The document discusses various concepts related to insurance including: 1. It defines a utility function and explains how it is used to represent consumer preferences and welfare for different consumption levels and probabilities of states occurring in insurance. 2. It discusses key features of the individual risk model including how consumption levels and probabilities are incorporated into the utility function. 3. It explains how the central limit theorem can be applied in insurance problems by allowing inferences about event probabilities to use normal approximations even if the underlying data is not normally distributed, which is useful for factors like determining claim probabilities and identifying changes over time.

1. 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 consumerhasreasonable preferencesaboutconsumptionindifferentcircumstances,then
we will be able to use a utility function to describe these preferences, just as we have done in
othercontexts.However,the factthat 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
actuallyoccur. Inotherwords,the rate at whichI am willingtosubstituteconsumptionif it rains
for consumptionif itdoesn'tshouldhave somethingtodowithhow likelyIthinkitisto rain.The
preferences for consumption in different states of nature will depend on the beliefs of the
individualabouthowlikelythose statesare.Forthisreason,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 actuallyoccurs.If the two .statesare 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 utility 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.

4. 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.

6. 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

7. 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

8. 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.

9. 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

10. 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

11. 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 presentsthe proportionsurviving,the cumulativehazardfunction,andthe hazard rates
of a large groupof subjectsfollowedovertime.The analysisaccountsfor subjects who die (fail) as
well assubjectswhoare 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
developedfirst,butthe Kaplan-Meiermethodhasbeenshowntobe 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 nowdefine eachof the columnsinthe 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 butTt+1 or [ , ) Tt Tt+1 , where t= 1, not includingTt+1. The intervalsare assumedto be fixed.
The intervals do not have to be of equal length, but it is often convenient to make them so. The
midpointof the interval,Tmt,isdefinedashalf waythroughthe interval.The widthof the interval
is t b where t Tt Tt b = +1 − . The width of the last interval, s b , is theoretically infinite, so items
requiringthisvalue will be leftblank.NumberLosttoFollow-UpThe 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 BenefitInsurance isa termlife insurance policy in
whichpremiums remainthe same throughoutthe term. Mostlevel termpolicieshave livesof 10 or 20ye
ars, but thisisnot alwaysthe case. The distinguishingfeature isthatboththe death
benefitandthe premiumare fixedforthe lifeofthe policy. Thismeansthatpremiums maybe more expe
nsive atfirst, but they will notincrease asthe policyholderbecomes
olderor 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

12. 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

13. 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.

14. 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.