Mortalite oranlarındaki sapmalar ve karma sigortanın sapma ortamındaki istikrarı

Abstract

. 68 MARMARA UNIVERSITY INSTITUTE FOR BANKING AND INSURANCE DEPARTMENT OF INSURANCE DEVIATIONS IN MORTALITY RATES AND THE STABILITY OF ENDOWMENT INSURANCE Prepared By Ali CANYÜREK ISTANBUL 1996. 69 CONTENTS FOREWORD / INTRODUCTION 1... FUNDAMENTALS of ENDOWMENT INSURANCE 2... MORTALITY DEVIATIONS. SAMPLE PRODUCT 24 - General Descriptions - Technical Base - Technical Interest - Premia and Loadings - Annual Net Premium - Loaded Annual Net Premium - Commissions - Annual Commercial Premium - Mathematical Reserves - Used Notation - Annual Net Premium Values - Selected Mathematical Reserve Values - Characteristic Ratios of Product BASES OF THE PRODUCT-REACTION AGAINST MORTALITY DEVIATIONS & CONCLUSION 38 - Product -Reaction By Different Deviation Rates ; Comparison Of Pre- Calculated Premia with Post-Deviational Premia APPENDIX 55 - General Descriptions To The Calculation Model - Spreadsheet Formulation70. ACTUARIAL ABBREVIATIONS Cx Dx MX Nx lo l«o lx l'x dx d'x qx px q'x k w SYPx LYPx L KOM(t) KOMBX TYPx YPx Vx(t) i n t q{x;10} ex{e} ex{k} : to age 0 reduced number of dyings at age x : to age 0 reduced number of livings at age x : sum of Cx values from ages x to w : sum of Dx values from ages x to w : radix of the mortality table used s radix of the mortality table modified ( which, reflects the real demographic situation ) number of livings at age x modified number of livings at age x number of dyings at age x modified number of dyings at age x dying probability at age x living probability at age x modified dying probability at age x (new mortality) deviation parameter ( coefficient for gx ) last age on mortality table annual net premium of an insurance contract, with `n` years insurance period starting at age x loaded annual premium premium loading commission paid in t'th year of insurance total commission factor annual commercial premium of an insurance contract, with `n` year insurance period starting at age x annual premium - in general mathematical reserve at the end of t. year technical interest rate total insurance period time past in years probability of dying in 10 years for people at age x life expectancy for males at age x life expectancy for females at age x i v.. *. v^ *.`; UV. 71 TABLES & GRAPHS - CHANGES IN TURKISH MORTALITY 10 - CHANGES IN TURKISH MORTALITY - GRAPH 11 - CHANGES IN AMERICAN MORTALITY - GRAPH 12 - NEW MORTALITY VALUES THROUGH DEVIATION 21 - SAMPLE PRODUCT / ANNUAL COMMERCIAL PREMIA 28 - MATHEMATICAL RESERVES WITHOUT PROFIT SHARE FOR SELECTED AGES & TERMS 31 - MATHEMATICAL RESERVES WITH PROFIT SHARE FOR SELECTED AGES & TERMS 33 - 35th AGE BASED PREMIUM INDEX 35 - SURRENDER / PAID RATIO - WITHOUT PROFIT SHARE 36 - SURRENDER / PAID RATIO - WITH PROFIT SHARE 37 - PRODUCT -REACTION TABLE [1.1]..TABLE [3.3] 42 - CHANGES IN WEIGHTED INDEX OF PREMIUM - GRAPH 50 `. * ' ' *>/72 CHAPTER 1. FUNDAMENTALS of ENDOWMENT INSURANCE Living and dying probabilities, separately and together, lead to three different insuring transactions. Products, which, are based on only living probability, are not widespread in Turkish life insurance market, except specific annuity insurances. Products, developed on only dying probability, are supplied from each insurer in form of `yearly term insurance`, but there is very small demand for these products on the market. Other variations ( like `cheap education guarantee insurance` / `cheap inheritance insurance` ) cannot be presented adequately, in spite of their likeliness to be bought. Third group of products, which are dealing with living and dying probability together, have the famous sub-item ; i.e., ENDOWMENT INSURANCE. Endowment insurance, has to pay the pre-defined `living capital` at the end of the period of insurance ( n ) or the `death benefit` at untimely death of the insured. In various applications, it is common, that the `living capital` is equal to the `death benefit`. There are other types of endowment insurances ( like double endowment or s emi -endowment ) in developed markets. But there is very small demand for these products also. In this study, we deal with an endowment insurance of which benefits ( `living capital` and `death benefit` ) are the same. Additionally, hereby studied endowment insurance has a profit-sharing rider, as unguaranteed policy element.. 