Do surviving homes' values tank after natural disasters?

Here are some pointers:

When natural disasters strike in these areas, the effects on the wider market are minimal, but the effects in that particular area are pronounced and long lasting. Prospective buyers may find the location less desireable after a disaster, and mortgage operations slow, both of which hurt home sales. The RealtyTrac study found that homes in low and very low risk areas experienced average home sale price increases of 6.6% and 9.5% respectively from 2005 to 2015, while high risk and very high risk areas have seen home sale prices decrease by 2.5% and 6.4% respectively in the same decade.

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Paradoxically, home values in high and very high risk areas tend to be higher than homes in low and very low risk areas. That’s because homes in higher risk areas tend to be more desirable places to live.

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While homes in higher risk areas may decrease in sale prices, they usually have a much higher starting point.

https://www.joinreal.com/blog/how-natural-disasters-affect-real-estate-markets

if I invest in protecting my house in this way, what happens to the value of the home if it survives, but the neighborhood does not?

That's an ambiguous question.

• Why did that house survive? Random chance, or the fortunes of geography?
• Will the government mandate that no new houses can be built?
• Will insurance companies continue to insure the property?

The answers to all these questions are highly location specific.

Do communities hit by serious disasters usually rebuild or relocate?

Look at New Orleans after Katrina: some (all of them poor) neighborhoods completely (or almost completely) evaporated as the (poor, mostly renting) evacuees decided to stay in Houston, Atlanta, etc.

My ex-wife bought a surviving house in one such (now gentrifying) neighborhood 14 years later, and it was still almost barren, with street lights that did not work.

HOWEVER, most people returned to most neighborhoods, repaired their houses and resumed living there.

The bottom line is to ask yourself whether you want to live there, even if the neighborhood does not quickly rebuild (because it will eventually rebuild if it is somewhere people want to live, since people need to live somewhere, and most Americans seem to dislike living in high-rise apartment blocks).