The chart above shows Xero (XRO.ASX) on a linear scale. The blue curve was drawn based on the price movement from 2014 to 2018, plotting the mean of the daily lows using polynomial regression. The curve was then extended (assuming that price would subsequently follow a parabolic curve) in an attempt to predict the mean of the lows from 2018 onwards.

It turns out that the predicted curve was reasonable, with the exception of the sharp drop occurring in March 2020. Given that the prediction was made based on price up to 2018, the outcome suggests that this might be a reasonable (although not perfect) approach.

Drawing curves on stock charts can be rather tricky. Why use the log scale? If there is a large price move, the semi log scale will give you a straight line which is much easier to deal with.

## Pitfalls of using the semi log scale on price charts

Shown above is the chart of Xero Ltd but this time on a semi-log scale. Yes, a straight line is present and appears to give a reasonable fit for the price action from 2017 onwards. However, note that we are looking at prices in retrospect. The earliest we could have drawn this trendline in 2019. Had a trendline been drawn prior to 2019 (eg a line joining the lows from 2015 to 2017), that line would have grossly underestimated subsequent lows.

Although looking at trendlines on the semi log scale is useful, price has to have moved by a reasonable amount and touched that line enough times to actually be able to draw that line to have any kind of confidence.

Another issue is that although the log scale is good for large price moves it is not effective when considering smaller price moves which can be parabolic. A recent example occurred in the silver price which exhibited a parabolic shaped move from June 2020 to August 2020. Because the numerical price move was relatively small, from a low of $17 to a high of $29, using a semi log scale does not alter the shape of the price move.

Shown above is the chart of silver priced in US dollars on the monthly scale from 1968 to 2012. In bull markets, the silver price appears to move initially forming a shallow curve that become increasingly steep, to end forming an almost vertical spike.

Focusing on the 1970s bull market in silver, the chart above shows silver USD on a semi log scale. Interestingly, the price action still have a curved shape. Although we can try and draw trend lines, early on in the price move the line will be too shallow for price moves further ahead in time. The steeper second trend line could not be drawn until later in the move. It is difficult to predict the trajectory of the price move using trend lines regardless of the type of chart used.

The semi log scale chart above shows the more recent bull market in silver in the early 2000s. The trend lines are not as steep as those observed in the 1970s bull market. Here, the price move occurred in two phases but interestingly the price action doesn’t have that curved appearance.

## Polynomial regression curves

Let’s return to polynomial curves and assess how useful these curves might be in predicting future price behavior. On the tradingview platform , it is possible to plug in the polynomial curve function for any chart. The bar duration can be adjusted so that it fits within the desired time, and the replay function may be used to go back and test your curves as if you were doing so prospectively.

There are two ways in which the yellow curve may be used. Once is to extend the curve upwards using the curve drawing tool, then move forwards in time using the play function, and observe how accurate the prediction was. Alternatively, we could allow the polynomial function to dynamically adjust the yellow curve. One drawback of this function is that the bar duration of the curve (diameter) remains constant, therefore the back and front of the curve move forwards in time, thereby losing the influence of preceding data.

Watch the video above. In this video, the yellow polynomial curve is the mean of the monthly low prices. Initially the curve does well in predicting where the lows are early on in the trend but as price moved higher, price lows moved further and further away from the curve. The curve became less and less accurate as price shot up. Why is that? What is going on here?

## Growth curves

Examples of different types of growth include:

- Linear growth
- Exponential growth
- Combinatorial growth

### Linear growth

Linear growth is where a value increases by the same amount with each unit of time. The image above shows that on the linear scale, linear growth appears as a straight line. However on the semi-log scale a curve is observed. Linear growth is described by the equation y=mx.

### Exponential growth

Exponential growth is where a value increases in proportion to its current value (e.g. doubles) with each unit of time. The above image shows that on a linear scale, exponential growth appears as a curve but on a semi-log scale, a straight line is observed. Exponential growth is described by the equation y=m^{x}.

### Combinatorial growth

Combinatorial growth is an explosive type of growth that dwarfs exponential growth. The above image shows that on a linear scale, combinatorial growth appears as a steep curve, and remains so on a semi-log scale (not shown). Combinatorial growth is described by the equation y=x!.

## Examples of combinatorial growth

Examples of combinatorial growth can be found in specific asset classes.

- Gold
- Silver
- Uranium
- Bitcoin

Check out historical charts for these asset classes and view them both on the linear and semi-log scales.

Why do such explosive price moves occur? One potential factor is fear, which can be a stronger motivator than greed. That is certainly true in the case of gold and silver. In the 1970s, there was fear of inflation. Another contributing factor might be could be price manipulation. A failure of price suppression would result in prices exploding upwards. Very strong supply and demand issues may also be a contributing factor. For a commodity that is either absolutely essential or highly desirable, explosives price moved may occur when demand greatly overwhelms supply.

Given that explosive price moves have occurred on multiple occasions in the aforementioned asset classes, there is every possibility that it could happen again. Note that something all these asset classes have in common is long periods where price is depressed, with low volatility. As price starts to take off from these depressed levels, many don’t see it coming and prices can move upwards very rapidly. As prices rise, volatility increases, shaking people out of their trades. Once the peak price is reached, the subsequent price drop is usually harsh.

## Conclusion

During a bull market in gold and silver, it is important to consider curves rather than straight lines when analyzing price charts. When looking for dips in the price action on the way up, relying on a straight trend line would overestimate how low price can go. A method of drawing and predicting combinatorial price curves would be useful. Explosive price moves are associated with high volatility and that is something that we need to be ready for as well.