Price determination for a collectible good: the case of rare U.S. coins.

I. Introduction

Researchers investigating markets for collectible goods such as paintings [1; 2; 6; 12], antique furniture [7], Stradivarius violins [14], wine [8; 10] and postage stamps [15] often report rates of return to holding collectibles that are both low [2; 6; 7; 8; 12] and volatile [6; 7; 8; 12]. Baumol [2] argues that the market for paintings of noted, deceased artists has no long-run equilibrium price, because the market is characterized by infrequent trades of

unique assets which are available in fixed supply. His empirical analysis supports the idea that paintings experience unpredictable price changes and highly dispersed rates of return with a mean near zero. In contrast Anderson [1) reports estimated equations explaining approximately 60 percent of the variation in (log) prices of paintings based on the size of painting, date of sale, and a constructed measure of repute of the artist. Other researchers have examined markets with more frequent transactions and greater substitutability among the goods traded. Schnitzel [15) finds that year-to-year changes in prices of postage stamps are predictable based on age of the stamp and quantity printed. Ross and Zondervan [14) report that variations in quality affect prices, but not rates of return, of Stradivarius violins, a result which is consistent with user benefits' being invariant with respect to quality.

Among markets for collectibles, the market for rare U.S. coins is active and well organized, and faces the potential for increased federal regulation. Annual sales are estimated at $5 billion [16]. Two Wall Street brokerages recently have formed multimillion-dollar limited partnerships to invest in rare coins, and some coins now are traded on electronic exchanges [5]. These innovations have come amid calls for increased regulation to guard against fraudulent grading of coins [5; 16]. A major determinant of the price of a collectible coin is its "grade," a measure of condition reflecting qualities such as sharpness of features. Regulators accuse coin dealers of defrauding collectors of millions of dollars annually by assigning higher-than-warranted grades to the coins dealers sell while downgrading coins they buy. For example, the Federal Trade Commission (FTC) filed a complaint against one dealer alleging $40 million in fraud and ordered another to return $400,000 to customers [5]. Independent, third-party grading firms claim to reduce the potential for fraud, yet the FTC recently obtained consent decrees from two of the three largest grading services, charging one with making "numerous false and misleading claims" [13].

Despite the growth of the coin industry and the potential for fraud and increased regulation, there has been no systematic investigation of price determination for collectible coins. This paper extends the literature on collectibles markets to examine price determination for rare U.S. coins. Specifically, we address three issues. First, we note that if the coin market attains an equilibrium, then coin prices should be predictably related to the characteristics of coins valued by collectors. There is general agreement among coin dealers and collectors on the characteristics of coins which affect price, and data are available on all relevant characteristics.(1) We use eight years' data on over 1300 coins of five denominations to quantify the relationship between price and characteristics and find that coin prices are largely predictable. We then compare conditions in the coin market with those in the market for paintings to offer additional insight on Baumol's explanations for the absence of a long-run equilibrium price in the market for a collectible good. Second, we examine the effect of grade on price by testing whether the market process generating prices is identical across two grades. We reject the hypothesis that the relationship between prices and characteristics is identical across grades, and make illustrative calculations of the effect of a change in grade on price, holding other characteristics equal. Results are interpreted in light of potential for fraudulent grading and increased regulation. Finally, we test whether coin characteristics affect rates of return. After correcting for unequal variances in rates of return, we find that rates vary with characteristics, suggesting that some coins provide collectors with greater user benefits than others.

II. The Coin Market: Background and Data

Collectors identify coins based on the following characteristics: (1) denomination, whether cent, nickel, dime, quarter- or half-dollar, or dollar; (2) vintage, or the date struck on the coin, which at least approximates the year minted; (3) type, or the image struck on the face of the coin; (4) the mint where the coin was produced; (5) the value of the silver contained in the coin; and (6) the condition or grade of the coin. An unusual feature of coins as heterogeneous goods is that the characteristics suffice to identify a coin uniquely, in the sense that two coins of the same denomination, vintage, type, mint and condition are essentially perfect substitutes.

These characteristics are expected to influence price because they affect the relative values collectors attach to different coins. While motives for collecting may vary, coin dealers distinguish two broad classes of numismatists, which we term "general" and "type" collectors. Other things equal, both classes prefer coins preserved in better condition, but the classes differ in how they value the other characteristics of coins.

