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Optimal Risky Portfolios- Asset Allocations. BKM Ch 7. Asset Allocation. Idea from bank account to diversified portfolio principles are the same for any number of stocks Discussion A. bonds and stocks B. bills, bonds and stocks C. any number of risky assets.

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Optimal risky portfolios asset allocations

Optimal Risky Portfolios- Asset Allocations

BKM Ch 7


Asset allocation

Asset Allocation

  • Idea

    • from bank account to diversified portfolio

    • principles are the same for any number of stocks

  • Discussion

    • A. bonds and stocks

    • B. bills, bonds and stocks

    • C. any number of risky assets

  • Bahattin Buyuksahin, JHU , Investment


    Diversification and portfolio risk

    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

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

    Bahattin Buyuksahin, JHU , Investment


    Figure 7 2 portfolio diversification

    Figure 7.2 Portfolio Diversification

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    Covariance and correlation

    Covariance and Correlation

    • 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

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    Two security portfolio return

    Two-Security Portfolio: Return

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    Two security portfolio risk

    = Variance of Security D

    = Variance of Security E

    = Covariance of returns for

    Security D and Security E

    Two-Security Portfolio: Risk

    Bahattin Buyuksahin, JHU , Investment


    Two security portfolio risk continued

    Two-Security Portfolio: Risk Continued

    • Another way to express variance of the portfolio:

    Bahattin Buyuksahin, JHU , Investment


    Covariance

    Covariance

    Cov(rD,rE) = DEDE

    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

    Correlation Coefficients: Possible Values

    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

    Bahattin Buyuksahin, JHU , Investment


    Table 7 1 descriptive statistics for two mutual funds

    Table 7.1 Descriptive Statistics for Two Mutual Funds

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    Three security portfolio

    Three-Security Portfolio

    2p = w1212

    + w2212

    + w3232

    + 2w1w2

    Cov(r1,r2)

    Cov(r1,r3)

    + 2w1w3

    + 2w2w3

    Cov(r2,r3)

    Bahattin Buyuksahin, JHU , Investment


    Asset allocation1

    Asset Allocation

    • 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)

    • 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)

    Bahattin Buyuksahin, JHU , Investment


    Table 7 2 computation of portfolio variance from the covariance matrix

    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

    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

    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

    Figure 7.4 Portfolio Standard Deviation as a Function of Investment Proportions

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    Minimum variance portfolio as depicted in figure 7 4

    Minimum Variance Portfolio as Depicted in 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

    Figure 7.5 Portfolio Expected Return as a Function of Standard Deviation

    Bahattin Buyuksahin, JHU , Investment


    Correlation effects

    The relationship depends on the correlation coefficient

    -1.0 << +1.0

    The smaller the correlation, the greater the risk reduction potential

    If r = +1.0, no risk reduction is possible

    Correlation Effects

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    Figure 7 6 the opportunity set of the debt and equity funds and two feasible cals

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

    Bahattin Buyuksahin, JHU , Investment


    The sharpe ratio

    The Sharpe Ratio

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

    The objective function is the slope:


    Optimal risky portfolios asset allocations

    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

    Figure 7.8 Determination of the Optimal Overall Portfolio

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    Asset allocation2

    Asset Allocation

    • 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)

    • Formally:

    Bahattin Buyuksahin, JHU , Investment


    Asset allocation 18

    Asset Allocation 18

    • formally (continued)

    Bahattin Buyuksahin, JHU , Investment


    Asset allocation 19

    Asset Allocation 19

    • 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

    Figure 7.9 The Proportions of the Optimal Overall Portfolio

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    Markowitz portfolio selection model

    Markowitz Portfolio Selection Model

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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 model1

    Markowitz Portfolio Selection Model

    • 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

    Bahattin Buyuksahin, JHU , Investment


    Markowitz portfolio selection model2

    Markowitz Portfolio Selection Model

    • 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

    Bahattin Buyuksahin, JHU , Investment


    Markowitz portfolio selection model3

    Markowitz Portfolio Selection Model

    • Finding the efficient frontier (cont’d)

