**Probability (means)**:

Random House a strong likelihood or chance of something. The relative possibility an event will occur (the ratio of the number of actual occurrences to the total number of possible occurrences).

A numerical measure of the chance or likelihood that a particular event will occur. Probabilities are generally assigned on a scale from 0 to 1. A probability near 0 indicates an outcome that is unlikely to occur, while a probability near 1 indicates an outcome that is almost certain to occur.

The probability of an event occurring is the chance or likelihood of it occurring. The probability of an event A, written P(A), can be between zero and one, with P(A) = 1 indicating that the event will certainly happen and with P(A) = 0 indicating that event A will certainly not happen.

Probability: (in other terms)

- the number of successful outcomes of an experiment
- the number of possible outcomes

If a coin were tossed, the probability of obtaining a head = ½, since there are 2 possible outcomes (heads or tails) and 1 of these is the ‘successful’ outcome.

In probability theory, the law (or formula) of total probability is a fundamental rule relating marginal probabilities to conditional probabilities. It expresses the total probability of an outcome which can be realised via several distinct events.

**kinds of probabilities**:

- Simple Probability:

The ratio of the number of outcomes favourable for the event to the total number of possible outcomes is termed as probability. In other words, a measure of the likelihood of an event (or measure of chance) is called probability.

It is the calculation of an outcome or the chance of an event ever happening. Insurance companies use probability statistics to determine the chances of having to pay out a claim. A simple probability is calculated by dividing a specific outcome by all the possible outcomes.

The probability of an event, like rolling an even number, is the lnumber of outcomes that constitute the event divided by the total number of possible outcomes. We call the outcomes in an event its “favourable outcomes”.

For instance, when flipping a coin, there are two outcomes: heads or tails. To find the probability of getting either heads or tails, divide one outcome (1) by the two possible outcomes (2). Dividing 1 by 2 results in .50 or 50%.

- Classical probability:

It is the statistical concept that measures the likelihood (probability) of something happening. In a classic sense, it means that every statistical experiment will contain elements that are equally likely to happen (equal chances of occurrence of something)

- Objective Probability:

It refers to the chances or the odds that an event will occur based on the analysis of concrete measures rather than hunches or guesswork. Each measure is a recorded observation, a hard fact, or part of a long history of collected data.

It refers to long-run relative frequency of an event based on the assumptions of an infinite number of observations and of no change in underlying conditions. It also ascertains that the occurrence of an event on the basis of already present information or observation or large portion of accumulated data.

- Subjective probability: (personal probability)

It is the individual’s personal estimate of the chance of loss and is based on your beliefs. The probability of an event is a “best guess” by a person making the statement of the chances that the event will happen. (e.g. 30% chance of rain)

It permits the analyst to calculate the probability of an outcome based on experience and their own judgement.

This type of probability is the perceived chance of a certain outcome happening. It is not an actual mathematical calculation of the odds, but rather a measure based on personal opinion, beliefs, prejudices, and emotions.

For example, a fan of the favourite game say their team has a 50% chance of winning the World Series, despite not having any statistical evidence. It is not based on the team’s record, but on the fan’s personal beliefs and confidence.

- Experimental or empirical probability:

It is expressed as a ratio of the number of times an event as occurred and the number of experiments performed. The method becomes increasingly accurate the more experiments are conducted, which translates into bigger and more revealing data.it is also based upon experiments.

This method helps insurers predict the likelihood of claims being filed so they can make informed decisions about premium rates. it is a statistical method for determining the frequency of an event by using actual data from experiments.

- Compound Probability:

It refers to a mathematical calculation that determines the possibility of two separate events happening at the same time. To calculate compound probability, multiply the possibility of the first even occurring with the probability of the second event occurring.

The insurance industry uses compound probability to analyse risks and determine premiums.

To use a simple example, suppose we knew someone who went for a morning jog 50% of the time and who also drove their car on average every other day. If we wanted to calculate the compound probability of them doing both activities on the same day, we would multiply both percentages, as follows:

0.5 x 0.5 = 0.25 This means there is a 25% chance that this person would both go for a jog and drive their car on a given day.

- Marginal probability:

It is the probability of an event irrespective of the outcome of another variable. It is the probability of occurrence of a single event. In calculating marginal probabilities, we disregard any secondary variable calculation. In our hypothetical example, we can calculate two marginal probabilities, we can

look at specific eating habits or we can look at commute times. In essence, we are calculating the probability of one independent variable. It is not conditioned on another event.

