Overview
Premiums are the fees that are charged by insurance companies on order to cover the payments that are required to make when a claim is made. The premium needs to be set high enough to actually cover the insured risks, but also to cover the costs of doing business and to provide some return to shareholders.
On the flip side, as premiums are increased, supply and demand would dictate that there is less demand for any given policy, especially in the presence of competitors. This creates a difficult balancing act for insurance companies.
Pure Premium
A pure premium (sometimes called an equivalence premium) is one that
{% \pi = \mathbb{E}(X) %}
where the premium is denoted as {% \pi %} and {% X %} is a random variable representing the total claim size from
an insurance contract. The pure premium is one that is equal to the expected claim size.
There are some considerations that must be applied to the pure premium principle. First, the premiums collected may themselves be random. For example, a person paying for a life insurance policy may die prematurely, having only made a few payments on the policy. In that sense, the principle should be stated to include the randomness of the premiums as
{% \mathbb{E}(\pi) = \mathbb{E}(X) %}
Second,
the
time value of money
means that the timing of the payments is important. In that sense, the payments must be
discounted.
{% \mathbb{E}(D_{\pi}\pi) = \mathbb{E}(D_{X}X) %}
Here, {% D %} is a generic discount factor.
An insurance company will have a portfolio of policies. The sum of the claims is typically denoted as {% S %}
{% S = \sum_i X_i %}
Because
expectations are linear,
the principle can be extended to all the policies in the portfolio.
{% Cash \, Flow = \sum_{i=1}^n \pi_i - \sum_{i=1}^n X_i %}
{% \mathbb{E}(Cash \, Flow ) = \sum_{i=1}^n \mathbb{E}(\pi_i) - \sum_{i=1}^n \mathbb{E}(X_i) %}
Factorization
In any given time period, the total number of claims, labeled {% N %} (this is less than the number of policies) is a random variable. Next, we assume that the claim size, given that there is a claim, are i.i.d. (independent and identically distributed), and the claim size distribution is independent of {% N %}. This implies that the expected value of {% S %} can be factorized as
{% \mathbb{S} = \mathbb{E}[X] \mathbb{E}[N] %}
(see Pitacco pg.71)
See the Factorized Model for more details.
Covering General Business Costs
In addition to the risks that are being covered, an insurance company must cover the general business costs that are needed to have marketed and sold the contract, as well as ongoing costs such as account administration. The challenge of determining these costs is the process of Cost Allocation.
The pure premium principle will then be addjusted to include a term {% ad_i %} representing the administrative cost of the ith account.
Economic Capital
Economic Capital is the amount of capital that the firm has set aside to cover unexpected losses. (That is losses over and above {% \sum_{i=1}^n \mathbb{E}(X_i) %}).
The capital that is set aside to cover losses is in some sense, capital owned by the shareholders that is idle, i.e. not invested in other productive equity investments. Shareholders will demand a return on that capital that is commensurate to the market value of the risks involved. This means that each policy should include an additional charge, here labeled {% rr_i %} which is the shareholder required return for capital allocated to that account. (see capital allocation)
When the shareholder required premium is added, the final premium equation becomes
{% \mathbb{E}(Cash \, Flow ) = \sum_{i=1}^n \mathbb{E}(\pi_i + ad_i + rr_i) - \sum_{i=1}^n \mathbb{E}(X_i) %}
Insurance and Choice
The economics of choice is a fundamental consideration when doing premium calculations.
- Adverse Selection - occurs when one party to a transaction has information that the other party does not have. The standard example is an individual buying health insurance, who has superior information about their health than the insurance company. This has the effect that people of poor health are more likely to buy health insurance than healthy people, making the insured population skewed in terms of health with respect to the actual population.
- Moral Hazard - is the phenomenon that individuals who are protected from risk, may engage in behaviours that exacerbate that risk. In the case of health insurance, individuals who have health insurance become more likely to engage in risky health behaviours after having been insured.