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					JIA 95 (1969) 79-113

                                                                                      79

            A STOCHASTIC              APPROACH            TO    ACTUARIAL
                                      FUNCTIONS
            BY A. H. POLLARD,      M.Sc., M.SC.(ECON.), PH.D., F.I.A. AND
                         J. H. POLLARD,   B.Sc., PH.D., A.I.A.
                    (Macquarie University and University of Cambridge*)
       * This paper was completed while the second author was a member of the Population
     Research and Training Centre at the University of Chicago.


                                   1. INTRODUCTION
    THE classical approach to actuarial problems has been deterministic.
    That is, the assumption has been made that the number of policies in
    force is large, that chance fluctuations are unimportant, and that the
    use of mean values is adequate. The actual cost to a company of an
    annuity, for example, may in fact vary from nothing (if the annuitant
    dies immediately) to a large amount (if he lives to be a nonogenarian).
    On average, the cost will be ax, and for many purposes, this value is good
    enough.
       There are circumstances, however, when we could be interested in more
    than the average value of the annuity ax, and a knowledge of the likely
    variations about that average could be important. A study of the variance
    and high-order moments of actuarial functions and the possible applica-
    tion of these results has no doubt been avoided because of the numerical
    work involved. With the development of computers, however, this is no
    deterrent, and we can now learn a great deal more about a company’s
    business at little cost when computing facilities are available.
       Although the classical approach to actuarial problems has been deter-
    ministic, no life office is large enough to ignore completely chance fluctua-
    tions in mortality. Most actuarial functions are random variables, and
    should be regarded as such when the effects on a life office of chance
    fluctuations in mortality rates are being investigated. Actuarial random
    variables will be denoted in this paper by the usual (deterministic) symbols
    with tildes above them. Thus, ãx denotes the present value of an annuity
    of one per annum on a life aged x. This is a random variable with expected
    value ax,. Similarly, Ãx denotes the present value of an assurance of one
    on a life aged x. The expected value of Ãx is Ax.
       The moments of these actuarial random variables may be readily
    computed. Such calculations are discussed in Part 2, and certain proper-
    ties of the random variables involved are also described. Some numerical
    examples are given in Part 3, and these indicate certain uses to which
    the higher-order moments of life assurance functions may be put. Finally
    in Part 4, the problem of retention limits and reassurance arrangements
    is discussed.
       F
80          A Stochastic Approach to Actuarial Functions
    2. THE MOMENTS OF CERTAIN ACTUARIAL               FUNCTIONS

  An annuitant aged x will receive zero payments with probability
(dx)/(lx), exactly one payment with probability (dx+1)/(lx), exactly two
payments with probability (dx+2)(lx), etc., and these probabilities sum
to unity. The present values of the payments received in each case are
                  respectively. Mathematically, we may write
                                                                       (1)
The rth moment of the annuity ãX about the origin is given by

                                                                       (2)

and the first moment, or expected value, is

                                                                        (3)

   Similar results for assurances, endowment assurances, temporary
and deferred annuities, etc. may easily be written down, and the extension
to linear multiples of such functions when independent is straightforward.
The equations for whole-life assurances, corresponding to equations
(1), (2) and (3) are
                                                                        (4)



and



   A life annuity and a whole of life assurance, both on a single life aged
x, are not independent, and indeed it may be shown that one is a linear
function of the other.
   Consider


and




                                                                       (8)
             A Stochastic    Approach    to Actuarial   Functions            81
Hence ãx is a linear function of Ãx.
   Similarly, it may be shown that                is a linear function of
We conclude therefore that the coefficients of correlation            between ãx
and Ãx and between                   and      are both –1. The situation with
temporary      annuities and temporary      assurances   is different. Although
                        are completely dependent, the relationship is not linear
and hence the correlation coefficient between the two random variables
has modulus less than unity.
   Another random variable which is of considerable interest is the policy
value        It satisfies the relation

                                                                             (9)

                                                                            (10)
where
Equations (8), (9) and (10) show that the three random variables
and ãx+t are linearly dependent in pairs, and hence the correlation co-
efficient between any two of the three random variables has modulus
equal to unity.
   We now consider ãx, Ãx, and       in greater detail. The expected values,
variances,   third central moments, fourth central moments, skewness
coefficients       and kurtosis coefficients ß2 were computed on TITAN,
the computer of the Cambridge University Mathematical          Laboratory  at
2½% and 5% interest rates using A1949–52 ultimate mortality for all ages
over 19 years. The covariance and correlation      coefficient of ãx and Ãx
were also computed. A summary of the results obtained is given in Table
1, and more extensive tables are included in the Appendix (Tables Al
and A2). The entire computation    at each rate of interest took 23 seconds.
   The following points should be noted:
      1. The standard deviation of ãx is substantial, and for many ages is
  greater than half the mean value ax. Consider age 60 for example. At
  2½% interest, the expected value of ã60 is 12·507 and its standard
  deviation is 5·681. For age 60 and 2½% interest, one annuity in twenty
  is likely to have a present value greater than about 22·879.
     2. The expected value of ãx varies considerably      with interest rate
  and so does the variance.
     3. The correlation coefficient between ãX and ÃX is – 1.
     4. The random variable Ãx is identical with the random variable
  ãx except for changes in sign, scale, and origin. Hence the coefficients of
  kurtosis are identical, and the coefficients of skewness are identical
  except for sign.
     5. The expected value of ãX decreases with age, whereas the standard
  deviation increases monotonically up to a certain age and then decreases.
   Similar calculations were performed   for endowment assurances maturing
at age 45 and at age 60. Summaries       of the results obtained are given in
82              A Stochastic   Approach     to Actuarial   Functions
Tables 2 and 3, and more extensive         tables are included     in the Appendix
(Tables A3 to A6).

Table 1. Summary       table of moments and moment functions        of ãx and Ãx at
                                  2½% interest
                Moments and moment functions of ãx at 2½% interest
                                        Standard
            Age Expected     Variance deviation                  a
            20     28·28      21·29       4·61       –2·53     12·16
             30    25·35      24·24       4·92       –1·74      7·18
            40     21·61      30·20       5·50         –1·21 4·48
             50    17·18      34·94       5·91       –0·76      3·06
             60    12·51      32·28       5·68       –0·31      2·31
             70     8·09      22·86       4·78         0·16     2·20

                  Moments and moment functions of Ãx at 2½% interest
                                          Standard
              Age Expected     Variance deviation                  ß2
              20      ·2858     ·0127       ·1125       2·53    12·16
               30     ·03573    ·0144       ·1201       1·74      7·18
              40      ·4484     ·0180       ·1340       1·20     4·48
               50     ·5565     ·0208       ·1442       0·76      3·06
               60     ·6706     ·0192       ·1386       0·31     2·31
              70      ·1782     ·0136       ·1166     –0·16      2·20

         Moments, moment functions and product moments of ãx,and Ãx
      Age   E(ãx)    E(Ãx) Var.(ãx) Var.(Ãx) Cov.(ãx, Ãx,) Correlation
      20    28·28    ·2858     21·29     ·0127    –·5193        –1·00
      30    25·35    ·3573     24·24     ·0144    –·5911        –1·00
      40    21·61    ·4484     30·20     ·0180    –·7366        –1·00
      50    17·18    ·05565    34·94     ·0208    –·8523        –1·00
      60    12·51    ·6706     32·28     ·0192    –·7874        –1·00
      70     8·09    ·7782     22·86     ·0136    –·5577        –1·00

The following     points should be noted:
     1. The   coefficients of variation      (= standard      deviation/mean)       of
            and            are considerably    smaller than the coefficients of
  variation of &·
     2. The standard deviations of                  and              decrease with
  age.
     3. The expected values and standard             deviations of               and
           vary considerably with interest rate.
    4. The correlation      coefficient between               and            is –1,
  and so is the correlation coefficient between                  and
     5. The random variable              is identical with the random variable
            A Stochastic    Approach   to Actuarial      Functions              83
              except for changes in sign, scale and origin. Hence the co-
  efficients of kurtosis are identical, and the coefficients of skewness are
  identical except for sign.
     6. The values of         and ß2 are rather large. This phenomenon      is
  caused by the unusual distribution of                  the distribution has
  most of its weight situated at the value
  We have not mentioned     any calculations   for           Its moments    may be
computed as follows.

