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Diamonds prices dont rise exponentially with size

Kran

Rough_Rock
Joined
Nov 28, 2017
Messages
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I am quite new to the forums and to diamonds in general. I am quite good at math, however :razz:

Why is it that everybody seems to claim that "diamond prices rise exponentially with size"? I see this claim repeated everywhere, from beyond4cs to Whiteflash. I am sure that what mean is only that a diamond twice as big will be much more than twice the price, all else being equal. However, that is goes up higher than linear does mean that it goes up exponentially.

If diamond prices raised exponentially with price, then we would have the following:

1ct diamond: Price 5.000$
2ct diamond: Price 20.000$ (4 times more expensive)
3ct diamond: Price 80.000$ (4 times more expensive than 2ct)
4ct diamond: Price 320.000$ (4 times more expensive then 3ct)
5ct diamond: Price 1.280.000$ (4 times more expensive than 4ct).

We know that 4ct and 5ct diamonds are expensive... but not that much. Instead, what I see is a very clear quadratic increase in prices!

1ct diamond: Price 5.000$ (5.000$ per carat)
2ct diamond: Price 20.000$ (10.000$ per carat)
3ct diamond: Price 45.000$ (15.000$ per carat)
4ct diamond: Price 80.000$ (20.000$ per carat)
5ct diamond: Price 125.000$ (25.000$ per carat)

I see that real diamond prices follow this curve almost perfectly. So a diamond two times bigger is 4x more expensive; a diamond three times bigger is 9x more expensive, and so on. A diamond X times bigger has a X times higher price-per-carat.

That works for small changes as well. A diamond that is 1% bigger seems to have an average of (1.01)^2 ~ 1.02 which means it is 2% more expensive.

So I think we can be more precise and start informing buyers that diamond prices rise quadratically with size!

I believe this is a very helpful way of thinking, as it allows a very useful statistic, in my opinion. Instead of talking about price-per-carat, we should start talking about price-per-squared-carat! Just calculate price / (carat * carat).

So if you are paying a higher price-per-squared-carat than you would for a similar stone in another size, you might be paying a premium. At the same time, looking for low price-per-squared-carat can help finding diamonds in a good pricing.
 
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Exponentially is not meant to be a precise term in that context. I assure you that you are the first person here to have interpreted it that literally! You can generalize even about quadratic, but that is absolutely not necessarily true, either. A lot depends on the specs and rarity of the particular stone.
 
... Why is it that everybody seems to claim that "diamond prices rise exponentially with size"? ...

Because most people are not very fluent in math.
I'm certainly not.
Next, most don't care.
It's been a couple generations since society expected everyone to have even basic knowledge of a range of academic subjects.
Now the mantra seems to be, learn the minimum to make money.

There's no entrance exam to join a forum so when you are unusually well-informed about a subject posts on that subject can drive you crazy.

I'm glad you posted this.
It's interesting.
If I've made this mistake I'll stop now.
(I'm going to say diamond prices rise logarithmically with size. :lol: )

Of course this thread will sink, and Pricescope and the world will not go study math.
 
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What Kenny said. People aren't using the word 'exponentially' correctly. To be fair, it's sort of true, just the exponents aren't integers and it's not consistent across the scale. A 2 carat stone IS exponentially more expensive than an otherwise similar 1 carat stone and a 3 carater is exponentially more expensive than that one, it's just a different exponent. It's a useless statement, rather like the way people tend to use the word 'fraction'.
 
I concur with the above, but your math is very interesting to see. I guess most people just don't do that math.
 
To be fair, it's sort of true, just the exponents aren't integers and it's not consistent across the scale.

Actually, it's not even sort of true, if we're going by the mathematical definition of the word "exponential"...

If P is price and C is carat weight, then an exponential dependence implies

P = a (b^C)

for some values of the constants a & b (where the symbol ^ denotes exponentiation, i.e., raising b to the power C).

