Introduction Sickness recovery intensities

Insurance: Mathematics and Economics 27 2000 365–396 Modelling the recent time trends in UK permanent health insurance recovery, mortality and claim inception transition intensities A.E. Renshaw, S. Haberman ∗ Department of Actuarial Science and Statistics, City University, Northampton Square, London EC1V 0HB, UK Received 1 March 1999; received in revised form 1 March 2000 Abstract Models representing the underlying trends in UK permanent health insurance PHI recovery, mortality and claim inception transition intensitives over the 20 year calendar period, from 1975 to 1994 inclusive, are proposed. The investigation of such trends is of special interest given that the three transition intensities, with stationary estimates based on the equivalent grouped data for the quadrennial observation window 1975–1978, form an important part of UK continuous mortality investigation CMI Bureau multiple-state model for PHI business and play an important role in the pricing and reserving for PHI sickness benefits. © 2000 Elsevier Science B.V. All rights reserved. Keywords: Permanent health insurance; Multi-state transition intensities; Trends

1. Introduction

The sickness recovery and inception transition intensities, together with the force of mortality when sick, which form the basis of UK continuous mortality investigation CMI Bureau’s multiple-state model, derive from the pooled experience of leading UK insurance companies in the observation window 1975–1978 CMI, 1991. The introduction of this model and the associated standard tables represents a significant step forward in the ways in which the premiums and benefits for UK permanent health insurance PHI contracts could be valued, with implications for pricing and reserving, and in which the underlying transition experience could be measured and monitored. In this paper, our objective is to search for any significant underlying time trends 1975–1994 in these three fundamental transition intensities by “curve” fitting, noting that these trends might have important impli- cations for current practice in terms of the pricing and reserving for PHI sickness benefits. Sickness recovery intensities and mortality from sickness intensities are modelled, respectively, in Sections 2 and 3. As explained in Section 4, since there are added complications, which require further assumptions when modelling sickness inception intensities as opposed to modelling the closely related claim inception intensities, we model the latter. The implications of doing this are also discussed in Section 4. The investigation is made possible by the re- cent consolidation of the information provided by contributing offices to the PHI experience into a suitable data base. ∗ Corresponding author. Tel.: +44-0171-477-8470; fax:+44-0171-477-8572. E-mail address: s.habermancity.ac.uk S. Haberman. 0167-668700 – see front matter © 2000 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 7 - 6 6 8 7 0 0 0 0 0 5 8 - 5 366 A.E. Renshaw, S. Haberman Insurance: Mathematics and Economics 27 2000 365–396 Fig. 1. Relationship between sickness inception and claim inception.

2. Sickness recovery intensities

2.1. Preliminaries In this section, we model the sickness recovery transition intensities. Numbers of recoveries with matching exposures, in the raw data, have been made available by individual weeks for sickness duration, individual years for age at sickness inception and by individual calendar years, 1975–1994. This applies for each of five deferred periods of 1, 4, 13, 26, 52 weeks DP1, DP4, DP13, DP26, DP52, for both males and females, see Fig. 1. Since the search for time trend patterns in the recovery intensities is a primary aim of the ensuing analysis, we have decided not to group the data by calendar year typically grouped by quadrennia by the CMI Bureau for the purposes of monitoring the experience relative to the standard tables. There is ample exposure to permit this, without materially increasing the proportion of data cells void of exposure. For grouping with respect to both sickness duration and age at sickness inception, we follow the approach used in recent CMI Bureau commentaries on parts of the data set see CMI, 1996, an arrangement which is judged to be satisfactory on the basis of trial and error. Hence, the complete cross-classification is summarised as follows: gender male, female deferred period 1, 4, 13, 26, 52 weeks duration z 1–, 2–, 3–, 4–, 8–, 13–, 17–, 26–, 30–, 39–51 weeks, 1–, 2–, 5–11 years age x 18–, 25–, 30–, 35–, 40–, 45–, 50–, 55–, 60–64 years period t 1975, 1976, 1977,. . . , 1994 coded 1–20 