# PRACTICAL APPLICATIONS OF RISK MEASUREMENT FOR INSURANCE

Now we begin our journey through the practical application of risk theory applied to insurance risk and portfolios. The purpose of the process is to optimize insurance placements and risk limits for a relevant organization. We will start with a basic understanding of terminology, knowledge, and skills needed for a proper analysis, and then dive into the details and calculations necessary for a robust study. In the end, we will establish that this process can transcend insurance and be used in alternative risk transfer, noninsurance settings.

For the purposes of working through a real-life example, we need to establish insurance equivalents for the portfolio theory formulas. What follows is a list of definitions that we will use throughout this chapter and the equivalent portfolio theory definition.

From the previous section, we bring forth the standard portfolio theory formulas for the optimal return and optimal variance using the capital asset line:

Here the expected risk spend on an insurance portfolio, *E(rs*p*),* replaces the expected return on an asset, E(rp). The expected risk spend is defined as the expected losses *not* transferred in the insurance contract plus the costs of the insurance contract. The expected risk spend is based on the insurance contract at hand, and will differ (often significantly) based on different contracts analyzed as part of the analysis.

The risk-free rate is replaced by an insurance portfolio with no risk transfer (i.e., an uninsured risk line/portfolio).

The intent here is to set the steady state at no insurance purchase and determine if insurance will actually lower the risk to the organization. If it does, then insurance should be purchased. If it does not, insurance should not be purchased. In other words, on the capital market line for a given level of risk, you want to buy a portfolio with the highest level of return, but here you want to put together a risk portfolio with the lowest level of losses outside of the insurance contract for a given level of risk. By minimizing the losses, you are maximizing your return.

Visually, in our insurance example, you want to pick the bottom of the portfolio efficient frontier and not the one on the capital asset line as in typical portfolio theory.

Tail value at risk, also known as *tail conditional expectation* (TCE) or conditional tail expectation (CTE), is a risk measure associated with the more general value at risk. It quantifies the expected value of the loss given that an event outside a given probability level has occurred.^{[1]}

- [1] Jerry A. Miccolis and Marina Goodman, "Next Generation Investment Risk Management: Putting the 'Modern' Back in Modem Portfolio Theory,"
*Journal of Financial Planning,*January 2012.