Introduction

Extending algebraic developments from Ednaldo Silva, I formalize a truncation of the perpetual‑inventory method (PIM) equation with explicit and tractable error bounds. This treatment of finite‑sample PIM truncation does not appear in standard asset‑valuation cookbooks.

Framework

Consider the case where:

g = \frac{X_t}{X_{t-1}} - 1

is the growth rate of CAPX for tangible assets accumulation, R&D, or XAD (advertising or marketing expense) for intangible assets accumulation. From the PIM identity:

K_t - K_{t-1} = X_t - \delta K_{t-1}

where δ is the depreciation or amortization rate depending on the type of asset. The infinite formula is obtained:

K_t = \sum_{j=0}^{\infty} \beta^j X_{t-j}

where:

\beta = \frac{1-\delta}{1 + g}

This infinite sum converges to:

K_t = \frac{1+g}{\delta+g} X_t

Approximation and Error Bounds

Consider now the partial sum:

K_t^N = \sum_{j=0}^{N} \beta^j X_{t-j}

This equals:

K_t^N = \frac{1-\beta^{N+1}}{1-\beta} X_t

The error bound then is given by in percentage terms:

\epsilon_t^N = \frac{K_t - K_t^{N}}{K_t} = \beta^{N+1}

From this follows the required depreciation rate δ needed to obtain an error bound for a partial sum up to N lags, such that:

\delta = 1 - (1+g) {\epsilon_t^N}^\frac{1}{N+1}

Example 1

Suppose g = 5%, and error is 2%, while sum goes for N = 10 lags. From this the depreciation (or amortization) rate is obtained:

\delta = 1 - (1+0.05) 0.02^{\frac{1}{11}} = 26.4\%

Example 2

Equivalently, given a depreciation rate δ, growth rate g and error ε, the number of lags required to attain that error bound is obtained:

N = \frac{\log{\epsilon}}{\log{\beta}} - 1

Suppose g = 5% and δ = 26.4% with error 2%, the number of lags is calculated:

N = \frac{\log{0.02}}{\log{\frac{1-0.264}{1+0.05}}} - 1 = 11.01 - 1 = 10.1

or N about 10 years.

References:

1. “Hard to Value Intangibles Sans Mystère”, by Ednaldo Silva, link: https://edgarstat.com/blog/hard-to-value-intangibles-sans-mystere/