Faustmann's formula is mathematically correct, for it brings into account in a logical and proper manner the various factors that have to be taken into consideration. It is nothing more or less than an algebraic form of expressing the plain and obvious fact that the income and the expenditure occurring at different times have all to be treated as if they were sums of money paid into a bank account and allowed to accumulate at compound interest-that is to say, they must be capitalised at compound interest and summarised with respect to a given point of time; whereas, in the previous method of fixing the rotation merely for the largest annual income, the revenue and expenditure at different times were all dealt with as if the precise time at which they occurred were of no consequence. There can be no doubt that, as a guiding principle, Faustmann's formula is as safe as anything of the sort can be; but, in practice, it cannot reasonably be taken as anything like a hard-and-fast rule for determining the best rotation of any given crop. There is not a single factor in the formula which one can be sure of estimating quite correctly for any given case; and one of the chief of all the factors, p, the percentage or rate of interest at which the business of forestry is supposed to be carried on, is almost an arbitrary quantity, selected more or less at will, except in the very few cases where it might be specifically determined from the returns yielded by woodlands which have been purchased as an investment for the production of timber. Even a difference of per cent in reckoning with p will perceptibly affect the results showing the most profitable rotation. The important effect of p, the rate per cent in such calculations, may be shown very simply. Suppose, for example, a woodland estate which produces a clear revenue of £1000 a-year is in the market. Any intending purchaser who desires to obtain 4 per 1000 cent from such an investment will fix its value at = £25,000. Another, content to 0.04 1000 get only 3 per cent, would reckon its capital value at =£33,333; while a third, satisfied with a return of only 2 per cent from such an investment in landed property, 1000 would assess its value at = £50,000. For other examples, see also vol. i. pp. 98, 99. 0.02 Again, with the gradual changes affecting the timber market from time to time, it is impossible to determine future values of wood with anything like accuracy, while at the same time some sort of practical standard must be adopted for determining the kind of crop and the rotation likely to prove most advantageous. And in both of these respects, even although absolute accuracy is not thus attainable, the results which Faustmann's formula gives are of great practical use as good general indications of the most advantageous crops and rotations for given situations. More especially will this be the case if the rate of interest be fixed not higher than 24, 24, or 3 per cent, according to local circumstances. Example.-A landowner has land suitable for planting with mixed conifer crops, which he thinks would probably pay best if worked with a rotation of 80 or 100 years. So far as he can ascertain from consideration of average yield tables and existing local market ever the tree has arrived at that period of its growth that the annual increase does not amount in value to the marketable interest of the money which, at the time, the tree would produce if cut down, then it would appear more profitable to cut it down than to let it stand. . . . A writer in the Gardener's Magazine states that Mr Larkin, an eminent purveyor of timber for shipbuilding, stated, when examined before the East India Shipping Committee, that in situations the most favourable for ship-timber (the Weald of Kent, for example), the most profitable time to cut Oak was at 90 years old; as, though the largest scantlings were produced at 130 years' growth, the increase in the 40 additional years did not pay 2 per cent (Gard. Mag., vol. xi. p. 690)." rates for wood, he may expect, for soil of the quality he takes his land to be, the following yield (net money value in each case) from the land per acre : 1. Intermediate Returns (Thinnings).—Thinnings at 40 years worth £4, at 50 years £5, at 60 years £6, at 70 years £7, at 80 years £8, and at 90 years £9. 2. Final Yield (Mature Fall).-Clear-felling at 80 years of age, £183; or at 100 years, £266. The cost of planting is £5 an acre, and the general expenses of supervision, protection, and rates and taxes annually amount to 5s. an acre. The rate of interest being taken as 3 per cent, which would prove the more profitable rotation? (1) For the 80 years' rotation, the present value of the ultimate income from all these future returns is [ (183 +13-0480 +12-1365 +10-8366+9-4073) – 53-2045 183+ (4 × 1·0340) + (5 × 1·0330) + (6 × 1·0320) + (7 x 1·0310) - (5 x 1·0380) 0.25 = -[(183 9.64 2045] 8.33 (2) For the 100 years' rotation it would be ( 266 + (4 × 1·036o) + (5 × 1·035o) + (6 × 1·031o) + (7 × 1·0330)+(8 × 1·0320) 1.031001 [(266 +23·5664 +21·9195+195720+16-9911+144488+12·0951) – 96·0930 ̄ = 18.22 0.25 0.03 -8.33 The 80 years' rotation would therefore be the more profitable. The method of The case is different when a landowner wishes to consider whether a crop that is 80 years old can more profitably be cut now, or 5, or 10, or 20 years hence. reckoning then employed will be subsequently explained (in section 6 below). It is easy to see that in the results given by the above formula, other factors remaining unchanged, the indicated capital value will be all the more, the greater the value of the thinnings and of the mature yield; and it will also be all the greater, the less the cost of regeneration and of the annual expenses for management and rates. It is equally clear, too, that the lower the rate of interest represented by p, the higher will be the estimated value of the capital invested in the woodlands. So far as concerns the question when it will be most advantageous to fell and regenerate wood-crops now approaching maturity, the matter is much simpler and less dependent on arbitrary factors. The present condition of the crop and the measurement of its rate of growth will give the necessary data for determining in a common-sense practical manner whether it be more profitable to harvest it now or to wait for some years before felling it and converting it into money. Where the rate of growth is still good, the present upward tendency in the price of timber will certainly point to advantage in permitting well-grown crops to develop into stems of large dimensions. 