No it is not some heavenly choir but an earth bound concept which may very well turn out to be the “Trojan horse” of finance upon which a bed of very subtle non-truths lie. Like ‘Don’t worry about any other risk, volatility is a good proxy for risk and we have that under control’ and ‘Diversify and your investments will be safe’. The reality is something different, volatility is not a good proxy for risk and diversification could very well be diworsification!
Where did reality and theory part ways? The answer might lie in the words “ergodic ensemble”. This is the idea that time averages and ensemble averages are the same thing. It is not an easy concept to intuitively understand so I will explain with a few examples.
- There is a tender from God and you are given a chance to give up a percentage of your wealth in order to replay one day in your life in the future e.g. one of your children gets run over by a car, you replay the day and avoid the situation. How much would you pay to replay?
- You are in the market for a new Land Cruiser and you get offered one at 60% discount but there is a thousand to one chance the brakes will fail. Would you take it?
- There is a coin tossing game on the street. You will be paid 50% profit of your initial stake if you win and make a 40% loss if you lose. The dealer must end the game. Would you play? How much of your initial bankroll would you pay to insure your initial stake?
Let’s look at each one in turn.
- How much to replay one day? Using ensemble or arithmetical average we would argue the chances of anything bad happening on any one day is so small it is worth ignoring. Time averaging says the following: Well if you are 40 and expected to live 40 years, you have 40*365 expected days left to live or 14,600 days. In your remaining life, the odds are high there will be an event worth replaying. The value of that event will be worth at least 50% of your wealth if not all of it so tender high for the right to replay. One says ignore, one says overpay. Which is right?
- The Land Cruiser problem. If there were 1000 Land Cruisers in front of you, 999 perfect and 1 with faulty brakes then you should probably go for the discount i.e. ensemble average as the odds on choosing the faulty one is so small (1000-1). If you buy a Land Cruiser and it is a time average i.e. one in one thousand times you use it (over time) then the brakes will most likely fail within 3 years. In which case you should probably not go for the deal unless you have a death wish! Which is right?
- The coin toss. You should never play even though the odds are in your favour. Risk of ruin is too high. After 1000 coin tosses 999 people out of 1000 would have gone to zero. But if you do decide to play, you should pay anything up to 99% of your original bankroll to insure against bankruptcy. But who would pay to insure a deal where the odds are in your favour?
We theorise about life using ensemble averages but live it experiencing time averages. Ditto for portfolio management. We are constantly surprised when something bad happens to us or our portfolio and we very rarely buy tail hedges because they seem too expensive whereas in reality they are cheap. We are like 6 foot tall people wanting to cross a river and being told it is a ‘constant 4 foot deep, without currents and only the odd pothole. You will make it safely to the other side.’ The reality is different; every couple of years time dishes up 20 foot waves and rip currents and large trenches that we fall into and drown. Never cross a river because it is 4 feet deep on average.
And we wonder why so many investors never make it intact to the other side?