8.4 & 8.5

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simple linear regression

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A simple linear regression is a statistical method that estimates a linear relationship between a dependent variable and a single independent variable. In a simple linear CAPM-based regression, the asset's excess returns are regressed against the market's excess returns. The slope coefficient measures the asset's beta, which is the sensitivity fo the asset's return to changes in the market portfolio returns. The intercept of the CAPM-based regression equals the incremental performance of the asset relative to the CAPM benchmark return, and is called the asset's alpha

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Ordinary least square (OLS)

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Ordinary least squares (OLS) is an estimation method that minimizes the sum of square regressional residues
* outliers have a disporportionately large effect on regressions due to the residual square process of the OLS regressions
* the OLS method assumes that regressions residuals are not correlated with their lagged values. Violation of the assumption is called serial correlation, which causes standard errors and t-statistics to be incorrectly calculated
* the OLS method assumes that the variance of the residuals is constant. Heteroskedasticity refers to a violation of the constant error variance assumption. Conditional heteroskedasticity is related to the level of the independent variables, and causes standard errors and t-statistics to be incorrectly calculated.

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Multi-factor regression models

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Multi-factor regresion models describe relationships betwee asset returns and the returns on multiple risk factors

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Fama French

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The Fama French model is a multi-factor model that regresses an asset's excess returns against the market's excess returns, firm size factor returns, ad book-to-market factor returns

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Multicollinearity

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Multicollinearity refers to the condition in which two or more of the independent variable are highly correlated. When independent variables are correlated, the intercept and slope standard errors are biased upward, which, in turn, biases the t-statistics downwards

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stepwise regression

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The stepwise regression method chooses independent variables based on each variable's explanatory power. The first independent variable chosen is the one with the highest t-statistic for its slope. Then, additional variables are added sequentially depending on the magnitude of their t-statistics

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Dynamic risk exposure models

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Dynamic risk exposure models examine non-linear relationships caused by actor risk exposures that change over time.

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Three dynamic risk exposure models are:

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1. dummy variable regression model
2. separate regressions model, and
3. the quadratic curve regression model

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Conditional correlation

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Alternative ivnestmetn returns are non-stationary, implying that means, variances, and/or correlations are not constant over time. Conditional correlation is the correlation between two variable relative to a specific set of circumstances. A positve conditional correlation exists when teh corerelation between a fund's returns and athe market index returns in higher in up-markets versus down-markets. A negative conditional correlation exists when teh correlation is lower during uup-markets than during down markets. A positve conditional correlation is indicative of a good market time.

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Rolling window

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Time-varying correlation adn regression estiamtes can be derived using a rollowing window approach, inw hich a moving window of time is used to derive periodic correlation and regression slope estimates

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Principal components analysis

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Principal components analysis is a multivariate statistical method that groups fund that correlate highly with each other. Studeies show that funds can be classified into one of five trading style groups: distressed, global/macro, value, opportunistic, adn trend-following

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Hedge fund replication

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Hedge fund replication identifies investment strategies mimicking a particular fund's returns. Int he fund replication method, a fund's return is explained by a specialized set of market-based 9as opposed to market-wide)factors.

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Multi-factor models

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Multi factor mdoels explain fund returns relative to:
1. returns of assets classes held by the fund
2. returns of funds with similar strategies
3e. market factors that drive asset returns, and
4. specialized market factors

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Performance Persistence

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Performance persistence can be exmained wtih regressions tests, emasures of skill tests, persistence of volatility tests, and serial correlation tests. Results of empirical testing regarding the performance persistence of hedge funds are mixed.

