1 Ris-based profit and loss attribution Oliver Locwood FIA 9 Louise Croft Birmingham B4 5NY United Kingdom Abstract The impending Solvency II regulatory regime in the European Union requires insurance undertaings to perform a regular attribution of profits and losses as one of the tests to gain approval for the use of an internal model in calculating capital requirements. Traditional analysis of surplus processes, depending on an arbitrary choice of order in which the riss affecting the undertaing are assumed to occur, may not be adequate to meet the standard of profit and loss attribution required by Solvency II. This paper proposes an approach to profit and loss attribution that does not depend on an arbitrary choice of order, which has the following further advantages over traditional analysis of surplus processes: Greater integration is possible between financial reporting and ris management within the business because items in the attribution correspond directly to ris factors considered in the internal model and/or in the Own Ris and Solvency Assessment (ORSA). Second (and higher) order impacts can be incorporated into the attribution in a systematic way. Each item in the attribution is more amenable to validation, because it is expressed as a percentage of a ris factor that has occurred multiplied by a figure for the sensitivity to that ris factor. The approach will enable riss not currently allowed for in the internal model or ORSA to be identified. The approach is illustrated by means of an example applying it to analyse the change in the value of a portfolio of annuity liabilities. Key words: Solvency II; Profit and Loss Attribution; Internal Models; Financial Reporting; Ris Management; Annuities
2 Ris-based profit and loss attribution Introduction. Under the impending Solvency II regulatory regime in the European Union, insurance undertaings may apply to their supervisor to use an internal model to calculate their Solvency Capital Requirement (SCR). Directive 009/38/EC specifies a number of tests that must be satisfied for the supervisor to approve the internal model. One of these tests is the profit and loss attribution test, specified in Article 3 of the Directive: Insurance and reinsurance undertaings shall review, at least annually, the causes and sources of profits and losses for each major business unit. They shall demonstrate how the categorisation of ris chosen in the internal model explains the causes and sources of profits and losses. The categorisation of ris and attribution of profits and losses shall reflect the ris profile of the insurance and reinsurance undertaings.. The Committee of European Insurance and Occupational Pensions Supervisors (CEIOPS) (now the European Insurance and Occupational Pensions Authority (EIOPA)) provided advice to the European Commission on implementing measures for these tests in CEIOPS (009). Section 7 of this CEIOPS paper covers profit and loss attribution..3 CEIOPS (009) contains relatively few recommendations for detailed requirements as to how profit and loss attribution should be carried out. It is therefore liely that many undertaings applying for internal model approval will use existing analysis of surplus processes as a starting point for meeting the profit and loss attribution test. However, there are a number of reasons to believe that such processes may be inadequate to satisfy the test..4 Paragraph 7.9 of CEIOPS (009) states the following: The Profit and loss attribution for each major business unit shall be as transparent as possible. The attribution shall enable the insurance or reinsurance undertaing to explain a large part of its annual profit and loss. Furthermore the Profit and loss
3 Ris-based profit and loss attribution 3 attribution has to be a tool for validating the internal model (Article 4) and for managing the business (Article 0)..5 The existing analysis of surplus process will typically have the following characteristics: The process will consist of a series of steps, in each of which one assumption or element of experience is changed from its opening to its closing value. There will typically be no specific justification for the choice of order of the steps, other than consistency with previous practice. The existing categorisation of ris factors in the process is unliely to be aligned with that in the proposed internal model. It may or may not be straightforward to amend the existing process to recategorise the ris factors. The process will typically not produce an expression for the impact of each step as the product of a figure for the percentage of the relevant ris factor that has occurred over the analysis period and a figure for the sensitivity to it. The process will, by default, report the impacts of interactions between ris factors along with whichever of the ris factors appears last in the analysis. Therefore, these impacts will not be reported separately and the place where they are reported will depend upon the arbitrary choice of order of the steps. Some elements of the analysis may be calculated as balancing items, to avoid the complexity of effectively performing a full valuation of the assets and liabilities for each step of the analysis..6 The following concerns that the existing process may fail to meet the standards of profit and loss attribution implied by the Directive and by the CEIOPS advice can therefore be identified: Both the Directive and the CEIOPS advice imply a need to mae the ris categorisation in the attribution consistent with that in the internal model. This is also important from the point of view of using the internal model to manage the business, the use test under Solvency II, as it ensures that the categorisation of riss used by the financial reporting function to explain movements in reported results is consistent with the categorisation used by the ris management function as per the internal model. Given other inadequacies
4 Ris-based profit and loss attribution 4 of the existing process, simply changing the categorisation in the existing process may not be the most appropriate method of achieving this consistency. An important means by which a profit and loss attribution provides validation of an internal model is by verifying consistency of sensitivities implied by the attribution with SCR stresses from the internal model. This suggests a need for an attribution methodology to generate such sensitivity data automatically. The lac of separate identifiability of the impacts of interactions between ris factors gives rise to a lac of transparency. Separate identification of these impacts is also important for monitoring these interactions as part of managing the business, particularly where riss are largely hedged away to first order. The inclusion of balancing items in a profit and loss attribution is unsuitable for demonstrating that it explains a large part of the undertaing s profit and loss, and for validating that the internal model captures all the material riss..7 The approach to profit and loss attribution proposed in this paper is to perform a Taylor series expansion of the profit over the attribution period in terms of the ris factors in the internal model. All the concerns noted in Section.6 can be addressed within this framewor. In particular, the first-order terms of the Taylor series expansion will be sensitivities to the ris factors multiplied by summary statistics for the weighted average percentages of the ris factors that occurred over the period..8 More generally, the proposed approach can be applied to analyse the change in any value metric in terms of any set of ris factors. Whilst the Solvency II requirement for profit and loss attribution refers specifically to an attribution in terms of the ris factors in the internal model, it is liely that management will also gain considerable benefit from an attribution in terms of ris factors considered in the Own Ris and Solvency Assessment (ORSA). This will include riss arising over the business planning horizon at a range of return frequencies, unlie the SCR which is specifically concerned with a one-year time horizon and a -in-00 return frequency..9 The remainder of this paper is structured as follows. Section applies the proposed approach to a concrete example, and Section 3 considers a refinement of the approach. Section 4 considers how the results of applying the approach can be communicated, and Section 5 gives suggestions for further research.
