The ``Fallacy of Large Numbers`` as formulated by Paul A. Samuelson in 1963. This is a very important and fundamental paradox in investment advice. Indeed, financial advisers typically advice risky investments for longer time horizons. So that if one for example has an investment horizon of one year typically only cash and cash-like assets are advised. Shares are typically advised if the investment horizon is longer. This seems to make sense, if one has 10K and wants to buy something of 10K on short term, investing in equities is risky as it would not be uncommon to loose 10% on such portfolio on a horizon of one year. The last few hundred years of capitalist history learns us that on longer horizons the probability that equities have a negative return is much lower. Therefore investment advisers will not advice equities on one year but on ten year it might be acceptable. This paradigm is fundamental to all investment advice, but in 1963 Samuelson wrote a short paper arguing that it is not rational to compound a mistake or in other words that "unfairness can only breed unfairness", meaning that it is not rational to accept a series of bets when one would not accept one of the atomic bets. This paradox was an important problem for every investor, investment advisor and academic that took his responsibility serious. The question if the rule of thumb that every advisor was using was a good one or not was open. Philippe De Brouwer published solved this puzzle and published a simple solution in 2001. The counter-example was based on an asymmetrical utility function. To some extent it can be argued that the utility function used by De Brouwer and Van den Spiegel was a simplified version of what Harry Markowitz describes in his 1952 paper "The Utility of Wealth". The utility function has a kink in the actual wealth and decreases faster for losses than it increases for profits. It seems that with such utility function it is actually natural to "accept a series of bets while one should be rejected". More profoundly this seems to be the natural shape for the utility function of a loss averse investor. Adding this to the fact that all investors are loss averse, this result is to be considered as an important step in responsible investment advice.
Once the Fallacy of Large Numbers falsified, Philippe De Brouwer could further investigate investment advice and try to come up with a coherent framework for investment advice. Till then the only available theory was Markowitz "Mean Variance theory". The model is a simply MCDA. The idea is that selecting the optimal asset mix one needs to optimize two functions: minimize "risk" and maximize "expected return". Because there will be no portfolios that has both the highest expected return and the lowest risk there is not one but rather an infinite set of "solutions". This means that there needs to be another principle to select one. Typically one uses a quadratic utility function, usually simplified to. While theoretically appealing for its consistency and logical coherence, in practice it is not easy to estimate the parameter. This is, according to Prof. De Brouwer because volatility is not even a risk measure, the utility function is wrong and the concept of one utility function does not even exist. Therefore, practitioners do not use this utility function at all but simply try to determine a "risk profile", which should be something as the "desired for that investor". This non-existent concept is the determined with the most primitive MCDA method, the Weighted sum model.. This observation becomes the key to Maslowian Portfolio Theory, which might be summarized as "investors who save to fulfil non-monetary life-goals should use a separate portfolio for each investment goal". These portfolios should be optimized with a coherent risk measure taking into account the parameters from that goal. De Brouwer develops a theory "Target Oriented Investment Advice" that provides the mathematical framework. necessary to put this theory into practice.
Other contributions
Noteworthy is De Brouwer's effort to make help practitioners understand the concept and importance of coherent risk measure and his contributions to apply theory into practice via risk management in financial institutions
