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 £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 | 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.

Mining Guide

Principles of Mining

Mine Valuation

Prospective value

Price of Metals

Amortization of Capital

Valuation of Mines

Development of Mines

Shape of Mining Shafts

Subsidiary Development

Methods of Ore-Breaking

Methods of Supporting Excavation

Mechanical Equipment

Drainage-Controlling Factors

Machine Drilling

Ratio of Output to the Mine

Labor Efficiency

Working Costs

Administrative Reports

Amount of Risk in Mining Investments

Mining Engineering

Principles of Mining

Mine Valuation

Prospective value

Price of Metals

Amortization of Capital

Valuation of Mines

Development of Mines

Shape of Mining Shafts

Subsidiary Development

Methods of Ore-Breaking

Methods of Supporting Excavation

Mechanical Equipment

Drainage-Controlling Factors

Machine Drilling

Ratio of Output to the Mine

Labor Efficiency

Working Costs

Administrative Reports

Amount of Risk in Mining Investments

Mining Engineering