5 Ris-based profit and loss attribution 5 Example. The approach proposed in Section will now be applied to analyse the change in the value of a hypothetical portfolio of annuity liabilities over the year 0. The following assumptions were used to generate the liability data: The annuities were assumed to be level single life immediate annuities payable in the middle of each future year. The opening ages of the annuitants were 40,000 randomly chosen integers uniformly distributed between 50 and 90 inclusive. The amounts of the annuities were 40,000 random observations from an exponential distribution with mean, % of the annuitants were assumed to be male and 30% were assumed to be female. 50% of the annuitants were assumed to suffer Heavy mortality and 50% were assumed to suffer Light mortality, as defined in Table below. There was assumed to be no correlation between the above characteristics of the annuities. No new annuitants were assumed to join the portfolio during 0.. The liabilities were discounted using the UK government nominal yield curves published by the Ban of England, available at statistics/documents/yieldcurve/unom(month_end).zip. As the yield curves from the Ban of England only extend to 5 years, an assumption was made that forward rates were constant beyond a term of 5 years. Figure shows the continuously compounded spot rates as at both 3 December 00 and 3 December 0.
6 Ris-based profit and loss attribution 6 Figure : Continuously compounded spot interest rates as at 3 December 00 and 3 December % 4.00% Spot rate 3.00%.00%.00% 3-Dec-0 3-Dec- 0.00% Term (years).3 Table shows the assumed base mortality, for males and females separately, for Heavy and Light annuitants separately and separately for the opening liability valuation, the closing liability valuation and experience over 0. The mortality tables PCMA00 and PCFA00 were published by the Continuous Mortality Investigation Bureau (CMIB) of the Institute and Faculty of Actuaries, and represent the mortality experience of UK insured group pension schemes that contributed to the CMIB s investigation over the period These tables can be found in CMIB (009). Table also assigns a class number to each combination of gender and Heavy or Light. In the remainder of this paper, the liability classes will be referred to by these numbers rather than by gender and Heavy or Light. Table : Base mortality assumptions Gender Heavy / Light Class Male Heavy Male Light Female Heavy 3 Female Light 4 Opening valuation 00% PCMA00 00% PCMA00 00% PCFA00 00% PCFA00 Closing valuation 05% PCMA00 85% PCMA00 05% PCFA00 85% PCFA00 Experience 0% PCMA00 80% PCMA00 0% PCFA00 80% PCFA00 The situation illustrated in Table shows experience emerging over 0 to indicate that Heavy annuitants suffer heavier mortality than Light annuitants. The opening liability valuation made no allowance for separate mortality assumptions for Heavy
7 Ris-based profit and loss attribution 7 and Light annuitants. The closing liability valuation does mae such an allowance, but it is assumed that the differentials between the assumptions are only half those in the experience as a result of limited credibility of the experience. The closing liability valuation also includes an overall strengthening of the assumptions by 5% of the PCMA/PCFA00 tables, introduced as a result of recent light mortality experience. This overall strengthening, however, turns out not to be borne out by the experience over 0..4 The PCMA00 and PCFA00 tables represent mortality as at 3 December 000, i.e. q x represents the probability that a life aged x exact as at 30 June 000 dies before 30 June 00. An allowance for mortality improvements since 3 December 000 is therefore required. In this example, these allowances were taen from the CMI Projections Model published by the CMIB. For the opening liability valuation, the 00 version of the Model, documented in CMIB (00), was used, and for the closing liability valuation, the 0 version of the Model, documented in CMIB (0), was used. For both the opening and closing liability valuations, the Model was parameterised with Core parameters, as per the terminology of these papers. For the opening liability valuation, the Long-Term Rate of Mortality Improvement was set to.5% p.a. For the closing liability valuation, the Long-Term Rate of Mortality Improvement was strengthened to % p.a., with the strengthening arising from a review of potential future improvement trends. For experience over 0, the same improvement allowances were made as the 0 allowances in the opening liability valuation..5 The opening value of the liabilities is: j OLDANNAMT ( OLDDISCFACMIDYR( OLDSURVPROBMIDYR( j,, () where: OLDANNAMT ( = the annual annuity amount payable to annuitant j as shown in the opening data file, OLDDISCFACMIDYR = exp( ( ) OLDSPOTMIDYR( ), ( OLDSPOTMID YR( = the opening continuously compounded spot interest rate for a term of ) years, and (
