458 VALUATION, DEPRECIATION AND THE RATE-BASE
life of both of these articles were in agreement with the group
expectancy of 40 years, $9.72 would have to be invested at 6 per
cent to produce $100 at the end of that period. The larger sum
of $17.41 is actually required for the article which fails at 30
years and only $5.43 for the article which lasts 50 years. These
figures show that the article which falls 10 years short of its
expected life has a present value $7.69 less than if it would continue
 in service 40 years and that the article which will serve 50
years exceeds the value which it would have if it were doomed to
fail at 40 years by $5.43. These figures clearly show the greater
effect of the years of service short of average life, than the effect
of the years in excess of average life upon the present value of
any group of articles all of which have the same expectancy.
Or, looking at the matter in another way, the aggregate number
 of service years which the short-lived articles are short of
normal must be equal (by definition) to the number of service
years by which the long-lived articles exceed the normal. For
every failure of an article m years before the average or normal
time of failure there will be a survival of some article m years
beyond this time. But itis obvious that the m years of deficiency
of service, being not so far in the future, represent a greater
deficiency of value than m years of excess service, farther in the
future, can make up. The aggregate number of years of deficiency
 of service, considered from the standpoint of present value,
will always outweigh a like number of years of service beyond
the normal term of life.
The actual present or remaining value of such a group of
articles will always, therefore, be smaller than the result found
by such formule as (40) provided that the average probable life
thereof has been correctly ascertained. Similarly the current
depreciation by formula will be below the correct amount. The
remedy for this defect, when sinking fund formule are to be used,
will obviously lie in making the forecast of probable life conservative
 — underestimating rather than overestimating the
same.