## Efficient Asset Management

Robert Michaud 1998

Robert Michaud is one of the few practitioners that gets it.

### Classic MVO

- MVO with equality constraints has a linear algebra solution similar to linear regression
- MVO with inequality constraint requires linear programming
- most important limitation is the noise amplification due to (1) inversion of a badly conditioned covariance matrix for large dimensions and (2) the difficulty to predict future returns

### Traditional Criticisms and Alternatives

- utility function based rather than variance based optimisation (this is a very academic discussion)
- multiperiod horizons: in practice, using $\mu-\frac{1}{2} \sigma^2$ instead of one period expected return $\mu$ changes the shape of the efficient parabola and makes it come down as variance increases.
- asset-liability planning: indeed, an optimal bond ladder can be worked when liabilities are known. The question of unbounded wealth maximisation remain.

### Mean-Variance Efficiency

- study by Jobson and Korkie show that MVO portfolio underperforms equal weight out of sample
- naive MVO is a bad allocation

### Resampled Efficient Frontier

- Michaud signature method is to do 500 resampled MVO and obtain weight distribution
- if weights follow a multinormal distribution, their confidence interval follows a Fisher distribution with p and n-p degrees of freedom, p = rank of covariance, n = number of samples
- minimum variance portfolio has a tighter confidence interval
- high return portfolio with positive weight constraints has active constraints on the higest return assets and very low degree of freedom, which makes the F distribution not normal.
- portfolio distance to MVO confidence region can be computed as a mahalinobis distance, which leads to a reco as to smallest weight change that puts the portfolio in the efficient confidence interval.

### Stein estimation

Charles Stein is a pioneer of multi-variate estimation who showed that the optimal mean estimator was not just necessarily the mean of each component if there is a prior.

For instance, if the prior is that the variables share the same mean, the James-Stein estimator is given by $$\hat{\mu}_i = \bar{x} + c_i (\bar{x}_i - \bar{x})$$

The Frost-Savarino (1986) estimator assumes the prio efficiency of the equal weight portfolio.

Ledoit (1994, 1997) covariance estimation is another form of Stein estimation.

Michaud considers Sharpe's 1963 CAPM as an adhoc method consist in factor projection of the historical return onto a single market component. Multi-factor components can be created by adding more factors and the question then lies in how to best select the factors.

### Return Forecasts

Using adhoc return forecasts while keeping covariance historical changes optimal allocation significantly while having the risk of inserting return assumptions that do not respect the internal correlation structure of the data.

Michaud refers to Theil and Goldberger (1961) method of mixing adhoc forecasts with historical data.

### Some references

- On Pure and Mixed Statistical Estimation in Economics H. Theil and A. S. Goldberger, 1961
- Estimation with Quadratic Loss James C, C Stein 1961
- Inadmissibility of the usual estimator for the variance of a normal distribution with unknown mean Charles Stein 1964
- An Empirical Bayes Approach to Efficient Portfolio Selection Peter A. Frost, James E. Savarino 1986
- Global Portfolio Optimization Black Litterman 1992