What does the FIFA 2022 World Cup and investment risk management have in common?

Bootstrapping to find less-probable scenarios

Instead of going with the most probable scenario, we artificially simulate that the World Cup is played 10,000 times, using these winning probabilities to determine the outcome of every single match. The method is well-known from both science and the financial industry, and the result is 10,000 different equally-probable outcomes for the World Cup. Some of these scenarios will have some of the underdogs as winners, such as this example from the playoff rounds, where Australia turns out as winners:

Only 2 of the 10,000 scenarios have Australia as winner, and as shown above, Australia will need events to go against the odds by beating Netherlands despite an estimated winning chance of 15%.

Combining all 10,000 scenarios, the estimated probability for each country winning the world cup is:

An overview like this reveals that the favourite scenarios with Brazil as winner is only likely in 1/4 of the cases – and that other lower-rated teams like Denmark and Uruguay have decent winning chances.

Tail scenarios in risk management

So how is this identical to investment risk management? Let say you invested in the S&P 500 index five years ago. Assuming that dividends were reinvested, the return would be 66 % on the investment which seems like a brilliant return. But one cannot mention returns without considering the associated risk of the investment. And the return is based on one realized path – but what if the macro events would have been different during the past five years? Would the investment still has been as good?

One way to quantify the risk is by applying bootstrapping to estimate all scenarios – not only the most probable one. Starting five years ago, the return of the S&P 500 index has in most of the weeks been around 0.5 %, and the return in half of the weeks has been between -1.1% and 1.7%:

There are however some rarely occurring weeks with very large losses or gains, exactly like the scenarios where Australia wins the World Cup. In the worst week, the S&P 500 index lost close to 15 % of its value. Thus when doing investment considerations, these so-called tail scenarios can have quite a large impact on the performance and thus cannot be ignored, despite the small probabilities.

A standard way to quantify the impact of these low-probability events is via the bootstrapping method. We do that by simulating 5 years of performance for 10,000 different scenarios. For each scenario we create an artificial path of 5 years performance by drawing weekly return from the histogram above. Thus some of the scenarios will due to the stochasticity contain a lot of the positive weeks, some the very negative events. The latter is exactly what is of importance for investors considering their risk profile. Below we show the 10,000 simulated scenarios which could occur with equal probability, including the actual cumulative performance indicated by the black line.

If we compare to events which (according to this basic model) have the same probability of occurring as the probability of Australia winning the world cup, i.e. 2 in 10,000, the lucky investor would have made 632 %. And the unlucky investor would have lost more than 72 % of the invested value in the worst 2 scenarios out of 10,000.

As for both the World Cup and financial investments, the least probable events are usually the most exciting ones, and bootstrapping is a simple way of quantifying these probabilities. When it comes to wealth management, tail-events can have severe impact on the performance. There are multiple ways of minimizing the impact from these tail events such as having diversified portfolios where the different assets have low correlations. Or by considering options which are far out of the money and only will be triggered in these tail events, although there is a premium associated with buying the option.

Note that the purpose of the bootstrapping exercise above is not to get the mathematical accuracies correct, but it should work as an illustration of how bootstrapping works when quantifying low-probability events. All simulated returns rely on historical data and should not be considered as indications for future returns. The used odds and Elo ratings were updated as of Nov 18, 2022.