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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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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 Stocks in the Portfolio

    Bahattin Buyuksahin, JHU , Investment


    Covariance and correlation
    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
    Two-Security Portfolio: Return Stocks in the Portfolio

    Bahattin Buyuksahin, JHU , Investment


    Two security portfolio risk

    = Variance of Security D Stocks in the Portfolio

    = 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 Stocks in the Portfolio

    • Another way to express variance of the portfolio:

    Bahattin Buyuksahin, JHU , Investment


    Covariance
    Covariance Stocks in the Portfolio

    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 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

    Bahattin Buyuksahin, JHU , Investment


    Table 7 1 descriptive statistics for two mutual funds
    Table 7.1 Descriptive Statistics for Two Mutual Funds Stocks in the Portfolio

    Bahattin Buyuksahin, JHU , Investment


    Three security portfolio
    Three-Security Portfolio Stocks in the 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 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)

    • 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

    Bahattin Buyuksahin, JHU , Investment


    Table 7 3 expected return and standard deviation with various correlation coefficients
    Table 7.3 Expected Return and Standard Deviation with Various Correlation Coefficients

    Bahattin Buyuksahin, JHU , Investment


    Figure 7 3 portfolio expected return as a function of investment proportions
    Figure 7.3 Portfolio Expected Return as a Function of Investment Proportions

    Bahattin Buyuksahin, JHU , Investment


    Figure 7 4 portfolio standard deviation as a function of investment proportions
    Figure 7.4 Portfolio Standard Deviation as a Function of Investment Proportions

    Bahattin Buyuksahin, JHU , Investment


    Minimum variance portfolio as depicted in figure 7 4
    Minimum Variance Portfolio as Depicted Investment Proportionsin Figure 7.4

    Bahattin Buyuksahin, JHU , Investment

    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 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 Effects

    Bahattin Buyuksahin, JHU , Investment


    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 and Two Feasible CALs

    Bahattin Buyuksahin, JHU , Investment

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

    Bahattin Buyuksahin, JHU , Investment


    Asset allocation2
    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)

    • Formally:

    Bahattin Buyuksahin, JHU , Investment


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

    • formally (continued)

    Bahattin Buyuksahin, JHU , Investment


    Asset allocation 19
    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
    Figure 7.9 The Proportions of the Optimal Overall Portfolio with the Optimal CAL and the Optimal Risky Portfolio

    Bahattin Buyuksahin, JHU , Investment


    Markowitz portfolio selection model
    Markowitz Portfolio Selection Model with the Optimal CAL and the Optimal Risky Portfolio

    Bahattin Buyuksahin, JHU , Investment

    • 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 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

    Bahattin Buyuksahin, JHU , Investment


    Markowitz portfolio selection model2
    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

    Bahattin Buyuksahin, JHU , Investment


    Markowitz portfolio selection model3
    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, …

    • 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

    Bahattin Buyuksahin, JHU , Investment


    Figure 7 10 the minimum variance frontier of risky assets
    Figure 7.10 The Minimum-Variance Frontier of Risky Assets with the Optimal CAL and the Optimal Risky Portfolio

    Bahattin Buyuksahin, JHU , Investment


    Markowitz portfolio selection model continued
    Markowitz Portfolio Selection Model Continued with the Optimal CAL and the Optimal Risky Portfolio

    Bahattin Buyuksahin, JHU , Investment

    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

    Bahattin Buyuksahin, JHU , Investment


    Markowitz portfolio selection model continued1
    Markowitz Portfolio Selection Model Continued Optimal CAL

    Bahattin Buyuksahin, JHU , Investment

    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 Optimal CAL

    Bahattin Buyuksahin, JHU , Investment


    Capital allocation and the separation property
    Capital Allocation and the Separation Property Optimal CAL

    Bahattin Buyuksahin, JHU , Investment

    • 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

    Bahattin Buyuksahin, JHU , Investment


    The power of diversification
    The Power of Diversification from the Efficient Set

    Bahattin Buyuksahin, JHU , Investment

    • 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

    Bahattin Buyuksahin, JHU , Investment


    Risk pooling risk sharing and risk in the long run
    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

