Finite-Time Ruin Simulation

Closed-form ultimate results are rare. Event-driven Monte Carlo can estimate

by checking surplus immediately after every claim—the only times a classical path can jump below zero.

1. Model and treaty

Use the Exponential example:

• $u=£100{,}000$;
• $\lambda=40$ claims/year;
• mean gross severity £5,000;
• gross premium rate £240,000/year.

Compare an unlimited £20,000 per-loss retention. For Exponential severity,

per ground-up claim. If illustrative reinsurance premium is 115% of expected ceded loss,

This simple premium rule excludes expenses, brokerage, capital cost, and market pricing; it is visible so the comparison is not “free cover.”

2. Run event-driven simulation

For comparison, the exact gross ultimate probability under these Exponential assumptions is about 2.97%. Each gross finite-horizon estimate should not exceed it beyond Monte Carlo noise.

3. Why endpoint simulation is wrong

Calculating only

misses paths that became negative before $T$ and later recovered. Store the running minimum or evaluate every downward jump.

4. Interpret the confidence interval narrowly

The displayed interval approximates uncertainty from a finite Bernoulli Monte Carlo sample, conditional on all model inputs. It excludes:

• estimation error in $\lambda$ and severity;
• tail-family and inflation uncertainty;
• catastrophe dependence;
• reinsurance pricing and default uncertainty;
• model omissions such as expenses and assets.

More paths narrow simulation error only.

5. Validation ladder

1. Verify $E[N(T)]=\lambda T$ and simulated severity mean.
2. Reconcile one-year aggregate moments with compound-Poisson formulas.
3. For Exponential severity, compare long-horizon gross simulation with the exact 2.97% result.
4. Increase path count and report stability.
5. Test heavy-tail and common-shock alternatives.
6. Compare net results with gross using a defensible reinsurance premium.

6. Scenario exercises

Predict direction, change one input, then run:

1. reduce initial capital from £100,000 to £60,000;
2. replace Exponential severity with a same-mean lognormal distribution;
3. add 5% annual severity inflation without repricing premium;
4. cluster claims in a catastrophe state;
5. delay reinsurance recoveries by six months.

The final two require a richer cash-flow process; do not fake them by changing only mean severity.

Practice

1. If estimated ruin frequency is 1% from 10,000 paths, approximate its Monte Carlo SE.
2. Why should gross and net simulations use common random numbers when comparing treaties?
3. Why can a same-mean heavy-tailed severity increase ruin probability?

Finish by integrating the full course in the Portfolio Risk Capstone.