March 14, 1913. _________________________________________________________________________________________ 542 THE COLLIERY GUARDIAN. THE VALUATION OF MINERAL PROPERTIES. What is Meant by a 10 Per Cent. Per Annum Security ?* By Daniel R. Stephen, M.A. For the purpose of the duties imposed by the Finance Act of 1909-10 the Commissioners of Inland Revenue required an estimate of the fair market value of the mineral portion of an estate in the South Yorkshire coalfield. An item occurred of anticipated rents on coal which it was expected would commence to be got in 36 years, and would be exhausted in 120 years from the present day. The rent was estimated at £3,750 per annum, lasting for 84 years, but deferred 36 years. In ascertaining the present value of this rent it was agreed to make the calculation as for an annuity at a security of 10 per cent, per annum. The calculation of the present value on behalf of the owner was made by the use of Inwood’s tables, and showed the present value of the annuity = £37,488 X 0-0323 = £1,211. The Superintending Valuer of Minerals for the district considered that the method of calculation and the present value of the annuity as thus calculated and amounting to £1,211, were wrong. He contended that in place of them the value of the annuity of £3,750 for 84 years should be calculated at 10 per cent, com- pound interest, and in addition a redemption fund at 3 per cent, per annum compound interest be provided for (Hoskold’s tables). This gave the present value of the annuity as £36,501 x 0 3450 == £12,337. The difference between the first result of £1,211 and the second result of £12,337 is so large and important that an investigation has been undertaken of the fore- going and other methods of calculation with the object of determining the correct one. It should be mentioned that the method No. 5 is stated to be approved by the Estate Duty Office. The present value, as obtained by this method, is £12,101, or ten times as large as by Inwood’s method, and the question at issue is therefore not merely one that will occur with isolated Govern- ment valuers in only a few cases. The additional Estate Duties put upon owners of unworked minerals will be charged not merely as fixed by authority of an Act of Parliament, but ten times the Parliamentary amount will be required to be paid owing to the methods of calculation adopted by the officers administering the Act. The methods and their results are summarised as follow ;—The value of £3,750 per annum, continuing for 84 years, and the present value of the same deferred for 36 years, calculated on the basis of a risk measured by a 10 per cent, per annum security, is:— Value with- Present out value on deferment, deferment. £ £ By method (1) Inwood ........... 37,488 ... 1.211 (2) South Yorkshire.... 36,501 ... 12,337 (3) Hoskoid and Gray ... 36 501 ... 1,179 (4) Birmingham (1906).. 37,488 ... 5,116 (5) Birmingham (Prac- tical 1912)....... 12 101 (6) Birmingham (Theo- retical 1912) .... (Approx.) 12,101 The problem, as stated above, is a definite and mathematical one, and is not’ a question of opinion or judgment. The large differences in the figures for the present value show that it is an important practical question seriously affecting all owners of unworked and unleased minerals. The difference between the methods occurs solely in consequence of different interpretations being put upon the word “ risk ” or “ security.” It is therefore needful to examine what is implied by the word“ risk,” and also* what is implied by “ a 10 percent, per annum security.” Here again there arises no question of opinion or judgment, but the answer can and should be found in a strictly mathematical way. The chapters that follow are devoted to an examina- tion of the six methods with a view to finding this answer. The difficulties of the problem can best be dealt with by considering “the probability of success ” of an investment, as is done in mathematics, rather than the more usual method of considering “ the security ” or “ the risk of failure.” It becomes necessary on this account to introduce a number of new terms and to define their meaning. When a large number of investments are considered, each of which is classed as a 10 per cent, security, an unknown number, a, will succeed (z.e., interest and capital will be repaid), and another unknown number, fe, will fail. Then, considering only the fact that a given investment is a 10 per cent, security, the pro- bability of its success is measured by the ratio of * Abstract of written contribution to the discussion on Mr. T. A. O’Donahue’s paper on the “ Valuation of Mineral Properties” (South Staffordshire and Warwickshire Institute of Mining Engineers, Birmingham, March 10, 1913). the number of successes a to the total number of investments (a + 6). That is, ______ The probability of success (p) = It is assumed that a rate of 3 per cent, interest is paid for the “ use ” of money where the probability of the repayment of the loan is considered “certain.” Any additional rate over and above 3 per cent, is an additional payment over the payment for the “use” on account of the risk. This view is also held by Mr. O’Donahue. The full value of an investment is the amount that a person would value it at in his own mind if he had secret information that its success was certain. If an investment of £100 for one year was held out as likely to pay 54J per cent., a person having secret information of its certain success would calculate the value of the investment at the end of the year as £1544, and discounting at 3 per cent., the charge for use, would reckon the present full value as £150; £154J would be the full value a year hence; and £150 the present full value (F). “ Probable value ” or “expectation ” is a term usedin mathematics to denote the full value multiplied by the probability of success. If a person has a chance of receiving a legacy of £1,000, and the probability of getting it is “ 4 ”—that is, he is as likely to get it as not —then his “ expectation,’ or the “ probable value,” is £1,000 x 4 = £500, and a speculator wishing to purchase it would value it in his own mind at one-half of its “full value.” In the form of an equation, this is expressed as V = F x p. The gross value, as defined in the Finance Act, 1909-10, is the value which a property might be expected to realise if sold in the open market by a willing seller in its then condition. This is also the fair market value. When a person is estimating the fair market value of a property he takes into account the “full value ” (F), and also the “ probability ” (p) of the “ full value ” being realised, and he estimates the fair market value as equal to F x p. That is, the fair market value is equal to V, the probable value. The value of p is now required for a 10 per cent, security in order to know exactly