As for MPT, given a portfolio of A, one would prefer B, C, or D as compared to A as shown in Exhibit 25.3. However, notice that the preferred portfolios are now below the stated portfolio, as the preference here is to lower the expected losses and premium dollars spent.

The replacement of the typical finance standard deviation is an important one. In most financial textbooks (and practical usage), the standard deviation is most often from a normal distribution. In our example, we may use any multivariate distribution that is applicable, but for practicality we have chosen closed-form log- normal/Pareto distributions, which are typically used in insurance. We have also made another significant variation in the use of the tail value at risk (TVaR) instead of the standard deviation. The intent of this replacement is that most insurance contracts are low-probability contracts, so the standard deviation does not completely describe the use or intent of the contract. By using the tail value at risk, we can focus on the main use of the insurance contract and allow for multiple distribution functions, which will better describe the underlying distribution for its intended use.

The given probability of the TVaR calculation is up to the user. We have selected a probability level of 95 percent, meaning that the worst 5 percent of outcomes are averaged to produce the TVaR figure at 95 percent.

The next complexity of selecting a TVaR calculation means that one will almost always be required to run a simulation model to determine the statistic. Only the most simplistic applications will allow for a closed-form expression of the measure of volatility. Therefore, we have chosen to use Monte Carlo simulation for our application of the efficient frontier for insurance portfolios.

The added benefit of using a simulation model is that we are now free to use multivariate distributions, complex correlations, copulas, and other transformations that may be too complex for most formulaic calculations. It is also important to note that most insurance portfolios contain more than seven to 10 different contracts/risks; so modeling is often a required component for any portfolio analysis.

We certainly do not want to gloss over the correlation concerns with insurance contracts, as there are many. It is becoming more common to use copulas (and different versions of copulas formulas – for example, a Gaussian copula or a Gumbel copula^{[1]}) to measure more complex correlations. The choice and use of correlations are critical elements of a proper model and should be reviewed with statisticians or actuaries versed in their use.^{[2]} For our purposes, we have assumed no correlations, for the simplicity of the calculations and translation of the results into knowledge.

It should be noted that TVaR is a simple method to allocate capital for insurance risk. The TVaR demonstrates the level of risk for a given insurance line or contract. Capital can thus be allocated based on that level of risk. Capital allocation theory is beyond the scope of this chapter, as there are many other variations upon this theme for allocating capital. It should be noted that the next step beyond the portfolio optimization is capital allocation.

One immediate question with the introduction of the TVaR as a risk measure is: "What is the right level of risk?" Or in simpler terms: "What is the largest loss I am willing to take?" Management should make a conscious decision on the level of risk to take through a formal enterprise risk management program. Risk setting is a critical step in any efficient frontier analysis and should not be overlooked. For our purposes, we have assumed that the organization will seek to minimize risk and minimize the annual costs to the budget (i.e., uninsured losses and insurance costs).

With some liberties taken in the usage of financial theory in the development of our risk transfer methods, we can now build a framework to analyze risk and optimize risk transfer spends (i.e., like insurance). The framework is intended for financial professionals versed in financial theory and its applications. With proper application, many organizations across the world could more efficiently allocate their risk spends and reduce the risk to their balance sheets.

[1] Ibid. Bodie, Zvi, Alex Kane, and Alan Marcus. Investments. 8th edition. New York: McGraw-Hill.

[2] For reference, a good article on copulas is available on the CAS website: casact.org/library/studynotes/feldblum-d ependency2013.pdf.

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