When economists speak about financial risk, and investor behavior the term “utility” comes up time and time again, and it seems safe to say that much confusion arises due to the fact that most readers do not speak or think in these terms. In this context the term “utility” relates to the idea that withdrawal amounts are of no real value for their own sake, but are converted to units of utility which is a rough proxy for “happiness”, or at least the aspects of happiness that money is related to. Numerically, this conversion is done via a preference function. In these functions some measure of wealth (w) is related to Utility (U). The simplest example would be U = w. However, the preference function will also account for the level of risk-aversion of the decision maker. For example, we may see U = ln(w). This classic form reflects the fact that an increase in w does produce more utility, but at a decreasing rate. Moving from $1000 to $2000 has a higher marginal impact than going from $1,000,000 to $1,001,000.
When we consider the lengthy research stream on retirement planning and portfolio theory, we find that when preference functions are formally considered, the vast majority of published works assume that the retiree exhibits constant relative risk aversion (CRRA). In simple terms, this is the assumption that if an investor or other retiree is indifferent between doing nothing and having a 50/50 chance at an increase of x percent versus a loss of y percent where x > y, then this evaluation is independent of the initial level of wealth. Moving from $1000 to $2000 is a much larger increase (in percentage) terms when compared to moving from $1,000,000 to $1,001,000. CRRA implies that if the utility increase from a 10-percent grain is equal and opposite to the utility decrease from a 5-percent loss, then this holds whether the investor starts with $1,000 or $1,000,000. Field experiments often find that this ratio of 2 to 1 or 3 to 1 are quite common. (See https://www.youtube.com/watch?v=vBX-KulgJ1o&t=30s as an example.) As a practical matter, this aspect of human behavior is often labeled as loss aversion. We feel losses more than we feel comparable gains.
Under this common assumption, it is easy to show that the only utility functions consistent with this story are transformations of 1/w ^ g where the coefficient g reflects the level of risk aversion. Let’s convert this function to simple English. Let’s start with a wealth level of 100 (or 100% if you like.) A 10% gain produces an increase of utility of 1/(10 ^ g) if your level of risk aversion corresponds a value of g = 2 the increase in utility is 1/10 = 0.1. If you look a loss of around 3.2% you have a negative utility change of – 1/ (3.2 ^ 2) = -0.1. In other words, a 10% gain produces about as much happiness as you would lose from around a 3% loss. When viewed in these naked mathematical terms, this may seem extreme, but psychologists and other researchers see this quite often, and retirement researchers consider a value of g equal to 2 to be a good starting point for a discussion of gains and losses with potential clients.
You almost certainly missed what I consider to be the most important aspect of the prior discussion. If a loss of money delivery a loss of utility described by – 1 / (w) ^ g and g is a positive constant greater than 1, I can ignore the precise value of g for the moment. The loss of utility is related to -1/w where w is the level of wealth. If w ever gets to 0, we do not have a utility level of 0. In stead we have a utility level proportional to -1/0 which is negative infinity. Let me repeat that in plain English. If your level of wealth ever gets to 0, that outcome is infinitely bad. Consider the expected utility from a strategy that has a 99.9999% chance to reach a value of 100 and a 0.00001 chance to reach a level of 0. This gives U = 99.9999 * 100 + 0.00001 * (Negative Infinity). The resulting value of simply negative infinity. Thus, any investment strategy that assigns a positive probability to such an outcome (no matter how small that probability is) has an expected utility of negative infinity. No rational decision maker would EVER implement such a strategy.
You need to let that sink in for a moment. NO ONE will EVER use such a strategy, if they are rational beings. You may have missed my slight of hand here. What I just said was that NO ONE will EVER use the 4% rule because it has this fatal flaw. Following such a rule assigns a positive probability to an infinitely bad outcome, which is why you do not use it. Your instinctive response is probably something like, “come on man, everybody uses the 4% rule. Just look at the thousands of references, articles, newscasts, blogs, videos, courses, books, and speakers.” Read carefully, I didn’t say no one will CLAIM to use the 4% rule. I just said no one will actually use the 4% rule. Let me explain. Anyone will use the rule, until it is time to stop using it. As long as the market goes up or doesn’t go down “too much” you can use it. But when the crash happens what will you do? You will adjust. Why? Because you are not an idiot. When you see yourself running of our money, you change the policy. You cut back, you get a reverse mortgage, your downsize, you get a part time job, etc.
My message is simply this. If you know in your heart you are not going to do something stupid when it really counts, can we all just stop talking about it as if it were ever a real rule to begin with? Think it over, and if you want more info and another strategy, pick up my paper on the topic here (https://papers.ssrn.com/sol3/papers.cfm?abstract_id=4011287) or on the document library page here http://cchambers1906.com/document.