RISK POOLING 101: Inside the Math of Insurance

We all know insurance—it's that thing our parents stress over and businesses stockpile like they’re prepping for an apocalypse. But what if I told you it’s not just some boring paperwork? Let’s try to understand how insurance really works and why it’s way more interesting than you think!

An insurance policy is just a contract where you pay regular premiums, and the insurance company promises to cover certain risks. Essentially, you pay up front, and they’ve got your back against financial hiccups.

So, if you experience a loss covered by your insurance, the company will help you recover even if your total premiums paid are less than the loss. For example, if you paid Rs. 15,000 in fire insurance premiums but suffered Rs. 1,00,000 in damage, the insurance will cover most of your loss. Pretty heroic, right?

The magic here is Risk Pooling. Essentially, by having many people buy insurance, individual risks combine into a collective one. The more policies there are, the lower the overall risk for everyone involved.

• Generally, Risk Pooling is understood by the fact that not all policyholders will file claims at the same time. Claims occur at different times, allowing insurance companies to handle them more easily. They maintain enough cash reserves to cover occasional claims without breaking a sweat.

This gives us a basic idea of risk pooling BUT leaves us with two key questions:

1. Is it accurate to assume that only a few claims will arise in a short period?

2. How do insurance companies manage risk pooling and set premiums amidst the uncertainty of claim frequency?

For this we'll try to understand this phenomena in terms of the probabilities of different situations. Let's say, the Probability of a Insurance Claim being made is y. Probability of the claim not made is x.

? x + y = 1

• This represents the sum of all probabilities for a claim being made or not made on a single insurance policy. (i.e. =1)

If there are n identical and independent insurance policies, the total probability of all possible outcomes—whether claims are made or not—will sum to 1 :

? (x + y)^n = 1^n (We multiply it n times because all claims, whether made or not, happen together)

Well now through Binomial Distribution we can expand this:

where 'k' is the no. of claims being made out of 'n' policies

• Simply put, this represents the sum of all probabilities of k claims being made out of n policies

Thus, the formula shows that the probability of k claims out of n policies is:

I get it—at first, the formula might look like it’s written in ancient code. But don’t worry, we’ll crack the code with a simple example!

• Let's say for a 0.5 probability of a claim and 5 policies in total, the probability of 2 claims using the above formula :-

Probability = 5C2 * (1-0.5)^3 * (0.5)^2

= 5C2 * (0.5)^3 * (0.5)^2 = 15 * 0.125 * 0.25

= 0.46875

With 6 policies, the probability of 2 claims drops to 0.23. With 7 policies, it drops further to 0.16. In general, as the number of policies increases, the probability of a claim decreases.

• However, the probability multiple claims is actually our main point of concern, so we can represent this in graph under different situations, i.e., when y>0.5, y=0.5, y<0.5 :-

*(Note: substitute p with y in the graph)

Here, X Axis represents the k ,i.e., the no. of claims

Y Axis represents the Probability of k claims out of n policies

Upon looking at graphs, you would notice:

• In Graph 1, when y>0.5, the probability for k claims out of n policies is higher when k is larger
• In Graph 2, when y=0.5, the shape is symmetrical, meaning probability for medium range values of k is greater
• In Graph 3, when y<0.5, the probability for k claims out of n policies is higher when k is smaller

In simple terms, if it’s less likely for a single claim to happen, it’s also less likely for multiple claims to happen. Conversely, if a single claim is more likely, then multiple claims are too.

It’s like a claim’s popularity contest!

HOW DO INSURERS TACKLE THESE PROBABILITIES?

Insurance companies have a few tricks up their sleeves for handling claim probabilities. Here’s a peek at how they juggle those numbers :-

1. LEGAL PRINCIPLES

Insurance contracts include strict terms and conditions, but they are also governed by key legal principles to ensure everything runs smoothly. Here’s a quick rundown:

• Principle of Utmost Good Faith : Think of this as the "no secrets" rule—both sides have to be honest and share all relevant information.
• Principle of Insurable Interest : You can’t insure your neighbor’s car just because it’s fancy. You need to have a legitimate interest in what you’re insuring.
• Principle of Indemnity : No getting rich off your insurance! You only get compensated for your actual loss, not a penny more.
• Principle of Contribution : If you’re double-covered, the insurers team up to pay the claim.
• Principle of Subrogation : After paying you, the insurer gets to play detective and recover the money from whoever caused the loss.
• Principle of Loss Minimization : If something goes wrong, you’re expected to act like a responsible adult and minimize the damage.
• Proximate Cause : The loss has to be directly caused by an insured risk—no blaming it on the neighbor's cat.

Without these safeguards, insurers would face moral hazard, where policyholders might take greater risks knowing they’re covered. This would lead to a spike in claims, undermining the stability of the insurance system.

2. SELECTION BIAS

Before issuing any insurance policy, companies thoroughly evaluate a person's history, profile, and suitability for the coverage. This process is known as Selection Bias. Essentially, it's the insurance company’s way of choosing who they want to insure based on their likelihood of making a claim.

For example, in life insurance, an applicant might undergo a series of medical tests, provide health records, and answer detailed questions about their lifestyle. This helps insurers assess the risk of the applicant potentially filing a claim. If someone has a history of serious health issues or engages in high-risk activities, they might be deemed a higher risk for the insurer.

By carefully selecting who they insure, companies aim to avoid high-risk individuals who are more likely to make claims. This helps them manage risk and maintain the financial stability of their insurance pool.

3. INCREASING PREMIUMS

• Another strategy for insurance companies is to increase annual premiums. This serves to either discourage high-risk individuals from purchasing insurance or ensure the company maintains sufficient reserves to cover future claims. By raising premiums, insurers can better manage their risk exposure and maintain financial stability.
• Additionally, if a person has previously made a claim or their policy is up for renewal, the premium often rises due to the increased likelihood of future claims. This adjustment reflects the higher probability of the insured making another claim, thus ensuring that the insurer remains financially prepared.

For example, consider flood insurance in the U.S. Homeowners in floodplains are required by the government to purchase flood insurance, which comes with high premiums due to the frequent risk of flooding. The elevated premiums serve two purposes: they deter individuals from building in flood-prone areas and ensure those who do can afford the higher costs associated with their increased risk.

#BALANCING RISK AND ACCESS

• Reducing claim probabilities is crucial for effective risk pooling, but it's also essential to ensure insurance is accessible to those in need and those less likely to make claims. The government's role is pivotal in achieving this balance through effective regulation and oversight of the insurance industry.

One example is the U.S. National Flood Insurance Program (NFIP), which was established to provide flood insurance coverage to communities at risk of flooding. This program helps manage risk while ensuring that people in flood-prone areas can obtain insurance coverage.

• Another example is Health Maintenance Organizations (HMOs), which offer managed care plans in the U.S. These plans provide not only coverage for medical services but also focus on preventive care and wellness, such as regular check-ups and health education, which can reduce the likelihood of more costly claims.

The Affordable Care Act (ACA) introduced the individual mandate, which required U.S. citizens to have health insurance or face a tax penalty. Although the mandate was repealed in 2019, it initially helped increase insurance coverage and improve access. Critics argued that it led to higher premiums for some, but it also contributed to expanding coverage to a broader segment of the population.

Government interventions and regulations play a crucial role in making insurance accessible to high-risk individuals, incentivizing healthier individuals to purchase insurance, and balancing the insurance pool. While there may be challenges in the short term, well-designed policies can lead to long-term benefits, including more affordable premiums and improved coverage by incorporating both high and low-risk individuals into the insurance system.