- Joint probability:

It is the joint probability which is the probability of two different events occurring at the same time or simultaneously .

It is the probability of the intersection of two or

more events. The probability of the intersection of A and B may be written p(A ∩ B). Example: the probability that a card is a four and red =p(four and red) = 2/52=1/26. (There are two red fours in a deck of 52, the 4 of hearts and the 4 of diamonds).

- conditional probability:

It is the probability that an event will occur given that another specific event has already occurred. For example, we would calculate the probability for some eating behaviour given that we know the commute times of the population. We say that we are placing a condition on the larger distribution of data, or that the calculation for one variable is dependent on another variable.Itis the probability of one event occurring in the presence of a second event.

Conditional probability is the probability of one event occurring in the presence of a second event.

p(A|B) is the probability of event A occurring, given that event B occurs. Example: given that you drew a red card, what’s the probability that it’s a four (p(four|red))=2/26=1/13. So out of the 26 red cards (given a red card), there are two fours so 2/26=1/13.

- Axiomatic Probability:

It is a type of probability that has a set of axioms

(rules) attached to it and a rule that the probability must be greater than 0%, that one event must happen, and that one event cannot happen if another event happens.

- Weighted probabilities:

Probabilities represent the chances that different events will occur. For example, if you were rolling a single six-sided die, you would have the same probability of rolling a one as rolling any other number because each number will come up one out of six times. However, not all scenarios have each outcome equally weighted. For example, if you add a second die to the mix, the odds of the dice adding up to two are significantly less than adding up to seven. This is because there is only one die combination (1,1) that results in two, while there are numerous die combinations–such as (3,4), (4,3), (2,5) and (5,2)–that results in seven.

Determine the total number of possible outcomes for the scenario. For example, with rolling two dice, there are 36 possible outcomes because there are six outcomes for each die so you would multiply six times six.

Determine how many ways the desired outcome can occur. For example, if you are playing a board game and will win if you roll an eight, you would need to determine how many ways an eight could be rolled, which is five: (2,6), (3,5), (4,4), (5,3) and (6,2).

Divide the number of ways to achieve the desired outcome by the number of total possible outcomes to calculate the weighted probability. To finish the example, you would divide five by 36 to find the probability to be 0.1389, or 13.89 percent.

- Game probability:

The probability of rolling a 6 is 1/6 (there are six numbers, so you roll a six one time out of every 6) The probability of not rolling a six is 5/6 (rolling any of the other five numbers)

- Playing card probability:

Let’s look at one probability in these two ways:

What is the probability of drawing a card from a deck and it being red and a face card?

For this probability, we need to look at which cards are both red and face cards. There are 6 of these: Jack of Hearts, Queen of Hearts, King of Hearts, Jack of Diamonds, Queen of Diamonds, and King of Diamonds.

P= 6/52 = 3/26 ≈ 11.5%

OR

This time the card can be red, or a face card, or both at the same time. There are 26 red cards (6 of which are also face cards). In addition, there are 6 more face cards that are not red: Jack of Clubs, Queen of Clubs, King of Clubs, Jack of Spades, Queen of Spades, and King of Spades. That is a total of 26 + 6 = 32 cards.

32/52 = 8/13≈61.5%

Be careful not to just add up the number of face cards (12) with the number of red cards (26). That would give a total of 38 cards, but it would count the red face cards twice.

Part.II

__Applicability of probability in insurance:__

__Principles of Insurance__:

The insurance is based upon (i) Principles of Co-operation and, (ii) Principles of Probability.

- Principles of Co-operation:

Insurance is a co-operation device. If one person is providing for his own losses, it cannot be strictly insurance because in insurance, the loss is shared by a group of persons who are willing to co-operate. In ancient period, the persons of a group were willingly sharing the loss to a member of the group. They used to share the loss to a member of the group.

They used to share the loss at the time of damage. They collected enough funds from the society and paid to the dependents of the deceased or the persons suffering property losses.

The mutual co-operation was prevailing from the very beginning up to the era of Christ in most of the countries. Lately, the cooperation took another form where it was agreed between the individual and the society to pay a certain sum in advance to be a member of the society.

The society by accumulating the funds, guarantees payment of certain amount at the time of loss to any member of the society. The accumulation of funds and charging of the share from the member in advance became the job of one institution called insurer.

Now it is became the duty and responsibility of the insurer to obtain adequate funds from the members of the society to pay them at the happening of the insured risk. Thus, the shares of loss took the form of premium. Today, all the insured give a premium to join the scheme of insurance. Thus, the insured are co-operating to share the loss of an individual by payment of a premium in advance.

