portfolio management ppt

Portfolio AnalysisPortfolio Analysis

Topic 13Topic 13

I. Markowitz Mean-Variance I. Markowitz Mean-Variance AnalysisAnalysis

A. Mean (Expost) vs.A. Mean (Expost) vs.Expected (Exante)Expected (Exante)

1. Mean (Expost) Returns are 1. Mean (Expost) Returns are statistically derived from historical statistically derived from historical observations.observations.

2. Expected (Exante) Returns are 2. Expected (Exante) Returns are statistically derived expected values statistically derived expected values from future estimates of observations.from future estimates of observations.

B. Expected Return of a B. Expected Return of a PortfolioPortfolio

1. The expected return of a portfolio is 1. The expected return of a portfolio is a weighted average of the expected a weighted average of the expected returns of its component securities, returns of its component securities, using relative market values as weights.using relative market values as weights.

Portfolio AnalysisPortfolio Analysis

Topic 13Topic 13

II. Diversification and MPTII. Diversification and MPT

A. The Dominance PrincipleA. The Dominance Principle

States that among all investments with States that among all investments with a given return, the one with the least a given return, the one with the least risk is desirable; or given the same level risk is desirable; or given the same level of risk, the one with the highest return is of risk, the one with the highest return is most desirable.most desirable.

Dominance Principle ExampleDominance Principle Example

SecuritySecurity E(RE(Rii)) ATWATW 7%7% 3%3%GACGAC 7%7% 4%4%YTCYTC 15%15% 15%15%FTRFTR 3%3% 3%3%HTCHTC 8%8% 12%12%

ATW dominates GACATW dominates GAC ATW dominates FTRATW dominates FTR

B. DiversificationB. Diversification

1. Normal Diversification1. Normal Diversification• This occurs when the investor combines This occurs when the investor combines

more than one (1) asset in a portfoliomore than one (1) asset in a portfolio

1 5 10 20 30

Risk

Systematic Risk

UnsystematicRisk

# of Assets

75% of Co.Total Risk

25% of Co.Total Risk

RiskRisk

Unsystematic RiskUnsystematic Risk• ... is that portion of an asset’s total risk ... is that portion of an asset’s total risk

which can be eliminated through which can be eliminated through diversificationdiversification

Systematic RiskSystematic Risk• ... is that risk which cannot be eliminated... is that risk which cannot be eliminated• Inherent in the marketplaceInherent in the marketplace

DiversificationDiversification

Superfluous or Naive DiversificationSuperfluous or Naive Diversification• Occurs when the investor diversifies in Occurs when the investor diversifies in

more than 20-30 assets. Diversification for more than 20-30 assets. Diversification for diversification’s sake.diversification’s sake.

• a. Results in difficulty in managing such a a. Results in difficulty in managing such a large portfoliolarge portfolio

• b. Increased costsb. Increased costs– Search and transactionSearch and transaction

3. Markowitz Diversification3. Markowitz Diversification

This type of diversification considers the This type of diversification considers the correlation between individual securities. correlation between individual securities. It is the combination of assets in a It is the combination of assets in a portfolio that are less then perfectly portfolio that are less then perfectly positively correlated.positively correlated.• a. The two asset case:a. The two asset case:

– Stk.Stk. AA Stk.Stk. BB– E(R)E(R) 5% 5% 15%15%– 10%10% 20%20%

3. Markowitz Diversification 3. Markowitz Diversification (continued)(continued)

Assume that the investor invests 50% of Assume that the investor invests 50% of capital stock in stock A and 50% in Bcapital stock in stock A and 50% in B• 1. Calculate E(R)1. Calculate E(R)

• E(RE(Rpp) = ) = xxii E(R E(Rii))

• E(RE(Rpp) =.5(.05) + .5(.15)) =.5(.05) + .5(.15)

• E(RE(Rpp) = .025 + .075) = .025 + .075

• E(RE(Rpp) = .10 or 10%) = .10 or 10%

i=1

n


2. Graphically2. Graphically

A

B

Portfolio AB

E(Rp)

15%

10%

5%

5 10 15 20 25


Portfolio Return of AB will always be on Portfolio Return of AB will always be on line AB depending on the relative line AB depending on the relative fractions invested in assets A and B.fractions invested in assets A and B.

3. Calculating the risk of the portfolio3. Calculating the risk of the portfolio• Consider 3 possible relationships between A Consider 3 possible relationships between A

and Band B– Perfect Positive CorrelationPerfect Positive Correlation– Zero CorrelationZero Correlation– Perfect Negative Correlation Perfect Negative Correlation

Perfect Positive CorrelationPerfect Positive Correlation

A and B returns vary in identical pattern. A and B returns vary in identical pattern. Hence, there is a linear risk-return Hence, there is a linear risk-return relationship between the two assets.relationship between the two assets.

