You are reading a page from An Elementary Treatise on Actuarial Mathematics by Harry Freeman (1932)
Part of the American Term Life Insurance History Project
Term Life Insurance
CHAPTER VIII
MISCELLANEOUS THEOREMS

DIFFERENCES OF ZERO    115 We have from the above
Anom=nm — n (n — 1)m+n2 (n — 2)m— ...
= n [nm-1 — (n — I)1 (n — 1)m-1 + (n — I)2 (n — 2)m-1 — ...]
=n[(I + n — 1)m-1— (n— 1), (1 +n—2)m-1
+ (n - 1)2 (I +n—3)m—1—. ]
...........................
we have, when n = m,
Anon=n(n — I) ... 2 (n — I)1
=n(n— 1)...2.1 n0
= n, whence    Qn+r0n = o,
i.e.    Anom = o, when n > in.
We can now build up a table of differences of zero by continued application of the above relation.
The table is
m
 Dom A20m
O' On'
  
O5 0'n
  
I
 I 0        
2
 I 2 0      
3
 I 6 6 o    
4
 I 14 36 24 0  
5
 I 30 150 240 120 0
6
 I 62 540 1560 1800 720

n [(E — I)n-1 xm-1]x=1
= n;,n-1 In'-1
= nLn-1 Eom-1
= nJn-1 (I + A) 0m-1
= n (On-10'n-i + QnO'n-1).
Again, since
Anon' = nAn-1 I m-1
= n (n — 1) On-2 2m-2
and so on.

I16    FINITE DIFFERENCES
An alternative method for obtaining the relation
An an = n (An-I am-1 + An orn-1) ,
depending upon the formula for the nth difference of the compound function uxvx, is given below (para. 4).
 THE COMPOUND FUNCTION uxvx    I17 First, if n = 1, we have
Aumvx = /[(I + Al) (1 + A2) — 1] uxvx
_ (AI + A2 + 01',2) uxvx
= (A7 + A2E1) uxvx
= Al uxvx + A2 E1 u, vs = vx Al ux + E1 ux . A2 vx
= vxAlux + ux+,A2vx, or, dropping the suffixes,
= vx Aux + ux+1 Avx ,
which is otherwise evident. Again,
An ux vx = (Al + El A2)n uxvx
= (A1n + fAln-1 E1 A2 + n2A1n-2 E12A22 + ...) uxvx,
which is easily seen to be
yr An u, + nAvx An-1 ux+1 + n2 A2vx An-2 ux+2 + ••••
If in the above expression we put ux = xm-1 and vx = x, we have An (xm-l.x) = xAnxm-1 + nAn-1 (x + 1)tn-1.
Let x = o ; then
Anom = nAn-1 1m-1 = nAn-1 Eom-1 = n (An-1 on,-1 + An 0'n-1),
which is the relation proved in para. 2.
5. The following is a further illustration of the application of the above method.
Example 1.
Prove that
En (uxvx) = uxEnvx — nAuxEn+lvx+1 + (n + 1)2A2uxEn+2vx+2
— (n + 2)3A3ux En}3vx+3 + ....
Now    Auxvx = [A2 + Al (1 + A2)] uxvx
En = A-n = [A2 + Al (1 + A2)]-n = A2-n (1 + A1A2 1E2)~. E.nuxvx = A2-n [1 — nA1A2 1E2 + (n + 1)2Al2A2 2E22
 I18    FINITE DIFFERENCES
Note. If n = I we have
Euxvx = ~2 1 (I + A1z 2_'E2)_luxvx
    0 A -1E .z    1l
Oz 1 (I — I     +1Q1A2-1E2) ux7Jx
= 0,0_1 '1z2-1E2 1 u vx
Oz+OlEz
/ x
A1A2 lE (A2_1         2I u
El E2 — 1/ x x
= ux Fivx — A—1 (Aux1vx+1) = ux Evx — E (DuxEvx+1),
the ordinary formula for summation by parts.
6. Functions of two variables.
In Chapter II it was stated that, when x and y are independent variables, ux,, f (x, y), ... represent functions which assume different values according to the values of x and y. For example, the function x2 + zxy + y2 + x + 3y, in which x and y both vary, may be written shortly as ux,,. If, further, y is a function of x, we may reduce uxv to the form vx and thus obtain a function depending on x alone.
Now suppose that x takes the value x + h while y remains conand that y takes the value y + k while x remains constant. Then the new value of the function is dependent on x + h and y + k. It is not necessary for both x and y to vary : x may become x + h while y remains constant or vice versa. This type of function is conveniently represented by the slightly different notation f (x :y) or ux_v.
If the values of the function proceed by equidistant intervals, we have the following scheme:
ux:v ux+h:v ux+2h:v tax+3h:v
ux:v+k ux+h:v+k ux+2h:v+k ux+3h:v+k
ux:v+2k ux-Ih:v+2k ux+2h:v+2k ux+3h:v+2k
ux:v+3k ux+h:v+3k ux+2h:vi.3k ux+3h :v+3k