In contrast, when we have a relationship of the following form, the dependence is a polynomial one:

P = a (C^b)

For example, a polynomial relationship in which b=2 is a quadratic dependence:

P = a (C^2)

The bottom line is that a polynomial relationship is fundamentally different from an exponential relationship. In a polynomial equation, one cannot find any value of the exponent b (either integer or non-integer) that will turn the polynomial dependence into an exponential dependence.
 
Actually, it's not even sort of true, if we're going by the mathematical definition of the word "exponential"...

If P is price and C is carat weight, then an exponential dependence implies

P = a (b^C)

for some values of the constants a & b (where the symbol ^ denotes exponentiation, i.e., raising b to the power C).

In contrast, when we have a relationship of the following form, the dependence is a polynomial one:

P = a (C^b)

For example, a polynomial relationship in which b=2 is a quadratic dependence:

P = a (C^2)

The bottom line isthat a polynomial relationship is fundamentally different from an exponential relationship. In a polynomial equation, one cannot find any value of the exponent b (either integer or non-integer) that will turn the polynomial dependence into an exponential dependence.


:lol:

That’s Greek to me!
 
An easy way to check for exponential and polynomial dependencies in data is to use logarithmic transformations.

Let's define a log-transformed price: Y = log(P), as well as a log-transformed carat weight: X = log(C).

If the relationship between price and carat weight is exponential (see previous post for definition), then the following holds:

Y = A C + B

This is the equation of a linear relationship, with slope A=log(b) and intercept B=log(a).

Likewise, if the relationship between price and carat weight is polynomial (see previous post for definition), then the following holds:

Y = b X + B

This is also a linear relationship, but note that the independent variable is now X instead of C. Another interesting result from this transformation is that the slope of the line (b) equals the exponent in the polynomial relationship. Thus, if there is a quadratic relationship between price and carat weight, then the slope of the line should be 2.

Note that in the first equation, we used the log-transformed price (Y) together with the actual carat weight (C), not the log-transformed weight (X). Plotting data in this manner is therefore referred to as a "semi-log" plot. Conversely, the second equation uses logarithmic transformations of both price (Y) and carat weigh (X). This type of plot is called a "log-log" plot.

We can now conclude that if the relationship appears to be linear on a semi-log plot, then the dependence is exponential. If the relationship appears to be linear on a log-log plot, then the dependence is polynomial. If, on a log-log plot, we get a linear relationship with a slope equal to 2 (i.e., Y rises by 2 units for every unit increase in the runner X), then the dependence between price and carat weight is quadratic.
 
Courtesy of @jyliu86, I happen to have on hand some old (2014) price vs carat data for MRBs.

Here I have graphed the data as a semi-log plot:
prices_semi-log.png

For comparison, here are the same data graphed as a log-log plot:
prices_log-log.png

So -- you be the judge: In which graph does the relationship look more linear?

As I explained in my previous post, if you see a linear relationship in the top plot (semi-log), then the dependence is exponential. On the other hand, if you see a linear relationship in the bottom plot (log-log), then the dependence is polynomial.
 
That’s Greek to me!

Let's try some numbers, then! If you'll indulge me, we'll try to estimate how a series of diamonds 1ct, 2ct, 3ct, 4ct, 5ct, etc. would be priced, all other factors being equivalent.

If the price increases quadratically with carat weight (as proposed by OP), we would get:

1ct: $2,000 = 2000 x 1ct x 1ct
2ct: $8,000 = 2000 x 2ct x 2ct
3ct: $18,000 = 2000 x 3ct x 3ct
4ct: $32,000 = 2000 x 4ct x 4ct

Can you guess the predicted price for a 5ct diamond based on this pattern? (spoiler alert: it would be $50,000).

Now, if the diamond price increases exponentially, we would get a completely different pattern:

1ct: $2,000 = 500 x (4)
2ct: $8,000 = 500 x (4 x 4)
3ct: $32,000 = 500 x (4 x 4 x 4)
4ct: $128,000 = 500 x (4 x 4 x 4 x 4)

Notice that in this pattern, the number of times that the factor 4 is repeated equals the carat weight. With this hint, you can probably figure out what the predicted price for a 5ct diamond would be (it should be about a half-million dollars).

