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 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.

Risk sometimes 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.

Subjective and Objective Probability

Subjective probability is a person's perception of the likelihood of an event. Subjective probability differs from objective probability, either because the person cannot calculate the actual probability or because the person feels lucky or unlucky, or because they think they can rig the game. Of course, if people had a better assessment of objective probability, few people would be playing the lottery or gambling, except for those individuals who are feeling lucky, or because they know how to obtain better odds, such as by counting cards at Black Jack.

Objective probability is the probability of an occurrence, calculated by either deduction or induction. A priori probability is a probability deduced by determining the ratio of a given outcome to finite possibilities of equal probability.

A Priori Probability = 1
Finite Number of Possible
Events of Equal Probability

Thus, there is a 50% chance that a perfectly balanced coin will come up heads if flipped, or a 1/6 chance that a 2 will come on top of a rolled, perfectly balanced and shaped, die.

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 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 for a representative sample.

Take a sample of 10,000 houses that were built many years ago, for instance. An expected frequency of fires can be calculated by learning how many houses burned each year, then averaging those numbers. If this average is 10, for instance, then this is the expected loss. However, it will be rare that exactly 10 houses burn each year. There may be years when none, 6, 12, or 17 of them burn.


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.

Subjective and Objective Risk

Subjective risk is what an individual perceives to be a possible unwanted event. Most people realize, for instance, that it's possible for them to have an accident, or a heart attack or some other health problem. Or that they will lose money buying lottery tickets. How much subjective risk people experience depends on their history and their expected possibility of its occurrence — subjective probability. Somebody who has lost a lot of money in the stock market will probably feel more risk investing in the market than someone who has profited handsomely. Subjective risk may alter the behavior of the risk taker if it is an undesirable risk. Thus, someone who was in a bad auto accident might drive much more carefully than someone who has never been in one.

Insurance is the transference of financial loss due to risk to a company or other organization, usually an insurance company. The company accepts this transference for a periodic premium, and profits by collecting more in premiums and making more from the investments of those premiums than it pays out in claims, which are payments to the insured for the losses they incurred.

Objective risk (aka degree of risk) is the actual losses for a sample in a given period, which can differ significantly from expected losses, and is inversely proportional to the square root of the sample size — the law of large numbers. For example, if you flip a coin 10 times, it is expected that 5 of those flips will yield heads and the other 5 will yield tails. However, in most sets of 10 flips the actual number of heads and tails will differ from this expectation, and it may differ significantly. It's possible that all will be heads, for instance. However, as the number of flips is increased, the number of heads and tails tends toward equality. In 1,000,000 flips, it is highly unlikely that they will all be heads or all tails, and, in fact, the number of both will be closer to the mean.

Because objective risk is the actual number of losses in a given time span for a given sample, it may differ significantly between different samples, even if those samples have the same expected losses. Statistically, objective risk is commensurate with the variance, and therefore, the standard deviation, of the sample.

Note that in the 2 hypothetical samples below, both have the same expected loss, but different objective risks. Because of the wider variation of losses from City 1, an insurance company would require greater reserves to pay for losses in City 1 than in City 2, where the number of losses are closer to the mean. Thus, even though both cities have the same expected loss, City 1 has a greater objective risk.

Sample Size: 10,000 Homes
Losses from Fire
City 1 City 2
Year 1 Losses 15 8
Year 2 Losses 5 12
Expected Loss
(Average Annual Loss)
10 10

The Degree of Risk Depends on Both Magnitude and Probability

The degree of risk depends not only on the probability of loss, but also on the magnitude of the loss. Hence, if 2 loss events have the same probability of occurrence, but one of the losses exceeds the other, then the greater potential loss is the greater risk. So, if there was an equal probability of losing $1 or $100, then the risk of losing $100 exceeds the risk of losing $1. Because insurance companies compensate for losses by paying money to the insured, it must calculate both the magnitude and probability of potential losses to set a premium that will cover those losses and earn the insurance company a profit. Thus, the main risk for an insurance company is that actual losses will exceed expected losses.