8 Ris-based profit and loss attribution 8 OLDSURVPRO BMIDYR( j, = the opening probability that annuitant j survives for a period of ) years. ( Similarly, the closing value of the liabilities is: NEWDISCFACMIDYR( NEWANNAMT (, () j: DTHIND( =0 NEWSURVPROBMIDYR( j, where: DTHIND ( = 0 if annuitant j survives 0, otherwise, NEWANNAMT ( = the annual annuity amount payable to annuitant j as shown in the closing data file, NEWDISCFACMIDYR = exp( ( ) NEWSPOTMIDYR( ), ( NEWSPOTMID YR( = the closing continuously compounded spot interest rate for a term of ) years, and ( NEWSURVPRO BMIDYR( j, = the closing probability that annuitant j survives for a period of ) years. (.6 The summations () and () can be evaluated more efficiently by noting that OLDSURVPRO BMIDYR( j, and NEWSURVPRO BMIDYR( j,, for fixed, depend only on the class and age of annuitant j. Thus the opening liability value is: OLDDISCFACMIDYR( OLDTOTANNAMT (, (3) (,, ) c x OLDSURVPROBMIDYR c x where: OLDTOTANNA MT ( = the total annual annuity amount payable to lives aged x in class c as shown in the opening data file, and OLDSURVPRO BMIDYR( = the opening probability that an annuitant aged x in class c survives for a period of ) years. ( This opening value evaluates to m in this example. The closing liability value is: NEWDISCFACMIDYR( NEWTOTANNAMT (, (4) ) c x NEWSURVPROBMIDYR( where: NEWTOTANNA MT ( = the total annual annuity amount payable to lives aged x in class c as shown in the closing data file, and
9 Ris-based profit and loss attribution 9 NEWSURVPRO BMIDYR( = the closing probability that an annuitant aged x in class c survives for a period of ) years. ( This closing value evaluates to m in this example..7 OLDSURVPRO BMIDYR( may be expressed in terms of the underlying base mortality and improvement assumptions as follows, with a similar expression for NEWSURVPRO BMIDYR( : OLDSURVPROBMIDYR( + OLDBASEMU ( x l ), (5) = exp l= OLDIMPFAC( x + l, l) * OLDBASEMU ( x + ) OLDIMPFAC( x +, where: OLDBASEMU ( y) = the opening base force of mortality, i.e. continuously compounded mortality rate, for an annuitant aged y in class c, calculated as log( OLDQ( y)), where OLDQ ( y) is the annually compounded mortality rate from the appropriate multiple of the PCMA/PCFA00 table, and OLDIMPFAC ( y, = the opening mortality improvement factor for an annuitant aged y in class c in the th year following the opening valuation date, as given by the CMI Projections Model. These factors are referred to as Mortality Reduction Factors in CMIB (00, 0) and the references therein. OLDBASEMU ( y) and OLDIMPFAC ( y, are here assumed to be constant between exact ages y and y +, and OLDIMPFAC ( y, is assumed to be constant over the th year. Alternatively: OLDSURVPROBMIDYR( = exp l= OLDMU ( x + l, l), (6) * OLDMU ( x +, where OLDMU ( y, is the opening assumed force of mortality for an annuitant aged y in class c in the th year following the opening valuation date..8 Differences between (4) and (3) may arise for the following reasons:
10 Ris-based profit and loss attribution 0 There may be inconsistencies in annuity amounts, classes and/or ages between the opening and closing data files. However, in this example, it is assumed that no such inconsistencies exist. The expected closing liability value, assuming that experience over 0 is in line with the opening assumptions, will in general be different from the opening liability value. The closing yield curve will in general be different from the opening yield curve rolled forward for a year. The closing base mortality assumptions will in general be different from the opening ones. The closing mortality improvement assumptions will in general be different from the opening ones. Mortality experience over 0 will in general be different from the opening assumptions. The expected closing liability value referred to in the second bullet above is: EXPDISCFACMIDYR( (, (7) ) OLDTOTANNAMT OLDSURVPROBMIDYR c x + c x (,, where: OLDDISCFACMIDYR( + ) EXPDISCFACMIDYR( =, OLDDISCFACENDYR() OLDDISCFACENDYR( ) = exp( OLDSPOTENDYR()), and OLDSPOTEND YR() = the opening continuously compounded spot interest rate for a term of one year. This expected closing value evaluates to m in this example. All differences between (4) and this arise from the last four bullets above, given the assumption stated in the first bullet..9 In this example, it is assumed that the internal model or ORSA considers the following four ris factors: A level shift in continuously compounded interest rates, relative to the expected position of the yield curve rolling forward.
11 Ris-based profit and loss attribution A level percentage change in continuously compounded base mortality assumptions, applied separately for each class so that the assumptions reflect the mortality experience of each class. A level percentage change in mortality improvement factors, applied across all classes together. A level percentage difference between continuously compounded mortality rates experienced over 0 and those expected from the opening assumptions, applied separately for each class. Thus a weighted average level percentage of each of these ris factors that has occurred over 0 will be derived. For interest rates, we write: NEWSPOTMID YR( = EXPSPOTMIDYR( + YLDRISE + YLDERRMIDYR(, (8) where: log( EXPDISCFACMIDYR( ) EXPSPOTMID YR( =, YLDRISE = the equivalent level increase in continuously compounded interest rates that has occurred over 0, and YLDERRMIDY R( = an error term. YLDRISE will be derived by equating to zero the sum of the impacts of these error terms upon the liability value across all durations. For base mortality, we write: NEWBASEMU( = ( + BASEMURISEPC( ) OLDBASEMU(, (9) + BASEMUERR( where BASEMURISE PC( is the equivalent level percentage increase in continuously compounded base mortality assumptions that has occurred over 0 for class c and BASEMUERR ( is an error term. For mortality improvements, we write: NEWIMPFAC( = ( IMPSTRESSPC) OLDIMPFAC( + ), (0) + IMPFACERR( where IMPSTRESSP C is the equivalent level percentage decrease in improvement factors, i.e. strengthening of improvement assumptions, that has occurred over 0 and IMPFACERR ( is an error term. For mortality experience over 0, we write:
