effect of reinsurance on ruin
An insurer worried about going broke has a powerful tool: it can hand off part of its risk to a reinsurer, paying a fee in exchange for the reinsurer covering some of the claims. This trims the size and variability of the claims the insurer keeps, which calms its surplus line — fewer deep plunges. But the reinsurance fee also eats into premium income, gentling the upward drift. Reinsurance and ruin is the study of that trade-off: does buying protection actually make the insurer safer, and how much should it buy?
Precisely, reinsurance changes both the claims the insurer retains and the premium it keeps. Under proportional (quota-share) reinsurance the insurer keeps a fraction of every claim and gives up the matching fraction of premium; under excess-of-loss reinsurance the insurer keeps each claim only up to a retention limit and pays a premium for the rest. Either way, the net surplus process is recomputed on the retained business, and the adjustment coefficient R is recalculated. The striking result: there is usually an optimal retention level that maximizes R — buy too little reinsurance and big claims still threaten you; buy too much and the fees starve your upward drift. The retention that maximizes R is the one that minimizes the Lundberg bound on ruin.
This matters because it gives reinsurance a precise solvency rationale beyond gut feel. By choosing the retention that maximizes the adjustment coefficient, an actuary directly minimizes the upper bound on ruin probability — turning 'how much reinsurance?' into an optimization. The honest caveat: the answer depends entirely on the reinsurer's pricing. If reinsurance is cheap relative to the risk it removes, heavy cession lowers ruin; if the reinsurer's loading is steep, ceding too much can paradoxically raise ruin by bleeding away premium faster than it removes risk. And as always, the clean R-based analysis assumes light-tailed claims; for catastrophe exposures the logic shifts toward simply capping the worst single-event loss.
An insurer keeps 100 percent of its risk and computes R = 0.000003. It buys quota-share reinsurance ceding 40 percent of claims (and 40 percent of premium, plus a small fee); recomputing on the retained 60 percent gives R = 0.000005 — a higher adjustment coefficient, hence a lower ruin bound. Ceding 80 percent, however, drops R back down as fees outrun the risk relief.
There is an optimal retention that maximizes R and thus minimizes the Lundberg bound on ruin.
Reinsurance does not always lower ruin: if the reinsurer's loading is too high, ceding more can drain premium faster than it removes risk and actually raise the ruin probability.