Notationally, we follow CMI 1991 in using z to indicate time and consequently use t to indicate years. Note that there is a maximum of 13 duration levels subject to the specific deferred period, with 10 levels for DP4, eight levels for DP13, and so on. For each of the five deferred periods and each gender 10 separate cases, based on this cross-classification, let r t xz = number of recoveries in cell t, x, z e t xz = exposure in years to the possibility of recovery in cell t, x, z as the case may be. Further, for each period t, the data are assumed to be located at the centroids of their respective sub-cells x, z, determined by weighted averages, with weights based on the relative exposures generated by the raw data. For the male experience, there is a relatively small number of cells with zero exposure located on the fringes of the resulting data grids, focused on ages 18–24 with sickness durations in excess of 26 weeks, and on ages 60–64 with sickness duration 5–11 years. Empty cells are somewhat more extensive than this for the female experience. Such cells are conveniently eliminated from the ensuing analysis by allocating zero weights to them. A.E. Renshaw, S. Haberman Insurance: Mathematics and Economics 27 2000 365–396 367 There is also a tendency for recoveries to be somewhat thinly spread across calendar years at sickness durations in excess of 2 years. For a specific deferred period and specific gender, the recovery intensity per year from sickness ρ t xz is targeted by modelling the numbers of reported recoveries in the various data cells as Poisson response variables r txz ∼ Poie txz ρ txz independently for all cells t, x, z for which m txz = Er txz = e txz ρ txz , Varr txz = V m txz = m txz , where V is the variance function of the associated generalised linear model GLM. The scale parameter is one. Hence the kernel of the log-likelihood is X txz −m txz + r txz log m txz = X txz −e txz ρ txz + r txz log ρ txz . Model fitting is implemented by optimising this expression using a log relationship log m txz = η txz = log e txz + log ρ txz linking the mean response m t xz to a parameterised linear predictor η t xz . The second term log ρ t xz in the predictor may be expressed as a variety of parameterised functions in t, x and z, which are linear in the unknown parameters. Since the first term in the predictor log e t xz does not involve any parameters, it is automatically subtracted from the predictor in the fitting process and is described as an “offset” term. This means that this part of the predictor remains constant throughout the fitting of the different parameterised formulae for ρ t xz . Since each deferred periodgender combination is modelled separately in this section, no notational provision for these two factors is required when formulating the model. The choice of a Poisson response GLM in this context is motivated by Sverdrup 1965. For a comprehensive discussion of GLMs, including applications, see McCullagh and Nelder 1989. For a review of actuarial applications of GLMs, see Haberman and Renshaw 1996. 2.2. Male, DP1 experience 2.2.1. Model selection Following exploratory analysis, centred mainly on investigating the relative merits of using low order poly- nomial structures in √ z , as opposed to z, for individual calendar years, we take as our starting point the model structure log ρ xz = β + β 1 √ z + β 2 z + β 3 x + β 4 x √ z + β 5 xz, where z is in weeks based on the metric of a 52.18 week year and x is in years. This model has been fitted previously Renshaw and Haberman, 1995, to an earlier version of the 1975–1978 data set, which was grouped into a single calendar quadrennium coupled with a less stringent grouping with respect to sickness duration. The structure is linear in the parameters and, as such, can be readily adjusted to allow for period effects. We begin by fitting this structure separately for each calendar year, viz. log ρ txz = β + β 0t + β 1 + β 1t √ z + β 2 + β 2t z + β 3 + β 3t x + β 4 + β 4t x √ z + β 5 + β 5t xz 2.1 subject to the constraints β 01 = β 11 = β 21 = β 31 = β 41 = β 51 = 0 where t = 1, 2, . . . , 20 is used to code the respective calendar years 1975, 1976, . . . , 1994. The structure involves a total of 120 unknown parameters. 