Vacation: Space Needle and Mt. Rainier

These chipmunks playing at Sunrise Lodge
were just about the cutest thing ever.

Simiao visited me for our 2-year dating anniversary, and to celebrate we visited the Chihuly museum and the Space Needle at Seattle City Center, and then drove out to Mt. Rainier National Park the next day to explore the area around Sunrise Lodge.  Sunrise Lodge is the highest elevation accessible by vehicle, and I hoped we'd get lucky and get a day where it wasn't too cloudy to see Mt. Rainier.  It turns out that it was mostly obscured, but the mists swirling around the trails made a great subject.  As you reach the ridge along the Sourdough Mountains, you can feel the strong wind resulting from the air being forced over the Cascades and can actually see the newly-formed clouds being formed on the mountaintops and whipping down into the valley below.  It's an area that really has to be seen in order to be appreciated, and I'm told that when the alpine meadows are in bloom that the area is a sight to behold.  Unfortunately that part will have to wait until July 2013 or so, but I've included some of our favorite pictures for you to appreciate now.

A unique view of the Space Needle from the Chihuly Museum.

No trees were harmed in the making of this photograph.

Western Pasqueflower in seed.  Obviously not my shot,
Simiao has an eye for composition.

Another shot by Simiao looking South from the Sourdough Trail

What are the Most Undervalued US Coins?

Since I was very young, I collected US coins.  It was fun to search through rolls of pennies and nickels and look for rare varieties or missing years and mint-marks for my set.  As an adult, I still add the occasional interesting piece to my collection.  Now as a data scientist, I wonder what the relationship between market price and rarity is.  Is there a very strong correlation between these two things?  How are they related?  Are there sets that lie largely above or below this curve?

All of my data is collected from the subscription-access database PCGS Coinfacts.  You can look at a few entries in this database for free, but after that trial you must purchase a subscription to see more data.  I will present the results here in a format that cannot be used to reconstruct the copyrighted data.  I want to get a few additional details out of the way:

• Please do not construe this article as an endorsement for investing in rare coins.  Rare coins are a terrible investment.  They do not pay dividends and their value is entirely subjective.  If you sell them at auction, you can expect the auction house to take about 15-20% of the buyer's price.  If you choose to collect coins as a hobby, you should do so out of historical and artistic interest and not as an investment.  Further, there is the risk of losing one's investment entirely as an enormous number of high-quality counterfeits are coming out of China.  Some are so good that even experts have a very hard time telling the real thing from frauds!
• PCGS Coinfacts has numerous, obvious errors in their database.  I will not attempt to correct these errors in my data, and instead try to make big-picture conclusions wherever possible.  
• I will only consider major varieties of US coins, and not rare minor variants.  

Let's start with a simple assumption, and a counterexample.  Assume that coin collectors will buy a coin for its melt value plus its numismatic value, which is a function only of its rarity.  Then, two coins of similar rarity should sell for the similar prices, plus the value of the metal they're made of.

Survival Estimate (MS60 or better)
2,000	55,833
Market Value (MS60)
$52	$1,890
Melt Value
($0.02)	($1,601.98)
Market Value Corrected for Melt (MS60)
$52	$288
Data and images © Collector's Universe reproduced for nonprofit educational purposes only.

So what does all of this mean?  It means that of the millions of 1913 Lincoln cents minted, only 2,000 are estimated to exist in mint state (MS60+) today.  On the other hand, of the 361,667 1907 St. Gaudens double eagles (low relief) minted, 55,833 of them are estimated to exist today in mint state.  This is important to know, because numismatic rarity is based on how many are still around today, not how many existed when they were made.  If I correct for the price of gold, I see that people are willing to pay 6 times more for the St. Gaudens that is actually 25 times more common.

Certainly, the double eagle is aesthetically more beautiful, and as such understandably more in demand than the Lincoln cent.  The antiques market is intrinsically subjective, and what each coin is worth is precisely what someone is willing to pay.  We will also see here that varieties that were only made for a brief period, such as this double eagle, tend to sell for far more than you might expect given the rarity.  Lincoln cents have been in production for over 100 years, so perhaps collectors are just as happy looking at the ones in their pocket as the ones from a century ago.  This example illustrates that there are emotional factors in play that sway collectors to pay much more for pieces that are in fact much more common.

