## Portfolio Optimization

It is useful to look back at the results of Markowitz on portfolio selection (1952). These results led to the formulation of Modern Portfolio Theory (MPT).

MPT considers how an investor should choose a portfolio with a good trade-off between risk and expected return. Markowitz showed that the set of possible expected returns and risks could be visualised in a 2D graph featuring expected return and risk (measured by standardd deviation). The graph below summarises his key result, which is that the optimal choice of portfolio lies on a hyperbola called the efficient frontier.

The challenge of the Markowitz portfolio selection procedure for practitioners are:

- the selection obtained is highly dependent on the prediction of future return, especially when requiring high returns
- the minimisation of risk is also subject to statistical estimation problems, especially when there are many assets as it involves a large covariance matrix input

In practice, when given hundreds of assets, the unconstrained optimiser will suggest an allocation featuring fewer than 10 assets with large weights which do not perform as well in a testing period as in the statistical estimation period.

## A pragmatic solution for stock pickers

The main challenge with optimisation is to obtain suitable inputs for it (and the way the problem is posed tells a lot about the persons trying to solve it).

A pragmatic step to obviate these problems is to cap the concentration of any single stock to 4% (tests show that this is an optimal value, this is corroborated by Elton and Gruber results on diversification), to validate the growth prospect of any stock selected by the optimiser through fundamental analysis for the return part, and to do the optimisation with differrent covariance matrix obtained with more or less weight on recent history, and check how the results differ.

Our approach here is evolutionary and pragmatic. The numerical solutions proposed here have been validated by backtesting by checking that the weights obtained with prior data keep delivering consistent index outperformance in the following three month, but the underlying assumption is that one is able to identify outperforming companies by checking out their financial statements.

## Some standard theory for non stock pickers: Capital Asset Pricing Model (CAPM) and Efficient Market Hypothesis (EMH)

In a nutshell, the efficient market theory is obtained by coupling markowitz model for optimising asset allocation with the assumption that all the market actors have the same access to market intelligence and act rationally on it.

This assumption means that all investors have the same variance and return expectations, acting rationnally on it leads them to use a mean variance optimiser. This results on everyone holding the optimal portfolio, which is therefore the capitalisation weighted market portfolio, albeit with different level of leverage depending on investor's risk aversion. since everyone is a rational investor acting optimally, every investor has a combination of cash earning the risk free rate (or borrowed at that rate) and of the tangency portfolio (shown in the MPT illustrative diagram), which is the same for everyone.

Indeed the way the problem is posed tells a lot about the persons trying to solve it. The EMH and CAPM are minimalist macroeconomic models. Rather than overanalyse these theories purported shortcomings, we'll see its principal success: According to this theory, buying a broad market ETF is the smartest thing to do. Although the assumptions are oversimplified, Burton Malkiel, a princeton economics professor, summed up the evidence for index investing in the classic investment book a random walk down wall street. Buying the index beats managed mutual fund investing on average and on median, as it is the simplest and most scalable and cost effective way to benefit from diversification.

However, as indexing becomes more popular, more investors are abdicating their ability to pick their investment, and go for the market consensus. It is ironic that more investors end up buying the market portfolio as if the consensus were truly optimal. This contrasts with value investors, who claim that stocks have values because they are shares in some business, not just because many other people have sunk money into it.