Читать книгу Pricing Insurance Risk - Stephen J. Mildenhall - Страница 13

Figure 1.1 shows the participants in the insurance pricing problem. Insureds, left, pay premiums to the insurer and in turn receive loss payments. The regulator, on top, observing the risk that the insurer is taking on, imposes asset requirements. Investors, right, provide capital and in turn receive the residual value (remaining assets) after losses are paid.

Figure 1.1 Players and their roles. The regulator applies a capital risk measure to determine required insurer assets. The pricing risk measure gives the cost of investors’ capital. Assets in excess of losses are paid to investors as the residual value of the business.

Insureds buy insurance because of their aversion to risk and because they are required to do so to drive a car, buy a house with a mortgage, etc. Regulators play a social policy role, addressing three principal concerns. First, to ensure mandated third-party insurance provides effective protection. Second, to manage the externality of losses exceeding assets. And third, to prevent insureds being fleeced by excessive premiums. The first concern is present in any tort-based system. We loosely identify the second as European and the third as American. We focus on the second concern, asset adequacy. Our development of technical premiums naturally aligns with the third fairness consideration if we assume that capital markets require fair returns.

Investors indirectly determine premiums because premiums plus capital add up to and fund assets, Figure 1.2. Investors’ willingness to provide capital to insurers translates into a pricing risk measure, which the insurer applies to the covered risks. Premium and asset levels are separate problems and need separate tools.

Figure 1.2 The different roles of capital and pricing risk measures.

Two important questions arise from insurance company promises to pay certain sums of money contingent on random events.

1 Are there sufficient assets to honor those promises?

2 Are investors being adequately compensated for taking on those risks?

Crucially, we need to talk about not one but two different risk measures to answer these questions.

Question 1 concerns risk tolerance and is usually answered by an economic capital model. It determines the assets necessary to back an existing or hypothetical portfolio at a given level of confidence. This exercise is also reverse engineered: given existing or hypothetical assets, what are the constraints on business that can be written?

We can imagine a regulator—interpreted broadly as an external authority—considering a portfolio of risks that the insurer proposes to cover. The regulator specifies the amount of assets the insurer must hold to cover the risk. If there is a shortfall after losses are realized, it will be made up by parties external to the insurer, e.g., a guarantee fund or other government entity, or the insureds themselves insofar as they are not reimbursed for claims. The regulator seeks to minimize the nonpayment externality, balanced with a desire for economical insurance.

A capital risk measure is applied to economic capital model output to quantify the level of assets the insurer must hold. Value at Risk (VaR) or Tail Value at Risk (TVaR) at some high confidence level, such as 99.5% or 1 in 200 years, are both popular, but other possible measures exist.

Question 2 concerns risk pricing or risk appetite. We must determine the expected profit insureds need to pay in total to make it worthwhile for investors to bear the portfolio’s risk. Regulated insurers are invariably required to hold capital on a regulated balance sheet. We generally assume a funding constraint where premium and investor supplied capital are the only sources of funds. Then, the pricing risk measure determines the split of their asset funding between premium and capital.

The capital and pricing risk measures should not be confused. Historically, capital risk measures have been studied in the context of finance and regulation, e.g., Artzner et al. (1999). In contrast, actuaries have studied pricing risk measures as premium calculation principles (Goovaerts, De Vylder, and Haezendonck 1984). The recent popularity and focus on coherent risk measures has overshadowed actuarial premium calculation principles and led to some confusion about the two distinct purposes of risk measures. Much of the recent literature implicitly or explicitly refers to the capital domain only. However, practitioners dealing with issues such as business unit performance, premium adequacy, and shareholder value are operating in the pricing domain. Taking a risk measure suitable for one use and applying it to the other invites unexpected and confusing results. Instead, we must understand how the capital and pricing measures work together in a complex, nonlinear manner to determine technical prices.

The top-down pricing process we have described may not seem commonplace, although those working in catastrophe reinsurance should find our process familiar. Most individual risk pricing actuaries can legitimately claim to use a bottom-up approach. Nevertheless, deep within almost every company lies a corporate financial model functioning exactly as we describe. It asks: How much capital is needed? What is the cost of that capital? What overall margin is necessary? And, how should it be allocated to each unit?