Amortization of Capital

REDEMPTION OR AMORTIZATION OF CAPITAL AND INTEREST.

It is desirable to state in some detail the theory of amortization
before consideration of its application in mine valuation.

As every mine has a limited life, the capital invested in it must
be redeemed during the life of the mine. It is not sufficient that
there be a bare profit over working costs. In this particular,
mines differ wholly from many other types of investment, such as
railways. In the latter, if proper appropriation is made for
maintenance, the total income to the investor can be considered as
interest or profit; but in mines, a portion of the annual income
must be considered as a return of capital. Therefore, before the
yield on a mine investment can be determined, a portion of the
annual earnings must be set aside in such a manner that when the
mine is exhausted the original investment will have been restored.
If we consider the date due for the return of the capital as the time
when the mine is exhausted, we may consider the annual instalments
as payments before the due date, and they can be put out at compound
interest until the time for restoration arrives. If they be invested
in safe securities at the usual rate of about 4%, the addition of
this amount of compound interest will assist in the repayment of
the capital at the due date, so that the annual contributions to
a sinking fund need not themselves aggregate the total capital to
be restored, but may be smaller by the deficiency which will be
made up by their interest earnings. Such a system of redemption
of capital is called "Amortization."

Obviously it is not sufficient for the mine investor that his capital
shall have been restored, but there is required an excess earning
over and above the necessities of this annual funding of capital.
What rate of excess return the mine must yield is a matter of the
risks in the venture and the demands of the investor. Mining business
is one where 7% above provision for capital return is an absolute
minimum demanded by the risks inherent in mines, even where the
profit in sight gives warranty to the return of capital. Where
the profit in sight (which is the only real guarantee in mine
investment) is below the price of the investment, the annual return
should increase in proportion. There are thus two distinct directions
in which interest must be computed,--first, the internal influence
of interest in the amortization of the capital, and second, the
percentage return upon the whole investment after providing for
capital return.

There are many limitations to the introduction of such refinements
as interest calculations in mine valuation. It is a subject not
easy to discuss with finality, for not only is the term of years
unknown, but, of more importance, there are many factors of a highly
speculative order to be considered in valuing. It may be said that
a certain life is known in any case from the profit in sight, and
that in calculating this profit a deduction should be made from
the gross profit for loss of interest on it pending recovery. This
is true, but as mines are seldom dealt with on the basis of profit
in sight alone, and as the purchase price includes usually some
proportion for extension in depth, an unknown factor is introduced
which outweighs the known quantities. Therefore the application of
the culminative effect of interest accumulations is much dependent
upon the sort of mine under consideration. In most cases of uncertain
continuity in depth it introduces a mathematical refinement not
warranted by the speculative elements. For instance, in a mine
where the whole value is dependent upon extension of the deposit
beyond openings, and where an expected return of at least 50% per
annum is required to warrant the risk, such refinement would be
absurd. On the other hand, in a Witwatersrand gold mine, in gold
and tin gravels, or in massive copper mines such as Bingham and
Lake Superior, where at least some sort of life can be approximated,
it becomes a most vital element in valuation.

In general it may be said that the lower the total annual return
expected upon the capital invested, the greater does the amount
demanded for amortization become in proportion to this total income,
and therefore the greater need of its introduction in calculations.
Especially is this so where the cost of equipment is large
proportionately to the annual return. Further, it may be said that
such calculations are of decreasing use with increasing proportion of
speculative elements in the price of the mine. The risk of extension in
depth, of the price of metal, etc., may so outweigh the comparatively
minor factors here introduced as to render them useless of attention.

In the practical conduct of mines or mining companies, sinking
funds for amortization of capital are never established. In the
vast majority of mines of the class under discussion, the ultimate
duration of life is unknown, and therefore there is no basis upon
which to formulate such a definite financial policy even were it
desired. Were it possible to arrive at the annual sum to be set
aside, the stockholders of the mining type would prefer to do their
own reinvestment. The purpose of these calculations does not lie
in the application of amortization to administrative finance. It
is nevertheless one of the touchstones in the valuation of certain
mines or mining investments. That is, by a sort of inversion such
calculations can be made to serve as a means to expose the amount
of risk,--to furnish a yardstick for measuring the amount of risk
in the very speculations of extension in depth and price of metals
which attach to a mine. Given the annual income being received,
or expected, the problem can be formulated into the determination
of how many years it must be continued in order to amortize the
investment and pay a given rate of profit. A certain length of
life is evident from the ore in sight, which may be called the
life in sight. If the term of years required to redeem the capital
and pay an interest upon it is greater than the life in sight,
then this extended life must come from extension in depth, or ore
from other direction, or increased price of metals. If we then take
the volume and profit on the ore as disclosed we can calculate the
number of feet the deposit must extend in depth, or additional tonnage
that must be obtained of the same grade, or the different prices of
metal that must be secured, in order to satisfy the demanded term
of years. These demands in actual measure of ore or feet or higher
price can then be weighed against the geological and industrial
probabilities.

The following tables and examples may be of assistance in these
calculations.

Table 1. To apply this table, the amount of annual income or dividend
and the term of years it will last must be known or estimated factors.
It is then possible to determine the _present_ value of this annual
income after providing for amortization and interest on the investment
at various rates given, by multiplying the annual income by the
factor set out.