- Principles and Theory of Probability:

The theory of probability (also known as probability theory or theoretical probability) is a statistical method used to predict the likelihood of a future outcome. This method is used by insurance companies as a basis for crafting a policy or arriving at a premium rate.It also aims to establish patterns for the occurrence of various types of events by using mathematical or statistical methods.

The loss in the shape of premium can be distributed only on the basis of theory of probability. The chances of loss are estimated in advance to affix the amount of premium. Since the degree of loss depends upon various factors, the affecting factors are analysed before determining the amount of loss. With the help of this principle, the uncertainty of loss is converted into certainty.

The insurer will have not to suffer loss as well have to gain windfall. Therefore, the insurer has to charge only so much of amount which is adequate to meet the losses. The probability tells what the chances of losses are and what will be the amount of losses.

The inertia of large number is applied while calculating the probability. The larger the number of exposed persons, the better and the more practical would be the findings of the probability. Therefore, the law of large number is applied in the principle of probability.

In each and every field of insurance the law of large number is essential. These principles keep in account that the past events will incur in the same inertia. The insurance, on the basis of past experience, present conditions and future prospects, fixes the amount of premium.

Without premium, no co-operation is possible and the premium cannot be calculated without the help of theory of probability, and consequently no insurance is possible. So these two principles are the two main legs of insurance.

The insurer charges only so much of amount which is adequate to meet the losses. Pooling of a large number of risks is very necessary for the successful operation of the theory of probability. The law of large numbers is a sub principle of the principle of probability.

__Concept of Probability Theory and Statistics:__

A branch of mathematics, predicting random events by analysing large quantities of previous similar events. Probabilities in statistics are the mathematical odds that an event will occur. To obtain a probability ratio, the number of favourable results in a set is divided by the total number of possible results in the set. The probability ratio expresses the likelihood that the event will take place. This ratio is significant to insurance providers.

- Statistical methods:

The first concept that insurance relies on is known to statisticians as “the Law of Large numbers” and it’s best explained by example.

Let’s say you’re sitting in your office, bored by whatever menial task is sitting in your inbox, and you decide to play a game. You pull out a coin and you see how many heads you can flip in a row. You flip one or two heads in a row easily. But you start to find that it’s much harder to keep getting heads (assuming you’re flipping fairly).

This is because the probability of you flipping a head is 1/2 or 50%. But 2 heads in a row is (1/2) x (1/2) or 1/4 or 25%. So statistically you’ll only flip 2 heads in a row once out of every 4 tries. That’s discouraging. The probabilities get lower and lower very quickly. The probability of flipping 6 heads in a row is 1/64 or 1.5%. You could try 100 times and have it happen only once or twice. Suddenly, your neglected paperwork seems much more friendly.

Let’s say you get discouraged and you decide you’re going to play a different game instead. Let’s also assume that you’ve grown up under a rock and you don’t know that the probability of flipping a head is 50%. So you decide to record the number of heads and tails you flip.

The first two coins you flip are heads and It appears as if the rule is that every time you flip a coin, you get a head. The probability of flipping a head is 100% according to your data and You are unsure though, so you keep flipping. Next is a tail. So you were wrong and the probability of flipping a head must be 2/3 or 66% right and Because that’s what you’ve flipped so far.

As you continue playing this game, we both know that your record will get closer and closer to 50% as you flip more and more. In other words, because of what you discovered with your first game, it gets harder and harder to ‘fudge’ the probability the more you flip. Mathematicians go one step farther to say that if you take any event and record lots and lots of trials, the results get closer and closer to the actual probability with each new trial, eventually getting so close that you can just accept the result as the actual probability.

So what does this have to do with insurance?we know that insurance agencies insure lots of people (they have to, or else it wouldn’t work, just like the coin). Every person pays a small amount of money each month and nothing happens to them. But every once in a while, an insurance company will have to pay lots of money to a single person.

An insurance company that insures 1000 people. Let’s say that 1 house will catch on fire per year. So the probability of a house catching on fire is (1/1000). Therefore the probability of a house not catching on fire is (999/1000)

If a house catches on fire, you have to pay Rs. 200,000

Every person pays you Rs. 20 a month, Rs. 240 per year

Let’s use our formula:

200,000x(1000) =200 +( 240x(999/1000)) = 200 +39.76 239.76

So you’ll make Rs. 39.76 on average per person insured. Therefore you’ll make 1000x(39.76) = Rs. 39,760.

