Our article Law of averages, which focused on the distinction between ‘time’ and ‘ensemble’ averages, provoked some interesting comments and feedback – both face-to-face and in the Twittersphere. This included a degree of scepticism over some of the maths involved in a potentially lethal game we called ‘Russian dice’ so we are going to revisit the subject with a different example.
So let’s play a game. Nothing so nasty as ‘Russian dice’ this time – you simply start off with a stake of, say, £100 in the pot and we toss a coin. Every time the coin comes up heads, you increase what is in the pot by 50%; every time it comes up tails, you lose 40% of whatever is in the pot. The coin is a fair one – so it is always a straight 50/50 chance – and there is not a firearm in sight. Would you like to play?
At first sight, there appears to be no reason not to play – after all, if on each coin toss the only two possibilities are going up 50% or falling 40%, then surely, on average, it is a winning game. And indeed it is. No matter how long you play it for, on average, the expected return is positive – but, as we argued in the previous article, the word ‘average’, if not defined very precisely, can be misleading.
In this particular example, the average obscures a pattern where the majority of people who play the game actually end up losing. Clearly, if you throw a succession of heads, you are going to end up very rich indeed but that is just one permutation and there are considerably more where it is the ‘house’ that ends up winning.
The game can be played indefinitely but, for the sake of space and simpler maths, we will confine ourselves to all the possible permutations involved if the coin is tossed four times in a row. On the bright side, if you throw four heads in a row, your £100 initial stake rises to £506.25; less happily, if you throw four tails in a row, you are down to just £12.96.
There is, of course, only one specific route to reach either of these extremes – in one case, all heads; in the other, all tails – but, as you can see from the table below, there are 14 other four-toss permutations. Four of these will take you to a positive outcome of £202.50 but six others will see your initial £100 stake reduced to £81 while four more will leave it down at £32.40.
Source: Schroders, September 2014
In other words, in the above example, 11 of the 16 possible sequences of coin tosses are losing permutations and, the longer the game goes on, the more money is lost. In a game where the only possible outcome each time is a 50% upside or a 40% downside, that would appear counter-intuitive so what is complicating matters? It is that difference between time and ensemble averages.
As a whole – that is, if one takes the ensemble average – this is a great game. If 16 people play the example above and each ends up throwing one of the 16 different permutations of four coin tosses, the ensemble average tells us that each £100 initial stake has ended up at almost £122 – the maths being (£506.25 x 1) + (£202.50 x 4) + (£81 x 6) + (£32.40 x 4) + (£12.96 x 1) or £1,944.81 ÷ 16 = £121.55.
So, in ensemble average terms, after four coin tosses our game has an expected return of close to 22% but the reality of our game is that time is an important factor and the time average shows that 11 out of the 16 players end up losing. What is more, the longer they play, the more they will lose – regardless of the fact that, as a whole, the game is a winning game.
By now we trust you would at least agree that ‘average’ is a very simple word that masks a very significant degree of complexity but there is no harm in hammering the point home with one brief investment-oriented example. It is ‘common wisdom’ that equities will over time average a return in the region of 8% a year but of course the reality is that average will include a few companies that have done spectacularly well and a lot more that have not. As investors, we should always be very careful about anything claiming to be ‘common wisdom’.