portfolio analysis - efficient frontiers

8/8/2019 Portfolio Analysis - Efficient Frontiers

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Portfolio AnalysisPortfolio Analysis Efficient FrontierEfficient Frontier

Alok Kumar


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Efficient FrontierEfficient Frontier

Two Conditions

1) Offer Maximum Return for varying levels of Risk,and

2) Offer Minimum Risk for varying levels of expected

return

All the feasible sets are not efficient unless it passes throughthis test


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Efficient Sets and Feasible SetsEfficient Sets and Feasible Sets

Feasible Sets

A

D

C

B


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How to form Efficient Frontier ?How to form Efficient Frontier ?


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2 Stock Case2 Stock Case

Stocks Expected Return Standard Deviation

A 5% 20%

B 15% 40%


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FormulaFormula

Expected Return of Portfolio = Xiri,where i range from 0to n.

and X is Proportion of total investment in ith security and riis expected return of the security.

Standard deviation of Portfolio =( Xi Xj ij)1/2

where i and j vary from 0 to n, and ij is covariance of iand j securities.

ij = iji j,where i & j is standard deviation of i and jrespectively.


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Expected Return for PortfoliosExpected Return for Portfolios

Portfolios Proportion in X Proportion in Y Return

A 1 0 5.00%

B 0.83 0.17 6.70%

C 0.67 0.33 8.30%

D 0.5 0.5 10.00%

E 0.33 0.67 11.71%F 0.17 0.83 13.30%

G 0 1 15.00%


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Standard Deviation of PortfolioStandard Deviation of Portfolio

Portfolios Lower Bound Upper Bound No relationship

A 20.00% 20.00% 20.00%

B 10.00% 23.33% 17.94%

C 0.00% 26.67% 18.81%

D 10.00% 30.00% 22.36%

E 20.00% 33.33% 27.60%

F 30.00% 36.67% 33.37%

G 40.00% 40.00% 40.00%


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Efficient FrontierEfficient Frontier

Upper and Lower Bounds to Portfolios

0.00%

2.00%

4.00%

6.00%

8.00%

10.00%

12.00%

14.00%

16.00%

0.00% 5.00% 10.00% 15.00% 20.00% 25.00% 30.00% 35.00% 40.00% 45.00%

Standard Deviations

E

x

ected

Return


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Market

Models

Market

Models


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Market ModelMarket Model

ri = iI + iI rI + iI

Where, ri = return on security i for given period

iI = intercept form

iI = slope form

rI = return on market index I for the same period

iI =random error


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Graphical Presentation ofMarket ModelGraphical Presentation ofMarket Model

ri = iI + iI rI


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BetaBeta

iI = iI

I2

iI = Covariance

I2 = Variance of Market Index


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Random ErrorRandom Error

Security A Security B

Intercept 2% -1%

Actual Return

on the Market

index X beta

10% X 2% = 12% 10% X 8% = 8%

Actual Return

on Security

9% 11%

Random Error 9%- (2% + 12%)

= -5%

11%- (-1% +8%) =

4%


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Graphical Presentation ofMarket ModelGraphical Presentation ofMarket Model

Infotech versus S&P 500: 1992-1996

-6.00%

-4.00%

-2.00%

0.00%

2.00%

4.00%

6.00%

8.00%

-15.00% -10.00% -5.00% 0.00% 5.00% 10.00% 15.00% 20.00%


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Securitys Total RiskSecuritys Total Risk

i2 =iI

2X I2 + i

2

Where ,

i2= variance of security i

iI2

X I2

= Market risk of security ii

2= Unique risk of security i


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Portfolios ReturnPortfolios Return

rp = Xi riWhere i range from o to n. and

Xi = proportion of investment in security i.ri = expected return of security i.

Also,

ri = iI + iI rI + iI

Hence rp = Xi (iI + iI rI + iI)

.....continued


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.....continued.....continued

rp = Xi (iI + iI rI + iI)

= Xi iI + (Xi iI ) rI + XiiI

= pI + pI rI + pI

Where i range from o to n.

Intercept Slope X independent

VariableRandom Error


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Portfolio RiskPortfolio Risk

2p =2pI

2I +

2p

Where ,

2pI = [Xi iI] 2 ----- Systematic Risk

2p = Xi2 2i ----- Unique Risk


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Risk and DiversificationRisk and Diversification

Unique Risk Market Risk

Total Riskp

N


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CalculationsCalculations

Stock

Portfolio

Weight Beta

Expected Return of

Stock

Variance of

Stock

A 0.25 0.5 0.4 0.07

B 0.25 0.5 0.25 0.05

C 0.5 1 0.21 0.07

Variance of

Market 0.06


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QuestionsQuestions

Residual Variance of each of the stocks?

Beta of the portfolio?

Variance of the Portfolio?

Expected Return on the portfolio?

Portfolio Variance on teh basis of Markowitz Variance Covariance

formula.

Covariance (A,B) = 0.020

Covariance (A,C) = 0.035

Covariance (B,C) = 0.035

portfolio analysis - efficient frontiers

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