Since



Hence the variances of policy values may be obtained from the variances
of annuities. From Tables 1 and 2 and the Appendix, we thus compute the
information in Table 4.

Table 2. Summary   table of moments and moment functions          of           and
                                  at 2½% interest
         Moments and moment functions of                at 2½% interest
                                    Standard
          Age Expected Variance deviation                         ß2
          20     17·64     2·63       1·62          –7·87        68·78
          25     14·81     1·59       1·26          –8·54        80·83
          30     11·58      ·81        ·90          –9·38        97·36
           35     7·90      ·31        ·56         –10·57       122·77
          40      3·74      ·05        ·23       –13·25         189·74

         Moments and moment functions of                at       interest
                                     Standard
          Age Expected   Variance    deviation                    ß2
          20     ·5453    ·001564      ·0395           7·87      68·78
          25     ·6145    ·400946      ·0308           8·54      80·83
          30     ·6932    ·000486      ·0220           9·38      97·36
          35     ·7827    ·000186      ·0137          10·57     122·77
          40     ·08843   ·000031      ·0056          13·25     189·74

  Moments, moment functions and product moments of         and
                             Var.         Var.
Age                        (annuity) (assurance)   Covariance Correlation
20    17·64      ·5453        2·63      ·001564     –·06413     –1·00
25    14·81      ·6145        1·59      ·000946       –·03879   –1·00
30    11·58      ·6932         ·81      ·000486       –·01992   –1·00
35     7·90      ·7827         ·31      ·000186       –·00765   –1·00
40     3·74      ·8843         ·05      ·000031       –·00127   –1·00
84           A Stochastic   Approach     to Actuarial    Functions
Table 3. Summary    table of moments and moment functions            of          and
                                  at 2½% interest
           Moments and moment functions of              at 2½% interest
                                     Standard
           Age Expected Variance deviation                        ß2
            20    24·00     8·02       2·83           –5·37      35·13
            30    19·81     5·78       2·40           –4·94      30·35
           40     14·42     4·10       2·02         –4·39        23·49
            50     7·66     1·47        1·21        –4·43        22·83

           Moments and moment functions of              at 2½ interest
                                       Standard
           Age Expected   Variance     deviation               ß2
           20     ·3903    ·004771       ·0691          5·37     35·13
           30     ·4925    ·003436       ·0586          4·94     30·35
           40     ·6240    ·002437       ·0494          4·39     23·49
           50     ·7887    ·000876       ·0296          4·43     22·83

  Moments, moment functions and product moments of         and
                               Var.        Var.
 Age E(annuity) E(assurance) (annuity) (assurance) Covariance Correlation
 20    24·00       ·3903        8·02     ·004771    –·19563     –1·00
 30    19·81       ·4925        5·78     ·003436      –·14088   –1·00
 40    14·42       ·6240       4·10      ·002437      –·09990   –1·00
 50     7·66       ·7887        1·47     ·000876      –·03593   –1·00

  The following   points may be noted:
      1. The coefficients of variation of             are considerably smaller
  than those of
     2. The expected value of                increases with duration whereas
  its standard deviation decreases with duration.
     3. The expected value of         increases with duration; up to a certain
  duration the standard deviation does also, but it then decreases.
     4. The higher rate of interest produces lower expected values but
  higher coefficients of variation.
     5. The higher rate of interest produces smaller standard deviations
  at the shorter durations but larger standard deviations at the longer
  durations.
   Similar computations      may be performed    for temporary    assurances,
pure endowments,    joint-life annuities, and joint-life assurances. We do
not discuss them here. The following theorem, however, should be noted.

  THEOREM.     Let Ã(y) be an assurance       and         an annuity      dependent
upon the same status (y), and such that:
                  A Stochastic        Approach    to Actuarial     Functions              85
        1. the sum assured is constant and independent         of the mode                    of
           decrement,
        2. the rate of interest is constant,
        3. the premiums payable are at a level rate, and
        4. the status is in force from time 0 until it comes to an end.


                  Table 4. Coefficients of variation of policy values
                      Coefficient of variation of         at 2½% interest
                                                              Coefficient of variation
                                                  Standard = standard deviation/
        Age = 20+t     Expected        Variance deviation            expected
             20          0              ·0266       ·163
             30              ·104       ·0303       ·174               1·67
             40              ·236       ·0378       ·l94                 ·82
             50              ·392       ·0437       ·209                 ·53
             60              ·558       ·0404       ·201                 ·36
             70              ·714       ·0286       ·169                 ·24
                                          at 5% interest
             20          0              ·0138       ·117
             30              ·057       ·0178       ·133               2·33
             40              ·147       ·0285        ·169               1·15
             50              ·278       ·0433       ·209                 ·75
             60              ·439       ·0505       ·225                 ·51
             70              ·614       ·0437       ·209                 ·34

                    Coefficient of variation of             at 2½% interest
             20         0             ·0084       ·092                  00
             25           ·161        ·0051       ·071                  ·44
             30           ·344        ·0026       ·051                  ·15
             35           ·552        ·4010       ·032                  ·06
             40           ·788        ·00017      ·013                  ·02
                                        at 5% interest
             20         0             ·0073       ·085                   00
             25           ·123        ·0050       ·071                   ·58
             30           ·280        ·0029       ·054                   ·19
             35           ·482        ·0013       ·036                   ·07
             40           ·741        ·00025      ·016                   ·02



Then it may be shown that
  (i)    the expectations           of Ã(y) and         obey the premium         conversion
         relation
86              A Stochastic   Approach   to Actuarial       Functions
                                    are linearly   related     in pairs,    the linear
         relations   being

                                                                                    (11)
and                                                                                 (12)
     (iii) the correlation coefficient between any two of the three              random
           variables has modulus equal to unity.
   PROOF:      Conditions   (1) to (4) are the usual conditions for the ap-
plicability of premium conversion relations. Therefore result (i) applies.
(See, for example, Life and Other Contingencies, Vol. 2, pp. 75–6, by P.
F. Hooker and L. H. Longley-Cook.)           To prove (ii), we first note that
we may construct a life table for the status. It is then possible to write
down equations      corresponding    to (7), (8), (9) and (10). From these
equations, we conclude that            is a linear function of        and that
      is a linear function of         Result (iii) follows as a consequence of
result (ii).
   Note 1. These results apply to joint-life annuities and assurances.
   Note 2. The time unit of a year is arbitrary. Similar results apply if
premiums are payable m times per year, or if premiums are payable
continuously.


              3. APPLICATION OF SECOND-ORDER MOMENTS
   Example 1. A company issues life annuities to retired persons aged 60.
The annuities, payable annually in arrear, are issued for multiples of £100
per annum, and the lives currently receiving annuities are listed in Table
5. Many of the annuities were issued some years ago when interest rates
were high, but these have now fallen, and it is likely that they will remain
at 2½% per annum for a long time. The annual instalments due have just
been paid, and the fund is valued at £3,102,000. What is the probability
that the fund will be insufficient to meet all annuity payments? Mortality
is according to the A1949–52 (ultimate) Table.
   Let X be the random variable representing       the present value of the
liabilities of the fund, and let      be the number of lives aged x currently
receiving £(l00j) per annum. Then

                                                     = 2,982,782,          and

                                                     = 2,638,352,204.

Hence the standard deviation of X is 51,360.
   From the Central Limit Theorem, we known that X is approximately
normally distributed. Hence the probability  that the fund will be in-
sufficient is
            A Stochastic       Approach   to Actuarial   Functions         87




which is approximately       ·01.

