I've heard that insurance appraisals for fine instruments are commonly inflated relative to replacement cost. Is that true? If so, typically by how much? For example if I see an insurance appraisal for $11,000, what should I be offering to buy that instrument?
Insurance appraisals are generally set at what they think a realistic replacement cost will be, which is not necessarily what the buyer paid for it. The appraisal is also not necessarily current. And the selling price of an instrument is relative to market.
As Lydia explains, there is much more that tells you about the value of an instrument than simply the insurance value. The condition, age, sound when played, do the strings need replacing, what are the quality of the strings, does the instrument appear symmetrical, is the bridge straight, are there visible wear marks on the instrument indicating a lot of use? Will the owner provide any documentation on the history of the instrument? I would want the answers to these questions before purchasing an $11,000 instrument. If you do not feel comfortable going alone, bring someone that has the experience necessary to help you assess the value of the instrument.
"Is the bridge straight?" Yeah, but thanks for the tip.
I don't think there's a useful algorithm for this, though 20% is a fair guess. You'd be better off trying to find "comps." Maybe look online and see what shops are asking for similar instruments. Auction prices are useful information but only as a real basement since auctions are wholesale.
I'd also stick to something around the 20 %. Did it that way with my violin, and a good friend (who happens to be also a good luthier) felt it was a good but fair price. You could also consult the Fuchs Taxe if the maker is known, but the book is quite expensive. Luthiers here mostly work with that when doing appraisals, not sure if it's the same in your country. I'd right away ask my luthier, and he'd give advice if he was the one I'd ask to do the restauration too.
for the record: using "do the strings need replacing, what are the quality of the strings" as criteria is equivalent to basing a house purchase on the color of the paint in the living room.
I don't think that anyone should care about the strings. Nor does the fact that the violin has been sitting for years affect value, although it might indicate that the sound will change after a playing-in period.