ruin and required capital

Every insurer has to answer a money question: how much capital should we hold so we are very unlikely to go broke? Ruin theory turns that vague worry into a clean recipe. Pick a small probability you are willing to tolerate — say a 0.5 percent chance of ruin — and then find the amount of starting capital that pushes the ruin probability down to that level. That amount is the required capital. The chosen probability is your risk appetite; the capital is what it costs to honor it.

Precisely, since the ruin probability psi(u) falls as initial surplus u rises, you invert the relationship: choose a target tolerance p and solve psi(u) = p for u. In the classical model the Lundberg bound gives a quick conservative answer — u at least equal to (negative natural log of p) divided by the adjustment coefficient R. So halving your tolerance (a stricter standard) adds a fixed chunk of capital equal to (log 2)/R, and a book with a larger R (safer mix, fatter loading) needs less capital to hit the same target. For example, with R = 0.000004 a 0.5 percent tolerance needs u of about (5.3)/0.000004 = 1,325,000.

This is the conceptual bridge from ruin theory to real-world capital and solvency regimes. The modern machinery — Value at Risk, economic capital, regulatory capital — is the same idea dressed up: choose a confidence level, hold enough capital that losses beyond it are rare. Two honest caveats. The classical 'required capital' uses an infinite-time, no-investment-income model, so real regimes use a one-year horizon and richer assumptions and will land on different numbers. And capital does not prevent bad outcomes — it makes the firm able to absorb them; tail events worse than the chosen percentile still happen, just rarely, which is exactly why the choice of tolerance is a judgment, not a fact.