ACST1052: Introduction to Actuarial Studies
WEEK 11 – GENERAL INSURANCE
Slightly less morbid
insurance
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ACST1052: Week 11
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Overview
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This week, we look at the General Insurance (GI) industry, which comprises a large variety of
insurance products (everything that is “not life” and “not private health”).
It is one of the most common areas of practice for actuaries in Australia (if not the most common).
29% of the Actuaries Institute members quoted GI as their “primary” area of practice in 2023.
We first give an overview of the main GI products sold, we describe some insurance “principles” and
we list some ways in which GI differs from Life Insurance.
We then present some bits of mathematics essential to GI.
Next week, we will provide some comments more specifically about the Australian GI industry.
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General insurance generalities
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“General Insurance” is a bit of a lazy term. In the USA and Canada, GI is called “Property and Casualty
Insurance”, which is a bit more descriptive:
― Property insurance refers to insurance for “stuff”: buildings and their contents, motor vehicles,
commercial aircraft or cargo, etc.
― Casualty insurance refers to insurance in the case you are found legally liable to provide
compensation to another party (e.g., negligent acts causing injury).
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There are significantly more lines of business in the general insurance industry compared to life
insurance, because so many more types of events can be insured.
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Usually, the GI industry is further split into two parts
― Personal lines: sold to individuals.
― Commercial lines: sold to businesses.
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Some of the main Personal Lines
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Motor Insurance
― Compulsory Third Party (CTP, call “Greenslip” in NSW)
▪ Covers death or injury compensation caused to a third party in a car accident, when you are at
fault.
▪ “Compulsory” because you cannot drive a car that does not have CTP insurance.
― Third Party Property Damage
▪ Covers your liability for damaging other people’s cars and property.
― Comprehensive
▪ Third Party Property Damage + covers your car being stolen or damaged (e.g., by weather).
Home and Contents
― Covers all kinds of damage to your home (and its contents).
― Often covers burglary as well.
Travel Insurance
― Covers losses related to travel, e.g., loss of baggage, trip cancellations, overseas medical costs.
Pet Insurance
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Some of the main Commercial Lines
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Commercial Motor
― Like personal lines but for business vehicles like taxis and trucks.
• Commercial Property
― Like “home and contents” but for business premises, such as shops, offices and factories.
― Often includes a “Business Interruption” component.
• Marine and Aviation
• Worker’s Compensation
― Employers purchase this to cover the cost of supporting workers who get injured at work.
• Public Liability
― Covers liability to members of the public on business property.
• Product Liability
― Covers products from businesses that may cause injury or death to customers.
• Professional Indemnity
― Covers liability for professionals who are sued for doing a “bad job” (e.g. negligent acts or advice).
(a bit more info here)
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Principle of Indemnity
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In general insurance, the amount payable to you for a claim is usually equal to the financial loss you
have suffered (subject to sum insured limits and deductibles, which we explain later).
― E.g. for comprehensive motor, a no-fault crash will mean the insurer pays to fix your car to its
original condition.
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The idea is to put the insured person in the same financial position as if the unexpected insured
event did not occur. The principle of indemnity is necessary to avoid moral hazard, which we
mentioned last week.
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There have been famous examples of moral hazard when over-insurance was allowed – consider the
following historical example of marine insurance.
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Marine Insurance and Samuel Plimsoll
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In the late 1800’s, UK ships were constantly being shipwrecked and hundreds of sailors died at sea
every year.
Samuel Plimsoll was a politician who campaigned heavily against “coffin ships” – these are ships that
were unseaworthy and often over-loaded, with significant insurance taken on them, usually above
their actual worth.
― This meant that if the ship was lost at sea, the ship owners would earn a profit!
― Plimsoll had quite a struggle getting bills passed on these issues – it turns out that many
parliamentary MPs were also ship-owners…
Even in the case of ship-owners not deliberately sinking their ships for the insurance money, the moral
hazard still existed because the ship-owners would be careless about maintenance, repairs and
crewing.
Bottom line: when the policyholder has control over the insured event, this generates moral hazard,
especially if over-insurance is allowed.
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The principle of “Insurable Interest”
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There are also limitations on what sort of events you can buy insurance on. For general insurance, an
insurable interest must exist at the time of the claim – this means that you can only make a claim if
you have suffered some financial or economic loss due to the insured event occurring.
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For example, you cannot buy home and contents insurance on your neighbour’s house.