The "general" collector chooses a particular type of coin, such as Barber quarters, and assembles a complete set of vintages and mints of Barber quarters. The "type" collector, on the other hand, assembles one specimen of each type of coin minted within some category, or one specimen of each type-mint or type-vintage combination. For example, a type collector might choose a denomination such as quarter and assemble one specimen from each mint for each type of quarter minted, without regard to vintage.

In any event, numismatists choose a manner of collecting based on individual tastes and budgets, and thus exert demands differentiated by denomination, type, vintage and mint. The market value of embodied silver may also influence the price of a coin, since coins may be preserved as collectibles while held jointly as stores of silver.

Data on coin prices and characteristics are taken from 1984 through 1991 editions of Yeoman, A Guide Book of United States Coins [19]. These publications list the quantities minted of various coins by denomination, vintage, type, and mint, as well as the silver content of coins and the previous year's prices for each coin by condition. The prices, which are estimated based on auction prices as well as data supplied by numismatic experts, thus pertain to the years 1983-90. Vintages used in this study range from the first regular U.S. mint issues of each denomination through 1964 for cents, nickels, dimes, quarter- and half-dollars. The final vintage (1964) was chosen as the last year prior to a major change in the metallic composition of U.S. coins: the silver content of the half-dollar was reduced by 50 percent in 1965, when silver was entirely eliminated from both the dime and quarter. Dollar coins are excluded because their metallic composition was changed 30 years before the other denominations: silver dollars were not minted for general circulation after 1935.

We include prices for two grades of coins in our data: "mint state" MS-60 and "extremely fine" EF-40. The MS-60 grade was chosen because it represents the average condition of newly minted coins. Although higher-grade coins exist, their price series are generally much less complete than the MS-60 series. The EF-40 grade in contrast represents a coin with all features sharp and well-defined but lightly worn on all surfaces.

Each denominational data set was assembled into cross sectional-time series format, where the year in which price is observed defines a point in time and the unique combination of denomination, vintage, type and mint defines a cross sectional unit. Table I reports the number of observations as well as sample means and standard deviations by denomination of all variables used in the empirical analysis. P refers to the price of the coin, while NETP refers to the price less the commodity value of the coin, computed as coin price less the product of silver content and silver price.(2) While prices of poorly preserved coins often equal market values of embodied silver, the silver value of coins in MS-60 and EF-40 condition accounts for a small percentage of mean prices. Both P and NETP are widely dispersed, with standard deviations ranging from 1.8 to 9.6 times as great as means. Mean prices and net prices do not necessarily increase with the face value of the coins.

ROR refers to the log of the price ratio between successive years in percentage terms: ROR = 100 X ln([P.sub.t]/[P.sub.t - 1]). This variable may be interpreted as an approximate financial rate of return to holding a coin for one year, based on the continuous compounding formula commonly applied in the collectibles literature [2; 6; 7; 14].(3) As shown in Table I, financial returns to holding coins recently have been low and variable: mean rates of return are under one percent per year for all denominations, with standard deviations ranging from 11 to 20 percent. Other researchers also have found low and volatile financial gains to investing in collectible goods. Baumol [2] reports a mean annual real rate of return to paintings, with no allowance for transactions or holding costs, of 0.55%, compared to an opportunity cost estimated at 2.5% annually. Frey and Pommerehne [6] extend Baumol's data and find a real rate of return of 1.5%, net of transactions costs, with a standard deviation of 5.0%. Mok et al. [12] similarly report a standard deviation of returns, net of transactions cost, to holding paintings by Chinese masters that is 35 percent greater than the mean return. Collectible goods other than paintings do not perform appreciably better from a financial perspective. For example, Ross and Zondervan [14] report mean rates of return on Stradivarius violins of 3.51% per year in nominal terms, while Graeser [7] finds a mean return of 6.97% on various types of American antique furniture over the period 1967-1986, compared to rates on 90-day Treasury bills of 7.31%. On average then, coins appear to be no better a financial investment than other collectibles, which as a group tend to perform poorly.