Now this is a really simple example (with really small numbers), but it’s the same math that insurance companies hire statisticians to calculate for them. In fact, there’s a whole branch of mathematics related to this concept called Actuarial Science and By collecting lots of small payments, insurers know that probability will protect them from the occasional loss and That’s why how the insurers are stay in business know that probability will protect them from the occasional loss.

__Applicability of probability to Insurance:__

Generally, rates will differ for policyholders contracting identical insurance policies depending on several analysable rating factors. Insurance providers have good reasons for this practice. As part of the analytical procedures, insurers study statistics to calculate and manage risk when evaluating policy applications and setting premium rates. The results show that, based on probability, some individuals simply pose a higher risk and are more likely to file claims.

- Health Insurance:

Insurance underwriters use probability theory when evaluating policy applications. For example, policyholders who smoke tobacco are at a higher risk for developing serious health problems. Statistics show that this often results in increased health insurance claims. The applicant’s age and geographic location also allow the underwriter to predict future claims based on probability.

- Life Insurance and Annuities:

Analysing mortality rates, the insurer considers where the policyholder lives and what socioeconomic factors apply to the policyholder’s current age and health. This analysis helps the insurer determine rates and options for life insurance policies and annuities using probability theory to predict the number of years a policyholder will live.

Insurance companies use this approach to draft and price policies. When issuing health insurance, for instance, the policy given to a smoker is likely more expensive than the one issued to a non-smoker. Statistical figures show a stronger association with a variety of health risks for habitual smokers or those with a history of smoking. Insuring a smoker, then, is a greater financial risk given their higher probability of serious illness and, hence, of filing a claim.

- Liability and Property:

Companies that provide property and liability insurance use probability to assess risks. Data show that the age and gender of the driver plays a role in the likelihood of an auto accident. The type of vehicle insured, the driver’s geographic location and the number of miles driven regularly are additional factors the insurer considers when setting premium rates based on probability.

The more miles a policyholder drives, for example, the greater the probability he’ll be involved in an accident. Setting rates for homeowners insurance also involves probability. Factors considered include the type of heating system in the home, the location and age of the property and any added security features it has.

__Law of probability in insurance :__

__Law of Large Numbers:__

__Law of large numbers:__

It enables insurers to predict future loss experience.

In probability theory, the law of large numbers is a theorem that describes the result of performing the same experiment a large number of times.

According to the law of large numbers, the average of the results obtained from a large number of trials should be close to the expected value, and will tend to become closer as more trials are performed.

Insurance companies use the law of large numbers to lessen their own risk of loss by pooling a large enough number of people together in an insured group. The size of the pool corresponds to the predictability of the losses, just like the more eggs we deal with, the more likely we are to know how many will be cracked.

For example, auto insurance may record and study the number of accidents caused by a very large population of 18-year-old males. They will be able to predict how many 18-year-old males will cause an accident in a given year. They will know that in a given year there is a high probability that X number of 18-year-old males will cause an accident. Knowing this, they partially can determine how much an 18-year-old male should pay for auto insurance (excluding other factors, such as the type of vehicle, region where the driver resides, etc.) This is how the law of large numbers helps insurance providers determine their rates, and why the rates vary from one type of individual to another.

__Law of Large Numbers Relates to Insurance:__

__(Understanding the Law of Large Numbers in Insurance)__

In the insurance industry, the law of large numbers produces its axiom. As the number of exposure units (policyholders) increases, the probability that the actual loss per exposure unit will equal the expected loss per exposure unit is higher. To put it in economic language, there are returns to scale in insurance production.

In practical terms, this means that it is easier to establish the correct premium and thereby reduce risk exposure for the insurer as more policies are issued within a given insurance class. An insurance company is better off issuing 500 rather than 150 fire insurance policies, assuming a stable and independent probability distribution for loss exposure.

First, all insurance companies are not equally adept at the business of providing insurance. This includes maintaining operational efficiency, calculating effective premiums, and mitigating loss exposure after a claim is filed. Most of these features do not impact the law of large numbers.

The law of large numbers is a statistical concept that relates to probability. It is one of the factors insurance companies use to determine their rates.

It means that the larger the number of units that are individually exposed to an event, the greater the likelihood that the actual results of that exposure will equal the expected results.

The Law of Large Numbers using eggs as an example, that for every three-dozen eggs sold by a grocer, an average of one of those eggs is cracked. Therefore, we expect that every time we buy three-dozen eggs, it is likely (though not guaranteed) we will find one cracked.The more eggs we buy, the more likely this is. If we buy 12-dozen eggs, the likelihood that one for every three-dozen will be cracked increases. If we buy 18-dozen eggs, the likelihood that one for every three-dozen will be cracked increases even more. The more eggs we deal with, the more likely we are to find that one out of every three-dozen is cracked.

