1 fin 2802, spring 10 - tang chapter 7: optimal investment portfolio fin 2802: investments spring,...

1FIN 2802, Spring 10 - TangChapter 7: Optimal Investment Portfolio

Fin 2802: Investments

Spring, 2010Dragon Tang

Fin 2802: Investments

Spring, 2010Dragon Tang

Lecture 18Optimal Investment Portfolio

March 30, 2010

Readings: Chapter 7Practice Problem Sets: 1-15, 17-21


Optimal Portfolio ChoiceOptimal Portfolio Choice

Objectives:

• Show how covariance and correlation affect the power of diversification

• Construct efficient portfolio

• Calculate the composition of the optimal risky portfolio

• Take risk wisely!


Diversification and Portfolio RiskDiversification and Portfolio Risk

• Market risk or beta risk

– Systematic or Nondiversifiable

• Firm-specific risk

– Diversifiable or nonsystematic


Portfolio Risk as a Function of the Number of Stocks

Portfolio Risk as a Function of the Number of Stocks


Portfolio Risk as a Function of Number of Securities

Portfolio Risk as a Function of Number of Securities


Two Asset Portfolio Return – Stock and BondTwo Asset Portfolio Return – Stock and Bond

ReturnStock

htStock Weig

Return Bond

WeightBond

Return Portfolio

rwrwr

S

S

B

B

p

rwrwr SSBBp


CovarianceCovariance

1,2 = Correlation coefficient of returns

1,2 = Correlation coefficient of returns

Cov(r1r2) = 12Cov(r1r2) = 12

1 = Standard deviation of returns for Security 12 = Standard deviation of returns for Security 2

1 = Standard deviation of returns for Security 12 = Standard deviation of returns for Security 2


Correlation Coefficients: Possible ValuesCorrelation Coefficients: Possible Values

If If = 1.0, the securities would be = 1.0, the securities would be perfectly positively correlatedperfectly positively correlated

If If = - 1.0, the securities would be = - 1.0, the securities would be perfectly negatively correlatedperfectly negatively correlated

Range of values for 1,2

-1.0 < < 1.0


Two Asset Portfolio St Dev – Stock and BondTwo Asset Portfolio St Dev – Stock and Bond

Deviation Standard Portfolio

Variance Portfolio

2

2

,

22222 2

p

p

SBBSSBSSBBp wwww


rp = Weighted average of the n securitiesrp = Weighted average of the n securities

p2 = (Consider all pair-wise

covariance measures)p

2 = (Consider all pair-wise covariance measures)

In General, For an n-Security Portfolio:In General, For an n-Security Portfolio:


Numerical Example: Bond and Stock

Numerical Example: Bond and StockReturns

Bond = 6% Stock = 10%

Standard Deviation

Bond = 12% Stock = 25%

Weights

Bond = .5 Stock = .5

Correlation Coefficient

(Bonds and Stock) = 0


Return and Risk for Example

Return and Risk for Example

Return = 8%

.5(6) + .5 (10)

Standard Deviation = 13.87%

[(.5)2 (12)2 + (.5)2 (25)2 + …

2 (.5) (.5) (12) (25) (0)] ½

[192.25] ½ = 13.87


Investment Opportunity Set for Stock and BondsInvestment Opportunity Set for Stock and Bonds


Investment Opportunity Set for Stock and Bonds with Various Correlations

Investment Opportunity Set for Stock and Bonds with Various Correlations

15

Table 7.1 Descriptive Statistics for Two Mutual FundsTable 7.1 Descriptive Statistics for Two Mutual Funds

FIN 2802, Spring 10 - TangChapter 7: Optimal Investment Portfolio

16

Table 7.3 Expected Return and Standard Deviation with Various Correlation Coefficients

Table 7.3 Expected Return and Standard Deviation with Various Correlation Coefficients


17

Figure 7.3 Portfolio Expected Return as a Function of Investment Proportions

Figure 7.3 Portfolio Expected Return as a Function of Investment Proportions


18

Figure 7.4 Portfolio Standard Deviation as a Function of Investment Proportions

Figure 7.4 Portfolio Standard Deviation as a Function of Investment Proportions


19

Figure 7.5 Portfolio Expected Return as a function of Standard Deviation

Figure 7.5 Portfolio Expected Return as a function of Standard Deviation


20

Table 7.4 Risk Reduction of Equally Weighted Portfolios in Correlated and Uncorrelated Universes

Table 7.4 Risk Reduction of Equally Weighted Portfolios in Correlated and Uncorrelated Universes



Portfolio SelectionPortfolio Selection

• Asset allocation

• Security selection

• These two are separable!


Asset AllocationAsset Allocation

• John Bogle: “Asset allocation accounts for 94% of the differences in pension fund performance”

• Identify investment opportunities (risk-return combinations)

• Choose the optimal combination according to investor’s risk attitude


Optimal Portfolio ConstructionOptimal Portfolio Construction

Step 1: Using available risky securities (stocks) to construct efficient frontier.

Step 2: Find the optimal risky portfolio using risk-free asset

Step 3: Now We have a risk-return tradeoff, choose your most favorable asset allocation

Step 4: Calculate optimal portfolio weights


Portfolios Constructed from Three Stocks A, B and CPortfolios Constructed from Three Stocks A, B and C


The Efficient Frontier of Risky Assets and Individual Assets

The Efficient Frontier of Risky Assets and Individual Assets


Optimal Capital Allocation Line for Bonds, Stocks and T-Bills

Optimal Capital Allocation Line for Bonds, Stocks and T-Bills


The Complete PortfolioThe Complete Portfolio


The Complete Portfolio – Solution to the Asset Allocation Problem

The Complete Portfolio – Solution to the Asset Allocation Problem


Discussion: Practical Portfolio RulesDiscussion: Practical Portfolio Rules

• Rule #1: do not be a amateur stock trader (don’t do it or do it full time!), choose to be a trader or investor first!

• Investment philosophy: define value! Be cost cautious!• Investment psychology: do not chicken out!

– Don’t get sentimental, history doesn’t matter– Stop loss and let your winner run– …

• Research, research, research!• Sector rotation, familiarity, estimation risk• Offense wins game, defense wins championship• Amateurs practice until they get it right, pros practice until

they can’t get it wrong.


Investor PersonalitiesInvestor Personalities

• Measured investors: Rich and greedy

• Reluctant investors: Rich and humble

• Competitive investors: Like to trade, which is hazardous

• Unprepared investors: Poor, greedy, and ignorant


Mistakes Investors MakeMistakes Investors Make

• Overconfident, underestimate market force

• Short-sighted, resulting in unnecessary transactions

• Mental accounting, do not see the big picture

• Can’t see “everyone is unique, just like everyone else”

• Disposition: holding on losers too long and selling winner too fast

• Averaging down in price rather than up in buying

• Buying on tips and rumors

• Speculating too heavily in options or futures wanting to get rich quick

• No investment strategy, or having one without persistence


SummarySummary

• Diversification

• Optimal risky portfolio and efficient frontier

• Allocation among risky and risk-free assets

• Next Class: Practical Portfolio Management

1 fin 2802, spring 10 - tang chapter 7: optimal investment portfolio fin 2802: investments spring,...

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