The Silver Singularity

We all know that a line on a log chart depicts steady growth, where price = ab^t, where t is time, and the price increase per unit of time is equal to 100(b-1) percent. 

But what about a parabolic curve on a log chart, such as the one below? (The chart depicts the $TNX/$SILVER ratio, which can be interpreted as the number of ounces of silver you get in return for lending the US Government $1000 for ten years.)

The above concave down parabola should take the form ab^-(t^2) + c = r (where r stands for the ratio the chart is measuring) (To see why, take the log of both sides to get a function of -t^2). The constants a,b,c can easily solved for by substituting points that touch the curve. The thing to keep in mind is that the first three points that touch the above curve (events that occurred a decade ago) determined its shape into infinity (the same way that connecting two points 10 years ago will determine a line to infinity). So theoretically I could've solved for the equation (and drawn the curve above) many years ago, well before it swooped down in an accelerated decline.

Another interesting thing about the curve above: the same way one duplicates and then shifts a trend line (without changing its slope) to make a trend channel, I only had to shift the above parabolic curve down (~40%) and to the left (~1.5 years) to complete a parabolic channel. In other words, I didn't have to change the important variables a and b in the formula ab^-(t^2) + c that determine the shape (and thus the rate of decline). 

Now, the astute reader might be thinking "Wait, as you say, the above curve takes the form ab^-(t^2) + c, but c is obviously negative, so after a certain amount of time, the ratio will cross zero and start to take on negative values . . . whereas, a log chart can never go negative!" Bravo, astute reader, you have anticipated the point of this post. The above parabolic curve, which caps the $TNX:$SILVER ratio, is now at 0.48 oz of silver (whereas it was at 13 oz of silver throughout most of the 1990's -- see the horizontal regression line in top left of first chart), and, as is the nature of a downwards parabola on a log chart, the "half life" of the ratio gets smaller and smaller at a faster and faster rate. As the value of the equation ab^-(t^2) + c gets very close to 0, the ratio begins to lose half its value every week, then every day, then every second, then KABOOM! And knowing the equation for the curve, we can predict the date of the singularity. It is right after New Years 2014. We can also calculate the minimum price of silver on any given date, for any value of the long bond yield:

w/ Yield =	Minimum Price of $SILVER on:
	Apr-13	Jul-13	Oct-13	Jan-14
0.05%	1.44	2.23	4.62	1212.83
0.50%	14.39	22.33	46.17	12128.35
1.00%	28.77	44.66	92.34	24256.7
1.50%	43.16	67.01	138.52	36385.05
2.00%	57.55	89.33	184.69	48513.39
2.50%	71.93	111.66	230.86	60641.74

Yup, if we assume that the above parabolic curve will continue to function as resistance forever (and how big an assumption is that, really?), then even if 10-year yields are virtually zero (0.05%), silver would still be at least $1212.83. 

I also fit a less concave "last support" curve that touches all 3 obvious "bottoms" over the past 12 years. It crosses the upper curve . . . right around January 2014 (see dotted vertical line). 

Looks like the Mayans were off by a year.