12 Ris-based profit and loss attribution DTHIND( = exp( ( + MORTEXPVAR( ) OLDMU ( )), () + HISTDTHERR( + HISTDTHSECONDORDEFFECT ( for annuitants j with a class in the opening data, OLDCLASS (, of c and an age in the opening data, OLDAGE (, of x, where MORTEXPVAR ( is the equivalent level percentage difference between the continuously compounded mortality rates observed over 0 for class c and those expected from the opening assumptions, and HISTDTHERR ( is an error term. The reason for introducing the additional term HISTDTHSEC ONDORDEFFECT ( is so that MORTEXPVAR ( can be derived by solving a linear equation, i.e. the equation given by setting the sum of HISTDTHERR ( over all j in class c to zero. HISTDTHERR ( and HISTDTHSEC ONDORDEFFECT ( are defined as follows: HISTDTHERR( = DTHIND( exp( OLDMU ( )), () + MORTEXPVAR( OLDMU ( ) exp( OLDMU ( )) HISTDTHSECONDORDEFFECT ( = exp( OLDMU ( )) MORTEXPVAR( OLDMU ( ) exp( OLDMU ( )) exp( ( + MORTEXPVAR( ) OLDMU ( )). (3).0 This gives the following expression for the closing liability value: c x OLDTOTANNAMT ( exp( ( + MORTEXPVAR( ) OLDMU ( )) + OLDANNAMT ( HISTDTHERR( j: OLDCLASS ( = OLDAGE( = x + OLDANNAMT ( HISTDTHSECONDORDEFFECT ( j: OLDCLASS ( = OLDAGE( = x EXPDISCFACMIDYR( exp( ( )( YLDRISE + YLDERRMIDYR( )), [( + BASEMURISEPC( ) OLDBASEMU ( x + l) + BASEMUERR( x + l)][( IMPSTRESSPC) l= OLDIMPFAC( x + l, l + ) + IMPFACERR( x + l, l)] exp [( + BASEMURISEPC( ) OLDBASEMU ( x + + BASEMUERR( x + ][( IMPSTRESSPC) OLDIMPFAC( x +, + ) + IMPFACERR( x +, ] (4). We now perform a Taylor series expansion of (4) in the following variables, up to second order:
13 Ris-based profit and loss attribution 3 YLDRISE YLDERRMIDY R(, for each term BASEMURISE PC(, for each class c BASEMUERR (, for each class c and each age x IMPSTRESSP C IMPFACERR (, for each class c, each age x and each term MORTEXPVAR (, for each class c HISTDTHERR (, for each annuitant j HISTDTHSEC ONDORDEFFECT (, for each annuitant j, which, as its name suggests, is treated as already being a second-order quantity. The constant term in the Taylor series expansion is equal to the expected closing liability value. Table shows the coefficients of YLDERRMIDY R(, BASEMUERR (, IMPFACERR ( and HISTDTHERR ( respectively. Table : Coefficients of the error terms in the Taylor series expansion Term in YLDERRMIDYR( BASEMUERR ( IMPFACERR ( Coefficient YLDERRMIDYRCOEFF( = ( ) EXPDISCFACMIDYR( OLDTOTANNAMT ( OLDSURVPROBMIDYR( + c x BASEMUERRCOEFF( OLDTOTANNAMT ( y) OLDIMPFAC( x y + ) EXPDISCFACMIDYR( x y) = OLDSURVPROBMIDYR( y, x y + ) y< x EXPDISCFACMIDYR( + > x y OLDSURVPROBMIDYR( y, + ) IMPFACERRCOEFF( = OLDTOTANNAMT( x OLDBASEMU( EXPDISCFACMIDYR( OLDSURVPROBMIDYR( x, + ) EXPDISCFACMIDYR( l) + l> OLDSURVPROBMIDYR( x, l + ) )
14 Ris-based profit and loss attribution 4 Table : Coefficients of the error terms in the Taylor series expansion (continued) Term in HISTDTHERR ( Coefficient HISTDTHERRCOEFF( = OLDANNAMT ( EXPDISCFACMIDYR( EXPSURVPROBMIDYR( OLDCLASS(, OLDAGE( +,. The criterion used in this example to determine YLDRISE, BASEMURISE PC(, IMPSTRESSP C and MORTEXPVAR ( is to equate the sum of the relevant error terms from Table to zero. This gives the expressions shown in Table 3 and the numerical results shown in Table 4. Table 3: Expressions for the amounts of the ris factors that occurred over 0 Quantity YLDRISE BASEMURISE PC( IMPSTRESSP C MORTEXPVAR ( x Expression YLDERRMIDYRCOEFF( [ NEWSPOTMIDYR( EXPSPOTMIDYR( ] YLDERRMIDYRCOEFF( BASEMUERRCOEFF( x [ NEWBASEMU( OLDBASEMU( ] BASEMUERRCOEFF( OLDBASEMU( j: OLDCLASS ( = c c x OLDIMPFAC( + )] IMPFACERRCOEFF( OLDIMPFAC( + ) HISTDTHERRCOEFF( [ DTHIND( c x j: OLDCLASS ( = c IMPFACERRCOEFF( [ NEWIMPFAC( exp( OLDMU ( OLDAGE(,))] HISTDTHERRCOEFF( OLDMU ( OLDAGE(,)) exp( OLDMU ( OLDAGE(,)) Table 4: Numerical results for the amounts of the ris factors that occurred over 0 Quantity Class Class Class 3 Class 4 YLDRISE -.6% BASEMURISE PC( 5.35% -5.83% 5.3% -5.74% IMPSTRESSP C 4.07% MORTEXPVAR(.86% -6.5% 5.89% 6.98%
15 Ris-based profit and loss attribution 5 The following observations can be made from the results in Table 4: The value of -.6% for YLDRISE indicates that interest rates decreased by an average of.6% p.a. over 0. This is a weighted average where the weights reflect the sensitivity of the closing liability value to interest rates at each term. The.6% decrease is measured relative to the position of interest rates increasing slightly over 0 to reflect the roll forward of the (upwardsloping) yield curve as at 3 December 00. Table suggests that the values of BASEMURISE PC( should be +5% for c =,3 and -5% for c =, 4, as these are the proportions the assumed percentages of the PCMA/PCFA00 tables have changed by. The actual values observed are consistent with this in sign, but slightly greater than 5% and 5% in magnitude as a result of the use of continuously as opposed to annually compounded mortality rates in the attribution. The continuously compounded mortality rates, not being bounded above by, increase more rapidly with age than the annually compounded rates at the oldest ages. The value of 4.07% for IMPSTRESSP C indicates that improvement factors decreased by an average of 4.07% over 0, i.e. that the change in improvement factors was equivalent to multiplying the mortality assumptions by 95.93% on average. This average is a weighted average where the weights reflect the sensitivity of the closing liability value to improvement factors at each age in each future year. Table suggests that the values of MORTEXPVAR ( should be approximately +0% for c =, 3 and -0% for c =, 4. The actual values observed are consistent with this in sign, except slightly for c = 4, but differ in magnitude as a result of random fluctuations in the experience and, to a much lesser extent, as a result of the use of continuously as opposed to annually compounded mortality rates in the attribution..3 The coefficients of the ris factors in the Taylor series expansion represent sensitivities of the closing liability value to the ris factors. Expressions for these sensitivities are shown in Table 5 and numerical results for the sensitivities are shown in Table 6. The following notation is introduced in Table 5 for brevity:
