BKM 6-8: Portfolio Theory Flashcards
Risk premium
Expected return in excess of the risk-free rate
Utility function
Risk-averse
A > 0
Risk-neutral
A = 0
Risk-seeking
A < 0
Mean-Variance Criterion
Portfolio A is preferred to portfolio B if the expected return of A >= expected return of B and risk of A <= risk B
Certainty-equivalent rate
Rate of return that would cause the investor to be indifferent between risky and risk-free investment
Indifference curve
X: standard deviation, Y: expected return
Expected return of complete portfolio (1 risky, 1 risk free)
Standard deviation of complete portfolio (1 risky, 1 risk free)
Capital allocation line (CAL)
Combinations of risk (x-axis) and expected return (y-axis) for complete portfolio: y-intercept: SD = 0 other point: y = 1
Sharpe ratio
Slope of the CAL Also called reward-to-risk ratio
Weight on risky portfolio, based on risk tolerance, in complete portfolio (1 risky, 1 risk free)
Capital Market Line
If risky asset is based on a broad index of common stocks (i.e. S&P 500) within a complete portfolio
Passive strategies (Indexing)
Choosing a risky portfolio to be large, well-diversified (i.e. S&P 500)
Why passive strategies make sense
Minimizes cost of information acquisition
Takes advantage of everyone else’s efforts to do so
Criticisms of passive strategies
Undiversified Top Heavy Chasing Performance You can do better (few active fund managers beat indices)
Portfolio variance with two risky assets
Risk and return, correlation = 1
Straight line
Risk and return, correlation = -1
Kinked line (sideways V) Will intercept y-axis
Minimum variance portfolio
Portfolio with the lowest variance that can be constructed from assets with a certain level of correlation
Optimal risky portfolio
Risky portfolio that produces the line tangent to the portfolio opportunity set
Hedge Asset
Has negative correlation with other assets in the portfolio
Minimum variance portfolio weight on A
When correlation between assets = -1
Perfectly hedged position can be obtained by setting weighted SDs equal to each other
Minimum variance portfolio SD
Must be smaller than that of either of the individual component assets
Diversification and correlation of assets
Lower correlation, diversification is more effective and portfolio risk is lower
Weights of optimal risky portfolio
Steps to arrive at complete portfolio
- Specify returns, variances, covariances
- Calculate optimal risky portfolio weights
- Allocate weights to optimal risky portfolio
- Calculate y*
- Distribute y* to weights of risky portfolio, (1 - y*) to risk-free
Minimum-variance frontier
Lowest possible variance that can be attained for a given portfolio expected return
Graph of: CAL
Indifference curve
Optimal risky portfolio
Complete portfolio
Portfolio opportunity set
Socially responsible investing
Cost of lower Sharpe ratio justifiably viewed as a contribution to underlying cause
Separation property
Two tasks:
- Determine optimal risky portfolio (technical)
- Capital allocation based on risk preference
Market vs. Firm-specific risk (graph)
Equally weighted portfolio variance
First term can be diversified away (firm-specfic risk)
Second term depends on covariances between returns (market risk)
Risk pooling
Merging uncorrelated, risky projects as a means to reduce risk
Results of pooling uncorrelated risks
Issues with risk pooling
Probability of loss declines, but overall standard deviation increases; does not allow shedding of risk
Risk sharing
Act of selling shares in an attractive risky portfolio to limit risk and maintain Sharpe ratio of resulting position
Risk sharing results, two uncorrelated assets
Pool tow assets and sell off half of combined portfolio
Two factors dampening process of risk sharing in insurance
Managing very large firms comes at a risk
Misestimating correlations can cause failure
Extending investment horizons for risk
Investing completely in risky asset for both periods analagous to risk pooling; investing half in risky asset for each period analogous to risk sharing
Issues with Markowitz model
Requires (n2 +3n)/2 total estimates
Errors in estimation of correlations can lead to nonsensical results
Decomposing rate of return
Single-factor model
Total risk of a security
Single-factor model, systematic risk of a security
Covariance between any pair of securities
Regression equation of single-index model
Single-index model
Uses market index to proxy for the common factor
Number of single index model estimates
n estimates of alpha
n estimates of beta
n estimates of firm-specific variances
1 estimate for market risk premium
1 estimate for variance of common factor
Security characteristic model
Straight line with intercept alpha and slope beta for a given security
Alpha
Nonmarket premium
Issues with relying on alpha
Past does not readily fortell the future; no correlation between estimates of one sample period to the next
Hierarchy of preparation of input list for single-index
- Macroeconomic analysis (risk premium)
- Statistical analyses (betas, residual variances)
3.
Initial weight in active portfolio, single-index model
Final weight in active portfolio, single-index model
Information ratio
The contribution of the active portfolio to the Sharpe ratio of the overall risky portfolio
Sharpe ratio of an optiamally constructed risky portfolio
Adjusted beta
2/3*sample beta + 1/3
As firm becomes more conventional, it will tend toward 1
Variables that help predict betas
Variance of earnings
Variance of cash flow
Growth in earnings per share
Firm size
Dividend yield
Debt-to-Asset ratio
Tracking portfolio
Designed to match the systematic component of a portfolio; must have same beta on index portfolio as P and as little firm risk as possible
Also called beta capture
Alpha transport
Separating search for alpha from the choice of market exposure
σ2(eP), formula
Σwi2σ2(ei)
