The economic addition of height in a building on a given site is that which will, upon a certain obtainable rental, return a market interest upon the value of the land and upon the cost of the structure, and it follows that the relation between the merchantable value of the site and the average rentals obtainable in its locality establishes, at any given rate of interest, not only the height, but the justifiable expense of construction of that height.

The net earnings may be conveniently considered in two parts, one of which is to provide the interest upon the assumed value of the site, and the remainder to make a suitable return upon the cost of the building.

As modern methods of construction are now well established, as regards both cost and space available for occupation in various types of buildings, it becomes practicable to lay down a mathematical relation between any land value and the economic building upon the site which will produce a predetermined rate of interest upon the site and upon the building. Such a method is given in the following formulas, by which for any value of land or any cost of building, and for any proportion of rented area to area of the site, and for any relation of net income to total rentals, the rate may be established for rentals per square foot per annum which must be obtained in order to maintain a given rate of interest upon the engagement of capital.

This method of computation is based on the "Elementary Building" shown in Fig. 2, a vertical section of a building one square foot in plan area, occupying the whole or a less part of the site, and having a certain proportion of each of its floors occupied as rentable space. This unit affords a direct relation between proportion of site, building, and rented areas and between their respective monetary relations, and these being established can be multiplied by the total of each item in any particular instance.

No. 1. Rate of rental per square foot of net rentable area required to produce interest on value of the site:

A=(10000 x V x i/r) / (f x n x p)

V = value of site per square foot.

i = rate of interest on investment.

r = ratio of net income to total income.

n = per cent. of occupied or net rentable area to gross area of building floors.

p = percentage of site occupied by building.

f = number of stories in building. A = rent per square foot required in dollars.

Example:

Question: What rental per square foot per annum is required to produce an interest of 4% on the investment in the site under the following conditions? Fig. 2 Elementary Building

 V = land value per square foot . (a) \$150 . (b) \$100 i = interest ....... per cent. 4 r = net income....... per cent. 45 f= 15 stories....... 15 n = net rentable or occupied area • • • per cent. 70 p = per cent, of site occupied by building . . 90

Then, substituting for the letters in the formula

A= (10000 x V x i/r) / (f x n x p) we have: (a)

A =(10000 x 150 x 4/15) / (15x70x90) = \$1.40 rental required per square foot net rentable area, or (b) For a land value of \$100,

A=10000x100x4/45/15x70x90=94 cents, being the rental required.

So that \$1.40 per square foot must be provided out of net income to afford interest on the site of a value of \$150 per square foot with a building of fifteen stories, and if a land value were assumed of \$100 per square foot, under the same conditions, \$0.94 would provide the interest. To this amount is now to be added the amount required to pay interest on the cost of the building, as follows:

No. 2. Rate of rental per square foot of net rentable area per annum required to produce interest on cost of building: a = 100xvxi/r/fxn h = average height of stories. f = number of stories.

H = h multiplied by f, or height of building in feet. c =cost in cents of construction per cubic foot of building. k = carrying expenses during construction in per cent, of cost of construction. V = value of building, or H multiplied by c plus k. n = per cent, of net rentable area to gross area per floor. i = rate of interest. r = ratio of net income to total rentals. a = rent per square foot required.

Note: Assuming a gross building area of I square foot, then the contents in cubic feet corresponds to the height (H) in feet. See Fig. 2.

Example:

Question: What rental per square foot per annum is required to produce an interest of 4% on the cost of the building?

 f = building of......, . stories 15 h = average height per story . . . feet 12.52 c = cost per cubic foot . . . . cents 50 k = carrying expense . . . , per cent. 10 n = net rentable or occupied area ... per cent. 70 i = interest......., per cent. 4 r = net income, of rentals per cent. 45

Then, substituting for the letters in the formula a = (100xvxi/r) / (fxn) we have: a =(100x10340x4/45) / (15x70)= 86.4 cents, rental required.

Note: The cost of basement is included in h by adding a proportion of its height to the average height per story, and thus the rental required on the stories f includes basement rental as a part of first-story rental.

The two foregoing results combined thus aggregate \$2.26, which is therefore the average rental per square foot per annum on all floors which is necessary to produce 4% on the investment in the land valued at \$150 per square foot and in the fifteen-story building which has been assumed to be erected thereon, or on land valued at \$100 per square foot the rental would be \$1.80 per square foot per annum.

It will be observed that if the land should increase in value after the foregoing conditions are established, then the average rentals thus ascertained must be raised, or the rate of interest upon the land part of the investment will decline.

Inversely, if the rate of interest on the entire investment be stationary or should it decrease, then any increment in the value of the land becomes unremunerative, in the same manner as vacant land would be, and from any assumed increased value of the land there must annually be deducted the interest thereon, at the rate which the land is actually earning.

The rate of annual net return upon the building should be such as to include in itself an amount which at compound interest will return the value of the building within some space of time, since appreciation of rentals cannot be assumed with certainty, or if assumed they may be accompanied by relative increase in cost of operation.

This is what occurred as a result of the shortage of new buildings during the disturbed period - 1917-20 - when the greatly enhanced cost of operation of buildings was accompanied by a large and not always proportionate increase in rentals, both being still further affected by the decrease in the purchasing value of the dollar and resulting rise in the price of money, reflected in higher rates of interest on mortgage and investment. "The rate of appreciation in value of any land is affected by a variety of exterior conditions"