Market maths: compound growth and the log scale

By Robert Vagg

“The most powerful force in the universe is compound interest” is a quote that has been attributed, perhaps unjustifiably, to Albert Einstein. Although surely something of an overstatement, it does hint at the wide occurrence and importance of compound (or exponential) growth that is evident throughout our natural world. This includes that related to the many human activities shaped by population growth, a major driver of growth in financial markets. This article looks at the importance to investors of understanding compound growth, particularly when represented numerically or in share price charts, and attempts to cover some of the mathematical basis for that understanding.

Growth of an investment

Compounding of an investment is a repetitive process by which new earnings are continually added to the total capital available so as to form a larger base on which future returns may accumulate. By this means, the reinvested earnings are able to generate their own earnings. Since public company earnings compound by this same process, so also will their associated share prices, as they grow by ever increasing increments. The meaningful interpretation of share price growth displayed in price charts therefore requires some understanding of the concept of compounding.
Examples of growth occurring at different compounding rates are shown in Table 1. This displays the effects of different compound growth rates (rates of return) on initial capital of $1,000 taken at different stages of a long-term investment.

Table 1. Growth of a $1,000 investment at different annual growth rates

The Table demonstrates the importance of maximising the compound growth rate for an investment. Viewed along the bottom line, the results of achieving an additional 5% annual growth over an investment lifetime are striking, particularly at the higher rates. They provide a compelling argument for the reinvestment of distributed earnings (i.e. dividends) by an investor, which would increase the compounding effect by approximately this amount, equivalent to compounding interest. That is, the result of capital growth of 15% p.a. plus a net 5% p.a. dividend consistently reinvested would be identical to that obtained from 20% p.a. capital growth alone. The data in the Table also demonstrate the importance of becoming an investor as early as possible so as to take advantage of compounding over an extended time period.

Charting price growth

The determination of long-term growth rates for financial data, whether they be price, earnings, dividends, etc, is crucial to informed investment decisions. This type of information generally may be obtained more readily from graphical representations of price plotted against time rather than from tables of numeric data, if available. In order to reach a decision based on the performance of two alternative investments, charts that allow ready comparison of each are almost indispensable. An informed decision requires that growth rates be readily determined from these charts.

An important alternative exists when choosing the most appropriate type of price chart. That choice, available in all charting software packages, is between the use of a linear or a log scale, and the latter is always appropriate for a value that displays compound growth. As a general guide, if it is not practical for the parameter being charted to have a negative value, then the correct choice is a log scale. This is necessary, in particular, when identifying or comparing trends in share prices and indices.

When the data contained in Table 1 are plotted on a normal linear scale (Figure 1A), very little information of value is obtained. In fact a false impression, that little or no growth in the first 25 years is followed by sudden rapid growth, is likely to result. Importantly, no quantitative determination may be made of each of the different growth rates, and of the lower ones in particular, simply from this chart. Growth at 5%, 10% or 15% rates is not even discernable when placed on this common scale. When plotted on a log scale, however, the six different growth rates may be readily compared (Figure 1B) and all data are observable.

The different growth rates now are apparent from straight-line plots whose gradients increase with each higher rate. The gradient of each line is determined solely by its growth rate. The mathematical reason for this is outlined in detail in the Appendix section later, but some discussion is included below. If this proves to be confusing, then readers may wish to skip forward to the section on Price Charts.

Price charts

Trendlines are very commonly included in charts of financial markets to quantify periods of (compound) growth. When doing so, it is important that the correct choice of a log scale be made. As demonstrated above, straight trendlines would have no theoretical basis unless they are drawn on log-scale plots. In fact, drawing a straight line on a chart with a linear scale may easily lead to misinterpretation of the price action represented. Drawn on a linear scale, a compounding trendline should not be a straight line but should show exponential curvature.

Figure 2 shows the All Ordinaries Index plotted on a linear scale (red). This provides very little information, and its shape easily might be likened to that of a hockey stick. No growth at all is apparent until the middle of the twentieth century, and an impression of sudden rapid growth with the possible formation of a massive bubble over the last few decades might be incorrectly concluded. Also shown are the same Index data plotted using a log scale (blue). The latter plot is far more informative. All Index values are observable, and trend growth and relative changes in volatility are apparent. Recent growth now is seen as being consistent with the Index’s 135-year history. A primary trendline, and its likely future direction, becomes obvious.

Similar caution is required when interpreting trends in the price history of individual shares. As an example, the 23-year price history of Wesfarmers Ltd shares is displayed in Figures 3A & 3B. If compound growth is not recognised, a trendline similar to that in Figure 3A could be drawn and likely be interpreted as representing long-term fair value for this stock. The period 2000-2007 then would be viewed as indicating considerable bubble formation that corrected with the GFC declines. A correct interpretation is permitted by Figure 3B, which shows that the company’s share price had been contained within a well-defined trend channel compounding at 17% p.a. until mid-2008, when it then plummeted well below that trend to levels from which it is yet to recover. The lack of validity of the linear trendline suggested in Figure 3A may be confirmed by realising that its extension back prior to 1986 would require the stock to have had a negative share price. The trend channel in exponential form also is shown here.

