Transcript The Relevance of Mathematics in Insurance Business

The Relevance of Mathematics in
Insurance Industry
(The Town and The Gown)
Being a Paper Presented by
Olufemi Ayankoya
(Managing Partner, Green UNCT Consulting Ltd)
at the Department of Mathematics,
Covenant University,
Cannanland Otta, Ogun State, Nigeria.
February, 2015
Content
1.
2.
Introduction
Overview of the Insurance Industry
1.
2.
3.
4.
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4.
5.
6.
7.
8.
The Concept of Insurance
Insurance Products
The Industry Operators and their Role
How an Insurance Company Operates
The Relevance of Mathematics – a career without boundaries
The Concepts of Mathematics in the Insurance Industry
Some Insurance Mathematics
Careers for Mathematics Majors in the Insurance Industry
Thank You
Questions
Introduction
History tells the story of how insurance began at a
coffeehouse in London called Lloyds, where merchant
shipmen sat around the table drinking coffee and discussing
their future voyages. One merchant explained that he was
afraid that he would lose everything if his next ship was lost at
sea, and another merchant offered to share in his risk – for
a fee. Around the table they went, dividing up the risk of loss
and the profits of the voyage, and giving birth to insurance. In
truth, insurance existed in many early forms long before that
conversation could have taken place at Lloyds coffeehouse.
This paper considers the factual basis for the history of
early insurance mechanisms and discusses the relevance of
mathematics .
Introduction
• The inevitability of risk in human life gave rise to the concept of insurance.
Insurance may comprehensively be described as a contract between two
parties (the Insurer and the Insured) whereby the Insurer undertakes to
indemnify the Insured or Assured against risk of loss, risk of damage and
risk of liability which may arise upon the uncertain occurrence of some
specified event, or in the case of life insurance, upon the occurrence of a
certain event (death) at an uncertain time.
• Before the advent of colonialism in Nigeria, some form of insurance
existed, though it did not conform strictu sensu with the definition as given
above. The insurance system that existed was one of shared responsibility
and that of a common pool, where members of a community contributed
resources in order to help lessen the burden of each other in the event of a
loss. It has been rightly described as a concept of mutual insurance; the
coming together of people to help each other.
Introduction
The Royal Exchange Assurance Company which was a London
based body opened a branch office in Lagos in 1921 and this
was the first insurance company in Nigeria headed by Late
George Golding. It enjoyed monopoly as the only insurance
company for a period of about twenty-eight years when three
others came up. (These three were the Legal and General
Assurance Society, Norwich Union Fire Insurance Society and
the Tobacco Insurance Company). By 1960, there were twentyfive insurance companies in Nigeria. Of the twenty-five
insurance companies in Nigeria by 1960, twenty-two were
foreign owned and three were indigenous. Patronage was
however, quite low as most Nigerians were not aware of
insurance and its importance
Operators As At 2014
Insurance Companies As At 2014
Life
16
Non-Life
30
Composite
11
Re-Insurance
2
Other Insurance Operators As At 2014
Brokers
577
Loss Adjusters
54
Agents
1900
Actuaries
The Concept Insurance
Insurance involves transfer and pooling
Risk transfer from the insured to the
insurer:
Insurer assumes financial responsibility for the loss
Insurer agrees to indemnify the insured in the event of a
covered loss
What is Insurance
Insurance is a method by which large groups of
people spread the risks they have in common.
PERSONAL INSURANCE provides protection for
the individual in different ways: car and bike
insurance, home and household contents,
travel insurance, that sort of thing.
What is Insurance
COMPANY INSURANCE enables companies to
meet their legal requirements and offer their staff
protection.
Different insurance companies write different 'lines'
of business, in which they often specialise. Life
insurance, satellite coverage and marine insurance
(which covers ships, passengers and ports), are
examples of lines of business. Insurance companies
accept the risks of
What is Insurance
COMPANY INSURANCE enables companies to
meet their legal requirements and offer their staff
protection.
