I think it’s time to talk more about Time.

In “What’s Your Future Worth?” I talk about the how to develop your own personal rate of discount and how important that is to making good decisions. It’s a hard concept and one that I find does not come naturally to most people. Unfortunately, the nature of Time (in this context) is also more complicated than I presented in the book. In the next few blog posts I will share some of my thoughts on how we humans experience the passage of time and how we should take that awareness into every important decision that we make. But for now, let’s start with one of the most important aspects of Time that we always need to consider and that is –when does time end?

Omega and the Actuarial Perspective on the End of Time

“There’s them that comes in (to the population) and them that goes out. There’s all kind a ways to get in, but there ain’t but one way to get out and that’s to die”
Professor Robert Batten, FSA — explaining the mathematics of Stationary Populations (a key part of “Life Contingencies”) to a class of actuarial students

The essence of actuarial mathematics (at least as it was taught in the 1980’s) and the “Pons Asinorum” for most actuaries of my generation was a subject called “Life Contingencies” explicated in a beautiful book of the same name by C.W. Jordan. Most theoretical mathematicians have never even heard of the subject let alone studied it, but almost all actuaries of a certain age will attest that it is every bit as hard, abstract and “pure” as any undergraduate math course they took in college, and for most of us, a well-thumbed and treasured copy of “Jordan” still resides on our bookshelves. Like most pure math, Life Contingencies has a pristine and pure elegance that richly rewards those who make the effort to learn its intricacies and master the deep and elegant theorems that are at its core. The only small problem that even those of us who loved the math had with Life Contingencies was that in order to become a licensed actuary we had to pass a 5 hour test on the subject that was diabolical in its construction, overwhelming in its scope and contained far more problems than could possibly be completed in the time allowed. Even those who studied for several hundred hours and thought they thoroughly understood the material would come out of the exam feeling like they had only scratched the surface in their preparation. To make matters worse, the Society of Actuaries (who administered the exam) used this particular exam to limit the number actuaries getting licensed and only passed about 25%-30% of those that took it. The test was so hard that most actuaries needed extra course work to pass it, and when my employer offered to send me to Georgia State University to attend a seminar given by Professor Robert Batten designed to help actuarial students like me get “over the hump” and on with our careers I jumped at the chance.

Even considering the wide range of characters inhabiting the profession, Professor Batten was not your typical actuary. A “good ole boy” from somewhere deep in the South, his neck was not just red, but bright red. A rabid Atlanta Braves fan he seemed far more comfortable downing a beer at the ballpark (where he took the class after the seminar ended) than explaining Lindstone’s theorem of annuity values to a bunch of aspiring actuaries. Yet his seminar was riveting. He attacked the subject from two sides. On the one hand, he provided colorful and cogent explanations of the math, almost always punctuated with pointed and often funny stories taken from his obviously less than academic personal history. Somehow his deep southern drawl and down home expressions made the subject approachable and less intimidating.

But the second and most important aspect of his teaching was his dissection of the actual problems from prior exams, where he taught us all kinds of ways to solve problems (or more specifically to get the correct answer) that were not in any book. There was one powerful technique he taught us to use when faced with a particularly inscrutable equation to solve. Specifically, he showed us that when we were faced with such a problem we could simply “special case it”. In his terminology this meant to consider what the expression (which was usually filled with all kinds of variables) would look like at the extreme values (eg if interest rates went to zero or if people became immortal and nobody ever died). Often surprising insights about the nature of the equation could be gained and/or most of the possible answers (it was a multiple choice exam) could be eliminated. In addition to being brutally pragmatic it was a mind expanding way to look at the fundamental equations that governed our profession. It was at this seminar and thinking about this technique that I became intimately familiar and finally understood on a deep level two important concepts that would ultimately inform my thinking about Time and its relationship to actuarial science. One (“Omega”) I will talk about now and the other (“Stationary Populations”) will be discussed in a future blog post.

