Fi8000Fi8000OptimalOptimal

Risky PortfoliosRisky Portfolios

Milind ShrikhandeMilind Shrikhande

Investment StrategiesInvestment Strategies

☺Lending vs. Borrowing (risk-free asset)Lending vs. Borrowing (risk-free asset)

☺ Lending: a positive proportion is invested in Lending: a positive proportion is invested in the risk-free asset (cash outflow in the the risk-free asset (cash outflow in the present: CFpresent: CF0 0 < 0, and cash inflow in the < 0, and cash inflow in the

future: CFfuture: CF1 1 > 0)> 0)

☺ Borrowing: a negative proportion is invested Borrowing: a negative proportion is invested in the risk-free asset (cash inflow in the in the risk-free asset (cash inflow in the present: CFpresent: CF0 0 > 0, and cash outflow in the > 0, and cash outflow in the

future: CFfuture: CF1 1 < 0)< 0)

Lending vs. BorrowingLending vs. Borrowing

rfC

A

B

0.0%

2.0%

4.0%

6.0%

8.0%

0.0% 1.0% 2.0% 3.0% 4.0%

STD(R)

E(R)

rf

A

LendBorrow

Investment StrategiesInvestment Strategies

☺A Long vs. Short position in the risky assetA Long vs. Short position in the risky asset

☺ Long:Long:

A positive proportion is invested in the risky A positive proportion is invested in the risky asset (cash outflow in the present: CFasset (cash outflow in the present: CF0 0 < 0, < 0,

and cash inflow in the future: CFand cash inflow in the future: CF1 1 > 0)> 0)

☺ Short:Short:

A negative proportion is invested in the risky A negative proportion is invested in the risky asset (cash inflow in the present: CFasset (cash inflow in the present: CF0 0 > 0, and > 0, and

cash outflow in the future: CFcash outflow in the future: CF1 1 < 0)< 0)

Long vs. ShortLong vs. Short

STD(R)

E(R)

B

A

Long A and Short B

Long A and Long B

Short A and Long B

Investment StrategiesInvestment Strategies ☺Passive risk reduction:Passive risk reduction:

The risk of the portfolio is reduced if we invest a larger The risk of the portfolio is reduced if we invest a larger proportion in the risk-free asset relative to the risky oneproportion in the risk-free asset relative to the risky one

☺The perfect hedge:The perfect hedge:The risk of asset A is offset (can be reduced to zero) by The risk of asset A is offset (can be reduced to zero) by forming a portfolio with a risky asset B, such that forming a portfolio with a risky asset B, such that ρρABAB=(-1)=(-1)

☺Diversification:Diversification:

The risk is reduced if we form a portfolio of at least two The risk is reduced if we form a portfolio of at least two risky assets A and B, such that risky assets A and B, such that ρρABAB<(+1)<(+1)

The risk is reduced if we add more risky assets to our The risk is reduced if we add more risky assets to our portfolio, such that portfolio, such that ρρijij<(+1)<(+1)

One Risky Fund and one Risk-free One Risky Fund and one Risk-free Asset: Passive Risk ReductionAsset: Passive Risk Reduction

rfC

A

B

0.0%

2.0%

4.0%

6.0%

8.0%

0.0% 1.0% 2.0% 3.0% 4.0%

STD(R)

E(R)

rf

A

Reduction in portfolio risk Increase of

portfolio Risk

Two Risky Assets with Two Risky Assets with ρρABAB=(-1):=(-1):

The Perfect HedgeThe Perfect Hedge

STD(R)

E(R)

B

AMinimum Variance is zero

Pmin

The Perfect Hedge – an ExampleThe Perfect Hedge – an Example

What is the minimum variance portfolio if we What is the minimum variance portfolio if we assume thatassume that

μμAA=10%; =10%; μμBB=5%; =5%; σσAA=12%; =12%; σσBB=6%=6% and and ρρABAB=(-1)=(-1)??

2

2 2

2

2 2

2

(6%) (12%)(6%)( 1) 1

(12%) (6%) 2(12%)(6%)( 1) 3

B A B ABA

A B A B AB

w

The Perfect Hedge – ContinuedThe Perfect Hedge – ContinuedWhat is the expected return What is the expected return μμminmin and the standard and the standard

deviation of the return deviation of the return σσminmin of that portfolio? of that portfolio?

min

2 2 2 2min

2 22 2

(1 )

1 2 2 10% 5% 6 %

3 3 3

(1 ) 2 (1 )

1 2 1 2 (12%) (6%) 2 12% 6% ( 1)

3 3 3 3

0%

A A A B

A A A B A A A B AB

w w

w w w w

Diversification: the Correlation Diversification: the Correlation Coefficient and the FrontierCoefficient and the Frontier

STD(R)

E(R)

B

A

ρAB=+1-1<ρAB<1ρAB=(-1)

Diversification: the Number of Diversification: the Number of Risky assets and the FrontierRisky assets and the Frontier

STD(R)

E(R)

B

A

C

Diversification: the Number of Diversification: the Number of Risky assets and the FrontierRisky assets and the Frontier

STD(R)

E(R)

B

A

C

Capital Allocation:Capital Allocation:n Risky Assetsn Risky Assets

State all the possible investments – how State all the possible investments – how many possible investments are there?many possible investments are there?

