This guy at NSider has the strategy guide for AC:WW, and here's what he said about Lyle's insurance -
"Lyle offers insurance, and if you sign up with him, he will schedule days redd comes to your town. some paintings you buy a redd's tent are counterfeit, and Lyle's insurance helps pay back on this, bee stings (which there is now medicine for)."
I thought it'd be for insuraing you against friend theft, but oh well.
![Stick out tongue :p :p](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
"Lyle offers insurance, and if you sign up with him, he will schedule days redd comes to your town. some paintings you buy a redd's tent are counterfeit, and Lyle's insurance helps pay back on this, bee stings (which there is now medicine for)."
I thought it'd be for insuraing you against friend theft, but oh well.