Insurers use the law of large numbers to estimate the losses a certain group of insureds may have in the future. An actuary looks at losses that have occurred in the past and predicts that in the future approximately two out of 100 policyholders will have a claim. Thus, if the insurers writes 100 automobile policies, it may expect to pay two claims. This is referred to as loss frequency.

Insurance companies must also determine the average cost of claims over time, or loss severity. If the average claim resulted in the company paying Rs.1,000, then the actuary will predict that total losses for the upcoming year will be Rs.2,000 (two claims at Rs.1,000 each).

The law of large numbers states that as the number of policyholders increases, the more confident the insurer is its prediction will prove true. Therefore, insurance companies attempt to acquire a large number of similar policyholders who all contribute to a fund which will pay the losses.

Statistics is used to determine what risk an insured poses to an insurance company, what percentage of policies is likely to pay out, and how much money a company can expect to pay out in claims.

The Law of Large Numbers theorises that the average of a large number of results closely mirrors the expected value, and that difference narrows as more results are introduced.

In insurance, with a large number of policyholders, the actual loss per event will equal the expected loss per event.

The Law of Large Numbers is less effective with health and fire insurance where policyholders are independent of each other.

With a large number of insurers offering different types of coverage, the demand for variety increases, making the Law of Large Numbers less beneficial.

__Probability Analysis in insurance:__

A technique used by risk managers for forecasting future events, such as accidental and business losses. This process involves a review of historical loss data to calculate a probability distribution that can be used to predict future losses. The probability analyst views past losses as a range of outcomes of what might be expected for the future and assumes that the environment will remain fairly stable. This technique is particularly effective for companies that have a large amount of data on past losses and that have experienced stable operations. This type of analysis is contrasted to trend analysis.

__Relative Probability to Risk:__

The risk, as defined in insurance, is the possibility of a loss. The obverse of this definition is that risk is the possibility of no loss. If there is no possibility of loss, then there is no risk. Likewise, if loss is a certainty, then again, there is no risk, even if the outcome is undesirable. Thus, the probability of a loss must be between 0 and 1, not inclusive. However, sometimes risk cannot be measured. Because insurance premiums are determined by expected losses, risks that cannot be measured cannot be insured.

Loss can be broadly defined as an undesirable outcome or as a less desirable outcome. If you have the choice to buy 2 stocks, and the one that you bought goes up less than the other one, then you did not suffer a loss, but you did incur an opportunity cost. On the other hand, if, as you cross the street, you get hit by a truck, then that is an undesirable outcome. Both of these cases illustrate a type of loss, but the opportunity cost is not insurable.

Only risk is insurable, but not every risk. Only economic loss that can be compensated by the payment of money is insurable, and only if expected losses can be ascertained.

It also denotes an object that is a cause of risk, or a person or property that would be risky to insure. Thus, a heavy drinker would be a risk as a driver, or a wooden building would be a poor risk for fire insurance. The profitability of any insurance company depends on how well it can predict losses; thus, assessing risk requires the accurate calculation of the probability of losses.

__Relative Probability to loss:__

However, most insurable risks cannot be calculated using deduction, because there are too many variables with varying degrees of influence on the probability of a loss. For these cases, only induction can be used to aHowever, most insurable risks cannot be calculated using deduction, because there are too many variables with varying degrees of influence on the probability of a loss. For these cases, only induction can be used to assess the objective probability of an insurable risk, by recording a large number of observations under a given set of conditions, where the actual number of losses are recorded against the number of possible losses .

Generally, the term loss is used to denote the absence of something previously possessed, such as lost time or lost opportunities as well as economic losses, such as a lost wallet. However, insurance uses only the more restricted definition of economic loss, since only economic losses are insurable risks. Hence, in insurance, a loss is the unexpected reduction in the economic value of one’s possessions. Insurance companies use this definition because they can only cover such a loss with the payment of money. A loss differs from an expense, which is an expected payment for a good or service. Thus, buying gas for your car is an expense, while a car accident is a loss.

Chance of loss is the probability that a loss will occur, which can either be an expected loss or an actual loss, divided by the number exposed to loss, or the sample population.

Chance of Loss = __Expected or Actual Loss__

Number of Possible Losses

Because the chance of loss is only an average, actual losses may differ significantly from expected losses, especially for small samples, but as the sample size increases, actual and expected losses tend to converge.