16 Ris-based profit and loss attribution 6 EXPTOTMUMIDYR( = OLDMU ( x + l, l + ) + l=. (5) * OLDMU ( x +, + ) The values in Table 6 have been divided by 00 so as to represent the impact of a % rather than a 00% change in each ris factor. Table 5: Expressions for the sensitivities of the closing liability value to the ris factors Ris factor Sensitivity ( ) EXPDISCFACMIDYR( OLDTOTANNAMT ( YLDRISE OLDSURVPROBMIDYR( + c x OLDTOTANNAMT ( BASEMURISE PC( EXPDISCFACMIDYR( x EXPTOTMUMIDYR( x +, OLDSURVPROBMIDYR( + ) OLDTOTANNAMT( IMPSTRESSP C EXPDISCFACMIDYR( c x EXPTOTMUMIDYR( x +, OLDSURVPROBMIDYR( + ) OLDTOTANNAMT( OLDMU ( ) MORTEXPVAR ( EXPDISCFACMIDYR( x OLDSURVPROBMIDYR( + ) Table 6: Numerical results for the sensitivities of the closing liability value to % of the ris factors ( m) Ris factor Class Class Class 3 Class 4 YLDRISE BASEMURISE PC( IMPSTRESSP C.38 MORTEXPVAR( Table 7 shows the sums of the numerical values of each type of term in the Taylor series expansion. The opening, expected closing and actual closing liability values have already been quoted, in Sections.6 and.8, but are repeated in Table 7 for ease of reference. The movement in liability value due to terms higher than second order in the Taylor series expansion can then be identified as a balancing item. )
17 Ris-based profit and loss attribution 7 Table 7: Sums of the numerical values of each type of term in the Taylor series expansion Terms in Value ( m) Opening liability value Expected closing liability value YLDRISE 7.84 YLDERRMIDY R( BASEMURISE PC( BASEMUERR ( IMPSTRESSP C IMPFACERR ( MORTEXPVAR ( HISTDTHERR ( (YLDRISE) 0.65 YLDRISE * YLDERRMIDYR( YLDRISE * BASEMURISEPC(.649 YLDRISE * BASEMUERR( YLDRISE * IMPSTRESSPC.90 YLDRISE * IMPFACERR( YLDRISE * MORTEXPVAR( YLDRISE * HISTDTHERR( 0.00 ( YLDERRMIDY R( ) YLDERRMIDY R( BASEMURISEPC( -0.4 YLDERRMIDY R( BASEMUERR( YLDERRMIDY R( IMPSTRESSPC YLDERRMIDY R( IMPFACERR( l), for l YLDERRMIDY R( MORTEXPVAR( YLDERRMIDY R( HISTDTHERR( ( BASEMURISE PC( ) BASEMURISE PC( BASEMUERR( BASEMURISE PC( IMPSTRESSPC -0.0 BASEMURISE PC( IMPFACERR( 0.4 BASEMURISE PC( MORTEXPVAR( BASEMURISE PC( OLDCLASS( ) HISTDTHERR( BASEMUERR ( BASEMUERR( y), for x y 0.00 BASEMUERR ( IMPSTRESSPC( BASEMUERR ( IMPFACERR( y, BASEMUERR ( MORTEXPVAR( 0.00 BASEMUERR ( OLDCLASS(, HISTDTHERR(, for x OLDAGE( ( IMPSTRESSP C) IMPSTRESSP C * IMPFACERR( 0.089
18 Ris-based profit and loss attribution 8 Table 7: Sums of the numerical values of each type of term in the Taylor series expansion (continued) Terms in Value ( m) IMPSTRESSP C * MORTEXPVAR( IMPSTRESSP C * HISTDTHERR( 0.00 IMPFACERR( IMPFACERR( y, + y, for x y 0.09 IMPFACERR ( MORTEXPVAR( 0.00 IMPFACERR ( OLDCLASS(, OLDAGE( +, HISTDTHERR( ( MORTEXPVAR ( ) HISTDTHSEC ONDORDEFFECT ( Terms higher than second order.884 Actual closing liability value The following comments should be made on Table 7: As required, the entries for YLDRISE, BASEMURISE PC(, IMPSTRESSP C and MORTEXPVAR ( are equal to the sensitivities to the ris factors from Table 4 multiplied by the percentages of the ris factors that have occurred over 0 from Table 6, summing over all classes c where necessary. The terms in YLDERRMIDY R(, BASEMUERR (, IMPFACERR ( and HISTDTHERR ( sum to zero. This is a consequence of how YLDRISE, BASEMURISE PC(, IMPSTRESSP C and MORTEXPVAR ( were derived. The left-hand column of Table 7 has been completed according to which terms in the Taylor series expansion are non-zero. For example, the table contains a row labelled ( YLDERRMIDY R( )) rather than YLDERRMIDY R( YLDERRMIDYR( l) because the terms in YLDERRMIDY R( YLDERRMIDYR( l) are zero for l. The significance of the positive entry for (YLDRISE), for example, is that a large decrease in interest rates has a greater impact than that estimated by prorating the impact of a small decrease, i.e. the entry is a convexity impact. The significance of the negative entry for YLDRISE * YLDERRMIDYR(, for example, is that the increase in liability value per unit decrease in interest rate is most sensitive to interest rates at the longest terms, where the interest rate decreases over 0 are relatively small and so YLDERRMIDY R( is
