VALUE OF COPYHOLDS. XV
which is thus found; take 23- of the square of a. If greater accuracy be required, add to the result one less than its 760th part, which is sure to make it correct within a single game. Thus if the number of his stakes be 100, 100 x 100 x 24 is '23750, the 760th part of which is 31, whence 23750+30, or 23780, is within one of the number of games required.
APPENDIX THE SECOND.
ON THE RULE FOR DETERMINING THE VALUE OF
SUCCESSIVE LIVES, AND OF COPYHOLD ESTATES.
THE rule given in the work is in a different form from that of any writer with whom I am acquainted, though it agrees with that given by Mr. Milne, as will be shown. This Appendix has been rendered necessary by the fact that no writer has solved the question of the value of copyhold estates with absolute correctness except Mr. Milne, whose solution is in a form of unnecessary diffiThe writers with whom I am acquainted, who give the old rule, or one involving an omission of the same kind, are be Moivre, Dodson, Thomas Simpson, Stonehouse, Morgan, Baily, and the French translator of the latter. Mr. Milne stands alone in proposing a somewhat different rule, which like many results of independent investigation, differs more than need have been the case from the form of preceding results.
Let there be an estate held on a single life, and renewable for ever upon payment of a fine of 11. Let it be a condition, that each renewal is to be made on the 1st of January next following the extinction of the previous life ; it is required to find the present value of all the fines.
Firstly, To find the value on the 1st of January, the moment after a fine has been received, and the best life which can be found has been put in. Let P be the
Y
xvi APPENDIX THE SECOND.
value of the life, or the value of an annuity upon it; r the rate of interest per pound, and F the value of all the fines. Then upon the new year's day next following the extinction of that life, the whole value of the fines will be 1+F, because the person who claims the fines will have one pound to receive, and will then have remaining the interest which we have called F. It foltherefore that F is the present value of 1+F to be received at the end of the year in which the life drops : or, by the well known formula
F 11+P(1+F)orF=E+P
where E is T, the value of a perpetuity of 11.
Secondly, Let the life already in possession be of the value A, the lives at each renewal having the value P, as before. Consequently, the present value of the fines is that of 1 t F to be received at the end of the year in which the present life drops : and this is
1—rA E—A 1+E E—A.
(1+F)or— or
1+r E +l 1+P 1+P
if an estate be held on any number of lives, with a fine of 11. on renewing each, it is precisely the same thing as if a similar number of estates were held on single lives, and the present value of all the fines, the present lives being worth A, B, C, &c., n in number, is
nE—(A+B+C+....)
t + P
the common rule divides by P instead of 1+ P.
A .mathematical rule, when erroneous, is best exposed in its extreme cases; let us then suppose the life P certain to drop in the year. that is, worth nothing. There will consequently be a fine to pay every 1st of January, and the present value of what I called F in
the preceding investigation, is ror E. The rule I have given, shows that F=E when P =0 ; that of De Moivre,
VALUE OF COPYHOLDS. Xvii
Simpson, &c., makes F infinite. Again, suppose the life P certain to last one year and to drop in the second ;
in which case its value is 1 +r, or EE 1, and F is a perpetuity of 11. receivable at the expiration of every two years,or2 E+ If A=P=E+1, the new formula becomes 2 E f 1, the old one becomes E, the value of
a perpetuity of 11. receivable at the end of every year.
I shall now show that the preceding rule agrees with that of Mr. Milne ; which is as follows: —Let P be worth an annuity certain of t years, and let v be the present value of 11. to be received a year hence. Then the pre-sent value of all the fines, according to Mr. Milne, is
A'+B'+C'+
1 —ve+I
where A', B', C', &c. mean the present value of 11., to be received at the end of the years in which the lives severally drop. Since P is the value of an annuity certain for t years, we have
P=E—v°E, 1+P=1 +r
r v
or 1+P=E -(I —v t+1) (E + 1) (1—v t-1-1)
E — A, whence A' E— A
A E+I 1—v,+i=+P
from which the coincidence of the two rules is manifest. The error of the old rule, by the Northampton Tables, and at 4 per cent. (the hest life being worth 17.25 years purchase) is, that the result is 5,'- per cent. too great.
The old rule, as Mr. Milne justly observes, is derived from the supposition that the new life is put in at the beginning of the year in which the old one drops, instead of at the end; which last was in the intention of those who formed the rule. It may be said however, that
y 2
and
XVIII APPENDIX THE SECOND.
the old rule is nearer the truth than the new; since one time with another, the renewal is made before the end of the year in which the old life drops. This objection must be valid to some extent; and I proceed to inquire how much weight must be allowed to it.
Let a be the fraction of a year allowed for renewal.:
it is clear then that - renewals (each accompanied by a
a fine) may take place in the year. Let the life A drop at the expiration of the fraction 0 of a year, or between 0 and 0 + dO, the chance of which is dO itself, if A be supposed equally likely to drop at any period of the year. At 0 + a, then, the new life is put in; and if this new life drop before 1—a of the year is gone, another fine must be paid, and another renewal is made, which again may drop before 1 —a, and so on. But since the chance of each additional renewal is very much smaller than that of the preceding, it will he sufficient to take the first only into consideration. Let it be supposed, then, that not more than one renewal shall take place within the year in which A drops.
