# There's another way to look at the risk/return trade-off

### Imagine a chance to win \$10,000!

Imagine there are fifty red and fifty black balls in an urn like the one pictured above. You are going to be allowed to select a color. One ball is going to be picked at random. Imagine, too, that if you pick correctly, you will receive \$10,000. That’s a 50/50 shot at winning a lot of money. How much would you pay for a one-off chance to play that game? Note your answer.

But, hold on. Suppose I now told you that the description above is not quite correct. Suppose I get to choose how many red and how many black balls there are and I’m not going to tell you what that mixture is. So now you don’t know the mix but, other than that, the game is the same. You still get to choose a color; a ball still gets chosen at random; it’s still a one-off game and if you’re right you still get \$10,000. Now how much would you pay to play?

I had the audience at the Russell Investments institutional summit go through that exercise earlier this week. If you are like over 90% of that audience, you’d pay less to play the second version of the game than you would to play the first.

## Some risks are scarier than others…

The exercise above may tell us something about where we should look for investment returns. We all know that risk and return are very closely connected. But there’s more than one type of risk, and the difference between the two versions of the urn game above does not lie in the odds: it lies in the type of risk involved.

First of all, let’s explain why the odds are no worse in the second version of the game. The ball that is chosen will still be either red or black; the two probabilities have to add up to 100%. If you think you can second-guess me and predict which color I’ll overweight in the urn, then you can maybe even get yourself odds better than 50% in the second version. But if you don’t want to try that, just pick a color at random and you know you’ve made it a 50/50 proposition. If you make a random choice between red and black then, no matter what the mix is and no matter what actually gets pulled out, you have a 50/50 chance of calling it right.

Yet even though the odds are no worse, most people will pay less to play the second game. That’s because it’s a different type of risk.

I’ve written before about Andrew Lo and Mark Mueller’s paper “WARNING! Physics Envy May Be Hazardous To Your Wealth!” in which they describe a hierarchy of five levels of uncertainty.  In the urn exercise, the second version of the game moved us from level 2 (known risk) of that hierarchy to level 3 (sort-of known risk – or fully reducible uncertainty as Lo and Mueller term it.) And most people don’t like that. They like the next level even less; that’s partially reducible uncertainty, or not-really-knowable risk.

## …and that means they may offer better returns

The fact that most people don’t like the second version of the game can translate into a bigger profit opportunity. And that idea extends into real investment markets: if you give markets a known distribution of possible outcomes, there’s an army of people who can analyze that and price it. But something that cannot ultimately be quantified is more difficult to price. Investors are more likely to steer clear. If other investors are steering clear, then that could enhance the return potential.

Lo and Mueller’s paper built upon Frank Knight’s famous work “Risk, Uncertainty, and Profit”, which documented the difference between risk (by which he meant risk that is measurable) and uncertainty (which is not) almost 100 years ago. Although that paper remains widely cited today, those citations rarely touch on Knight’s main argument, which was that “it is this ‘true’ uncertainty, and not risk… which forms the basis of a valid theory of profit”. In other words, Knight argued that it is only the unquantifiable uncertainty that carries any worthwhile profit opportunity. Some would argue that that’s overstating it. And it would certainly not be right, of course, to suppose that every instance of unquantifiable uncertainty necessarily delivers a positive expected return.

But there does seem to be a good case to be made that return opportunities and risk premia may be more likely to be found in unquantifiable uncertainty than in simpler, more comfortable, types of risk.

Appendix: Mueller & Lo’s five levels of uncertainty¹

Level 1: Complete Certainty

Level 2: Risk without Uncertainty

Level 3: Fully Reducible Uncertainty

Level 4: Partially Reducible Uncertainty

Level 5: Irreducible Uncertainty

¹Source: Lo, A. and M. Mueller (2010) WARNING! Physics Envy May Be Hazardous To Your Wealth!