March 14, 1913.

THE COLLIERY GUARDIAN.

539

logic, the question was not one that only touched the
Government claims for duty; that was a minor matter
compared with the buying and selling of mineral estates,
as whichever principle was right referred to both.
Mr. Stephen had put before them a tangible proposition,
in the case he had taken, and to arrive at the valuation
desired there were certain points to be considered. As
they were dealing with a mineral area the commercial
success was not the main element, as the physical
difficulties which might rain one concern might be
conquered by another, and that would be covered by the
rate of interest taken. The 10 per cent, under all the
circumstances seemed reasonable, but to deal with it on
the Inwood tables, assuming a one rate compounded for
the whole period, surely could not be right. It struck
him that Mr. Stephen went wrong in assuming a
recurring risk instead of a continuing one. The climax
or loss could only take place once, and the risk was not
cumulative. Finally the speakers, upon the proposition
they were considering, was convinced the owner of
proved minerals, with the prospect of receiving £3,750
per annum for 84 years, although deferred 36 years,
would not accept anything like so small a sum as
£1,211, and further the colliery owners with a prospect
of a further life of 84 years, with a saving in royalty of
£3,750 per annum for the period, would probably give
more than the £12,101, especially if there was an open
market and competition.	*

Mr. J. C. McLaren (Birmingham University) said
that he was only able to give the facts as they appeared
to him as a mathematician, the technical points being
outside his range. He had read Mr. O’Donahue’s paper
and Mr. Stephen’s pamphlet, and it appeared to him
that the latter’s arguments were entirely sound and his
methods logical and properly worked out. A 10 per
cent, basis might not be a fair one at all, but once they

necessary indication of approximate value, they were to
a very great extent only a means to an end. The broad
view that he took was that no person would pay
anything approaching the sum suggested looking at the
thing as a person would have to look at it calculated
upon the table. No person would think of paying down
£12,000 with the prospect of getting at the end of the
period only three times that amount, seeing that the
prospect was subject to all the factors represented by
possible legislation, changes in the relations between
capital and labour and in taxation and all the other
things which might have to be taken into account.
Therefore it seemed to him perfectly clear that the risk
and the discounting which, independently of the tables,
had to be taken into consideration, formed a factor
which must be taken as extending throughout the whole
of the deferred period as well as through the active
time of the realisation of the annuity.

Mr. S. L. Thacker thought the question of the
probability of the coal and all considerations affecting
the prospective yield and value of the property should
be settled before the mathematical calculation was
commenced. When one considered an investment to
be deferred 36 years and considered market values, he
did not think 10 per cent, would be a sufficient return.
That was the whole point as to the general value of
property, but that did not affect the mathematics of the
problem at all. The whole question was, “ What do you
mean when you say you must have a certain term, and
you must have that at simple interest or compound
interest ? ”

Mr. O’Donahue was then called upon to reply. He
said he was pleased to have an opportunity of replying
to Mr. Daniel Stephen’s criticism of his paper, but he
regretted that the criticism was first published in a
manner by which he was precluded from replying

The result was follows:—

where + A = present value of an annuity of 1 for
n years deferred d years.

Alw -f- d = amount of an annuity of 1 for n + d
years at rate s.

Rlw + d = amount of £1 for n + d years at rate s.

Aw = amount of an annuity of 1 for n years
at rate Vr.

Aw + d = amount of an annuity of 1 for n + d
years at rate Vr.

r = accumulative rate of interest.

s = speculative rate of interest.

He might be allowed to explain that the difference
between the rule which he had previously given and the
one above was the same as that between the dual rate of
interest method and the single rate of interest method
as applied to an immediate annuity. The former allows
the higher rate of interest on the original capital for the
whole period, whereas the latter allows the higher rate of
interest on outstanding capital only. As he previously
explained, either method might be applied; the second
method showed the true rate of interest allowed, and
with the former method the effective rate of interest was
somewhat higher than the nominal rate.

Mr. Stephen’s logical premises were quite sound—his
application of them was quite the opposite. He failed
to appreciate the fact that the year was merely a period
arbitrarily fixed for converting interest, and that it
could not be assumed that such a period represented an
operation in an investment analogous to the tossing of
a coin. His own method of dealing with deferred
annuities was to allow the same rate of interest whether
the annuity was deferred or immediate.

