Here is how each claim breaks down:

Auto accident ($400,000): The underlying auto policy absorbs its $250,000 limit, leaving $150,000 for the umbrella to cover. That falls comfortably within the $1 million per-occurrence limit.
Slip-and-fall ($600,000): The homeowners policy pays its $300,000 limit, and the umbrella picks up the remaining $300,000 — again within the per-occurrence cap.
Defamation suit ($300,000): Because no underlying policy applies, the policyholder first pays the $1,000 SIR, reducing the umbrella's share to $299,000. However, that full $299,000 cannot be paid because when added to the earlier claims it would exceed certain per-occurrence boundaries, resulting in $149,000 from the umbrella on this claim.

Adding it all together: $150,000 + $300,000 + $149,000 = $599,000 in total umbrella payments — well under the $2 million aggregate limit, and each individual claim also stays within the $1 million per-occurrence limit. This exercise highlights why evaluating your aggregate limit matters just as much as the per-occurrence limit: a year with several large liability events can steadily erode that aggregate ceiling and leave you personally exposed if another claim arises.

One question. Every day.

Step one: The primary policy tied to the type of loss pays up to its limit. In an auto accident, that's your auto liability coverage — which here pays its full 300,000 dollar per-accident limit.
Step two: Once the primary policy is exhausted, the umbrella policy activates and covers the remaining balance — in this case, 500,000 dollars — bringing the total paid to 800,000 dollars.