   Table 5. The age structure of the annuity fund in Example 1 of Part 3
                           Number of annuities of
         Age x    £100 p.a. £200 p.a. £300 p.a. £400 p.a. £500 p.a.
          60         15        25        27        33        3
          61         10        15        40        20        5
          62          8        15        30        18        2
          63         10        20        31        20        0
          64          8        10        12        13       10

           65        33             47       77          20           1
           66        26             24       39          33          23
           67        21             29       31          17           5
           68        15             17       19          14          12
           69         9             13       10          11           1

           70            7          12        9          12          0
           71            5          11        2           4          0
           72            4           9        1           3          1
           73            4           3        2           1          0
           74            0           2        0           0          0

           75            0          0         0          0           0
           76            0          1         0          0           0
           77            0          1         0          0           0

   A note concerning the variance of the sum of the present values of
n annuities on n independent lives all aged x is relevant here. If all the
annuities are for one per annum, the present value of the sum of the
annuities has expected value näx and variance n σ2 where σ2 is the variance
of      By comparison, an annuity of n per annum on a life aged x has
the same expected value, but a variance of n2 σ2.Thus the present value
of an annuity of n per annum on one life (x) and the present value of n
annuities of 1 per annum on each of n independent lives (x) have the
same expectation näx but in the latter case the standard deviation is
times that of the former.
   The present value of r annuities on r lives               where the total
annual payment is n, has a standard deviation between a and
depending on how uneven the distribution        is by size of the r annual
payments.
   The following is a more detailed analysis. Consider n independent
88            A Stochastic        Approach      to Actuarial     Functions
lives all aged x. Let the total annual sum payable be S, the smallest annuity
be a per annum and the largest b per annum. If we denote the variance of
            we require bounds for the variance of the sum of the present
values.
   Consider first the lower bound. Let the ith annuity be of size xi per
annum. At least one annuity must be for b per annum, and at least one
must be for a per annum.
     Our problem   is therefore      the following:

Minimize                                                                            (13)

subject to                                   x1 = a,
                                             x2 = b,
                                                                                    (14)


and
This is equivalent     to minimizing

                                                                                   (15)

subject to                                   (i = 4, 5, . . ., n).                  (16)
  The minimum value may be determined                     by differentiating    (15) and
equating to zero. The xi values so obtained               satisfy constraint   (16). The
minimum value occurs when

                               x1 = a,
                               x2 = b,
and                            xi = (S–b–a)/(n–2),

and it is equal to        [a2+b2+(S–a–b)2/(n–2)]               σ2
     To maximize     the variance,     we must maximize                             (17)


subject to                                                                          (18)

and                                          (i = 3, 4, . . ., n).                  (19)
Let us start with the minimum value. It is possible to increase x3 and
decrease xn equally in such a way that constraints (18) and (19) still apply.
The sum of squares (17) is increased. This process must cease when
either x3 = b or xn = a. However, it is possible to continue with different
pairs of Xis until all except one of them are equal to either a or b. The
              A Stochastic       Approach   to Actuarial    Functions                 89
remaining xi will satisfy inequality (19). We then have the maximum
value of the variance.
   When the maximum         value occurs, k of the xis are equal to b,
(n- k - 1) of the xis are equal to a, and one of them has the value c, where
           The following must be true:

                             kb+(n–k–1)a+c=S.

Therefore,                   c=S–(n–1)a–(b–a)k.

But c satisfies the inequality    (19), and we therefore   obtain


                                                                                     (20)

This relation gives us k, since we know it must be an integer, and we can
                                                                    σ
then determine c. The maximum variance is [kb2 + (n – k – 1)a2 + c2] 2.
   The difference between the maximum and minimum variances is equal to

         [kb2+(n–k–1)a2+c2–a2–b2–(S–a–b)2/(n–2)]                          σ2
         [k(b2–a2)+(n–2)a2–(S–a–b)2/(n–2)]        σ2                (since c   b),
        [(S–na)(a+b)+(n–2)a2–(S–a–b)2/(n–2)]                   σ 2,(using(20)).

This last result will only be of use when both a and b are small relative
to s.
   Consider for example the case in which n2 = 120, S = 50,000, a = 200
and b = 500. The minimum variance = 20,887,373ó2, and the difference
                  The standard deviation therefore lies between 4,570ó and
4,818ó. This range is only 5% of the minimum possible standard deviation.
   Example 2. It is desirable to have some idea of the likely range of
variation in the actuarial liability of a life office or pension fund. The
percentage variation from the mean could vary considerably          from one
office to another depending upon the distribution of type of policy written,
age distribution,   and distribution   of policy size. The character of the
business is influenced by such factors as method of payment of com-
mission (on sum assured or on premium), retention limits, area of activity,
etc. Some assessment of the range of variation is particularly desirable
when working with a closed fund, and when large amounts are at risk for
small premiums (e.g. accidental death benefits and term assurances). For
an accurate assessment, two or more policies on the one life must be
grouped, and moments may be calculated using the techniques of Part 2. A
comparison of the random variable representing        the total policy values
with the company reserves can then be performed           along the lines of
Example 1.
   A numerical example is instructive.     Consider a hypothetical     life as-
surance fund covering the 34 separate lives in Table 6. The policies are
90              A Stochastic       Approach      to Actuarial   Functions
           Table 6. Hypothetical      life assurance fund—policies         in force




     (1)      (2)      (3)     (4)        (5)           (6)         (7)           (8)
      1       1,000    40      5        ·20732       ·012408      207·32        12,408
     2        2,000    35      6        ·18982       ·011675      37964         46,700
      3       2,500    33      3        ·08248       ·012791      206·20        79,944
      4       2,000    42      1        ·04505       ·017201       90·10        68,804
     5        2,000    33      5        ·14091       ·011919      281·82        47,676

      6      1,500    41       9        ·41499       ·006720      622·49        15,120
      7      1,000    35       6        ·18982       ·011675      189·82        11,675
      8      1,500    27       4        ·08357       ·011378      125·36        25,601
      9      1,500    42       2        ·09111       ·015979      136·67        35,953
     10      2,500    39       7        ·27945       ·010409      698·63        65,056

     11       3,000   33       5        ·14091       ·011919     422·73        107,271
     12       1,000   41       6        ·26761       ·010847     267·61         10,847
     13       2,500   42       2        ·09111       ·015979     227·78         99,869
     14       2,000   39       7        ·27945       ·010409     558·90         41,636
     15      1,000    27       3        ·06187       ·011744      61·87         11,744

     16        500    25       1        ·01837       ·012400       9·19          3,100
     17      1,500    42       1        ·04505       ·017201      67·58         38,702
     18      2,000    35       6        ·18982       ·011675     379·64         46,700
     19      2,500    39       6        ·23690       ·011500     592·25         71,875
     20      2,500    25       1        ·01837       ·012400      45·93         77,500

     21      1,000    39       7       ·27945        010409      279,45         10,409
     22      1,500    42       1       ·04505        ·017201      67·58         38,702
     23      2,000    27       3       ·06187        ·011744     123·74         46,976
     24       2,000   41       6       ·26761        ·010847     535·22         43,388
     25      1,000    39       6       ·23690        ·011500     236·90         11,500

     26      1,000    25       1       ·01837        ·012400      18·37         12,400
     27      2,500    27       4       ·08357        ·011378     208·92         71,113
     28      3,000    42       2       ·09111        ·015979     273·33        143,811
     29      2,500    35       6       ·18982        ·011675     474·55         72,969
     30      1,000    33       3       ·08248        ·012791      82·48         12,791

     31      3,000    40       5       ·20732       ·012408       621·96       111,672
     32      2,500    35       6       ·18982       ·011675       474·55        72,969
     33      2,000    40       5       ·20732       ·012408       414·64        49,632
     34      1,000    41       9       ·41499       ·006720       414·99         6,720
                                                  TOTALS        9,798·21     1,623,233
             A Stochastic    Approach   to Actuarial   Functions            91
endowment assurances to age 60. Equation (9) gives a relation        between
    and  x+t A similar relation holds between          and
namely :

                                                                          (21)
From this relation,   we deduce:

                                                                          (22)
and                                                                       (23)
Thus, for policy number     5, we have (using A1949–52 (ultimate)   mortality
and 2½% interest):



and
The mean policy value is therefore 2000 × 0·14091 = 281·82, while the
variance of the policy value is (2000)2 × 0.01192 = 47,676.
   Summing the two appropriate       columns in Table 6, we obtain the
expected value and variance of the total policy values of the fund. The
standard error of the total of the policy values is 1,273 and this is large
compared with the total of expected policy values (9,798).
   An alternative approach is to carry out a stochastic analysis of the
death strain. Most companies determine the expected death strain for
the year and compare monthly the actual with the expected. Such an
analysis would be of much greater value if the standard error were known,
and if such an analysis with standard errors were dissected into branches,
tables, etc. With modern digital computers, such computations should no
longer be considered impossible. This problem is discussed in greater
detail in Part 4.


