120 VALUATION, DEPRECIATION AND THE RATE-BASE
Remaining Value and the Replacement Requirement according
to the Law of Probability. — Table ¢ has been prepared to show
for numerous articles, all of which have a probable life term of
10 years, the probable annual replacement requirement on the
assumption that failures occur according to the law of probabili-
ties, all articles going out of use within 20 years. There is also
shown in the table the remaining value of these articles if esti-
mated by the Equal Annual Payment Method, and also the
expectancy of a single article with a ro-year probable life and its
remaining value if computed by the Straight Line Method and
by the Equal Annual Payment Method.
Computation of Annuities to Meet Actual Replacement Re-
quirement. — The computation of the annuities which would re-
place each lot of annually failing articles, if the same be assumed
to fail on any hypothesis similar to those already suggested, can
readily be made and will prove instructive. It will be found
that in every case the sum of all such annuities will exceed the
annuity computed in the ordinary way from the average or prob-
able life. If the computation be then extended to cover all
articles remaining in service from year to year and to include
also the new articles which have been added to replace the fail-
ures, it will be found that in the early years the sum of the
annuities is larger than the annuity computed by the use of
probable life, in the ordinary way, that after a period in excess
of the probable life term, the sum of the annuities will be a
minimum and somewhat less than that computed in the ordi-
nary way, and that thereafter it will increase again to about the
amount computed by the Straight Line Method.
It will be unnecessary to introduce a complete calculation to
illustrate this point and only the results for articles with a prob-
able life of To years will be briefly referred to.
Let it be supposed that of 10,000 articles, which all have a
probable life of 10 years, 100 fail at the end of the first year;
200 at the end of the second, 300 at the end of the third and so
on to Toco at the end of the tenth, goo at the end of the elev-