6. Sylvicultural and Actuarial Considerations affecting the Management of Woodlands.-Owing to the uncertainty affecting, in a greater or less degree, each of the various factors with which the calculation must be made, VOL. II. it is impossible to determine beforehand with accuracy what rotation will really prove most profitable. Where a particular rotation has been customary for any given kind of wood, there are usually good grounds for its having been adopted; and careful consideration should be given to the benefits likely to be derived before changing it into any higher or lower rotation. To lengthen the rotation means to increase, out of savings made in the annual falls, the capital invested as the growing-stock; and to diminish it is to obtain a larger annual yield temporarily, while decreasing the amount of timber required as the normal capital in wood. The temporary advantage obtained in the latter case is nothing more than liquidating part of the capital. That this is not self-evident is solely due to the fact that in forestry the bulk of the capital (wood) is indistinguishable per se from the mature product (timber) which it yields (see p. 221). One of the main things to be considered, both as to forestry in general in Britain and with regard to fixing the rotation with which woods should be worked, is, of course, the probable tendency of future prices. The demands for timber are gradually increasing, more quickly than its place can be taken by other materials (iron, &c.), while the available supplies which can be brought to market with profit are diminishing; hence prices naturally show an upward tendency. In some parts of Britain the rise in the local rate for timber has recently been as much as 25 per cent; and the present condition of the existing sources of supply being considered (chiefly the Baltic countries, Scandinavia, and Canada), it seems very probable not only that this recent rise will be permanent, but also that the market value of timber in Britain will very probably soon be considerably higher than it has ever been before. In view of this upward tendency it therefore seems certain that the most profitable systems of management must be such as will produce clean, straight, well-grown timber, and the most profitable rotations will be those that can yield timber of the dimensions best satisfying the requirements of the local markets; because timber is so bulky and heavy a raw material that the advantages of the higher prices of far distant centres of consumption can, in the majority of cases, only be made available when there is cheap transport by water (drifting and rafting). But even when what seems the most profitable rotation has been considered and determined, circumstances may afterwards occur to make a deviation from it desirable. Thus, if any crop should be damaged by wind, insects, &c., sylvicultural considerations may necessitate its being felled prematurely; while, if the prices for large timber are specially good, the fall will naturally be delayed for several years if the wood is in vigorous growth and the annual fall can more conveniently be made in some of the other woods. Besides the condition of the local market, the other chief matters to be considered are the soil and situation generally, the kind of crops, the distribution of the various age-classes, and the actual condition of the growing-stock. Woods growing on good soil and in a favourable situation can be worked profitably with a longer rotation than those on poor land, whether in a sheltered or a high and exposed position. Here the increment culminates and begins to sink sooner than on good soil, and the rotation must be fixed so as to obviate danger of deterioration and consequent decline in productivity. The kind of crop selected should, of course, be such as will best suit the market. Thus, though well-grown Spruce crops yield about 15 per cent more wood than Scots Pine, yet in Britain the advantage will clearly be on the side of the latter, because its timber commands a considerably higher price. Douglas Fir produces larger crops than Spruce, Pine, or Larch, and yields wood ranking between Larch and Scots Pine in quality; and there can be no doubt that this is, under judicious treatment, likely to prove the most remunerative of all the woodland crops that can be grown in Britain. Again, Ash commands a good and ready sale from the size of poles upwards, and this should be the strongest possible argument for encouraging its growth on a large scale among all hardwood crops, whether treated as highwoods, copse, or coppice. And the rotation, as well as the sylvicultural treatment in thinnings, &c., must of course accommodate itself to the requirements of whatever particular tree is intended to be the ultimate crop. In mixed woods where Oak forms the main crop, the various operations for promoting its development must be conducted with a view to the special requirements of the Oak rather than of the trees grown in admixture with it; and so also as regards fixing the rotation. In Beech-woods the rotation must consider the maturity and the regeneration of the Beech itself rather than that of the Oak, Ash, Maple, Larch, &c., sprinkled throughout it to increase the profit yielded. And the same holds, too, in mixed conifer crops of Douglas Fir, Larch, Pine, Spruce, &c., the rotation being determined solely by considerations as to the main portion of the crop. And, of course, any sylvicultural operations