Which is the evidence of contract between policyholder and insurance company?

6.7.1.2.2.4 Life Insurance Companies

Life insurance companies receive premiums from policyholders in return for a promise to pay their heirs a settlement upon death. The insurance company charges the policyholder a premium each year that includes an amount to cover the pure mortality risk as well as an amount that is invested on behalf of the policyholder. The investment portion of the policy is expected to grow over time, creating a source of funds for the policyholder upon retirement.

Life insurance policies are long-term liabilities, and insurance companies are always searching for low-risk, long-tenure investments to match their liabilities. Real estate mortgages and equity investments are both used to fund policy liabilities.

Insurance companies also provide investment services to corporate pension investors who create special investment accounts with the insurance company to help fund their pension liabilities.

3.3.2 Life Settlement Market and Its Welfare Effects

A life settlement is a financial transaction in which policyholders sell their life insurance policy to a third party—the life settlement firm—for more than the cash value offered by the policy itself. The life settlement firm subsequently assumes responsibility for all future premium payments to the life insurance company and becomes the new beneficiary of the life insurance policy if the original policyholder dies within the coverage period.ay The life settlement industry is quite recent, growing from just a few billion dollars in the late 1990s to about $12–15 billion in 2007, and, according to some projections, it is expected to grow to more than $150 billion in the next decade (see Chandik, 2008).

The opportunity for the life settlement market results from two main features of life insurance contracts. First, most life insurance policies purchased by consumers, either term or whole life, have the feature that the insurance premium stays fixed over the course of the policy. Because policyholders’ health typically deteriorates over time, the fixed premium implies that policyholders initially pay a premium that is higher than actuarially fair, but in later years the same premium is typically actuarially favorable. This is the front-loading phenomenon we described earlier. Front-loading implies that policyholders of long-term life insurance policies, especially those with impaired health, often have locked in premiums that are much more favorable than what they could obtain in the spot market. This generates what has been known as the actuarial value of the life insurance policy (see Deloitte Report, 2005). Second, as we mentioned earlier, the CSV for life insurance policies is either zero for term life insurance or at a level that does not depend on the health status of the policyholder. Because the actuarial value of a life insurance policy is much higher for individuals with impaired health, the fact that the CSV does not respond to health status provides an opening for gains of trade between policyholders with impaired health and life settlement companies.az Life settlement firms operate by offering policyholders, who are intending to either lapse or surrender their life insurance policies, more cash than the CSV offered by the insurers.

The emerging life settlement market has triggered controversies between some life insurance companies who oppose it and the life settlement industry who supports it. The views from the two opposing camps are represented by Doherty and Singer (2002) and Singer and Stallard (2005) on the proponent side, and the Deloitte Report (2005) on the opponent side. Doherty and Singer (2002) argued that a secondary market for life insurance enhances life insurance policyholders’ liquidity by eroding the monopsony power of the carrier. This will increase the surplus of policyholders and in the long run will lead to a larger primary insurance market. On the other side, life insurance companies, as represented by the Deloitte Report (2005), claim that the life settlement market, by denying them the return on lapsing or surrendered policies, increases the costs of providing policies in the primary market. They allege that these costs will have to be passed on to consumers, which would ultimately make consumers worse off.

A key issue in the contention between the opposing sides is the role of lapsing or surrendering in the pricing of life insurance in the primary market (see Daily, 2004). Policyholders may choose to lapse or surrender in a variety of situations. First, the beneficiary for whom the policy was originally purchased could be deceased or no longer need the policy; second, the policyholder may experience a negative income shock (or a large expense shock) that leads him to favor more cash now over a bequest.ba In the absence of the life settlement market, when a health-impaired policyholder chooses to lapse or surrender its policy, the life insurance company pockets the intrinsic economic value of the policy, which potentially allows the life insurance company to offer insurance at a lower premium. In the presence of the life settlement market, these policies will be purchased by the life settlement firms as assets; thus the primary insurance company will always have to pay their face value if the original policyholder dies within the coverage period.

