# The Relevance of Mathematics in Insurance Industry (The Town and The Gown) Being a Paper Presented by Olufemi Ayankoya (Managing Partner, Green UNCT Consulting.

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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.Introduction 2.Overview of the Insurance Industry 1.The Concept of Insurance 2.Insurance Products 3.The Industry Operators and their Role 4.How an Insurance Company Operates 3.The Relevance of Mathematics – a career without boundaries 4.The Concepts of Mathematics in the Insurance Industry 5.Some Insurance Mathematics 6.Careers for Mathematics Majors in the Insurance Industry 7.Thank You 8.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 twenty- five 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 Life16 Non-Life30 Composite11 Re-Insurance2 Other Insurance Operators As At 2014 Brokers577 Loss Adjusters54 Agents1900 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  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’  Formation  Investments  Operations  Prices  Policy forms  Sales practices

Regulatory Function The National Insurance Commission regulates the following;  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.

6-27 Insurance Company Structure  Rate making  Underwriting  Production  Claim settlement  Reinsurance  Investments & Finance  General Administration

6-28 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

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

6-30 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

6-31 Underwriting  Underwriting starts with the agent in the field  Information for underwriting comes from:  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

6-32 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

6-33 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

6-34 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

Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 6-35 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

6-37 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

6-38 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

6-39 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

6-40 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

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 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 Relevance of Mathematics Insurance?

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 n A typical distribution used for these loss is Pareto Typical Loss distribution

The Concept of Mathematics in Insurance Industry The alpha depends on the type of risk. F(x) Pareto loss alpha = 2 alpha = 1

The Concept of Mathematics in Insurance Industry Very commonly used is the Poisson distribution Poisson works fine if events are rare and independent. Frequency

The Concept of Mathematics in Insurance Industry 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 Outcome

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  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) = 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 Reserves also?

The Concept of Mathematics in Insurance Industry Correlations! Example : Pandemic will not only trigger many life insurances, but the stock market will go down, too! Research Question

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. P 1   1  x100%  P0 P0  Equation 1 where P1P0P1P0 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. GRP 1 x100%   GWP 0   Equation 2 where GRP 1 is the gross written premium from renewals in the current accounting period; and GWP 0 is the total gross written premium for the same class of insurance in the previous corresponding accounting period.

Other Insurance Mathematics Mechanisms P REMIUM GROWTH RATES R ENEWAL R ATES C HANGES IN ‘P ER R ISK ’ P REMIUMS REINSURANCE C ESSION RATES N ET R ETENTION R ATES M AXIMUM E VENT R ETENTIONS R EINSURANCE R ECOVERIES PROFITABILITY – LEVEL, QUALITY AND SOURCE C LAIMS R ATIO E XPENSE R ATIO C OMBINED R ATIO I NVESTMENT I NCOME R ATIO O PERATING R ATIO P ROFIT R ATIOS Q UALITY OF THE I NVESTMENT R ESULT TECHNICAL PROVISIONS U NEARNED P REMIUMS U N - EXPIRED R ISK C LAIMS P AID TO C LAIMS P ROVISIONS IBNR TO REPORTED PROVISIONS C LAIMS DEVELOPMENT OVER THE YEAR A LTERNATIVE C LAIMS E STIMATES ASSETS A SSET MIX I NVESTMENT ASSETS AND OTHER ASSETS I NADMISSIBLE ASSETS L IQUIDITY SOLVENCY AND CAPITAL P REMIUMS AND C APITAL T ECHNICAL P ROVISIONS AND C APITAL S OLVENCY M ARGIN S OLVENCY COVERAGE Q UALITY OF C APITAL THE VIEW OF OTHER STAKEHOLDERS R ETURN ON C APITAL

Careers for Mathematics Majors  Actuarial – pricing  Actuarial - reserving  Aggregate Modelling  Broking  Capital Modelling  Catastrophe Modelling  Claims  Compliance  Risk Management  Business Analysis  Quatitative Analyst  Research

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