      • optimization constraints

        • portfolio weights sum up to 1

        • no short sales, dividend yield, asset restrictions, …

    • 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

    Figure 7.10 The Minimum-Variance Frontier of Risky Assets

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    Markowitz portfolio selection model continued

    Markowitz Portfolio Selection Model Continued

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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

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

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    Markowitz portfolio selection model continued1

    Markowitz Portfolio Selection Model Continued

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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

    Figure 7.12 The Efficient Portfolio Set

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    Capital allocation and the separation property

    Capital Allocation and the Separation Property

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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

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

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    The power of diversification

    The Power of Diversification

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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

    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

    Risk Pooling, Risk Sharing and Risk in the Long Run

    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

    Risk Pooling and the Insurance Principle

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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

    Risk Pooling and the Insurance Principle Continued

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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

    Risk Sharing

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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

    An Asset Allocation Problem

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    An asset allocation problem 2

    An Asset Allocation Problem 2

    • Perfect hedges (portfolio of 2 risky assets)

      • perfectly positively correlated risky assets

        • requires short sales

      • perfectly negatively correlated risky assets

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    An asset allocation problem 3

    An Asset Allocation Problem 3

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    Chapter 8

    CHAPTER 8

    Index Models


    Factor model

    Factor Model

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    • Idea

      • the same factor(s) drive all security returns

  • Implementation (simplify the estimation problem)

    • do not look for equilibrium relationship

      • between a security’s expected return

      • and risk or expected market returns

  • look for a statistical relationship

    • between realized stock return

    • and realized market return


  • Factor model 2

    Factor Model 2

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    • Formally

      • stock return

        • = expected stock return

        • + unexpected impact of common (market) factors

        • + unexpected impact of firm-specific factors


    Index model

    Index Model

    Bahattin Buyuksahin, JHU , Investment

    • Factor model

      • problem

        • what is the factor?

  • Index Model

    • solution

      • market portfolio proxy

      • S&P 500, Value Line Index, etc.


  • Advantages of the single index model

    Reduces the number of inputs for diversification

    Easier for security analysts to specialize

    Advantages of the Single Index Model

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    Single factor model

    ßi = 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 Model

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    Single index model

    Single-Index Model

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    Regression Equation:

    Expected return-beta relationship:


    Single index model continued

    Single-Index Model Continued

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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

    Index Model and Diversification

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    • Portfolio’s variance:

    • Variance of the equally weighted portfolio of firm-specific components:

    • When n gets large, becomes negligible


    Optimal risky portfolios asset allocations

    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

    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

    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

    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

    Figure 8.4 Excess Returns on Portfolio Assets

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    Alpha and security analysis

    Alpha and Security Analysis

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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

    Alpha and Security Analysis Continued

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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

    Single-Index Model Input List

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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

    Optimal Risky Portfolio of the Single-Index Model

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    • Maximize the Sharpe ratio

      • Expected return, SD, and Sharpe ratio:


    Optimal risky portfolio of the single index model continued

    Optimal Risky Portfolio of the Single-Index Model Continued

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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

    The Information Ratio

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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

    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

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

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    Index model industry practices

    Index Model: Industry Practices

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    • Beta books

      • Merrill Lynch

        • monthly, S&P 500

    • Value Line

      • weekly, NYSE

  • etc.

  • Idea

    • regression analysis


  • Index model industry practices 2

    Index Model: Industry Practices 2

    Bahattin Buyuksahin, JHU , Investment

    • Example (Merrill Lynch differences, Table 8.3)

      • total (not excess) returns

        • slopes are identical

        • smallness

    • percentage price changes

      • dividends?

      • S&P 500

  • adjusted beta

    • beta = (2/3) estimated beta + (1/3) . 1

    • sampling errors, convergence of new firms

  • exploiting alphas (Treynor-Black)


  • Table 8 3 merrill lynch pierce fenner smith inc market sensitivity statistics

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

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    Table 8 4 industry betas and adjustment factors

    Table 8.4 Industry Betas and Adjustment Factors

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    Using index models

    Using Index Models

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    Using index models 2

    Using Index Models 2

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    Using index models 3

    Using Index Models 3

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    Using index models 4

    Using Index Models 4

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