1. Explain utility function. State its uses in insurance

Utility function is a mathematical tool which ranks alternatives according to their utility

to an individual. It is a function that specifies the utility (well being) of a consumer for all combinations goods consumed (and sometimes other considerations) represents both

their welfare and their preferences. If the consumer has reasonable preferences about consumption in different circumstances, then we will be able to use a utility function to describe these preferences, just as we have done in other contexts. However, the fact that we are considering choice under uncertainty does add a special structure to the choice problem. In general, how a person values consumption in one state as compared to another will depend on the probability that the state in question will actually occur. In other words, the rate at which I am willing to substitute consumption if it rains for consumption if it doesn't should have something to do with how likely I th ink it is to rain. The preferences for consumption in different states of nature will depend on the beliefs of the individual about how likely those states are. For this reason, we will write the utility function as depending on the probabilities as well as on the consumption levels. Suppose that we are considering two mutually exclusive states such as rain and shine, loss or no loss, or whatever. Let Cl and C2 represent consumption in states 1 and 2, and let 7rl and 7r2 be the probabilities that state 1 or state 2 actually occurs. If the two .states are mutually exclusive, so that only one of them can happen, then 7r2 = 1 -7rl. But we'll generally write out both probabilities just to keep things looking symmetric. Given this notation, we can write the utili ty function for consumption in states 1 and 2 as U (Cl,C2,7rl,7r2). This is the function that represents the individual's preference over consumption in each state.

2. Discuss the salient features of the individual risk model.

See pdf attached

3. Explain the use of central limit theorem in solving insurance problems.

In other words, regardless of the underlying distribution of the sample observations, if the

sample is sufficiently large (generally > 30), the sample mean will be approximately

normally distributed with mean μ and standard deviation σ/√N.

The implication of Central Limit Theorem is that Inferences about probabilities of events

based on the sample mean can use the normal approximation even if the data themselves

are not drawn from a normal population.

Insurance is a complicated business. The Central Limit theorem finds application in Insurance

industry.

Insurance is sold to the N members of a ‘pool’ of purchasers, any one of which

may experience the ‘adverse event’ being insured against.

P = ‘premium’ = the price of the insurance against the adverse event

F = ‘payout’ = the amount that is paid if the adverse event occurs

= the probability that a member of the pool will experience the adverse event.

The expected profit to the insurance company is N[P - F]

The company sets P based on . If P is set too high, the company will make lots

of money, but competition will drive rates down. If P is set to low, the company

loses money.

The industry uses Law of Large Numbers and Central Limit theorem to answer basic questions

such as -

What is the probability of an adverse event occurring (θ)?

What if changes over time. How does the company find out?

4. Derive survival function from the force of mortality.

5. Gompertz Law

A Gompertz curve or Gompertz function, named after Benjamin Gompertz, is

a sigmoid function. It is a type of mathematical model for a time series, where growth is

slowest at the start and end of a time period. The right-hand or future value asymptote of

the function is approached much more gradually by the curve than the left-hand or lower

valued asymptote, in contrast to the simple logistic function in which both asymptotes are

approached by the curve symmetrically. It is a special case of the generalised logistic

function.

Formula

where

a is an asymptote, since

b, c are positive numbers

b sets the displacement along the x axis (translates the graph to the left or right)

c sets the growth rate (y scaling)

e is Euler's Number (e = 2.71828...)

Derivation

The function curve can be derived from a Gompertz law of mortality, which states the rate of

mortality (decay) falls exponentially with current size. Mathematically

where

is the rate of growth.

k is an arbitrary constant.

The Gompertz–Makeham law states that the human death rate is the sum of an age-

independent component (the Makeham term, named after William Makeham) and an

age-dependent component which increases exponentially with age. In a protected

environment where external causes of death are rare (laboratory conditions, low

mortality countries, etc.), the age-independent mortality component is often negligible. In

this case the formula simplifies to a Gompertz law of mortality. In 1825, Benjamin

Gompertz proposed an exponential increase in death rates with age.

6. ELEMENTS OF AN INSURANCE CONTRACT

A valid insurance contract requires both an offer and an acceptance. A blank application

provided to the potential insured is typically not considered an offer; but once the application is

completed and returned to the insurer, it becomes an offer for an insurance contract. The

elements of insurance are:

Offer: A completed application for an insurance policy is an offer to make a contract and is

subject to the prospective insurer‘s acceptance or rejection. Basic policy or application forms for

insurance must be preapproved by the insurance regulator.