To argue that the value of such a property was

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Fig. 2.—Diagram No. 1.

started with it they must not introduce a small rate of
/interest in order to calculate for the redemption of
capital. The reason was obvious. All returns from the
colliery, whether they called them redemption of capital
or profit, were loaded with the same risk. The 10 per
cent, allowance was not really accumulative interest at
all; it was simply a method.

Mr. W. Bentley (Birmingham) said as far as his
little experience in that district went they had always
worked on Inwood’s tables. When they were buying
an annuity they expected to get back their capital
during the time the annuity was running, together with
interest at the agreed rate on the amount of capital
from time to time at risk. If that was the right view
of it, all questions of double rates of interest, remunera-
tive rate and accumulative rate went by the board,
because as soon as the money went into their pockets it
was out of risk altogether. What they had to get back
again was their principal, not at the end of the trans-
action, but from time to time by instalments as the
annuity ran on. Several persons to whom he had
spoken on the subject of valuations seemed to him to be
paying rather too much attention to the question of
Estate Duty. He knew that that was the old valuer’s
idea for probate ; but there was the question of incre-
ment duty, and he hoped they would carefully consider
the effect of increment duty on very small areas which
would work out in a few years after the coal had been
•entered upon.

Mr. Arthur Sopwith said that they had heard a
great deal of the question of the mathematical working
out of valuations, but the conclusion he had come to
was that while the tables, whether Hoskold’s or Inwood’s
-or any others, were a very good, valuable and absolutely

immediately. The rules he suggested would mean
higher values for Estate Duty. He wondered if the
writer of the review would have so readily condemned
his methods had he realised that increased increment
value duties would fall to be paid on unworked minerals
if the old rules were adopted. The capital values were
now being settled, and his difficulty—despite his rules—
was to make valuations high enough to satisfy owners,
as the majority of them were prepared to risk the
relatively lower rate of duty payable for probate rather
than the higher increment value duty. Mr. Stephen
purported to deal with three of his methods. He could
not have given his 1906 paper very careful attention, or
he would have known that he gave no such rule in 1906
as the one he quoted. He gave that particular rule in
his last paper, with the qualification that it could not
logically be applied in the manner in which he had applied
it. Mr. Stephen next stated that in 1912 he extended
the Birmingham 1906 method to obtain the practical
rule which he (Mr. O’Donahue) advocated. He did
nothing of the kind. It should be quite clear to anyone
who had read both papers that the rule which he gave
in 1906 for deferred annuities was identical with the one
advocated in his last paper. It was a difficult matter to
meet a criticism which contained so many misstatements
on points of fact, and most of the ninepins which
Mr. Stephen knocked down were of his own manu-
facture.

With regard to the “theoretical” rule, he (Mr«
O’Donahue) had something to add to what he stated in the
last paper. He had not then succeeded in deducing for
that method a formula which was simple enough for
ordinary application. By approaching the problem in
another way he had latterly been more successful.

Fig. 3.—Diagram No. 2.

£1,211, was equivalent to saying that it was probable
that on the average 30 properties of similar risk would
be required to realise the revenue estimated to be
derived from the one. Mr. Stephen, sen., appeared to
have a suspicion that the £1,211 was too low, and no
practical man would hesitate in coming to that
conclusion.

It should be noted that deferment of realisation had
no effect theoretically so long as interest was accu-
mulating. The ordinary individual was naturally
desirous of realising within his lifetime, and con-
sequently competition for the purchase of long-
deferred payments was restricted, but to the State
or to corporate bodies deferment mattered nothing,
and they would make a very fine investment
if they could purchase all unworked minerals
on the 10 per cent, compound interest basis.
He had been asked if he could not present the problem
in such a manner as to clear it from the cloud of
complicated mathematical calculations, and it had
occurred to him that the accompanying charts might
possibly serve the purpose. The horizontal lines of the
charts represented years, a period of 30 years, and the
vertical lines represented pounds. The curve A Z
showed the accumulation of an annuity of £100 a year
for 20 years at 3 per cent. The curve B Z showed the
accumulation of an annuity of £56 48 per annum for
30 years at 3 per cent. The curve C Z showed that a
present sum of £1,107, invested at 3 per cent., accumu-
lated to £2,687 in 30 years, just as each of the two
annuities amounted to the same value in 30 years.
Assume that the two annuities, instead of being secured
properties, were estimated to be derived from mineral
estates. The “ full ” value at the end of the 30 years