The notion of who bears risk for various sorts of failures, circumstances, or losses is a prevalent one in the financial world, and many institutions make all of their money accepting risks.
To accept a risk is to bear the burden of loss or replacement if an event occurs that causes an asset to lose value or disappear. There is a bright side to this, however. There is a real and theoretical “risk premium” due to those who accept a risk.
Insurance companies accept a literal premium for this task, and they have underwriters and actuaries computing the probability of loss constantly. Sometimes insurance cannot be purchased to cover a risk, and a risk is transferred through a sale or other arrangement.
In the world of loans, collateralized debt obligations and other forms of securitized loans have shifted the risk of default from the lending institution who made the loan to the investors and investment banks that created the collateralized debt obligation note out of pools of mortgage cash flows.
Risks can be accepted, but “hedged” with financial instruments with an inverse correlation to the event or circumstance that might trigger a loss. An example of this in the investing world might be a short position or a derivatives contract. Often the acceptance of a specific risk is spelled out in a contract document, and sometimes it is implied.
An investor who purchases stock in an IPO from a company without a very high credit rating and a volatile price history has accepted a risk that the money invested will be lost. The company in that example has not borrowed anything from the investor, and they have effectively shifted the risk of losing capital to the investor.
You may have heard higher risks come with higher (potential) returns. This is of course true with equities as well. Small cap stocks have the most room to grow, and have returned more than the large cap stocks over the long term. They are among the riskier investments a person can make, of course.
Self-insuring is to be financially stable enough, and to have enough liquidity, for certain losses to be within an acceptable range.