Acceptance: When a statute or the insurer requires an application, the completed application

often is considered a mere offer that the insurer must accept in order to complete the insurance

contract. This acceptance may be conditional. For example, an application may provide that the

insurance will not take effect until the applicant receives the policy and the first premium is paid

in full during the applicant‘s lifetime and good health.

Execution and Delivery of Policy: Insurers frequently impose the condition that the contract of

insurance will not be effective before the execution, or signing or countersigning, and delivery of

the policy. In Oregon, a single life or health insurance policy may not be made on an individual

without a written application or written consent by the individual insured. ORS 743.027 (with

some exceptions).

Statutes Regulating Content or Form: When a statute regulates the content of an insurance

policy, the statutory requirements are deemed to be part of the policy, adding to or displacing

provisions of the policy itself. Any policy provisions that are less favorable to the insured than

the provisions required by statute are unenforceable.

Effect of Change in the Law All statutes in force at the time the policy is issued are

incorporated into the policy, even if the statutes are later repealed during the policy period.

Conversely, statutes that are enacted after a policy is issued do not become part of the policy

absent expression of a clear legislative intent to the contrary.

Effect of Conformity Clause: A conformity clause, which incorporates all applicable statutes

into a policy, extends only to statutes directly applicable to the issuance and General Principles

of Insurance / Chapter 1 1-23 2011 Edition content of insurance policies.

Requirements for Validity: In order to have a valid oral contract of insurance, all requirements

of an ordinary contract must be met, although some of those requirements may be implied. Those

requirements include competent contracting parties, consideration, the existence of a subject

matter, and an agreement on all essential elements of the contract.

Essential elements of the contract are as follows: (1) Identity of insured; (2) Identity of

insurer; (3) Subject matter to be insured; (4) Risk insured against; (5) The commencement

and period of risk; (6) Amount of insurance; and (7) Amount of premium and time in

which it is to be paid.

8. Endowment Insurance

An endowment policy is a life insurance contract designed to pay a lump sum after a specific

term (on its 'maturity') or on death. Typical maturities are ten, fifteen or twenty years up to a

certain age limit. Some policies also pay out in the case of critical illness. Policies are

typically traditional with-profits or unit-linked. Endowments can be cashed in early (or

surrendered) and the holder then receives the surrender value which is determined by the

insurance company depending on how long the policy has been running and how much has been

paid into it.

Traditional With Profits Endowments

An amount is guaranteed to be paid out, called the sum assured, and this can be increased on the

basis of investment performance through the addition of periodic (e.g. annual) bonuses. Regular bonuses (reversionary bonuses) are guaranteed at maturity and a further non-guarantee bonus

may be paid at the end known as a terminal bonus. During adverse investment conditions, the encashment value or surrender value may be reduced by a 'Market Value Reduction' (MVR)

Unit-linked endowment

Unit-linked endowments are investments where the premium is invested in units of a unitised

insurance fund. Units are encashed to cover the cost of the life assurance. Policyholders can

often choose which funds their premiums are invested in and in what proportion. Unit prices are

published on a regular basis and the encashment value of the policy is the current value of the

units. This is the simplest definition.

Full endowments

A full endowment is a with-profits endowment where the basic sum assured is equal to the death

benefit at start of policy and, assuming growth, the final payout would be much higher than the

sum assured.

Low cost endowment

A low cost endowment is a medley of: an endowment where an estimated future growth rate will

meet a target amount and a decreasing life insurance element to ensure that the target amount

will be paid out as a minimum if death occurs (or a critical illness is diagnosed if included). The

main thing of a low cost endowment has been for endowment mortgages to pay off interest only

mortgage at maturity or earlier death in favour of full endowment with the required premium

would be much higher.

9. Deferred Insurance means that the insured is covered for the length of the term, but should

death occur within the first two policy years, the death benefit payment is limited to a return-of-

premium plus interest.

e.g. Industrial Alliance Deferred Term Policy (Canada)

The policy offers insurance coverage for 20 years The death benefit payout if death occurs by non-accident in the first two policy years is limited

to a return-of-premium plus 5 per cent interest. The plan is non-convertible and non-renewable.

The annual premium for a 40-year-old, male non-smoker is $475 a year.

10. Net Premium & Pure Premium

Insurance pricing is the determination of what rates, or premiums, to charge for insurance. A rate is the price per unit of insurance for each exposure unit, which is a unit of liability

or property with similar characteristics. For instance, in property and casualty insurance, the exposure unit is typically equal to $100 of property value, and liability is measured in

$1,000 units.

