Workbook for Volume 1 – Part IV - Section #12: Calculating Projections” with the “Tools” the 19th Century (3 of 4)
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For returning readers and subscribers: This post introduces a Revised Version for Volume 1 – Part IV - Section #12: Calculating Projections” with the “Tools” the 19th Century (3 of 4)
Summary:
Volume 1 – Part IV - Section #12: Calculating Projections” with the “Tools” the 19th Century (3 of 4)- Sections #10 through #13 present a selection of “Calculated Projections” with the tools of 19th Century. These “Calculated Projections” show the foundational presence of alternatives to “Expected Value”, including “Time Average”. Step 1, in Section #10, converted prices into rates of return. Step 2, also in Section #10 explained the difference between two methods for the averaging of rates of return. Step 3, in Section #11, showed the differences between each method’s form of calculation. Step 4, also in Section #11, made the link between rates of returns and multipliers. The fifth Step, in this Section #12, makes the connection between interest rates expressed as annual percentage rates and compounding multipliers. Continuous compounding with the exponential function leads to the following question: Can we recover the annual percentage rate from the continuous effective annual rate? Fortunately, the natural logarithm function recovers the annual percentage rate from the continuously compounding multiplier because: (i) The natural logarithm function works as the inverse of the exponential function, and (ii) The logarithm linearizes the non-linear exponential function to recover the annual percentage rate. This means that financial models based on the combination of the exponential function & the logarithm function can use simple, single-period rates of return as input, convert them into multipliers, calculate growth rates, and use the calculus of the mathematics of change to recover the simple rates of return in the end. This property models the necessary internal consistency to makes an explicit connection between periodic “Expected Returns”, and the “Growth Rate” of a trajectory of prices. This leads to a final question that did not have a generalized answer until the beginning of the 21st. Century: What other functions, besides the Exponential Function and the Logarithm Function, can build other internally consistent models of growth?
Developing…
”CTRI by Francois Gadenne” writes a business book in three volumes, published serially on Substack for public peer-review. The book connects the dots of life-enhancing practices for the next generation, free of controlling algorithms, based on the lifetime experience of a retirement age entrepreneur, & continuously updated with insights from reading Wealth, Health, & Statistics research papers on behalf of large companies as the co-founder of CTRI.