FUNCTIONS OF TWO VARIABLES    I19
or, if our origin be (o, o) and h = k = 1,
7. If we are to apply the processes of finite differences as hitherto defined we must distinguish between an increase in the value of x and an increase in the value of y. We therefore write Ex to denote the operation of increasing the value of x by a unit difference while y remains constant, and Ey similarly for y while x remains constant.


That is,    Ex uo:0 = u1:o and Equo:0 = uo:1,


so that    Oxuo:o=u1:o—uo:o and Dyuo:o=uo:1—uo:o• Again, Ox1 yuo:o = Ox (A,uo:0)


_ Ax (uo:m — uo:o)


= 0xuo:1 — Oxu0:0
= u1:1 — uo:1 — U1:0 + U0:o,


and    Ax2Ayuo:o = 0x2 (u0:1 — uo:o)
_ Ax2 uo:1 — x2uo:o


(U2:1 — 2111:1 + uo:1) — (u2:0 — 2U1:o + uo:o).
The general formula corresponding to the advancing difference formula for one independent variable is
m
um:n = (1 + Ox) I + v)n u0:0
= (1 +mOx+m20x2+...) (1 +nty+n20y2+...)uo:o (1 +mLx+m21 2+m3Qx3+ ...


+ nLy + mnLxL0 + m2nLx20y + ...
+ n20y2 + mn20x0.y2 + ...


+n3Ay3 + ...) 210:0


= uo:o + (mtx + nAy) 110:0 + (m2Ax2 + mnLx&y + n2Ly2) uo:o + (m3&3 + m2n&x20y + mn2AxA,,2 + n3Ay3) 140:0 + ....