Since insurance is a business, rate charged must cover losses and expenses, and earn some profit, after meeting regulatory & legal requirements. The main business objective is to

charge an adequate premium to cover losses, expenses, and allow for a profit; otherwise the insurance company would not be successful. Pure premium is the premium determined

by actuarial studies and consists of that part of the premium that is necessary to pay for losses and loss related expenses. Loading is the part of the premium necessary to cover other expenses, particularly sales expenses, and to allow for a profit. The gross rate is the

pure premium and the loading per exposure unit. Gross premium is the premium charged to the insurance applicant, and is equal to the gross rate multiplied by the number of

exposure units to be insured. The ratio of the loading charge over the gross rate is the expense ratio.

Gross Rate = Pure Premium + Load

Gross Premium = Gross Rate × Number of Exposure Units

Expense Ratio = Load / Gross Rate

The main regulatory objective when setting rates is to protect the customer. A corollary of this is that the insurer must maintain solvency in order to pay claims. Thus, the 3 main regulatory requirements regarding rates is that:

1. they be fair compared to the risk; 2. premiums must be adequate to maintain insurer solvency; and 3. premium rates are not discriminatory—the same rates should be charged for all members of

an underwriting class with a similar risk profile.

The main problem that many insurers face in setting fair and adequate premiums is that actual losses and expenses are not known when the premium is collected, since the premium pays for insurance coverage in the immediate future. Only after the premium period has

elapsed, will the insurer know what its true costs are. Larger insurance companies maintain their own databases to estimate frequency and the dollar amount of losses for each

underwriting class, but smaller companies rely on rating bureaus for loss information.

Pure Premium = (Actual Losses + Loss – Adjusted Expenses)/No. of Exposure units

If actual loss ratio differs from the expected loss ratio, then the premium is adjusted

according to the formula:

Rate change = (Actual Loss Ratio – Expected Loss Ratio) / Expected Loss Ratio

Rate making for life insurance is simpler as mortality tables are used to tabulate no. of deaths (part of the population) for each age. Age is the most important factor in life

expectancy, followed by gender and smoking habit. Thus a reasonable estimate can be made for the average age of death for a group. Hence a single net premium that covers the death

claim, without covering expenses or profit is possible to calculate.

Net Premium is the present value of the death benefit.

11. Survival Function & Life Table Function

A life table presents the proportion surviving, the cumulative hazard function, and the hazard rates of a large group of subjects followed over time. The analysis accounts for subjects who die (fail) as well as subjects who are censored (withdrawn). The life-table method competes with the Kaplan-Meier product-limit method as a technique for survival analysis. The life -table method was developed first, but the Kaplan-Meier method has been shown to be superior and with the advent of computers is now the method of choice. However, for large samples, the life -table method is still popular in that it provides a simple summary of a large set of data.

A life table is constructed from a set of grouped or ungrouped failure data. The columns of the table are created using a set of formulas. The rows of the table represent various time intervals. We will now define each of the columns in the life table. Note, however, that because of the large number of columns required to display all of the items, there will be several output reports produced. Time Interval Each time interval is represented by Tt ≤ T < ,s. The interval is from Tt up

to butTt+1 or [ , ) Tt Tt+1 , where t = 1, not including Tt+1 . The intervals are assumed to be fixed. The intervals do not have to be of equal length, but it is often convenient to make them so. The midpoint of the interval, Tmt , is defined as half way through the interval. The width of the interval is t b where t Tt Tt b = +1 − . The width of the last interval, s b , is theoretically infinite, so items requiring this value will be left blank. Number Lost to Follow-Up The number lost to follow-up, t l , is the number of individuals who were loss to observation during this interval, so their survival status is unknown. Number Withdrawn Alive The number withdrawn alive, wt , is the number of individuals who had not died (failed) by the end of the study.

12. Level Benefit Insurance

Level Benefit Insurance is a term life insurance policy in which premiums remain the same throughout the term. Most level term policies have lives of 10 or 20years, but this is not always the case. The distinguishing feature is that both the death benefit and the premium are fixed for the lifeof the policy. This means that premiums may be more expensive at first, but they will not increase as the policyholder becomes older or if he/she suddenly becomes ill.

If the policy is guaranteed renewable, the isured can extend coverage for an additional term without

having to qualify again, though the annual premium will increase with renewed policy.

Although the cost of insurance in the first few years will probably be higher for a level term than an increasing term policy, the total cost of a

level term with the same benefit is usually less. As with all term policies, you don't build up a cash reserve and your coverage ends at the

end of the term or at any time you stop making payments.

15. Assurance

Assurance service is an independent professional service, typically provided by Chartered

or Certified Public Accountants, with the goal of improving the information or the context of the

information so that decision makers can make more informed, and presumably better, decisions.

