Thuyết trình môn kinh tế lượng optimal risky portfolio

OPTIMAL RISKY PORTFOLIO

Giảng viên hướng dẫn: TS. Trần Thị Hải Lý
Nhóm 02

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Group members

Investment
Nguyễn Thành Trung
Vũ Thanh Tùng

Phạm Minh Tuấn

Chapter
Chapter 77

Đinh Xuân Minh
Nguyễn Thành Việt

www.themegallery.com

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Contents

1

Diversification and Portfolio Risk

2

3

Portfolios of Two Risky Assets

Asset Allocation with Stocks, Bonds and Bills

The Markowitz Portfolio Optimization Model

4

5

Risk Pooling, Risk Sharing and the Risk of Longterm Investment

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The Investment Decision

Top-down process with 3 steps:





Capital allocation between the risky portfolio and risk-free asset
Asset allocation across broad asset classes
Security selection of individual assets within each asset class

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Diversification and Portfolio Risk



Market risk





Systematic or nondiversifiable

Unique risk




Diversifiable or nonsystematic
Unique and firm-specific

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Diversification and Portfolio Risk

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Diversification and Portfolio Risk

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Portfolios of Two Risky Assets

 The rate of return on portfolio:
r p = w D. r D + w E. r E

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Portfolios of Two Risky Assets



Covariance and Correlation

Portfolio risk depends on the correlation between the returns of the assets in the portfolio
Covariance and the correlation coefficient provide a measure of the way returns of two assets
vary

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Portfolios of Two Risky Assets



The Expected Return of Two-Security Portfolio:

E (rp ) = wD E (rD ) + wE E (rE )
wr

+ wEr E

rp

=

rP

= Portfolio Return

D

D

wD = Bond Weight
rD

= Bond Return

wE = Equity Weight
rE
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= Equity Return
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Portfolios of Two Risky Assets



The Risk of Two-Security Portfolio:

σ = w σ + w σ + 2wD wE Cov( rD , rE )
2
p

2
D

2
D

Cov( rD , rE )

2
E

2
E

Covariance of returns for
Security D and Security E

σ P2 = wD wDCov(rD , rD ) + wE wE Cov (rE , rE ) + 2wD wE Cov (rD , rE )
Cov(rD,rE) = ρ DEσ Dσ E
ρ DE : Correlation coefficient of returns

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Portfolios of Two Risky Assets



Correlation Coefficients: Possible Values

- 1.0 ≤ ρ ≤ +1.0



When ρDE = 1, there is no diversification

σ P = wEσ E + wDσ D


When ρDE = -1, a perfect hedge is possible

σD
wE =
= 1 − wD
σ D +σ E
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Portfolios of Two Risky Assets

0.72
0.3

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Portfolios of Two Risky Assets

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Portfolios of Two Risky Assets



Concept: Portfolio of Three Risky Asset

E (rp ) = w1 E (r1 ) + w2 E (r2 ) + w3 E (r3 )
σ p2 = w12σ 12 + w22σ 22 + w32σ 32
+ 2w1w2σ 1, 2 + 2 w1w3σ 1,3 + 2 w2 w3σ 2,3

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Portfolios of Two Risky Assets

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Portfolios of Two Risky Assets

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Portfolios of Two Risky Assets



The minimum variance portfolio is the portfolio composed of the risky assets that has the
smallest standard deviation, the portfolio with least risk.



The minimum-variance portfolio has a standard deviation smaller than that of either of the
individual component assets.

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Portfolios of Two Risky Assets

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Portfolios of Two Risky Assets




The amount of possible risk reduction through diversification depends on the correlation.
The risk reduction potential increases as the correlation approaches -1.



If r = +1.0, no risk reduction is possible.



If r = 0, σP may be less than the standard deviation of either component asset.



If r = -1.0, a riskless hedge is possible.

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Portfolios of Two Risky Assets



The Sharpe Ratio

Maximize the slope of the CAL for any possible portfolio, P.
The objective function is the slope:

SP =
The slope is also the Sharpe ratio.

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E (rP ) − rf

σP

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Asset Allocation with Stocks, Bonds and Bills



Determining the weights associated with the optimal risky portfolio P (consisting of a stock
fund and bond fund)



Determining the optimal proportion of the complete portfolio (consisting of an investment
in the optimal risky Portfolio P and one in a risk free component (T-Bills)) to invest in the
risky component

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Asset Allocation with Stocks, Bonds and Bills

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Asset Allocation with Stocks, Bonds and Bills

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Asset Allocation with Stocks, Bonds and Bills

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