All of these terms can be used interchangeably. Capacity, capital, supply, surplus, & equity are all basically about how much surplus an insurance company has -- and the resulting amount of premium it can write.
So, surplus is a measure of underwriting capacity. It is analogous to Owners Equity or Net Worth in non-insurance organizations.
Capacity could also be used in terms of a company's appetite or ability to assume risk. Sometimes that would be a discussion of the largest risk it could take (as in a large property risk). More generally, it is the aggregate premium that the insurance company can assume. Given risk based capital analyses and business mix, that would vary considerably from company to company. In addition, reinsurance can expand capacity by allowing a company to use someone else's surplus to write business. In that sense, a company can have more capacity (for gross premium but probably not for net premium) than it has surplus.
Policyholder surplus is the amount of money remaining after all liabilities are subtracted from assets - essentially an insurance company's net worth. Policyholder surplus is a financial cushion that protects a company’s policyholders in the event of unexpected or catastrophic losses. In other industries it is known as “net worth” or “owners equity.” It is a measure of underwriting capacity because it reflects the financial resources (capital) that stand behind every policy written by the insurer. A weakened surplus can lead to ratings downgrades or insolvency.
Insurance companies are required to have a minimum level of capital and policyholder surplus before starting business and must maintain certain levels relative to the business they assume. In a stock company, policyholder surplus consists of retained earnings and capital paid in by shareholders. In mutual companies, it consists of retained earnings and amounts paid by policyholders and others to meet licensing requirements.
Cost of capital
Julian James at Lloyds of London, was quoted in National Underwriter, November 4, 2002 that the "industry's cost of capital is 12% - in other words, we need to achieve a 12% return on investment in order to retain and attract capital. Since 1991, the U.S. industry's rate of return has fallen well short, averaging just 5.3% and with a return of just over 3% for the first half of 2002, it isn't even in the ballpark."