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Optimal Risky Portfolios- Asset AllocationsPowerPoint Presentation

Optimal Risky Portfolios- Asset Allocations

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### CHAPTER 8 Correlated and Uncorrelated Universes

Asset Allocation Discussion

- Idea
- from bank account to diversified portfolio
- principles are the same for any number of stocks

- A. bonds and stocks
- B. bills, bonds and stocks
- C. any number of risky assets

Bahattin Buyuksahin, JHU , Investment

Diversification and Portfolio Risk

- Market risk
- Systematic or nondiversifiable

- Firm-specific risk
- Diversifiable or nonsystematic

Bahattin Buyuksahin, JHU , Investment

Figure 7.1 Portfolio Risk as a Function of the Number of Stocks in the Portfolio

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Figure 7.2 Portfolio Diversification Stocks in the Portfolio

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Covariance and Correlation Stocks in the Portfolio

- 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 two assets vary

Bahattin Buyuksahin, JHU , Investment

Two-Security Portfolio: Return Stocks in the Portfolio

Bahattin Buyuksahin, JHU , Investment

= Variance of Security D Stocks in the Portfolio

= Variance of Security E

= Covariance of returns for

Security D and Security E

Two-Security Portfolio: RiskBahattin Buyuksahin, JHU , Investment

Two-Security Portfolio: Risk Continued Stocks in the Portfolio

- Another way to express variance of the portfolio:

Bahattin Buyuksahin, JHU , Investment

Covariance Stocks in the Portfolio

Cov(rD,rE) = DEDE

D,E = Correlation coefficient of

returns

D = Standard deviation of

returns for Security D

E = Standard deviation of

returns for Security E

Bahattin Buyuksahin, JHU , Investment

Correlation Coefficients: Possible Values Stocks in the Portfolio

Range of values for 1,2

+ 1.0 >r> -1.0

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

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

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Table 7.1 Descriptive Statistics for Two Mutual Funds Stocks in the Portfolio

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Three-Security Portfolio Stocks in the Portfolio

2p = w1212

+ w2212

+ w3232

+ 2w1w2

Cov(r1,r2)

Cov(r1,r3)

+ 2w1w3

+ 2w2w3

Cov(r2,r3)

Bahattin Buyuksahin, JHU , Investment

Asset Allocation Stocks in the Portfolio

- Portfolio of 2 risky assets (cont’d)
- examples
- BKM7 Tables 7.1 & 7.3
- BKM7 Figs. 7.3 (return), 7.4 (risk) & 7.5 (trade-off)
- portfolio opportunity set (BKM7 Fig. 7.5)

- examples
- minimum variance portfolio
- choose wD such that portfolio variance is lowest
- optimization problem

- minimum variance portfolio has less risk
- than either component (i.e., asset)

- choose wD such that portfolio variance is lowest

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Table 7.2 Computation of Portfolio Variance From the Covariance Matrix

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Table 7.3 Expected Return and Standard Deviation with Various Correlation Coefficients

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Figure 7.3 Portfolio Expected Return as a Function of Investment Proportions

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Figure 7.4 Portfolio Standard Deviation as a Function of Investment Proportions

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Minimum Variance Portfolio as Depicted Investment Proportionsin Figure 7.4

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Standard deviation is smaller than that of either of the individual component assets

Figure 7.3 and 7.4 combined demonstrate the relationship between portfolio risk

Figure 7.5 Portfolio Expected Return as a Function of Standard Deviation

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The relationship depends on the correlation coefficient Standard Deviation

-1.0 << +1.0

The smaller the correlation, the greater the risk reduction potential

If r = +1.0, no risk reduction is possible

Correlation EffectsBahattin Buyuksahin, JHU , Investment

Figure 7.6 The Opportunity Set of the Debt and Equity Funds and Two Feasible CALs

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The Sharpe Ratio and Two Feasible CALs

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Maximize the slope of the CAL for any possible portfolio, p

The objective function is the slope:

Figure 7.7 The Opportunity Set of the Debt and Equity Funds with the Optimal CAL and the Optimal Risky Portfolio

Bahattin Buyuksahin, JHU , Investment

Figure 7.8 Determination of the Optimal Overall Portfolio with the Optimal CAL and the Optimal Risky Portfolio

Bahattin Buyuksahin, JHU , Investment

Asset with the Optimal CAL and the Optimal Risky PortfolioAllocation

- Finding the optimal risky portfolio: II. Formally
- Intuitively
- BKM7 Figs. 7.6 and 7.7
- improve the reward-to-variability ratio
- optimal risky portfolio tangency point (Fig. 7.8)

- Intuitively
- Formally:

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Asset Allocation 18 with the Optimal CAL and the Optimal Risky Portfolio

- formally (continued)

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Asset Allocation 19 with the Optimal CAL and the Optimal Risky Portfolio

