Growth

 

I’ve Got The Power

I've got the power (Power, power)
(You could break my heart, you could break my heart apart)
(You could break my heart, you could break my heart apart)
I've got the power (Power, power)
(Oh-oh-oh-oh)
I've got the power (Power, power)

You don’t WANT the power! You need an exponential.

Thinking about the future, forecasting/planning, has a long history in ….history. The simplest approach to forecasting, is to assume that it is like the past. This can take the form of trend analysis, but it is also why forecasting models are VALIDATED, based on historical data, to ensure that the present is consistent with the past, and those models are “backcast” to see how well a model “predicts” the past.

The simplest form of forecasting is to assume that there is a straight line from the past to the present to the future, where a simple Annual Growth Rate, AGR is used.

Future=Present+(AGR)*Years_{(from present to future)}

This does not acknowledge the fact that the years between the present and the future should also be considered, i.e. that compounding should occur. One of the songs in Mary Poppins, “Fidelity Fiduciary Bank” https://www.youtube.com/watch?v=XxyB29bDbBA goes into the glories of compounding. When loans or investments are made, it is typical to state a Compound Annual Growth Rate, often abbreviated as a CAGR. Compounding can be decibed mathematically as a power function.

Future=Present *(100%+CAGR)^years_{from present to future}

which can be restated as a power function of the form, y=a*b^x, where y is the future, a is the present, 100% + CAGR is the base, b, and the x is the number of periods, years, which have elapsed between the present and the future. This also means that it is possible to calculate a CAGR if you have a present value, a future value, and the number of years, periods, between these values.

CAGR=(Future/Present) ^(1⁄(years _{from present to future}⁾⁾ -100%

This assumes that the rate of growth is constant. An alternative function to growth is an exponential function which only requires that the growth be continuous, NOT continuous and constant. E.g. it is quite easy to grow in the early years, but very hard to grow in the future years.

Future=Present*exp (λ*years _{from present to future} )

The base in this case, exp(x), is the function of the special irrational number, e, the Euler number. This  can also be written as a power function, y=a*e^x, where y is the future, a is the present, e is the Euler number, and x is λ times the period, years, from the present to the future. In the special case when λ=1 and CAGR=e-100%, the power and exponential functions are identical.

This equation might be more familiar as a decay function when λ is less than 1, which if the future is considered to be event 1 and the present is event 2, is equivalent to saying the elapsed years between event 1 and event 2 are negative, i.e. the future occurs after the present. A half-life is given to the term  in a radioactive decay function and is proposed as the name given to λ, even when the elapsed time is positive, growing and not decaying. The “half-life” can be simply computed as

λ=ln (Future/Present)/years _{from present to future}

Exponential growth is an outcome of maximizing entropy. Maximizing entropy is implicit in many of the tools that Cambridge Systematics, the firm at which I work, uses: e. g.

·       Frataring;

·       Trip Distribution ( i.e. A.G. Wilson’s Gravity Model);

·       Mode Choice ( e.g. logit models);

·       ODME ( a digression ODME is the trade name given by Caliper. The generic name is Maximum Entropy Matrix Estimation);

·       Stochastic assignment (i.e. use of a logit equation to select among efficient paths);

·       bridge deck and pavement deterioration;

·       capacity investment decisions;

·       reliability.

Cambridge Systematics’ motto is “Think Forward” where those two words are separated by a stylized CS. (another digression. I have always translated this as “Thinking Ahead” inside of a diamond safety warning sign). If Cambridge Systematics is thinking ahead, I hope in the future we consistently always do it as an exponential rather than as a power function. Do not use a CAGR. Compute the half life. Growth is not constant, it must be resilient. That is why Ponzi schemes do not work.