19 Ris-based profit and loss attribution 9 positive. It should be noted that the significance of this and similar entries is liely to be less readily communicable to users of the attribution than for other entries. The refinement to the attribution methodology developed in Section 3 should be considered as a means of addressing this. The significance of the positive entry for YLDRISE * BASEMURISEPC(, for example, is that the overall strengthening of the base mortality assumptions gives a larger increase in liability value after taing account of the fact that interest rates also decreased. The significance of the positive entry for YLDRISE * IMPFACERR(, for example, is that the larger strengthenings of improvement factors, i.e. the more negative values of IMPFACERR (, occur at the longer terms, where the liability value has a relatively high sensitivity to a level interest rate decrease. The entry for ( YLDERRMIDY R( ), for example, arises from the relatively simple method of determining YLDRISE used in this example, involving solving a linear equation. If this and similar entries are significant, then the refinement to the attribution methodology developed in Section 3 should be considered. The significance of the negative entry for YLDERRMIDY R( IMPFACERR( l), for l, for example, is that the larger reductions in improvement factors, i.e. the more negative values of IMPFACERR ( l), occur at the longer terms l, where the interest rate decreases over 0 are relatively small and so YLDERRMIDY R( is positive for l. The terms higher than second order are reasonably small in aggregate. The second bullet in Section 5, on suggestions for further research, suggests an approach that could be used to reduce the higher-order terms further, without performing a full Taylor series expansion to higher than second order. 3 Refinement 3. The approach developed in Section derives YLDRISE, for example, by equating the sum of the terms in YLDERRMIDY R( in the Taylor series expansion to
20 Ris-based profit and loss attribution 0 zero. A more refined approach would be to equate the sum of the terms in YLDERRMIDY R(, YLDRISE * YLDERRMIDYR( and ( YLDERRMIDY R( )) to zero and then solve a quadratic rather than a linear equation for YLDRISE. section pursues this approach, in respect of YLDRISE only for brevity. This 3. Table 8 shows the coefficients of YLDERRMIDY R(, YLDRISE * YLDERRMIDYR( and ( YLDERRMIDY R( )) in the Taylor series expansion, for each term. The coefficient of YLDERRMIDY R( has already been shown in Table but is repeated in Table 8 for ease of reference. Table 8: Coefficients of YLDERRMIDY R(, YLDRISE * YLDERRMIDYR( and ( YLDERRMIDY R( ) in the Taylor series expansion Term in YLDERRMIDYR( YLDRISE * YLDERRMIDYR( ( YLDERRMIDY R( ) Coefficient YLDERRMIDYRCOEFF( = ( ) EXPDISCFACMIDYR( OLDTOTANNAMT ( c x OLDSURVPROBMIDYR( + ) YLDRISEYLDERRMIDYRCOEFF( = ( ) EXPDISCFACMIDYR( OLDTOTANNAMT ( c x OLDSURVPROBMIDYR( + ) YLDERRMIDYRYLDERRMIDYRCOEFF( = ( ) EXPDISCFACMIDYR( OLDTOTANNAMT ( OLDSURVPROBMIDYR( + ) c x 3.3 The quadratic equation to be solved for YLDRISE is: + YLDERRMIDYRCOEFF( [ NEWSPOTMIDYR( EXPSPOTMIDYR( YLDRISE] + YLDRISE YLDRISEYLDERRMIDYRCOEFF( [ NEWSPOTMIDYR( EXPSPOTMIDYR( YLDRISE] YLDERRMIDYRYLDERRMIDYRCOEFF( [ NEWSPOTMIDYR( EXPSPOTMIDYR( YLDRISE] = 0. (6) This reduces to: A ( YLDRISE) + B * YLDRISE + C = 0, (7)
21 Ris-based profit and loss attribution where: A = B = YLDERRMIDYRYLDERRMIDYRCOEFF( YLDRISEYLDERRMIDYRCOEFF( YLDRISEYLDERRMIDYRCOEFF( [ NEWSPOTMIDYR( EXPSPOTMIDYR( ] YLDERRMIDYRYLDERRMIDYRCOEFF( [ NEWSPOTMIDYR( EXPSPOTMIDYR( ] YLDERRMIDYRCOEFF( YLDERRMIDYRCOEFF( C = [ NEWSPOTMIDYR( EXPSPOTMIDYR( ]. YLDERRMIDYRYLDERRMIDYRCOEFF( + [ NEWSPOTMIDYR( EXPSPOTMIDYR( ] This quadratic equation is satisfied by -.58% (and by 3.% which clearly does not appropriately represent the movement in interest rates over 0). YLDRISE is therefore taen to be -.58%. The slight reduction in magnitude of YLDRISE compared with the value of -.6% shown in Table 4 is consistent with the negative value of the terms in YLDRISE * YLDERRMIDYR( shown in Table 7 being larger in,, magnitude than the positive value of the terms in ( YLDERRMIDY R( )). 3.4 Table 9 shows the entries in Table 7 which change as a result of the reestimation of YLDRISE. Table 9: Changes in the sums of the numerical values of each type of term in the Taylor series expansion following the re-estimation of YLDRISE Terms in Revised value Change from Table 7 ( m) ( m) YLDRISE YLDERRMIDY R( (YLDRISE) YLDRISE * YLDERRMIDYR( YLDRISE * BASEMURISEPC( YLDRISE * BASEMUERR( YLDRISE * IMPSTRESSPC YLDRISE * IMPFACERR(
22 Ris-based profit and loss attribution Table 9: Changes in the sums of the numerical values of each type of term in the Taylor series expansion following the re-estimation of YLDRISE (continued) Terms in Revised value Change from Table 7 ( m) ( m) YLDRISE * MORTEXPVAR( YLDRISE * HISTDTHERR( ( YLDERRMIDY R( ) YLDERRMIDY R( BASEMURISEPC( YLDERRMIDY R( BASEMUERR( YLDERRMIDY R( IMPSTRESSPC YLDERRMIDY R( IMPFACERR( l), for l YLDERRMIDY R( MORTEXPVAR( YLDERRMIDY R( HISTDTHERR( Terms higher than second order The following observations can be made from Table 9: The terms in YLDRISE and (YLDRISE) have reduced in magnitude as a result of the reduction in absolute value of YLDRISE from.6% to.58%. The terms in YLDERRMIDY R(, YLDRISE * YLDERRMIDYR( and ( YLDERRMIDY R( )) now add to zero when combined. It will therefore no longer be necessary to explain what these terms represent when communicating the attribution. The changes in the remaining second (and higher) order terms are not material but have been included in Table 9 for completeness. 4 Communication of results 4. For communication purposes, it will often be necessary to summarise the ey features of a profit and loss attribution, without listing all the second-order terms of the Taylor series expansion as was done in Table 7. It will also often be necessary to provide evidence that the results of the attribution are reasonable, without undertaing a full review of the detailed calculations. This section suggests techniques that could be used to achieve this. 4. Table 0 is similar to Table 7 except that: Only the first-order terms in the ris factors are shown explicitly.
