Compound growth

Method 1

Using this method is more mathematically precise than Method 2. In finance, there is often rounding, and the resulting yields/growth rates depend on the exact number of days between periods of whichever balance (stock or flow) one prefers to measure. This formula takes this factor into account.

Where CAGR is the Compound annual growth rate (expressed as %):

\begin{aligned}
CAGR_{1} &= \left( \frac{FV}{PV} \right) ^{\frac{1}{n}} - 1
\end{aligned}

and:
FV = future value (balance)
PV = present value (balance)
n = number of years between present and future values

Method 2

Using this method is less mathematically precise than Method 1; however, this method allows for easier long time series comparisons across a sovereign with multiple currency histories (i.e. Deutsch mark progresses to EU euro), or currency defaults (Brazilian cruzeiro real progresses to Brazilian modern real). In these cases where there are breaks in the series, the break month is simply ignored (when zeroes are slashed from a country’s money supply, for example). Then all the remaining monthly growth rates can still be averaged for a wholly “systemic” performance of the sovereign.

First, analyze the time series with the lowest available frequency growth rate; typically, in our work, the lowest available frequency growth rate is monthly growth rate % from a monthly dataset. 

Where r is the Monthly growth rate (expressed as %):

\begin{aligned}
r &= \left( \frac{FV}{PV}\right) - 1
\end{aligned}

and:
FV = future value (balance)
PV = present value (balance)

Second, find the average monthly growth rate across the dataset. The least available frequency of any dataset is most ideal for this method; typically, in our work, the lowest available frequency growth rate is monthly growth rate % from a monthly dataset.

Where r̄ is the Mean of all monthly growth rates in the range (expressed as %):

\begin{aligned}
\bar{r} &= \frac{1}{n} \sum_{i=1}^{n} r_i
\end{aligned}

and:
n = total number of months in range
r = each monthly growth rate (expressed as %) in range
i=1 means start summing range with first monthly growth rate available

Yet another reminder, as noted above, when dealing with breaks/defaults in a dataset; for example, when one currency defaults and another is created (say by slashing zeroes), you simply need to ignore the reset monthly growth rate r in this case, as it will not make sense. 

Finally, the monthly mean growth rate r can be compounded to annual growth. In these examples (and typically with our datasets), as we are transforming our monthly mean growth rates into a compounded annualized figure, raising to 12th exponent is required.

Where CAGR is the Compound annual growth rate (expressed as %):

\begin{aligned}
CAGR_{2} &= \left( 1 + \bar{r} \right) ^{12} - 1
\end{aligned}

and:
r̄ = monthly mean growth rate (expressed as %)

Doubling time

Method 1

There is a simple, back-of-the-envelope way, and a more precise, mathematical way, to determine doubling time. This is the more precise and mathematical way. To determine doubling time, you must have already computed the compound annual growth rate explained above. Either method will satisfy, but as noted CAGR1 will be more mathematically precise across longer time series.

Using this first method, we need to work with the log function:

\begin{aligned}
Doubling Time_{1} &= \frac{log(2)}{log(1 + CAGR)}
\end{aligned}

The result computed will be the amount of years it will take for the PV Present Value (Initial balance) to double at the stated compound annual growth rate.

Method 2

This is the back of the envelope method. Note that for this method, it is most ‘accurate’ when the compound annual growth rate itself is closest to 10%. The further away the compound annual growth rate is from 10%, the less accurate this method will be:

\begin{aligned}
Doubling Time_{2} &= \frac{72}{CAGR \times 100}
\end{aligned}

Notice that in the denominator, the CAGR rate in our research is always expressed as a %. In other words, one would simply need to ‘drop’ the % sign before carrying out the equation. As a simple example, the PV Present Value (Initial balance) would double every ~7.2 years at a 10% compound annual growth rate.

A caution to keep your terms straight when computing these values, and don’t confuse the crucial difference between simple growth rates (r) and the compound annual growth rate (CAGR).

Compound annual growth rates and doubling times are related in a power curve. Have a look at our compounding exhibit to see how these compound growth rates vs. doubling times play out in real life.

Regression and Trendlines

Regressions are also very helpful in determining the nature and growth of things, organisms, and technology in our world. Four basic trendlines that I often discuss in my work are described below.

I am using consistent a and b variable nomenclature for the regression coefficients here, but they do have slightly different meanings, depending on the regression type.

Linear regression

The linear regression is the classic, straight-line trend analysis. Coefficient a denotes the y-intercept (the value when x = 0), and coefficient b denotes the slope, or rate of change of the trendline’s y values as you move across x.

\begin{aligned}
y &= a + bx
\end{aligned}

Logarithmic regression

The first of the basic, non-linear regressions. Coefficient a in this case is not the y-intercept, but a base coefficient. It shifts the curve vertically, and is always the y value when x = 1. Coefficient b is the slope, or rate of change coefficient. It controls the shape of the curve. Function ln(x) is the natural logarithm function. It maps each positive real number x to the unique number y so that e^y = x. The constant e is the base of natural logarithms, approximately 2.718.

\begin{aligned}
y &= a + b  \text{ln}(x)
\end{aligned}

Exponential regression

The second of the basic, non-linear regressions. Coefficient a here is identical to linear regressions, denoting the y-intercept (when x = 0). Coefficient b is the slope, or rate of change coefficient, an exponent. It controls the shape of the curve; in this case, the rate of exponential growth. The constant e is the base of natural logarithms, a special number in exponential growth. It is approximately 2.718.

\begin{aligned}
y &= ae ^{bx}
\end{aligned}

Power regression

The third of the basic, non-linear regressions. Coefficient a in this case is similar to the logarithmic regression, in that it is not precisely the y-intercept, but a base coefficient. It shifts the curve vertically, and is the y value when x = 1. It scales the curve, influencing the magnitude of y across all values of x. Coefficient b is the slope, or rate of change coefficient. It controls the shape of the curve. Similar to exponential, variable b is in the exponent; however, in this case b is a standalone factor. Coefficient b determines the rate at which y changes as x changes.

\begin{aligned}
y &= ax ^{b}
\end{aligned}

Axes

The above are basic variations of potentially complex formulas. There is one variable we are solving for each time, and that is variable y. This is called the dependent variable. This can be solving for price, market cap, volume, or any other trendline based on a set of underlying data.

Variable x is called the independent variable. As insights can be gleaned from studying trends over time, x is typically, unsurprisingly, a time factor such as days or months.

My findings and discussions of various trendlines I have observed in the markets can be found in this playlist on my YouTube channel.