Clearly, with an exponential dependence, the price increase is very dramatic. This is why the word "exponential" is such a popular hyperbole when one wants to describe very drastic increases. However, I think you will agree that taken literally, price predictions using an exponential formula lead to somewhat exaggerated results.
 
1.2 is a perfectly valid exponent. For that matter, so is 1.00001.
 
1.2 is a perfectly valid exponent. For that matter, so is 1.00001.

The confusion comes from the fact that polynomial relationship does feature an exponent (b):

P = a (C^b)

The fact that there is an exponent in this equation (whether b=2, b=1.2, b=1.00001, etc.) does not make this an exponential relationship (at least not in the way that this word is used by mathematicians, scientists, and engineers).

To have an exponential dependence on C, the quantity C itself must be in the exponent:

P = a (b^C)

This results in a much more rapid increase than any polynomial equation. In fact, for any values of the constants a & b, the exponential relationship will always produce a larger value than does the polynomial relationship, as C becomes larger.
 
I am enjoying this thread :geek:
 
I am enjoying this thread :geek:
+1

I like that one can rely on @drk14 to bring some cool science to the party :)

Those graphs are interesting, though - one can clearly see how cutters are cutting to hit those psychological (and therefore more expensive) whole- and half-carat weights!
 
I seem to think that what Whiteflash was meaning when they wrote the word was that the diamond increased in size all over its surface, e.g. depth, width on the pavilion and not just face up size of width. Did they not have pictures of diamond profiles showing this alongside each other.

Also what about the other C's, an E, VVS1, XXX, 1 carat is more expensive than a 2 carat H, VS1, xxx
 
So -- you be the judge: In which graph does the relationship look more linear?

No one wants to play? Or perhaps you're keeping your conclusions to yourselves (to better game the market!). :mrgreen:

I'll offer my own thoughts on the data...

Based on the semi-log plot (top plot in Post #10), the initial trend (until ~1.25ct) looks somewhat linear, so perhaps one can argue that the dependence of price on carat weight is in fact exponential for diamonds up to 1.25ct or so. But clearly, an exponential relationship does not adequately describe the trend over a larger range of carat weights.

The log-log plot (bottom plot in Post #10), to me, looks approximately linear across almost the entire data set (with the possible exception of a slight flattening of the trend for <0.75ct). This implies that the dependence of price on carat weight can be described by a polynomial relationship. But what is the exponent? Is the dependence quadratic (exponent b=2), as suggested by @Kran ? Let's do a linear regression to the log-transformed data and find out!

prices_log-log_fit.png

Here, I have used Excel to fit a linear model (Y = bX + B) to the log-transformed data. I included only data in the range (0.75ct-2.75ct), because the original data set is confounded by truncation above and below this range (very expensive and very inexpensive diamonds were omitted).

As shown in the figure above, the slope of the best-fit line (which equals the polynomial exponent) is

b = 1.72​

Thus, as a very rough approximation, one could consider 1.72 to be approximately equal to 2 (the difference is only 15%), and call the relationship quadratic. However, more exactly, the polynomial relationship is given by the following equation:

P = ($5370) x (C^1.72)​

As was noted by @denverappraiser , non-integer exponents are perfectly valid. However, if your calculator doesn't handle fractional exponents, you could approximate 1.72 as 1.75, and get a pretty close result using the following formula:

C^1.75 = sqrt[sqrt(C x C x C x C x C x C x C)]​

That is, multiply C by itself seven times, then take the square root, and finally take the square root one more time.

One final thing to note is that the above polynomial only gives you an estimate of the mean price (for those who care -- the geometric mean) for a given carat weight, and there is obviously a lot of variance in the data. Nonetheless, following the suggestion made by OP, if you want to compare the "value" (V) of two diamonds that have different weight, you could compute the quantity

V = P/(C^1.72)​

to allow you to determine which diamond is more discounted.
 
My head is spinning ... how did I ever make through college calculus?

@denverappraiser

How about some real numbers. Currently what are the prices for D-FL and H-SI1 for
1 ct
2 ct
3 ct
4 ct
5ct ?
 
