'value of appreciation' would not work here because it refers price, not the type of 'appreciation' corresponding to 欣赏。For example, 'your house is appreciated at' means: the price of your house is...
I'd put aesthetic value for 欣赏价值，but perhaps other colleagues have something better in mind.

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Note added at 2 days8 hrs (2008-03-09 15:18:44 GMT)
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I agree with our colleague, Lai An, on 'music appreciation'. However, appreciation in this sense is only used on general subjects such as music, art, sport..音乐欣赏，艺术欣赏，体育欣赏..
When speaking of appreciation value of something tangible (even an art work from the above 'art'), it refers to price. Please see the following links for example:
http://saltlakecity.craigslist.org/jwl/567589928.html
http://www.pricescope.com/idealbb/view.asp?topicID=2991 (second posting 3) appreciation)
http://www.perrywinkles.com/right.php/pid/7/sid/34 (second to last paragraph)

'Enjoyment' usually refers to 享受…的乐趣 or 享有，享受. Something that gives you pleasure or is useful to you. The enjoyment value of a car means how useful it is to you, the enjoyment value of a day on the beach means how much pleasure it gave you...with diamond from your sentence, while it does give people pleasure to look at, it is not the goal. The goal is really to learn about the beauties of a diamond. For enjoyment value, see example:
http://beeradvocate.com/forum/read/574710