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The principle of insurable interest also applies to life insurance (but you only have to show insurable
interest at the time of purchasing the policy). This might be reasonable for the life of a spouse, parent,
or even a business partner/employee. However, this would not apply to a random stranger (or
celebrity?), as their death would not cause you a financial hardship.
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Insurable Risks
Not all risks are insurable – general insurers can be pickier about their products and so will only insure
things they deem likely to be profitable. The following principles are good “rules of thumb” for what
constitutes insurable risks:
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The loss must be fortuitous (i.e., a matter of chance).
The insurer must be able to define the risk.
The insurer must be able to assess the risk.
The premium must be affordable.
The insurance must not be contrary to the public interest.
There must not be excessive risk / accumulation of risk (although you might get around this with
reinsurance).
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Can you think of an insurance product that would be “contrary to the public interest”?
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Example of “being contrary to the public interest”
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It usually isn’t possible to insure yourself against actions you take that are against the public interest
(e.g., criminal activity).
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For example, you couldn’t buy insurance against getting a speeding ticket – the moral hazard
generated may result in the temptation to drive fast and endanger the lives of others.
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Another example comes from the USA where “Carry Guard” policies were sold by the National Rifle
Association (NRA). These have been labelled as “murder insurance” because the policies provided an
upfront payment to cover legal fees if the policyholder is involved in a shooting – even if the
policyholder ends up pleading guilty or is convicted of a crime. But,
― “After two years of state investigations and numerous lawsuits, the National Rifle Association has
stopped offering its Carry Guard self-defense insurance products”.
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Examples of “Accumulation of Risk”
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Insurers are often reluctant to cover events that might cause a large number of claims all at the same
time, as these claims could potentially ruin the insurance company. Some examples include:
― Flood and Cyclone insurance in Australia
― Earthquake insurance in NZ
― Terrorism insurance
Insurers MAY be willing to cover these risks if they are able to purchase reinsurance – but reinsurance
premiums will be high and reinsurers may also be reluctant to provide such reinsurance arrangements.
When insurance is not available on the market but would be societally beneficial, governments
sometimes step in to provide insurance themselves. For example,
― New Zealand has the Earthquake Commission (EQC) which covers the costs of certain natural
disasters. Premiums are paid into the scheme whenever someone buys a home insurance policy.
― Australia has a Cyclone Reinsurance Pool:
“The Government has implemented a reinsurance pool for insurance companies to transfer their
risk for cyclones and cyclone-related flood damage, called the Cyclone Reinsurance Pool (cyclone
pool), which is backed by a $10 billion Government guarantee.”
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Claims Development Process
Claims Development Process
• The development process of a claim is generally much more variable for general insurance than for
life insurance.
• Some important features:
• Reporting Delay – time between Time of Occurrence and Time of Report
• Settlement Delay – time between Time of Report and Time of Settlement
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Reporting and Settlement Delays
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Reporting delay can be very long
― Example: Medical malpractice (professional indemnity)
― Example: Worker’s compensation
▪ Famous case: Asbestos (banned in 2003, discovered to be carcinogen).
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Settlement delay can be very long
― Example: CTP may require lengthy legal procedures
― Example: Worker’s compensation may require long rehabilitation
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Short vs Long Tail Lines of Business
Often in practice, you will see lines of business divided into short vs long tail
• Short tail lines are those where claims are reported and settled quickly
― Because the whole process doesn’t take very long for most of the claims, the risk of having lots of
large claim payments in the future is not very high
― Remember that insurers must hold capital as buffer for future uncertainty.
▪ Because short-tailed lines have less risk and uncertainty (usually), they will have to hold lower
amounts of capital to maintain appropriate levels of solvency.
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Long tail lines may take years or even decades before most of the claims are settled
― Lots of risk and uncertainty (e.g., Asbestos case)
― You must hold lots of capital for these LoBs
― It is difficult to assess the value of outstanding claims liabilities… this valuation of liabilities is
one of the important tasks actuaries do!
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Key differences between LI and GI
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Life insurance insures specific mortality / disability events whereas general insurance products insure a
much larger variety of products.
Life insurance products generally have a shorter reporting and settlement delays (except for IDII).
Life insurance products have a defined Sum Insured (and a claim payment is usually for the full sum
insured), this is not always the case for general insurance (so the magnitude of claims is not known
ahead of time).
― For example, CTP has no SI amount – the claim cost is however much money is required to
compensate the claimant for damages, and this can be HUGE (medical costs and rehabilitation,
pain and suffering, future economic loss).