Carbon trading markets

Many market-traded securities whose prices are influenced by population growth display compound growth, of particular relevance to commodities and futures markets. A very topical example, of considerable importance to investors, is the variety of proposed carbon-trading schemes arising from the anthropologically-derived emission of carbon into the atmosphere.

Figure 4 displays carbon emissions over the last two centuries³. The linear plot (red) again gives the impression of little growth during the nineteenth century followed by particularly rapid increases in growth rate in the second half of the twentieth. From a value of ca 1,500 MtC p.a. reached in 1950 the growth rate of emissions appears to have exploded suddenly, resulting in a six-fold increase by 2008.⁴ Presented as a log plot (blue) however, a less alarming overall impression results. This shows the growth rate of emissions generally to be decreasing with time. Average growth in 1800-1910 emissions is represented by a well-defined trendline of gradient corresponding to a growth rate of 4.4% p.a. For the period 1910-2010 this growth rate averages 2.3% p.a. For the last four decades growth has averaged 1.8% p.a., consistent with global population growth⁵. Similar to share price charts, different perceptions may be determined by the provider’s choice of a linear or a log scale. The latter is the more informative.

It is intended that in a future article the basis for the well-known ‘Rule of 72’ also will be ascribed to “the most powerful force in the universe”.

Appendix: mathematical background

The following description is provided for readers wishing to look further into the mathematical basis of compound growth, and its graphical representation.

Compound price growth occurs when price increases by application of a growth multiplier at constant time intervals. For normal compound growth the value of the growth multiplier, and its inherent growth rate, remains constant throughout.

If the initial price (P₀) of a financial asset were to increase over a year by a measured effective growth rate (E), then the annual growth multiplier (G) is equal to 1+E and the new price at the end of the year (P₁) is calculated simply as P₁ = P₀ G or P₀ (1+E).  Growth for a second year would be determined by applying the same growth multiplier to the new price P₁, such that P₂ = P₁ G = P₁(1+E).  Alternatively, P₂ may be expressed as either P₀(1+E)(1+E) or as P₀(1+E)² or P₀G². Similarly, after a third year of growth the new price P₃ could be calculated as P₂(1+E) or P₀(1+E)(1+E)(1+E) or P₀(1+E)³. Thus, compound price growth after n years may be described by the general expression (a1):

(a1)    P_n   =  P₀ Gⁿ  =  P₀ (1+E)ⁿ             where     n       =   number of years

                                                                              P₀     =   initial price

                                                                              P_n     =   price after n years

                                                                              G      =   annual growth multiplier

                                                                and        E       =   effective annual growth rate.

This mathematical description (a1), employing an effective growth rate, is applicable if the prices achieved are considered only at the end of each year. In practice, price changes on financial markets, and hence growth, is essentially continuous. The relevant time interval for price change therefore approaches zero, and continuously compounding growth occurs. In such case, growth is exponential and is better described by application of a nominal growth rate, as described by the expression (a2):

(a2)        P_n   =  P₀ e^Nn     where         n   =   the growth period expressed in years                  

                                                            N   =   nominal compounding annual growth rate

                                             and         e    =   the natural growth base (2.718281828...)⁶.

The relationship between the effective annual growth rate E and the nominal rate N is given by (a3):

(a3)        E   =  e^N – 1  .

The differences between these two growth rate descriptions may be explained by example. A security priced at $10.00 and displaying annual compound growth for three years at an effective rate of 15.0% p.a. would result in a price of $15.21 { $10.00 x 1.15³ }. This is equivalent to a continuous nominal rate of 14.0% p.a. Alternatively, continuous compounding at a nominal 15.0% p.a. rate would see the price increase to $15.68 over this period, an effective annual rate of 16.2% p.a.

Compound growth rates may be determined graphically by mathematical manipulation of the formulae above. Since relative price growth remains constant throughout, the plot requires price to be displayed on a relative scale. This may be achieved simply by plotting price on a logarithmic scale against time. In logarithmic form, relationship (a1) becomes

(a4)         log₁₀P_n   = log₁₀P₀ +  n log₁₀G  = log₁₀P₀ + n log₁₀(1+E),

whereas formula (a2) becomes either

(a5)         lnP_n     =  lnP₀ + N n       

or      (a6)         log₁₀P_n   = log₁₀P₀ +  2.3026 N n  .

Each of these relationships (a4-6) has the common linear equation form of  y = ax + b   which, when plotted, generates a straight line of gradient (annual increment) a and a y-axis intercept of b.  Using (a4), as an example, a linear plot of log₁₀P_n ( = y ) against n ( = x ) has an intercept of log₁₀P₀ ( = b ) and gradient of log₁₀(1+E) ( = a ).  Taking the antilogarithm of the measured gradient gives 1+E, thereby allowing determination of the effective annual compound rate E.  A similar plot from equation (a6) would have a measurable gradient corresponding to 2.3026N, allowing easy calculation of the nominal compounding rate of return N. Either analysis may be applied in practice, with equation (a3) allowing inter-conversion between the resultant nominal and effective rates of return.

Robert Vagg is a member of the AIA. (Contact email: rsvagg@gmail.com)

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