Different insurance companies write different 'lines'
of business, in which they often specialise. Life
insurance, satellite coverage and marine insurance
(which covers ships, passengers and ports), are
examples of lines of business. Insurance companies
accept the risks of large numbers of people and
bundle them together, to spread the risk.
What is Insurance
We buy insurance so that we can rest easy at
night, knowing that our lives and those of our
loved ones are covered, or that we have taken
every step possible to minimize our chances
of not being able to pay for life's more costly
moments, such as home repairs, doctor's bills,
or our kids going to college.
What is Insurance
What insurance companies sell is a promise.
In return for the premium you have paid, the
company promises to be there when you need
help, as they said they would, during the life
of the contract. In many cases, that can be a
long time. Insurance companies are built to
be solid, because they are our defence
against what life can throw at us.
Insurance Industry Operators
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Insurance
Insurance Brokers
Insurance Agents
Insurance Loss Adjusters
Actuaries
Reinsurance
Industry Regulation
The National Insurance Commission in Nigeria (NAICOM) is
charge by the constitution with the responsibility of regulating
the operations of the industry in conjunction with the Central
Bank of Nigeria.
Why Regulate?
The rationale for regulation includes;
 Insurers collect payment in advance
 Insurance transactions are often complex and opaque
 To protect the insured, the government monitor and regulate
the industry operators’
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Formation
Investments
Operations
Prices
Policy forms
Sales practices
Regulatory Function
The National Insurance Commission regulates the following;
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Solvency
Accounting Standard
Rates
Policy Forms
Sales Practices
How Insurance Company Operates
Insurance companies are like all other
businesses in that their primary objective is
to make a profit. Any business makes a profit
when revenues exceed expenditures. In the
case of an insurance company, this means when
premiums are greater than the combined cost of
paying claims and the cost of doing business, the
company makes a profit.
How Insurance Company Operates
Let’s look at the earlier example of a N1,000
premium for N200,000 of property coverage in
a homeowner’s policy. It may seem
impossible for the insurance company to
make money. If there is a total loss of the
home within a 200-year span. The insurance
company loses money. That is why insurance
companies write a lot of different homes in
different areas.
How Insurance Company Operates
The operation of an insurance company is based
on two basic concepts:
• the concept of independent losses, and
• the concept of spreading risk
Under the concept of independent losses, an
insurance company looks for one loss that is not
likely to affect a large number of different
policyholders.
How Insurance Company Operates
An insurance policy is a type of contact. And in this
contract both parties are expected to act in “good
faith.”
This means that the policyholders should not file
false or fraudulent claims and the insurance
company should pay claims promptly and
accurately. In order for any insurance company,
or the entire insurance industry, to be profitable,
all parties must act in good faith. Insurance is,
above all, a contract of good faith
How Insurance Company Operates
An example of this is robbery/burglary. One
policyholder may suffer a large burglary loss, but
this affects only the one policyholder. So the
cost of this one loss spread among a large
group of policyholders. Under the concept of
spreading risk, the insurance company tries
to “spread” the risk among a large number of
policyholders so that one type of loss won’t
affect the entire group of policyholders.
How Insurance Company Operates
This is why an insurance company does not write
all of their homeowner’s coverages in a
coastal area. Otherwise, one hurricane could
destroy all of their policyholder’s homes,
resulting in huge payouts for the insurance
company. Again, the losses experienced by a
few policyholders will be paid for by the
majority of policyholders who do not
experience any losses.
How Insurance Company Operates
These two concepts are closely related. By
successfully achieving both of these on a large
scale, premiums can be kept low, and the
insurance company can make a profit. It is
important to understand how insurance
companies operate so that you can better
understand the reasons behind why they do
what they do.
