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Efficient Diversification II

Efficient Diversification II. Efficient Frontier with Risk-Free Asset Optimal Capital Allocation Line Single Factor Model. Eff. Frontier with Risk-Free Asset. With risky assets only No portfolio with zero variance GMVP has the lowest variance With a risk-free asset

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Efficient Diversification II

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  1. Efficient Diversification II Efficient Frontier with Risk-Free Asset Optimal Capital Allocation Line Single Factor Model

  2. Eff. Frontier with Risk-Free Asset • With risky assets only • No portfolio with zero variance • GMVP has the lowest variance • With a risk-free asset • Zero variance if investing in risk-free asset only • How does it change the efficient frontier?

  3. Optimal CAL • Mean-variance with two risky assets • w in security 1, 1 – w in security 2 • Expected return (Mean): • Variance • What happens when we add a risk-free asset? • A riskfree asset with rf= 5% • What is achievable now?

  4. E[r] CAL (P) M M P P CAL G F  P P&F M Eff. Frontier with Risk-Free Asset

  5. Eff. Frontier with Risk-Free Asset • CAL(P) dominates other lines • Best risk and return trade-off • Steepest slope • Portfolios along CAL(P) has the same highest Sharpe ratio • No portfolio with higher Sharpe ratio is achievable • Dominance independent of risk preference • How to find portfolio (P)?

  6. Optimal Portfolio • How much in each risky asset? • The expected return and standard dev. • Sharpe Ratio

  7. Eff. Frontier with Risk-Free Asset • What’s so special about portfolio (P)? • P is the market portfolio • Mutual fund theorem: An index mutual fund (market portfolio) and T-bills are sufficient for investors • Investors adjust the holding of index fund and T-bills according to their risk preferences

  8. Optimal Portfolio Allocation • Two Step Allocation • Step 1: Determine the optimal risky portfolio • Get the optimal mix of stock and bond • Optimal for all investors (market portfolio) • Step 2: Determine thebest complete portfolio • Obtain the best mix of the optimal risky portfolio and T-Bills • Different investors may have different best complete portfolios Investment Funds y 1-y P T-Bills w 1-w Bond Stock T - Bills Bond Stock 1 - yy×wy×(1 - w)

  9. Single Factor Model • Quantifies idiosyncratic versus systematic risk of a stock’s rate of return • Factor is a broad market index like S&P500 • The excess return is • : stock’s excess return above market performance • : stock’s return attributable to market performance • : return component from firm-specific unexpected event • Example: a statistical analysis between the excess returns of DELL and market shows that  = 4.5%,  = 1.4. If expected market excess return is 17%, what is the expected excess return for DELL? • Solution:

  10. Single Factor Model • Security Characteristic Line Dell Excess Returns (i) . Security Characteristic Line . . . . . . 28.3% . . . . . . . . . . . 23.8% . . . . . . . . . . ß = 1.4 . . . 4.5% . . . . . 17% Excess Returns on market index

  11. Single Factor Model • Meaning of Beta () • Indicator of how sensitive a security’s return is to changes in the return of the market portfolio. • A measure of the asset’s systematic risk. • Example: market portfolio’s risk premium is +10% during a given period, and  = 0%. •  = 1.50, the security’s risk premium will be +15%. •  = 1.00, the security’s risk premium will be +10% •  = 0.50, the security’s risk premium will be +5% •  = –0.50, the security’s risk premium will be–5%

  12. Single Factor Model • Beta coefficients for selected firms • Question: • What are the betas of market index and T-bills?

  13. Single Factor Model • Systematic Risk • Risk related to the macro factor or market index • Non-diversifiable/market risk • Unsystematic Risk • Risk related to company specific problems • Diversifiable/Firm-specific/Idiosyncratic risk • Total risk = Systematic + Unsystematic % of variance explained by the market

  14. Single Factor Model • Example • Given the following data on Microsoft, analyze the systematic risk, unsystematic risk and percentage of variance explained by systematic risk. (σi= 0.25, σM= 0.15, Cov[Ri,RM]=0.0315) • Solution

  15. Diversification in a Single Factor Security Market • A portfolio of three equally weighted assets 1, 2, and 3. • The excess return of the portfolio is • Risk of the portfolio is

  16. Wrap-up • What does the efficient frontier look like with the presence of a risk-free asset? • What are the two steps of asset allocation? • What is a single index model? • What are the meaning of systematic and unsystematic risks?

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