A recent report on risk management mentions near the top that risk and reward have a fundamental relationship. But experience tells us that just is not at all true in most situations.

The first person (that RISKVIEWS can find) to comment on that relationship was the great economist Alfred Marshall:

“in all undertakings in which there are risks of great losses, there must also be hopes of great gains.”

1890 Principles of Economics

That seems to be a very realistic characterization of the relationship – one of hope. But his statement has been heavily distorted through the years. Many have come to believe that if you increase risk then you also, automatically, increase reward. Or that if you want increased reward that you must increase risk.

Perhaps the risk reward relationship is a simple arithmetic statement. Made by those who believe that all economic actors are rational. And by rational, they mean that they make choices to maximize expected value.

So if all of the choices that you actively consider have a positive expected value, then those with higher risk will have to have higher rewards to keep the sum positive. (Alternately, risks would have much lower likelihood than gains – but this hardly seems to fit in with the concept of higher risks.)

So perhaps the “relationship” between risk and reward is this:

For opportunities where the risk and reward can be reliably determined in both amount and likelihood, then among those opportunities with a positive expected value, those with higher risk will have higher reward.

But isn’t that the rub? Can we reliably determine risk, reward and their likelihood for most opportunities?

But then there is another issue. For a single opportunity, the outcome will either be a loss or a gain. If there is higher risk, the likelihood or amount of loss is higher. So if there is higher risk, there is a higher chance of a loss or a higher chance of a larger loss.

So by definition, an opportunity with higher risk may just produce a loss. And either the likelihood or amount of that loss will, by definition, be higher. No reward – LOSS.

Now, you can reduce the likelihood of that loss by creating a diversified portfolio of such opportunities. And by diversified, read unrelated.

So the rule above needs to be amended…

For opportunities where the risk and reward can be reliably determined in both amount and likelihood, then among those opportunities with a positive expected value, those with higher risk will have higher reward. To reliably achieve a higher reward, rather than more losses, it is necessary to choose a number of these opportunities that are unrelated.

Realize here that we are talking about Knightian risk here. Risk where the likelihood is knowable. For Knightian Uncertainty – where the likelihood is not knowable – this is much more difficult to achieve. Investors and business people who realize that they are faced by Uncertainty will usually Hope for even greater gains. They require higher potential returns. And/or set higher prices.

The issue is that in many cases, humans will make mistakes when assessing likelihood of uncertainty, risk and reward (see Restaurant failure rate). There are quite a number of reasons for that. One of my favorites is survivor bias in our data of comparables (They just don’t make them like they used to). We also overestimate our chances of success because we overrate our own capabilities. (see Lake Wobegone, above average children). And to achieve that portfolio diversification effect, we need to be able to also reliably assess interdependence (see mortgage interdependence, 2008).

The real world problem is that aside from lottery tickets, there are very few opportunities where the likelihood of losses is actually knowable. So risk and reward are not necessarily related. Except perhaps in the way that all humans are related . . . through Adam (or Lucy if you prefer).