__Probabilities of insurance claim:__

For an individual, the probability of having an accident in a period of 24 hrs (and therefore the probability of making a claim) is 0.00037. Claims on successive days are independent, and a person cannot have more than two accidents in one day.

What is the premium should the insurer charge for each or policy to assure or ensure that the premium income will cover the cost of all the claims.

A car insurance company has 2,500 policy holder and the expected

claim paid to a policy holder during a year is 1,000

with a standard deviation of 900

What premium should the company charge each policy holder to assure that with probability 0.999

the premium income will cover the cost of the claims.

And we need to look at the aggregate claims random variable. The idea is that the insurer needs to collect sufficient premiums to cover the aggregate claims with at least 0.999 probability.

Simple probability is the calculation of an outcome or the chance of an event ever happening. Insurance companies use probability statistics to determine the chances of having to pay out a claim.

Insurance companies must take into account many more than two outcomes, as life is not as simple as a 50/50 coin toss. However, they use the same basic formula to determine how much they are likely to pay out to sets of policyholders who have the same type of policy. This result gives them an idea of how much money they need to collect to cover their potential losses (the amount they pay out).

Going back to the coin example, on the first two flips, the 50/50 split that one would expect with a 50% probability may not occur and it may land on heads twice, then tails, then heads again. This makes the probability of getting one outcome less predictable. This is where the law of large numbers comes in. The law of large numbers states that the more data points there are, the more accurate an outcome prediction will be. Continuing to flip the coin and record the results (say, 100 times or more), the probability will become closer to 50%.

__Pricing mechanism of insurance:__

Pricing is one of the most essential components of an insurance company. It is the process by which an insurance company sets up the premium that needs to be charged from policyholder by considering various risk factors such as age, mortality, gender, location, etc. These are some of many factors used by the insurance sector for calculating premium and will vary with the nature of the product being priced. It is one of the task of an Actuary to perform pricing in General Insurance company.

To calculate the premium amount, a very important concept is used called the “Generalised Linear Models” which is used by the Pricing team for a really precise premium calculation since it can accommodate many factors. As more factors are added, more accuracy will be there in premium calculation but one should remember that every additional factor should have a significant effect in terms of explaining the variability of the dependent variable, which in this case will be the premium to be charged. The significant effect for each added factor is important because each additional factor is associated with heavy cost in terms of modelling time and analysis which the insurance company has to bear.

In this highly competitive world, if one has to survive in the market it is important to keep the price of products competitive to what other companies are charging. And to do so, premium is charged lowered than the actual calculated one which leads underwriting losses also and that is why, pricing in General Insurance requires an Actuary to focus on each and every aspect of the product being made.

Under insurance parlance the premium is an amount paid periodically to the insurer by the insured for covering his risk.

In an insurance contract, the risk is transferred from the insured to the insurer. For taking this risk, the insurer charges an amount called the premium.

The premium is a function of a number of variables like age, type of employment, medical conditions, etc. The actuaries are entrusted with the responsibility of ascertaining the correct premium of an insured. The premium paying frequency can be different. It can be paid in monthly, quarterly, semiannually, annually or in a single premium.

The company goes through a lot of decision making before reaching the final premium and in each step premium decreases. The steps are as follows :

1)BENCHMARK PREMIUM – This premium form the basis for the calculation of actual premium. The benchmark premium constitutes of mainly

- profit margin for the company for covering the risk,
- expected claim cost and
- claim related expenses

2)UNDERWRITING ADJUSTMENTS – The next step involves making underwriting adjustments in the benchmark premium. These adjustments differ for different groups of homogeneous policy holders depending upon many factors of the policy. These arrangements are pre-decided according to different policyholders needs that will be shown in proposal forms. These adjustments are later fixed by underwriter based on the client.

3) TECHNICAL PREMIUM – After doing the underwriting adjustments, the actuary calculates the technical premium. This premium is calculated considering the past experiences of the claim.

4) PREMIUM ALIGNMENT – The next step is to align the premium with the sum assured in such a way that they remain parallel at all levels of sum assured. This step helps the agents and insured to understand the different premium rates.

5) TARGET PREMIUM – Due to fierce competition in the insurance sector, the actuary lowers the premium by lowering the overall premium. The target premium is such that it constitutes mostly the claim cost.

6) WALKAWAY PREMIUM – Due to the motive of sustaining in market, the company still doesn’t charge the target premium. So, a walkaway premium is decided by lowering the target premium is leading some risk of underwriting loss.