Daily et al. (2008) and Fang and Kung (2010) studied the effect of life settlement on the primary life insurance market, using a dynamic equilibrium model of life insurance similar to Hendel and Lizzeri (2003), under the assumption that the lapsation of policyholders is driven by loss of bequest motives. Fang and Kung (2010) showed that the life settlement market affects the equilibrium life insurance contracts in a qualitatively important manner: with the settlement market, risk reclassification insurance will be offered in the form of premium discounts, rather than in the form of flat premiums as is the case without a settlement market, which we discussed in the previous section. This may lead to a smaller degree of front-loading in the first period. They also show a general welfare result that the presence of the settlement market always leads to a decrease of consumer welfare relative to what could be achieved in the absence of the settlement market. They also provide conditions under which the life settlement market could lead to a complete collapse of reclassification risk insurance as a result of unraveling. If one relaxes the assumption that prohibits endogenously chosen CSVs, Fang and Kung (2010) found that whether or not CSVs can be made health-contingent has crucial implications. If CSVs are restricted to be nonhealth contingent, Fang and Kung (2010) show that an endogenous CSV is an ineffective tool for primary insurance companies to counter the threat of the life settlement industry. Fang and Kung (2012b), however, shows that if policyholders’ lapsation is driven by income or liquidity shocks, then a life settlement market may potentially improve consumer welfare.

The intuition for the difference in the welfare result is as follows. Life insurance is typically a long-term contract with one-sided commitment in which the life insurance companies commit to a specified death benefit provided that the premium payments are made, whereas the policyholder can lapse anytime. Because the premium of life insurance policies is typically front-loaded, the life insurance company pockets the lapsation profits whenever policyholders lapse their policy after holding it for several periods, which is factored into the pricing of the life insurance policy to start with because of competition (see Gilbert and Schultz, 1994). The key effect of the settlement firms on the life insurers is that the settlement firms will effectively take away the lapsation profits, forcing the life insurers to adjust the policy premiums and possibly the whole structure of the life insurance policy, since lapsation profits can no longer exist. In the theoretical analysis, we showed that life insurers may respond to the threat of life settlement by limiting the degree of reclassification risk insurance, which certainly reduces consumer welfare. However, the settlement firms are providing cash payments to policyholders when the policies are sold to the life settlement firms. The welfare loss from the reduction in the extent of reclassification risk insurance has to be balanced against the welfare gain to the consumers when they receive payments from the settlement firms. If policyholders sell their policies because of income shocks, then the cash payments are received at a time when the marginal utility of income is particularly high, and the balance of the two effects may result in a net welfare gain for the policyholders. If policyholders sell their policies as a result of losing bequest motives, the balance of the two effects on net results in a welfare loss. Thus, to inform policy-makers on how the emerging life settlement market should be regulated, an empirical understanding of why policyholders lapse is of crucial importance.

8.2.2.3.1 Health insurance

Health insurance, also called medical insurance, helps policyholders protect themselves and family from expensive or unexpected health care-related expenses. Health insurance are designed to estimate the overall risk of health-related expenses and supplement health care-related costs. Most insurance policies do not cover all healthcare costs. These include co-payment and co-insurance costs. Co-payments, premiums, and out-of-pocket expenses depend on the type of health insurance. A plan called a PPO (preferred provider organization) tends to have more out-of-pocket costs than an HMO (health maintenance organization). But PPOs offer more flexibility when choosing a doctor and other services.

Disability insurance is a type of medical coverage that pays part of the income for a policyholder if he or she becomes ill or injured and requires an extended period of time to recover. Medicare is another type of health insurance program provided to people over the age of 65 with certain health conditions. Medicaid is a type of federal health insurance that pays healthcare costs for low-income citizens of all ages. Long-term care insurance helps clients cover costs for long-term care assistance.

1 The Social Value of Insurance

There is an added value to insurance only because policyholders are risk-averse, that is they dislike zero-mean risk on wealth. Consider an agent facing a random loss X to their wealth. An insurance contract stipulates a premium P and an indemnity schedule I(.) that determines the indemnity I(X) for each possible loss X. There is full coverage if I(.) is the identity function. The actuarial value of the contract is the expected indemnity EI(X). The insurance premium is said to be actuarially fair if it is equal to the actuarial value of the contract: P=EI(X). Suppose that it is the case. Then, the purchase of a full insurance contract at an actuarially fair premium has the effect of replacing a random loss X by its expectation P=EX. The private value of such a contract is equal to the risk premium attached to the initial risk by the policyholder. It is increasing with the policyholder's degree of risk aversion and with the riskiness of the loss. As a first approximation, the riskiness of the loss can be measured by its variance.