Table I. Variable Means and Standard Deviations, by Denomination

Variable          Cent     Nickel     Dime     Quarter     Half
P(a)              1.675    9.794      5.294    8.696       6.495
                  9.033    17.97      51.02    72.46       24.95
NETP(a)           1.675    9.794      5.289    8.684       6.470
                  9.033    17.97      51.02    72.46       24.95
ROR(b)            0.209    0.436      0.166   -0.088      -0.098
                  13.11    20.41      11.52    17.38       13.57
QUANT(c)          137.5    41.74      30.14    13.94       4.827
                  323.4    116.0      101.0    53.43       14.30
AGE(d)            83.06    60.54      86.20    84.90       92.82
                  44.54    25.39      41.46    39.55       43.03
MS                0.497    0.515      0.490    0.496       0.498
                  0.500    0.500      0.500    0.500       0.500
MM1               0.637    0.532      0.463    0.466       0.475
                  0.481    0.499      0.499    0.499       0.499
MM2               0.182    0.256      0.153    0.160       0.131
                  0.386    0.437      0.360    0.366       0.337
MM3               0.180    0.212      0.264    0.236       0.239
                  0.385    0.409      0.441    0.425       0.426
MM4               --(e)    --(e)      0.018    0.023       0.029
                  --       --         0.133    0.151       0.169
MM5               --(e)    --(e)      0.101    0.114       0.126
                  --       --         0.303    0.318       0.332
TYPE1             0.006    0.092      0.006    0.003       0.003
                  0.077    0.289      0.077    0.056       0.055
TYPE2             0.022    0.187      0.024    0.011       0.002
                  0.147    0.390      0.153    0.104       0.039
TYPE3             0.014    0.357      0.054    0.049       0.015
                  0.118    0.479      0.226    0.215       0.123
TYPE4             0.080    0.365      0.318    0.330       0.106
                  0.272    0.482      0.466    0.470       0.308
TYPE5             0.084    --(f)      0.223    0.232       0.337
                  0.278    --         0.417    0.422       0.473
TYPE6             0.012    --(f)      0.231    0.119       0.225
                  0.109    --         0.421    0.324       0.418
TYPE7             0.216    --(f)      0.144    0.257       0.197
                  0.412    --         0.351    0.437       0.398
TYPE8             0.529    --(f)      --(f)    --(f)       0.111
                  0.499    --         --       --          0.315
TYPE9             0.012    --(f)      --(f)    --(f)       0.003
                  0.109    --         --       --          0.055
TYPE1O            0.024    --(f)      --(f)    --(f)       --(f)
                  0.153    --         --       --          --
Observations(g)    3992    2872       5336     5112        5192

a. P and NETP measured in dollars; figures in the table are divided by 100.

b. ROR = 100 X ln([P.sub.t]/[P.sub.t - 1]) is measured in percent and approximates the annual financial rate of return to holding a coin.

c. QUANT measured in units; figures in the table are divided by 1,000,000.

d. AGE measured in years.

e. Neither Carson City (MM4) nor New Orleans (MM5) mints produced cents or nickels.

f. Only four types of nickels, seven types of dimes and quarters, and nine types of half-dollars produced prior to 1965.

g. A price for each coin is observed in each of eight years, 1983-90.

Table I also indicates that quantities minted (QUANT) are widely dispersed, with all standard deviations exceeding means by factors of at least two. Mean quantities minted decline as face value increases. Data on current stock by coin condition are unavailable. AGE refers to the approximate age of the coin in 1983, computed as 1983 less the date struck on the coin.(4)

The dummy variable MS takes the value unity if the observed price corresponds to a coin graded MS-60, and zero if the price corresponds to a coin graded EF-40. The variables prefixed with MM in Table I are categorical variables reflecting the location where the coin was minted. For each denomination, the Philadelphia mint (MM1) has been the largest producer, accounting for between 46 and 63 percent of production. Cents and nickels have been minted only at the Philadelphia, Denver (MM2) and San Francisco (MM3) mints. The Carson City (MM4) and New Orleans (MM5) mints produced the smallest quantities of any of the five mints for all five denominations. Each mint except the one in Philadelphia identifies its coins by striking a small letter or "mint mark" on the face of the coins it produces. The TYPE variables listed in Table I also are 0-1 dummy variables categorizing the image on the face of the coin and are not comparable across denominations. For example, TYPE1 for cents refers to the Liberty Cap Type and TYPE2 to the Draped Bust Type, while for nickels TYPE1 refers to the Shield Type and TYPE2 to the Liberty Head Type. As shown in Table I, different numbers of types have been minted by denomination.