A simple illustration would be that of a mine earning a profit of
$200,000 annually, and having a total of 1,000,000 tons in sight,
yielding a profit of $2 a ton, or a total profit in sight of $2,000,000,
thus recoverable in ten years. On a basis of a 7% return on the
investment and amortization of capital (Table I), the factor is
6.52 x $200,000 = $1,304,000 as the present value of the gross
profits exposed. That is, this sum of $1,304,000, if paid for the
mine, would be repaid out of the profit in sight, together with
7% interest if the annual payments into sinking fund earn 4%.

TABLE I.

Present Value of an Annual Dividend Over -- Years at --% and Replacing
Capital by Reinvestment of an Annual Sum at 4%.

=======================================================
Years | 5% | 6% | 7% | 8% | 9% | 10%
-------|-------|-------|-------|-------|-------|-------
1 | .95 | .94 | .93 | .92 | .92 | .91
2 | 1.85 | 1.82 | 1.78 | 1.75 | 1.72 | 1.69
3 | 2.70 | 2.63 | 2.56 | 2.50 | 2.44 | 2.38
4 | 3.50 | 3.38 | 3.27 | 3.17 | 3.07 | 2.98
5 | 4.26 | 4.09 | 3.93 | 3.78 | 3.64 | 3.51
6 | 4.98 | 4.74 | 4.53 | 4.33 | 4.15 | 3.99
7 | 5.66 | 5.36 | 5.09 | 4.84 | 4.62 | 4.41
8 | 6.31 | 5.93 | 5.60 | 5.30 | 5.04 | 4.79
9 | 6.92 | 6.47 | 6.08 | 5.73 | 5.42 | 5.14
10 | 7.50 | 6.98 | 6.52 | 6.12 | 5.77 | 5.45
| | | | | |
11 | 8.05 | 7.45 | 6.94 | 6.49 | 6.09 | 5.74
12 | 8.58 | 7.90 | 7.32 | 6.82 | 6.39 | 6.00
13 | 9.08 | 8.32 | 7.68 | 7.13 | 6.66 | 6.24
14 | 9.55 | 8.72 | 8.02 | 7.42 | 6.91 | 6.46
15 | 10.00 | 9.09 | 8.34 | 7.79 | 7.14 | 6.67
16 | 10.43 | 9.45 | 8.63 | 7.95 | 7.36 | 6.86
17 | 10.85 | 9.78 | 8.91 | 8.18 | 7.56 | 7.03
18 | 11.24 | 10.10 | 9.17 | 8.40 | 7.75 | 7.19
19 | 11.61 | 10.40 | 9.42 | 8.61 | 7.93 | 7.34
20 | 11.96 | 10.68 | 9.65 | 8.80 | 8.09 | 7.49
| | | | | |
21 | 12.30 | 10.95 | 9.87 | 8.99 | 8.24 | 7.62
22 | 12.62 | 11.21 | 10.08 | 9.16 | 8.39 | 7.74
23 | 12.93 | 11.45 | 10.28 | 9.32 | 8.52 | 7.85
24 | 13.23 | 11.68 | 10.46 | 9.47 | 8.65 | 7.96
25 | 13.51 | 11.90 | 10.64 | 9.61 | 8.77 | 8.06
26 | 13.78 | 12.11 | 10.80 | 9.75 | 8.88 | 8.16
27 | 14.04 | 12.31 | 10.96 | 9.88 | 8.99 | 8.25
28 | 14.28 | 12.50 | 11.11 | 10.00 | 9.09 | 8.33
29 | 14.52 | 12.68 | 11.25 | 10.11 | 9.18 | 8.41
30 | 14.74 | 12.85 | 11.38 | 10.22 | 9.27 | 8.49
| | | | | |
31 | 14.96 | 13.01 | 11.51 | 10.32 | 9.36 | 8.56
32 | 15.16 | 13.17 | 11.63 | 10.42 | 9.44 | 8.62
33 | 15.36 | 13.31 | 11.75 | 10.51 | 9.51 | 8.69
34 | 15.55 | 13.46 | 11.86 | 10.60 | 9.59 | 8.75
35 | 15.73 | 13.59 | 11.96 | 10.67 | 9.65 | 8.80
36 | 15.90 | 13.72 | 12.06 | 10.76 | 9.72 | 8.86
37 | 16.07 | 13.84 | 12.16 | 10.84 | 9.78 | 8.91
38 | 16.22 | 13.96 | 12.25 | 10.91 | 9.84 | 8.96
39 | 16.38 | 14.07 | 12.34 | 10.98 | 9.89 | 9.00
40 | 16.52 | 14.18 | 12.42 | 11.05 | 9.95 | 9.05
=======================================================
Condensed from Inwood's Tables.

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 Ł1, 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 | Number of years of life required to yield--% interest, and in
Rate of | addition to furnish annual instalments which, if reinvested at
Dividend.| 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.

As at this increased rate of production there is in the ore in
sight a life of five years, the extension in depth must be depended
upon for 1.8 years, or only 360,000 tons,--that is, 360 feet of
extension. Similarly, the present value of the ore in sight is
$268,000 greater if the mine be given double the equipment, for
thus the idle money locked in the ore is brought into the interest
market at an earlier date. Against this increased profit must be
weighed the increased cost of equipment. The value of low grade
mines, especially, is very much a factor of the volume of output
contemplated.