In order to determine the social value of insurance, the effect of the transfer of the risk on the insurer's welfare should also be measured. Consider an insurer selling n fair full insurance contracts to n policyholders, each of them bearing a random loss Xi, with X1, …,Xn being identically and independently distributed. The common wisdom is that the insurer does not bear any risk in the aggregate if it is able to sell enough such insurance contracts to cover independent risks, i.e., if n is large enough. This is not true, as explained by Samuelson (1963). It is a fallacious interpretation of the Law of Large Numbers which leads people to believe that accumulating several independent risks generates diversification. Indeed, the aggregate indemnity to be paid by the insurer is the sum of the Xi. Its riskiness measured by its variance equals nσ2, where σ2 is the variance of each individual risk Xi. If the insurer is risk-averse, the risk transfer is not Pareto-improving since it makes the insurer worse off. The classical view of modern finance on this problem is to recognize that insurance companies are not owned by a single person, but rather by a large set of shareholders. If insurable losses are uncorrelated with financial markets returns, the riskiness of insurance companies has no adverse effect on shareholders. In such a situation, the risk transfer to insurance companies is Pareto-improving, and the social value of insurance companies can be measured by the sum of the risk premia associated by policyholders to their random loss.

An alternative insurance scheme is to organize a mutual arrangement among the n risk-averse agents. Consider an arrangement in which the pool guarantees to each of its member a complete coverage of losses against an ex-post contribution equaling the average loss in the pool. In such a scheme, the random loss Xi is replaced by (X1+…+Xn)/n, whose variance is 1/n the variance of Xi. In consequence, this mutual arrangement is Pareto-improving. At the limit, when n tends to infinity, individual risks are completely washed out by diversification.

The Social Value of Insurance

There is an added value to insurance only because policyholders are risk-averse, that is they dislike zero-mean risk on wealth. Consider an agent facing a random loss X to their wealth. An insurance contract stipulates a premium P and an indemnity schedule I(.) that determines the indemnity I(X) for each possible loss X. There is full coverage if I(.) is the identity function. The actuarial value of the contract is the expected indemnity EI(X). The insurance premium is said to be actuarially fair if it is equal to the actuarial value of the contract, P = EI(X). Suppose that it is the case. Then, the purchase of a full insurance contract at an actuarially fair premium has the effect of replacing a random loss X by its expectation P = EX. The private value of such a contract is equal to the risk premium attached to the initial risk by the policyholder. It is increasing with the policyholder's degree of risk aversion and with the riskiness of the loss. As a first approximation, the riskiness of the loss can be measured by its variance.

In order to determine the social value of insurance, the effect of the transfer of the risk on the insurer's welfare should also be measured. Consider an insurer selling n fair full insurance contracts to n policyholders, each of them bearing a random loss Xi, with X1, …, Xn being identically and independently distributed. The common wisdom is that the insurer does not bear any risk in the aggregate, if it is able to sell enough such insurance contracts to cover independent risks, i.e., if n is large enough. This is not true, as explained by Samuelson (1963). It is a fallacious interpretation of the law of large numbers, which leads people to believe that accumulating several independent risks generates diversification. Indeed, the aggregate indemnity to be paid by the insurer is the sum of the Xi. Its riskiness measured by its variance equals nσ2, where σ2 is the variance of each individual risk Xi. If the insurer is risk-averse, the risk transfer is not Pareto-improving, since it makes the insurer worse off. The classical view of modern finance on this problem is to recognize that insurance companies are not owned by a single person, but rather by a large set of shareholders. If insurable losses are uncorrelated with financial markets returns, the riskiness of insurance companies has no adverse effect on shareholders. In such a situation, the risk transfer to insurance companies is Pareto improving, and the social value of insurance companies can be measured by the sum of the risk premia associated by policyholders to their random loss.

