The 4 C’s of Diamonds

Diamond Database

Analysis of Diamond Prices

Diamond Price Calculator

“Better a diamond with a flaw than a pebble without one.”

-- Chinese Proverb

During the course of many of the projects I’ve worked on, as part of the analysis I have designed different types of “calculators”.  Each of my calculators is a model that calculates some value, where the user can easily change various assumptions or inputs to see how the final value changes.

About a year and a half ago (January 2009), a colleague and I (along with about a zillion other people) thought about getting into the iPhone apps business.  Given my past experiences, I figured a natural type of app for us to offer would be calculators.  One of the ideas I came up with was a diamond calculator, where users could input the “4 Cs” of a diamond (carats, color, cut, clarity) and get an estimate of an average or reasonable price for that diamond.  The situation I was seeking to address was that of a customer walking into a diamond store, picking out a diamond, and having the clerk quote him a price.  I think it’s safe to say that most people don’t know enough about diamonds to tell whether or not a quote they get for any particular diamond is reasonable.  So my idea was to provide a tool to give the unknowing buyer some independent, 3^rd party estimate of the value of that type of diamond to help determine how good of a deal he might be getting.

A major issue with the concept of any pricing calculator involves the vintage of the pricing data used to come up with the equations that are used in the calculator to estimate price, and whether or not the equations will still work over time as prices change.  As this issue relates to diamonds, one possibility is that the structure of the prices of different qualities of diamonds relative to one another remains constant over time.  In this case, a high quality diamond of a given size (number of carats) would be worth, say, twice the value of a moderate quality diamond, regardless of the absolute level of prices.  Then, over time, the underlying equations used in the calculator would still be valid, and you would just have to adjust the estimated price accordingly by the change in the price level.  In other words, if, after two years the CPI had increased by 10%, then you could use the calculator to estimate the price of a diamond based on prices from two years ago, and then adjust the estimated price upward by 10%.

The other possibility is that the structure of the prices of different qualities of diamonds relative to one another changes over time.  In this case, higher quality diamonds might increase in value relative to lower quality diamonds.  For example, a higher quality diamond might have been worth twice the amount of the same size moderate quality diamond two years ago, but now the higher quality diamond is worth three times the value of the moderate quality diamond.  If the pricing structure changes over time, then the equations used as the basis of the calculator will become obsolete after a period of time and will need to be re-calculated based on current data in order to continue to provide valid estimates.

At the time (January 2009), I hypothesized that the situation would be the former, that is, that the structure of diamond values relative to one another wouldn’t change over time.  Of course, I had a vested interest in this hypothesis being true, because then I would only need to do the analysis for my calculator once and it would be valid into the future, even as prices changed with inflation.

I was able to find some good data to use, and I came up with a pretty cool calculator.  Unfortunately, we never got it turned into an app.

Now that I’m teaching myself to program, I decided to return to my diamond calculator as a good project to help me learn programming.  Since my original database was over a year and a half old, I had to update the prices.  As an economist (normal people probably don’t think this way) I was excited at the prospect of actually being able to test my hypothesis about whether or not the structure of diamond prices had changed.

I just updated the analysis and redid my calculator, which you can find posted here:  diamond price calculator.

The rest of this blog entry describes the diamond price analysis I undertook.

The 4 C’s of Diamonds

Harold Weinstein Ltd., Gemmological Laboratory  http://www.hwgem.com/diascale.htm

A diamond's cost is based on the characteristics known as the "4 C's". Clarity, Colour and Cut (proportion) are the quality elements which together with the Carat Weight determine the value of a stone. The closer a diamond grades to the left of one or all of these scales the rarer and the more costly it will be.

This section provides a brief description of the 4 C’s of diamonds.  The following text was taken from The Four C’s of Diamonds website, while some of the pictures were taken from this site and others were taken from the Harold Weinstein Ltd. site.

