Portfolio Selection (PPT)

Markowitz: Portfolio Selection

FRM

Markowitz: Portfolio Selection 

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Markowitz provided a comprehensive theoretical framework for analysis of the investment portfolio Harry M. Markowitz, “Portfolio Selection,” The Journal of Finance, March, 1952, pp. 77 - 91

Markowitz: Portfolio Selection 

Portfolio of securities is an integrated whole, each security complementing the other

Markowitz: Portfolio Selection 

Consider both the characteristics of the individual securities and the relationships between those securities

Markowitz: Portfolio Selection 

Investors like return and dislike risk

Markowitz: Portfolio Selection 

Find the set of portfolios that  – 

 –   – 

Provides the minimum risk for every possible level of return The ‘efficient’ set Investor selects from the ‘efficient’ set the single portfolio that meets his/her needs

Markowitz: Portfolio Selection 

Maximize the expected return E(R)

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Minimize the variance V(R)

Markowitz: Portfolio Selection 

Expected return of a portfolio is the weighted sum of the expected return from each of those securties

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n

 E ( R ) =

∑

 xi ei

i =1

Markowitz: Portfolio Selection 

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To compute the variance of a portfolio, we need to know more than the variance of the individual investments We need to know the covariances

Markowitz: Portfolio Selection   

Xi is the proportion invested in the ith stock Vi is the variance of the ith stock Cij is the covariance between the ith and jth stocks

Markowitz: Portfolio Selection

n

V ( R ) =

∑

2  xi vi

i

cij = σ i σ  j ρ ij

n

+

n

∑∑

 xi x  j cij , i ≠  j

i =1  j =1

vi =

2 σi

Markowitz: Portfolio Selection 

Minimize n

σ( R) = 

n

∑

2  xi

σ +

∑∑

 xi x  j ρ ij σ i σ  j

i =1  j =1

i

Subject to

2 i

n

n

n

∑ x i =1

i

=1

 xi ≥ 0

∑ e  x

i i

i =1

(k is a minimum acceptable expected return)

≥ k 

Markowitz: Portfolio Selection 

Limitations  – 

Assumes that deviations both above and below the level of expected return are equally undesirable

Markowitz: Portfolio Selection 

Assumes that the only investment objectives are the acquisition of return and the avoidance of risk  – 

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Type of returns (dividends vs. capital gains) are important Timing of realization of income is important

Markowitz: Portfolio Selection 

Assumes that historical returns will be repeated in the future  – 

The decision is how long a time period to include in the data set is an important one