Obviously, the price of a gold coin does indeed go up when the price of gold goes up. if it didn't, then the coins would all be melted as soon as the bullion value went above the numismatic market value.
I prefer to think of it in this sense: all coins have a "total value" comprised of three components: face value (the amount of money you'd get for it at the bank), the "bullion/scrap premium" (the difference between what a bank will give you and what a jeweller or scrap metal merchant will give you) and numismatic premium (the difference between what a scrap metal merchant will give you and what a coin collector will give you). Add the three together and you obtain the total value to a coin collector.
For ancient coins, the face value is of course zero, since no country today still uses ancient coins as legal tender. The bullion/scrap premium goes up and down with metal prices. The numismatic premium goes up and down with collector supply and demand.
For most ancient gold coins, the numismatic premium is still two or three times the bullion value. The price of a typical aureus will go up and down a little, but not measurably so. However, some Byzantine and early Islamic gold coins can still be obtainable with a numismatic premium so low it is comparable to a modern bullion-grade coin like a sovereign.
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The question I fail to understand is how people benefit by melting things that have any numismatic identification. When held as a coin, the item is still melt plus numismatic value. Once melted, it is simply melted and forever limited in value to spot. In a thousand years what will be the relative value of a coin (1964 quarter or Phocas solidus) and their equivalent weight in metal?
The vast majority of Westerners would recognize that an ancient coin is worth far more as an intact ancient coin, or even as an identifiable ancient coin in a piece of jewellery, than as a piece of scrap metal. I suspect most of the melting of ancient coins that happens these days is happening in countries where owning and selling ancient coins is illegal. If, say, a peasant farmer in Lebanon finds a gold coin in their field, they have three options:
- Do what Lebanese law says they're supposed to do and hand it over to the government. Low risk, but low returns: I'm not entirely sure what compensation they would be entitled to under the new 2008 Movable Cultural Property laws since I can't find a reliable English translation of the laws, but it's likely it will be minimal, if not zero. The Lebanese government certainly can't afford to pay full and fair market value for all the antiquities found on their soil.
- Keep it for themselves, and/or try to sell the coin illegally on the black market. This has the highest possible return, since they'd likely get close to full and fair market value, but also the highest risk, of jail time if they get caught.
- Take the safe middle path: destroy the coin and sell it for scrap. Once melted, no-one can tell it used to be a coin and it isn't a crime in Lebanon to sell scrap gold for scrap gold value. Before the civil wars tore their country apart, the gold markets of Lebanon were world-famous. They'd only be in legal trouble if they happened to get caught in the act of melting.
Don't say "infinitely" when you mean "very"; otherwise, you'll have no word left when you want to talk about something really infinite. - C. S. Lewis