Now let's explore the relationship between price and scarcity.  I have collected scarcity and market value for all major varieties of US coins using PCGS Coinfacts.

Figure 1:  Market price (adjusted for melt value) versus scarcity for US coins in mint state.  Copper coins are shown in brown, silver in gray, and gold in yellow.  The regression line shown is ln(price) = -0.6716ln(survival) + 10.4208, with R² = 0.7614.  In order to reduce the influence of outliers, the regression was performed after taking the natural log of both price and survival.  Modern gold and silver Eagles do not appear on this curve as their numismatic value does not exceed their melt value.

Figure 1 shows market price versus scarcity in mint-state coins.  While I collected data for lower grades, I found the survival estimates to be more plausible for mint state coins than the population at large.  The first interesting thing about this plot is that the number of surviving specimens and the market price have a linear relationship on a log-log plot, and it's worth a bit of discussion with regard to why this is so remarkable.

The linear relationship on the log-log plot indicates to me that the fact that other collectors can't have a specimen is a substantial part of the appeal of having a specimen.  This exclusivity manifests itself in the following way: the fewer coins of a variety exist, the more the total value all specimens of that variety have!  For that reason, on a plot with linear axes, this data looks like an "L" where a coin is either very rare and very valuable, or too common for anyone to value it at all.

I also notice a few trends.  The most overvalued coins tend to be gold.  Copper coins tend to be more undervalued.  Modern gold and silver Eagles could not be placed on figure 1, since their numismatic
value did not exceed the melt value and there is no zero on a log-log plot.

My goal with this study is not to chide people for collecting what they like.  I recognize that the numismatic hobby is not about investment, but rather about artistic and historical interest.   I want to see was which series of US coins are undervalued and which are overvalued per rarity, and suggest to myself and other collectors what might be the most neglected series to focus on to build an interesting collection.  In order to do this, I will find the median percent difference from coins in each series to the regression line ln(price) = -0.6716 ln(survival) + 10.4208.  In this way I will minimize the effect of noise and outliers in each set.