How about some real numbers. Currently what are the prices for D-FL and H-SI1 for

Just going by the PS Diamond Search tool, and looking at H-SI1 MRBs with no other filters applied, there is a range of $2906-$7633 for 1.00-ct diamonds, $11444-$20498 for 2.00-ct diamonds, and $31134-$34974 for 3.00-ct diamonds. There was only one diamond in the data base that hit exactly 4.00 ct, and none at exactly 5.00 ct.

Because of the spread in prices, and because of the inherent logarithmic distribution of price points at a given carat weight, the geometric mean (not the conventional, arithmetic mean) should be used if one wants a single representative number. Without downloading the price data for each diamond, I will estimate the geometric mean of each sample by taking the geometric mean of the minimum and maximum: thus, for 1.00-ct diamonds, the mean price can be estimated by taking sqrt(2906x7633) = 4709. In this manner, I get the following trend:

1.00 ct: $4,709
2.00 ct: $15,316
3.00 ct: $32,998

Now, according to the little formula I came up with in my previous post, I would predict that the "value" metric (V) should be very similar for the mean price at each carat weight. Let's try it:

1.00 ct: V = $4,709/(1ct ^ 1.72) = 4709
2.00 ct: V = $15,316/(2ct ^ 1.72) = 4649
3.00 ct: V = $32,998/(3ct ^ 1.72) = 4987

If I may say so myself, those numbers are pretty darn close to each other! Put another way, if I only knew the mean price of 1-ct diamonds ($4709), I could predict the mean price for the larger sizes within 2%-6%:

1.00 ct: P = 4709 x (1ct ^ 1.72) = $4,709
2.00 ct: P = 4709 x (2ct ^ 1.72) = $15,513
3.00 ct: P = 4709 x (3ct ^ 1.72) = $31,159

Based on this model, I'll even go out on a limb and predict that 4.00-ct and 5.00-ct diamonds (H-SI1 MRB) should have mean price points of around $51k and $75k, respectively.
 
I suppose in beginning the "Size" was diameter , not mass.

so price is 6th degree from linear dimension that for most people looks as exponential .
 
this is interesting. I'll have to show this to my daughter so she could explain it to me lol! In graduate school one of my first projects was looking up old articles on forgetting, collecting data points, and then seeing if there was a way to create a function that predicted forgetting over time. All I remember is if we transformed one variable to a log, that it created a linear function, but that there was a non-log function that also fit... That's all my brain remembers. As math is not my strength I moved to a different project as soon as I could.
 
I thought I was doing pretty good at maths until this thread :D lol
 
@OoohShiny . . . you and me both....I was ok for about 3 posts . . . but once the charts started popping up i got lost.
 
Very interesting, @drk14 can you split the data by colour, donate different constants to each colour and incorporate it into the formula, although the price difference between colours is not linear and relationship of cost per price difference may change based on clarity. It probably wouldn't be as accurate. Unless you can form a trend between colour and one clarity, maybe vs2/ si1, then find the trend across clarity, for a certain colour and see if they can be made into one formula to estimate cost on carat, colour and clarity.
 
In graduate school one of my first projects was looking up old articles on forgetting, collecting data points, and then seeing if there was a way to create a function that predicted forgetting over time. All I remember is if we transformed one variable to a log, that it created a linear function

@partgypsy , yes, this is the same idea.
 
@OoohShiny . . . you and me both....I was ok for about 3 posts . . . but once the charts started popping up i got lost.

@metall Sorry, I was hoping the graphs would help not hurt! I prefer to think visually, that's why I included the graphical representations of the data set.

For anybody who's lost (@OoohShiny ? @cflutist ? @AprilBaby ), if there is any specific point or concept that you have a question about, I can try to clarify. I may have gone overboard with the multiple verbose posts, but data analysis is one of my passions, so I couldn't help myself lol!

If you take away only one* message from this thread, let it be that exponential functions and polynomial functions are fundamentally different from each other (even though exponents are used in both), and that the common-parlance usage of "exponentially" does not necessary match the mathematical definition of "exponentially."

*OK, so two messages.
 
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