368 A.E. Renshaw, S. Haberman Insurance: Mathematics and Economics 27 2000 365–396 Table 1 Deviance profile, Eq. 2.1: male, DP1 experience Model terms Deviance Degrees-of-freedom Difference Degrees-of-freedom Mean deviance β 87386 2113 4009 19 211.0 +β 0t 83377 2094 76167 1 76167 +β 1 √ z 7209.7 2093 281.1 19 14.8 +β 1t √ z 6928.6 2074 904.8 1 904.8 +β 2 z 6023.8 2073 149.5 19 7.9 +β 2t z 5874.3 2054 3175 1 3175 +β 3 x 2699.6 2053 32.6 19 1.7 a +β 3t x 2667.0 2034 12.3 1 12.3 +β 4 x √ z 2654.7 2033 27.7 19 1.5 a +β 4t x √ z 2627.0 2014 13.7 1 13.7 +β 5 xz 2613.3 2013 78.3 19 4.1 +β 5t xz 2535.0 1994 a Judged not to be statistically significant. It is possible to assess the significance of the various sets of parameters by fitting these sequentially and conducting an analysis of deviance. One such possibility based on the order in which the terms appear on the RHS of Eq. 2.1, is reported in Table 1. Firstly, note that the order in which the parameters were fitted, may be deduced by reading down the first column. Then, for a Poisson GLM, as here, the model deviances reported in the second column of the table, are given by e.g. McCullagh and Nelder, 1989, p. 34 X t,x,z ω txz 2 r txz log r txz ˆ m txz − r txz − ˆ m txz = X t,x,z ω txz d txz , 2.2 where ˆ m txz denote the predicted number of recoveries fitted values under the specific predictor structure and d t xz is the contribution to the model deviance from the data cell defined by t, x, z, with weight ω txz = 0 if a data cell has zero exposure, and weight ω txz = 1, otherwise. The number of degrees-of-freedom reported in the third column, is equal to the sum of these weights the number of non-empty cells minus the number of effective parameters included in the linear predictor structure. Columns 4 and 5 of Table 1 are constructed by differencing the respective model deviances and matching degrees-of-freedom and the mean deviance based on the ratios of these differences are recorded in the last column. The entries in the last three columns of Table 1 monitor the significance of the parameters identified by the matching adjacent lower entry in the first column. Then, under the assumption that the difference in the deviances column 4 are approximately distributed as χ 2 exact in the case of Gaussian response GLMs, we set the parameters {β 3t } and {β 4t } equal to zero on the basis that they are not statistically significant. We remark that the values of these difference statistics are moderately sensitive to the order in which the associated parameters are added to the structure. However, this source of variation is relatively small and is generally not an issue of practical consequence. Thus, for each case in question, we have experimented by changing the model fitting sequence in various ways, and each time the conclusion is unaffected. For an interesting example A.E. Renshaw, S. Haberman Insurance: Mathematics and Economics 27 2000 365–396 369 Fig. 2. Parameter plots: Eq. 2.1a. involving an analysis of deviance relating to car insurance claims and which is similar to the present case, albeit based on a gamma response GLM, see Section 8.4.1 of McCullagh and Nelder 1989. We, therefore, fit the simplified model structure log ρ txz = β + β 0t + β 1 + β 1t √ z + β 2 + β 2t z + β 3 x + β 4 x √ z + β 5 + β 5t xz 2.1a and search for possible time trends by plotting each of the resulting sets of parameter estimates { ˆ β 0t }, { ˆ β 1t }, { ˆ β 2t } and { ˆ β 5t } against time t. These plots are reproduced in Fig. 2. Focusing first on the bottom right hand frame which relates to {β 5t }, on the basis of this visual evidence that the plotted points are not supportive of any trend; coupled with the information that only two out of the 19 parameters differ significantly from zero established by individual t -statistics, we have decided to further simplify the model by also setting the parameters {β 5t } equal to zero. We note in addition, that one out of the 2114 contributing non-empty data cells contributes a disproportionate amount of 18.2 to the value of 78.3 in the deviance difference reported in the penultimate row of Table 1. By way of explanation, this 370 A.E. Renshaw, S. Haberman Insurance: Mathematics and Economics 27 2000 365–396 Fig. 3. Residual plots: Eq. 2.4. cell t = 1986, x = 55–59, z = 5–11 years corresponds to the extreme case of two acknowledged poor quality data cells involving sickness duration 5–11 years. We have decided to retain such data cells and note their appearance as outliers in the ensuing residual plots e.g. Fig. 3. Secondly, on the basis of the three remaining estimator plots of Fig. 2, which exhibit straight line trends with relatively little dispersion in two cases, we have decided to trade some local variation within the context of the heavily parameterised model 2.1a, for the nine parameter model log ρ txz = β + β 1 √ z + β 2 z + β 3 x + β 4 x √ z + β 5 xz + β 6 t + β 7 t √ z + β 8 tz 2.3 generated by substituting β 0t = α + β 6 t, β 1t = α 7 + β 7 t and β 2t = α 8 + β 