Why Firm Age Is Often Transformed as log(1 + Age)

Summary

The note discusses the common transformation of firm age as the logarithm of one plus age, rather than the logarithm of age alone. Adding one permits observations with age zero to be included, since the logarithm of zero is undefined. This is useful when a dataset counts age from zero and the transformation should cover newly established firms as well as older ones.

The answer also points to convenience in expressing log(1 + x) as a Taylor series around zero. It frames the shift as a way to start the age scale at zero while retaining a logarithmic transformation. The discussion is brief and does not compare model fit, predictive performance, or alternative transformations, so it offers a mathematical and data-handling rationale rather than empirical evidence that this transform is best for every finance application.

Key ideas

• Using log(1 + age) avoids the undefined value that log(age) has at zero.
• The shifted transform supports datasets that count firm age from zero.
• The answer notes that log(1 + x) has a convenient Taylor series expansion.
• The document gives no empirical comparison of this transform with alternatives.

Tags

# Why we should use log(1+Age) rather than log(Age)?


# Why we should use log(1+Age) rather than log(Age)?












Normally, when calculating the firms age in finance, I saw that people usually use

firm age= log(1+age).

Apart from the reason documented by Loderer, 2009 that we avoid the age of zero that log(0) makes no sense, is there any other reason?

## Answer by Mr. N (score 2, accepted)

https://quant.stackexchange.com/a/68545

As some people have already answered,

- It's more comfortable to expand it in Taylor Series, since:

\begin{equation} \ln(1+x) = - \displaystyle\sum \frac{(-x)^n}{n}, -1 < x \leq 1 \end{equation}

On the other hand, we would end up with

\begin{equation} \ln(x) = - \displaystyle\sum \frac{(-(x-1))^n}{n}, -1 < (x-1) \leq 1 \end{equation}

- Most of the time we would like to "begin counting" at zero. In other words, we would like to have

\begin{equation} D(\textrm{Age}) = \mathbb{R} \setminus \mathbb{R}^- \end{equation}

Which is only possible when we translate the logarithmic function to the left. Otherwise, we would end up at $\ln(0)$, as you previosuly mentioned to be avoided.

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