456 VALUATION, DEPRECIATION AND THE RATE-BASE 
maining value if the cost of replacement at the end of its period 
of usefulness may be estimated at $2500? 
On page 465 in the 20-year life 6%, section of Table 33 at 11 years 
in the right-hand column there are found in the column with the 
bottom heading “ Present Value” the figures 68.7614 which 
represent percentage. The remaining value estimated from cost 
would be 0.687614 X 3000 = $2062.84; consequently the accrued 
depreciation, estimated from cost, would be 3oco — 2062.84 
= $037.16. The remaining value estimated from the cost to 
replace will be 0.687614 X 2500 = $1719.04. 
The actual accrued depreciation, therefore, will be 
3000 — 1719.04 = $1280.96. 
Demonstration of Error in Formule as Ordinarily Used in 
Estimating Current Depreciation 
That the ordinary depreciation formule based on expectancy 
or probable remaining life must be in error, even if it be supposed 
that the expectancy has been correctly ascertained, and that they 
should be regarded only as approximation formula, can be demon- 
strated as follows: 
Let it be assumed that the failures of a large number of articles, 
each representing an original investment of $100, which are all in 
the same life class and of the same age, can be grouped in couples 
such that for each article failing or going out of use s years before 
the end of the term of the originally predicted probable life, there 
will be another that fails s years after the end of this term. 
If the probable life as originally determined was # years, and 
some years of this life have already elapsed, the remaining actual 
years of service of the first of the two articles under consideration 
would be e — s years and of the second e -} s years, where e repre- 
sents the expectancy of all articles still in service and s is the de- 
parture in years from this expectancy by each of the two articles 
separately considered. 
If the actual failure of these two articles could be correctly 
forecast the current depreciation thereof would be estimated 
from the following formule based on equation (41).