There can be multiple claims on a GI policy.
GI policies are generally short term, e.g. you buy your car insurance every year, covering the next year.
Regulations in Life Insurance mean that life insurance pricing is generally much more regulated,
and you can only take a small number of factors into account (e.g., age, sex, medical history). This is
less the case for GI.
GI companies can adjust premiums selectively and cancel unprofitable policies.
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General Insurance Claims Modelling
Motivation
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When we did calculations for life insurance contracts, the claim “amount” was not random (only the
timing of payments was random).
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A common feature of general insurance products is that claim amounts are not known ahead of time
(e.g., if you claim on your comprehensive motor insurance, it could be for a few 100’s dollars of repairs
up to the total value of the car).
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Also, contrary to LI, on GI policies an insured is usually allowed to make multiple claims on a single
policy. E.g., if you are very unlucky (or maybe a very bad driver), you could have 1,2,3, etc. car
accidents per year (while you can only die once).
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So, we are now in a context where it is relevant to model both frequency (how many claims) and
severity (how big are the claims) as random variables.
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Modelling general insurance claims
Claim Frequency
• First, let 𝑁 be a random variable representing the number of claims.
― For example, this could represent the yearly number of claims on a single policy or on a whole
portfolio of policies.
Claim Severity
• Let 𝑋1 , 𝑋2 , … , 𝑋𝑁 be the sizes of the 𝑁 claims.
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We make the simplifying assumption that the claim amounts are i.i.d. So, we can without confusion call
“𝑋” the generic severity random variable.
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The total claim amount 𝑆 is then
𝑁
𝑆 = ෍ 𝑋𝑖
𝑖=1
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Claim frequency distributions
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It makes sense to have 𝑁 be a discrete random variable. In Week 9, we had the “number of deaths” 𝐷
be Binomially distributed. This makes less sense in the context of general insurance claims modelling
(what would 𝑛 and 𝑝 represent?).
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Instead, a common assumption for frequency is the Poisson distribution. Recall the PMF for the
Poisson distribution is
Pr 𝑁 = 𝑛
𝑒 −λ λ𝑛
=
𝑛!
for 𝑛 = 0,1,2,3 …
Properties you (should) already know:
• σ∞
𝑛=0 Pr(𝑁 = 𝑛) = 1.
• E 𝑁 = 𝜆.
• Var 𝑁 = 𝜆.
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Additive property of independent Poisson rvs
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The Poisson has a cool property: if 𝑁1 ∼ Poisson 𝜆1 , 𝑁2 ∼ Poisson 𝜆2 , … , 𝑁𝑛 ∼ Poisson(𝜆𝑛 ) are
independent Poisson random variables (with possibly different lambdas). Then, the sum:
𝑛
𝑁 ∗ = ෍ 𝑁𝑖
𝑖=1
follows a Poisson(𝜆∗ ) distribution!
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Question for a Champion: what is 𝜆∗ ?
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This property is useful as it allows us to model the total claims at the insurer portfolio level (under
simplifying assumptions).
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Additive property: Josephine’s love life
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On any given day, the number of “Likes” Josephine gets on a dating app is a Poisson random variable.
The average number of likes on weekdays is 1, while it is 2 on Saturday and Sunday. The numbers of
likes received on different days are independent.
What is the probability she gets at least 4 likes in any given week?
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Solution
• Call 𝑁 the weekly number of likes. We have 𝑁 = 𝑁1 + 𝑁2 + ⋯ + 𝑁7 , hence
𝑁 ∼ Poisson 1 + 1 + 1 + 1 + 1 + 2 + 2 = Poisson(9)
• So,
Pr 𝑁 ≥ 4 = 1 − Pr 𝑁 < 4
= 1 − (Pr 𝑁 = 0 + Pr 𝑁 = 1 +Pr 𝑁 = 2 + Pr 𝑁 = 3 )
= 1 − 𝑒 −9
92 93
1+9+ +
2! 3!
= 1 − 𝑒 −9 ⋅ 172
= 0.978774
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Additive property: Insurance Claims Example
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There are 20,000 policies in a portfolio. The number of claims on any one policy is Poisson distributed
with parameter 0.01, and each claim costs $5,000.
What are the expected value and standard deviation of the total claim cost?
