hold because of the (1-1Tw) term!

Again, this should be clear from the picture. Not allowing shorting of risky assets and the risk free asset suffers from the same problem.

01)1( frrww

L

0)1( pfT

f rrrwr


Appendix 3:Lagrange Multipliers and the

Objective Function



),( uxf

cuxg ),( 0),( cuxg

),( uxf

0

duu

fdx

x

fdf

0

dcduu

gdx

x

gdg

dxx

gdcdu

u

g

dxx

gdc

u

gdu

1

Min:

s.t.:

Optimality Condition

Solve for du

Why is the derivative of the objective with respect to the constraint?



dxx

gdc

u

g

u

fdx

x

fdf

1

dcu

g

u

fdx

x

g

u

g

u

f

x

fdf

11

So:

0

u

g

u

f 1

u

g

u

f

dcdxx

g

x

fdf

But:

dc

df

(L10)Markowitz_ns

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