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Mean Variance otpimization

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Mean-variance optimization derives efficient portfolioes,w hich maximize expected return withi risk class. The efficent frontier comprises the set of efficient portfolios. in theory, all investors should choose the efficient portfolio that matches their risk target

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deriving MVO portfolios

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MVO portfolios are derived by maximizing portfolio expected returns subject to portfolio risk constraints. The efficient frontier is developed by optimizing portfolios for variosu targeted risk levels. Therefore , each portfolio maximizes expected return within its risk class

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two-fund separation theorem

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according to the two-fund separation theorem, if all capital market theory assumption hold, then investors maximize their risk-return preferences by investing in a combination of two funds
1. a risk free fund and
2. the broadly diversified fund

Front 18

Describe the concept of mean-variance optimizer as "error maximizers."

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Mean variance optimization is highly sensitive to estimation error. MVO portfolio allocations are high for assets with high return-to-risk estimates, but high return estimates tend to be associated with large estimation errors. there MVO tends to be an error maximizer.

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Describe how mean-variance optimizers ignore higher moments

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The MVO solution is derived solely based on two paramters of the portfolio: mean and variance. higher moments, such as skewness and kurtosis, are ignored, which is a limitation of MOV application for non-noraml alternative investments returns. The MVO solution migh produce portfolios with desirable mean and variance but with undesirable kew and kurosis

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Three suggestions to the way mean variance optimizers ignore higher moments

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1. the optimizer can be adjusted to include higher moements. This transforms the optimizer into a multiple objective optimizer rather than a mean-variance optimizer
2. skewness and kurtosis can be included amon the constraints of the MOV
3. Explicit constraints (e.g., weight less than or equal to 5%) can be imposed the MOV solution for assets with undesirable skew and kurtosis

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Why is estimation of variance difficult for alternative investments and what can be done about

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Estimation of variance is particularly difficult for alternative investments, such as private equity and real estate, where valuations are based on smoothing processes from appraisals. The smoothing process leads to underestimation of variances and co-variances, which in turn, lead the high MVO weightings for the affected assets. The recommendation is to unsmooth rates of return before estimating variances and covariance for MVO. Unfortunately unsmoothing is not a perfect science and, often does not solve the programs associated with estimation error. In particular, the number of estimates that must be derived is daunting. For instances, estimates must be derived for N expected returns, N variances and N(N-a1)/2 covariances. For a portfolio of 10 assets, a total of 65 estimates are needed, 55 of which are variances and covariances that are susceptible to smoothing effects.

Factor models can be used

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three extension of MOV models

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1. black litterman approach
2. shrinkage techniques and
3 additional constraints approaches

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Black litterman

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The black litterman method modifies MVO by derivign expected return that are consistent iwth market equilibrium. the apporach assumed that hte market is in equilibrum, implying that market value weights are ooptimal. The equilibrium expected return for an asset is teh expected return associated with its market value weight

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Shrinkage estimates

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shrinkage estimator modify MVO by adusting abnormal covariance toward the average covariance. By reducing the extreme values, the shrinkage estimation technique reduces the effects of estimation errors

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additional constraints apporach

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the additional constraints approach imposes furhter constratins, such as limits on tracking error of the MVO portfolio relative to the benchmark, departure of the MOV weights form market value weights, skewness and kurtosis of the MOV protfolio, and on the range of the MVO weights

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Risk budgeting

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Risk budgting is the process of defining an acceptable amount of risk and then selecting portfolio weights based on the budgeted risk. Risk budgeting relies on specification of a risk target for any number of risk candidates, such as standard deviation, beta, tracking error; or value at risk

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Risk parity

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Risk parity is a risk-budgeting strategy that allocates risks equally across asset classes int eh portfolio.

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Three steps to aply teh risk parity approach are

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1. define the total risk of the portfolio
2. calculate the margin contribution of each asset class to the total risk of the portfolio.
3. determine risk parity portfolio weights.

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What other ways are there to construct a low-volatility portfolio.

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Other alternative include:
equal weights, minimum variance weights, and volatility weights. In the volatility weights method, the weight for any asset class equals the reciprocal of the asset class volatility relative to the sum of the reciprocals for all asset classes int he portfolio.