Brainstorms

It's About Time

by Stephen W. Philbrick

One of the themes of the Social Security Privatization debate is the notion that stocks may be risky in the short term but the risk is reduced over longer timeframes. This notion has some validity. Essentially, we can think of this as a time diversification. Returns that may look unacceptably risky over a short time horizon become more acceptable over a longer planning horizon.

The concept of diversification is basic also to insurance and financial professionals, although we are more apt to discuss diversification across items, rather than over time. In financial securities, we look to Markowitz for motivation of diversification. The law of large numbers motivates the existence of insurance—the willingness of a company to write a portfolio containing many risks, each of which individually has a potential loss many times its expected value.

We formally express this in our not-yet-finished actuarial principles. The Statistical Regularity principle in the joint committee's working version (Principle 1.1) states:

Phenomena exist such that, if a sequence of independent experiments are conducted under conditions that are substantially similar to a set of specified conditions, the proportion of occurrences of a given event converges as the number of experiments becomes large.

As DFA tools progress, we are learning to quantify risk relationships. We intuitively knew that adding geographically diverse auto exposures improved diversification more than adding a cluster of new homes in an earthquake or hurricane region. Now, we can calculate (okay, estimate) correlations to get a quantification of diversification across risks.

DFA tools also help us look at risk over a planning horizon. It is reasonably obvious that the risk profile of a company will look different over a five-year horizon than over a one-year horizon. We anticipate that typical risk measures, such as the ratio of the standard deviation to the mean, will depend on the time frame of the model. DFA tools will help us quantify the differences, as a prelude to improved strategic planning.

What is less obvious and more interesting (and finally, the point of this column) is that the relative mix of risk components will vary considerably depending on the time horizon. I won't go into all of the risks that an insurance company faces, but concentrate on an important source of risk—underwriting risk. I will break it into two components: real claims cost and the effect of inflation on claims costs. Normally, these two components are combined, but there is merit to distinguishing the two. In this discussion, I will refer to the two components as real underwriting risk (the risk that actual real claim payments differ from expected claims payments) and inflation risk (the risk that actual inflation differs from expected inflation).

Teasing the inflation risk component from the overall underwriting risk isn't trivial, but it can be done by accessing a DFA model that explicitly considers the inflation sensitivity of liabilities. We need to choose a variable of interest, and a risk measure. In this example, the variable of interest is end-of-period surplus, and the risk measure is the variance of the end-of-period surplus. We can decompose the total variance into components by running a model three times; once with stochastic modeling of both real claims cost and inflation, and once with each of these elements modeled deterministically while the other is modeled stochastically.

In a one-year horizon, real underwriting risk dominates the contribution to the total risk. Not surprisingly, the inflation risk could have only a modest impact. In a short time period, adverse underwriting scenarios (such as hurricanes) can have a material impact on the value of the company, but adverse inflation scenarios have much smaller impact. The results are markedly different over a five-year horizon. Adverse underwriting shocks in one year are often followed by better than expected underwriting results in a subsequent year (although this will be dampened, and, in extreme cases nonexistent, depending on the size of parameter risk, such as a mispricing of the entire portfolio). To be sure, some scenarios produce several years of adverse underwriting results, but scenarios with five consecutive years worse than expected results are quite rare. (Note that this does not simply mean a five-year soft cycle, but five years worse than the expected, where the expectation might be poor results.) In contrast, inflation results tend to be highly correlated over time. A scenario where modeled inflation exceeds expected inflation is likely to be followed by another year with higher than expected inflation. As a result, adverse inflation tends to "accumulate" rather than "diversify."

In a particular example of a realistic property casualty company, we found that the real underwriting risk was four times the inflation risk (in a one-year horizon). However, over a five-year period, the relationship was almost the opposite. The contribution of inflation was almost four times the contribution of underwriting. (For simplicity, I'm ignoring the contributions of other types of risk, such as asset risk.) Not surprisingly, the exact values are heavily dependent on assumptions, such as the sensitivity of insurance cash flows to inflation. However, over a broad range of reasonable values, the relative contributions of risk were dramatically different for different horizons. This has major implications for planning. A company with a one-year outlook will view real underwriting risk as a major concern, and will focus on ways to control this risk, such as underwriting standards, limit profiles, and reinsurance. A company with a longer time horizon, however, will conclude that inflation is a major risk and will concentrate on ways to control inflation exposure, such as contract terms or asset selection.

After looking at the results, it is easy to conclude that they are obvious, but I haven't seen much written on the subject. Has anyone else done any work in this area, or read any research into this subject?