The Other Measure of Risk Aversion

Same letter. Different objects.

The range from the tickets

In the last post I showed an interval that measured risk aversion on a quiz. A clean switch on the worked example gave. That was my book. The higher the interval, the longer you stayed on the tighter sleeve after expected dollars had already flipped; the higher your risk-aversion. That interval, however, is only one measure. There is another coefficient that uses the same letter. We need a way to distinguish the two.

There is a formula that turns A into a share

As we go through our series on risk aversion, it would help if we define the two different coefficients of risk version.

The first one we encountered, the one based on the Holt-Laury lab experiment, we will refer to as A_HL. In this post, we will be discussing the second one.

The second one sits inside a short formula that turns a coefficient into a share of the book in the risky asset. We will call it A_M. It labels a mix once you already have one. It does not measure the tickets.

Same letter in the model. Different objects in the data.

The formula is simple. It takes A_M, the excess return on the hundred-year price sample, and the variance of that sample.

Call the excess return μ. On this book it is about 4.63 percentage points (8.43% which is the market average minus the 3.8% risk-free cash stand-in).

The variance is about 0.0345 (standard deviation of ~18.57%).

The formula is:

Omega $(\omega)$ is the share of the book in the risky asset. The formula is a label: given an $A_M$ you already accept, it returns the share that matches it. It is not a menu that writes you a portfolio from the quiz.

Now watch what happens if you ignore that and plug the quiz in anyway.

Take the middle of the worked-example interval, about 0.54. Feed that number and the sample moments into the formula. It prints a number well above 1.

Above 1 means more than 100% of the book in stocks. Leverage.

Even at the high end of the interval, about 0.68, the formula still prints a number above 1. The tickets do not produce a weight you can hold without borrowing.

We do not leverage the book on this instrument.

$A_{HL}$ measures appetite on the ten rows; how hard you worked to keep the $8,000 floor after expected dollars had already flipped. $A_M$ is the number that goes with a mix you already chose. Change the mix and the number changes. It does not measure the tickets.

They share a letter. They are not interchangeable, that’s why we use subscripts, so we can tell the difference.

Next we look at ordinary mixes — 80/20 down to 20/80 — and the $A_M$ that labels each one.

Stay tuned.