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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

Efficient Diversification II

Efficient Frontier with Risk-Free Asset

Optimal Capital Allocation Line

Single Factor Model

eff frontier with risk free asset
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?
optimal cal
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?
eff frontier with risk free asset2
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)?
optimal portfolio
Optimal Portfolio
  • How much in each risky asset?
  • The expected return and standard dev.
  • Sharpe Ratio
eff frontier with risk free asset3
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
optimal portfolio allocation
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)

single factor model
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:
single factor model1
Single Factor Model
  • Security Characteristic Line

Dell Excess

Returns (i)

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Security

Characteristic

Line

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28.3%

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23.8%

.

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.

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ß = 1.4

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4.5%

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17%

Excess Returns

on market index

single factor model2
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%
single factor model3
Single Factor Model
  • Beta coefficients for selected firms
  • Question:
    • What are the betas of market index and T-bills?
single factor model4
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

single factor model5
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
diversification in a single factor security market
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
wrap up
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?