8 t in Eq. 2.1a. A.E. Renshaw, S. Haberman Insurance: Mathematics and Economics 27 2000 365–396 371 Table 2 Parameter estimates, standard errors, t-statistics, Eq. 2.4: male, DP1 experience Parameter Estimate Standard error t -Statistic β 2.76175 4.195E −1 6.6 β 1 −9.9429E−1 5.115E −2 −19.4 β 2 3.2281E −2 4.515E −3 7.2 β 3 1.6973E −1 3.009E −2 5.6 β 4 5.6928E −3 9.830E −4 5.8 β 5 −3.4555E−4 8.531E −5 −4.1 β 6 4.2984E −2 2.769E −3 15.5 β 7 −2.6338E−2 1.642E −3 −16.0 β 8 1.3632E −3 1.336E −4 10.2 β 9 −4.4399E−3 7.019E −4 −6.3 β 10 2.9982E −5 5.298E −6 5.7 Finally, noting that the RHS of Eq. 2.3 is now a polynomial in the three variates t, x and √ z , we experiment by introducing additional polynomial terms in the three variates, using as a criterion a significant reduction in the model deviance coupled with the retention of a complete set of statistically significant parameter estimates. On this basis, Eq. 2.3 is modified to include additional parameterised terms in x 2 and x 3 , a feature which induces a further reduction of 110.4 in the deviance for the loss of 2 degrees-of-freedom. This leads finally to the adoption of the model structure log ρ txz = β + β 1 √ z + β 2 z + β 3 x + β 4 x √ z + β 5 xz + β 6 t + β 7 t √ z + β 8 tz + β 9 x 2 + β 10 x 3 . 2.4 The parameter estimates, together with their standard errors and t-statistics, are presented in Table 2. Separate investigations not reported here demonstrate the credibility of this model structure. For example, grouping the data by four year calendar periods, it transpires that, for each of 1975–1978, 1979–1982, . . . , 1991–1994, a model of the type 2.4, with the time dependent parameters β 6 , β 7 , β 8 pre-set to zero, provides a satisfactory fit to the recovery transition intensities for males for deferred period equal to 1 week. 2.2.2. Model interpretation Diagnostic checks of the model structure are conducted using deviance residuals, defined by signr txz − ˆ m txz ω txz p d txz , where d t xz are the components of the model deviance, defined in Eq. 2.2. A variety of residual plots have been examined. For example, deviance residuals plotted against the index or counter indext, z ′ = z ′ + z ∗ t − 1 based on duration categories z ′ = 1, 2, . . . , z ∗ z ∗ = 13 for DP1, z ∗ = 10 for DP4, etc. serialised by calendar year t = 1, 2, . . . , 20, for each separate age category, provides a compact way of viewing the residuals plotted against sickness duration, arranged end-on for each period. Thus, for a specified age, the first z ∗ points on the index represent the full range of possible sickness durations, arranged in increasing order, for 1975, then for 1976, and so on. By way of illustration we reproduce three such plots in Fig. 3. Note in particular the outlier described earlier, together with a further outlier. The structured model 2.4 may be interpreted as a three-dimensional surface in t, x and √ z for values of z less than 5 years so as to avoid durations where the data are particularly sparse. This model may be viewed from a number perspectives, which include the following: Perspective 1: log ρ txz = A xz − B xz t, 372 A.E. Renshaw, S. Haberman Insurance: Mathematics and Economics 27 2000 365–396 where A xz = β + β 1 √ z + β 2 z + β 3 x + β 4 x √ z + β 5 xz + β 9 x 2 + β 10 x 3 , B xz = −β 6 − β 7 √ z − β 8 z. Perspective 2: log ρ txz = A tx + B tx √ z + C tx z, where A tx = β + β 6 t + β 3 x + β 9 x 2 + β 10 x 3 , B tx = β 1 + β 7 t + β 4 x, C tx = β 2 + β 8 t + β 5 x. Perspective 3: log ρ txz = A tz + B tz x + C tz x 2 + D tz x 3 , where A tz = β + β 1 √ z + β 2 z + β 6 t + β 7 t √ z + β 8 tz, B tz = β 3 + β 4 √ z + β 5 z, C tz = β 9 , D tz = β 10 . Perspective 1 focuses on the model prediction that, for fixed x, z, the log-recovery intensity has changed linearly in time over the 20 year calendar period. Note that the behaviour of the signs of B xz over the x, z grid is of special interest, since these dictate the cells for which the predicted recovery intensities have increased or decreased over the period concerned. On noting that B xz is independent of x and is a quadratic in √ z c.f. a √ z 2 + b √ z + c, with negative a and positive discriminant b 2 − 4ac 0, see Table 2 for parameter values, it follows that B xz is positive, and hence the predicted recovery intensities decrease, for values of √ z between the roots 1.80 and 17.52 of the quadratic. In essence, these imply that recovery intensities have increased over time for sickness durations of 3 weeks and under and for durations 307 weeks and over, but otherwise have been in decline during the period of investigation. We return to the quadratic form