Insurance Company Structure
 Rate making
 Underwriting
 Production
 Claim settlement
 Reinsurance
 Investments & Finance
 General Administration
6-27
Rate making
Rate making refers to the pricing of insurance
 Total premiums charged must be adequate for
paying all claims and expenses during the policy
period
 Rates and premiums are determined by an
actuary, using the company’s past loss experience
and industry statistics
6-28
Underwriting
 Underwriting refers to the process of selecting, classifying, and
pricing applicants for insurance
 The objective is to produce a profitable book of business
 A statement of underwriting policy establishes policies that are
consistent with the company’s objectives, such as
 Acceptable classes of business
 Amounts of insurance that can be written
 A line underwriter makes daily decisions concerning the
acceptance or rejection of business
Underwriting
There are three important principles of underwriting:
 The underwriter must select prospective insureds
according to the company’s underwriting standards
 Underwriting should achieve a proper balance within each
rate classification
 In class underwriting, exposure units with similar loss-producing
characteristics are grouped together and charged the same rate
 Underwriting should maintain equity among the
policyholders
6-30
Underwriting
 Underwriting starts with the agent in the field
 Information for underwriting comes from:
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The application
The agent’s report
An inspection report
Physical inspection
A physical examination and attending physician’s report
MIB report
 After reviewing the information, the underwriter can:
 Accept the application
 Accept the application subject to restrictions or modifications
 Reject the application
6-31
Production
Production refers to the sales and marketing activities
of insurers
 Agents are often referred to as producers
 Life insurers have an agency or sales department
 Property and liability insurers have marketing departments
 An agent should be a competent professional with a
high degree of technical knowledge in a particular
area of insurance and who also places the needs of
his or her clients first
6-32
Claim Settlement
The objectives of claims settlement include:
 Verification of a covered loss
 Fair and prompt payment of claims
 Personal assistance to the insured
Some laws prohibit unfair claims practices, such as:
 Refusing to pay claims without conducting a reasonable
investigation
 Not attempting to provide prompt, fair, and equitable
settlements
 Offering lower settlements to compel insureds to institute
lawsuits to recover amounts due
6-33
Claim Settlement
 The claim process begins with a notice of loss
 Next, the claim is investigated
 A claims adjustor determines if a covered loss has occurred
and the amount of the loss
 The adjustor may require a proof of loss before the
claim is paid
 The adjustor decides if the claim should be paid or
denied
 Policy provisions address how disputes may be resolved
6-34
Reinsurance
Reinsurance is an arrangement by which the primary
insurer that initially writes the insurance transfers to
another insurer part or all of the potential losses
associated with such insurance
 The primary insurer is the ceding company
 The insurer that accepts the insurance from the ceding
company is the reinsurer
 The retention limit is the amount of insurance retained by
the ceding company
 The amount of insurance ceded to the reinsurer is known as
a cession
Copyright © 2008 Pearson
Addison-Wesley. All rights
reserved.
6-35
Reinsurance
Reinsurance is used to:
 Increase underwriting capacity
 Stabilize profits
 Reduce the unearned premium reserve
 The unearned premium reserve represents the unearned portion of
gross premiums on all outstanding policies at the time of valuation
 Provide protection against a catastrophic loss
 Retire from business or from a line of insurance or territory
 Obtain underwriting advice on a line for which the insurer
has little experience
Copyright © 2008 Pearson
Addison-Wesley. All rights
reserved.
