1.1.1 What is insurance?

Insurance is an instrument designed to protect against a (potential) financial loss. Formally, it is a risk transfer mechanism whereby an individual or an organization pays a premium to another entity to protect against a loss due to the occurrence of an adverse event. Insurance in a broad sense thus includes typical insurance policies (life, homeowner’s), pension plans and other social security systems.

Example 1.1.1 Homeowner’s insurance

In a basic homeowner’s insurance policy, the insurance company commits to repairing or rebuilding the house, and to buying new furniture, if a fire occurs. Therefore, the homeowner has transferred the fire risk to the insurance company. In exchange, the owner agrees to pay a fixed monthly fee, i.e. an insurance premium. ◼
 

Instead of taking the risk of making one large and random payment, if for example a fire destroys their house, individuals are willing to make small and fixed payments to an insurance company (or a pension plan sponsor) in exchange for some protection.

Example 1.1.2 Pension plan

A pension plan can also be viewed as a risk transfer mechanism set up by an employer, known as the pension plan sponsor, for its employees in order to provide them with revenue during retirement. Each employee faces the risk of having insufficient savings for retirement because one cannot predict investment returns, nor when one will die. The pension plan thus provides protection against these risks and is funded by contributions, similar to fees or premiums, jointly paid by the employer and its employees. ◼
 

1.1.2 Actuarial liabilities and financial assets

As a result of selling insurance protection, an insurance company or a pension plan sponsor:

• receives premiums or contributions that are invested in the financial markets;
• reimburses claims and/or pays out death and survival benefits.

Therefore, insurance companies and pension plans have important assets and liabilities. Generally speaking, an asset is what you own and a liability is what you owe while the equity is the difference between the two, i.e. your net worth. We have the fundamental accounting equation:

(1.1.1)

Examples of assets for ordinary people range from a bank account (and other savings) to a house or a car. Typical financial assets for an insurance company mostly consist of investments such as stocks, bonds and derivatives.

Common liabilities for most people are loans, e.g. a student loan, and a mortgage, which is a loan on an asset provided as collateral. Liabilities of an insurance company are contractual obligations tied to the insurance policies sold to its policyholders. For pension plans, the benefits promised to the participants constitute the single most important liability. To differentiate between liabilities tied to insurance contracts and pension plans from other types of liabilities, such as loans or accounts payable, we will refer to insurance and pension obligations as actuarial liabilities.

1.1.3 Actuarial functions

As the actuary is involved in the management of financial assets and actuarial liabilities, typical actuarial functions include:¹

• Pricing: computing the cost of an insurance protection (or of a specific pension plan design), designing policies and protections that are best for the customer (or employee) and the company (or pension plan sponsor), participating in the price determination.
• Valuation: given the current market conditions (interest rates, returns on financial markets) and a set of assumptions, computing the current value of actuarial liabilities, i.e. of contractual obligations. Valuation is useful to determine the amount of money to reserve, an actuarial function known as reserving, and to determine capital requirements.
• Investments: designing investment strategies and finding financial assets to mitigate the risks related to the organization’s actuarial obligations. Not as common for property and casualty (P&C) insurance companies.

Example 1.1.3 Pricing in P&C insurance

The process of finding the appropriate premium for a property and casualty insurance policy is often known as ratemaking. Actuaries use a large history of claims to better understand the risks. They use factors such as age, sex, characteristics of the car or house, etc. to determine the appropriate premium (rate) for a specific policyholder. P&C actuaries are also involved in the design of insurance policies through determining deductibles, limits and exclusions. ◼
 

Example 1.1.4 Valuation in life insurance

Interest rates affect the value of life insurance policies. The valuation actuary determines the reserve reflecting the company’s current mortality experience and the level of interest rates. If, in the future, interest rates go down significantly, the actuary will have to increase the reserve. ◼
 

Example 1.1.5 Liability-driven investments

One of the roles of the investment actuary is e.g. to find the appropriate allocation between stocks and bonds to meet actuarial obligations in the future. This is known as a liability driven investment (LDI) strategy. The actuary can also manage the investment portfolio on a day-to-day basis or advise the pension plan administrators on appropriate investment strategies. ◼
 

As time passes, interest rates and financial returns will evolve and so will the company’s claim and/or mortality experience. Therefore, insurance companies and pension plans will make gains or suffer losses. To mitigate the impact of important losses, the actuarial obligations (liabilities) and the investment portfolio (assets) should be aligned, as much as possible. The process of managing assets and liabilities together is known as asset and liability management (ALM). It is a key concept and it will be discussed throughout the book under the wording risk management, replication and hedging.