for B xz in Section 2.6. Predicted log-recovery intensities, plotted against calendar period, for specific ages at sickness inception are reproduced in the upper frame of Fig. 4. Perspective 2 focuses on the log-recovery intensity, viewed as a function of sickness duration, for fixed t, x. It is a quadratic in √ z , with C t x , the coefficient of √ z 2 , positive for all values of t, x within the domain defined by the data grid. Hence, the quadratic is convex, with a turning point that is a minimum. A typical family of quadratics for each t, fixed x, is illustrated in the bottom left frame of Fig. 4, in which the turning point lies “off the chosen scale”. Perspective 3 focuses on the prediction that log-recovery intensities change as a cubic function in age x. This feature is illustrated in the bottom right frame of Fig. 4, in which a typical family of cubic profiles, for different calendar periods and specific policy duration, is presented. Note that calendar period effects impact only on the coefficient A t z , thereby inducing parallel cubic curves. Further, over the relevant age range, 18–65 years, it can be shown that the cubic expression in x has a maximum at age 49.36 − 1.96 q √ z − 8.24 2 + 75.20 with a point of inflexion at age 49.36. 2.3. Male, other deferred periods 2.3.1. Model selection A different model is needed for each of the other deferred periods. We take as our starting point the model structure log ρ xz = β + β 1 x + β 2 z + β 3 z − z + + β 4 z − z 1 + + β 5 xz − z + , A.E. Renshaw, S. Haberman Insurance: Mathematics and Economics 27 2000 365–396 373 Fig. 4. Log-recovery rates vs. various perspectives: Eq. 2.4. where z − z j + = z − z j if z z j , and z − z j + = 0 if z ≤ z j . This has been fitted previously Renshaw and Haberman, 1995, to versions of the grouped 1975–1978 quadrennium data sets for DP4, DP13 and DP26. We again begin by fitting the structure separately for each calendar year, so that log ρ txz = β + β 0t + β 1 + β 1t x + β 2 + β 2t z + β 3 + β 3t z − z + +β 4 + β 4t z − z 1 + + β 5 + β 5t xz − z + 2.5 subject to the constraints β 01 = β 11 = β 21 = β 31 = β 41 = β 51 = 0. With the exception of DP52, we set z equal to the length of the DP in question plus 4 weeks. This is done Renshaw and Haberman, 1995 in recognition of the reported Section 3, Part B of CMI 1991, pp. 25–31 sluggish recovery 374 A.E. Renshaw, S. Haberman Insurance: Mathematics and Economics 27 2000 365–396 Table 3 Deviance profiles for knot settings, Eq. 2.5: male experiences z 1 weeks DP4 DP13 DP26 52.18 1795.9 1522.9 1034.8 56.53 1790.2 1503.4 1017.3 60.88 1788.1 a 1491.1 1007.8 65.22 1790.4 1486.0 a 1005.2 a 69.57 1796.9 1487.7 1007.3 73.92 1807.4 1495.6 1012.2 78.27 1821.8 1508.8 1018.5 a Denotes a minimum. Table 4 Deviance profile, Eq. 2.5 Model terms Deviance Degrees-of-freedom Difference Degrees-of-freedom Mean deviance a Male, DP4 experience, z = 8, z 1 = 60.88 β 16186 1663 12285 1 12285 +β 2 z 3901.2 1662 503.0 1 503.0 +β 1 x 3398.3 1661 1.7 1 1.7 a +β 3 z − z + 3396.5 1660 1106 1 1106 +β 4 z − z 1 + 2290.1 1659 32.7 1 32.7 +β 5 xz − z + 2257.4 1658 315.0 19 16.6 +β 0t 1942.5 1639 25.3 19 1.3 a +β 2t z 1917.0 1620 30.7 19 1.6 a +β 1t x 1886.3 1601 16.3 19 0.9 a +β 3t z − z + 1870.1 1582 27.2 19 1.4 a +β 4t z − z 1 + 1842.8 1563 54.7 19 2.9 a +β 5t xz − z + 1788.1 1544 b Male, DP13 experience, z = 17, z 1 = 65.22 β 6684.0 1355 3440 1 3440 +β 2 z 3243.6 1354 554.8 1 554.8 +β 1 x 2688.8 1353 44.0 1 44.0 +β 3 z − z + 2644.8 1352 886.9 1 886.9 +β 4 z − z 1 + 1757.9 1351 A.E. Renshaw, S. Haberman Insurance: Mathematics and Economics 27 2000 365–396 375 Table 4 Continued Model terms Deviance Degrees-of-freedom Difference Degrees-of-freedom Mean deviance 11.5 1 11.5 +β 5 xz − z + 1746.4 1350 135.9 19 7.2 +β 0t 1610.5 1331 40.1 19 2.1 a +β 2t z 1570.4 1312 17.4 19 0.9 a +β 1t x 1552.9 1293 18.7 19 1.0 a +β 3t z − z + 1534.2 1274 20.0 19 1.1 a +β 4t z − z 1 + 1514.2 1255 28.3 19 1.5 a +β 5t xz − z + 1486.0 1236 c Male, DP26 experience, z = 30, z 1 = 65.22 β 2416.2 994 624.6 1 624.6 +β 2 z 1791.6 993 336.7 1 336.7 +β 1 x 1454.9 992 23.3 1 23.3 +β 3 z − z + 1431.6 991 207.8 1 207.8 +β 4 z − z 1 + 1223.8 990 3.9 1 3.9 +β 5 xz − z + 1219.9 989 64.5 19 3.4 +β 0t 1155.4 970 40.2 19 2.1 a +β 2t z 1115.1 951 25.8 19 1.4 a +β 1t x 1089.3 932 18.7 19 1.0 a +β 3t z − z + 1070.6 913 28.2 19 1.5 a +β 4t z − z 1 + 1042.4 894 37.2 19 2.0 a +β 5t xz − z + 1005.2 875 d Male, DP52 experience, z 1 = 78.27 β 425.83 415 15.83 1 15.8 +β 2 z 410.45 414 35.30 1 35.3 +β 1 x 375.15 413 6.66 1 6.7 +β 4 z − z 1 + 368.48 412 24.31 19 1.3 a +β 0t 344.18 393 15.98 19 0.8 a +β 2t z 328.19 374 14.59 19 0.8 a +β 1t x 313.60 355 36.92 19 1.9 a +β 4t z − z 1 + 276.68 336 a Judged not to be statistically significant. 