    Bahattin Buyuksahin, JHU , Investment

    Consider the following:


    Risk pooling and the insurance principle
    Risk Pooling and the Insurance Principle Correlated and Uncorrelated Universes

    Bahattin Buyuksahin, JHU , Investment

    • 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 Correlated and Uncorrelated Universes

    Bahattin Buyuksahin, JHU , Investment

    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 Correlated and Uncorrelated Universes

    Bahattin Buyuksahin, JHU , Investment

    • 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 Correlated and Uncorrelated Universes

    Bahattin Buyuksahin, JHU , Investment


    An asset allocation problem 2
    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

    Bahattin Buyuksahin, JHU , Investment


    An asset allocation problem 3
    An Asset Allocation Correlated and Uncorrelated UniversesProblem 3

    Bahattin Buyuksahin, JHU , Investment


    Chapter 8

    CHAPTER 8 Correlated and Uncorrelated Universes

    Index Models


    Factor model
    Factor Model Correlated and Uncorrelated Universes

    Bahattin Buyuksahin, JHU , Investment

    • 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 Correlated and Uncorrelated Universes

    Bahattin Buyuksahin, JHU , Investment

    • Formally

      • stock return

        • = expected stock return

        • + unexpected impact of common (market) factors

        • + unexpected impact of firm-specific factors


    Index model
    Index Model Correlated and Uncorrelated Universes

    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 Correlated and Uncorrelated Universes

    Easier for security analysts to specialize

    Advantages of the Single Index Model

    Bahattin Buyuksahin, JHU , Investment


    Single factor model

    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 Model

    Bahattin Buyuksahin, JHU , Investment


    Single index model
    Single-Index Model Correlated and Uncorrelated Universes

    Bahattin Buyuksahin, JHU , Investment

    Regression Equation:

    Expected return-beta relationship:


    Single index model continued
    Single-Index Model Continued Correlated and Uncorrelated Universes

    Bahattin Buyuksahin, JHU , Investment

    • 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 Correlated and Uncorrelated Universes

    Bahattin Buyuksahin, JHU , Investment

    • 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

    Bahattin Buyuksahin, JHU , Investment


    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

    Bahattin Buyuksahin, JHU , Investment


    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

    Bahattin Buyuksahin, JHU , Investment


    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

    Bahattin Buyuksahin, JHU , Investment


    Figure 8 4 excess returns on portfolio assets
    Figure 8.4 Excess Returns on Portfolio Assets Hewlett-Packard

    Bahattin Buyuksahin, JHU , Investment


    Alpha and security analysis
    Alpha and Security Analysis Hewlett-Packard

    Bahattin Buyuksahin, JHU , Investment

    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 Hewlett-Packard

    Bahattin Buyuksahin, JHU , Investment

    • 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 Hewlett-Packard

    Bahattin Buyuksahin, JHU , Investment

    • 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 Hewlett-Packard

    Bahattin Buyuksahin, JHU , Investment

    • 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 Hewlett-Packard

    Bahattin Buyuksahin, JHU , Investment

    • 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 Hewlett-Packard

    Bahattin Buyuksahin, JHU , Investment

    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

    Bahattin Buyuksahin, JHU , Investment


    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

    Bahattin Buyuksahin, JHU , Investment


    Index model industry practices
    Index Model: Industry Practices Full-Covariance Models

    Bahattin Buyuksahin, JHU , Investment

    • 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 Full-Covariance Models

    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

    Bahattin Buyuksahin, JHU , Investment


    Table 8 4 industry betas and adjustment factors
    Table 8.4 Industry Betas and Adjustment Factors Market Sensitivity Statistics

    Bahattin Buyuksahin, JHU , Investment


    Using index models
    Using Index Models Market Sensitivity Statistics

    Bahattin Buyuksahin, JHU , Investment


    Using index models 2
    Using Index Models 2 Market Sensitivity Statistics

    Bahattin Buyuksahin, JHU , Investment


    Using index models 3
    Using Index Models 3 Market Sensitivity Statistics

    Bahattin Buyuksahin, JHU , Investment


    Using index models 4
    Using Index Models 4 Market Sensitivity Statistics

    Bahattin Buyuksahin, JHU , Investment


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