what is meant by a 10 per cent, risk, and it may be found by the considera- tion of the investment of £100. For purchase a year hence:—The full value of the £100 is £110—that is, F = £110. The fair market value of the £100 is £103— that is, 103 = 110 X p, or p = That is, the proba- bility of success of a mining investment rated at 10 per cent, per annum, and extending over one year, is The result may be expressed in another way, as follows ••—The probability of success per annum is therefore the probability of failure per annum -- 1 __ 103 - _ 7,_ — X 110 — 110* The probability of the investment of the £100 continuing successful during each of two successive years is:— TTU X TTU = °r_p2, and similarly for n successive years is pn. This result destroys Mr. O’Donahue’s contention that the proba- bility of success does not decrease with the increase in the number of years. The erroneous idea which Mr. O’Donahue appears to have followed is that to part with £100 in a 10 per cent, per annum security implies a certain probability of recovery of capital (namely, 4° J), and when once the money is paid over represents the probability of recovery, however long the period may be before recovery is to be obtained. When the £10 interest from the first year is received, this again, if put out at 10 per cent., incurs a further probability of or a total of Grvo)2, and is accordingly .facing a smaller probability (or greater risk) than the £100 does during the first year. This proves another of Mr. O’Donahue’s contentions. On Mr. O’Donahue’s erroneous assumption the £10 of interest for the first year cannot be calculated at a smaller total probability than (|y^), which is shown to be the amount already involved in the first year. He has, therefore, in order to maintain the probability at suppose the £10 invested in a security without risk at 3 per cent, for all the succeeding years. It has been shown, however, that the £100 itself incurs a probability of (fyj)2 if let out at risk for two years, and therefore, on Mr. O’Donahue’s basis, to keep the probability of recovery as large as ^or the whole period, it would be necessary to allow 10 per cent, interest on the first year’s capital, and only 3 per cent, compound interest on all capital and interest in the succeeding years. Instead of this, the method allows 10 per cent, simple interest for all the years, and is therefore inconsistent in itself. It is as though a person were to allege that a risk of 10 per cent, per month represents the same thing as a risk of 10 per cent, per annum, or this again the same as a risk of 10 per cent, for any indefinite number of years, so long as only one investment is made, and the capital is not repaid and re-invested again. It appears that if capital were invested in a 10 per cent, per annum security, and withdrawn and re-invested annually, then Mr. O’Donahue would be willing to calculate 10 per cent, per annum on both capital and interest. To sum up, Mr. O’Donahue takes a 10 per cent, security as meaning that the probability of recovery of capital is the same for the whole period of an invest- ment, however long the period is to extend before recovery. This is erroneous; a 10 per cent, security means that the probability of recovery decreases with each additional year considered, and is represented by He then uses the erroneous interpretation of a 10 per cent, security as a reason for compounding the interest at 3 per cent., whereas, if the risk is properly understood, the interest must be compounded at the same rate as the capital (10 per cent.) Returning now to the consideration of No. 2 method,, in it interest is taken at 10 per cent., and the redemption fund at 3 per cent. This is the error introduced by Mr. Hoskoid, and requires no further discussion. The deferment is taken at a. discount of 3 per cent., and the valuer supports his views as follows :— “ The rate for this period should be taken at an investment rate, 3 per cent., as it is manifestly inaccu- rate to allow a high rate of interest (10 per cent.) when estimating the present value for the immediate annuity, and then again to allow the same high rate (10 per cent.) during the deferred period when the investor is taking no risk.” As thus expressed, it implies that the success of the venture is a certainty during the deferred period, but that when the coal begins to be worked the probability of the success falls to (|t^)84, but it is difficult to* believe that such is the real intention. The erroneous idea underlying the position appears to be that when a man parts with money in a 10 per cent, security he anticipates a probability of recovery of (|4u) for the whole period. The valuer allows 10 per cent, during the- annuity, and considers that the probability of recovery is then sufficiently allowed for. He fails to recognise' that the probability is recurrent every year, and that each year’s probability has to be allowed for. The- “ whole risk ” cannot be recovered by an allowance for only a part of the period. Finally, because the probability of recovery for one- year is only as large as p (= fyf), 10 per cent, is- allowed on the capital of £100. In the second year the- probability is again p on all the capital of £110, and 10 per cent, is therefore due on that amount. The- additional £10 of capital also receives 10 per cent, for the probability p, just as the original £100 does for its new probability p. That is, the capital must be taken to accumulate at a single rate of compound interest (10 per cent.). When this is done the present value of the annuity of £3,750 due for 84 years deferred 36 years is equal to the present value of £3,750 due 37 years hence, and if R”is the amount of £1 in n years at 10 per cent, compound interest, the present value of £1 due n years hence at 10 per cent, compound interest is ^L. Hence the present value of the annuity is 3,750 X f , 1 _ , , ______ (R37 ~ R38 Jjsy = 3,750 x f 1 + 1 + 1 +------------+ 1 1 R36 (R R2 R3 R8*5 _ 3.750 f1 ~ R81} ~ R»«) 1_ \ X 1 ~ R _ 3,750 fi — —L-. (l-l)33. __________ 1 R~ R3S ( B — 1 = 3,750 x (0 0323) x (9 9967) = £1,211 as found by Inwood’s tables. Method No. 1 is the only one of the six which is correct both in logic and in method. Mr. Hoskoid has shown that his own method, if the redemption is taken at 10 per cent., gives the same result as Inwood. It is an alternative method, but involves two errors in logic. Any other method which may be devised to give a different result from Inwood must be wrong both in logic and in method. ______________________________ The will has now been proved of the late Mr. J. W. Summers, of the firm of John Summers and Sons Limited, Hawarden and Stalybridge, and who was member of Parliament for the county of Flint, and justice of the peace for the counties of Flint and Denbigh. The deceased gentleman’s estate is valued at £170,483.