I know that Greek naval trade was insured. Rich would insure many of sailor-traders making money and spreading the risk. While a single trader would not go bankrupt when a storm or pirates would ruin/steal/sink their goods and ship.

Though possibly you could assume that the Mesopotamian and other cultures did the same thing when they started central governments who taxed the farmers and kept reserves for in times of bad harvest.

Ironically one of the suggested reasons from numismatics for the invention of coinage/money is the opposite. This way governments/land lords/... could tax money instead of a part of the harvest. Securing the income of the elite, but making life difficult for the farmers themselves. Though personally I'm not sure we should give this theory to much credibility, especially not as a general rule in history of numismatics.

I also seem to remember that many temples in certain ancient near eastern cultures acted some way as insurance companies. Though one could even go further and see offerings and religion as the first means of an insurance.

Anyway these are the first things I could come up with that are really early examples. Though without a doubt the insurance business got a boom in the late medieval period and early modern period, when the financial world started to evolve drastically. Much to credit of long distance naval trading (cfr. above). First from the east Mediterranean to Italy, To North/West Europe and later with the chartered companies and colonies. These companies for one had investors who could share and spread their financial risks. But anyway that's not really my period so I'll leave that to someone else.