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Fi8000 Optimal Risky Portfolios

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Fi8000OptimalRisky Portfolios

Milind Shrikhande

- 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)

A

A

Lend

B

Borrow

C

rf

rf

- 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:

E(R)

Long A and Short B

Long A and Long B

A

Short A and Long B

B

STD(R)

- 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)

A

A

Reduction in portfolio risk

B

Increase of portfolio Risk

C

rf

rf

E(R)

A

Minimum Variance is zero

Pmin

B

STD(R)

What is the minimum variance portfolio if we assume that

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

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

E(R)

A

ρAB=(-1)

-1<ρAB<1

ρAB=+1

B

STD(R)

E(R)

A

C

B

STD(R)

E(R)

A

C

B

STD(R)

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.

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

E(R)

The Frontier

i

STD(R)

E(R)

The Efficient Frontier

i

STD(R)

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}

Optimization:

Find the minimum variance portfolio for a given expected return

Constraints:

A given expected return;

The budget constraint.

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.

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

E(R)

The Frontier

i

STD(R)

E(R)

The Frontier

i

rf

STD(R)

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

μ

m

i

rf

σ

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)

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}

Optimization:

Find the minimum variance portfolio for a given expected return

Constraints:

A given expected return;

The budget constraint.

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%

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

Mathematics of Portfolio Theory:

Read and practice parts 11-13.