In actuarial science, Omega, is defined as the limit of the human lifespan. Actuaries need this concept because we use mortality tables that assign a probability of death to every age (eg a 65 year old man has about a 1% chance of dying before turns 66 while approximately one in ten 90 year old women die before reaching age 91) and we need to pick an age where the table ends and the probability of dying within the year is 100%. Whether such an age exists is an interesting philosophical/scientific question, and maybe one day we will know the answer, but for actuaries the concept of Omega is merely the solution to the practical problem of how to keep the calculation of present value tractable. The mathematicians among you may say that a specific Omega is not necessary if one defines mortality as a function of age and in fact there were many early attempts by actuaries to define such a function using ideas like the “force” of mortality (based on an assumed basic law of the universe whereby the “life force” of a human naturally and predictably diminishes over time). Unfortunately none of those attempts were successful in closely modeling the observed mortality rates upon which so much, including the financial solvency of insurance companies, often depends. In the future l may talk more about some of those attempts in the context of how actuaries have historically tried to quantify Risk, but for now suffice it to say that history has shown that the best way to develop a mortality table is to observe the ages at which millions of people actually die and develop tables based on those statistics. This is not as simple as it sounds and in fact there is a whole sub branch of actuarial science called “Mortality Table Construction” that is devoted to that effort.

But what interested me was the broader implications of assuming an age where essentially time stops. Consider an Insurance company that issues an annuity to a 75 year old man that promises to pay him $1000 per month for as long as he lives. Currently most Insurance Companies use an Omega of 110. If we assume the reality of Omega, then the Insurance Company and our 75 year old are not entering into a permanent contract which might end at any time, but rather one that they both assume will end in 35 years. This tension between a timeline with a fixed (certain) endpoint and one that theoretically could go on forever comes up again and again in the actuarial world. It is the fixed timeline that we most often use (sometimes implicitly), but in fact many of the promises made are explicitly indeterminate, and “eternal” in nature. This disconnect has always troubled me and continues to trouble me to this day. I truly believe that no one will live forever, and as a practical matter the cost of the annuity described above won’t change whether we assume a 100% probability of death at 110 or assume a vanishingly small (but still positive) probability of living to 175 or beyond. But what about the Insurance Company that issued that contract? Can we really assume that they will be around to make good on their promise 100 years from now? Is there an Omega for Insurance Companies as well that we should be considering? It might not make a difference from a cost perspective, but it does seem odd to me that we consider the “natural” lifespans of one party to the transactions we value, but blithely ignore the lifespan of the other.

The above is particularly troubling (and relevant to my work as retirement actuary) when we consider the case of a Company sponsored traditional Pension Plan where the Plan promises to pay a fixed pension for life to any employee who renders sufficient service with the Company and then retires. For a 25 year old newly hired employee that promise could extend another 80 years (or even longer if she takes on a younger husband and elects a “joint” annuity). Can anyone really be sure that the Company promising this pension will be around to provide it? That the government that “insures” this promise in the event the company goes bankrupt will make good?

We actuaries pride ourselves on being long term thinkers and in fact have helped keep a few large Life Insurance companies in business for more than a century with our prudent advice on pricing and managing time, risk and money, but the time horizons implicit in our calculations have always struck me as far too long for us to have any real confidence that the promises based on what we are calculating today will have any real meaning 50-80 years from now when some of the consequences of our projections will be fully realized.
The situation is even more acute when we think about our own lives and the decisions we make every day. Beyond the obvious issue of how long we will live, almost every decision we make involves others; either people, institutions, and/or the environment itself. All of those entities have their own “omega” that needs to be taken into account. Sometimes the time horizon is obvious (eg when we consider Climate Change, a subject of a future blog, we need to consider that the earth essentially has no time horizon, or at least none that will matter to us human beings), but in many others it is not.

When we think about a legacy we are leaving to our children and grandchildren or a particular institution, how far in the future should we consider? When we think about investing in our education, our community or even our personal relationships, how long will that knowledge/educational benefit, relationship or community last? We tend to think the answer is “forever” (or at least as long as we and those we are in relationship with are alive), but I would suggest that that is an overestimate and all of us would do well to consider that everything is impermanent, transitory and will eventually cease to exist or transform into something else. We will never know when the end will come, but betting on the permanence and immortality of anything is usually a losing proposition. Be sure to consider this the next time you use Present Value to make a decision.