Assurance services provide independent and professional opinions that reduce the information

risk (risk that comes from incorrect information)

A Temporary Insurance Agreement typically includes certain conditions. For instance, if the

applicant for a life policy dies during the application process, the company may provide

coverage only if the underwriting process eventually determines that he would have been eligible

for permanent coverage had he lived. If the applicant was struck and killed by a car due to no

fault of his own, for example, the company would honor the agreement. Even though a

temporary insurance is only meant to provide coverage for a short period of time, it still is a

significant document. For example, if a claim occurs during the agreement period, the insurer

may still be liable to pay the full amount of a claim unless the agreement specifies other

conditions. A TIA may be used in most types of insurance, such as during the life insurance

application process. In auto insurance, an agent may issue a binder that provides temporary

coverage if the applicant is currently uninsured but needs to drive immediately.

Time Frame

Depending on the line of insurance for which it is issued, a TIA may last from several days to a

few months. In the case of life insurance e.g. a TIA could be in force for as long as 90 days.

Considerations

Certain situations may make a TIA null and void or reduce the amount of benefits paid if a claim

occurs while it is in force. For example, if the insured was found to have lied about not having a

certain medical condition, the insurer may have the right to rescind coverage and not pay the

claim.

EXTRA

What are the different types of life insurance policies?

If a person wants to take out insurance to ensure his/her partner or family is provided for in the

event of his/her death, there are a number of types of life-insurance policies to choose from.

Whole-of-life cover

As the name suggests, this type of policy will guarantee your dependants a payment irrespective

of when you die. Other types of cover (see below) will only pay out if you die before a specified

date. This can be appropriate, for example, if the insurance is only needed to ensure mortgage

payments – which end after 25 years typically – are met. Because whole-of-life policies are

guaranteed to pay out at some point in the future, it will generally cost more than other types of

cover. If you are looking for cheap life insurance, you may be better off considering term

insurance.

Term insurance

Term assurance, or term life insurance as it is also known, guarantees your family a payment if

you die within a specific time period. People often take out life insurance because they want their

dependants to be able to cover housing costs, for example, if the worst happens.

But given that the typical mortgage is paid off after 25 years, it may not be necessary to extend

life cover beyond this. Equally, policyholders may want to be covered only while their children

are living at home or in full-time education. Limiting the life insurance policy term in this way

means that premiums will be lower than with whole-of-life cover. This type of cover can also be

called level-term assurance or insurance if the payout would be the same no matter when the

policyholder died during the term.

Decreasing-term insurance (also known as mortgage life insurance)

An option for those buying term life insurance is to have the potential payout fall year after year.

This is most commonly to reflect the fact that mortgage debts are likely to be falling as more is

paid off. For example, you could take out a 25-year life insurance policy to cover £150,000 – the

same amount as you have borrowed on a 25-year mortgage – in the event of your death.

However, after 15 years, for example, the mortgage is likely to have shrunk considerably so you

could find yourself “over-insured” and paying more than is necessary in premiums as a result.

Decreasing-term insurance deals with this issue and, as you would expect, premiums will be

lower than with normal term insurance.

Increasing-term insurance

Alternatively, you may wish to have your potential payouts rise every year, perhaps to reflect

increasing inflation. With an index-linked policy you can choose to link your payout directly to

an inflation measure such as the Retail Prices Index (RPI) or Consumer Prices Index (CPI), or

you can simply arrange for the extent of cover to rise by a fixed percentage every year. If the

cover is scheduled to rise every year, your premiums will be higher than for level-term and

decreasing-term insurance.

Renewable term insurance

This is a policy that provides cover for a fixed period, but which can be extended when that

period comes to an end without you having to undergo further medical checks. The premiums

may increase based on your age at this point, but if you have suffered any health problems since

the original policy was taken out, these will not be taken into account or reflected in the new cost

of the policy.

Joint life insurance

If you are part of a couple, you could consider taking out a single policy that will pay out in the

event of one of you dying. This can be cheaper than paying the premiums on two separate

policies, but bear in mind that joint policies only pay out on the first death – after that the cover

ends. If you had two separate policies, the second policy would remain in force even after a

claim had been made on the first.

Death-in-service benefits

Many companies offer their staff’s families a lump-sum payment if the employee dies while they

are employed by the firm (althought this doesn’t mean the death has to be at the workplace or in

any way related to the job done). And members of company pension schemes may also be

entitled to payments from the pension if they die before they retire. It is worth bearing these

benefits in mind when you consider life insurance, but generally speaking, death-in-service

payments are equal to three or four years’ salary and may not provide all the cover you and your

family need. And remember, this cover may end as soon as you leave the company.