1.2.1 Insurance market

To enter (buy) an insurance policy, regulations require individuals to have an insurable interest. If you have an accident with a car you own, you suffer losses and therefore you have an insurable interest in that car. You obviously also have an insurable interest in yourself due to the temporary or permanent injuries you might suffer from an accident. But unless you can prove it, you do not have an insurable interest in the life of your neighbor.

The insurance market is tightly regulated. It is composed of competing chartered insurance companies (registered to a government agency) which are the sole sellers of insurance policies. Individuals and organizations buying insurance policies are not allowed to sell them. If you want to get rid of your insurance policy, it is not permitted to (re-)sell it to another individual or company. Insurance policies are not tradable assets. The best a policyholder can do is to terminate the contract and pay the penalty.

Typical insurance policies sold in the insurance market are:

• Life insurance: a fixed or random benefit is paid, upon the death of the policyholder, to a beneficiary. Classes of policies are permanent and term insurance, universal life, etc.
• Annuity: contract providing a stream of income until death or until a predetermined date. Classes of policies are life or term-certain annuities, fixed and variable annuities, etc.
• General insurance: protection covering specific hazards (fire, accident, flood, etc.) to an owned property (car, house, etc.). Includes car insurance, homeowner’s insurance, etc.
• Health and disability insurance: protection covering the costs of treatments, hospitalization, doctors and/or the loss of income following an injury, disease, etc.
• Group insurance: life, health and disability insurance sold to a group of employees.

In practice, policies are sold directly by the insurance company or through a broker, i.e. an intermediary between an individual and an insurance company. Even the broker is not allowed to buy back insurance policies from policyholders.

1.2.2 Financial market

The financial market is composed of thousands of knowledgable investors, ranging from individuals to institutional investors, such as investment banks, insurance companies and pension plans. In between, there are market makers, i.e. intermediaries selling to investors willing to buy and buying from investors willing to sell.

The financial market is not as tightly regulated as the insurance market. Consequently, investors are allowed to buy and sell securities quite easily. In practice, however, only large institutional investors can easily access the breadth of securities available.

The range of goods traded over the financial market is extremely large. There are securities, i.e. financial assets (such as stocks, bonds and derivatives), commodities (such as aluminum, wheat, gold) and currencies. Even carbon emissions are now traded on specially-designed markets.

Formally, a (financial) security represents a legal agreement between two parties. It can be traded between investors. The most common financial securities traded on the financial market are:

• Bond: type of loan issued by an entity such as a corporation or government;
• Stock: share of ownership of a corporation;
• Derivative: financial instrument whose value is derived from the price of another security or from a contingent event. Examples include futures and forwards, swaps and options.

These financial assets will be discussed in more detail in Chapter 2 for bonds and stocks, in Chapter 3 for forwards and futures, in Chapter 4 for swaps, in Chapter 5 for options and in Chapter 7 for other derivatives.

Other important assets are exchanged on the financial market:

• Commodity: tangible or non-tangible good such as crude oil, gold, coal, aluminum, copper, wheat, electricity (non-tangible).
• Currency: money issued by the government of a given country.

1.2.3 Insurance is a derivative

According to its definition, a derivative is a financial instrument whose value is derived from a contingent event. Therefore, insurance can be viewed as a derivative based on the occurrence (or not) of a risk. For example, life insurance is a derivative based on the life of an individual.

