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       Market risk, including interest rate, equity, real estate, spread, currency, concentration, and illiquidity risks

       Operational risk.

      Some Enterprise Risk Management frameworks categorize risks as:

       Health and safety risks

       Reputational risks

       Operational risks

       Strategic risks

       Compliance risks

       Financial risks.

      Insurance-focused ERM lists may add:

       Asset risk

       Credit risk

       Market risk

       Underwriting risk

       Reserving risk

       Catastrophe risk.

      This book focuses on pure underwriting, reserving, and catastrophe risks. For property-casualty insurance, the others tend to be background risks that are hard to distinguish between units and therefore not relevant to pricing.

      The next three sections categorize risk according to the following dimensions:

       Diversifiable (idiosyncratic) vs. systematic (including catastrophe)

       Systemic vs. nonsystemic

       Objective vs. subjective probability and uncertainty.

      3.3.1 Diversifiable Risk

      Insurance is based on diversification, where the risk of the sum is less than the sum of the risks. It is important to understand whether a risk is diversifiable, also known as idiosyncratic. Risks diversify when each unit is small relative to the total and their losses exhibit a material degree of independence from one another. A diversification benefit occurs when adding independent units to a portfolio increases its risk by much less than what the standalone risks represent. The central limit theorem ensures that pooling is an effective mechanism to manage diversifiable risk.

      The opposite of diversifiable risk is nondiversifiable, also known as systematic risk. The failure to diversify usually means that there is a common underlying cause or other source of dependence risk to multiple unit losses, or there is a single unit heavily influencing the total loss. Catastrophes affecting multiple units simultaneously are an example of the former. A catastrophe line of business with outsize losses compared to the other lines is an example of the latter. The presence of systematic risk means there is less diversification benefit than in its absence.

      Dependence risk between units can manifest itself in different ways, some more dangerous than others. It is easy to identify in a simulation context: where are you sharing variables? Variables resimulated in each iteration for each unit diversify, at least to some extent. Any variable whose value is shared between units introduces dependence and systematic risk. Weather and loss trend assumptions are examples of shared variables.

      Remark 2 In finance, systematic risk usually refers to the common variation of stock prices over time whereas idiosyncratic risk refers to the deviation of individual stock prices from the common movement. By adding many stocks to a portfolio, idiosyncratic risk—but not systematic risk—can be diversified away. This leads to pricing principles where only systematic risk matters because the well-diversified investor can make idiosyncratic risk “go away.” In Section 12.4 we will see that this simplification does not apply to the problem of pricing insurance risk.

Example 3 You live in a Dystopian Dictatorship and are a member of a group of 100 prisoners. Evil Leader decides to execute 1% of the prisoners. EL considers two approaches. Execute each prisoner with a 1% probability or with 1% probability execute all prisoners. The number of deaths is diversifiable in the former approach and nondiversifiable in the latter. The expected number of executions is the same for each approach.

      Remark 4 The discounting impact of timing remains even when amount risk diversifies. Timing risk tends to be quite tame since payout patterns follow a predictable claim settlement process, regulated by the cadences of medicine and law. The historical development of insurance pricing reflects this distinction: in many cases amount risk is largely irrelevant but estimating the appropriate discount rates and investor and insured cash flows remains paramount; see the discounted cash flow and internal rate of return models in Chapter 8.

      3.3.2 Systemic Risk

      Systemic risk affects a financial system consisting of many interacting agents or firms and is created by that system’s operation or structure. It occurs when an event causes a chain reaction of consequences. So-called systemically important financial institutions or SIFIs—those deemed too big to fail—generate systemic risk. They are so interconnected and interwoven in the financial system that the failure of one would have dire consequences for all. Systemic risks are often triggered by exogenous shocks such as the oil crisis of the 1970s or the failure of Long-Term Capital Management. The October 1987 Black Monday crash and Global Financial Crisis (GFC) of 2008 are two examples where losses from systemic risk emerged suddenly but endogenously. Systemic risk is linked to complex webs of contracts, collateral valuation, and duration transformation, where the system relies on a fragile confidence that can quickly erode. Systemic risks are by nature nondiversifiable (Brunnermeier and Oehmke 2013)

      Property-casualty insurers are not usually regarded as systemically important financial institutions, although some large life insurers and AIG were designated as SIFIs by the Financial Stability Board after the GFC. A large, highly connected reinsurer could generate systemic risk if, for example, its failure would cause knock-on insolvencies. Insurers themselves are generally not regarded as systemically risky because they have liquid assets and illiquid uncallable liabilities (Chen et al. 2013).

      The interaction of rating plans as another example of systemic risk for insurers. Adverse selection against a rating plan can cause realized rates to be lower than expected, through a combination of adverse selection and winner’s curse. The effect is on premium rather than loss.

In a catastrophe, many insureds are subject to the same occurrence. Such risks are systematic (nondiversifiable) but not systemic. The risk arises as the result of an outside event and is not caused or exacerbated by the operation of the system.

      Example 5 The October 1987 Black Monday stock market crash was a largely unexpected and sudden global decline in share prices. The Dow Jones Industrial Average fell 22.6% in one day. It was partially caused by the widespread use of portfolio insurance, which triggered sell orders in a declining market. Long-Term Capital Management was a hedge fund that failed in 1998 due to a combination of high leverage and exposure to the 1997 Asian financial crisis and 1998 Russian financial crisis. Its history is told in Lowenstein (2000).

      3.3.3 Types of Uncertainty

      Probabilities can be objective or subjective. Objective probabilities are amenable to precise determination. They are usually based on physical symmetry (coin toss, dice roll) or repeated observations. Objective probability applies the law of large numbers, the central limit theorem, Bayesian statistics, and credibility theory to make precise predictions about samples. Insurance is largely based on objective probabilities from repeated observations (loss data).

      Subjective probabilities provide a way of representing a degree of belief. They follow the same rules as objective probabilities and have proven a powerful way to impose order and consistency in economics and finance. Subjective probabilities are applied to nonrepeatable events: an election, a horse race, or an economic

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