Is optimization really a tool that can help a (re)insurer improve their portfolio? Are these types of exercises a futile attempt to create a theoretical position a company could be in? In truth, the question of whether an optimization approach to planning (algorithmic or otherwise) will be achievable or not comes down to the question of whether you, as a company, are able to express your corporate goals and constraints with the right level of detail. Here we will review several things a company may need to consider in attempting to “ask the right question” for an optimization exercise.

## Optimization in General Terms

Optimization is the notion of determining the “best”, or “most efficient” option among a set of alternatives. In different disciplines this may describe different attributes. Mathematical optimization describes the process of either finding the maximum or minimum values for a function – in particular, finding the global maxima or minima. In general the financial industry is concerned with this type of mathematical optimization.

Mathematical optimizations can be broadly broken out into two categories: 1) Continuous optimization and 2) Combinatorial optimization. Each of these categories can be separated into various “problem types”, of which there are two umbrella types: 1) Single objective optimization and 2) Multi-objective optimization. The former is concerned with optimization problems that only incorporate a single objective function while the latter is concerned with cases where two or more objective functions are to be optimized simultaneously.

## What are my options for optimizing a (re)insurance portfolio?

At one level, a company may wish to do strategy planning as an “expert opinion”, or so-called “gut feeling”, exercise. In the (re)insurance space this generally involves the pricing teams collectively deciding which directions they can/wish to take contracts in. Sometimes this is accompanied by a “proforma analysis” to compare the current position with the planned position.

More algorithmic techniques include linear programming and iterative and heuristic algorithms. The linear and iterative methods tend to be powerful on a subset of optimization problems. These techniques tend to fail on more complex problems where the solutions need to be found in a non-smooth environment. In addition, these techniques are more difficult to apply to the multiple objective problems seen in the (re)insurance industry. Heuristic methods were designed to overcome these issues but can only provide estimates of efficient solutions.

## Am I explaining myself correctly?

In order for any optimization exercise to go beyond the realms of providing theoretical answers, the problem has to be expressed in a way that is neither too simple nor too complex. Optimization definitions that are too simple for the problem at hand tend to give results that do not fit the goals of the business. Generally, these are met with reactions such as “That is too good to be true.” or “That is nice but in reality…”. Thus, results of these optimizations may not offer actionable insights. On the opposite spectrum, definitions that are too complex tend to more difficult to interpret and require more time and resources to create a viable solution. In some cases, this also means insights that can be gained from a less complex problem, are missed. Essentially, planning gets “lost in the details” instead of looking for more general solutions that can be further tailored.

This is sometimes called a mapmaker problem. The goal of the mapmaker is to design a representation of reality that is detailed enough to direct a traveler to the destination while not being so detailed that the map is too complex to interpret. The following are a few concepts and questions to keep in mind when defining the reinsurance optimization strategy:

## Concepts

##### Objectives

These are measurements that are used to make decisions on, such that solutions with “better” measurements will generally be selected in favor of other solutions. The optimization, however, has the ability to move these to their best possible values without being hindered.

Objectives may be measurements such as expected returns and tail value at risk, where the goal is simply to increase or reduce them by as much a possible.

##### Constraints

These are measurements that need to be in a particular range, which once within this range, have little to no effect on what is considered a better solution.

Constraints may be measurements such as capital or an aggregate limit amount that has to be within a set of values. However, once these measurements are within these values, the portfolio is considered to be writable, but the best performing portfolio are determined by the objective of the optimization.

##### Constrained-Objectives

These are a combination of the above two and may also be known as targets. They are measurements that must fall in a particular range and once met, they are also used to determine with solutions are better than other solutions.

For example, a company may require their return on equity to have a minimum value and after meeting this minimum, portfolio with a larger return on equity are still considered to be better performing portfolios.

## Questions

##### What is it that I want to achieve?

##### Do I actually have another objective?

Is there another metric that makes me change my mind, given that all the other metrics are equal?

##### Did I miss input from anyone?

It is important to include input from the parties who need to act on any strategies designed. Generally their input will help craft the universe of what is possible (for example maximum and minimum values for parameters). This usually leads to the universe of the optimization being more realistic than a case where analysts make some assumptions about certain parameters and metrics.

##### Am I adding objectives and constraints that are not needed?

In many cases, there are objectives and constraints that will either never be breached or will be met naturally due to other objectives/constraints. This includes measurements that may be highly correlated with each other, in which case using only one of them may be sufficient.

##### Does this actually reflect how my company is going to make its decisions?

After the first attempt at creating the optimization problem, this is an important question to ask. Here you should be able to see most of the key decision drivers in your optimization, such as, pricing decisions may need to include some notion of capital or the company’s cost structure is approximated in the model.

Interested in how Analyze Re can help you ensure that you are able to express your optimization problems correctly? Drop us a line, we are always happy to share our thoughts.