Set	Median Percent Overvalued MS60+	Comment
4 Stella	3822%	Only 425 of these were made and sold to Congressmen.
Flowing Hair Dollar	2925%	Ultra-rare, with only some 100 of them surviving.
Draped Bust $10	2204%	Far overvalued given its rarity.  Consider the draped bust half cent or Liberty head Eagle.
Modern Commemorative Gold	1423%	Modern commemoratives made for collectors are sold for a premium not justified by rarity.
Sacagawea Dollar	1220%	These are very inexpensive to begin with. Millions sit unwanted in the treasury.
Draped Bust $5	1118%	Consider the draped bust half cent for a more cost effective option.
Capped Bust $5	1109%	Consider the capped bust quarter for a more cost effective option.
Draped Bust $2.50	897%	Consider the draped bust half cent for a more cost effective option.
Flowing Hair Half Dime	872%	Consider the flowing hair large cent for a more cost effective option.
Liberty Cap Half Cent	723%	Only made for a few years.  If you like this, consider the undervalued classic head half cent.
50 States Quarter	707%	These are very inexpensive to begin with, but are far overvalued in mint condition.
Capped Bust $2.50	650%	Consider the capped bust quarter for a more cost effective option.
Commemorative Gold	637%	Commemorative coins start their lives in the collectors' market and never tend to leave.
Flowing Hair Half Dollar	615%	Consider the flowing hair large cent for a more cost effective option.
St. Gaudens $20	506%	The most beautiful of US coins comes with a high premium.  Consider modern gold eagles.
Flowing Hair Large Cent	434%	The least overvalued coin of the flowing hair set is still quite expensive given its rarity.
Draped Bust Dime	424%	Consider the draped bust half cent for a more cost effective option.
Draped Bust Dollar	416%	Consider the draped bust half cent for a more cost effective option.
Modern Lincoln Cents	393%	These are very inexpensive to begin with, so the percent overvalued is misleading!
Draped Bust Quarter	359%	Consider the draped bust half cent for a more cost effective option.
Draped Bust Half Dime	345%	Consider the draped bust half cent for a more cost effective option.
Twenty Cent	309%	If you like this variety, take a look at the three cent nickels or two-cent piece.
Indian $2.50	282%	This set is probably valued more highly for being the only incuse set of US coins.
Liberty Head $20	236%	The Liberty Head $10 and $2.50 are far less overvalued for their rarity.
Trade Dollar	233%	Aesthetically similar to Seated Liberty, except far more overvalued.
Peace Dollar	224%	If you like large silver coins, you should take a look at Franklin half dollars.
Indian $10	217%	The Indian $5 is slightly less overvalued if you must have coins bearing this design.
Classic Head $2.50	201%	The Classic Head half cent is far less overvalued given its rarity.
Three Dollar	192%	Take a look at the gold dollars which are similar in design but much less overvalued.
Classic Head $5	187%	The Classic Head half cent is far less overvalued given its rarity.
Liberty Seated Dollar	181%	The Liberty Seated half dollar or quarter are much more cost-effective alternatives!
Indian $5	177%	The least overvalued of the Indian eagles in terms of its rarity.
Draped Bust Half Dollar	176%	The draped bust half cent is far more cost effective if you like the draped bust series.
Flying Eagle Cent	146%	Stunningly overvalued for such an unattractive coin.  Consider the undervalued Indian cent.
Walking Liberty Half Dollar	142%	The same design appears on modern silver eagles without the high premium.
Morgan Dollar	138%	One of the most popular US coins to collect, comes with a premium to match.
Ike Dollar	119%	It's not immediately clear why collectors like Ike so much, but consider the Franklin half.
Classic Head Cent	104%	The Classic Head Half Cent is far less overvalued given its rarity.
Draped Bust Cent	56%	The draped bust half cent is far more cost effective if you like the draped bust series.
Capped Bust Half Dollar	37%	The capped bust quarter has a more favorable price given its rarity.
Washington Quarter	31%	Many are quite inexpensive, so the fact that it is overvalued won't set you back much.
Commemorative Silver	25%	Coins made explicitly for collecting have a propensity to never disappear from the market.
Liberty Head $5	24%	Consider the Liberty Head $2.50 for a less overvalued option.
Susan B. Anthony Dollar	24%	Quite inexpensive and very common, the fact that it is overvalued won't cost you much.
Barber Half Dollar	14%	Slightly overvalued given its rarity.  Maybe a good stand-in for overvalued Morgan Dolllars.
Gold Dollar	14%	One of the least overvalued US gold coins that circulated.
Liberty Head $10	13%	One of the least overvalued US gold coins, and quite large for those who prefer larger ones.
Liberty Head $2.50	13%	The least overvalued set of US gold coins that circulated.  Great choice if you must have gold!
Capped Bust Dime	12%	The capped bust quarter, on the other hand, tends to be undervalued given its rarity.
Modern Silver Eagles	0%	If you like large, silver coins, consider the Franklin half.  Worth little beyond melt.
Modern Gold Eagles	0%	Worth almost nothing beyond melt. Good stand-in for St. Gaudens $20!
Liberty Seated Half Dollar	-17%	A great choice for those who prefer large silver coins.
Capped Bust Quarter	-20%	If you like the capped bust design, the half dime is even less expensive given its rarity.
Buffalo Nickel	-22%	A charming design and lots of interesting varieties, and a great value for its rarity!
Standing Liberty Quarter	-22%	The most undervalued coin of the standing liberty set.
Three Cent Silver	-23%	Cousin to the also undervalued three cent nickel.
Liberty Seated Quarter	-23%	Another great choice for those who prefer large silver coins.
Draped Bust Half Cent	-30%	The most undervalued coin in the draped bust set.
Jefferson Nickel	-33%	Probably suffers a lack of interest for being the standard nickel for such a long time.
Barber Quarter	-35%	Barber coinage is exceptionally charming but somewhat undervalued given its rarity.
Liberty Nickel	-36%	A design similar to the Morgan Dollar in a much less overvalued series.
Roosevelt Dime	-38%	The most undervalued coin that is still ubiquitous.  Also quite inexpensive!
Coronet Head Cent	-38%	An attractive, affordable and undervalued series with lots of interesting varieties.
Shield Nickel	-41%	A less attractive engraving may contribute to these being rather undervalued.
Two Cent	-45%	A really interesting and short-lived piece which doesn't get much attention from collectors.
Barber Dime	-49%	A very attractive series, which is nonetheless quite undervalued.
Liberty Seated Dime	-49%	Another Barber coin that is rather undervalued for its rarity.
Three Cent Nickel	-50%	An interesting and short-lived series that is grossly undervalued for its rarity.
Braided Hair Half Cent	-56%	Quite undervalued alongside its one cent cousin. Still, quite rare and expensive.
Braided Hair Cent	-56%	The most undervalued coin in the Braided Hair / Coronet set.
Liberty Seated Half Dime	-57%	The most undervalued coin in the Liberty Seated set.
Capped Bust Half Dime	-58%	While still quite expensive, the most cost effective of the capped bust series.
Kennedy Half Dollar	-64%	Modern half dollars seem to be off the radar of collectors and the population at large.
Classic Head Half Cent	-66%	While these are quite expensive, rare and interesting, collectors neglect this variety.
Lincoln Cent Wheat Reverse	-71%	The modern Lincoln cent's predecessor may suffer for having the same obverse.
Indian Cent	-80%	Perhaps people are dissuaded by its tendency to turn brown with age.
Mercury Dime	-84%	This is a beautiful coin!  Perhaps people don't like it for its small size.
Franklin Half Dollar	-84%	The most undervalued set of US coins by rarity. These can be had very inexpensively to boot!