376 A.E. Renshaw, S. Haberman Insurance: Mathematics and Economics 27 2000 365–396 rates in the first 4 weeks immediately after sickness benefit becomes payable, for the male experience for the 1975–1978 quadrennium. This feature has been further investigated and found to persist throughout the 20 year period under investigation. The details are available from the authors on request. In addition, it may be further assessed by referring to the degree of statistical significance associated with the parameters β 3 and β 5 in the ensuing analysis. The separate analysis for DP4, DP13 and DP26 follows a near identical pattern, subject to decreasing overall exposure with increasing fixed deferred period. Firstly, the optimum position of the second knot z 1 is determined by the repeated fitting of model structure 2.5, under incremental changes of 1 month based on a 12 month year, each of length 4.35 weeks in the position of z 1 . The resulting deviance profiles are reported in Table 3, and the optimum value of z 1 is selected accordingly. For DP52, it is necessary to pre-set β 3 = β 3t = β 5 = β 5t = 0 since the associated data are not amenable to the specific investigation of the 4 week “run-in” period. Neither are these data amenable to the construction of a finely graded deviance profile as above, due to the necessary broad grouping of the raw data. Consequently, we have set the knot at the centre of the first sickness band. Next, the deviance profiles, as the terms on the RHS of Eq. 2.5 are added sequentially into the structure of the model, are reported in Table 4 a–d. Here, as implied by the first column of each of these tables, we have introduced all the parameters which do not display period effects in the first instance, and then added in the parameters representing the period effects. On the basis of the statistical significance, or otherwise, of the deviance differences column 4 when referred to an approximate χ 2 distribution, the conclusions are similar in all four cases, leading to the adoption of the simplified model structure log ρ txz = β + β 0t + β 1 x + β 2 z + β 3 z − z + + β 4 z − z 1 + + β 5 xz − z + 2.5a for DP4, DP13, DP26, and with additionally β 0t = 0 β 3 = β 5 = 0 for DP52. Finally, since the ensuing {β 0t } time patterns, reproduced in Fig. 5, exhibit essentially straight line trends, we trade some local variation, within the context of the model 2.5a, by setting β 0t = α + β 6 t and absorbing the parameter α into β and adopt the model structure log ρ txz = β + β 1 x + β 2 z + β 3 z − z + + β 4 z − z 1 + + β 5 xz − z + + β 6 t 2.6 for DP4, DP13, DP26 with additionally β 3 = β 5 = β 6 = 0 for DP52. 2.3.2. Model interpretation The parameter estimates, together with their standard errors and t-statistics, for all four cases, are presented in Table 5. Note that we have retained the period effects out of interest in the case of DP52, in spite of the fact that they are not statistically significant, a feature confirmed by the value of the t-statistic of the relevant parameter. Diagnostic checks are again based on deviance residual plots, typical examples of which are reproduced in Fig. 6. Eq. 2.6 may be viewed from the same three perspectives as Eq. 2.4, namely Perspective 1: log ρ txz = A xz − B xz t, where A xz = β + β 1 x + β 2 z + β 3 z − z + + β 4 z − z 1 + + β 5 xz − z + , B xz = −β 6 . Perspective 2: log ρ txz = A tx + B tx z + C tx z − z + + D tx z − z 1 + , where A tx = β + β 1 x + β 6 t, B tx = β 2 , C tx = β 3 + β 5 x, D tx = β 4 . A.E. Renshaw, S. Haberman Insurance: Mathematics and Economics 27 2000 365–396 377 Fig. 5. Parameter plots: Eq. 2.5a. Perspective 3: log ρ txz = A tz − B tz x, where A tz = β + β 2 z + β 3 z − z + + β 4 z − z 1 + + β 6 t, B tz = −β 1 − β 5 z − z + . Under Perspective 1, B xz is positive for all four deferred periods, so that the predicted log-recovery intensities decrease linearly, for fixed x, z, over the period under investigation. This feature is illustrated in the upper frame of Fig. 7. 