Insurance Premiums
Overview
Premiums are the fees that are charged by insurance companies to cover a given a risk. 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 seom return to shareholders.
On the flip side, as premiums are increased,
supply and demand
would dictate that there is less demand for any given contract, especially in the presence of competitors. This
creates a difficult balancing act for insurance companies.
Pure Premium
A pure 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. Because
expectations are linear,
the principle can be extened 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) %}
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) %}). Economic capital is capital that is owned by the shareholders, which is not
being deployed in the market. Shareholders will demand a return on that capital that is commensurate to the market
value of the risks involved. This means that the eah policy will includel an additional charge, here
labeled {% rr_i %} which is the shareholder required return for capital allocated to that account.
(see
capital allocation)
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) %}