Data used in this study contrast sharply, both in amount and detail, with data available for many investigations of collectibles markets. In the case of paintings of noted artists or of Stradivarius violins, for example, the relatively small number of works produced and the long holding periods for the assets severely limit the number of observed transactions. Also, observing a price only when a transaction finally occurs creates the potential for sample selection bias [6; 12]. The large number of coins minted and the frequency of transactions, on the other hand, permit observation of a price for virtually every coin in the data for the full eight years under consideration.

III. Empirical Results

We begin empirical analysis of price-determination in the market for collcctible coins by estimating a hedonic regression in logarithmic form.(5)

ln([NETP.sub.it]) = [[Beta].sub.0] + [[Beta].sub.1]
ln([QUANT.sub.i]) +
[[Beta].sub.2] ln([AGE.sub.i]) + [[Beta].sub.3][MS.sub.i]

+ [[Beta][prime].sub.4][MM.sub.i] +
[[Beta][prime].sub.5][TYPE.sub.i] +
[[Beta].sub.6][YEAR.sub.t] + [[Epsilon].sub.it]. (1)

To focus more clearly on the numismatic value of coins, we remove the influence of fluctuations in silver prices by defining prices net of silver values as the dependent variable; however, similar results are obtained if P, rather than NETP, is used.(6) In equation (1), the [Beta] denote unknown parameters to be estimated, the [[Epsilon].sub.it] represent mean zero normal random variables; MM and TYPE denote vectors of mint and type dummies respectively conformable with [[Beta][prime].sub.4] and [[Beta][prime].sub.5]; YEAR is a time counter taking the value zero for 1983 observations and incrementing by one in each subsequent year; other symbols have been previously defined, and i indexes coins while t indexes years. Equation (1) is estimated after pooling all time series within each denomination. Owing to the large sample sizes which result, we adopt the one percent significance level for testing hypotheses and also make use of Bayesian sample-size-adjusted critical values based on Leamer [11].

In the pooled cross section-time series data, the error terms are likely to be serially correlated within each time series and heteroskedastic between cross sections. Statistical tests based on preliminary estimates of equation (1) reject the hypothesis of no first-order serial correlation and the hypothesis of equal variances for all cross sections at the one percent significance level.(7) As a result, the equation is estimated using the three-step feasible generalized least squares (GLS) procedure outlined by Kmenta [9] for the cross-sectionally heteroskedastic and timewise autoregressive model.

Feasible GLS results are presented in Table II. Estimated coefficients confirm that net prices are higher for coins minted in smaller quantities, older coins, coins in better condition, and coins not minted at the Philadelphia mint. Also, significant variation in log net price exists according to the image struck on the face of the coin. Estimated elasticities of price with respect to quantity minted are significant at one percent, ranging from -.33 to -.58. Age elasticities are large, positive, and significantly different from zero at one percent.

It is widely held in numismatic circles that mint-marked coins are generally worth more than coins minted in Philadelphia, because in any given year the Philadelphia mint usually produced greater quantities than other mints. The coefficients of the mint dummies in Table II represent in contrast the partial effects of mint-marks, holding quantity minted constant. These coefficients are significantly greater than zero in all cases except for the MM3 dummy for cents and dimes. Thus, estimated equations confirm the existence of mint-mark premiums but refute the explanation that they arise from quantity effects alone. An alternative explanation for the mint-mark premium is the demand exerted by the "type" collector for mints independently of dates.

Similarly, the t-statistics for the null hypothesis of zero TYPE effects indicate that there is significant variation in log net prices by the image struck on the face of the coin. There appears to be no common pattern in the relative valuations of coins by type. For example, the excluded category for each denomination is the most recently minted type, but coefficients of TYPE dummy variables are not uniformly positive or negative across denominations, nor do there appear to be uniform differences in coin prices for coins bearing images of presidents compared to other coins. An alternative explanation for price variation by type is again the demand of the "type" collector, who purchases types independently of other characteristics.