Solution: Because the claim cost is fixed at $5,000, the expression for total claim cost is simply
𝑆 = 5000𝑁
where 𝑁 is known to be a Poisson(200). Hence,
E 𝑆 = 5000 ⋅ E 𝑁 = 1,000,000
S. D 𝑆 =
ACST1052: Week 11
50002 ⋅ Var 𝑁 = 70,711
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Claim severity distributions
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The previous example was oversimplified. As argued before, for GI the claim size would usually be
random as well.
There are many options to model such random claim sizes, and we usually prefer to use continuous
distributions.
There are many choices available – we often consider the shape and tail of the distribution when
examining the optimal distribution to fit to the data. Common choices are
― Lognormal
― Gamma
― Pareto
― Burr
Unfortunately, in this course we don’t really have the mathematical tools yet to derive the full
distribution of 𝑆… unless we are in the simple case of a discrete claim size 𝑋.
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Distribution of 𝑺: Example
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Assume that both 𝑵 and 𝑿 are discrete, with PMFs given below.
Question: what is Pr 𝑺 = 200 ?
ACST1052: Week 11
𝑛
0
1
Pr(𝑁 = 𝑛)
0.90
0.07
𝑥
100
200
Pr(𝑋 = 𝑥)
0.5
0.3
2
3
0.02
0.01
300
0.2
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What is Pr 𝑺 = 𝟐𝟎𝟎 ?
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We need to list all the possibilities which make the total claim size exactly 200.
1. One claim of size 200
2. Two claims of size 100
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Thus, Pr 𝑆 = 200 = 0.07 × 0.3 + 0.02 × 0.52 = 0.026.
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To get the full distribution of 𝑆, you would need to do this for every possible value of 𝑆
0,100,200,300, … , 900 . It’s tedious to do by hand, although not terrible to do via Excel or R.
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Even though we can’t (yet) derive the full distribution of 𝑆 in the case 𝑋 is continuous, there is one more
thing we can do, which is to derive a general expression for the expected value of 𝑆.
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Expected value of 𝑺
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As before, denote the (random) number of claims by 𝑁 and the (random) claim amounts by
𝑋1 , … , 𝑋𝑁 (assumed to be i.i.d. and independent of 𝑁).
Then, the excepted value of the total claim amount 𝑆 = σ𝑁
𝑖=1 𝑋𝑖 is simply:
E 𝑆 = E 𝑁 ⋅ E[𝑋]
law of total expectation
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Proof. Recall the law of total expectation: E 𝑆 = E[E[𝑆|𝑁]]. This is the main ingredient of the proof, as
we then have
𝑁
E 𝑆 = E[E[𝑆|𝑁]] = E E ෍ 𝑋𝑖 |𝑁
= E 𝑁 ⋅ E[𝑋] = E 𝑁 ⋅ E 𝑋 .
𝑖=1
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What is the variance of 𝑆 ? We don’t cover it here, but there is a “simple” formula for it, which is the
topic of a tutorial exercise.
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Example
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Assume that an insurer’s total number of claims for a certain line of business is a Poisson(200) random
variable. In addition, the size of any individual claim is modelled with the following density function:
𝑥
1
−
𝑓 𝑥 =
𝑒 5000 ,
5000
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𝑥≥0
What is the expected value of the total claim amount?
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Deductibles
Deductibles in general insurance
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In general insurance, policies often require the policyholder to cover the first portion “$𝑑” of a claim.
This is called a deductible or excess.
Example: if you have a motor policy with a deductible of 𝑑 = $700 and the repairs after an accident
cost $3,000, then you must pay $700, and the insurer pays the other $2,300. Of course, this means
that if you are in a minor accident and the repairs cost only $500, then the insurer doesn’t pay anything.
Question: What are some advantages of deductibles?
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Advantages of a deductible
1. Reduces moral hazard
― People are incentivized to be careful if they know that they will have to bear some part of any loss.
2. Eliminates a lot of small claims
― Saves administration costs since people won’t submit claims that are less than the deductible.
3. Reduces the premium rates
― From the insurer’s PoV, a deductible means a lower expected value for the claims, hence they can
charge less for the cover. This effect is compounded by the efficiency gained from not having to
handle lots of small claims.
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Calculating a risk premium allowing for a deductible
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Consider the previous example but we now allow for a deductible of $150 per claim.
