Calculus and Probability for Actuarial Students

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Term Life Insurance

CHAPTER IX
FINITE DIFFERENCES. FUNCTIONS OF TWO VARIABLES

1.
Questions involving functions of two variables arise frequently in actuarial practice. Thus the tabulated values of functions (e.g. annuities) dependent upon two lives may be given only for comof quinquennial ages in order to economise space. If the value corresponding to any other combination of ages is required, resort must be had to methods of interpolation.

In considering the problem of the changes induced in the

value of

f (x, y) by

finite changes in the values of

and

we must

consider

and

as being independent of each other. Clearly, if

were a function of

the expression

f (x, y)

could

be made to assume the form of a function of x alone and it could be dealt with by the methods already developed in previous chapters.

Thus x may vary while y remains constant, so that, if x changes to x + h, the value of the function becomes f (x + h, y); or y can vary while x remains constant, giving a value f (x, y + k); or both x and y can vary independently, giving a value for the function of f(x+h, y+k).

3.
We shall proceed first to discuss the problem where the values of the function are given for combinations of successive equidistant values of x and y.

Thus we may have

f (x, y)f (x + h, y)f (x + 2h, y)... f (x + mh, y)

f(x,y+k)

f(x+h,y+k)

f(x+2h,y+k)

...f(x+nth, y+k)

(x, y + nk) f (x+ h, y+nk) f (x+ 2h, y + nk) ... f (x + nth, y+ nk)

As has already been seen in the case of functions of one variable (Chapter III, 6), this scheme can be simplified, for the origin can be placed at the point (x, y), and the unit of measurement can be taken as h in the case of the variable x and k in the case of the variable y.

FUNCTIONS OF TWO VARIABLES    55
The scheme then becomes
f(0, 0)    f (¹, ⁰)    f(2, 0)     f(m, 0)
f(⁰, ¹)    f(¹, ¹)    f(², 1)     f(in, 1)

f(⁰, ⁿ)    f(¹, ⁿ)    f(², n)     f(m, n)

4. Since x and y may vary independently, a fresh scheme of notation must be introduced to express the variations which may arise. Thus A_z will be used to denote the operation of differencing with respect to x, y remaining constant, a corresponding significance attaching to A_y, so that

^Az_,f (⁰, 0) ⁼.f (¹, 0) ^—f (⁰, 0),
A_yf (0, 0)=f(0, 1)—f(0, 0), or, using the method of separation of symbols,

f (¹, ⁰) = (¹+ ^Az)J (⁰, ⁰)^.
Accordingly, we have
f(m, n) = (1 +A_z)^m(1 +^Ay)^"f(⁰, ⁰)

=(1+mOz+(2)o'z+...)(1+n0y+(2)O'y+...)(0,0)
=f(0, 0) + ^mzzf(⁰,⁰)⁺(2) ^A'z f (⁰, 0) + (3) ^A3zf (0, 0) + ...
+nAyf (0,0)+mn A z f (⁰,⁰)+ (2) nA²_zA,f (0, 0)+...

+ (2).Vy f (⁰, ⁰) + ^m(2) AzA2yf (0,0) + ...
+ () Q3f (0, 0) + ...

(1).

Here 0'_z, A'_z, ... can be written down by differencing the rows of the table of the function; similarly A'_y, ,'_y, ... are the differences of the columns of the table.

56 FINITE DIFFERENCES To find ~xOy, 6,²_x A_y, etc. we have
^Ax^Ayf(⁰, 0) = ^Ax [f(⁰, ¹)-f(⁰, ⁰)]

=f(1,1) — f (0, 1) — f (1, 0) + f (0, 0),

^O,

yf(

⁰

)

^=O2

x[f(

⁰

)

^—

⁰

)]

=f(2,1)—2f(1, 1)+f(0, 1)—f(2, 0)

+2f(1,0)--f(0,

0),

and so on.

Example 1.

Table XVI of the "Short Collection of Actuarial Tables.

To find

_A31:83,

having given

A31,,so

•

11669,

35:so

•

13190,

₄

_i80

•

15494,

3Uc85

•

09809,

₃

•

11039,

3o:70 =

.07812.

Here

= *,

and

•

01521,

= •00783,

_Ox

= — .00291,

6,y=—•01860,

=—•

00137.

Whence

₃

_i83

•

10776,

the correct value being

•10773.

5. An obvious method of procedure involving only first differences is as follows. Obtain the value off (0, n) by interpolation between the values of f(0, 0) and f (0, 1). Similarly, obtain the value of f (1, n) from the values off (1, 0) and f (1, 1). Finally find f (na, n) by interpolation between f (0, n) and f (1, n). Thusf(0, n) = 1 —nf(0, 0)+nf(0, 1), f(1, n)=1—nf(1, 0)+nf (1, 1), f(m, n)=1—nmf(0, n)+mf(1, n)

= f (0, 0) + m

f (0, 0) + nOyf (0,

mnA

y.f

(

⁰

=1—ml—nf(0,0)+n.1—mf(0,1)

+m.1—nf(1, 0)+mnf(1, 1) ...(3).

