Yes, You Should Gamble (Sometimes)

A few years ago, I transferred-in an account for a client. As I looked through the positions to prepare recommendations about which positions to sell and which to keep, I noticed a handful of penny stocks. Actually, to call them penny stocks would be an exaggeration. They were each worth fractions of a penny and, of course, only traded over-the-counter.

I assumed that these were positions-gone-bad—stocks that had fallen far from grace, trophies to amateur overconfidence. I called my client to discuss removing them.

“…Oh, and one more thing. I’ll send you a form to remove these stocks from your account since they don’t trade and aren’t worth anything.”

“What?! No, don’t do that!” was his urgent reply. “Those are my lottery tickets! I put about a hundred bucks into each of them and I want to see if they pay off!”

I chuckled. “Alright, no problem, we’ll leave them, but I’m not going to follow them, okay? Just let me know if you change your mind.”

I didn’t know it then, but I gave him terrible advice that day. In fact, I should have been the one to tell him to put some money in those micro-penny stocks.

* * *

Before you excommunicate me as a heathen, at least hear me out. Let’s take a step back and remember where the advice “never gamble” comes from.

A standard utility function taught in the CFA Program curriculum (sometimes called quadratic utility) determines an investor’s happiness from her portfolio’s expected return, minus the variance (volatility) of those returns, times her risk aversion parameter. The more averse to risk, the more unhappy she is with variance (volatility).

In this model, all else equal, higher volatility is always bad. In this model we would never expect an investor to choose a high volatility, low-return portfolio (i.e., a gambling portfolio) when low-volatility, high-return portfolios are on offer. We have this expectation because this model assumes that the thing our investor wants to avoid is volatility.

By contrast, goals-based theories of choice take a different approach. Rather than define risk as volatility, goals-based utility defines risk as “not having the money you need when you need it,” to quote my friend Martin Tarlie. Risk, in goals-based investing, is not volatility, but the probability that you fail to achieve your goal. 

Running with this more intuitive definition yields some surprising results because it changes the math of the portfolio choice problem. We move from an equation in which return and volatility are the only two variables, to a probability equation of which return and volatility are inputs, but not the only inputs.

All the variables which define our goal (minimum wealth level, time horizon, current wealth, etc), are also inputs in the probability equation. Lastly, when we remove the inexplicable academic assumption that investors can borrow and sell short without limit, then we find that the efficient frontier has an endpoint, the last efficient portfolio.

Here’s the catch: sometimes, investors have return requirements that are greater than what the last efficient portfolio can offer. When that happens, her probability of achievement is maximized by increasing variance rather than decreasing it, even if returns are lower.

And so we enter the world of rational gambles.

Rational gambles are those portfolios to the right of and below the last efficient portfolio, but for which the probability of achievement continues to rise. Irrational gambles are those for which the probability of achievement begins to fall. The plot below illustrates the point.