An alternative insurance scheme is to organize a mutual arrangement among the n risk-averse agents. Consider an arrangement in which the pool guarantees to each of its member a complete coverage of losses against an ex-post contribution equaling the average loss in the pool. In such a scheme, the random loss Xi is replaced by (X1 + ⋯ + Xn)/n, whose variance is 1/n the variance of Xi. In consequence, this mutual arrangement is Pareto-improving. At the limit, when n tends to infinity, individual risks are completely washed out by diversification.

Mutuals, Self-Insurance and Captives

A mutual insurer is an insurer that is owned by its policyholders. The main advantage of mutual insurance is that the policy holders have full information on the levels of premium charged for the cover and the extent of cover provided under the policies. The use of mutual insurance for project financing creates some additional complexity. The creditworthiness of mutual insurers can be problematic and lenders may not accept the financial risk associated with particular mutual policies. In addition, mutual insurers are often much less flexible in agreeing to the additional lender requirements, which are often needed to ensure that an insurance programme is bankable.

Self-insurance is a method of insuring whereby businesses insure themselves by retaining contingency funds that can then be used when a particular risk event occurs. Self-insurance is often effected through the use of a company owned by the business and established to manage the funds set aside. Special insurance subsidiaries like this are known as captive insurance companies and many large companies have established captive companies to manage their self-insurance activities. It is common for a company to only partially insure risks through a captive and take out an insurance policy to the full value in the commercial market.

Like mutual insurance, self-insurance and captive insurance can raise additional challenges in the structuring of insurance programmes for project finance. Although project finance insurance programmes have included non-commercial insurers the negotiations can become protracted and challenging if the relevant insurer is not experienced in dealing with project finance lender requirements.

5.10.2 Market Discipline

The scope of market discipline provided by policyholders and capital markets is a key element affecting capital decisions in many models. Factors affecting overall market discipline include (1) potential loss of franchise value, which arises from insurers’ upfront investments in infrastructure and distribution, underwriting expertise and information, and reputation; (2) the risk sensitivity of demand, which depends on the ability of policyholders or their representatives to assess insolvency risk, the scope of any government guarantees of insurers’ obligations, and the scope of the judgment proof problem, where some buyers may rationally seek low prices regardless of insolvency risk; (3) the risk sensitivity of insurance intermediaries (agents and brokers), who are exposed to increased costs or reduced revenues from insurer financial distress; and (4) risk sensitive debt-holders, which provide debt finance primarily at the insurance holding company level (Harrington, 2004b; also see Zanjani, 2002a).

A variety of studies provide empirical evidence on market discipline. Consistent with risk sensitive demand, studies that employ measures of property/casualty insurers’ premiums in relation to realized claim costs as proxies for the price of coverage have found that prices are negatively correlated to insolvency risk (Sommer, 1996; Phillips et al., 1998; Phillips et al., 2006). Zanjani (2002a) provides evidence that life insurance policy termination rates are greater for insurers with lower A.M. Best financial strength ratings, although terminations are not related to rating changes. Epermanis and Harrington (2006) estimate abnormal premium growth surrounding changes in A.M. Best financial strength ratings for a large panel of property/casualty insurers during 1992–1999. They report evidence of significant premium declines in the year of and the year following rating downgrades. Consistent with greater risk sensitivity of demand for customers with more sophistication and less guaranty fund protection, premium declines were concentrated among commercial insurance. Premium declines were greater for firms with low pre-downgrade ratings, and especially pronounced for firms falling below an A-rating, a traditional industry benchmark for many corporate buyers. Eling and Schmit (2012) conduct similar tests for German insurers and obtain qualitatively similar results, although the responses to rating changes appear smaller in magnitude than in the United States.

