Reinsurance
Reinsurance is a means by which an insurance company can protect itself against the risk of losses with other insurance companies.  Individuals and corporations obtain insurance policies to provide protection for various risks (hurricanes, earthquakes, lawsuits, collisions, sickness and death, etc.).  Reinsurers, in turn, provide insurance to insurance companies.

# Functions of reinsurance
There are many reasons why an insurance company would choose to reinsure as part of its responsibility to manage a portfolio of risks for the benefit of its policyholders and investors :

## Risk transfer
The main use of any insurer that might practice reinsurance is to allow the company to assume greater individual risks than its size would otherwise allow, and to protect a company against losses. Reinsurance allows an insurance company to offer higher limits of protection to a policyholder than its own assets would  allow. For example, if the principal insurance company can  write only $10 million in limits on any given policy, it can reinsure (or cede) the amount of the limits in excess of $10 million.
Reinsurances highly refined uses in recent years include applications where reinsurance was used as part of a carefully planned hedge strategy.

## Income smoothing
Reinsurance can help to make an insurance companys results more predictable by absorbing larger losses and reducing the amount of capital needed to provide coverage.

## Surplus relief
An insurance company's writings are limited by its balance sheet (this test is known as the solvency margin). When that limit is reached, an insurer can either stop writing new business, increase its capital or buy "surplus relief" reinsurance. The latter is usually done on a quota share basis and is an efficient way of not having to turn clients away or raise additional capital.

## Arbitrage
The insurance company may be motivated by arbitrage in purchasing reinsurance coverage at a lower rate than what they charge the insured for the underlying risk.

# Types of reinsurance

## Proportional
Proportional reinsurance (the types of which are quota share & surplus reinsurance) involves one or more reinsurers taking a stated percent share of each policy that an insurer produces ("writes").  This means that the reinsurer will receive that stated percentage of each dollar of premiums and will pay that percentage of each dollar of losses.  In addition, the reinsurer will allow a "ceding commission" to the insurer to compensate the insurer for the costs of writing and administering the business (agents' commissions, modelling, paperwork, etc.).
The insurer may seek such coverage for several reasons.  First, the insurer may not have sufficient capital to prudently retain all of the exposure that it is capable of producing.  For example, it may only be able to offer $1 million in coverage, but by purchasing proportional reinsurance it might double or triple that limit. Premiums and losses are then shared on a pro rata basis. For example, an insurance company might purchase a 50% quota share treaty; in this case they would share half of all premium and losses with the reinsurer. In a 75% quota share, they would share (cede) 3/4 of all premiums and losses.
The other form of proportional reinsurance is surplus share or surplus of line treaty. In this case, a retained line is defined as the ceding company's retention - say $100,000. In a 9 line surplus treaty the reinsurer would then accept up to $900,000 (9 lines). So if the insurance company issues a policy for $100,000, they would keep all of the premiums and losses from that policy. If they issue a $200,000 policy, they would give (cede) half of the premiums and losses to the reinsurer (1 line each). The maximum underwriting capacity of the cedant would be $ 1,000,000 in this example. Surplus treaties are also known as variable quota shares.

## Non-proportional
Non-proportional reinsurance only responds if the loss suffered by the insurer exceeds a certain amount, called the retention or priority. An example of this form of reinsurance is where the insurer is prepared to accept a loss of $1 million for any loss which may occur and purchases a layer of reinsurance of $4m in excess of $1 million - if a loss of $3 million occurs the insurer pays the $3 million to the insured(s), and then recovers $2 million from its reinsurer(s). In this example, the reinsured will retain any loss exceeding $5 million unless they have purchased a further excess layer (second layer) of say $10 million excess of $5 million.
The main forms of non-proportional reinsurance are excess of loss and stop loss.
Excess of loss reinsurance can have three forms - "Per Risk XL" (Working XL), "Per Occurrence or Per Event XL" (Catastrophe or Cat XL), and "Aggregate XL". In per risk, the cedants insurance policy limits are greater than the reinsurance retention. For example, an insurance company might insure commercial property risks with policy limits up to $10 million and then buy per risk reinsurance of $5 million in excess of $5 million. In this case a loss of $6 million on that policy will result in the recovery of $1 million from the reinsurer. 
In catastrophe excess of loss, the cedants per risk retention is usually be less than the cat  reinsurance retention (this is not important as these contracts usually contain a 2 risk warranty i.e. they are designed to protect the reinsured against catastrophic events that involve more than 1 policy). For example, an insurance company issues homeowner's policies with limits of up to $500,000 and then buys catastrophe reinsurance of $22,000,000 in excess of $3,000,000. In that case, the insurance company would only recover from reinsurers in the event of multiple policy losses in one event (i.e., hurricane, earthquake, flood, etc.).
Aggregate XL afford a frequency protection to the reinsured. For instance if the company retains $1m net any one vessel, the cover $10m in the aggregate excess $5m in the aggregate would equate to 10 total losses in excess of 5 total losses (or more partial losses). Aggregate covers can also be linked to the cedant's gross premium income during a 12 month period, with limit and deductible expressed as percentages and amounts. Such covers are then known as "Stop Loss" or annual aggregate XL.

