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Fi8000 Optimal Risky Portfolios. Milind Shrikhande. Investment Strategies. Lending vs. Borrowing (risk-free asset) Lending: a positive proportion is invested in the risk-free asset (cash outflow in the present: CF 0 < 0, and cash inflow in the future: CF 1 > 0)

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Fi8000 Optimal Risky Portfolios

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Fi8000 optimal risky portfolios l.jpg

Fi8000OptimalRisky Portfolios

Milind Shrikhande


Investment strategies l.jpg

Investment Strategies

  • Lending vs. Borrowing (risk-free asset)

    • Lending: a positive proportion is invested in the risk-free asset (cash outflow in the present: CF0 < 0, and cash inflow in the future: CF1 > 0)

    • Borrowing: a negative proportion is invested in the risk-free asset (cash inflow in the present: CF0 > 0, and cash outflow in the future: CF1 < 0)


Lending vs borrowing l.jpg

Lending vs. Borrowing

A

A

Lend

B

Borrow

C

rf

rf


Investment strategies4 l.jpg

Investment Strategies

  • A Long vs. Short position in the risky asset

    • Long:

      A positive proportion is invested in the risky asset (cash outflow in the present: CF0 < 0, and cash inflow in the future: CF1 > 0)

    • Short:

      A negative proportion is invested in the risky asset (cash inflow in the present: CF0 > 0, and cash outflow in the future: CF1 < 0)


Long vs short l.jpg

Long vs. Short

E(R)

Long A and Short B

Long A and Long B

A

Short A and Long B

B

STD(R)


Investment strategies6 l.jpg

Investment Strategies

  • Passive risk reduction:

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

  • The perfect hedge:

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

  • Diversification:

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

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


One risky fund and one risk free asset passive risk reduction l.jpg

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

A

A

Reduction in portfolio risk

B

Increase of portfolio Risk

C

rf

rf


Two risky assets with ab 1 the perfect hedge l.jpg

Two Risky Assets with ρAB=(-1):The Perfect Hedge

E(R)

A

Minimum Variance is zero

Pmin

B

STD(R)


The perfect hedge an example l.jpg

The Perfect Hedge – an Example

What is the minimum variance portfolio if we assume that

μA=10%; μB=5%; σA=12%; σB=6% andρAB=(-1)?


The perfect hedge continued l.jpg

The Perfect Hedge – Continued

What is the expected return μmin and the standard deviation of the return σmin of that portfolio?


Diversification the correlation coefficient and the frontier l.jpg

Diversification: the Correlation Coefficient and the Frontier

E(R)

A

ρAB=(-1)

-1<ρAB<1

ρAB=+1

B

STD(R)


Diversification the number of risky assets and the frontier l.jpg

Diversification: the Number of Risky assets and the Frontier

E(R)

A

C

B

STD(R)


Diversification the number of risky assets and the frontier13 l.jpg

Diversification: the Number of Risky assets and the Frontier

E(R)

A

C

B

STD(R)


Capital allocation n risky assets l.jpg

Capital Allocation:n Risky Assets

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

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

Present your results in the μ-σ (mean – standard-deviation) plane.


The expected return and the variance of the return of the portfolio l.jpg

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

wi = the proportion invested in the risky asset i (i=1,…n)

p = the portfolio of n risky assets (wiinvested in asset i)

Rp = the return of portfolio p

μp= the expected return of portfolio p

σ2p= the variance of the return of portfolio p


The set of possible portfolios in the plane l.jpg

The Set of Possible Portfoliosin the μ-σ Plane

E(R)

The Frontier

i

STD(R)


The set of efficient portfolios in the plane l.jpg

The Set of Efficient Portfoliosin the μ-σ Plane

E(R)

The Efficient Frontier

i

STD(R)


Capital allocation n risky assets18 l.jpg

Capital Allocation:n Risky Assets

The investment opportunity set:

{all the portfolios {w1, … wn} where Σwi=1}

The Mean-Variance (M-V or μ-σ ) efficient investment set:

{only portfolios on the efficient frontier}


The case of n risky assets finding a portfolio on the frontier l.jpg

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

Optimization:

Find the minimum variance portfolio for a given expected return

Constraints:

A given expected return;

The budget constraint.


The case of n risky assets finding a portfolio on the frontier20 l.jpg

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


Capital allocation n risky assets and a risk free asset l.jpg

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

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

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

Present your results in the μ-σ (mean – standard-deviation) plane.


The expected return and the variance of the return of the possible portfolios l.jpg

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

wi = the proportion invested in the risky asset i (i=1,…n)

p = the portfolio of n risky assets (wiinvested in asset i)

Rp = the return of portfolio p

μp= the expected return of portfolio p

σ2p= the variance of the return of portfolio p


The set of possible portfolios in the plane only n risky assets l.jpg

The Set of Possible Portfoliosin the μ-σ Plane (only n risky assets)

E(R)

The Frontier

i

STD(R)


The set of possible portfolios in the plane risk free asset included l.jpg

The Set of Possible Portfoliosin the μ-σ Plane(risk free asset included)

E(R)

The Frontier

i

rf

STD(R)


N risky assets and a risk free asset the separation theorem l.jpg

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

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

1.Find the Mean Variance efficient frontier

for the risky assets

2.Find the Capital Allocation Line with the

highest reward to risk ratio (slope) - CML


The set of efficient portfolios in the plane26 l.jpg

The Set of Efficient Portfoliosin the μ-σ Plane

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

μ

m

i

rf

σ


The separation theorem consequences l.jpg

The Separation Theorem: Consequences

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

1.Decide on the optimal portfolio of risky assets m

(the stage of risky security selection is identical for all the

investors)

2.Decide on the optimal allocation of funds between

the risky portfolio m and the risk-free asset rf –

choice of portfolio on the CML (the asset allocation stage is

personal, and it depends on the risk preferences of

the investor)


Capital allocation n risky assets and a risk free asset28 l.jpg

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

The investment opportunity set:

{all the portfolios {w0, w1, … wn} where Σwi=1}

The Mean-Variance (M-V or μ-σ ) efficient investment set:

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


N risky assets and one risk free asset finding a portfolio on the frontier l.jpg

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

Optimization:

Find the minimum variance portfolio for a given expected return

Constraints:

A given expected return;

The budget constraint.


N risky assets and one risk free asset finding the market portfolio l.jpg

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


N risky assets and one risk free asset finding the market portfolio31 l.jpg

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


A numeric example l.jpg

A Numeric Example

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

μA=10%; μB=5%; σA=12%; σB=6%; ρAB=(-0.5) and rf=4%


Example continued l.jpg

Example Continued


Example continued34 l.jpg

Example Continued


Practice problems l.jpg

Practice Problems

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

Mathematics of Portfolio Theory:

Read and practice parts 11-13.


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