uo:o
uo:1
uo:2
U1:0 u1:1 u1:2
U3:0
u3:1 u3:2

I20    FINITE DIFFERENCES
 Here the interval of differencing is 5. Changing the origin to the point (o, o) and the unit to I, the data are given for the points (o, o),
(I, o), (2, o); (o, I), (o, 2); (I, I). The value required is u.s..4. Differencing downwards for values of Ox 110:0 , etc., we have
Axu0:0 = '099; ',x2110:0 = — •ool.
Differencing across for values of D, u0i 0 , etc., 6.yu0:o = — 1'529; Dy211o:0 = '658.
Also    0x0yuo:o=111:1-uo:1-111:0+u0:o='013.
11.6:.4 = (I + •60x — •12Ax2 ...) (I + '4L , — .12Ay2 ...) 119:0 (I + •60x + •4~y — •12L,x2 + •2q.AxAy — .I204,2) 1/0:0
= 5'319.
 FUNCTIONS OF TWO VARIABLES    I21
Two-variable functions are of great frequency in actuarial work. Tables of annuity-values (ax:,,,) depending upon joint lives are often available for quinquennial values only of x and y, and when values at ages other than those tabulated are required recourse must be had to methods of interpolation. Although the formulae given above are of general application special methods can be found to meet the requirements of the problem to be solved.
For example, if quinquennial values of ax,, are available, and if the two ages concerned are such that their sum is a multiple of 5, we may choose our origin and interval of differencing so that x + y = I. We have then, from the general formula for ux:y,
ux • 1 _x — 160:0+ [x& + (I — x) + ix (x— I) (,x2 — 20x + A,2)] u0 ,o ;
i.e.
Again, if x + y = 2, this formula becomes
ux:2_x= 2x (x— 1)U2:0—x (x—2)u1:1+ 2 (x— I) (x— 2)u0:2,
or, on changing the origin,
ux _x = .y (x — I) (x — 2) uo:o — x (x — 2) 111:_1 + nx (x    I) 162:—2r
for which the data required are uo:o, ul:_1 and U2:__2. The problem is thus reduced to a single variable interpolation.
This second formula is very useful in practice. As a rule we can choose our data within wide limits, and it has been found that with certain functions the three-term formula gives as good approximations to the true results as do formulae involving higher orders of differences (see Spencer, J.I.A. vol. xL, pp. 293-301).
The general second difference formula of which the above is a particular example is
ux:rx = mu0:0 + uul:r + PU2:2r,
and in the note referred to above, Spencer gives a table showing the application of this formula according as r takes the values a, I,—Ior2.
Another form of the formula for an interpolated value of ux;y when
four values are given is ux:y = (duo: o + yuro: + x (71ul: o +yul:I), where
x and y are both less than unity and x + = y + = I. The second difference formula can be written as
ux: = {1 — (klcsx2 + k28y2)} { (nuo: o + yuo:1) + x 6111: o +

I22    FINITE DIFFERENCES
where 8x2u and 8,2u are second central differences with respect to x and y respectively and k1 and k2 are constants depending upon the values of x and y (Buchanan, T.F.A. vol. x, pp. 329, 330).
Example 3.
Values of the joint-life annuity ax;y for quinquennial ages being avail-able, find a value for a44:51
 FRASER'S GRAPHIC METHOD    123
work. The general theory follows the same lines as that for single-variable interpolation. It will be remembered that the ordinary advancing difference formula may be applied to the expansion of us in terms of the differences of us on the assumption that y = ux is a rational integral function of x. In these circumstances we may represent the function graphically, and the successive values of x and y will be points on the plane curve y = ux. When we are considering a function of two variables x and y we assume similarly that we may represent z = ux:, as a surface. Now in Chapter vi (para. Io) it was proved that the effect of including higher differences in the expansion for us does not necessarily give better results than if they are neglected. In the same way it may be shown that by choosing more points on which to work we may produce a result farther from the true value z on the surface z = ux:,, than we should obtain by relying on fewer data.
With regular data the formulae with x, y in the central area of the given points are usually preferable. In the space for which x, y are both positive and less than a simple central difference formula is
ux:v = 110:0 + $x (111:0 — u—1:0) + 2Y (110:1 — uo:—1)•
This is based on five points.
This formula and the six-point formula consisting of the same terms with the addition of
1x2 (U_i: — 2110:0 + 111:0) + Y2 (uo:l — 2110:0 + 110:—1)
+ xy (u, a. + 110:0 — 111:0—110:1)
are probably the most useful interpolation formulae for ordinary actuarial purposes (Todhunter, J.I.A. vol. LIII, p. 89).
11. Central difference formulae: Fraser's diagrams.
No demonstration of central difference formulae would be comwithout reference to Fraser's graphic method. In this method the ordinary differences of a function of x are combined with the relation (x + I)r = xr + xr_1 in diagrammatic form so that by adopting certain conventions any finite difference formula can be written down immediately (Fraser, J.I.A. vol. XLIII, pp. 235 et seq.).
We have    (x + t + 1)r = (x + t)r + (x +
or    (x + t + I)r — (x + t), = (x +
A relation similar to the fundamental finite difference identity