# Contracts
Most of the above examples concern reinsurance contracts that cover more than one policy (treaty). Reinsurance can also be purchased on a per policy basis, in which case it is known as facultative reinsurance. Facultative reinsurance can be written on either a quota share or excess of loss basis. Facultative reinsurance is commonly used for large or unusual risks that do not fit within standard reinsurance treaties due to their exclusions. The term of a facultative agreement coincides with the term of the policy.  Facultative reinsurance is usually purchased by the insurance underwriter who underwrote the original insurance policy, whereas treaty reinsurance is typically purchased by a senior executive at the insurance company.
Reinsurance treaties can either be written on a continuous or term basis. A continuous contract continues indefinitely, but generally has a notice period whereby either party can give its intent to cancel or amend the treaty within 90 days. A term agreement has a built-in expiration date. It is common for insurers and reinsurers to have long term relationships that span many years.

# Markets
Most reinsurance placements are not placed with a single reinsurer but are shared between a number of reinsurers. For example a $30,000,000 xs of $20,000,000 layer may be shared by 30 or more reinsurers. The reinsurer who sets the terms (premium and contract conditions) for the reinsurance contract is called the lead reinsurer; the other companies subscribing to the contract are called following reinsurers.
About half of all reinsurance is handled by reinsurance brokers who then place business with reinsurance companies. The other half is with direct writing reinsurers who have their own production staff and thus reinsure insurance companies directly. In Europe reinsurers write both direct and brokered accounts.
Using game-theoretic modeling, Professors Michael R. Powers (Temple University) and Martin Shubik (Yale University) have argued that the number of active reinsurers in a given national market should be approximately equal to the square-root of the number of primary insurers active in the same market.  Econometric analysis has provided empirical support for the Powers-Shubik rule.
Reinsureds tend to choose their reinsurers with great care as they are exchanging insurance risk for credit risk. Risk managers monitor reinsurers' financial ratings (S&P, A.M. Best, etc.)and aggregated exposures regularly.

# Retrocession
Reinsurance companies themselves also purchase reinsurance and this is known as a retrocession. They purchase this reinsurance from other reinsurance companies.  The reinsurance company who sells the reinsurance in this scenario are known as retrocessionaires. The reinsurance company that purchases the reinsurance is known as the retrocedent.
It is not unusual for a reinsurer to buy reinsurance protection from other reinsurers. For example, a reinsurer that provides proportional, or pro rata, reinsurance capacity to insurance companies may wish to protect its own exposure to catastrophes by buying excess of loss protection. Another situation would be that a reinsurer which provides excess of loss reinsurance protection may wish to protect itself against an accumulation of losses in different branches of business which may all become affected by the same catastrophe. This may happen when a windstorm causes damage to property, automobiles, boats, aircraft and loss of life, for example.
This process can sometimes continue until the original reinsurance company unknowingly gets some of its own business (and therefore its own liabilities) back. This is known as a spiral and was common in some specialty lines of business such as marine and aviation. Sophisticated reinsurance companies are aware of this danger and through careful underwriting attempt to avoid it.
In the 1980s, the London market was badly affected by the creation of reinsurance spirals. This resulted in the same loss going around the market thereby artificially inflating market loss figures of big claims (such as the Piper Alpha oil rig). The LMX spiral (as it was called) has been stopped by excluding retrocessional business from reinsurance covers protecting direct insurance accounts.
It is important to note that the insurance company is obliged to indemnify its policyholder for the loss under the insurance policy whether or not the reinsurer reimburses the insurer. Many insurance companies have experienced difficulties by purchasing reinsurance from companies that did not or could not pay their share of the loss (these unpaid claims are known as uncollectibles). This is particularly important on long-tail lines of business where the claims may arise many years after the premium is paid.