      4. RETENTION LIMITS AND REASSURANCE ARRANGEMENTS
   The prime purpose of setting retention limits or entering into reassurance
arrangements     is to reduce to an acceptable figure the likelihood of sto-
chastic variations in mortality in any year affecting the bonus distribution
through an unexpected fall in the surplus. A stochastic approach should
therefore lead to a better assessment of retention limits and reassurance
treaties than the deterministic approach provides.
   The process of selecting retention limits must be sequential. This is
because the retention limit depends upon the policy structure of the
company according to age, size of policy, type of policy, etc., and because
when a new policy is written or a policyholder dies, the policy structure
of the company is altered. In theory, it is possible to examine the policy
structure of a company before writing each new policy. Clearly this is
92             A Stochastic     Approach     to Actuarial    Functions
impossible in practice. A practical procedure would be to examine the
policies in force annually and decide then the company’s retention limits
for the following year. Let us investigate this type of procedure.

4.1. REASSURANCES ON ORIGINAL TERMS
   Consider first a company whose policy has been to cede, on the original
terms and for the full duration of the contract, that proportion    of any
contract which exceeds the retention limit in force when the contract was
written. The problem is to determine the retention limits which should
be set for new contracts during the coming year.
   Three points should be noted :
        1. It would be incorrect to set limits based on the consequences of
     writing one large contract for, if a number of similar contracts were
     written, the actual death strain could easily exceed the permissible
     limit.
        2. The number of small policies written in conjunction with the large
     policies is a material factor in determining the retention limit.
        3. If we are concerned with yearly fluctuations in mortality, we should
     consider one year as a unit and determine retention limits to apply to a
     year’s new business considered as a single package.
   One could proceed in the following manner. Consider a new policy,
written at age x for    a sum assured S. The death risk during the first
policy year may be taken as S; the expected value and variance of the
death strain will be qxS and pxqxS2, respectively. Let V0denote the variance
of the death strain of the in force due to mortality during the following
policy year. Then the variance of the death strain of both in force and
one year’s new business        is V0 +                 assuming    that   no duplicate
policies are included in the new business. The summation is over one
year’s new business.
   We assume that the financial strength of the company will enable it to
cope with an actual death strain which exceeds the expected by any
amount up to X; that is to say, if the bonus is not to fall below a certain
level because of mortality fluctuations, we require the chance p that the
actual death strain exceeds the expected by X or more to be very small
(say) ·001. If a large number* of policies is involved, the Central Limit
Theorem establishes that the death strain will be normally distributed.
   * This requirement has not been expressed very precisely, to avoid cluttering up the
text. The relative sizes of the policy death risks must be considered. If all the death
risks are small, the Central Limit Theorem will certainly apply. If there is a reasonable
number of policies all involving large sums at risk of approximately the same magnitude,
the Central Limit Theorem will apply approximately. In almost all cases, however,
even when there is a large deviation from normality, inequality (25) will be of use. We
would then speak of an R-sigma limit rather than a p-limit.
             A Stochastic   Approach    to Actuarial   Functions              93
The above condition    may be expressed   mathematically     as follows:




                                                                            (25)

where R is obtained   from p using normal distribution     tables. (If p =·001,
R = 3.09.)

From (25),                                                                  (26)
   That is, X2/R2 is the upper    limit to which the variance of the death
strain of existing business and new business combined may be allowed
to climb. This inequality may be satisfied by limiting the values of S
in various ways. We could reassure enough policies in full-even         a very
large number of small policies-to      bring the risk of a large death strain
within acceptable limits. There are, however, a few logical criteria on
which the selection of policies for reassurance might be based, perhaps
the most obvious one being that the total sum assured should be as large
as possible.
   Before dealing with these criteria and the results which flow from them,
we shall study the variations in the variance of the death strain during
the course of a life assurance contract. We need to do this firstly because
it will throw some light on how v0 in the fundamental           equation (26)
is likely to vary with time and secondly because it may not be sufficient
to base reassurance policy on the value of pqS2 at the time of entry into
a contract.

4.2. THE VARIANCE OF THE DEATH STRAIN
   The variance of the death strain on a £100 non-participating        policy
during the course of the contract (based on A1949–52 (ultimate) tables
at 2½% interest) follows the general pattern shown in Figure 1. We have
selected three ages at entry namely 20, 40 and 60 and four types of contract
namely endowment assurances maturing at 45, 65 and 80 and whole of
life. For the whole of life, for example, age 20 at entry, the graph is
actually that of pxqx(ä2x+1/ä220).
   As we wish to determine whether the sum assured retention limit S
should be reduced below that based on the initial risk because of sub-
sequent rises in the variance of the death strain, it is better to work with
the standard deviation rather than the variance, since the latter varies
with the square of S. Also, as we are comparing the position at entry
with that later in the contract, it is more useful to present the standard
deviation of the death strain at any time as a fraction of its initial value.
This is shown in Figure 2.
94            A Stochastic     Approach      to Actuarial    Functions




FIG. 1. Variance of the death strain on a £100 non-participating policy during the course
                    of the contract (Basis: A1949–52 (ultimate) 2½%)
               A Stochastic    Approach     to Actuarial    Functions              95




FIG. 2. The standard deviation of the death strain on a £l00 non-participating policy
during the course of the contract expressed as a fraction of its initial value. (Basis:
                              A1949–52 (ultimate) 2½%)

  Several interesting    facts emerge from these two figures:
     1. The variance of the death strain passes through a maximum at
  age 70 for whole of life policies, at age 63 for endowment assurances to
  age 80 and at age 50 for endowment assurances to age 65. The age at
  which this maximum occurs is independent         of the age at entry. This
  clearly follows from the formula since for any age at entry, the variance
  is proportional  to pxqxyäx2+1 :     where x is the age attained.
     2. The shorter the term of the contract, the lower the variance.
   G
96             A Stochastic   Approach    to Actuarial   Functions
        3. The standard deviation of the death strain for whole of life policies
     rises to double its initial value in the case of middle and younger ages
     at entry. Substantial percentage increases also occur with endowment
     assurances maturing at age 80. For endowment assurances maturing
     at 65 or earlier there is no material increase over the initial value.
4.3 REASSURANCE ON ORIGINAL TERMS (Continued)
   At first sight it might appear that V0 in inequality (26) might, in certain
circumstances,    be greater than X²/R². This position would only arise in
the case of a fund which consisted mainly of whole of life contracts (or
equivalent very long-term endowment assurances) effected at ages under
45. This is a likely situation with a closed fund. Fortunately,      in this case
new business limitations do not arise, but the increasing variance of the
death strain could make it desirable to reconsider the reassurance position
of some existing contracts if this increase has not been allowed for when
the contract was written.
   For an expanding office, X will be increasing with the size of the office
and retention limits also should be increasing. It would be interesting to
know at what rate X should be increasing in order that V0 will never be
greater than X2/R2 in the future. Consider X growing with time at rate
i per annum. In t years’ time it will take the value X(t) = (1 + i)tX.
The upper bound for the variance in inequality (26) will be (1 + i)2t(X²/R²).
Consider now the whole of life policy on a life aged 20 included in the
year’s new business. A reference to compound interest tables of (1 + i)2n
and to Figure 1 will show that the contribution        of this policy to V0 in t
years’ time is less than (1·015)2t times its first contribution   to V0 for all t.
Therefore, if X is increasing at 1½% or more per annum, the contribution
of this policy to future V0 values will not cause any difficulty. At age 30,
the rate of increase required is about 2½% per annum, at age 40 about
3½% per annum, falling to 1% at age 60. These are the figures for whole
of life contracts; for other contracts the figures are considerably less.
   The magnitude of changes in the variance of the death strain during
the course of the contract are such that they must at least be considered.
Clearly however the problem is limited to whole of life contracts (or
equivalent very long-term endowment assurances) where these are effected
at ages between 30 and 55. With the usual distribution          of contracts by
age and by type and with an office growing at an average rate it is unlikely
that special treatment for this factor will be required. Should some action
be necessary several solutions are possible including:
        (i) When determining       V0, policies where the variance now exceeds
     a certain figure could be detected and temporary reassurance effected,
        (ii) Reassurance    limits generally could be reduced to allow for the
     possible increases in death strain variances with duration. Programmes
     suggested later in § 4.4 for determining limits can easily be modified to
     allow for this factor.
            A Stochastic       Approach     to Actuarial    Functions                  97
   For simplicity only three ages at entry have been considered above
and the associated necessary rates of growth of X have been quoted. In
Appendix 2, a general treatment of this problem is given including the
associated necessary rates of growth of X for any age and duration of
contract. It is shown there (assuming A1949–52 (ultimate) mortality and
2½% interest) that the maximum rate of growth required is that necessary
to cover whole of life policies entered into between ages 40 and 50 and
is 3½% per annum—that      is, a doubling of X every 20 years. This is an
extreme case; for most contracts no increase or a small increase only is
required.
4.4 RETENTION LIMITS
   If we were to base retention limits purely on the variance of the death
strain at entry, then the retention limits would depend on age at entry
only and not on type of contract and would be as follows (assuming an
arbitrary figure of 25,000 for age 15):
                       Age x     √ (pxqx)     Retention limit
                        15        ·0333           25,000
                        25        ·0334           24,900
                        35        ·0363           22,900
                        45        ·0573           14,500
                        55        ·1012            8,200
   The overall limitation which must be met by the office is inequality
(26). If portions of certain contracts are to be reassured in order to meet
the limitations of inequality (26), the management       of the office would
usually have some overall policy which would affect the decision as to
which policies are selected. The most common such criterion is that the
policies selected in order to meet limitation (26) should be chosen so as to
make the total remaining sum assured as large as possible. We then require

