A portfolio dominates all others if no other equally risky portfolio has a higher expected return, or if no portfolio with the same expected return has less risk
Example: MU and INTC vs. MU and MOT
Which portfolio dominates?
The Role of Uncorrelated Securities
15.
The Efficient Frontier : Optimum Diversification of Risky Assets
16.
The Efficient Frontier : The Minimum Variance Portfolio
The right extreme of the efficient frontier is a single security; the left extreme is the minimum variance portfolio .
expected return risk (standard deviation of returns) single security with the highest expected return minimum variance portfolio
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The Efficient Frontier : The Minimum Variance Portfolio
Note that the minimum variance portfolio is not the security with the lowest variance
In general, the further you move to the left of the efficient frontier (less risk), the greater the number of securities in the portfolio
How to determine the minimum variance portfolio ( 2 security case )
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The Efficient Frontier : The Minimum Variance Portfolio Insert Figure 16-6 here.
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The Efficient Frontier : The Effect of a Risk-free Rate
When a risk-free investment complements the set of risky securities, the shape of the efficient frontier changes markedly.
M = Market portfolio R f = Risk-free rate risk (standard deviation of returns) Efficient frontier: R f to M to C expected return dominated portfolios impossible portfolios M R f C E
20.
The Efficient Frontier : The Effect of a Risk-free Rate
The straight portion of the line is tangent to the risky securities efficient frontier at point M and is called the capital market line .
The In theory, all rational investors hold some combination of the market portfolio (M) and the risk-free asset. In equilibrium, M should contain ALL risky assets and should be the only risky portfolio that exists.
The only risk that matters for an individual security is the risk that it brings to the market portfolio M
Beta measures this risk
The market portfolio M contains a percentage of all investable assets in proportion to their market cap.
In practice, the S&P 500 index serves as a proxy for M
Since buying a Treasury bill amounts to lending money to the U.S. Treasury, a portfolio partially invested in the risk-free rate is often called a lending portfolio .
Buying on margin involves financial leverage, thereby magnifying the risk and expected return characteristics of the portfolio. Such a portfolio is called a borrowing portfolio .
22.
The Efficient Frontier with Borrowing Efficient frontier: the ray from R f through M
If it is possible to buy stocks on margin, then the efficient frontier gets expanded again
Efficient frontier : The ray from R f through M and beyond expected return risk (standard deviation of returns) dominated portfolios impossible portfolios M R f lending borrowing
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The Efficient Frontier : Different Borrowing and Lending Rates
Most of us cannot borrow and lend at the same interest rate, this leads the efficient frontier to change again (R B = borrowing rate > R L = lending rate)
expected return dominated portfolios impossible portfolios M R L N Efficient frontier : R L to M, the curve from M to N, then the ray from N risk (standard deviation of returns) R B
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The Efficient Frontier : Naive Diversification
As portfolio size increases,
total portfolio risk, on average, declines. After a certain point, however, the marginal reduction in risk from the addition of another security is modest.
Naive diversification is the random selection of portfolio components without conducting any serious security analysis.
total risk Nondiversifiable risk (market risk) number of securities 20 40
The remaining risk, when no further diversification occurs, is pure market risk.
Research shows that of a single security’s total risk, about 75% is unsystematic and 25% is systematic (i.e. most risk can be diversified away)
Market risk is also called systematic risk and is measured by beta .
A security with average market risk has a beta equal to 1.0. Riskier securities have a beta greater than one, and safer securities have a beta less than 1.0
Capital Market Theory indicates that investors are only rewarded for bearing necessary (unavoidable) risk in the form of additional expected return
This implies that investors should always diversify, since diversification eliminates a substantial portion of portfolio risk (namely diversifiable risk)
Three main results from Evans and Archer:
Total risk declines as the number of securities increases
Increasing the number of portfolio securities provides diminishing benefits as the number of securities increases
In large portfolios, the benefits of additional diversification may be out-weighted by the additional transaction costs
In order to determine portfolio variance, a pair-wise comparison of the thousands of securities in existence would be an unwieldy task. To get around this problem, the single index model compares all securities to a benchmark measure, the market portfolio.
The single index model relates security returns to their betas, thereby measuring how each security varies with the overall market. (Instead of how each security varies with respect to each other)
Using Beta, we only need to calculate a beta for each security instead of covariances
The Efficient Frontier : The Single Index Model
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The Efficient Frontier : The Single Index Model
Beta is the statistic relating an individual security’s returns to those of the market index.
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The Efficient Frontier : The Single Index Model
The relationship between beta and expected return is the essence of the capital asset pricing model (CAPM), which states that a security’s expected return is a linear function of its beta.
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Security Market Line (SML) Insert Figure 16-11 here.
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The Efficient Frontier : The Single Index Model Insert Figure 16-12 here. Beta can be estimated from historical data using the market model - Linear Regression of Market Proxy (S&P 500) and Security excess returns
The intercept from the linear regression (the market model) is know as alpha (also know as the Jensen Index), and is sometimes used as a measure of (risk-adjusted) performance
Positive alpha: return earned is greater than expected based on risk borne
Negative alpha: return earned is smaller than expected based on risk borne
More useful when evaluating portfolios (like mutual funds)
In efficient markets, the expected return and the required rate of return will be equal.
CAPM can be used to obtain the shareholder’s required rate of return in the Dividend Discount Model (DDM)
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