Assuming you can use the Mean-Variance Assuming you can use the Mean-Variance (M-V) rule, which investments are M-V (M-V) rule, which investments are M-V efficient?efficient?

Present your results in the Present your results in the μμ--σσ (mean – (mean – standard-deviation) plane.standard-deviation) plane.

The Expected Return and the Variance of The Expected Return and the Variance of the Return of the Portfoliothe Return of the Portfolio

wwii = the proportion invested in the risky asset = the proportion invested in the risky asset ii ( (ii=1,…n)=1,…n)

p = the portfolio of n risky assets (p = the portfolio of n risky assets (wwii invested in asset invested in asset ii))

RRpp = the return of portfolio p = the return of portfolio p

μμpp = the expected return of portfolio p= the expected return of portfolio p

σσ22pp = the variance of the return of portfolio p= the variance of the return of portfolio p

1 1 2 21

1

2

1 1

...

( )

( )

n

p n n i ii

n

p p i iin n

p p i j iji j

R w R w R w R w R

E R w

V R w w

The Set of Possible PortfoliosThe Set of Possible Portfoliosin the in the μμ--σσ Plane Plane

STD(R)

E(R)

i

The Frontier

The Set of Efficient PortfoliosThe Set of Efficient Portfoliosin the in the μμ--σσ Plane Plane

STD(R)

E(R)

i

The Efficient Frontier

Capital Allocation:Capital Allocation: n Risky Assetsn Risky Assets

The investment opportunity set:The investment opportunity set:

{all the portfolios {{all the portfolios {ww11, … w, … wnn} where } where ΣΣwwii=1=1}}

The Mean-Variance (M-V or The Mean-Variance (M-V or μμ--σσ ) ) efficient investment set:efficient investment set:

{only portfolios on the efficient frontier}{only portfolios on the efficient frontier}

The case of n Risky Assets:The case of n Risky Assets:Finding a Portfolio on the FrontierFinding a Portfolio on the Frontier

Optimization:Optimization:

Find the minimum variance portfolio for a Find the minimum variance portfolio for a given expected returngiven expected return

Constraints:Constraints:

A given expected return;A given expected return;

The budget constraint.The budget constraint.

The case of n Risky Assets:The case of n Risky Assets:Finding a Portfolio on the FrontierFinding a Portfolio on the Frontier

1{ ,... }1 1

1

1

. .

1

n

n n

i j ijw w

i j

n

i i pi

n

ii

Min w w

S t w

w

Capital Allocation: n Risky Capital Allocation: n Risky Assets and a Risk-free AssetAssets and a Risk-free Asset

State all the possible investments – how State all the possible investments – how many possible investments are there?many possible investments are there?

Assuming you can use the Mean-Variance Assuming you can use the Mean-Variance (M-V) rule, which investments are M-V (M-V) rule, which investments are M-V efficient?efficient?

Present your results in the Present your results in the μμ--σσ (mean – (mean – standard-deviation) plane.standard-deviation) plane.

The Expected Return and the Variance of The Expected Return and the Variance of the Return of the Possible Portfoliosthe Return of the Possible Portfolios

wwii = the proportion invested in the risky asset = the proportion invested in the risky asset ii ( (ii=1,…n)=1,…n)

p = the portfolio of n risky assets (p = the portfolio of n risky assets (wwii invested in asset invested in asset ii))

RRpp = the return of portfolio p = the return of portfolio p

μμpp = the expected return of portfolio p= the expected return of portfolio p

σσ22pp = the variance of the return of portfolio p= the variance of the return of portfolio p

0 1 1 2 2 01

01

2

1 1

...

( )

( ) 0

n

p n n i ii

n

p p i ii

n n

p p i j iji j

R w rf w R w R w R w rf w R

E R w rf w

V R ww

The Set of Possible PortfoliosThe Set of Possible Portfoliosin the in the μμ--σσ Plane (only n risky assets) Plane (only n risky assets)

STD(R)

E(R)

i

The Frontier

The Set of Possible PortfoliosThe Set of Possible Portfolios

in the in the μμ--σσ Plane Plane (risk free asset included)(risk free asset included)

STD(R)

E(R)

i

rf

The Frontier

n Risky Assets and a Risk-free n Risky Assets and a Risk-free Asset: The Separation TheoremAsset: The Separation Theorem

The process of finding the set of Mean-The process of finding the set of Mean-Variance efficient portfolios can be separated Variance efficient portfolios can be separated into two stages:into two stages:

1.1. Find the Mean Variance efficient frontierFind the Mean Variance efficient frontierfor the risky assetsfor the risky assets