124    FINITE DIFFERENCES
ux+n — ux = Aux exists therefore between these coefficients. If we carry the analogy still further we can construct a table of values of (x + t)r corresponding to a difference table.
The two tables are set down thus :
u—2              
(x + 3)4
 
Du--2
          
(x + 2)3
  
u—1   02112      
(x + I)2
  
(x + 2)4
 
Du_1
  
A3u_2
  
xi
  
(x + I)3
  
u0   02u—1  
04u_2
  
x2
  
(x + 1)4
 
Au,
  
A3u_1
  
(x — 1)1
  
x3
  
ui   A2u,      
(x — I)_
  
x4
 
Dui
          
(x — I)3
  

u2    (x — 1)4
If now these two tables be combined, we have the following scheme :

(x — I)o>    >OZIo     x2(
    \    0311_1
— I)1<.    >A2uo     .


(x — 2)0<    >Aui—(x — I)2<


>112--(x — 2)1<
where for any one of the hexagons we may write in general


Oru_t    —(x + t)r+1
(x + t — 1)r/    \Ar+lu—t \Oru_ t+1 (x + t — I )r+i/
Now
(x + t — I)rr ru—t + (x + t)r+i Xr+1u_t — (x + t — 1~r+1,r+l u_t
— (x + t — 1)rAru_t+1
>u_2(x + 2)1<    jA2u_3—(x + 3)3
(x + Ip// >,u_2 (x + 2)2<\    z,A3u—3
>U_1—(X + I)1<    >,2u_2 (x + 2)3<


x0\    >Au—1 (x + I)2<    >A311_2


>uo     x1(    >A2 u 1 (x 1)3(


FRASER'S GRAPHIC METHOD(    125
(x + t — I)r [Oru-t — L1ru-t+i] + Qr+1u-t [(x + t)r+i
— (x + t — I )r+i] = (x + t — 1)r [— Or+1 u-t~ + L1 r-1 a-t (x + t — I )r
= O.
A relation is therefore established between the constituents of the various hexagons. If we make the following assumptions : (i) the oblique lines denote multiplication and the horizontal lines addition ; (ii) a line taken in a clockwise direction gives the product a positive sign, and in the opposite direction a negative sign, we can say that the sum of the operations performed in travelling round any hexagon is zero. It follows easily that the sum of the operations in travelling round any closed circuit is also zero.
It is evident from a consideration of the diagram that if we travel from any value of (x + t)k to any difference Omu,, the result will be the same whatever route be taken. For example, from (x — 1)0 through uo, xt,Au,ix2to A'no and back along (x — I)2, Dui, (x—2)i, u2i (x — 2)0i ul to (x — 1)0 again enables the following identity to be established :
(x — I)0u0 + x1L1u0 + x2L12u0 — (x — 1)2L12u0 — (x — 2)1`)1[1
- (x - 2)0u2 + (x - 2)01[1 - (x - I)01[1 = o.
Re-writing this, we have
u0 + xL1u0 + Zx (x — I) A2up    (x — I) (x — 2) !12u0
— (x— 2)(111—u2+u1—u1=0,
or
1[0 + xL u0 + Z x (x — I) A 2 u0 = u2 + (x — 2) L1 u1
+ (x — 2) (x — I),2u0.
Exactly the same result will be obtained by proceeding along an alternative route
(x — IL, u0 , xi , L1 u0 , x2, 21[0
and back through
(x — I)2, Au,, (x — I)i, ui, (x — 1)0.
The identity will be
(x - 1)0 1[0 + x(11[0 + x2021() - (x - I)2 t2u0 - (x - I)1 L u1
— (x — 1), u1 = O,

I26    FINITE DIFFERENCES
or uo+xAuo+Zx(x—I)A2uo=(x—I)(x—2)A2uo
+ (x— I)Alll+uli
i.e.    =2(x—I)(x—2)A2uo+(x—2)Atli +u2, the same result as before.