Table II. Log Net Price Equations: GLS Estimates(a) (Estimated
t-statistics in
parentheses)

Variable     Cent         Nickel       Dime     Quarter     Half

CONSTANT    -2.896      -4.415       6.312      .873        .518
           (-5.311)    (-9.220)     (14.33)    (3.200)     (.770)
ln(QUANT)    -.561       -.580      -.539       -.339      -.365
           (-40.10)    (-41.08)     (-53.56)   (-77.56)  (-36.63)
ln(AGE)       3.162       3.762      1.481       1.880      2.553
            (46.13)     (43.03)      (15.71)   (28.72)    (26.68)
MS            1.944       1.791       1.816      1.596      1.283
             (60.32)     (74.61)      (79.28)   (94.01)    (45.67)
MM2            .263       1.319         .094      .070       .152
             (8.407)    (40.492)      (2.967)   (3.197)    (4.091)
MM3           -.040        .537        -.135      .0849      .204
             (-.987)    (15.07)      (-5.204)   (5.047)    (8.214)
MM4            --(b)     --(b)        1.083       .655      1.144
               --        --          (8.748)    (7.080)   (16.66)
MM5            --(b)     --(b)         .0675      .191       .221
               --        --          (2.324)    (9.029)    (7.836)
TYPE1(c)      3.458       -.773        .227      2.963       .056
             (2.504)    (-5.282)      (.641)    (6.528)     (.080)
TYPE2(c)      3.604        .587        .233      1.276      1.148
             (7.428)     (6.283)     (1.088)    (6.591)    (1.553)
TYPE3(c)      1.951       1.111       -.196       .254     -1.574
             (6.500)    (16.76)     (-1.069)    (2.393)   (-2.536)
TYPE4(c)      -.823      --(b)        -.593      -.079     -1.808
            (-4.405)     --         (-3.846)   (-1.055)   (-2.963)
TYPE5(c)       .671      --(b)         .290       .584     -2.159
            (-4.904)     --          (2.427)   (11.11)    (-3.587)
TYPE6(c)       .840      --(b)         .526      1.087      -.707
             (2.136)     --          (5.030)   (23.69)    (-1.198)
TYPE7(c)       .277      --(b)        --(b)     --(b)       -.603
             (2.497)     --           --        --        (-1.032)
TYPE8(c)       .146      --(b)        --(b)     --(b)       -.210
             (2.110)     --           --        --         (-.360)
TYPE9(c)      -.018      --(b)        --(b)     --(b)       --(b)
             (-.070)     --           --        --          --
YEAR           .009      -.001        -.040     -.021       -.032
             (1.542)   (-.215)     (-9.440)   (-7.049)    (-6.645)
RHO(d)         .980      .977         .964       .949        .953
             (314.5)     (244.3)     (264.6)    (214.7)    (225.8)
SER(e)         .014      .010         .021       .021        .031
[R.sup.2](e)   .99       .99          .99        .99         .99
Observa-
 tions(f)     3493      2513          4669      4473        4543

a. The dependent variable is the natural log of the net price.

b. Mint or type dummy not defined for this denomination.

c. The TYPE variables are not comparable across denominations.

d. Estimated first-order autocorrelation coefficient.

e. SER denotes standard error of the regression, computed using the transformed data; the [R.sup.2] is computed using original data.

f. The first year of each time series is deleted in the feasible GLS estimation procedure.

The Effect of Grade on Price

Coins graded MS-60 are worth one-and-one-third to two-and-one-half times more than otherwise identical coins graded EF-40. Expressed in dollar terms, the effect of a change in grade from EF-40 to MS-60, evaluated at denominational means, ranges from nearly $500 for cents to about $2000 for nickels; at sample maximum prices, the estimated effect ranges from tens of thousands of dollars for cents to hundreds of thousands of dollars for other denominations. These calculations illustrate how a difference in grade can have a substantial impact on the value of a coin, but they are not necessarily representative of price variations that would occur due to intentional or unintentional misgrading. Reasonably knowledgeable collectors can distinguish between coins belonging to EF-40 and MS-60 grades; fraudulent or erroneous grading typically would involve less noticeable differences in grade.