7)ACTUAL PREMIUM – Due to competition, the underwriter still wants to charge less premium.This is done with motive of increasing sales but this leads to underwriting losses,

8)Insurance companies earn from two ways.

- Underwriting Income
- Investment Income

The underwriting income is nothing more than the difference between premium received and the expenses and claim amount. paid. The net difference which remains is underwriting income for company.

On the other hand, every insurance company invest part the collected premium in different profitable businesses. It also invests in share market, mutual funds and other financial instruments. The profit or interest generated on these investments is the investment income for company.

__Functional aspects of actuary or Actuarial aspect:__

An actuary is a professional statistician who calculates the risks associated with insurance coverage and the likelihood that claims will be filed or that benefits will have to be paid out. Using relevant statistical data, actuaries also compute dividends and decide premium rates.

An insurance policy offers coverage from certain risks. In case the event insured against occurs, the policyholder or a third party (e.g., health insurance) notifies the insurance company and provides any relevant documentation, if applicable. Insurance claims comprise a variety of benefits, such as death benefits in life insurance; medical expenses in health insurance and restoration expenses and replacements of damaged items or cash value of damaged items in property insurance.

Insurance companies hire actuaries, highly trained professionals in probability statistics and data analysis, to determine the likelihood of an event happening. Actuaries study and train for at least 6–10 years to more accurately predict outcomes.

Based on the probability of the event, the insurance company will determine the likelihood of having to pay out on a certain type of claim (outcome). Looking at these results, the company determines how much money they will need to gather in order to pay out the claims that are made for that year. They will collect this money through insurance premiums.

If it is not likely to happen, the insurance premium for one coverage will be less expensive than a claim for something that is more likely. For example, if a town historically gets a lot of hail, the premium for that coverage will likely be high. If that same town is far from large bodies of water and waterways, flood coverage will likely be cheap.

Another concept actuaries use is weighted probability. Because there are usually more than one or two factors (not simply heads or tails) when it comes to predicting life events, actuaries have to factor in not only the possible outcomes, but the desired outcomes and how many ways one can get to those outcomes.

For example, rolling one dice, the probability of rolling a two would be the same as rolling a four: 1/6. But by rolling two dice, there would be a greater chance of rolling a score of four than of two. This is because there is only one combination to get two when using two dice (1, 1), but four can be scored with more than one combination (1, 3 or 2, 2 or 3, 1). This means the probability of scoring four is weighted heavier than two.

To determine weighted probability, determine the total number of possible outcomes (total values) in the scenario. Then, determine how many ways the outcome can occur. Divide the number of ways to achieve the outcome by the number of possible outcomes.

Going back to the example of the two dice, when trying to determine the weighted probability (%) of rolling a score of four with two dice, the total number of possible outcomes is 36 (6 sides × 6 sides = 36 outcomes). There are three ways to achieve this roll (1, 3 or 2, 2 or 3, 1). The final calculation is as follows:

3 preferred outcomes ÷ 36 possible outcomes = 0.083 or 8.3%

Again, the insurance companies factor in a huge number of outcomes and ways to achieve those outcomes when deciding what is or is not covered, which is why they rely on highly trained professionals to crunch the big numbers. The bottom line is that insurance companies do not just randomly decide what is and is not covered. They do the math to make sure they can sustain their financial health so they can protect their clients

__Actuarial functions:__

__Actuarial science a__nd insurance can work hand in hand or hand in glove to protect companies and individuals in unforeseen circumstances. Actuaries use their expertise in finance and statistics to asses risk in insurance, finance and other industries. They then advise businesses and individuals of the amount they would need to set aside to tackle risks and costly events that may happen in the future.

__Life Insurance:__

Actuaries have worked with companies that provide life insurance, pensions etc… from the start and it’s more of the traditional area actuaries build a career in. In this field, actuaries are involved in all stages of the produce development, pricing, risk assessment and marketing of the products. They also can work in financial management by developing plans from their analysis to ensure customers get a good return.

__General Insurance:__

Actuaries normally are found working with insurance companies and consultancies and in some instances can work in general insurance also. General insurance includes providing cover for personal insurance such as home and motor insurance, as well as insurance for large commercial risks such as a natural disaster. Actuaries are employed by insurance companies to work in reinsurance and broking operations as well as assist with their financial management this is done by analysing statistics about claims severity and frequency to help insurance companies invest wisely to ensure they maximise income and pay out potential claims. Actuaries also analyse different types of risk depending on the different groups of people and circumstances to help design and price policies correctly.