12. Application of the hexagon diagram.
The above example gives a formula for u2 in terms of u0, Au), A2uo and Au,, and if we put x = z we have a well-known identity. A similar process will give a formula for un, and since we may take various routes a number of different expansions of u„ will arise, all giving exact expressions for un. It should be further obthat when an nth difference has been reached by travelling along the upper route the terms other than u„ in the lower route will be zero, and it follows that by travelling round any circuit we obtain expressions involving an initial term u„ and terms of lower degree than n. This is seen to he so by considering A"uo: all the coefficients along the lower route will contain (x — n) as a factor and will therefore vanish when x = n.
We have therefore from the diagram the following expansions :
 THE OPERATOR V    127
the ordinary algebraic laws. By omitting the function u the various processes can be applied to the operators alone, with a resultant simplification of procedure. It will be seen later that the method can be adapted to the needs of the infinitesimal calculus, but before that stage is reached it is proposed to demonstrate the use of the method in other operations connected with finite differences.
14. The operator V.
In Chapter v attention was drawn to certain symbols of operawhich may be considered as supplementary to the A and E which are the basic operators in finite differences. These symbols —namely 8 and p,—may also be assumed to follow the normal algebraic laws (with the usual limitations), and the method of separation of symbols may be applied to them equally with A, E and A further symbol has been introduced connecting u,. with the next lower value ux_1 instead of with the more usual value ux+1. This symbol is V, and Vu, is defined as ux — ur_1 .
Corresponding to Dux = (E — i) ua., we have therefore    Vux = (1 — E-1) ur. Thus, for example,
Vn ux = (1 — E_1)n ux
= nx — nnx-1 + n221r-2 — n3ux-3 + ....
In addition to the familiar
Ox"m) = nnx(m-1),
there is a similar relation

Ox(-m) = nnx(-m+1)where x(-'n) - (— 1)'n (— x)(ni) and not the inverse factorial defined on p. 39; and if we denote the product
x (x + in — 1) (x -} in — 2) ... (x — in + 1)
by x[n] it is easy to show that
8x[m] = mx[m-1]
(Steffensen, Interpolation, pp. 8, 9).
No new principle is involved in dealing with these further symbols of operation ; their introduction simply enables us to

I28    FINITE DIFFERENCES
develop expansions and to write down formulae for interpolation with an economy of labour.

15. "Summation n."
An interesting example of the development of a series of operations by the method of separation of symbols occurs in the theory of graduation. One of the objects of graduation is to obtain a smooth series of numbers instead of the rough series given by the actual data. A method for the solution of the problem consists in replacing each term of the series by the arithmetic mean of the n successive terms of which the given term is the central term. The operation of summing these successive terms is generally denoted by [n] (" summation n ").
For example [5] U0 = u_2 + u_1 + u0 + u, + u,,
[n] uo = u_n—1 + u_n—3 + ... + Un—3 + un—I .
2    2    2    2
Consider a simple summation : [3] U0.
By definition    [3] up = u_1 + U0 + U1,
and if we write vo for [3] U0 we may operate again on vo to obtain
[3] vo.
In that event we shall have
[3] vo = [3] u—1 + [3] uo + [3] U1
= u_2 + 2u_1 + 3u0 + 2u, + U2
= uo+ (u_1+uo+u1)+ (u_2+u_1+u„ + u1+ u,)
= [I] uo + [3] uo + [5] uo.
If therefore we denote the double operation [3] [3] uo by [3]2uo, we have the symbolic identity
[3]2 = [1] + [3] + [5].
Similarly    [5]2 = [I] + [3] + [5] + [7] + [9],
and    [n]2=[I]+[3]+[5]+[7]+...+[2n—I], where n is odd.
The identity between [3]2 and [I] + [3] + [5] can be seen at once by writing down the terms in diagrammatic form :

"SUMMATION n"    I29 [3]2110 = [3] u—i + [3] uo + [3] 111
(Fraser.)
16. We can express [n] in terms of the ordinary finite difference symbols thus:
For a simple value of n, say 3,
[3]110=11_1+110+111