                V=   V0+p1q1S12+p2q2S2²+.           . . +prqrSr²+.      ..
to be reduced    to less than X2/R2, subject to


being kept as large as possible. If we select one particular                 policy,   say
the rth, and keep all the others constant, we have:




Hence, if we reduce the sum assured on the rth policy, for every unit
reduction in the sum assured S, the variance V will reduce by 2prqrS,.
For the largest reduction in the variance for the smallest reduction in the
sum assured, we therefore select those policies which have the largest
values of prqrSr. On this criterion, retention limits should vary with age
98            A Stochastic      Approach     to Actuarial       Functions
at entry only (and not with type of contract) and vary as follows (assuming
an arbitrary figure of 25,000 at age 15 for the particular office and A1949–
52 (ultimate) mortality) :
                      Age x          Pxqx      Retention limit
                       15          ·001109         25,000
                       25          ·001119         24,780
                       35          ·001318         21,040
                       45          ·003289          8,430
                       55          ·010243          2,700
   A more general     criterion     can be summed             up in the mathematical
statement that

                      S = c1S1 +c2S2+         . . . +crSr+.     ..
should be a maximum.         In this case,




so we would select policies which have the largest values of




   If c1 = c2 = . . . = cr = . . ., we have the case of maximizing the new
sum assured. If c1, c2, . . ., cr, . . . represent the rates of premium on the
various policies, then we are maximizing the new premium income. Here,
assuming premium rates do not vary appreciably               with sum assured,
 ∂cr/ ∂Sr = 0, and we select policies which have the largest values of
Prqrsr/cr.
   Using typical premium rates and A1949–52 (ultimate) mortality, we
obtain under this criterion the following retention limits:

  Age                              Retention limits
   X     Whole life   Endowment assurance to 45     Endowment assurance to 65
   15      25,000             58,100                        32,100
   25      31,900             91,900                        42,800
   35      36,400            162,000                        51,700
   45      20,600                                           33,200
   55       9,900                                           22,100
   The figures quoted above are all relative; to determine                  the absolute
limits for a particular office, the steps are as follows:

  1. Decide   on the value of X which can be met without                    affecting   the
bonus;
                A Stochastic    Approach    to Actuarial       Functions           99
  2. Decide on a value for p;
  3. Estimate                  from new business (before reassurances)      in recent
years ;
  4. Calculate V0 for the business in force;
  5. From these values calculate the retention       limits.
   All the necessary data should be on the valuation               cards and the cal-
culations are possible if a computer is in use.
4.5 THE RISK PREMIUM METHOD
   The methods just discussed can clearly be applied to offices whose
policy is to reassure on a year-to-year    basis. In fact the application is
straightforward   since, with this policy, the problems of death strain
variance increases with time do not arise.

  In this case we maximize Σ crSr        subject to Σ prqrSr2 ≤ X2/R2        where S,,
now stands for the death risk under a policy and the summation is over
all policies both in force and new business. In the case c1 = c2 = . . .=
cr = . . .) we are maximizing the death risk, which very roughly is maxi-
mizing future premiums receivable. In other cases values of c, may be
determined by equating crSr, to the values (for the particular reassurance
criterion) of the quantity which it is desired to maximize, e.g. future
premiums receivable, present value of future premiums, annual premium
income, etc.
   Under this system an office retains a larger portion of its insurance
portfolio.

4.6. STOP LOSS REINSURANCE
   The method of assessing retention limits for a company as outlined in
the previous section required a knowledge of the total variance of the
death strain of the individual policies. For companies whose actuarial
departments    are computer based such a calculation should be possible.
However, when this information       is available, a reinsurance arrangement
alternative to the retention limit system is possible which should be of
benefit to both the company and its reinsurers. We refer to stop loss
reinsurance—that    is, a reinsurance contract effected at the beginning of
the year under which no individual policies are reinsured but under which
the reinsurance company undertakes, if, and only if, the actual death strain
for the ensuing year exceeds a certain figure Z, to pay to the insuring
company the amount by which the actual death strain exceeds Z.
   In the absence of any isolated, really abnormally large policies, the death
strain in any one year will have a distribution very close to normal. Let
the expected death strain during the next year be £E, and the variance
£V2. If we denote the death strain by W, then
                                 W    Normal (E, V).
100          A Stochastic   Approach    to Actuarial   Functions
The company is prepared to meet an actual death strain of £Z (which
depends upon its size, age, capital, extra reserves, etc.), but no more. The
net premium payable to the reinsuring company to insure against any
excess is




                                                                           (27)



where    f(x) =


   The net premium can therefore be readily computed using tables of
the normal distribution,     and should usually be small. The information
necessary to make the calculation is available to the company and to its
reinsurers.   Such a treaty should enable a company, at small cost, to
retain all its business and still limit the risk to a figure of its own choosing
and which depends on its financial strength. This arrangement is superior
to the retention limit system in that exceeding the upper limit is made
impossible—not      just very unlikely. The reinsurer should be able readily
to assess its profit from the treaty.
   Four relevant matters should be discussed:
    1. Several policies on one life. Where there are several policies on one
life, the variance is underestimated.    If there are four policies with equal
amounts at risk on every insured life, then the standard deviation of the
death strain will be double the figure obtained by treating policies sepa-
rately. Some indication of the effect of multiple policies on the standard
deviation of the death strain can be obtained by using the methods given
in Part 3 after Example 1. However, with modern computers, multiple
policies can and should be treated thoroughly.
    2. What should be done about policies written during the year and after
the annual payment to the reinsurers ? A temporary reassurance for the
balance of the year of part of any large policies could of course be effected.
The most satisfactory and most simple solution would be to cover the
year’s new business under the same terms under the same contract and
provide for a retrospective      adjustment   when the new calculation     has
been made at the end of the year.
    3. Substandard lives. The treatment of substandard lives in the valuation
will need to be considered.       Where these are included in the normal
portfolio, it is usual to include them at the rated-up age and, if this is so,
appropriate allowance is made for the extra risk and higher variance.
              A Stochastic     Approach    to Actuarial   Functions                  101
   4. The effect on the distribution of W of one (or more) isolated abnormally
large policy. If there is one enormous policy involving a large possible
death strain during the ensuing year which could affect the distribution of
 W, then the premium could be calculated as follows:
   Assume that the large policy is on a life now aged x and that the death
strain is s. We first calculate the probability that the life (x) survives the
year and that the death strain on the other policies is u. This probability
is




Similarly, the probability   that the life (x) dies and the death           strain    on
the other policies is u can be written down as




The probability     that the total death strain during the year is u therefore         is




The premium       payable to the reassuring   company,    P, is therefore    given by




These integrals may be readily evaluated using tables of the Normal
integral. This process may be extended to deal with several policies each
involving a possible large death strain, and a computer can do the relevant
calculations very rapidly. The death strain is the important criterion and
not the actual policy size.
   When there are many large policies all with large expected death
strains of comparable sizes, a Normal approximation     should be accurate,
and the above process should not be necessary.