Employing this formula in the example given above we find

^Aslss = 10822.

The method is suitable

only a rough approximation is required,

but cannot be depended upon to give an accurate value.

FUNCTIONS OF TWO VARIABLES 57

Obviously the method can be extended by taking higher orders of differences. The disadvantage of this procedure is that it involves the calculation of further values of the function correto a given value of x as a preliminary to applying the interpolation formula to find the value of f (x, y). The process thus becomes laborious and moreover we do not necessarily obtain identical values for f (x, y) if we interpolate first with regard to x and then with regard to y, or vice versa.

As in the case of functions of one variable, we shall expect to obtain the best results when the principles of central differences are applied, i.e. when the required term occupies as nearly as posa central position among the terms employed in the formula. The difficulty is that, in dealing with functions of two variables, we cannot adapt our formulas to any system of values which may be given. Thus an inspection of the advancing difference formula (1) shows that it involves points whose coordinates form a trianguplan which may be illustrated thus:

(0, 2)

(0, 1) (1, 1)

(0, 0) (1, 0) (2, 0)

000

This illustrates the formula where two orders of differences are taken into account, the black dot representing the interpolated term. It will be seen that the scheme is hardly satisfactory from the point of view of central differences. For most practical purhowever, where ordinary actuarial functions are involved, formula

(1) will give satisfactory results.

Formulas embodying the principles of central differences can conveniently be obtained by an adaptation of Lagrange's forThis formula applied to functions of two variables has not the same wide application as the ordinary formula of Lagrange previously given in Chapter IV, but, as will be seen below, it gives expressions for f (x, y) in terms of the neighbouring values.

58 FINITE DIFFERENCES

9.General _.formula for 4 points.

Taking all combinations of two terms except those which give rise to

and

let

f(x,

y)=A(x—/3)(y—b)+B(x—0)

(y— a) + C (x – a) (y — a)

+ D (x — a) (y —

b),

then

f (a, a) = A (a — 0) (a — b),

f(a,b)=B(a–/3)(b–a),

f(13,a)=D(/3–a)(a–b),

f(/3,b)=C(,(3— a)(b–a),

whence, substituting for

A, B,

C and

in the original formula,

(

, y)=f

(

a)(x—0)(y—b)+

f(a,

b)(x—13)(y—a)

(a—/3)(a—b)

(a—/3)(b—a)

(R,

b)(a—a)(b—a)+f (Q, a)($—a)(a—b)... (4).

10.General formula for 6 points.

Taking all combinations of two terms, let

f(x,

y)=A(x–/3)(y–b)+B(x–/3)(y–a)+C

(x – a) (y – a)

+D(x–a)(y–b)+E(x–a)(x--0)+F(y-

-a)(y–b).

Taking the points

a :a, a : b, a : c, /3 : a, 0 : b

and

y :

and pro

ceeding as before, we arrive at the result

.f(x,y)=.f(a,a){(x—l3)(y—b)+(x—a)(x—/3)+(y—a)(y—b)l

(C3—a)(b—a) (/3—a)(7 —a) (b — a) (c — a)

— .f(

)

((x—S)(y—a)+(y—a)(y—b)~

((

—a)(b—a)

(b —a)(c—b)

+.f(a,c)(y—a)(y—b)

(c—a)(c—b)

—f(R,a) l(/3—a)(b—a)+(R—a) (y — R)l

f(R,b'(0—a)(b–a)

^+.f(7,a)

(

-a)(y—

(5).

FUNCTIONS OF TWO VARIABLES 59

11.General formula for 9 points. Taking all combinations of two terms, each involving x, with two terms each involving y, let

f (x, y) = A (x –

(x —

₇

)

(y —

b) (y – c)

+B(x—4)(x—y)(y—a)(y—c)+C(x—(x—y)(y—a)(y—b)

+ D (x — a) (x —

₇

)

(y — b) (y — c) + E (x — a) (x —

₇

)

(y — a) (y — c)

+F(x–a)(x–y)(y–a)(y–b)+G(x–a)(x–/3)(y–b)(y–c)

+ H(x – a) (x -4)(y – a)(y – c) + I (x – a) (x –

(y –

a)(y – b).

Whence, proceeding as before, we have

.f(x,y)=f(a,a)(x—i3)(x—y)(y—b)(y—c)

(a — 4)(a –

₇

)

(a — b) (a — c )

f (ab)

(

—

(

-7)(y

—

)

(

—

)

(^a—8) (^a— 7) (^b— ^a) (^b— c)

(x—/3)(x—y)(y—a)(y—b)

+f (a, c) (a—4)(a—y)(c—a)(c—b)

f(R,a)(x—a)(x-7)(y—b)(y—c)

(4–

a) (J3—

₇

) (a — b) (a — c)

+f(13,b)(x-a)(x—7)(y—a)(y—c)

(R—

)

($—7)(b—a)(b—c)

+.f(8,c)(x—a)(x—7)(y—a)(y—b)

^3—

)($

^—y

) (c — a)

(cb)

_J(7

a)(x—a)(x—(~')(y—b)(y—c)

(7 —a)(y—4)(a—b) (a—c)

+.f (7, b)

(

—

)

(

—

^s)

(

—

)

^—a

)(7

^—

(b—a) (b—c)

(x —

a)(x

—

)(y —

)

+.f(7,c) (7 —

)

(7 —

de)

(c —

)

(c—b)

(G).