To further investigate the effect of grade on price, we test the hypothesis that the market process generating coin prices is identical across grades after allowing for a constant percentage difference in price. Specifically, log net price equations were re-estimated with feasible GLS using as additional explanatory variables the cross-products of the MS dummy and all other coin characteristics. The null hypothesis that coefficients of the additional variables are jointly zero is rejected even by the stringent Leamer-Bayes criterion in each denomination. Thus, slopes of log net price equations appear to differ between grades (even after allowing for different intercepts), indicating structural change between grades in the market process generating coin prices.

The substantial impact of grade on price, together with the likely asymmetry of information between the dealer and collector, highlights the incentives for fraudulent grading and suggests the potential for adverse selection. While these results do not resolve whether regulation is warranted, they underline the importance of consistent and reliable grading of coins in promoting an efficiently functioning market.

Predictability and Stability of Coin Prices

In contrast to Baumol's [2] results for paintings of noted artists, estimates in Table II suggest that prices of collectible coins are largely predictable: the proportion of variation explained is high, as are estimated autocorrelation coefficients, and coefficients of explanatory variables are plausibly signed and significant.(8) These results are consistent with the attainment of equilibrium in the coin market. Baumol's explanation for the absence of a long-run equilibrium price rests primarily on four features of the market for paintings of noted, deceased artists: (1) the stock of paintings of a given artist is fixed, so there is no supply response to increased price; (2) each painting is essentially unique, implying that there are no perfect substitutes and the seller has monopoly power; (3) transactions occur infrequently, and (4) the "true" equilibrium price of a painting is not known even in principle (i.e., there is no analogy to the minimum of long-run average cost for a competitive industry, or the share of discounted earnings for a security). The market for collectible coins shares only the first of these features: the total stock of a high-grade coin cannot be increased. In most cases there are a number of coins within each denomination-vintage-type-mint grade; these coins are homogeneous with many potential sellers, and transactions are far more frequent than in the market for paintings. Also, while the equilibrium price of a rare coin may not be known in advance, prices evidently are largely determined by coin characteristics.

In light of this comparison of coin and picture markets, it appears that the absence of a supply response alone is not sufficient to preclude equilibrium in the market for a collectible good. In fact, economic theory illustrates how equilibrium may be attained in markets with fixed supplies [17, 89-91]. Thus it would appear that one or more of Baumol's additional conditions (or some other condition) are necessary if equilibrium is not to be attained.

Despite the predictability of coin prices, the data provide some evidence that the market process generating coin prices is not temporally stable. To test for temporal stability, equation (1) was re-estimated by feasible GLS after replacing YEAR with six dummy variables indexing the year of observation as well as interactions of these dummies with all other coin characteristics. The hypothesis that coefficients of the interaction terms are jointly zero (allowing for intercept shifts between years) is rejected at one percent (though not by the Leamer-Bayes criterion) for each denomination except nickel.(9) In other words, the slopes of the log net price equations appear to shift over time, a result which may reflect changes in underlying demands for characteristics of coins.

Rates of Return

To further investigate temporal features of the coin market, we estimate regressions to quantify the relationship between the annual rate of return and coin characteristics. In light of probable shifts in coefficients of log price equations over time, the year-to-year difference in log prices is regressed on the same explanatory variables appearing in equation (1), to investigate the predictability of rates of return and whether returns vary with characteristics.(10) Based on initial estimates of the equation, the hypothesis of no first-order serial correlation is rejected at one percent significance, with negative serial correlation found in all five denominations. Recent stock market research also reports evidence of negative serial correlation of returns, and it is well known that serial correlation alone need not imply market inefficiency [3]. Also, the hypothesis of equal variances across coins is rejected at the one percent level in each denomination, a result which is consistent with variations in risk of holding different coins.

Feasible GLS results are presented in Table III. The low [R.sup.2] values indicate that rates of return to holding coins are difficult to predict, notwithstanding the serial correlation and the significant (at one percent) relationship between covariates and returns. Estimated coefficients in Table III are interpreted as incremental effects on annual percentage rate of return. For example, the -0.27 coefficient of MS in the cent regression indicates that cents graded MS-60 are estimated to earn a return approximately 1/4 of one percent lower than otherwise identical cents graded EF-40.