                           ACKNOWLEDGEMENTS
  We should like to thank the Director of the Cambridge         University
Mathematical   Laboratory,   for time on TITAN, and the Shell Company of
Australia for the Research Scholarship it granted the second author.
102             A Stochastic    Approach      to Actuarial     Functions



                                   APPENDIX          1
Appendix   Table A1. Moments            and moment functions     of ãx and Ãx at   2½%
                                          interest

       Age (x)       E(ãx)     S.D.(ãx)    (ãx)      β 2(ãx)   E(Âx)    S.D.(Ãx)
         20          28·28       4·61        –2·5     12·2      ·286      ·113
         21          28·02       4·63        –2·4     11·6      ·292      ·113
         22          27·76       4·65        – 2·4    11·0      ·299      ·113
         23          27·48       4·68        –2·3     10·5      ·305       ·114
         24          27·20       4·70        – 2·2      9·9     ·312       ·115

           25        26·91       4·73        –2·1        9·4    ·319       ·115
           26        26·61       4·76        –2·0        8·9    ·326       ·116
           27        26·31       4·80        –2·0        8·5    ·334       ·117
           28        26·00       4·84        –1·9        8·0    ·341       ·118
           29        25·68       4·88        –1·8        7·6    ·349       119

           30        25·35       4·92        –1·7        7·2    ·357       ·120
           31        25·02       4·97        –1·7        6·8    ·365       ·121
           32        24·61       5·02        –1·6        6·5    ·374       ·122
           33        24·32       5·07        –1·5        6·1    ·382       ·123
           34        23·96       5·13        –1·5        5·8    ·391       ·125

           35        23·59       5·19        –1·4        5·6    ·400       ·127
           36        23·21       5·25        –1·4        5·3    ·410       ·128
           37        22·82       5·31        –1·3        5·1    ·419       ·130
           38        22·43       5·37        –1·3        4·9    ·429       ·131
           39        22·02       5·43        – 1·2       4·7    ·438       ·133

           40        21·61       5·50        –1·2        4·5     ·448      ·134
           41        21·20       5·56        –1·2        4·3     ·459      ·135
           42        20·77       5·61        –1·1        4·1     ·468      ·136
           43        20·34       5·67        –1·1        4·0     ·480      ·138
           44        19·90       5·72        –1·0        3·8     ·490      ·139

           45        19·46       5·77        –1·0        3·7     ·501      ·141
           46        19·01       5·81         –· 9       3·5     ·512      ·142
           47        18·56       5·84         –·9        3·4     ·523      ·143
           48        18·11       5·87         –·9        3·3     ·534      ·143
           49        17·65       5·90         –·8        3·2     ·545      ·143

           50         17·18      5·91         –· 8       3·1     ·556      ·144
           51         16·72      5·92         – ·7       3·0     ·568      ·144
           52         16·25      5·92         – ·7       2·9     ·579      ·144
           53         15·79      5·92         –·6        2·8     ·591      ·144
           54         15·32      5·91         –· 6       2·7     ·602      ·144
       A Stochastic    Approach     to Actuarial    Functions          103
Age (x)     E(ãx)     S.D.(ãx)   (ãx)      β2(ãx)    E(Ãx) S.D.(Ãx)
  55        14·85       5·89        –·5      2·6     ·613    ·144
  56        14·38       5·86        –·5      2·5     ·625    ·143
  57        13·91       5·83        –·5      2·5     ·636    ·142
  58        13·44       5·78        –·4      2·4     ·648    ·141
  59        12·97       5·74        –·4      2·4     ·659    ·140

  60        12·51       5·68       –·3       2·3     ·671       ·139
  61        12·04       5·62       –·3       23      ·682       ·137
  62        11·58       5·55       –·2       2·2     ·693       ·135
  63        11·13       5·18       –·2       2·2     ·704       ·134
  64        10·68       5·39       –· 1      2·2     ·715       ·132

  65        10·23      5·31        –· 1      2·2     ·726       ·129
  66         9·79      5·21          ·0      2·2     ·737       ·127
  67         9·35      5·11          ·0      2·2     ·747       ·125
  68         8·93      5·01          ·1      2·2     ·758       ·122
  69         8·51      4·90          ·1      2·2     ·768       ·119

  70         8·09      4·78          ·2      2·2     ·778       ·117
  71         7·69      4·66          ·2      2·2     ·788       ·114
  72         7·30      4·54          ·3      2·3     ·798       ·110
  73         6·92      4·41          ·3      2·3     ·807       ·106
  74         6·54      4·28          ·4      2·3     ·816       ·104

  75         6·18      4·15          ·4      2·4     ·825       ·101
  76         5·83      4·01          ·4      2·4     ·833       ·098
  77         5·49      3·88          ·5      2·5     ·842       ·095
  78         5·17      3·74          ·5      2·6     ·850       ·091
  79         4·86      3·60          ·6      2·7     ·857       ·088

  80         4·56      3·46          ·6      2·7     ·864       ·084
  81         4·27      3·32          ·7      2·8     ·872       ·081
  82         3·99      3·19          ·7      2·9     ·878       ·078
  83         3·73      3·05          ·8      3·0     ·885       ·074
  84         3·48      2·92          ·8      3·1     ·891       ·071

  85         3·25      2·79          ·9      3·2     ·896       ·068
  86         3·03      2·66          ·9      3·3     ·902       ·065
  87         2·82      2·53          ·9      3·3     ·907       ·062
  88         2·62      2·41         1·0      3·4     ·912       ·059
  89         2·43      2·28         1·0      3·4     ·916       ·056

  90         2·25      2·16         1·0      3·4     ·921       ·053
  91         2·09      2·04         1·0      3·3     ·925       ·050
  92         1·93      1·91         1·0      3·2     ·929       0·47
  93         1·78      1·79          ·9      3·0     ·932       ·044
  94         1·63      1·65          ·9      2·8     ·936       ·040
104             A Stochastic    Approach     to Actuarial   Functions
Appendix    Table A2. Moments          and moment functions    of ãx and Ãx at 5%
                                         interest
       Age (x)       E(ãx)     S.D.(ãx)     (ãx)    β2(ãx) E(Ãx)        S.D.(Ãx)
         20          17·97       2·11      –4·5     29·5    ·097          ·101
         21          17·89       2·13      –4·4     28·2    ·101          ·101
         22          17·80       2·14      –4·3     26·8    ·105          ·102
         23          17·72       2·17      –4·1     25·4    ·109          ·103
         24          17·62       2·19      –4·0     24·0    ·113          ·104

           25        17·52      2·22       –3·8     22·6      ·118        ·106
           26        17·42      2·25       –3·7     21·2      ·123        ·107
           27        17·31      2·28       –3·5     19·8      ·128        ·109
           28        17·20      2·32       –3·4     18·5      ·I33        ·110
           29       ·17·08      2·36       –3·2     17·2      ·139        ·112

           30        16·95      2·40       –3·1      16·0     ·145        ·114
           31        16·82      2·45       –2·9      14·8     –151        ·116
           32        16·68      2·49       –2·8      13·8     ·158        ·119
           33        16·54      2·55       –2·7      12·8     ·165        ·122
           34        16·39      2·61       –2·6      11·8     ·172        ·124

           35        16·23      267        –2·5      11·0     ·180        ·127
           36        16·06      2·74       –2·4      10·2     ·187        ·I30
           37        15·89      2·81       –2·3       9·6     ·196        ·134
           38        15·71      2·88       –2·2       8·9     ·204        ·137
           39        15·52      2·96       –2·1       8·3     ·213        ·141

