An introduction to Expected Value

In the classic family game of Russian Roulette, you have a revolver with 6 chambers with a bullet in one.

There is a 5 in 6 chance of pulling the trigger with no adverse consequences.

So in the majority of spins, playing Russian Roulette is harmless.

The problem is obviously that one time out of 6 (on average) the player suffers a catastrophic loss, an event that takes them out of the game.

The investing equivalent of this is bankruptcy and ruin, a situation where you lose so much that it’s damn near impossible to come back from over any meaningful period of time.

Russian Roulette also illustrates another feature of randomness: you can’t tell the quality of a decision from the outcome. Someone who played Russian Roulette and got away with it didn’t make a good decision, they just got lucky.

We want to make good decisions in investing and in life. What we are looking for is a way to identify good decisions with positive expected outcomes. Good decisions that take into account the chances of things going very well or very badly. This is the concept of Expected Value.

“Take the probability of loss times the amount of possible loss from the probability of gain times the amount of possible gain. That is what we’re trying to do. It’s imperfect but that’s what it’s all about.”

Warren Buffett

The Expected Value (EV) is the weighted average net benefit of a given choice. It’s the average outcome if the decision was made many times.

We can put numbers to this. The formula is:

Expected Value = (Amount of gain x % probability of gain) – (Amount of loss x probability of loss)

Russian Roulette is a dumb game to play because there is more downside than upside. Or, to use the lingo, it involves asymmetric pay-offs skewed to the downside.

But once we think in these terms, we can start looking for asymmetric pay-offs skewed to the upside.

An investment with a low probability of success can still carry a high expected value. Most of these investments are failures but occasionally something amazingly good happens that more than makes up for it. This is the opposite of Russian Roulette.

Take venture capital for example.

Most venture capital investments fail.

It is the small number of successful “unicorns” (start ups that grow to a value > £1 billion) that make up for all the failures and drive the overall returns of the fund and venture capital as an asset class.

This is why it’s smart to diversify with a fund.

It’s the same with the stockmarket. Most small cap companies fail or stay small. It’s the handful that grow into mega caps (e.g. Microsoft, Google etc) that make up for the failures and drive the overall positive returns of the index and equities as an asset class. This is why it’s smart to diversify with a fund.

Or take crypto as another example. Many of you are sceptics: I get it. So let’s all agree for the purposes of this simplified example that Bitcoin will probably fail at some point.

What if we estimate that Bitcoin has a 60% chance of total failure (it goes to zero due to hack, government attack or other catastrophic failure) but a 40% chance of success where it ends up replacing gold as a store of value?

Gold has a market cap of roughly $11 trillion versus Bitcoin’s of roughly $1.1 trillion currently. What if Bitcoin has a 40% chance of replacing gold and increasing in value by a factor of 10x?

Unrealistic? Remember that Bitcoin has already compounded at 224% per year for an entire decade. At that rate, you 10x your £100 into £1,000 in just 2 years.

With these odds, putting £100 into Bitcoin is a bet with a positive expected value. The calculation of expected value is : (40% x £1,000) – (60% x £100) = £340

I’m not recommending Bitcoin or saying these numbers are right or wrong (we can never be sure).

I’m saying that investors should understand the concept of Expected Value!

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  1. […] An Introduction To Expected Value – The Escape Artist […]

  2. A useful discussion of expected value, though for the Bitcoin example my feeling is that it’s closer to 95/5 that it will fail (or be replaced with a different cryptocurrency that offers most of the same benefits with fewer of the major flaws, which amounts to the same thing).

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