𝑛
0
1
Pr(𝑁 = 𝑛)
0.90
0.07
Loss
100
200
Net 𝑥
0
50
Pr(𝐍𝐞𝐭 𝑋 = 𝑥)
0.5
0.3
2
3
0.02
0.01
300
150
0.2
To determine the expected value of the claim costs, we just need to calculate the average frequencies
multiplied by average net claim size (allowing for the deductible):
E 𝑆 = E 𝑁 × E[Net 𝑋]
= 0.14 × 45 = 6.30
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Claim size caps and large losses
Setting claim caps
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Sometimes, general insurers can limit their exposure to large losses by capping the amount payable.
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For example, if you own a $50M house, the insurer can limit the amount you are able to insure with
them on a home and contents insurance policy.
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But in many cases, this can contradict the purpose of insurance
― Worker’s compensation – the point is to pay compensation and costs for rehabilitation until the
person can get back to work (however long this takes).
― Motor CTP – similar to the above.
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So, if an insurer does not put a cap on the amount payable, how can they limit their risk?
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Reinsurance
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A common way for insurers to manage their risks is through reinsurance. The two main types of
reinsurance contracts are:
― Quota Share
― Excess of Loss
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Note that even if an insurer sells policies with only small sums insured, they can still be exposed to
high total claim costs if there are catastrophe losses.
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Let’s do a quick example involving Excess of Loss reinsurance. Excess of loss reinsurance is
essentially an insurance company buying insurance with a deductible (from a reinsurer).
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Insurer’s expected claims cost with XOL reinsurance
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Consider the same example as before, but the reinsurer pays the excess of claims cost over $180 on
individual claims.
To calculate the expected value of claims for the insurer, we have that
E 𝑆 = E 𝑁 × E[Net 𝑋]
= 0.14 × 140 = 19.60.
ACST1052: Week 11
𝑛
0
Pr(𝑁 = 𝑛)
0.90
Loss
100
Net 𝑥
100
Pr(Net 𝑋 = 𝑥)
0.5
1
0.07
200
180
0.3
2
0.02
300
180
0.2
3
0.01
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Reinsurer’s expected claims cost
From the reinsurer’s perspective, we have that
𝑛
0
1
Pr(𝑁 = 𝑛)
0.90
0.07
Loss
100
200
𝑦
0
20
Pr(𝑌 = 𝑦)
0.5
0.3
2
3
0.02
0.01
300
120
0.2
In the above, we have let 𝑌 represent the amount paid by the reinsurer for each claim.
E 𝑆 = E 𝑁 × E 𝑌 = 0.14 × 30 = 4.20
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Gross Premiums
Terminology
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We call “risk premium” the expected present value of the insurance benefits (only) on a given
insurance policy (which typically covers a 1-year period).
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We call “pure premium” the expected present value of insurance benefits and all other costs and
expenses associated with selling the policy.
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We call “gross premium” (or “office premium”) the premium actually charged to the customer.
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Often, for GI, we won’t worry about “present value” because all costs are assumed to arise in a short
period of time, so we may ignore discounting. If nothing is specified in a question, you may ignore
discounting (i.e., assume the interest rate is 0%).
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Note those terms are (unfortunately) not universal (you may see elsewhere people calling “pure
premium” what we call “risk premium”).
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Gross Premiums
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As we had previously discussed, insurers need to charge more than just the expected value of the
claims, otherwise they would almost certainly become insolvent.
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We previously discussed that “loadings” are applied on top of the risk premium to get the final price
charged:
― Expense loading
― Profit loading
― Safety loading
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Additionally, insurance policies are often sold through insurance brokers, who are paid a commission
(which we must include in the calculation of the final “gross premium”). Usually, commissions are
stated as a percentage of the gross premium.
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Risk Premium Example
•
Going back to our example (with no deductible or reinsurance)… what would be the risk premium?
𝑛
0
1
2
3
•
Pr(𝑁 = 𝑛)
0.90
0.07
0.02
0.01
𝑥
100
Pr(𝑋 = 𝑥)
0.5
200
300
0.3
0.2
Answer: $23.80.
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Gross Premium Example
•
The risk premium for one policy is $23.80, but now let’s assume that:
― The broker commission is 5% of the gross premium
― Claims handling expenses are $10 per claim made
― Other expenses are $2 per policy
― The gross premium needs to provide $4 in safety/profit loading
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What is the gross premium (denoted 𝐺)?
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Gross Premium = Risk Premium + Expected CHE + Expenses + Profit Loading + Commission
𝐺 = 23.80 + 10 × 0.14 + 2 + 4 + 0.05𝐺
𝐺 = 32.85.
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