The extent to which partial state government guarantees of U.S. insurers’ obligations dull policyholders’ incentives and/or induce excessive risk taking from moral hazard has been explored in a number of studies that exploit cross-state variation in insurance guaranty fund characteristics or adoption dates to test whether guarantees have increased risk taking. Lee et al. (1997) provide evidence that asset risk increased for stock property/casualty insurers following the introduction of guaranty funds (also see Lee and Smith, 1999, and Downs and Sommer, 1999). Brewer et al. (1997) provide evidence that life insurer asset risk is greater in states where guaranty fund assessments against surviving insurers are offset against state premium taxes and thus borne by taxpayers, which may reduce financially strong insurers’ incentives to press for efficient regulatory monitoring.56

Many studies and numerous anecdotes for particular insolvencies document unusually large premium growth prior to the insolvencies of some property/casualty insurers (e.g., A.M. Best Company, 1991; Bohn and Hall, 1999; also see Harrington and Danzon, 1994), which could in some cases be plausibly related to underpricing and excessive risk taking.57 The 1990–1991 insolvencies of First Executive Corporation, First Capital Corporation, and Mutual Benefit Life in the United States suggest some degree of ex ante excessive risk taking, but the evidence is not sharp. Mutual Benefit ended up meeting virtually all of its obligations. Both First Executive and First Capital might have remained viable without regulatory intervention despite the temporary collapse of the junk bond market, and their experience demonstrated that demand was risk sensitive: as more bad news surfaced, more policyholders withdrew their funds (DeAngelo et al., 1995, 1996). Consistent with market discipline by equity holders, Fenn and Cole (1994) and Brewer and Jackson (2002) provide evidence that life insurer stock price declines during the high-yield bond and commercial real estate market slumps of 1989–1991 were concentrated among firms with problem assets.

10.9.3.2 Al wakala model

In this model, the takaful operator acts as an agent or representative of the participants or policyholders. The takaful operator earns a fee for the services as a wakeel or agent and does not participate in the profits or share in the underwriting results. Under this model, the surplus or deficit belongs to the participants. The takaful operator may charge a fund management fee or a performance incentive fee. The takaful operator assumes the business risks in developing and operating the takaful business on behalf of participants but does not participate in the mutual underwriting losses.

Partial loss

In the case of partial loss, the primary issue becomes the repair. Repair work can be done by the policyholder or it can be transferred to a third party. If the policyholder performs the repair, then the amount is estimated as a sum of all materials, wages to employees, and any necessary transportation costs. However, supervisory wages cannot make up more than 50% of the labor expense. If a third-party performs the repairs, then the provider pays all costs, including transportation. In other words, the insurance company simply pays the bill sent from the repair facility. Again, any deductible stated in the aircraft hull insurance is carried by the policyholder. Note that in both cases of partial loss, there is no place for depreciation.

Oftentimes, the repair cost for partial loss can be quite high. This is common for old aircraft with parts that are priced at the same level as those for new aircraft. Because of this, insurance contracts will state that the total amount of partial loss cannot exceed the amount paid for a total loss. Policies also state that when transportation is needed, it must be the least expensive means, regardless of whether it involves moving the damaged aircraft or securing spare parts and materials. Damaged aircraft must be transported to the most practical repair facility, and materials should be secured from the nearest source. Transportation costs include the expense of taking the aircraft from the place of accident to the repair facility and the expense of returning it to the place of accident or to the insured’s airport, whichever is closer or less expensive.

10.7 Actuarial Application: Markov Chains

We end this chapter with a brief discussion of an application of matrices to the insurance industry. The following discussion assumes some familiarity with basic probability theory and the reader is invited to review Chapter 9 if the terminology proves difficult.

Consider a random process that returns one of a finite number of possibilities at each discrete time step t = 0,1,2,…. That is, a process that returns some realization, Xt, of a random variable at each t. While this wording may appear unfamiliar, it is sufficient to understand that we are simply defining a process that, as time moves on in steps, returns one of a finite number of possible results, each with a particular probability. For example, we might be repeatedly rolling a dice. In this case, each discrete time step t is considered as a new roll and Xt is the result of that roll. Xt therefore returns one outcome from the discrete state space

. One particular run of the process for 9 time steps might return, for example,

That is,

.

A general random process is said to possess the Markov property if the future development of the process depends only on the current state. We can express this property mathematically in terms of conditional probabilities,

(10.33)PXt=x|X0=x 0,X1=x1,…,Xt−1=xt−1=PXt=x| Xt−1=xt−1

This expression is interpreted as stating that, given the history of the random process from time 0 to time t − 1, the probability that the event at time t returns some particular value depends only on the most recent result at time t − 1. We might summarize the Markov property with the simple phrase “history does not matter.” A process that has the Markov property is often called a Markov process or a Markov chain.