Cut

When we speak of cut we are more interested in the proportions of the diamond as opposed to its shape (Round Brilliant, Marquise, Pear, Princess, etc.) Every diamond regardless of its shape gets it brilliancy and scintillation by cutting and polishing the diamond facets to allow the maximum amount of light that enters through its top to be reflected and dispersed back through its top. A correctly cut, "well made", stone is pictured in diagram 1. As you can see if the angles are correct the light that enters is dispersed properly back through the diamond's top facets. When a stone is cut too shallow (diagram2) or too deep (diagram3) the light that enters through the top is allowed to escape through the diamond's bottom and does not allow the maximum beauty of the diamond to be realized.

Color

Diamonds come naturally in every color of the rainbow. However most people are concerned with diamonds in the white range. The Gemological Institute of America (GIA) rates the body color in white diamonds from D (colorless) to Z (light yellow).

The best color for a diamond is no color at all. A totally colorless diamond allows light to pass through it easily, resulting in the light being dispersed as the color of the rainbow. Colors are graded totally colorless to light yellow. The differences from one grade to the other are very subtle and it takes a trained eye and years of experience to color grade a diamond.

Clarity

The clarity of a diamond is determined by the amount and location of flaws, or blemishes, in the diamond when viewed under 10 power (10x) magnification. GIA rates clarity grades in diamonds from Flawless to Imperfect 3 (see chart below)…

Most diamonds contain very tiny birthmarks known as "inclusions." An inclusion can interfere with the light passing through the diamond… A diamond that is free of inclusions and surface blemishes is very rare…and therefore very valuable.

Diamond Database

Price Scope Consumer Advocate Site contains a database of diamonds offered by various retailers.  A prospective user of the database enters

• a diamond shape,
• size (carat) range (sizes are specified to 2 decimal points),
• color range
• clarity range, and
• HCA cut rank.

The site will then list all diamonds in the database by source of offering for that diamond specification.  The site will enable you to view up to 400 diamonds for any particular specification.

From this database, I collected data on round diamonds, up to 400 diamonds for each size by combination of:

• size (all diamonds between 0.1 and 4.0 carats by 0.1 carat increment),
• color (all diamonds with colors D through I), and
• clarity (all diamonds with clarity IF through I3).
• HCA cut

If there were more than 400 diamonds for a particular specification, I collected the 200 least expensive and 200 most expensive diamonds listed.  After collecting the data, I checked for duplicates based on all 15 specifications provided in the PriceScope database and deleted duplicates.  My reasoning for these deletions was that given the number of specifications (15), there is a high probability that any duplicates are duplicate listings of the same diamond, rather than duplicate diamonds with the exact same specifications.

Since there are only 15 0.1 carat diamonds, I excluded those from the analysis.  Also, HCA cut information was provided for less than 2% of the diamonds, so I couldn’t use this as a distinguishing characteristic.  What I’m left with, then, is a database of round diamonds for sizes 0.2 – 4.0 carats by 0.1 increment, each with a color (D – I) and a clarity (IF – I3) ranking.

The counts of diamonds in my de-duped database, as compared with the number of diamonds listed on the site are:

Graphs 1 and 2 display the distributions of diamonds in my database by size, color and clarity:

Graph 1

Graph 2

As you might expect, since large, high quality diamonds are rare, there are larger portions of higher quality diamonds (higher color and higher clarity) in the smaller sizes than in the larger sizes.

Analysis of Diamond Prices

Estimates of Diamond Prices

Now that I had my dataset put together, the first thing I did was to graph the price per carat by size, color, and clarity. By using price per carat, instead of price per diamond, I can do comparisons across all diamonds, not just those for the same size.  The results are displayed in Graph 3.

Graph 3

Yes, this graph is a bit, um, busy.  What I’m looking for is general patterns in price per carat across diamond sizes, colors, and clarities.  What the graph says is that the patterns fall into four distinct groupings:

1. 0.2 – 0.4 carats
2. 0.5 – 0.9 carats
3. 1.0 – 1.9 carats
4. 2.0 – 4.0 carats

Graph 4 shows you what Graph 3 looks like for 0.2, 0.5, 1.0, and 2.0 diamond sizes:

Graph 4

In Graph 4, it looks like the lines for the larger diamonds are steeper than those for the smaller diamonds, in which case quality commands a higher price premium for the larger diamonds than the smaller ones.  But its hard to tell from the graph above, due to scaling issues.  I wanted a better look at the differences in the prices of the higher versus lower quality diamonds for the different sizes of diamonds.  To this end, I calculated all prices for each diamond size relative to a low quality diamond of that size, of color I and clarity SI2.  These normalized prices will better reveal the different price premiums for quality across the different diamond sizes.  The results are displayed in Graph 5.

