# Survivorship bias: an investment decision trap

### Nicolás Forteza

#### 21/02/2018

Survivorship bias is one of the most common biases in finance, and it’s easy to fall victim to it. Let’s find out how to remain vigilant and overcome this hurdle.

“History is written by the victors”. – Winston Churchill

A cognitive bias is a consequence of subjective judgement. When it comes to finance, would you say that this famous British politician was right?

We have lots of human biases in finance (or life), and a very common one is the Survivorship bias. It arises when you go back and try to reconstruct history, as in backtesting or during performance studies. More specifically, it will lead to a selection bias error type, which is the error that comes from sub-selecting assets. How so? We tend to look to what persists in time. Stocks can get acquired and be merged with others. Mutual funds can close. Therefore, not taking into account these dead assets can lead to overstated conclusions. Moreover, it affects every agent in the finance market: a portfolio manager, an analyst, and even investors. Not considering the risks properly may lead to wrong decisions! Please, allow me to explain it carefully with an example…

## The distortion of reality

Try to answer this simple question: what has been the average return of a stock index’s components? As you may know, a stock index is a weighted average (sometimes other measurements are computed) of the movements in prices of the market constituents. You may also know that the most common criteria for entering into the index is to have a big market capitalisation (at the same time, this depends on the performance of the stock):

$$market cap = P \times N$$

where $$N$$ is the number of outstanding shares of that company. Let’s keep this formula in mind and see the next graph (total return computations were made as in this fantastic post!).

As you can see, the mean return of all stocks that were part of the index in the given year is lower than the mean return of the survivors at the end of that year. This is due to the market capitalisation: as these stocks start performing poorly, its price falls, and they are removed from the index because of the minimum market capitalisation criteria (regardless the eventual M&A’s that could ever happen). In the previous chart, we saw how the EuroStoxx criteria (which, by the way, ensembles minimum liquidity and market cap conditions) and the FTSE 100 (top 100 companies by market cap of the London Stock Exchange) constituents change over time, although the number of constituents remains constant. In effect, different constituents are swapped in and out of the index according to their appropriateness.

Survivorship bias can distort the figures when evaluating a strategy. In our blog we have shown some examples of non-biased strategies. Not having treated the Survivorship bias effect properly would have led those figures to go much higher, since we would have been taking into account the “best performer” stocks. It’s not the same to select stocks from a smaller subsample (i.e., from the survivors) than from the whole universe (i.e. all the index constituents of that year). The absolute error representation of the phenomena would take the following form:

$$\epsilon_t = \| M(S_t) – M(A_t) \| \quad ; \forall S_t \in A_t , \forall t \in T$$

where $$S$$ is the set of the survivors, $$A$$ is the set of all the constituents and $$M$$ is a measure.