This is done where actuaries use statistical techniques to create a statistical model which will then be used extensively in the analysis of large amounts of data. This analysis is then used to understand the risks and to make sure that the amount paid for insurance are adequate to meet the eventual settlement of insurance claims. Actuaries have been integrally involved in estimating ultimate costs for unforeseen circumstances such as terrorist attacks, natural disasters and industrial diseases.

__Risk Management:__

Risk management is a great career path for actuaries as it is well suited with their skills and expertise. Another key skill that actuaries need is to explain their findings to businesses to understand so they can implement the actuary’s findings in their future decisions. As well as analysing specific risks from this they develop models for businesses to help minimise their own future risks.

__Basis of RateMaking:__

Insurance is a business. An insured, or a policyholder, pays premiums to transfer his/her risks to the insurance companies, or the insurer. The goal of the insurer is to collect enough premiums to cover for losses should they occur as well as to make a marginal profit. In order to do so, they must price accurately, and the pricing of insurance is called ratemaking.

To evaluate if the ratemaking is optimal, measurements like loss ratios are used.

The most basic ones are

Loss ratio=__loss adjust expenses__

**Earned premium**

This measures how much of the premiums received are used for settling claims

Expenses Ratio=__underwriting expenses__

** Earned premium **

This calculates how much of the premiums received are used for issuing new policies

Combined loss and expenses ratio =

** Loss ratio plus expenses ratio**

The sum of the two helps an insurance company gauge its actual cost of writing new policies and covering losses for a given period. The lower it is indicates more profits for the company. However, there is not one ideal loss ratio. Depending on industry subsidies and company objectives, companies’ target loss ratios vary greatly from 40% to sometimes over 100%.

When the actual loss ratio differs from the expected loss ratio, a rate change is needed to correct the difference. Though a simple percentage surcharge or discount can be applied to the overall premiums, a rate change usually involves evaluating the entire ratemaking algorithm and revising individual rating variables. This process requires actuarial skills and can take weeks or months of time.

__Concepts of Rating;__

In general insurance, claims due to physical damage (to a vehicle or building) or theft are often reported and settled reasonably quickly. However in other areas of general insurance, there may be considerable delay between the time of a claim inducing event and the determination of the actual amount the company will have to pay in settlement. When an incident leading to a claim occurs, it may not be reported for some time. In employer liability insurance, the exposure of an employee to a dangerous or toxic substance may not be discovered for a considerable amount of time. In the case of an accident the incident may be quickly reported, but it may be a considerable amount of time before it is determined actually who is liable and to what extent.

Clearly an insurance company needs to know on a regular basis how much it should be setting aside in reserves in order to handle claims arising from incidents that have already occurred, but for which it does not yet know the full extent of its liability. Claims arising from incidents which have already occurred but which have not been reported to the insurer are termed IBNR (incurred but not reported) claims. Claims which have been reported but for which a final settlement has not been determined are called Outstanding. Claims reserving is a challenging exercise in general insurance, and one should never underestimate the knowledge and intuition that an experienced claims adjuster uses in establishing reserves and estimating ultimate losses. However mathematical models and techniques can also be very useful, and give the added advantage of laying a basis for simulation.

__ Problems Faced by Insurers:__

__ __

- Probabilities may change through time
- Policy holders may alter probabilities (moral hazard)
- Policy holders may not be representative of population from which probabilities were derived
- Insurance Company’s portfolio faces risk.
- Probability of Ruin is the percentile of the probability distribution corresponding to the point at which capital is exhausted. Typically, a minimum acceptable probability of ruin is specified, and economic capital is derived therefrom.

__CONCLUSION__:

The general insurance actuary needs to know the essentials of decision and game theory

in the market of general insurance. An understanding of probability and statistical distributions is necessary to absorb and evaluate risk and ruin when balancing claims, reserves and premiums. In introducing and developing new products, credibility theory and statistics play a role in evaluating sample and collateral information. Generalised methods are essential tools in finding risk factors for premiums calculations. Time series methods are used in various ways to predict trends, and simulation methods are crucial to understanding the many models considered for anything from new products to revisions in rating schemes. Is it no wonder that the general insurance actuary must be a practicing statistician.

Any actuary must keep abreast of investment and economic trends, and therefore should be familiar and comfortable with the basic concepts of univariate time series including stationarity. A knowledge of the basic theory of random walks and co-integration of times series is also essential together with an ability to apply these concepts to investment models.