Example 1.2.1 Homeowner’s insurance

You buy a house for $300,000. The building itself and your belongings are worth $200,000. In the event of a fire, the value of your home would drop considerably. Your homeowner’s insurance policy will restore your home’s value to what it was prior to the fire by providing enough money to repair or even rebuild your house, buy new furnitures and clothes, etc. It can therefore be viewed as a derivative contingent upon the occurrence of a fire and paying the amount of damages, up to a value of $200,000. ◼
 

Therefore, insurance companies sell actuarial derivatives, i.e. insurance, to individuals in the insurance market, similar to investment banks selling financial derivatives to investors in the financial market. But, as discussed above, there are many aspects in which the insurance market and the financial market differ.

There are two additional differences between insurance policies and typical financial derivatives. Insurance policies have long maturities: decades for life insurance and pension plans, 1–2 years for general insurance, whereas most financial derivatives mature within a few months. Moreover, insurance policies are paid for over the life of the contract with periodic premiums whereas financial derivatives usually require a single premium paid up-front.

In conclusion, pricing actuarial derivatives in the insurance market is different from pricing derivatives in the financial market. We will come back to this topic in Chapter 2.

1.4.1 Illustrative example

A systematic risk affects many or most individuals, if not all. For example, if we have 100 eggs to carry from point A to point B, then we can make the decision to put them all in a single basket and have one carrier bring them to their destination. In this case, if the carrier drops the basket, many or all eggs may break. However, if the basket is not dropped, they will all be intact. This extreme example is not that far from reality: there are many real-life examples of (close to) systematic risks such as the risk of a stock market crash or the risk of a natural catastrophe in a given region.

At the exact opposite, a non-systematic risk or a diversifiable risk for one individual does not affect the same risk for another individual. For example, if we have instead 100 people each carrying a basket containing one egg, then it is nearly impossible to break all 100 eggs or to bring them all safely to their destination. We can reasonably expect that only a few eggs will be broken, as some clumsy carriers will drop their basket. In this case, the risk of dropping eggs is diversified away over 100 carriers. We can of course imagine other situations such as having 25 people each carrying a basket with four eggs. There is also a wide range of real-life risks that are (close to being) diversifiable that we will discuss later in this section. The benefit of diversification is to reduce the uncertainty of the aggregate outcome. It is at the core of actuarial science.

1.4.2 Independence

One important assumption above was the independence between risks: the clumsiness of an egg carrier does not affect the skills of other egg carriers. In general, risks are independent if they do not influence each other.² In particular, independent risks do not depend on a common source of risk.

Example 1.4.1 Mortality risk

In most cases, death or survival of an individual does not affect the death or survival of other individuals. Therefore, mortality risk is usually assumed to be independent from one individual to another, except in the following circumstances:

• Epidemics, wars, etc.: those are events that can cause the death of many people in the same period and thus create statistical dependence. Death from these causes is usually excluded in life insurance policies.
• Family members: spouses often have similar living habits (eating habits, physical activities, etc.) and death can be explained by similar factors. There is also the well-known broken heart syndrome where the death of one of the spouses can accelerate death of the other. ◼
   

Example 1.4.2 Car accidents

Reasons for a car accident are not necessarily related to those of another accident. Most accidents are caused by individual factors: driver distraction or tiredness, speeding, driving while impaired, local weather, mechanical failure, etc. Like many other types of claims in P&C insurance, the risks associated with car accidents are assumed to be independent between policyholders. ◼
 

Example 1.4.3 Natural hazards

In a given region, say a state or a small country, all insureds might be exposed to a natural hazard such as an earthquake or a hurricane. Therefore, losses resulting from these natural catastrophes are not independent.

1.4.3 Framework

In the rest of this section, we will compare diversifiable and systematic risks with (copies of) a generic die. Consider a die with six possible outcomes:

The die is assumed to be well balanced, i.e. outcomes are equally likely. In other words, each outcome has a probability of of appearing on any given throw of the die.