which increase the quantity and value of large-sized timber, such as partial clearance with simultaneous formation of underwood, will require special consideration when the practical question of clearing and regenerating has to be decided. The actual distribution of the age-classes in woodlands is in so far of influence that, where the crops are mostly young, a low rotation may seem desirable in order to give the landowner, as soon as possible, the full advantages of a regularly sustained annual yield; while, on the other hand, an excess of old woods beyond what are required for the normal capital or growing-stock must either lead to a long rotation being fixed on, or to the capital being reduced by annual cuttings in excess of the actual annual increment throughout all the crops. But as this would mean increased income for several years, till the excess of actual growing-stock over normal capital is consumed, no landowner would be likely to besitate in such a matter. It is not a case that will be general, though it may not infrequently be found with regard to large wooded estates on which plantations have been made at various times during the early part of the past century, without following any fixed scheme of management or regular plan of felling and regenerating or replanting from year to year or period to period. The actual condition of the woods is a matter which must exert a very important influence in determining the fall and harvesting of most of the woods now to be found in Britain. In consequence of injudicious thinning, most of the crops now reaching maturity are so branching, incomplete, and irregular in many respects as to make it desirable that they should be regenerated and cleared-or cleared and replanted, as the case may be as early as convenient, so as to make way for crops to be managed on more business-like principles than hitherto. Excessive branch development, badly-formed boles, unproductive blank spaces, and general want of density in the cover or leaf-canopy overhead, are characteristic of most of the British woodlands; and where such conditions obtain, they can certainly not be the most profitable system of management for the woodlands in question. It is only when the clearance of such crops is being considered that there will be any necessity for fixing the future rotation, as must, of course, be done in drawing up a scheme of management to form the general instruction for practical work. But the actual condition of the existing woods must be the chief thing to be considered with regard to harvesting those of them which should be felled within the next ten or twenty years. And this particular consideration can, of course, only be properly given in connection with the local market rates for timber and the general circumstances of the estate. In the great majority of cases in Britain the owners of woodlands are content to work their forests without any regular sort of management, or they prefer to treat them by casual or sporadic falls (so-called selection fellings) for æsthetic reasons, or desire to regard them mainly as game-coverts. In all such circumstances, whatever annual income is obtained must, of course, be less than would be received under management on commercial principles, while the method of working adopted must adapt itself chiefly to the specific object desired by the proprietor. So far as the practical question regarding the fall of woods approaching maturity is concerned, the rate of growth ascertained for each crop will show whether it has a satisfactory increment or not. If this is not quite so good as it ought to be, the next thing is to consider whether the crop can be made more profitable by means of partial clearance and underplanting, or if it is not best to clear the timber and form a new crop. The percentage of increment in cubic contents is easily ascertained by dividing the amount of the current annual increment by the whole cubic contents of the crop and multiplying the result by 100 The percentage of increment in value, or current rate of interest afforded by the crop as an investment, is also easily calculable from year to year by means of the simple formula Fm+1 - Fm p=100 (F+L+G as is explained on opposite page. Now, if p in this case is greater than the rate of interest the woods are supposed to bring in, it must be profitable to let the crop continue growing; if it is equal to that, it will be equally profitable to clear it or to let it grow on ; and if it is less than the arbitrary rate of interest, the crop can no longer be grown with profit, but should be felled and utilised. Example.-An 80-year-old Larch crop, with 7500 cb. ft. to the acre, shows a current increment of 150 cb. ft. per acre: what is the percentage of increment? 150 If the present net value of the crop is £300 an acre, and it is estimated that its net value next year would probably be £307, 10s., what will be the rate of interest it would yield during the year, the capital value of the land (L) being taken as £20 an acre, and (G) being £13? the capital sum required to cover the annual general expenses (op Another method-the evolution and precise signification of which will shortly be described (p. 247)—gives p as an indicating percentage in a simpler formula, which can easily be used in practical work in the forest for determining whether woods of marketable dimensions should be felled or allowed to stand for 5 or 10 years longer. According Fm+n- Fm 200 to itP=Fm+n+Fm ቤ Here Fm signifies the net money value of the crop or fall at present, and Fm+n represents its net value n years hence-i.e., next year in this case. If this formula be used for the case in the above example, then the "indicating percentage" would be 307-300 200 _ 7 × 200 _ 3000 P=3073+300 1 607 1215 2'46 per cent. For exact actuarial calculations made with great circumstantiality a much more scientific system has, of course, been evolved in Germany and embodied in formulæ. Thus Faustmann's formula can be employed for the purpose, the present and the future |