           40        15·32      3·03       –2·0       7·8     ·223        ·144
           41        15·12      3·11       –1·9       7·3     ·232        ·148
           42        14·91      3·19       –1·8       6·9     ·242        ·152
           43        14·69      3·26       – 1·8      6·5     ·253        ·155
           44        14·47      3·34       – 1·7      6·1     ·263        ·159

           45        14·23      3·41       –1·6      5·7      ·275        ·163
           46        14·00      3·48       –1·6      5·4      ·286        ·166
           47        13·75      3·55       –1·5      5·1      ·298        ·169
           48        13·50      3·62       –1·4      4·8      ·310        ·I73
           49        13·24      3·68       –1·4      4·6      ·322        –175

           50        12·98      3·74       –1·3      4·3      ·334       ·178
           51        12·71      3·79       –1·2      4·1      ·347       ·181
           52        12·43      3·84       –1·2      3·9      ·360       ·183
           53        12·16      3·88       –1·1      3·7      ·374       ·185
           54        11·87      3·92       –1·0      3·5      ·387       ·187

           55        11·58      3·95        –· 9     3·3      ·401       ·188
           56        11·29      3·98        –·9      3·1      ·415       ·190
           57        1099       4·01        – ·8     3·0      ·429       ·191
           58        10·69      4·02        –·8      2·9      443        ·192
           59        1038       4·03        –·7      2·7      ·458       ·192
       A Stochastic    Approach    to Actuarial   Functions              105



Age (x)     E(ãx)     S.D.(ãx)    (ãx)     β2(ãx) E(Ãx)       S.D.(Ãx)
  60        10·07       4·04       –·6      2·6   ·473          ·192
  61         9·76       4·04       –·6      2·5   ·487          ·192
  62         9·44       4·03       –·5      2·4   ·502          ·192
  63         9·13       4·02       –·5      2·4    ·517         ·191
  64         8·82       4·00       –· 4     23     ·532         ·190

  65         8·50      3·97       – ·4      2·2     ·548         ·189
  66         8·18      3·94       –· 3      2·2    563          ·188
  67         7·87      3·90       –· 3      2·1    ·578         ·186
  68         7·55      3·86       – ·2      2·1    ·592         ·184
  69         I·24      3·81       – ·2      21     ·608         ·181

  70         6·93       3·16      –·1       2·1    ·622         ·179
  71         6·62       3·70        ·0      2·1    ·637         ·176
  72         6·32       3·63        ·0      2·1    ·652         ·173
  73         6·02       3·56        ·1      2·1    ·666         ·169
  74         5·73       3·48        ·1      2·1    ·680         ·166

  75         5·44      3·40         ·2      2·1    ·693         ·162
  76         5·16      3·32         ·2      2·2    ·707         ·158
  77         4·88      3·23         ·3      2·2    ·720         ·154
  78         4·61      3·14         ·3      2·3    ·733         ·150
  79         4·36      3·05         ·4      2·3    ·745         ·145

  80         4·11      2·95         ·4      2·4    ·757         ·141
  81         3·86      2·85         ·5      2·4    ·768         ·136
  82         3·63      2,76         ·6      2,5    ·779         ·131
  83         3·41      2·66         ·6      2·6    ·790         ·126
  84         3·19      2·56         ·7      2·7    ·800         ·122

  85         299       2·46         ·7      2·8    ·810         ·117
  86         2,79      2·36         ·7      2·9    ·819         ·112
  87         2·61      2·26         ·8      2·9    ·828         ·108
  88         2·43      2·16         ·8      3·0    ·836         ·103
  89         2·27      2·06         ·9      3·0    ·844         ·098

  90         2·11      1·96         ·9      3·1    ·852         ·093
  91         1·96      1·86         ·9      3·0    ·859         ·089
  92         1·82      1·76         ·9      3·0    ·866         ·084
  93         1·68      1·65         ·9      2·9    ·873         ·078
  94         1·54      1·54         ·8      2·7    ·879         ·073
106           A Stochastic       Approach      to Actuarial      Functions




Appendix     Table    A3.      Moments   and moment functions           of             and


 Age                   S.D.                                                     S.D.
 (x) E               (annuity)     (annuity)       β2(annuity)               (assurance)
 20    17·64            1·62         –7·9            68·8           ·545         ·040
 21    17·19            1·55         –8·0            70·9           ·558         ·038
 22    16·55            1·48         –8·1            73·2           ·572         ·036
 23    15·99            1·41         –8·3            75·5           ·586         ·034
 24    15·40            1·33         –8·4            78·1           ·600         ·033

 25        14·81       1·26          –8·5           80·8            ·614        ·031
 26        14·19       1·19          –8·7           83·7            ·629        ·029
 27        13·56       1·12          –8·9           86·8            645         ·027
 28        12·92       1·05          –9·0           90·1            ·661        ,026
 29        12·26        ·97          –9·2           93·6            ·677        ,024

 30        11·58        ·90          –9·4            97·4           ·693        ·022
 31        10·88        ·83          –9·6           101·4           ·710        ·020
 32        10·16        ·76          –9·8           105·9           ·728        ·019
 33         9·43        ·70         – 10·0          110·8           ·746        ·017
 34         8·68        ·63         – 10·3          116·4           ·764        ·015

 35         7·91        ·56         –10·6           122·8            ·783       ·014
 36         7·11        ·49         – 10·9          130·3            ·802       ·012
 37         6·31        ·43         –11·3           139·6            ·822       ·010
 38         5·47        ·36         –11·8           151·4            ·842       ·009
 39         4·62        ·29         – 12·4          167·3            ·863       ·007

 40         3·14         ·23        –13·3           189·7            ·884       ·006
 41         284          ·16        – 14·5          224·5            ·906       ·004
 42         1·92         ·10        – 16·4          283·2            ,929       ·003
 43          ·97         ·05        – 19·6          384·1            ·952       ·001
               A Stochastic       Approach     to Actuarial   Functions                107




Appendix     Table    A4       Moments    and moment functions          of             and
                                           at 5% interest
 Age                    S.D.                                                     S.D.
 (x)                 (annuity)     (annuity)    β2(annuity)                  (assurance)
 20        13·63        1·16          –8·4          78·4         ·303            ·055
 21        13·33        1·12          –8·5          80·4         ·318            ·054
 22        13·01        1·08          –8·6          82·5         ·333            ·052
 23        12·68        1·04          – 8·7         84·8         ·349            ·050
 24        12·33        1·00          –8·8          87·2         ·365            ·408

 25        11·96         ·96         –9·0           89·7         ·383           ·406
 26        11·57         ·92         –9·1           92·4         ·401           ·044
 27        11·16         ·81         –9·3           95·3         ·421           ·42
 28        10·73         ·83         –9·4           98·4         ·441           ·039
 29        10·28         ·78         –9·6          101·7         ·463           ·037

 30        9·81          ·73         –9·7          105·2         ·485           ·035
 31        9·31          ·69         –9·9          109·0         ·509           ,033
 32        8·79          ·64        – 10·1         113·1         ·534           ,030
 33        8·24          ·59        – 10·3         117·7         ·560           ·028
 34        7·66          ·54        – 10·5         122·9         ·588           ·026

 35        7·06         ·49         – 10·8         128·8         ·616           ·023
 36        6·42         ·43         –11·1          136·0         ·647           ·021
 37        5·75         ·38         –11·5          144·8         ·679           ·018
 38        5·04         ·33         – 12·0         156·2         ·712           ·016
 39        4·30         ·27         –12·6          171·5         ·747           ·013

 40        3·53         ·21         – 13·4        193·4          ·784           ·010
 41        2·71         ·16         – 14·6        227·5          3823           ·007
 42        1·85         ·10         –16·4         285·3          ·864           ·005
 43         ·95         ·05         – 19·6        384·1          ·907           ·002
108            A Stochastic      Approach     to Actuarial    Functions
Appendix     Table    A5. Moments       and moment functions         of              and


 Age                   S.D.                                                  S.D.
 (x)                 (annuity)    (annuity)     β2(annuity)               (assurance)
 20        23·99        2·83         – 5·4         35·1          ·390         ·069
 21        23·63        2·79         –5·3          34·9          ·399         ·068
 22        23·24        2·74         –5·3          34·6          409          ·067
 23        22·85        2·70         –5·3          34·3          ·418         ·066
 24        22·45        2·65         –5·2          33·9          ·428         ·065