73 Parameters of this product are the same with the commercial parameters of Turkish life insurance market. In Turkey, during 1985 - 1995, a type of endowment insurance ( say `average age based 'death + accumulating` life insurance` or in short `accumulating life` ) had a great success in the market. This product has a negative trend in sales nowadays. Below we will discuss the main differences between `real endowment insurance` and modified version of endowment insurance, i.e., accumulating life insurance. COMPARISON OF TWO KINDS OF ENDOWMENT INSURANCE - MAIN DIFFERENCES - 1. Principally, endowment insurance is based on exact age, where accumulating life insurance has the `estimated average age of the company's portfolio` as its base ( accumulated life insurance premiums are formed of `savings premium part`, `risk premium part` and expenses ). This specification enables the customers to buy the life insurance product for a `fair price` ( i.e., more for old people and less for young people ). Another result of this specification is the decision- flexibility of the insurer in reinsurance relations by `growing old of the portfolio` : Changing the reinsurer will be more costly when the average age of the portfolio is grown. 2. In endowment insurance, benefits are combined. On the other hand, in accumulating life insurance, benefits ( accumulated found benefit and death indemnity ) are operated separately. ' ' %/. 74. The mechanism of endowment insurance has in fact two opposite working insurance components. First one of these components works like `term insurance`, but has a decreasing charge in premium with the period past ( t ) in insurance. In time, most of the guaranteed value ( living capital ) is collected via premia paid. So, there is very small amount necessary to complete the death benefit, which becomes expensive in further ages. There is also no need to reinsure the death benefit if a proper retention level is chosen. Because of this structure, getting an accumulated personal found through endowment insurance is easier than trough `accumulated life` insurance. 3. In endowment insurance, the definition of the benefit, which is to pay at the end of the insurance period, is more clearly expressed. That specification helps to re- sale of new / additional life insurances related to the changed financial situation of the policyholder. However, in accumulating life, insureds have to pay a `definite` premium for an `indefinite` period of time. So, the benefit `looses` its meaning partially.. 75 CHAPTER 2. MORTALITY DEVIATIONS There is a conceptual distinction between mortality deviations and various mortality adjustments. Mortality adjustments create the difference between raw mortality measurements and mortality table values : Principally used methods are `graduation` ( fitting to a curve ) and `adding safety margins `. The significant agents, which affect the mortality are `age` ; `sex` ; `marital status` ; `occupation` ; `climate` ; `race` and `living style`. Because of instability of these agents, the mortality values change in time, in spite of their validity at the phase of product -development. These time based changes in pre-defined mortality values are hereby called as `mortality deviations`. Main reasons, which raise mortality deviations are `becoming obsolescent of the mortality table used` ; `diversion of the risk composition of the insurance portfolio` ; `selected mortality table's weakness to represent the sub -population to be insured` and ` extraordinary developments `. BECOMING OBSOLESCENT OF THE MORTALITY TABLE In this case, generally, there is a mortality decrease to observe : The mortality table used is based often on the previous generation, furthermore, new medical methods and technological improvements lead to higher surviving rates. DIVERSION OF THE RISK COMPOSITION OF THE INSURANCE PORTFOLIO In this case, there are two mortality deviation types to observe : ?»!*` -f*,*1.-If, `S>.76. The balance of policyholder's sex can. vary from tlie theoretical ratio or the product may draw the attention of married people more than single people, which have different mortalities respectively. SELECTED MORTALITY TABLE'S WEAKNESS TO REPRESENT THE SUB -POPULATION In this case, there are two mortality deviation types to observe : Community or sub -population mortality is very different and often stays at a lower level than the mortality of whole population. In case of mass marketing to middle-upper economic class, for example, the mortality table based on population measurements will lead the insurance company to a commercially inconvenient situation. EXTRAORDINARY DEVELOPMENTS Extraordinary developments are very dangerous factors with low probabilities. There are many circumstances, which are not noted as exclusions in many country's `life insurance's general conditions` : An indefinite and critical/lethal epidemic event ( like a genetic mutant of Lassa, Ebola or other virus ) may lead to a collapse of mortality interpretations and disorder various demographic indicators. AN APPROACH TO THE FORMULATION OF DEVIATIONS - FORMULATION of MORTALITY DEVIATION - See pages 17 for mathematical expressions. The formulation is based on only one coefficient ( k ), VVN'İ?.