- Example (BKM7 Fig. 7.8)
- 1. plot D, E, riskless
- 2. compute optimal risky portfolio weights
- wD = Num/Den = 0.4; wE = 1- wD = 0.6

- 3. given investor risk aversion (A=4), compute w*
- bottom line: 25.61% in bills; 29.76% in bonds (0.7439x0.4); rest in stocks

Bahattin Buyuksahin, JHU , Investment

Figure 7.9 The Proportions of the Optimal Overall Portfolio with the Optimal CAL and the Optimal Risky Portfolio

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Markowitz Portfolio Selection Model with the Optimal CAL and the Optimal Risky Portfolio

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- Security Selection
- First step is to determine the risk-return opportunities available
- All portfolios that lie on the minimum-variance frontier from the global minimum-variance portfolio and upward provide the best risk-return combinations

Markowitz Portfolio Selection Model with the Optimal CAL and the Optimal Risky Portfolio

- Combining many risky assets & T-Bills
- basic idea remains unchanged
- 1. specify risk-return characteristics of securities
- find the efficient frontier (Markowitz)

- 2. find the optimal risk portfolio
- maximize reward-to-variability ratio

- 3. combine optimal risk portfolio & riskless asset
- capital allocation

- 1. specify risk-return characteristics of securities

- basic idea remains unchanged

Bahattin Buyuksahin, JHU , Investment

Markowitz Portfolio Selection Model with the Optimal CAL and the Optimal Risky Portfolio

- finding the efficient frontier
- definition
- set of portfolios with highest return for given risk
- minimum-variance frontier

- take as given the risk-return characteristics of securities
- estimated from historical data or forecasts
- n securities ->n return + n(n-1) var. & cov.

- use an optimization program
- to compute the efficient frontier (Markowitz)
- subject to same constraints

- definition

Bahattin Buyuksahin, JHU , Investment

Markowitz Portfolio Selection Model with the Optimal CAL and the Optimal Risky Portfolio

- Finding the efficient frontier (cont’d)
- optimization constraints
- portfolio weights sum up to 1
- no short sales, dividend yield, asset restrictions, …

- optimization constraints
- Individual assets vs. frontier portfolios
- BKM7 Fig. 7.10
- short sales -> not on the efficient frontier
- no short sales -> may be on the frontier
- example: highest return asset

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Figure 7.10 The Minimum-Variance Frontier of Risky Assets with the Optimal CAL and the Optimal Risky Portfolio

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Markowitz Portfolio Selection Model Continued with the Optimal CAL and the Optimal Risky Portfolio

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We now search for the CAL with the highest reward-to-variability ratio

Figure 7.11 The Efficient Frontier of Risky Assets with the Optimal CAL

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Markowitz Portfolio Selection Model Continued Optimal CAL

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Now the individual chooses the appropriate mix between the optimal risky portfolio P and T-bills as in Figure 7.8

Figure 7.12 The Efficient Portfolio Set Optimal CAL

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Capital Allocation and the Separation Property Optimal CAL

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- The separation property tells us that the portfolio choice problem may be separated into two independent tasks
- Determination of the optimal risky portfolio is purely technical
- Allocation of the complete portfolio to T-bills versus the risky portfolio depends on personal preference

Figure 7.13 Capital Allocation Lines with Various Portfolios from the Efficient Set

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The Power of Diversification from the Efficient Set

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- Remember:
- If we define the average variance and average covariance of the securities as:
- We can then express portfolio variance as:

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

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Risk Pooling, Risk Sharing and Risk in the Long Run Correlated and Uncorrelated Universes

Loss: payout = $100,000

p = .001

No Loss: payout = 0

1 − p = .999

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Consider the following:

Risk Pooling and the Insurance Principle Correlated and Uncorrelated Universes

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- Consider the variance of the portfolio:
- It seems that selling more policies causes risk to fall
- Flaw is similar to the idea that long-term stock investment is less risky

Risk Pooling and the Insurance Principle Continued Correlated and Uncorrelated Universes

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When we combine n uncorrelated insurance policies each with an expected profit of $ , both expected total profit and SD grow in direct proportion to n:

Risk Sharing Correlated and Uncorrelated Universes

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- What does explain the insurance business?
- Risk sharing or the distribution of a fixed amount of risk among many investors

An Asset Allocation Problem Correlated and Uncorrelated Universes

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An Asset Allocation Correlated and Uncorrelated UniversesProblem 2

- Perfect hedges (portfolio of 2 risky assets)
- perfectly positively correlated risky assets
- requires short sales

- perfectly negatively correlated risky assets

- perfectly positively correlated risky assets

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An Asset Allocation Correlated and Uncorrelated UniversesProblem 3

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Index Models

Factor Model Correlated and Uncorrelated Universes Implementation (simplify the estimation problem) look for a statistical relationship

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- Idea
- the same factor(s) drive all security returns

- do not look for equilibrium relationship
- between a security’s expected return
- and risk or expected market returns

- between realized stock return
- and realized market return

Factor Model 2 Correlated and Uncorrelated Universes

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- Formally
- stock return
- = expected stock return
- + unexpected impact of common (market) factors
- + unexpected impact of firm-specific factors

- stock return

Index Model Correlated and Uncorrelated Universes Index Model

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- Factor model
- problem
- what is the factor?