Graph 5

It becomes clearer in Graph 5 that, indeed, larger diamonds with better color and clarity command higher price premiums than do smaller diamonds with better color and clarity. In particular, for 0.2 carat diamonds, the highest quality diamonds (color D and clarity VVS1) are priced at about 3.6 times the price of the lowest quality diamonds (color I and clarity SI2).  In contrast, for 2.0 carat diamonds, the highest quality diamonds (color D and clarity VVS1) are priced at about 5.4 times the price of the lowest quality diamonds (color I and clarity SI2).

The patterns in diamond prices across different levels of color and clarity are different for different sizes of diamonds (as per the four groups defined above).  This means that I have to use separate equations to estimate pricing relationships for each of the different groups.  If, instead, I lump all diamonds together and estimate one set of relationships, then my calculator estimates of prices will tend toward the average.  More specifically, my equations will tend to under-estimate prices of higher quality, large diamonds and over-estimate prices of higher quality, small diamonds.  To provide the most accurate estimates of price as possible, my calculator will use four sets of equations to estimate diamond prices, one for diamonds in the 0.2 – 0.4 size range, one for diamonds in the in the 0.5 – 0.9 size range, one for diamonds in the in the 1.0 – 1.9 size range, and one for diamonds in the in the 2.0 – 4.0 size range.

I performed regressions on diamond prices per carat using dummy variables for color and clarity. It turns out that inclusion in the estimates of the lab that performed the diamond grading is an important determinant of price, especially for larger sizes of diamonds.  Graph 6 displays the distribution of labs for my database:

Graph 6

87% of diamonds in my database were rated by GIA (Gemological Institute of America ) and 9% were rated by EGL (European Gemological Laboratory).  GIA diamonds tend to command higher prices than diamonds rated by other labs.  As such, if I exclude the lab from my equations, my calculator will tend to under-estimate the price of GIA diamonds and over-estimate the price of other (non-GIA) diamonds.  Since the lab that did the diamond rating is a commonly available piece of information, I include it as a user input in my calculator.  The good news is that if the user does not know the lab, they can use GIA as the default value, since in the overwhelming majority of cases, this will be the lab that did the diamond’s rating.

So finally, here are the results of my regression analyses:

The regressions indicate that the vast majority (82 – 93%) of the variation in prices across different diamonds (see Adjusted R² in the table) can be attributed to size (number of carats), color, clarity, and lab.  Also, due to the large number of observations (N), the standard errors of the estimates are small, on the order of a couple of dollars.  This means the equations provide accurate estimates of the average price of diamonds of particular size, color, and clarity.  Yet, there is still plenty of variation in price.  This means that when comparing the price of a given diamond with that of its estimated average price using the equations derived herein, one should really consider a range of prices, on the order of 25% around the mean, to get a better estimate of the reasonable range of prices for diamonds with particular characteristics.

Diamond Price Premiums for Size and Quality

Next, I wanted to better understand the patterns in price premiums for color and clarity across different diamond sizes.  To do this, I used my regression estimates to estimate the price for each combination of color and clarity of different sizes of diamonds (0.2, 0.5, 1.0, and 2.0 carats) (holding lab constant at GIA).

Graph 7 presents the ratio of the estimated average price for each diamond size, color, and clarity to the average price of a diamond of the same size and color, but with clarity held fixed at the low level, SI2. This graph enables me to compare price premiums for diamond color and clarity separately for each level of size and across all levels of size.