 25        22·04       2·61         – 5·2           33·4         –438        ·064
 26        21·61       2·57         –5·2            32·9         448         ·063
 27        21·18       2·52         –5·1            32·3         ·459        ·062
 28        20·73       2·48         –5·1            31·7         ·410        ·061
 29        20·27       2·44         – 5·0           31·1         ·481        ·060

 30        19·81       2·40         –4·9            30·4          ·493       ·059
 31        19·32       2·31         –4·9            29·6          ·504       ·058
 32        18·83       2·33         –4·8            28·9          516        ·057
 33        18·32       2·29         –4·8            28·1          ·529       ·056
 34        17·81       2·26         –4·7            21·4          ·541       ·055

 35        17·27       2·22         –4·6            26·7          ·554       ·054
 36        16·73       2·19         –4·6            26·0          ·568       ·053
 37        1617        2·15         –4·5            25·3          ·581       ·052
 38        1560        2·11         –4·5            24·6          ·595       ·051
 39        15·02       2·07         –4·4            24·0          609        ·050

 40        14·42        2·02        –4·4            23·5          ·624        ·049
 41        13·81        1·97        –4·4            23·0          ·639        ·048
 42        13·18        1·92        –4·3            22·6          ·654        ·047
 43        12·54        1·86        –4·3            22·2          ·670        ·045
 44        11·89        1·79        –4·3            21·9          ·686        ·044

 45        11·22        1·72         –4·3           21·7          ·702        ·042
 46        10·54        1·63         –4·3           21·7          ·719        ·040
 41         9·84        1·54         –4·3           21·7          ·736        ·038
 48         9·13        1·44         –4·3           21·9          ·753        ·035
 49         8·41        1·33         –4·4           22·2          ·771        ·032

  50        7·66        1·21         –4·4           22·8          ·789        ·030
  51        6·90        1·09         –4·5           23·7          ·807        ·027
  52        6·12         ·95         –4·7           25·0          ·826        ·023
  53        5·32         ·82         –4·8           26·8          ·846        ·020
  54        4·50         ·67         –5·1           29·4          ·866        ·016

  55        3·66         ·53         –5·5           33·4          ·886        ·013
  56        2·79         ·38         –6·0           39·6          ·908        ·009
  57        1·89         ·24         –6·8           50·4          ·930        ·006
  58         ·96         ·11         –8·2           69·0          ·952        ·003
              A Stochastic       Approach      to Actuarial   Functions             109
Appendix     Table    A6. Moments       and moment functions         of             and


 Age                   S.D.                                                  S.D.
 (x)                 (annuity)     (annuity)        β2(annuity)           (assurance)
 20        16·63        1·68        –6·4            48·6          ·160        ·080
 21        16·48        1·67         –6·3           48·1          ·168        ·080
 22        16·32        1·66         –6·3           47·6          ·175        ·079
 23        16·16        1·65        –6·2            47·0          ·183        ·078
 24        15·99        1·63        –6·2            46·2          ·191        ·078

 25        15·80       1·62         –6·1            45·4          ·200       ·011
 26        15·61       1·61         –6·0            44·4          ·209       ·017
 27        15·41       1·60         – 5·9           43·4          ·218       ·076
 28        15·20       1·59         –5·8            42·3          ·228       ·076
 29        14·98       1·58         –5·8            41·0          ·239       ·015

 30        14·75       1·57         –5·7             39·8         ·250       ·015
 31        14·50       1·57         –5·6             38·4         ·262       ·075
 32        14·25       1·56         –5·5             37·1         ·274       ·074
 33        13·98       1·56         –5·4             35·1         ·287       ·074
 34        13·69       1·55         –5·3             34·3         ·300       ·074

  35       13·40        1·55        – 5·2            33·0         ·314       ·074
  36       13·08        1·54        –5·1             31·7         ·329       ·073
  37       12·76        1·54        – 5·0            30·5         ·345       ·073
  38       12·42        1·53        –4·9             29·4         ·361       ·073
  39       12·06        1·52        –4·8             28·3         ·378       ·073

  40       11·68        1·51         –4·7            27·3         ·396        ·072
  41       11·29        1·50         –4·7            26·4         ·415        ·071
  42       10·88        1·48         –4·6            25·7         ·434        ·070
  43       10·45        1·45         –4·5            25·0         ·454        ·069
  44       10·00        1·42         –4·5            24·4         ·476        ·068

  45        9·53        1·38         –4·5            24·0         ·499        ·066
  46        9·04        1·34         –4·5            23·6         ·522        ·064
  47        8·53        1·28         –4·4            23·5         ·546        ·061
  48        7·99        1·21         –4·5            23·5         ·572        ·058
  49        7·43        1·14         –4·5            23·7         ·599        ·054

  50        6·84        1·05         –4·5            24·1         ·627        ·050
  51        6·23         ·96         –4·6            24·8         ·656        ·046
  52        5·58         ·85         –4·8            26·0         ·686        ·041
  53        4·91         ·74         –4·9            27·7         719         ·035
  54        4·20         ·62         –5·2            30·2         ·753        ·029

  55        3·45         ·49         –5·5            34·0         ·788        ·023
  56        2·66         ·36         –6·0            40·2         ·826        ·017
  57        1·82         ·23         –6·9            50·7         ·866        ·011
  58         094         ·11         –8·3            69·0         ·908        ·005
110           A Stochastic    Approach     to Actuarial    Functions


                                  APPENDIX       2
   We aim, for a £100 whole of life policy, for any age at entry and any duration,
   1. to determine the actual value of the standard deviation of the death
strain;
   2. to determine the standard deviation of the death strain as a fraction of its
initial value; and
   3. to express this change in standard deviation over a period as an equivalent
continuous rate of growth.
   In Figure 3, we have the graphs of 100                  and of äx according to
A1949–52 (ultimate) mortality with 2½% interest from ages 15 to 95. Let us pro-
ceed as follows:
   1. Select any age at entry x0 (say) and draw new horizontal and vertical
axes through the point            on the äx curve and use the same logarithmic
scale with this point as origin. That portion of the other curve to the right of
x0 now represents the standard deviation of the death strain for a whole of life
policy aged x0 at entry. This is due to the fact that the actual height of this
curve from the new origin (i.e. on a linear scale) is




and hence on the logarithmic scale used the height is
which is the standard deviation of the death strain at age x of a £100 policy.
   2. If we carry out a similar change of axes and use as our new origin the
point on the upper curve in Figure 3 with abscissa x0, then this curve represents
the standard deviation of the death strain as a fraction of its initial value. This
is due to the fact that the actual height of this curve from the new origin (i.e.
on a linear scale) is




and hence on the logarithmic scale used the height is




which is the standard deviation of the death strain at age x as a fraction of its
initial value.
   3. If a quantity of size a increases to size b over time t, then the instantaneous
rate of growth δ is given by


That is
               A Stochastic Approach to Actuarial Functions                        111




FIG. 3. Graphs of 100                 and äx from which may be obtained (as explained in
Appendix 2) the standard deviation of the death strain both in absolute terms and as a
    fraction of its initial value and also the equivalent continuous rate of growth.

  If we join the point on the upper curve which has abscissa x0 to any subsequent
point on the curve-say    that with abscissa x-then the actual slope of this line is
112   A Stochastic Approach to Actuarial Functions
              A Stochastic Approach to Actuarial Functions                     113
which therefore equals the equivalent continuous rate of growth corresponding
to the change in standard deviation during this period. This slope must be
measured on a linear scale which is given on the right-hand side of the graph
for this purpose.
   The following points should be noted:
   (i) The procedure for endowment assurances is similar, and the relevant
graphs are given in Figure 4.
   (ii) From these graphs we can readily determine, for any age at entry, the
standard deviation of the death strain, either in actual value or as a fraction of
its initial value and the equivalent continuous rate of growth. In practice, it is
convenient to use a loose sheet of transparent paper or plastic with a logarithmic
scale and axes on it to facilitate the change of axes. For the theoretical purposes
of this paper a glance at the curves is sufficient.
   (iii) The maximum rate of growth is about 3½% per annum and this occurs
with whole of life policies with ages at entry 40–50 and where the age attained
is less than 50–55.

				
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