182 VALUATION, DEPRECIATION AND THE RATE-BASE 
throughout the probable life term of the pump for each $100 
of its cost. The agreement will provide that he gets this sum 
annually so long as the pump remains an efficient appliance and 
that the payment shall cease as soon as the pump becomes use- 
less. The present value of this pump to Jones for each $100 
of its cost, at any time of its life, will be the present value of 
an annuity of $7.82 for the expectancy or remaining life of the 
pump. This will be true if he is under no obligation to replace 
it and if scrap value be disregarded. 
When the pump is new, the probable life or expectancy is 25 
years and the present value will be roo per cent. 
When the pump is 10 years old, its expectancy may be about 
17 years and the remaining value at that time $82 for each $100 
of first cost or 82 per cent. 
When the pump is 25 years old and still in good condition, its 
expectancy may be about g years and the remaining value of 
each $100 of original cost at that time would be the present 
value of an annuity of $7.82 for g years or $53 or when compared 
with the total original cost, 53 per cent thereof. 
If the pump is defective or is put out of service by some 
accident before it has served 25 years, Jones will be a loser. If 
it serves beyond the term assumed as the limit of its usefulness 
or beyond 25 years, Jones will be the gainer. If Jones is in the 
business of supplying pumps in large numbers on these terms 
he will find, if probable life has been correctly determined, that 
his capital is yielding 6 per cent per annum. 
Further Comparison of Methods with Tables and Diagrams. 
— A further comparison of the various methods of procedure 
which have found favor when rates for the output of public 
service properties are to be fixed, is given in the following pages 
with tables and diagrams where practicable. These comparisons 
are strictly theoretical and are based in each case as noted on 
the impossible assumption that the actual life of every item 
which goes to make up the property will coincide with the prob- 
able life of that item. The comparison nevertheless shows where 
these methods, if consistently applied from the beginning, would