Table III. Rate of Return Equations: GLS Estimates(a) (Estimated
t-statistics
in parentheses)

Variable       Cent     Nickel      Dime       Quarter     Half

CONSTANT      .445       .097       -.508       -.799      .405
             (.748)     (.260)     (-.638)   (-1.028)     (.301)
ln(QUANT)    -.003       .011       -.091       -.066     -.133
            (-.198)     (.831)    (-4.630)   (-3.571)   (-6.470)
ln(AGE)      -.054      -.060        .621        .572      .014
            (-.572)    (-.765)     (3.076)    (2.876)     (.092)
MS           -.270      -.102        .230       -.225     -.487
           (-5.974)   (-2.686)     (4.997)   (-4.286)   (-7.302)
MM2           .013       .109       -.081       -.049     -.104
             (.295)    (1.640)    (-1.087)    (-.683)   (-1.504)
MM3         -.0362      .111       -.220       -.045     -.013
            (-.605)    (2.058)    (-3.809)    (-.829)    (-.251)
MM4        --(b)       --(b)         .146       .290       .172
           --          --           (.508)    (1.132)    (1.211)
MM5        --(b)       --(b)        -.162     -.0202     -.009
           --          --         (-2.531)    (-.272)    (-.146)
TYPE1(c)     4.925       .202      -1.505      6.943      4.015
            (2.065)    (1.527)    (-2.446)    (5.744)    (2.228)
TYPE2(c)     5.342       .0484     -1.403       -.078     5.060
            (12.01)      (.639)    (-2.840)    (-.189)    (3.180)
TYPE3(c)     2.959       .047       -1.915      -1.447     1.197
            (9.006)     (.838)     (-4.694)   (-3.921)     (.956)
TYPE4(c)      .862       --(b)      -1.020       -.955     1.629
            (3.620)      --       (-3.130)    (-3.587)   (1.325)
TYPE5(c)     -.232       --(b)      -.448       -.914     1.521
           (-1.385)      --       (-1.972)    (-5.147)   (1.254)
TYPE6(c)     1.945       --(b)       .116       -.395     1.263
            (2.839)      --         (.746)    (-2.724)   (1.052)
TYPE7(c)     -.279       --(b)      --(b)       --(b)     1.271
           (-1.976)      --         --          --       (1.069)
TYPE8(c)     -.197       --(b)      --(b)       --(b)     1.987
           (-2.019)      --         --          --       (1.664)
TYPE9(c)     -.370       --(b)      --(b)       --(b)      --(b)
            (-.663)      --         --          --         --
YEAR          .014      -.015       -.056        .001      .030
            (1.561)   (-2.602)    (-4.599)      (.114)   (2.654)
RHO(d)       -.538      -.141       -.253       -.377     -.211
          (-37.68)    (-7.149)   (-17.86)    (-27.19)   (-14.55)
SER(e)       1.274       .525       1.639       1.788      1.480
[R.sup.2](e)   .23        .02         .04         .13        .07
Observa-
  tions(f)    2994       2154        4002        3834       3894

a. The dependent variable is the difference in log prices between successive years, multiplied by 100: ROR = 100 X [ln([P.sub.it]) - ln([P.sub.i,t - 1])].

b. Mint or type dummy not defined for this denomination.

c. The TYPE variables are not comparable across denominations.

d. Estimated first-order autocorrelation coefficient.

e. SER denotes standard error of the regression, computed using the transformed data; the [R.sup.2] is computed using original data.

f. The initial year of each time series is lost in first-differencing the log prices, and the feasible GLS estimation procedure drops the first year of the resulting series.

With the exception of cents and nickels, coins minted in smaller quantities earn significantly higher financial rates of return; older coins also earn higher returns in the three larger denominations with significant differences found for dimes and quarters. Schnitzel [15] similarly found that quantity and age were significant determinants of rates of price change for postage stamps. To the extent that older coins and coins minted in smaller quantities are rarer, these results offer at least some support to the common advice offered to beginning investors in coins to "focus on rarity" [4].

While evidence in Table II suggests that collectors are willing to pay higher prices for mint-marked coins, there appears to be no significant mint-mark premium in rates of return: only two of the 16 mint-mark coefficients are significant at one percent. Estimated coefficients and t-statistics for TYPE variables, on the other hand, suggest that returns vary according to the image struck on the coin.