6-36
Reinsurance
There are two principal forms of reinsurance:
 Facultative reinsurance is an optional, case-by-case method
that is used when the ceding company receives an application
for insurance that exceeds its retention limit
 Treaty reinsurance means the primary insurer has agreed to
cede insurance to the reinsurer, and the reinsurer has agreed
to accept the business
 Under a quota-share treaty, the ceding insurer and the reinsurer agree
to share premiums and losses based on some proportion
 Under a surplus-share treaty, the reinsurer agrees to accept insurance
in excess of the ceding insurer’s retention limit, up to some maximum
amount
 An excess-of-loss treaty is designed for catastrophic protection
 A reinsurance pool is an organization of insurers that underwrites
insurance on a joint basis
6-37
Reinsurance Alternatives
 Some insurers use the capital markets as an alternative to
traditional reinsurance
 Securitization of risk means that an insurable risk is
transferred to the capital markets through the creation of a
financial instrument, such as a futures contract
 Catastrophe bonds are corporate bonds that permit the issuer
of the bond to skip or reduce the interest payments if a
catastrophic loss occurs
6-38
Investments
 Because premiums are paid in advance, they can be invested
until needed to pay claims and expenses
 Investment income is extremely important in reducing the cost
of insurance to policy-owners and offsetting unfavorable
underwriting experience
 Life insurance contracts are long-term; thus, safety of principal
is a primary consideration
 In contrast to life insurance, property insurance contracts are
short-term in nature, and claim payments can vary widely
depending on catastrophic losses, inflation, medical costs, etc
6-39
Other Insurance Company Functions
 The electronic data processing area maintains information on
premiums, claims, loss ratios, investments, and underwriting
results
 The accounting department prepares financial statements and
develops budgets
 In the legal department, attorneys are used in advanced
underwriting and estate planning
 Property and liability insurers provide numerous loss control
services
6-40
The Relevance of Mathematics
Insurance?
The process of managing risk is highly mathematical
and quantitative. The insurance, pension and social
insurance industry employs certified professionals
called actuaries with the specific skills required to
address risk management. These skills include
advanced analytical and mathematical expertise,
problem solving abilities, and general business
acumen.
Mathematic is the mechanism for the accurate
calculation of insurance premiums and provides the
foundation for the insurance transaction.
The Relevance of Mathematics
Insurance?
 The first step in understanding the insurance
mathematics is to calculate the net single premium.
 insurance contracts calls for monetary payments from
the insurer . Therefore, the insurance company needs
to calculate very quickly the amount of money needed
to deliver its future promises to policyholders.
 Premium charged is a function of the event insured.
The mathematical models(such as Mortality Tables)
are used to predict the relative frequency of the
insured event and to calculate the premium required
to insure that event.
The Relevance of Mathematics
Insurance?
 The insurance contracts often last for many years and the
time value of money is very important.
 The time value of money plays an important role in life
insurance mathematics .
 Money received can be invested for long periods of time
until funds are needed to pay claims.
 During this accumulation phase , the insurance company
relies on compound interest to provide sufficient funds to pay
the benefits called for by the contract.
 The basis for operating a successful life insurance company
is the calculation of the proper insurance rate.
The Concept of Mathematics in Insurance
Industry
Consider the following scenario;
How much should be charge to insure a friends car?
How often per year does he have an accident? (=f)
How much money does it usually cost to repair his car? (=X)
The average loss per year: S = f*X
Because this is business, so add a profit percentage (=p).
Price = f*l*(1+p)
The Concept of Mathematics in Insurance
Industry
This seems easy!
But, how much money would you need to keep
aside (=reserves) to pay your friend, in case he
has an accident?
For one car, you will have to have reserves up to
the maximum possible loss, in other words, the
value of the car.
The Concept of Mathematics in Insurance
Industry
If we want to insure many cars.
The yearly loss now is (X is the loss, N the number of losses):
It is obvious, that S will not be the same for every year, but has a
distribution. The challenge is to find distributions for X~F(x) and N~P.
The Concept of Mathematics in Insurance
Industry
Typical Loss distribution
n
A typical distribution used for these loss is Pareto
The Concept of Mathematics in Insurance
Industry
Pareto
The alpha depends on the type of risk.
F(x)
alpha = 2
alpha = 1
loss
The Concept of Mathematics in Insurance
Industry
Frequency
Very commonly used is the Poisson distribution
Poisson works fine if events are rare and independent.
The Concept of Mathematics in Insurance
Industry
The Outcome
We now have a distribution for the loss size and loss number to represent S.
The aggregated cdf is usually calculated with Monte Carlo methods:
- draw the number of losses per year
- draw the loss amounts and add them up.
