2. Kritzman (2011)- Single Index Model
2. Kritzman (2011)- Single Index Model SIM: An alternative to the Markowitz full-covariance optimisation method by decomposing the risk of an individual security into a firm-specific component and market-wide component. It also reduces the number of required inputs that the full covariance method requires. Assumption: All stock returns are related to a single influence (market return), ππ = π¬(πΉπ ) + π + ππ , whereby M denotes unanticipated macoreconomic surprises
Rit ο½ Ξ±i ο« Ξ²iRmt ο« Ξ΅it
, where return on asset = return on the factor *sensitivity of asset to factor + constant + residual component not explained by the factor ο·
To calculate the mean + variance, just take expected value + take the variance of the term
This factor is usually the market index β residual assumed to be uncorrelated to the factor return and has a mean of zero + variance (constant) ο· ο·
π 2 2 2 βπ βπ ππ2 = ππ π π π€π π€π π½π π½π + βπ π€π πππ , variance is here broken down into the sum of the total covariance and the individual variances 2 Covariance is calculated as = ππ π₯ π½π π₯ π½π , as the only thing that connects these two securities is their exposure to market risk, so therefore beta is a good proxy to calculate their covariance, whereas usually idiosyncratic risk is used
Diversification: SIM says that using equally weighted portfolios, diversification benefits are still realised, assuming firm-specific variance is not correlated β however, there is a degree of correlation esp. in interindustry correlation that is not accounted for Mean Variance Optimisation (Benefits vs Disadvantages) Advantages It is too unrealistic an expectation to hold that it is a perfect model β evidence that equally weighted portfolios beat optimised portfolios is flawed (they chose short samples of data) (Evidence that constraints and objectives of client can be inputted into portfolio by adjusting formula and that it has been more successful) There is empirical evidence that it does not make much difference on return β MV optimisation not hypersensitive as although there are large errors, this doesnβt affect return β it only affects the allocation between similar assets only and therefore returns distributions of correct and incorrect portfolios will be quite similar Avoids ad hoc weighting of information by analysts MV depends on false assumptions β returns follow a normal distribution + investors have quadratic utility
Disadvantages Garbage in, Garbage out: All inputted assumptions are based on historical data and therefore as biased ; expecting garbage to be converted into value (Forecast error due to historical inputs) MV Optimisation is highly sensitive to input errors: It is biased towards assets that have positive error (higher return) and negative error for variance (less risky)- therefore overstating return and understating risk ο therefore not really optimising, it is instead a false portfolio (High return low variance stocks are those likely to contain more errors) May create a black box β too quantitative MV only requires one of these to be true and will fail when they are jointly + significantly violated; however, this seldom occurs
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Overall, MV Optimisation is far from retirement β criticism of garbage in, garbage out β no more indictment of mean-variance optimisation than it is of calculators/ cooking recipes (inputs bad, therefore model bad? Not really) Hypersensitive to small input errors is vastly overstated β small input errors lead to large misallocations across assets that are close substitutes for one another, but hardly reduces efficiency MV only requires one of these to be true and will fail when they are jointly + significantly violated; however, this seldom occurs
Rit ο½ Ξ±i ο« Ξ²iRmt ο« Ξ΅it
, where return on asset = return on the factor *sensitivity of asset to factor + constant + residual component not explained by the factor ο·
To calculate the mean + variance, just take expected value + take the variance of the term
This factor is usually the market index β residual assumed to be uncorrelated to the factor return and has a mean of zero + variance (constant) ο· ο·
π 2 2 2 βπ βπ ππ2 = ππ π π π€π π€π π½π π½π + βπ π€π πππ , variance is here broken down into the sum of the total covariance and the individual variances 2 Covariance is calculated as = ππ π₯ π½π π₯ π½π , as the only thing that connects these two securities is their exposure to market risk, so therefore beta is a good proxy to calculate their covariance, whereas usually idiosyncratic risk is used
Diversification: SIM says that using equally weighted portfolios, diversification benefits are still realised, assuming firm-specific variance is not correlated β however, there is a degree of correlation esp. in interindustry correlation that is not accounted for Mean Variance Optimisation (Benefits vs Disadvantages) Advantages It is too unrealistic an expectation to hold that it is a perfect model β evidence that equally weighted portfolios beat optimised portfolios is flawed (they chose short samples of data) (Evidence that constraints and objectives of client can be inputted into portfolio by adjusting formula and that it has been more successful) There is empirical evidence that it does not make much difference on return β MV optimisation not hypersensitive as although there are large errors, this doesnβt affect return β it only affects the allocation between similar assets only and therefore returns distributions of correct and incorrect portfolios will be quite similar Avoids ad hoc weighting of information by analysts MV depends on false assumptions β returns follow a normal distribution + investors have quadratic utility
Disadvantages Garbage in, Garbage out: All inputted assumptions are based on historical data and therefore as biased ; expecting garbage to be converted into value (Forecast error due to historical inputs) MV Optimisation is highly sensitive to input errors: It is biased towards assets that have positive error (higher return) and negative error for variance (less risky)- therefore overstating return and understating risk ο therefore not really optimising, it is instead a false portfolio (High return low variance stocks are those likely to contain more errors) May create a black box β too quantitative MV only requires one of these to be true and will fail when they are jointly + significantly violated; however, this seldom occurs
ο·
ο·
ο·
Overall, MV Optimisation is far from retirement β criticism of garbage in, garbage out β no more indictment of mean-variance optimisation than it is of calculators/ cooking recipes (inputs bad, therefore model bad? Not really) Hypersensitive to small input errors is vastly overstated β small input errors lead to large misallocations across assets that are close substitutes for one another, but hardly reduces efficiency MV only requires one of these to be true and will fail when they are jointly + significantly violated; however, this seldom occurs