Graph 7

Notice that for each color, the set of lines for clarity as diamond size increases swivel out as clarity and size increase.  That is, the lines in each color grouping look like this:

The fact that each line slopes upward as you move from low clarity to high clarity means that diamond prices increase with clarity.  The fact that the lines for larger diamonds are steeper than those for smaller diamonds means that the price premium for clarity (that is, the increase in price as clarity improves) is larger for larger diamonds. In particular, the price for 0.2 and 0.4 carat diamonds increases by about 100 to 150% as you move from lowest clarity (SI2) to highest clarity (IF).  However, the price for 1.0 carat diamonds increases by about 100% to 200% as you move from lowest to highest clarity, while the price for 2.0 carat diamonds increases by about 175% to 300% as you move from lowest to highest clarity.

Now let’s see how prices change as you increase the quality of color.  Graph 8 displays the price of each diamond size, color, and clarity relative to the price of a diamond of the same size and clarity, but with one level lower of color.  For example, the color F, clarity IF point on the graph for 0.5 carat diamonds is the estimated price of a 0.5 carat color F, clarity IF diamond divided by the estimated price of a 0.5 carat color G, clarity IF diamond.  What the graph tells me is that

• There is a larger price premium for color for larger (1.0 and 2.0 carats) diamonds than smaller diamonds (0.2 and 0.5 carat).
• There is a larger price premium for color when you move up from lower levels of color than when you move up from moderate or high levels. That is, there is a bigger price increase when you move from color I to H, color H to G, or color G to F than there is for moving from color F to E or color E to D.
• There is a large price increase for moving the highest level of color and clarity, that is, for moving from color E, clarity IF to color D, clarity IF.

Graph 8

One of the big questions I wanted to answer with this analysis was:  If you want to spend a bit more money on a diamond, are you better off moving up one level of color or one level of clarity?

I assume that price increases with value, that is, you pay a higher price for more valuable diamonds.  In this case, the question becomes:  is there a larger price increase from moving up one level in color or one level in clarity?

In Graph 9 I compare the price increase associated with moving up one level in color versus moving up one level in clarity.  As in Graph 8, Graph 9 displays the price of each diamond size, color, and clarity relative to the price of a diamond of the same size and clarity, but with one level lower of color (solid circles) and the price of each diamond size, color, and clarity relative to the price of a diamond of the same size and color, but with one level lower of clarity (rings).

Graph 9

The graph says that for all but the highest color and clarity diamonds, there is a larger increase in price (value) by moving up one level in clarity than moving up one level in color.  In contrast, when the choice is between moving to the highest level of clarity (from VVS1 to IF) or to the highest level of color (from E to D), price (value) increases more with a move up in color as opposed to clarity. In other words, if you want to invest in a higher quality diamond, you’re generally better off moving up a level in clarity rather than moving up a level in color.

Diamond Prices over Time

So what happened to the structure of diamond prices between January 2009 and August 2010?

Here is the graph for January 2009 diamond prices that corresponds to Graph 7 above for August 2010 prices:

Graph 10

The graph shows that the basic relationship between prices for various levels of diamond color, clarity, and size have not changed over the past year and a half.

A comparison of average actual August 2010 versus January 2009 prices for diamonds of specific size, color and clarity can be seen in the following set of graphs.

Graph 11

Graph 12

Graph 13

Graph 14

Graphs 11 through 14 suggest that the diamond price differences between January 2009 and August 2010 are larger for the larger diamonds.  More specifically, diamond prices have generally changed by about ±10% for 0.2 – 0.9 carat diamonds, about ±20% for 1.0 – 1.9 carat diamonds, and about ±30% for 2.0 – 4.0 carat diamonds.

A direct comparison of January 2009 and August 2010 estimated prices for specific combinations of diamond size, color, and clarity is displayed in the following graph:

Graph 15

The price comparisons indicate that generally speaking, the average prices for most sizes ad color of diamonds have increased by 0 to 5%.  The exceptions are

• The prices for small, low clarity diamonds, which have increased by 15 – 20%
• The prices of 1.0 carat diamonds with moderate levels of clarity, which have increased by about 10%
• The prices of 1.0 and 2.0 carat diamonds with the highest levels of color, which have increased by about 10 - 30%.

This tells me that the calulator should work fairly well over time by inflating prices by the CPI -- that is, it should give reasonable estimates for ballpark figures.  However, it could definitely stand to be fine-tuned every so often.

So, while the final diamond price calculator could look a little sharper, it works!