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Maths and gambling

You could be gambling with your future not because you want to, but because you haven't thought the maths through

Maths and gambling

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The other day on Twitter, I came across a thread where the topic was the difference between aggregating point-in-time experiences vs aggregating the same experiences over a period of time. That sounds like something exotic but it's actually a very simple idea and every investor should understand it.

The original tweet asked if you were offered an investment that had a 50 percent chance of returning 0.6x (40% loss) and a 50 percent chance of returning 1.5x (50%) gain, should you take it? The answer would appear to be an obvious yes because (0.5 * 0.6) + (0.5 * 1.5) = 1.05. You can, on average, expect a 5 percent gain so why not?

However, the plot of this story is thicker than that, because if you were to repeatedly do this, you would eventually go bankrupt. That's right, two options that average out to 1.05, if taken repeatedly, would cause you to lose money, and continue to lose it. For all but the most mathematically challenged, the reason should be evident. The above equation executes both the arms of the experiment simultaneously. However, if you first lose 40 percent, and then gain 50 percent, you are left with a 10 percent loss. Obviously, the same thing happens when the gain comes first and the loss come later. So when averaged, this gets you 1.05x, but when done sequentially, it's 0.9x.

At one level, this is a simple trick of maths, worth no more than a witty tweet. You could say that all this signifies is that to compensate for a 0.6x loss, you need a 1.667x gain and that's it. However, there's a point here which in real life is the cause of a lot of losses for investors. As your loss increases, the gain needed to compensate for it goes up exponentially. If, instead of 0.6x, you got 0.5x, then the 1.67x shoots up to 2. At 0.4x, it has shot up to 2.5x. At 0.1x (something that happens to lots of stocks), you'll need 10x to compensate!

The moral of the story is that it's very hard to make up for losses, and it gets disproportionately harder even with modestly larger losses. Averages are fine for writing articles and for looking at a large mass of investments. However, the average (or any other kind of aggregate) may not reflect the experience of the individuals that make that set of investments.

To understand my point in a more colourful story, read this thought experiment that Nassim Nicholas Taleb conducts in his book 'Skin in the Game': First case, one hundred persons go to a casino to gamble a certain set amount. Some may lose, some may win, and we can infer at the end of the day what the "edge" is, that is, calculate the returns simply by counting the money left with the people who return. ... Now assume that gambler number 28 goes bust. Will gambler number 29 be affected? No. You can safely calculate, from your sample, that about 1% of the gamblers will go bust. And if you keep playing and playing, you will be expected to have about the same ratio, 1% of gamblers go bust over that time window. Now compare to the second case in the thought experiment. One person goes to the Casino a hundred days in a row, starting with a set amount. On day 28 he is bust. Will there be day 29? No. No matter how good he is, you can safely calculate that he has a 100% probability of eventually going bust.

Let me not scare you, but for the individual investor who is trying to meet his life's financial goals, it's better to err on the side of conservatism than adventure. Our life is a series of sequential actions and therefore we are the second case in the above story. As to how much conservatism and how to decide that, we'll talk about that next week.