Best Way to Ship & Insure Small, High Value items?

This is not a post about how best to pack & ship big, heavy amps & speakers & such (lot of great posts on A'gon on that subject). Rather, my question is which shipper is most reliable, & likely to pay an insurance claim, on a high value item, say.......$2,000 to $5,000 (or more). i.e.; how best to ship a relatively small but valuable item that can easily be shipped by air, & the weight & shipping cost is not an issue, & you're not as much worried about shipping damage as it possibly getting lost? I'm guessing, insured USPS priority or express mail? Thanks in advance for any feedback!
i would go with ups as they always pay off,ups seems to pay me faster where as the post office with their lagging senority driven employee's took 6 months to pay off.

not to mention that the uspo's traking system is abysmal.

i wouldnt ship a bunch of carrots with fedex,ever.

USPS without question. I have made two claims in six years, and had no problem with either.

They do not seem to have a strategy designed to cause you to quit pursuing a claim as UPS evidently does.

I may be wrong, but I believe I have read that the Hope Diamond was once transported via the U.S. Post Office before it became the USPS. Just a factoid.
United States Registered Mail is the best way to ship high value items. It provides protection and security for valuables or important mail. It also provides evidence of mailing and optional proof of delivery. Registered Mail is the most secure service offered by the United States Postal Service. Registered Mail is separated and does not travel enroute with First Class Mail because of the high degree of security. This extra security can delay mail by 24 to 48 hours over other First Class delivery objectives. Registered Mail must be taken by the mailer to a post office for acceptance.