378 A.E. Renshaw, S. Haberman Insurance: Mathematics and Economics 27 2000 365–396 Table 5 Parameter estimates, standard errors, t-statistics, Eq. 2.6: males DP4 DP13 DP26 DP52 z 8 17 30 z 1 60.88 65.22 65.22 78.27 β 2.32278 – −3.94600 2.019978 1.109E −1 1.9573 7.005E −1 20.9 −2.0 2.9 β 1 −2.1163E−2 −3.4500E−2 −6.0664E−2 −5.6383E−2 1.222E −3 2.029E −3 3.776E −3 9.131E −3 −17.3 −17.1 −16.1 −6.2 β 2 8.3032E −2 1.5841E −1 2.2873E −1 −2.5353E−2 1.385E −2 6.012E −3 6.642E −3 8.420E −3 6.0 26.3 3.4 −3.0 β 3 −1.3916E−1 −2.0988E−1 −2.9583E−1 – 1.488E −2 7.237E −3 6.844E −2 −9.4 −29.0 −4.3 β 4 6.1605E −2 5.2470E −2 6.0614E −2 2.3846E −2 1.720E −3 1.636E −3 4.060E −3 9.161E −3 35.8 32.1 14.9 2.6 β 5 −2.7710E−4 −1.2102E−4 7.7527E −5 – 4.660E −5 3.431E −5 4.359E −5 −5.9 −3.5 1.8 a β 6 −3.3735E−2 −3.4516E−2 −3.6173E−2 −2.1258E−1 2.176E −3 3.524E −3 5.952E −3 1.672E −2 −15.5 −9.8 −6.1 1.3 a a Judged not to be statistically significant. Under Perspective 2, recalling that z − z j + = z − z j if z z j but is zero otherwise, it follows that the gradients of the three line segments for each of DP4, DP13, DP26 are given by B tx = β 2 , z ≤ DP + 4, B tx + C tx = β 2 + β 3 + β 5 x, DP + 4 z ≤ z 1 , B tx + C tx + D tx = β 2 + β 3 + β 4 + β 5 x, z z 1 . It then follows trivially from Table 5 that the predicted log-recovery intensities first increase in the so-called 4 week “run-in” durational period, then decrease linearly with increasing sickness duration, for fixed t, x, and generally continue to decrease but less rapidly so for durations in excess of z 1 weeks. The lower frame in Fig. 7 illustrates this. Under Perspective 3, B tz 0 and the predicted log-recovery intensities decrease linearly with age at sickness inception, for fixed t, z. 2.4. Female, DP1 experience In the absence of any reported previous analysis of UK female PHI experience e.g. from the 1975–1978 investi- gation, we follow the same approach as that adopted in Section 2.2 for the male experience, while noting that the female experience is much more sparse than the corresponding male experience. The deviance profile associated with the sequential fitting of the terms on the RHS of expression 2.1 is reported in Table 6. While the approximate χ 2 test applied to the deviance differences is supportive of all five parameters β 1 , β 2 , β 3 , β 4 and β 5 being non-zero, A.E. Renshaw, S. Haberman Insurance: Mathematics and Economics 27 2000 365–396 379 Fig. 6. Residual plots DP13: Eq. 2.6. the experience is not supportive of any statistically significant period effects in the sets of parameters with the possible exception of the {β 0t }. The parameter estimates { ˆ β 0t } under the fitted model structure log ρ txz = β + β 0t + β 1 √ z + β 2 z + β 3 x + β 4 x √ z + β 5 xz are reproduced in the upper frame of Fig. 8, in which five of the parameters differ significantly from zero on the basis of the individual t-statistics. On the basis of this visual evidence, the parameters are not supportive of any 380 A.E. Renshaw, S. Haberman Insurance: Mathematics and Economics 27 2000 365–396 Fig. 7. Log-recovery rates vs. various perspectives: Eq. 2.6. obvious time trend. This is further supported by fitting the model structure log ρ txz = β + β 1 √ z + β 2 z + β 3 x + β 4 x √ z + β 5 xz + β 6 t 2.7 in which we have set β 0t = α + β 6 t in the foregoing equation and replaced β + α by β . Eq. 2.7 can, in turn, be rewritten as Perspective 1: log ρ txz = A xz − B xz t, where A xz = β + β 1 √ z + β 2 z + β 3 x + β 4 x √ z + β 5 xz, B xz = −β 6 . A.E. Renshaw, S. Haberman Insurance: Mathematics and Economics 27 2000 365–396 381 Table 6 Deviance profile, Eq. 2.1: female, DP1 experience Model terms Deviance Degrees-of-freedom Difference Degrees-of-freedom Mean deviance β 9239.4 1684 138.6 19 7.3 +β 0t 9100.9 1665 7018 1 7018 +β 1 √ z 2082.5 1664 14.9 19 0.8 a +β 1t √ z 2067.6 1645 108.7 1 108.7 +β 2 z 1958.9 1644 14.5 19 0.8 a +β 2t z 1944.4 1625 233.8 1 233.8 +β 3 x 1710.6 1624 19.3 19 1.0 a +β 3t x 1691.3 1605 18.6 1 18.6 +β 4 x √ z 1672.7 1604 21.6 19 1.1 a +β 4t x √ z 1651.1 1585 9.8 1 9.8 +β 5 xz 1641.2 1584 29.4 19 1.5 a +β 5t xz 1611.9 1565 a Judged not to be statistically significant. The parameter estimates, standard errors, and t-statistics are reported in Table 7. These indicate that the period effect is not statistically significant, but that, since B xz 0, there is some weak evidence to support the general statement that recovery intensities have increased, if anything, over the 20 year period concerned. The quality of the fit has again been assessed by the residual plots associated with the fitted structure, two of which are reproduced in the lower frames of Fig. 8. 2.5. Female, other deferred periods Structure 2.6, fitted to the equivalent male experiences in Section 2.3, has also been fitted to the female DP4, DP13 and DP26 experience but with β 5 pre-set to zero, as the product term xz − z + proved to be statistically non-significant in this case. The optimum position of the knot in each case, is determined by reference to deviance Table 7 Parameter estimates, standard errors, t-statistics, Eq. 2.7: female, DP1 experience Parameter Estimate Standard error t -Statistic β 6.03457 1.717E −1 35.1 β 1 −1.33404 1.012E −1 −13.2 β 2 4.9429E −2 8.177E −3 6.0 β 3 −4.1072E−2 4.117E −3 10.0 β 4 1.0026E −2 2.385E −3 4.2 β 5 −5.1722E−4 2.025E −4 −2.6 β 6 4.9795E −3 2.992E −3 1.7 a a Judged not to be statistically significant. 