A few additional things you should do if you ship high dollar value items (above the maximum insurance levels) with FedEx or UPS. The company I worked for prior to my retirement used to ship high value metals all the time and FedEx/UPS provided guidelines for us. (Some shipments were approaching $1M USD. 
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- Use descrete packages, i.e. blank boxes. Don't do anything to advertise what the contents are.
- As @gregrolka mentioned, you need to seal every edge, corner, etc. And use "tamper proof" printed tape. It makes it a bit more difficult to slit the box open and then retape it. You can get it on Amazon.
- If handing the package directly to the driver, make sure he/she is aware of the value. Their protocols should have them segregate the package in the truck so it can be locked up when it is unloaded at the depot.
- Otherwise, take the package to the actual FedEx or UPS desk at their respective shipping centers, not at the UPS Store or Walgreens, etc. One, you will get a better rate, and more importantly, if you let them know the high value, the supervisor will lock them in the "cage" until ready to be placed on the truck/plane.
- Ship overnight or 2 day delivery instead of ground if possible. Fewer transfer points and hands on the package. Less likely to get misplaced or "lost" in transit.