Table 1:  Median percent overvalued by variety and suggested alternatives.  The median percent overvalued is obtained by the following procedure:  First, the coin's value in MS60 is corrected by its melt value.  Second, its expected price for its rarity is obtained from the regression line seen in Figure 1.  Third, the percent overvalued for each coin in the series is found by taking the ratio of the expected price to its actual market price and subtracting 100%.  Finally, the median of these values for the set is reported as the median percent overvalued in MS60.  

As can be seen in the Table 1, gold coins tend to be far more overvalued for their rarity even after correcting for melt value than their silver or copper counterparts.  People may have an instinctive reaction to the fact that something has both numismatic and intrinsic rarity and respond by making an offer higher than they would for an equally numismatically rare coin and a lump of gold of the same weight.  The most undervalued coins given their rarity are the Indian cent, the Mercury dime, and the Franklin half dollar.

Rare coins are not a good investment unless you are a dealer buying from estates and flipping to collectors, an auctioneer who is acting as an agent to sell other people's coins, or an individual hoarding pre-1982 pennies for their melt value.  You should be sure that you are collecting for the right reasons: artistic or historic interest.  The value of antiques is so subjective that the fact that something lies below the price versus rarity curve means nothing unless you can get someone to buy the piece for a price tag closer to that curve.  If you are a collector who wishes to build an interesting and rare collection for the best value possible, the most undervalued sets are Indian cents, Mercury dimes, and Franklin half dollars.  This doesn't imply that you will make any more money collecting these, but it does mean that you can stretch your budget further.

In particular, you can determine what the expected price for any mint-state coin (MS60) should be based on rarity and melt value using this formula:

price  = exp(-0.6716 ln(survival estimate in MS60+) + 10.4208) + melt value

It would be foolish to use this formula to determine what you should actually bid or ask for a particular coin.  As I said above, individual pricing is highly subjective.  The history of a particular piece and its aesthetic value have a profound impact on its market price.  I've seen coins with beautiful toning sell at auction for 10 or more times what other coins technically in the same grade sell for.  This equation's utility is therefore limited to evaluating whether a coin is undervalued based on its rarity alone in the current market.  It's possible that in the future collectors' tastes will change and the undervalued coins will fetch a price more in line with their actual rarity.  It's also possible that they won't.  So, if you wish to use this result, you should use it as a guideline and not as gospel.

In my case, I like to use the recent auction sale prices at Heritage Auctions to determine what I am likely to actually be able to buy a coin for, then compare that price to the rarity-based price.  If my equation gives a higher price than the recent sale prices for a similar coin, I call that a good value.