23 Ris-based profit and loss attribution 3 The refinement to the calculation of YLDRISE developed in Section 3 has been implemented. The terms in BASEMURISE PC( and MORTEXPVAR ( are shown separately for each of the four classes c. Each impact on the liability value is expressed as the product of the percentage of the relevant ris factor that has occurred over 0, as shown in Table 4, and the sensitivity to the ris factor, as shown in Table 6. The significance of this is that it will often be possible to validate the figures for the percentages of the ris factors that have occurred and the sensitivities by reference to a highlevel understanding of events over the year and of the nature of the business. Table 0: Summary of attribution, showing the percentages of the ris factors that have occurred over 0 and the sensitivities to the ris factors Ris factor % Sensitivity to % of ris Impact occurred factor ( m) ( m) Opening liability value Expected closing liability value YLDRISE -.58% BASEMURISE PC() 5.35% BASEMURISE PC() -5.83% BASEMURISE PC(3) 5.3% BASEMURISE PC(4) -5.74% IMPSTRESSP C 4.07% MORTEXPVAR ().86% MORTEXPVAR () -6.5% MORTEXPVAR (3) 5.89% MORTEXPVAR (4) 6.98% Non-linear responses to ris factors Closing liability value The following comments should be made regarding Table 0: The references to the notation used in this paper for the different ris factors could readily be replaced by descriptions of the ris factors for communication purposes. Some users of the attribution may not require the split of the terms in BASEMURISE PC( and MORTEXPVAR ( by class c. In this case the
24 Ris-based profit and loss attribution 4 presentation of Table below would be more appropriate. The % occurred figures in Table for BASEMURISE PC( and MORTEXPVAR ( are weighted averages of those in Table 0, where the weights are the Sensitivity to % of ris factor figures. As the two ris factors BASEMURISE PC(, across all classes c, and IMPSTRESSP C are both defined as level percentage shifts in mortality rates, in opposite directions, the Sensitivity to % of ris factor figures for these ris factors are equal and opposite. Table : Simplification of Table 0, omitting the split by liability class Ris factor % Sensitivity to % of ris Impact occurred factor ( m) ( m) Opening liability value Expected closing liability value YLDRISE -.58% BASEMURISE PC( -5.3% IMPSTRESSP C 4.07% MORTEXPVAR ( 5.43% Non-linear responses to ris factors Closing liability value Users are liely to require further explanation of the impact of non-linear responses to the ris factors. An approach to explaining this impact is suggested in Section 4.3. It should also be noted that adopting the proposal in the second bullet of Section 5, on suggestions for further research, would prevent the large impact from arising here in the first place. 4.3 The largest non-linear response to the ris factors in this example is the term in (YLDRISE), i.e. the interest rate convexity impact. It is therefore instructive to show a graph showing how the increase in liability value for a level decrease in interest rates varies with the amount of that decrease, taing account of both the term in YLDRISE, i.e. the linear interest rate impact, and this convexity impact. This graph is shown in Figure.
25 Ris-based profit and loss attribution 5 Figure : Impact of a level decrease in interest rates upon the liability value Impact ( m) % 0.6% 0.3% 0.47% 0.63% 0.79% 0.95%.0% Decrease in interest rates.6%.4%.58% Convexity impact Linear impact Figure shows the convexity impact remaining small for decreases in interest rates up to around %, but then becoming more significant as a percentage of the linear impact for larger decreases, reaching 3.7% of the linear impact for a.58% decrease which is what was observed over 0. It is also instructive to show the split of the impact of the interest rate movements by term, both for a level.58% decrease in interest rates and for the actual movement at each term, and split between the linear impact and the convexity impact. This is shown in Figures 3 and 4 respectively. Figure 3: Impact of a level.58% decrease in interest rates upon the liability value Impact ( m) Linear impact Linear impact + Convexity impact Year from closing valuation date
26 Ris-based profit and loss attribution 6 Figure 4: Impact of the actual movement in interest rates at each term upon the liability value Impact ( m) Linear impact Linear impact + Convexity impact Year from closing valuation date The ey comments to be made regarding Figures 3 and 4 are as follows: Both Figures 3 and 4 have been cut off at a term of 5 years because there were no particular features of note in the tails beyond 5 years there was simply a gradual decay of the impacts. Figure 4 has a sharper pea than Figure 3. This is because the actual decrease in continuously compounded spot rates over 0, after allowing for the expected roll forward of the yield curve, was less than.58% at the shortest terms and at the longest terms, but greater than.58% at intermediate terms (years 3-8 inclusive), reaching a pea of.07% in year 7. In both Figures 3 and 4, the convexity impacts increase as a percentage of the linear impacts with increasing term, as they depend on the square of the term rather than on the term itself. In both Figures 3 and 4, the sum of the heights of the light turquoise bars, including terms above 5 years, is equal to the sum of the terms in YLDRISE and in (YLDRISE) in the Taylor series expansion, which is equal to 7.444m m = 8.8m from Table 9. The sum of the heights of the plum bars is equal to 7.444m in Figure 3, but is equal to the figure of 7.84m from Table 7 in Figure 4, as the latter figure was based on an equivalent level interest rate fall derived without taing the convexity impacts into account.