2.2. Find the Capital Allocation Line with theFind the Capital Allocation Line with thehighest reward to risk ratio (slope) - CMLhighest reward to risk ratio (slope) - CML

The Set of Efficient PortfoliosThe Set of Efficient Portfoliosin the in the μμ--σσ Plane Plane

σ

μ

i

The Capital Market Line: μp= rf + [(μm-rf) / σm]·σp

rf

m

The Separation Theorem: The Separation Theorem: ConsequencesConsequences

The asset allocation process of the risk-averse The asset allocation process of the risk-averse investors can be separated into two stages:investors can be separated into two stages:

1.Decide on the optimal portfolio of risky assets 1.Decide on the optimal portfolio of risky assets mm (the stage of risky security selection is identical for all the(the stage of risky security selection is identical for all the investors)investors)

2.Decide on the optimal allocation of funds between2.Decide on the optimal allocation of funds between the risky portfolio the risky portfolio mm and the risk-free asset and the risk-free asset rf –rf – choice of portfolio on the CML (the asset allocation stage ischoice of portfolio on the CML (the asset allocation stage is personal, and it depends on the risk preferences ofpersonal, and it depends on the risk preferences of the investor)the investor)

Capital Allocation: n Risky Capital Allocation: n Risky Assets and a Risk-free AssetAssets and a Risk-free Asset

The investment opportunity set:The investment opportunity set:

{all the portfolios {{all the portfolios {ww00, w, w11, … w, … wnn} where } where ΣΣwwii=1=1}}

The Mean-Variance (M-V or The Mean-Variance (M-V or μμ--σσ ) ) efficient investment set:efficient investment set:

{all the portfolios on the Capital Market Line - {all the portfolios on the Capital Market Line - CML}CML}

n Risky Assets and One Risk-free n Risky Assets and One Risk-free Asset: Asset: Finding a Portfolio on the FrontierFinding a Portfolio on the Frontier

Optimization:Optimization:

Find the minimum variance portfolio for a Find the minimum variance portfolio for a given expected returngiven expected return

Constraints:Constraints:

A given expected return;A given expected return;

The budget constraint.The budget constraint.

n Risky Assets and One Risk-free n Risky Assets and One Risk-free Asset: Asset: Finding the Market PortfolioFinding the Market Portfolio

1{ ,... }1 1

1 1

. . 1-

n

n n

i j ijw w

i j

n n

i i i pi i

Min ww

S t w w rf

n Risky Assets and One Risk-free n Risky Assets and One Risk-free Asset: Asset: Finding the Market PortfolioFinding the Market Portfolio

1

1 11 2 12 1 1

1 21 2 22 2 2

1 1 2 2

Solve the following system of equasions and

find the proportions { ,... } invested in the risky assets

...

......

...

Scale the pr

n

n n

n n

n n n nn n

w w

w w w rf

w w w rf

w w w rf

1

1 2

oportions: for 1,...

and { , ,... } is the market portfolio.

ii n

jj

n

wz i n

w

m z z z

A Numeric ExampleA Numeric ExampleFind the market portfolio if there are only two risky assets, Find the market portfolio if there are only two risky assets, A and B, and a risk-free asset rf.A and B, and a risk-free asset rf.

μμAA=10%; =10%; μμBB=5%; =5%; σσAA=12%; =12%; σσBB=6%; =6%; ρρABAB=(-0.5) and rf=4%=(-0.5) and rf=4%

2

To find the proportions { , } invested in assets A and B

use the system of equations for two risky assets:

Using our data we get two equations:

(12%)

A B

A AA B AB A

A BA B BB B

A

w w

w w rf

w w rf

w

2

(12%)(6%)( 0.5) 10% 4%

(12%)(6%)( 0.5) (6%) 5% 4%

B

A B

w

w w

Example ContinuedExample Continued

2

2

If we solve the two equations:

(12%) (12%)(6%)( 0.5) 10% 4%

(12%)(6%)( 0.5) (6%) 5% 4%

we get the proportions { , } {0.06481,0.09259}.

Now we

A B

A B

A B

w w

w w

w w

have to scale the proportions

0.064810.41176 and 0.58824.

0.06481 0.09259

and { , } {0.41176,0.58824} is the market portfolio.

AA B

A B

A B

wz z

w w

m z z

Example ContinuedExample Continued

2 2 2 2

2

The expected return of the market portfolio is

0.41176 10% 0.58824 5% 7.06%

The standard deviation of the return of the market portfolio is

2

(0.41176) (1

m A A B B

m A A B B A B A B AB

z z

z z z z

2 2 22%) (0.58824) (6%)

2 0.41176 0.58824 12% 6% (-0.5)

4.41%

Practice ProblemsPractice Problems

BKM Ch. 8: 1-7, 11-14BKM Ch. 8: 1-7, 11-14

Mathematics of Portfolio Theory:Mathematics of Portfolio Theory:

Read and practice parts Read and practice parts 11-1311-13..