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

2. Investments 102 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. Investments 103 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 r f = 5%  What is achievable now?

4. Investments 104 Eff. Frontier with Risk-Free AssetE[r]CAL CAL (P) M P F PP&F M G P M 

5. Investments 105 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. Investments 106 Optimal Portfolio  How much in each risky asset?  The expected return and standard dev.  Sharpe Ratio

7. Investments 107 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. Investments 108 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 the best complete portfolio  Obtain the best mix of the optimal risky portfolio and T-Bills  Different investors may have different best complete portfolios Investment Funds PT-Bills BondStock w 1-w T - Bills Bond Stock 1 - y y×w y×(1 - w) y1-y

9. Investments 109 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. Investments 1010 Single Factor Model  Security Characteristic Line Dell Excess Returns (i) SecurityCharacteristicLine........................................................................ Excess Returns on market index ß = 1.4 4.5% 23.8% 28.3% 17%

11. Investments 1011 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. Investments 1012 Single Factor Model  Beta coefficients for selected firms  Question:  What are the betas of market index and T-bills?

13. Investments 1013 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. Investments 1014 Single Factor Model  Example  Given the following data on Microsoft, analyze the systematic risk, unsystematic risk and percentage of variance explained by systematic risk. (σ i = 0.25, σ M = 0.15, Cov[R i,R M ]=0.0315)  Solution

15. Investments 1015 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. Investments 1016 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?