382 A.E. Renshaw, S. Haberman Insurance: Mathematics and Economics 27 2000 365–396 Fig. 8. Parameters plot: Eq. 2.5a and residual plots: Eq. 2.7. profiles as in Section 2.3, constructed by varying the position of the knot by monthly increments in the model 2.6, the results of which are reported in Table 8. We can again write the model structure as Perspective 1: log ρ txz = A xz − B xz t, where A xz = β + β 1 x + β 2 z + β 3 z − z + + β 4 z − z 1 + , B xz = −β 6 A.E. Renshaw, S. Haberman Insurance: Mathematics and Economics 27 2000 365–396 383 Table 8 Deviance profiles for knot settings, Eq. 2.6: female experiences z 1 weeks DP4 DP13 DP26 52.18 1354.7 993.0 640.7 56.53 1353.4 990.8 637.1 60.88 1352.8 a 989.8 637.0 a 65.22 1353.2 989.7 a 638.7 69.57 1354.5 990.5 641.5 73.92 1356.8 992.2 644.7 78.23 1360.0 994.6 647.9 a Denotes a minimum. Table 9 Parameter estimates, standard errors, t-statistics, Eq. 2.5: females DP4 DP13 DP26 z 8 17 30 z 1 60.88 65.22 60.88 β 1.26347 – −8.62066E−1 2.383E −1 4.2509 5.3 −2.0 β 1 −1.2125E−2 −3.3972E−2 −5.2882E−2 2.595E −3 4.552E −3 6.906E −3 −4.7 −7.5 −7.7 β 2 1.5836E −1 1.5034E −1 3.7466E −1 2.930E −2 1.427E −2 1.439E −1 5.4 10.5 2.6 β 3 −2.2256E−1 −2.0043E−1 −4.4379E−1 3.052E −2 1.507E −2 1.479E −1 −7.3 −13.3 −3.0 β 4 5.7243E −2 4.425E −2 6.6455E −2 3.537E −3 3.888E −3 8.650E −3 16.2 11.4 7.7 β 5 – – – β 6 −2.5772E−2 −3.4997E−2 −3.9576E−2 4.812E −3 8.269E −3 1.251E −2 −5.4 −4.2 −3.2 in order to highlight the dependence on t. Details of the parameter estimates are reported in Table 9. Additionally, we have pre-set β to zero for DP13 since, in the event, this term also proves to be statistically insignificant. Since the estimated value of β 6 is negative for each deferred period, there is an implied deterioration in the recovery rates over the 20 year period concerned. Perspectives 2 and 3 are similar in outline to the equivalent male experiences. 2.6. Discussion of results It is of interest to compare the magnitudes of B xz , the rates by which the predicted log-recovery intensities change with time, across all deferred periods where feasible see Perspective 1 in each case. With the exception of the male DP1 experience, these rates of change with time are particularly simple, with B xz = −β 6 , a constant for all 384 A.E. Renshaw, S. Haberman Insurance: Mathematics and Economics 27 2000 365–396 x, z. Consequently, it is a simple matter to compare rates in these cases by tabulating the values of −B xz = β 6 , say, taken from Tables 5, 7 and 9. Thus DP1 DP4 DP13 DP26 DP52 Males a −0.034 −0.035 −0.036 −0.021 b Females 0.0050 b −0.026 −0.035 −0.040 c a See comments below. b Not statistically significant. c Depleted experience. with a negative sign implying decreasing recovery intensities over time. The similarity of the results for the male DP4, DP13 and DP26 experience, together with the female DP13, DP26 experience, and to a lesser extent the female DP4 experience, is noteworthy. The situation is somewhat more complex for the male DP1 experience, with B xz dependent on z. For this experience, recovery intensities decrease with time for durations in excess of 4 weeks but less than 6 years. Discussions with reinsurers and direct insurers active in the PHI market in UK suggest that a possible explanation for the increase in recovery rates with time at these short sickness durations for DP1 policies could be the improved management of claims by insurers over this period, together with a move to active intervention in newly admitted claims. These changes may have led to fewer marginal cases being accepted as new claims; to accepted claims being managed more actively at the short durations, leading to increased short duration recoveries; and to the residual claims, surviving this initial, active management stage, being the more problematic and long term cases. These effects would be particularly noticeable for shorter deferred policies, as these would provide greater scope for early intervention in any newly notified claims. For longer deferred period policies, claims would be notified somewhat later and tend to be more established by the time that the insurer’s claims management process could intervene. These changes could explain the difference in the nature of the results for DP1 compared to the longer deferred periods.

3. Mortality from sickness intensities