Ordered by loss amount of the year one can calculate the aggregated CDF.
The average of these outcomes returns the expected loss.
The Concept of Mathematics in Insurance
Industry
Aggregated CDF
Probability
Agg Loss
The Concept of Mathematics in Insurance
Industry
But there is more to consider..
 Long term/short term claims
 Capital costs
 Liquidity
 Profit margin
 Brokerage
 Recovery
 Internal costs
 Taxes
The Concept of Mathematics in Insurance
Industry
The Reserves also?
 How much money do we have to reserve now?
 To hold the MPL for all contracts would be way too expensive!
 Therefore we hold reserves cover two 99% shortfall years:
The shortfall is defined as: shf(S) = <S¦ S>Q(99%)>
 We calculate distribution of the losses versus the capital we hold for the
whole Swiss Re group.
 There is a possibility that we go bankrupt! Otherwise we would be way too
expensive.
The Concept of Mathematics in Insurance
Industry
Research Question
Correlations!
Example : Pandemic will not only trigger many life insurances, but
the stock market will go down, too!
Some Insurance Mathematics
Premium growth rates:
These ratios can be calculated on the basis of written premium or earned
premium or both, gross and net of reinsurance, and by class of business,
business segments (for example, retail compared to commercial business
lines) or for the total portfolio. If information is available, premium growth
rates can also be calculated in terms of new business only.
 P1 
 1 x100%
 P0 
Equation 1
where
P
1
P
0
is the premium in the current accounting period; and
is the corresponding premium in the previous accounting
period.
Some Insurance Mathematics
(2) Renewal Rates
These ratios are more usually calculated on the basis of gross written premiums, and by class of
business, business segments or for the total portfolio.
As renewal rates reflect the relationship between the company and its customers, it is not usual
to examine renewal rates using premium information net of reinsurance.
This ratio is of less relevance for reinsurance business as this business involves a smaller number of
large contracts so it can be expected that the ratio will be volatile in normal circumstances.
 GRP1  x100%
 GWP 
0

Equation 2
where
GRP1 is the gross written premium from renewals in the current accounting period;
and
GWP0 is the total gross written premium for the same class of insurance in the
previous corresponding accounting period.
Other Insurance Mathematics
Mechanisms
PREMIUM GROWTH RATES
RENEWAL RATES
CHANGES IN ‘PER RISK’ PREMIUMS
REINSURANCE
CESSION RATES
NET RETENTION RATES
MAXIMUM EVENT RETENTIONS
REINSURANCE RECOVERIES
PROFITABILITY – LEVEL, QUALITY AND SOURCE
CLAIMS RATIO
EXPENSE RATIO
COMBINED RATIO
INVESTMENT INCOME RATIO
OPERATING RATIO
PROFIT RATIOS
QUALITY OF THE INVESTMENT RESULT
TECHNICAL PROVISIONS
UNEARNED PREMIUMS
UN-EXPIRED RISK
CLAIMS PAID TO CLAIMS PROVISIONS
IBNR TO REPORTED PROVISIONS
CLAIMS DEVELOPMENT OVER THE YEAR
ALTERNATIVE C LAIMS ESTIMATES
ASSETS
ASSET MIX
INVESTMENT ASSETS AND OTHER ASSETS
INADMISSIBLE ASSETS
LIQUIDITY
SOLVENCY AND CAPITAL
PREMIUMS AND CAPITAL
TECHNICAL PROVISIONS AND CAPITAL
SOLVENCY MARGIN
SOLVENCY COVERAGE
QUALITY OF CAPITAL
THE VIEW OF OTHER STAKEHOLDERS
RETURN ON CAPITAL
Careers for Mathematics Majors
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Actuarial – pricing
Actuarial - reserving
Aggregate Modelling
Broking
Capital Modelling
Catastrophe Modelling
Claims
Compliance
Risk Management
Business Analysis
Quatitative Analyst
Research