To reduce the risk that the results are portfolio-specific, we construct the maximum number of combinations of portfolios of different sizes (ranging between two and 100 assets per portfolio) using daily returns. For each of these sets of portfolios, we forecast VaRs for the period 2005–2012 using the EWMA and GARCH models. To keep things simple, these models are all estimated over a common sample period. The expected losses can then be compared with actual losses, allowing “violation ratios” for portfolios of differing sizes to be calculated. The results are summarised in Chart 19. For very simple portfolios of two or three assets, the performance of the simple and complex models is not so different. As the number of assets increases, however, the simple model progressively out-performs the complex one. That is a direct result of the uncertainties associated with over-fitting the complex model relative to the simple one. The message from these experiments is clear and consistent. Complexity of models or portfolios generates robustness problems when understanding a complex financial system over plausible sample sizes. More than that, simplicity rather than complexity may be better capable of solving these robustness problems. (7)  Public policy – more or less? In forgone output, financial crises can be as costly as wars. The public policy issue, then, is whether the war on crises is best waged with the weapons of the past. Einstein wrote that: “The problems that exist in the world today cannot be solved by the level of thinking that created them”.
Drawing on experience from a variety of real world systems – nuclear power plants, oil rigs, aircraft navigation systems – Tim Harford (2011) illustrates how complex control of a complex environment has often been calamitous. The general message here is that the more complex the environment, the greater the perils of complex control. The optimal response to a complex environment is often not a fully state-contingent rule. Rather, it is to simplify and streamline (Gigerenzer (2010)). In complex environments, decision rules based on one, or a few, good reasons can trump sophisticated alternatives. Less may be more. In Isaiah Berlin’s famous essay “The Hedgehog and the Fox” (Berlin (1953)), the fox knew many things, the hedgehog one big thing. Philosophers continue to debate their relative merits. But sports fans, doctors and detectives have made their choice. They think the hedgehog has the upper paw. Selective unclogging of the cognitive inbox can make for better decisions. (c)  Weighting may be in vain  John von Neumann and Oskar Morgenstern established that optimal decision-making involved probabilistically-weighting all possible future outcomes (von Neumann and Morgenstern (1944)). Multiple regression techniques are the statistical analogue of von Neumann-Morgenstern optimisation, with behaviour inferred by probabilistically-weighting explanatory factors. 3  Gigerenzer and Kurzenhäuser (2005). 4  Snook et al (2005). 5  DeMiguel et al (2007). 6  Wübben and von Wangenheim (2008)). 4  BIS central bankers’ speeches  In an uncertain environment, where statistical probabilities are unknown, however, these approaches to decision-making may no longer be suitable.