Table II is practically a compound discount table. That is, by it can be determined the present value of a fixed sum payable at the end of a given term of years, interest being discounted at various given rates. Its use may be illustrated by continuing the example preceding.

TABLE II.

Present Value of $1, or £1, payable in — Years, Interest taken at —%.

Years 4% 5% 6% 7%
1 .961 .952 .943 .934
2 .924 .907 .890 .873
3 .889 .864 .840 .816
4 .854 .823 .792 .763
5 .821 .783 .747 .713
6 .790 .746 .705 .666
7 .760 .711 .665 .623
8 .731 .677 .627 .582
9 .702 .645 .592 .544
10 .675 .614 .558 .508
11 .649 .585 .527 .475
12 .625 .557 .497 .444
13 .600 .530 .469 .415
14 .577 .505 .442 .388
15 .555 .481 .417 .362
16 .534 .458 .394 .339
17 .513 .436 .371 .316
18 .494 .415 .350 .296
19 .475 .396 .330 .276
20 .456 .377 .311 .258
21 .439 .359 .294 .241
22 .422 .342 .277 .266
23 .406 .325 .262 .211
24 .390 .310 .247 .197
25 .375 .295 .233 .184
26 .361 .281 .220 .172
27 .347 .268 .207 .161
28 .333 .255 .196 .150
29 .321 .243 .184 .140
30 .308 .231 .174 .131
31 .296 .220 .164 .123
32 .285 .210 .155 .115
33 .274 .200 .146 .107
34 .263 .190 .138 .100
35 .253 .181 .130 .094
36 .244 .172 .123 .087
37 .234 .164 .116 .082
38 .225 .156 .109 .076
39 .216 .149 .103 .071
40 .208 .142 .097 .067
Condensed from Inwood's Tables.

If such a mine is not equipped, and it is assumed that $200,000 are required to equip the mine, and that two years are required for this equipment, the value of the ore in sight is still less, because of the further loss of interest in delay and the cost of equipment. In this case the present value of $1,304,000 in two years, interest at 7%, the factor is .87 X 1,304,000 = $1,134,480. From this comes off the cost of equipment, or $200,000, leaving $934,480 as the present value of the profit in sight. A further refinement could be added by calculating the interest chargeable against the $200,000 equipment cost up to the time of production.

TABLE III.

Annual Rate of Dividend. Number of years of life required to yield—% interest, and in addition to furnish annual instalments which, if reinvested at 4% will return the original investment at the end of the period.
% 5% 6% 7% 8% 9% 10%
6 41.0
7 28.0 41.0
8 21.6 28.0 41.0
9 17.7 21.6 28.0 41.0
10 15.0 17.7 21.6 28.0 41.0
11 13.0 15.0 17.7 21.6 28.0 41.0
12 11.5 13.0 15.0 17.7 21.6 28.0
13 10.3 11.5 13.0 15.0 17.7 21.6
14 9.4 10.3 11.5 13.0 15.0 17.7
15 8.6 9.4 10.3 11.5 13.0 15.0
16 7.9 8.6 9.4 10.3 11.5 13.0
17 7.3 7.9 8.6 9.4 10.3 11.5
18 6.8 7.3 7.9 8.6 9.4 10.3
19 6.4 6.8 7.3 7.9 8.6 9.4
20 6.0 6.4 6.8 7.3 7.9 8.6
21 5.7 6.0 6.4 6.8 7.3 7.9
22 5.4 5.7 6.0 6.4 6.8 7.3
23 5.1 5.4 5.7 6.0 6.4 6.8
24 4.9 5.1 5.4 5.7 6.0 6.4
25 4.7 4.9 5.1 5.4 5.7 6.0
26 4.5 4.7 4.9 5.1 5.4 5.7
27 4.3 4.5 4.7 4.9 5.1 5.4
28 4.1 4.3 4.5 4.7 4.9 5.1
29 3.9 4.1 4.3 4.5 4.7 4.9
30 3.8 3.9 4.1 4.3 4.5 4.7

Table III. This table is calculated by inversion of the factors in Table I, and is the most useful of all such tables, as it is a direct calculation of the number of years that a given rate of income on the investment must continue in order to amortize the capital (the annual sinking fund being placed at compound interest at 4%) and to repay various rates of interest on the investment. The application of this method in testing the value of dividend-paying shares is very helpful, especially in weighing the risks involved in the portion of the purchase or investment unsecured by the profit in sight. Given the annual percentage income on the investment from the dividends of the mine (or on a non-producing mine assuming a given rate of production and profit from the factors exposed), by reference to the table the number of years can be seen in which this percentage must continue in order to amortize the investment and pay various rates of interest on it. As said before, the ore in sight at a given rate of exhaustion can be reduced to terms of life in sight. This certain period deducted from the total term of years required gives the life which must be provided by further discovery of ore, and this can be reduced to tons or feet of extension of given ore-bodies and a tangible position arrived at. The test can be applied in this manner to the various prices which must be realized from the base metal in sight to warrant the price.

Taking the last example and assuming that the mine is equipped, and that the price is $2,000,000, the yearly return on the price is 10%. If it is desired besides amortizing or redeeming the capital to secure a return of 7% on the investment, it will be seen by reference to the table that there will be required a life of 21.6 years. As the life visible in the ore in sight is ten years, then the extensions in depth must produce ore for 11.6 years longer—1,160,000 tons. If the ore-body is 1,000 feet long and 13 feet wide, it will furnish of gold ore 1,000 tons per foot of depth; hence the ore-body must extend 1,160 feet deeper to justify the price. Mines are seldom so simple a proposition as this example. There are usually probabilities of other ore; and in the case of base metal, then variability of price and other elements must be counted. However, once the extension in depth which is necessary is determined for various assumptions of metal value, there is something tangible to consider and to weigh with the five geological weights set out in Chapter III.

The example given can be expanded to indicate not only the importance of interest and redemption in the long extension in depth required, but a matter discussed from another point of view under "Ratio of Output." If the plant on this mine were doubled and the earnings increased to 20% ($400,000 per annum) (disregarding the reduction in working expenses that must follow expansion of equipment), it will be found that the life required to repay the purchase money,—$2,000,000,—and 7% interest upon it, is about 6.8 years.