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‘Expected value’ is the notion of trying to assess the value of anything with an uncertain monetary worth – be it a share in a company or a lottery ticket
Here in the UK, your odds of winning the National Lottery do not change. When, for example, you could pick from 49 numbers, your chances of scooping the jackpot – that is, of your six numbers exactly matching the six drawn at random by ‘Arthur’ or one of the other three machines – were always 1 in 13,983,816. Since 10 extra numbers were added in 2015, they now always stand at a tantalising 1 in 45,057,474.
How MIT students made a fortune
Not all lotteries operate in the same way, however, and an unusual feature of the Massachusetts state lottery allowed a syndicate of MIT students to win millions back in the middle of the last decade. Having spotted that, if the main jackpot was not claimed, it was ‘rolled down’ to increase the lesser prizes, they calculated that on roll-down days the mathematical value of a $2 ticket could actually be as high as $5.53.
Providing they could buy enough tickets to make the maths work, they would make a fortune. The syndicate started manually filling out thousands of tickets, the maths worked and they duly made a fortune. All of which was completely legal – the students were just playing the odds in their favour using the idea of ‘expected value’ or ‘expected return’, which we also touched upon in the context of the gameshow Deal or no Deal.
The MIT story is highlighted in How Not to be Wrong: The Hidden Maths of Everyday Life and we were interested to see its author Jordan Ellenberg also took the opportunity to discuss the origins of expected value. After all, the notion of trying to assess the value of anything with an uncertain monetary worth – be it a lottery ticket or a share in a company – has huge resonance for us, here on The Value Perspective.
Expected value
The concept of expected value as we know it today, Ellenberg explains, really began to take shape in the late 1600s when, to fund its various military endeavours in Europe, the government of William III decided to start selling life annuities to the UK’s population. In other words, in return for an initial lump sum, the Crown would pay a guaranteed annual pay-out for the rest of the annuitant’s life.
With actuarial science still in its infancy, however, the cost of the annuity was set without reference to the annuitant’s age – meaning a 20-year-old could pay the same price as a 40-year-old despite the former theoretically being able to look forward to two extra decades of annual payments. That may seem absurd to us now but it took the then Astronomer-Royal to illustrate why age was an important part of this equation.
He was Edmond Halley – of comet fame – whom we met recently in A range of possible outcomes. “By using birth and death statistics, Halley was able to estimate the probability of various lifespans for each annuitant, and thereby to compute the expected value of the annuity,” Ellenberg writes. And, as a result: “Grandpa, with his shorter expected lifespan, pays less for an annuity than Junior.”
'Average value' a better name?
Expected value, as Ellenberg also points out, is one of a number of important mathematical notions whose names do not quite capture their meaning. If you bet £10 on a dog at odds of 10 to 1, then you are really only expecting one of two outcomes – £100 or nothing. But since your bet assumed the dog had a 10% chance of winning its race, your expected value was (10% x £100) + (90% x £0) = £10.
And, as Ellenberg notes, that is not even a possible outcome of the bet, let alone the expected one. He adds: “A better name might be ‘average value’ – for what the expected value of the bet really measures is what you would expect to happen if you made many such bets on many such dogs.”
Applying the same notion to stock valuation, if our estimated value of a company is £10 a share but we pay £12 a share, we are very likely to lose money in the long run. If we can pay £5 a share, however, we should try to buy as many as we can. As we have said many times before, value investing is all about keeping “on the right side of the averages” – waiting for the market to offer you a pound’s worth of assets for 50p.
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