Perfect Positive Correlation Perfect Positive Correlation (continued)(continued)

A

B

AB

E(Rp)

15%

5%

10 15 20 p

Perfect Positive Correlation Perfect Positive Correlation (continued)(continued)

Therefore, the risk of portfolio AB is simply Therefore, the risk of portfolio AB is simply the weighted value of the two assets’ the weighted value of the two assets’ ..• In this case:In this case:

pp = x = xAA22 AA

22 + x + xBB22 BB

22 + 2 x + 2 xAAxxBBAABBABAB

pp = .25(.10) = .25(.10)22+.25(.20)+.25(.20)22+2(.5)(.5)(.10)(.20)+2(.5)(.5)(.10)(.20)

pp = .15 or 15% = .15 or 15%

Zero CorrelationZero Correlation

A’s return is completely unrelated to B’s A’s return is completely unrelated to B’s return. With zero correlation, a return. With zero correlation, a substantial amount of risk reduction can substantial amount of risk reduction can be obtained through diversification.be obtained through diversification.

Zero Correlation (continued)Zero Correlation (continued)

A

B

AB

E(Rp)

15%

10%

5%

10 11.2 20 p

p = .25(.10)2+.25(.20)2

p 11.2%

Negative CorrelationNegative Correlation

A’s and B’s returns vary perfectly A’s and B’s returns vary perfectly inversely. The portfolio variance is inversely. The portfolio variance is always at the lowest risk level always at the lowest risk level regardless of proportions in each asset. regardless of proportions in each asset.

Negative Correlation (continued)Negative Correlation (continued)

A

B

AB

E(Rp)

15%

5%

5 10 20 p

10%

p = .25(.10)+.25(.20)+2(.5)(.10)(.20)(-1)

p = .05 or 5%

Markowitz DiversificationMarkowitz Diversification

Although there are no securities with Although there are no securities with perfectly negative correlation, almost all perfectly negative correlation, almost all assets are less than perfectly assets are less than perfectly correlated. Therefore, you can reduce correlated. Therefore, you can reduce total risk (total risk (pp) through diversification. If ) through diversification. If we consider many assets at various we consider many assets at various weights, we can generate the efficient weights, we can generate the efficient frontier.frontier.

Efficient Frontier GraphEfficient Frontier Graph

E(Rp)

p

M

EfficientFrontier

Efficient FrontierEfficient Frontier

The Efficient Frontier represents all the The Efficient Frontier represents all the dominant portfolios in risk/return space.dominant portfolios in risk/return space.

There is one portfolio (M) which can be There is one portfolio (M) which can be considered the market portfolio if we considered the market portfolio if we analyze all assets in the market. Hence, analyze all assets in the market. Hence, M would be a portfolio made up of assets M would be a portfolio made up of assets that correspond to the real relative that correspond to the real relative weights of each asset in the market.weights of each asset in the market.

Efficient Frontier (continued)Efficient Frontier (continued)

Assume you have 20 assets. With the Assume you have 20 assets. With the help of the computer, you can calculate help of the computer, you can calculate all possible portfolio combinations. The all possible portfolio combinations. The Efficient Frontier will consist of those Efficient Frontier will consist of those portfolios with the highest return given portfolios with the highest return given the same level of risk or minimum risk the same level of risk or minimum risk given the same return (Dominance given the same return (Dominance Rule)Rule)


4. Borrowing and lending investment 4. Borrowing and lending investment funds at R to expand the Efficient funds at R to expand the Efficient Frontier.Frontier.• a. We keep part of our funds in a saving a. We keep part of our funds in a saving

accountaccount– Lending, ORLending, OR

• b. We can borrow funds for a greater b. We can borrow funds for a greater investment in the market portfolioinvestment in the market portfolio


E(Rp)

p

M

EfficientFrontier

B

A

Lending

BorrowingCML

RF

Portfolio A: 80% of funds in RF, 20% of funds in MPortfolio B: 80% of funds borrowed to buy more of M, 100% or own funds to buy M


By using RBy using RFF, the Efficient Frontier is now , the Efficient Frontier is now

dominated by the capital market line dominated by the capital market line (CML). Each portfolio on the capital (CML). Each portfolio on the capital market line dominates all portfolios on market line dominates all portfolios on the Efficient Frontier at every point the Efficient Frontier at every point except M.except M.

5. The Portfolio Investment5. The Portfolio Investment

E(Rp)

p

M

EfficientFrontier

CML

RF

Mutual Fund Portfolioswith a cash position

Investors’ indifference curves are based on their degreeof risk aversion and investment objectives and goals.

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