- problem

- solution
- market portfolio proxy
- S&P 500, Value Line Index, etc.

Reduces the number of inputs for diversification Correlated and Uncorrelated Universes

Easier for security analysts to specialize

Advantages of the Single Index ModelBahattin Buyuksahin, JHU , Investment

ß Correlated and Uncorrelated Universesi = index of a securities’ particular return to the factor

m = Unanticipated movement related to security returns

ei = Assumption: a broad market index like the S&P 500 is the common factor.

Single Factor ModelBahattin Buyuksahin, JHU , Investment

Single-Index Model Correlated and Uncorrelated Universes

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Regression Equation:

Expected return-beta relationship:

Single-Index Model Continued Correlated and Uncorrelated Universes

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- Risk and covariance:
- Total risk = Systematic risk + Firm-specific risk:
- Covariance = product of betas x market index risk:
- Correlation = product of correlations with the market index

Index Model and Diversification Correlated and Uncorrelated Universes

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- Portfolio’s variance:
- Variance of the equally weighted portfolio of firm-specific components:
- When n gets large, becomes negligible

Figure 8.1 The Variance of an Equally Weighted Portfolio with Risk Coefficient βp in the Single-Factor Economy

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Figure 8.2 Excess Returns on HP and S&P 500 April 2001 – March 2006

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Figure 8.3 Scatter Diagram of HP, the S&P 500, and the Security Characteristic Line (SCL) for HP

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Table 8.1 Excel Output: Regression Statistics for the SCL of Hewlett-Packard

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Figure 8.4 Excess Returns on Portfolio Assets Hewlett-Packard

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Alpha and Security Analysis Hewlett-Packard

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Macroeconomic analysis is used to estimate the risk premium and risk of the market index

Statistical analysis is used to estimate the beta coefficients of all securities and their residual variances, σ2 ( e i )

Developed from security analysis

Alpha and Security Analysis Continued Hewlett-Packard

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- The market-driven expected return is conditional on information common to all securities
- Security-specific expected return forecasts are derived from various security-valuation models
- The alpha value distills the incremental risk premium attributable to private information

- Helps determine whether security is a good or bad buy

Single-Index Model Input List Hewlett-Packard

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- Risk premium on the S&P 500 portfolio
- Estimate of the SD of the S&P 500 portfolio
- n sets of estimates of
- Beta coefficient
- Stock residual variances
- Alpha values

Optimal Risky Portfolio of the Single-Index Model Hewlett-Packard

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- Maximize the Sharpe ratio
- Expected return, SD, and Sharpe ratio:

Optimal Risky Portfolio of the Single-Index Model Continued Hewlett-Packard

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- Combination of:
- Active portfolio denoted by A
- Market-index portfolio, the (n+1)th asset which we call the passive portfolio and denote by M
- Modification of active portfolio position:
- When

The Information Ratio Hewlett-Packard

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The Sharpe ratio of an optimally constructed risky portfolio will exceed that of the index portfolio (the passive strategy):

Figure 8.5 Efficient Frontiers with the Index Model and Full-Covariance Matrix

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Table 8.2 Comparison of Portfolios from the Single-Index and Full-Covariance Models

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Index Model: Industry Practices Full-Covariance Models etc. Idea

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- Beta books
- Merrill Lynch
- monthly, S&P 500

- Merrill Lynch
- Value Line
- weekly, NYSE

- regression analysis

Index Model: Industry Practices 2 Full-Covariance Models adjusted beta exploiting alphas (Treynor-Black)

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- Example (Merrill Lynch differences, Table 8.3)
- total (not excess) returns
- slopes are identical
- smallness

- total (not excess) returns
- percentage price changes
- dividends?
- S&P 500

- beta = (2/3) estimated beta + (1/3) . 1
- sampling errors, convergence of new firms

Table 8.3 Merrill Lynch, Pierce, Fenner & Smith, Inc.: Market Sensitivity Statistics

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Table 8.4 Industry Betas and Adjustment Factors Market Sensitivity Statistics

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Using Index Models Market Sensitivity Statistics

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Using Index Models 2 Market Sensitivity Statistics

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Using Index Models 3 Market Sensitivity Statistics

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Using Index Models 4 Market Sensitivity Statistics

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