More precisely, we will denote by X the result of a throw, i.e. the number of points appearing on the face of the die (). Said differently, X = 1 when the outcome is , X = 2 when the result is , and so on. Clearly, X is a random variable uniformly distributed on {1, 2, …, 6}. We can easily verify that

and

1.4.4 Diversifiable risks

Understanding diversifiable risks and their impact in insurance can be illustrated with dice. Suppose there are n policyholders each throwing their own die. Each die behaves as the generic one described above. More precisely, let X_i be the result for policyholder/die number i = 1, 2, …, n. As we also assume that the throws do not influence each other, the random variables X₁, X₂, …, X_n are independent and identically distributed. In particular, for each i = 1, 2, …, n, we have

Let us now look at the average of the n throws:

Under the above assumptions, we know that

This means that, no matter how many die we consider, the expectation of will always be 3.5. Yet the variance of will decrease if the number of die increases, which means that the possible values of will be more and more concentrated around 3.5. In fact, according to the law of large numbers, if n is large enough, then

i.e. the average of the n throws will be almost equal to the expected result of a single throw, with a very high probability.

For the sake of illustration, assume that the i-th policyholder faces a loss amount of X_i for a given event. Of course, this is an over-simplified situation as the loss can only take the values 1, 2, …, or 6 (e.g. hundreds or thousands of dollars). For each policyholder, the loss amount X_i is unknown at the beginning of the year or, said differently, it is random.

From the perspective of each policyholder, it is a risky situation. Indeed, no one knows in advance whether the loss they will suffer will be zero, small or large. Consequently, a risk-averse person would prefer to transfer this risk to an insurance company in exchange for the payment of a non-random premium.

From the point of view of the insurance company, if the number of policyholders is large enough, then the realization of its average loss over the whole portfolio will be close to 3.50, no matter which policyholder suffers large or small losses. By aggregating a large number of individual risks, the insurance company has diversified away the risk of suffering large losses. This is how pure diversification works for the benefit of individuals, and for the company as well. In this setting, the insurer could charge each policyholder the value of this pure premium of 3.50 and break even,³ with a very high probability.

Diversification is the cornerstone of traditional insurance and, as we have seen, it is based upon the law of large numbers. It does not really matter who exactly claims a small or a large loss, what matters is the number of insureds in the portfolio so that approaches 3.50 as much as possible. As highlighted in Examples 1.4.1 and 1.4.2, typical actuarial risks such as mortality, fire, theft, vandalism, etc. are independent and as such are considered diversifiable risks.

However, in practice, there is a lot of heterogeneity in insurance portfolios, i.e. policyholders represent different risks with different probability distributions. For example, in example 1.4.1, a 55-year-old male smoker does not represent the same mortality risk as a 25-year-old non-smoker female. Even if those two lives can be considered independent, they are not identically distributed. Similarly, for car insurance risk as in example 1.4.2, the probability of a car accident in any given year depends on individual risk factors such as age and sex of the driver. Again, even if the independence assumption is reasonable, car accident risk is not identically distributed from one insured to the other.

Insurance companies use several characteristics to distinguish between individuals. In fact, they use buckets of insureds, in which policyholders represent similar risks. Those sub-portfolios are known as risk classes. Even if there are fewer policyholders in a given risk class, in many cases the diversification principle described above still applies within a risk class. Whenever the number of policies within a class is not large enough, margins for adverse deviations are added to the premium.

Finally, it might not always be possible to add more people in a portfolio. Take the case of a pension plan. The capability of the pension plan to diversify mortality risk depends on the size of the plan, which in turn depends on the number of employees of the sponsor. Therefore, some significant mortality risk might remain in the portfolio.

1.4.5 Systematic risks

When dealing with diversifiable risks, if we have more policyholders (assumed to be independent and representing similar risks), then the diversification benefit is stronger, i.e. aggregated losses are closer to the expected value. Things are drastically different for systematic risks.

In this direction, let us look at an entirely different setup. Suppose now that we draw a single die and that the random variables X₁, X₂, …, X_n are all linked to this unique throw. More precisely, let X be the result of this throw. Then, we set X₁ = X₂ = ... = X_n = X. This is illustrated in Figure 1.1.

In this situation, the average of the n throws is

The average number of points is always (in all scenarios) equal to X, no matter how small or large n is. This means that has a uniform distribution over {1, 2, …, 6}. This is completely different from the situation in Section 1.4.4.