Let a be the chance that the life P drops in a year after nomination, in which case we may call x a the chance that it drops in any fraction x of a year. Then d 0 (1—0—a) a is the chance that the life A drops between B and B + d 9, and that the next life drops within the year, in which case another fine is to be paid at the beginning of the next year. Consequently, negthe interest of the fine in a fraction of a year, the lessee has the chance d 8 (1 -9 — a) a of having a second fine to pay, upon the contingency of A dropping between 0 and 9 + d O. Integrate this expression from
0=0 to 9= 1—a, and we have (1—a)2 for the
chance of a second fine: which, with the fine certain upon the death of A, shows that l + ! a (1— a)2 is the mathematical expectation of the fines to be paid, when the probability of one renewal within the year is conand another at the end, if necessary.
A more complicated process, proceeding on the sup-
VALUE OF COPYHOLDS xis position that any given number of renewals may take place within the year in which A dies, gives the follow
ing terms for the total probability of one or more retaking place before the end of the year : — a, the chance that there shall be no renewal ;
a (1 -2a)2
1—a — 2 the chance of one only ;
a (1—2a)2 a 2 (1 -3a)2 the chance of two only;
2 2.3
and so on, the series being continued as long as the requisite multiples of a are less than unity. Hence the chance that the number of renewals shall not exceed n, is
an (1—(n+1)a)n+l
1
2.3. //
. n.n +1
which, in the case most against us, that is, supposing instantarenewals, or a =0, is
1— an
2. 3.... n. n+1
Let the life P be one of seven years old, in which case the Northampton table gives a= ,.V. = 0186. Neglecting interest for the fractions of the year, and remembering that the number of renewals is the numof fines, the mathematical expectation of all the fines is the sum of
a(l—2a)2~ 2 (a(1—2a)2_a2(I—3a)'\ 1—a 2 l 2 2.3
To this add a, the chance that the renewal fine certair on the death of A, shall outrun the year, and we find ( 1 — 2 + a2 (1 -3a)2
I+ 2 2.3
for the mathematical expectation of all the fines which shall be paid in the year in which the present life drops, including the chance of that life dropping too late to renew it within the year, and of its being therefore rewithin a period not exceeding a of the year folIf a—0, this becomes
Y 3
xx APPENDIX THE SECOND.
Ea—1 , E being 2.7182818 a
and when a= . 0186 , a 1 019
Hence 019 _ 0186 being 1.02, it appears that I 02(1 +r) is an enormously exaggerated representation of the fine which must be substituted for l l in the amended rule to make the correction which might be suggested by the advocates of the old rule. Allowing r= 035, it may be granted that the old rule is correct, in the particular case before cited, if we suppose : 1. That no time whatever is allowed for renewal ; 2. That the best life which can be found is such, that 1 out of 59 of such lives drop in a year, and ; 3. That the lessor is to receive his fines at the beginning of the year succeeding that in whichA drops, with 31 per cent. interest, reckoned from the beginning of the preceding year. 'To take a more rational supposition, let us allow six months for renewal. Here a= and only the first term of the series can be taken, which, with the interest, gives 1+'.
It appears from the preceding, that the advantage of the lessor over that which the rule gives him, is trivial, except in this, that at the time when the rule supposes him to receive one pound, he may have received and improved one pound during a fraction of a year. There is also another advantage which he has, and which neither rule allows him. The renewal may have been made before the time supposed in the rule, in which case the existing life will be somewhat worse than that supposed. To take all these circumstances into account, suppose the life A to drop at the end of 8 of the year, in which case the lessor is in possession of 1 +r— (0 + a) r at the end of the year, and the lessee has a life r< worth
P — €1— (0 + a) l P, where L P is the decrement
* Let it be remembered that the renewal may take place after the beginning of the year, which is equivalent to having a better life than that supposed. Algebra, as in other cases, strikes the balance of positive and negative quantities without the necessity of introducing several formula.
VALUE OF COPYHOLPS. Xxi
of the value of the life in one year. Multiplying by dB, the chance of this occurrence, and integrating from 0—0 to 0=1, we find 1 + r— a r for the mean sum, and P — ?- P -1 a A P for the mean life. If the time allowed for renewal be more than six months, the rule (without this correction) gives an advantage to the lessor; if six months, to neither ; if less than six months, to the lessee. Substitute 1+r—ar and P— AP+a6. P for 1l fine and a life P, and we have
E—A
x l+=r—ar 1+P—1AP+a.aP
which I believe to be the most correct rule that can be given for the present value of all the fines upon a single life renewable for ever, the value of the present life being A, the tabular value of the best life P, that of a life one year older P—. P, the fraction of a year allowed for renewal a, r the interest of one pound for one year, and E the perpetuity. By substituting nE—(A +B+ C } ) for E — A, the value of all the fines from an estate held on any number of lives is found. Similar considerations apply to the present value of Il. to be received in a of a year after the death of a life A. The offices frequently pay in three months after the death is proved, whereas the tables are calculated for the end of the year of death. Again, they rate all lives as they will be at the next birthday, the parties being one with another half a year younger. To a party who dies at the end of 0 of a year, the office has paid by the end of the year 1+ (1 —a— 0) r which, treated as be-fore, gives 1+ r—ar. And the present value of ll., which is computed by the office from
E—A(I+k) +1
(where k is the proportion of profit demanded) may be more strictly computed from
E — A —iAA x (1 + r—ar) 1