27 Ris-based profit and loss attribution There is also a potential need to provide graphs to illustrate how the percentage of a ris factor that has occurred has been derived. To this end, we shall here illustrate how the figure of 4.07% for IMPSTRESSP C was determined so that: IMPSTRESSP C was derived. = c x IMPFACERRC OEFF( IMPFACERR( 0. (8) Therefore if a surface is plotted with x and as the independent variables and c IMPFACERRC OEFF( IMPFACERR( as the dependent variable, then the volume between the surface and the horizontal plane will be zero. This surface is plotted in Figure 5. Figure 5: Impacts upon the liability value of the differences between the closing improvement factors for each age and year and ( )% of the opening improvement factors 0,000 5,000 Sum over c of IMPFACERRCOEFF ( * IMPFACERR( ( ) 0-5,000-0,000-5,000-0,000-5, Year from closing valuation date, Age, x 5 5,000-0, ,000-5, ,000--5,000-5,000--0,000-0,000--5,000-5,000--0,000 It should be noted that Figure 5 shows zero impacts away from a diagonal band starting from the age range 5-9 in year from the closing valuation date. This is because all the annuitants in this example are in the age range at the opening valuation date and no new annuitants join the portfolio. The fact that the volume between the surface in Figure 5 and the horizontal plane is zero demonstrates that the equivalent level strengthening of 4.07% in improvement factors has been calculated correctly. However it is also instructive to examine the drivers of the shape of the surface, in terms of the differences between the closing improvement factors and ( )% of the opening improvement factors, IMPFACERR (, and the sensitivities of the closing liability value to those
28 Ris-based profit and loss attribution 8 differences, IMPFACERRC OEFF(. To this end, Figure 6 shows the weighted averages of IMPFACERR ( across the different classes c, c of c IMPFACERRCOEFF( IMPFACERR(, and Figure 7 shows the values IMPFACERRCOEFF( c IMPFACERRC OEFF(. Figure 6: Weighted average differences between the closing improvement factors for each age and year and ( )% of the opening improvement factors 6.00% 4.00% Weighted average of IMPFACERR(.00% 0.00% -.00% -4.00% -6.00% -8.00% 3 34 Year from closing valuation date, Age, x %-6.00%.00%-4.00% 0.00%-.00% -.00%-0.00% -4.00%--.00% -6.00%--4.00% -8.00%--6.00% Figure 7: Sensitivities of the closing liability value to the differences between the closing improvement factors for each age and year and ( )% of the opening improvement factors 0-00,000 Sum of IMPFACERRCOEFF ( ( ) -00, , , , , , Year from closing valuation date, Age, x 5-00, , , , , , , , , , , , ,000 The following are the ey observations from Figures 6 and 7:
29 Ris-based profit and loss attribution 9 As for Figure 5, Figures 6 and 7 show zeros away from a diagonal band starting from the age range 5-9 in year from the closing valuation date. Figure 6 shows positive values in the early years of the projection. This is because the increase in the Long-Term Rate of Mortality Improvement from.5% to % p.a. has relatively little effect upon improvement factors in the early years. In later years of the projection, the values in Figure 6 become negative as the effect of the increased Long-Term Rate of Mortality Improvement builds up. At the oldest ages, the values in Figure 6 become positive again. This is because the Core parameterisation of the CMI Projections Model incorporates a tapering of long-term mortality improvements from the assumed Long-Term Rate of Mortality Improvement at age 90 to zero at age 0. The sensitivities of the closing liability value in Figure 7 are all negative, because an increase in an improvement factor will always increase mortality rates and so reduce the liability value. The largest negative sensitivities in Figure 7 occur towards the top of the age range in the early years of the projection. In this region, the population is relatively high, mortality rates are relatively high so a given change in improvement factors has a relatively large impact upon mortality rates in absolute terms, and the annuity payments are only being discounted for a relatively short period. 5 Suggestions for further research The following suggestions for further research can be identified. They all relate to areas where there is a need to extend the theory developed in this paper to apply the methodology: The example in this paper considered only a level shift in interest rates as a ris factor. In practice, insurance undertaings are exposed to changes in the shape as well as in the level of the yield curve. It would therefore be valuable to introduce ris factors to the attribution allowing a wider range of movements in the yield curve. Some techniques for determining which shapes of yield curve movement to consider to most effectively capture the range of riss are detailed in Section 7 and Appendix C of Franland et al. (009).
30 Ris-based profit and loss attribution 30 The example in this paper had a relatively large second-order term in the Taylor series expansion, in (YLDRISE), which was a convexity impact. However, it would be more convenient in terms of communicating the results of the attribution if such large second-order terms did not arise. In addition, economic capital calculations often assume as a first approximation that the capital position varies linearly with the ris factors, with an additional allowance for non-linear responses to the ris factors being calculated as a separate exercise which may be updated less frequently than the main capital calculation. For these reasons, it would be valuable to define the ris factors in such a way that the financial impact of x % of a ris factor occurring was approximately proportional to x. The levels of the second-order terms in the Taylor series expansion would be used to determine a transformation to apply to the ris factor to achieve this. This technique could also be extended to higher orders than the second to improve the accuracy of the transformation. A number of value metrics considered in actuarial wor do not vary smoothly with the underlying ris factors. For example, an insurance undertaing s capital requirement might be the greater of a calculation based on a formula prescribed by the supervisor and a calculation that taes account of the undertaing s specific ris profile. This does not vary smoothly with the ris factors when the biting calculation changes. To accommodate this situation within the methodology of this paper, it would be necessary to introduce firstorder sensitivities that are step functions of the ris factors and second-order sensitivities that are Dirac delta functions. References CEIOPS (009), CEIOPS Advice for Level Implementing Measures on Solvency II: Articles 0 to 6: Tests and Standards for Internal Model Approval. CMIB (009), Continuous Mortality Investigation Report Number 3. CMIB (00), The CMI Mortality Projections Model, CMI_00, Woring Paper 49. CMIB (0), The CMI Mortality Projections Model, CMI_0, Woring Paper 55. Franland R., Smith A.D., Wilins T., Varnell E., Holtham A., Biffis E., Eshun S., Dullaway D. (009), Modelling extreme maret events, British Actuarial Journal, 5(), 99-7.
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