Still interpreting X_i as the amount of loss for the i-th policyholder, we deduce that there is absolutely no diversification benefit from pooling risks together. Indeed, even if n is large, still has a uniform distribution over {1, 2, …, 6}. Sometimes, it will be equal to 2, with probability 1/6, while in other scenarios it will be equal to 6, with the same probability. It will not get closer to 3.5, with a high probability, even for a very large value of n. In this case, adding more policies to the (sub-)portfolio will not generate diversification as risks are systematic.

In the portfolio of policyholders of an insurance company or in a pension plan, the most important systematic risk is usually financial risk. Indeed, all premiums and employee contributions are managed and invested in the financial market with different kinds of investments. The returns earned on the company's or plan’s investments are the same for all policyholders and employees. Therefore, uncertainty tied to the time value of money and thus the returns required to meet the company’s commitments are systematic risks. Having more insureds or employees will not diminish the overall risk.

Example 1.4.4 Interest rates

For many years after the financial crisis of 2007–2009, interest rates have reached levels close to zero in many industrialized countries, including the United States, Canada, Germany, France and Japan. This has significantly increased the present value of cash flows tied to life insurance and annuities (including pension), increasing in turn required premiums and pension contributions of millions of savers.

Because the interest rate level is common to all contracts, again the insurance company clearly does not benefit from underwriting more policies. Interest rate risk is therefore a systematic risk. ◼
 

Example 1.4.5 Natural catastrophes

A local Californian P&C insurer offers earthquake risk protection in its homeowner’s insurance policies. If an earthquake occurs, it is exposed to an important systematic risk. Indeed, in this case, many policyholders may claim simultaneously a random amount according to their losses.

Earthquake risk is thus a systematic risk for this local insurance company and the insurer clearly does not gain by underwriting more policyholders in the same region. The insurer's solvency can even be at stake if an earthquake occurs, so it should manage this risk appropriately. There are several possibilities and reinsurance is a popular one. ◼
 

Another very important example of actuarial risk that is systematic is longevity risk. Reasons that explain increased human longevity are common to all: overall quality of the healthcare system, medical research, better living habits, etc. The uncertainty as to how many years humans will live in the long run is a systematic risk that life insurance companies and pension plans need to bear.

1.4.6 Partially diversifiable risks

In reality, most risks fall somewhere between being purely diversifiable or purely systematic. In these situations, there are benefits to diversification but they will be limited. To illustrate how partially diversifiable risks behave, we will use dice once more.

Suppose we have n + 1 dice. Let Y_i be the number of points on the throw of the i-th die for i = 1, 2, …, n and let Z be the number of points on the extra die. Then, define:

for i = 1, 2, …, n. In other words, X_i is the sum of an individual component Y_i and a common element Z. Clearly, the X_is are not independent random variables, but they have the same probability distribution.

Increasing the number n in the portfolio, we can diversify the individual components away but not the common shock Z. If Z is small (large), it will be small (large) for all Xs, potentially generating small (large) losses for all Xs. Figure 1.2 shows for example that when Z = 6, then most Xs will be relatively large: we have X_i ≥ 7. The factors that are common to many risks (Z in this illustration) are known as the systematic components whereas the individual parts are known as the diversifiable or idiosyncratic components.

Let us now analyze which actuarial and financial risks are partially diversifiable. Risks tied to natural hazards can be systematic or partially diversifiable depending on the insurer’s ability to geographically diversify its risks. As discussed in Example 1.4.5, a company that operates in a U.S. state does not have the same diversification capabilities as a large global insurer. Natural hazards are better diversified globally as earthquake risk in for example California is not related to earthquake risk in the Middle East. The same applies for hurricanes in the North Atlantic (East Coast of the U.S.) and West Pacific (typhoons in Asia).

Returns in a portfolio of stocks are also an example of a partially diversifiable risk. Stock returns are affected by macroeconomic (e.g. economic growth), country- and sector-specific factors. For example, oil companies, auto makers and software companies can be affected by the general growth of the economy but the crude oil price will not affect oil drilling companies in the same way as auto makers, and certainly will have a negligible impact on most tech firms. Again, due to the presence of common factors explaining stock returns, there is an important limit to diversification benefits we can achieve by increasing the number of stocks in a portfolio.