Natural Disaster Risk Reinsurance Program Act of 2023 This bill establishes the Natural Disaster Risk Reinsurance Program within the Department of the Treasury. States may voluntarily participate in the program. The program provides payments to states for damages from a natural disaster not covered by the National Flood Insurance Program for amounts in excess of trigger amounts. These trigger amounts are calculated by the National Academy of Sciences. States must present a plan that ensures the coverage of losses by insurers not exceeding the trigger amount. States must also pledge to repay the full amount provided by the program within 10 years.