Formula (4) is the general formula corresponding to the method of 5; by altering the notation the identity of the two formulas (3) and (4) is apparent.

Formula (5) includes six values of the function, the co-ordinates being related in the manner shown. It will be seen that the formula

60 FINITE DIFFERENCES
can be applied to any of the following groups of values, the black dot representing the interpolated value:

(0, 2) (0, 1) (1, 1) (0, 1)

o0 0 0

(0,

(1,

1)(-1,

(0,

(1,

0)(-1, 0) (0, 0) (1, 0)

q0 0 0 0 0 0 0

(0, 0)(1, 0)(2, 0)(0, -1)(

-1,

-1) (0,

-I)

q0 o O o 0

(i)(ii)

(iii)

(0,

1)(I, I)

(1, 1)

000

( —

1 ,

0)(0, 0)(1, 0)(-- 1, 0) (0, 0) (1, 0)

000

(1,

)(-

, -1)

^(1,

-1)

000

(iv)

(v)

Other systems could be written down, but the above are suffor purposes of illustration.

System (i) is obviously the same as the advancing difference formula. It is obtained by writing a = 0, = 1, 7 = 2, a = 0, b =1, c = 2 in formula (5).Example 2. Find A₃1_i from the data of Example 1, using the Lagrange formula applicable to system (i). It will be found that the same result is obtained.System (ii) should be expected to give a formula which will be more accurate than the advancing difference formula, since the interpolated term will occupy a more central position in relation to the terms employed. The formula is obtained by putting a = 0, ,Q =1, y = -1, a = 0, b = 1, c = - 1 in formula (5). Working on the same example as before and taking the origin at the point (30, 60), the values of the function entering into the formula are

,=	= •10080,	1₃ _i_6;_ •09809,	A ₃1.= •11669,
	^A3/.L⁼•¹⁴⁰⁰⁶,	A₃;.=•11039,	A₃.=•13190.

We obtain as a result A₃ :₈₃= •10770. The degree of approxithough close, has not, in this instance, been improved.

FUNCTIONS OF TWO VARIABLES 61
System (iii) is an inversion of system (ii) and should be useful for interpolation where x and y have negative values.
The lack of symmetry of systems (iv) and (v) suggests that they are not likely to yield good results in practice.
13. When nine points are used, as in formula (6), the system is represented by the following diagram :

(-¹, ¹⁾_(0,¹⁾(¹, ¹)

(-1, 0) (0, 0)• (1, 0)

0 0

(-1,

-1) (0, -1) (I,

-1)

q0 0

It will be seen that this scheme embodies all the principles of central differences and should therefore give good results.

Taking the previous example with the origin at the point (30, 60) the six values entering into the formula for system (ii) are used together with the following additional values :

, = •

08435,A

₂

•

12132,A„1,,, = •15972.

Making use of formula (6) the interpolated value is found to be •10771, a slightly better approximation to the true value than those obtained previously.On general reasoning we should expect a somewhat better result by taking the origin at the point (30, 65) so that the interpolated value would occupy a more central position. The values A₂1_:5,, A, : _bo, A2_5: , entering into the immediately preceding calculation are excluded, and the following values introduced :

₂

₇₀

= •

06642,A

₃

₇

•07812,

:70 = •

08756.

On working out the result, however, we arrive at the value ^.10848, which is a worse approximation than the value obtained by the rough method of 5.14. This apparent inconsistency illustrates one of the chief difficulties of interpolating between functions of two variables, namely, that one does not necessarily obtain a better degree of approximation by proceeding to a higher order of differences or by employing more terms in a formula. Changes in the value of f (x, y) occasioned by alterations in the values of and y may be so con-

62 FINITE DIFFERENCES

siderable that distant terms may have such a disturbing effect upon the formula used as to upset the agreement between the approximate interpolation surface and the true surface which represents f (x, y).
It is thus difficult to say what will be the degree of approximaof a given formula, but an inspection of the course of the differences will be some guide as to the advisability of introducing further terms into the calculation.
15. Other devices may sometimes be adopted which enable the interpolation to be reduced to the work of a single variable inter

Thus, if the sum of x and y is a multiple of 5, by suitably selecting the origin we may write

f(x, —x)=f(0, 0)+x[f(1, -1)—f(0, 0)]
+ (z) [f (2, — 2)—2f(1, -1)+f (0, 0)]
x—lx—2
= ₂ f(0, 0)—x.x—2f(1, -1)
x.x-1

By referring to the point diagrams on previous pages it will be seen that the process is equivalent to interpolating along a diagonal line running through the various points. The formula is of the advancing difference type ; the corresponding central difference formula would preferably be employed in practice.

J. Spencer has given (J.I.A. Vol. 40, pp. 296—301) examples of the use of several ingenious methods of this character.