Returning to the dice-rolling process, the probability that the 10th roll is, say,

is not affected by the previous rolls and so it is a Markov process. This of course follows from the independence of each roll and is such that Eq. (10.33) can be simplified yet further,

You may be familiar with no-claims discount policies for motor insurance. Under these policies the insurance premium for a policyholder is reduced if no claims have been made in the previous year. A simple example of such a policy has three “states” of discount,

State 1: 0% discount

State 2: 10% discount

State 3: 50% discount

A policyholder moves up a state (or remains in State 3) as a result of making no claims in the previous year, and moves down a state (or remains in State 1) as a result of making one or more claims in the previous year. Making a claim is a random event and the state that a policyholder finds himself in at time t is Xt ∈{1,2,3}. We therefore see that the states form a discrete state space for a discrete-time random process. If the probability of making no claims in any particular year is 0.9, the probability of making at least one claim is 0.1. That is, the probability of moving up a state (or remaining in State 3) is 0.9, and the probability of moving down a state (or remaining in State 1) is 0.1. This process is illustrated in Figure 10.1 and we can confirm that it is a Markov process.

Example 10.32

You are given that a particular policyholder’s 14-year discount history is

1,2,1,2,3,2,3,3,2,1,1,1,2,3

where the number refers to a particular discount state, as defined above. Determine the probability that the policyholder will be in the following states at t = 15.

State 1

State 2

State 3

Is the process Markov?

Since this no-claims discount process is a Markov process, it is possible to define a 1-step transition matrix, P = [pij]. Each element pij is the probability that the policyholder will move to State j at time t + 1 given that he is in State i at time t. In our motivating example, this matrix is given by

(10.36)P=0.10.900.100.9 00.10.9

Note that the 3rd row is formed from the probabilities calculated in Example 10.32; the other rows can be calculated in a similar way. The matrix P can be thought of as containing all the information about a single step of the random process: it indicates which transitions are impossible (pij = 0) and gives the probability of those that are possible (pij≠0)

One-step transition matrices are very useful for determining the distribution of possible events after multiple time steps. For example, the transition matrix that applies after two time steps, that is from time t to time t + 2, is given by

P(2)=P×P=0.10.9 00.100.900.10.90.1 0.900.100.900.10.9=0.10.090.810.010.180.810.010.090.9

We see that the probability that a policyholder starts in State 1 and ends in State 1 after two time steps is given by element p11(2)=0.1. Furthermore, the probability that a policyholder starts in State 2 and ends in State 3 after two time steps is p23(2)=0.81.

Example 10.33

Confirm that p11(2)=0.1 by considering all possible transitions over two time steps.

The two-stage transition matrix P(2) for a general Markov process is given by

P(2)=P×P=p11p12p13p 21p22p23p31 p32p33p11 p12p13p21p 22p23p31p32p33

and the particular element p11(2) is then

p11(2)=p11p11+p12p21+p13p31

Given that each pij represents the probability of an independent transition, this expression can be interpreted as stating that the probability of starting and ending in State 1 is given by

the probability of the chain State 1 → State 1 → State 1, plus

the probability of the chain State 1 → State 2 → State 1, plus

the probability of the chain State 1 → State 3 → State 1.

A little work should convince you that it is possible to generalize the above expressions to give that the n-step transition matrix

P(n)≡Pn

This relationship stems directly from the properties of matrix multiplication, as demonstrated above for n = 2.

Example 10.34

Obtain an expression for p32(2) in terms of elements of the 1-step transition matrix P for a general Markov process. Interpret your expression.

Example 10.35

Obtain an expression for p11(3) in terms of elements of the 1-step transition matrix P for a general Markov process. Interpret this expression.

Example 10.36

Use the transition matrix in Eq. (10.36) to determine the probabilities of all possible transitions within the no-claims discount system over 4 years.

Example 10.37

An insurance company has 1000 policyholders distributed over the three states of the no-claims discount process with transition matrix given by Eq. (10.36). The initial distribution is,

100 in State 1

600 in State 2

300 in State 3

Determine the distribution of policyholders 4 years later.

Example 10.38

Determine whether there is a long-term, stable distribution of the 1000 policyholders in Example 10.37.