With the exception of dime, coins graded MS-60 earn a significantly lower financial rate of return than coins graded EF-40. This result contrasts with that of Ross and Zondervan [14], who found for Stradivarius violins that quality differences affect prices, but not rates of return. The result obtained here is consistent with the idea that coins in better condition offer collectors greater user benefits in compensation for lower rates of financial gain.

IV. Conclusions

This paper has shown that prices of rare coins are predictably related to characteristics of coins valued by collectors, a result consistent with the attainment of equilibrium prices for coins. This result suggests that the fixed supply of a collectible good is not by itself sufficient to preclude equilibrium: some other condition is necessary, perhaps one or more of those given by Baumol [2] including uniqueness of the asset, monopoly position of sellers, infrequent trades, or unknown determinants of value.

In light of recent FTC allegations of fraudulent grading and calls for increased regulation of the coin industry, tests were performed which confirmed the importance of grade in determining price. A change of grade appears to cause a structural shift in the market process generating coin prices. Illustrative calculations highlight the incentives for fraudulent grading and underline the need for consistent and reliable grading to promote an efficiently functioning market. In addition, estimation results indicate that prices are higher for coins minted in smaller quantities, older coins, and coins not minted at the Philadelphia mint. In contrast to widely held views among numismatists, mint-mark premiums do not arise from quantity differences between mints but appear to reflect instead demands of "type" collectors.

Rates of return to holding coins during the sample period appear to be both low and volatile, a result common to studies of collectible goods. Although the financial performance of coins as a group is poor, rarer coins appear to be somewhat better investments. In equilibrium, given the low and volatile returns, rational collectors must invest in coins primarily for the pleasure of collecting rather than for monetary gain. In fact, results confirm that higher-quality coins earn lower rates of return, suggesting that they offer greater user benefits to collectors.

1. Our data were not obtained from the grading services under FTC investigation.

2. Data on silver prices were taken from the U.S. Bureau of Mines [18]. Silver prices peaked at $11.44 per troy ounce in 1983 and fell as low as $5.00 in 1990.

3. No adjustment is made for transactions or holding costs.

4. The date struck on the coin approximates year minted because historically mints did not always change the date each year. Regressions include a time trend to capture price changes over calendar time.

5. Note that quantity is not jointly determined with price. Also, in preliminary regressions, the net price equation was estimated by maximum likelihood with a common Box-Cox transformation applied to net prices, quantity and age. Although the null hypothesis that the transformation parameter equals zero was rejected at one percent in four of five denominations, estimated values of the parameter were less than 0.1 in all denominations and less than 0.01 in three denominations. These results suggest that the logarithmic transformation used in equation (1) is a reasonably close approximation to the true functional form.

6. Results from estimating equation (1) with P as the dependent variable, as well as results from all other unrepotted regressions discussed in the text, are available from the authors on request.

7. Initial ordinary least squares estimates of equation (1) were used to test for serial correlation, and the null hypothesis of no first-order serial correlation was rejected at less than one percent significance in each denomination. Residuals obtained from reestimating equation (1) after correcting for serial correlation then were used to test for group-wise heteroskedasticity, and the null hypothesis of equal variances for all cross sections also was rejected at less than one percent significance for each denomination.

8. Estimates of serial correlation coefficients are close to unity, suggesting that log net prices may follow a random walk. In other words, log net prices are predictably related to characteristics, but changes in log net prices may be more difficult to predict. Note that the dependent variable in equation (1) cannot be cointegrated with any explanatory variables in the regression, since all independent variables (except YEAR) are fixed over time.

9. Similarly, the hypothesis that coefficients of the interaction terms and the dummy variables are jointly zero also is rejected at one percent in the same four denominations.

10. The dependent variable is defined as 100 X [ln[P.sub.it] - ln[P.sub.i,t - 1]]. Prices, rather than net prices, are used to capture the full change in monetary value. No adjustment was made for transaction or holding costs. Also the test for temporal stability was repeated with the natural log of price, rather than net price, as the dependent variable, and the hypothesis that slope coefficients are constant over time was rejected at the one percent level. If coefficients were temporally stable, first-differencing equation (1) would leave a constant term as the only regressor.

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