>VV- :/77. In order to get a more precise model of deviation, it is possible to use different coefficients regarding the shape of the death-probability function : 1. layer -> ages between 1 and 10 2. layer -> ages between 11 and 20 3. layer -> ages between 21 and 40 4. layer -> ages between 41 and 60 5. layer -> ages between 61 and w On the other hand, using only one coefficient reflects the real commercial state adequately, keeping in mind that reinsurance agreements ( commissioning ) have the similar esprit. The deformations of the mortality values in the last 5 ages are in fact unimportant, in such a manner that the sum of entry-ages and duration/period is less than 71. For a list of new mortality values produced by k = 0,7 see page 20. ;.:J((( ^-`S- ? !.. V^,V. «*,;. ?.....»' A. 78. CHAPTER 3. SAMPLE PRODUCT This chapter surveys a sample product with, profit-sharing mechanism. Parameters of this product are the same with the commercial parameters of Turkish life insurance market. The insurance contract has a minimum period ( or term ) of 5 years and a maximum period of 20 years. Entry age is between 18 and 50. The formulas of the product development phase are given in pages 23 to 26. ANNUAL COMMERCIAL PREMIA See page 28 for the list. As noted, premium values are given where the capital is equal to 100 000 unit. MATHEMATICAL RESERVES WITHOUT PROFIT SHARE - SELECTED VALUES - See page 31 for this guaranteed policy element, MATHEMATICAL RESERVES WITH PROFIT SHARE - Most Common Profit Sharing System in Turkey - For the first year, in Turkish life insurance market usually, the insured does not obtain profit share. The profit share is the result of investing the previous year's mathematical reserve and separating the interest obtained by the investment. 'j c t t* /rx79 Beginning by the second year, the mathematical reserve of the previous year is invested and a profit is obtained. This profit is decreased by the previous year's technical interest and 95 % - in general - (policyholder's share of the investment income} is distributed to the insured's account. MATHEMATICAL RESERVES WITH PROFIT SHARE - SELECTED VALUES - See page 33 for this unguaranteed policy element. The list is based on a regular annual investment income/ which gives an extra net profit rate of 4 % annualy. CHARACTERISTIC RATIOS 35 AGE BASED PREMIUM INDEX This index shows the changes in premium level, compared to the amount paid by the mode ( in statistical mean ) of consumers. The mod-consumer ( is 35 years old ) pays always 100 units for each term / period. x. 20 ( age ) 25 30 35 40 45 ^OT^l'^'V 50 iFMW ( n : term ). 10 15 2080 SURRENDER / PAID RATIOS - WITHOUT PROFIT SHARE - WITH PROFIT SHARE These ratios show the total amount of surrender value ( number of units ), whereby total payment during the insurance period is expressed as 100 units. See page 36 for values computed without profit share and page 37 with profit share. 2 * [ft** / *? %. 81 CHAPTER 4. BASES OP THE PRODUCT- REACTION AGAINST MORTALITY DEVIATIONS SIMULATION & RESULTS This chapter makes a comparison between pre-defined premium rate and post-deviational premium rate of an endowment insurance at different mortality deviations ( symbolized with, `k` ) and technical interest rates. The stability of premiums of the endowment insurance depends on three components in its formulation : ANNUAL NET PREMIUM SYPx SYPX = Mx - Mx+n Dx Dx+n Dx Dx Nx - Nx+n ANALYSIS OF THE STABILITY A. COMPONENT 1 There will be less `current` deaths in insurance period than the expected death number. So, the relation... M'x - M'x+n Mx - Mx+n < D'x Dx...causes to collect overpremia insurance company. positive effect for the '< ;'.' ?. Jw- 1.' % ' :. ?.,V- )... S..Ç `><%***..->.]h M. 82. B. COMPONENT 2 There will be more survivors ( insureds ) at the end of insurance period than expected. So, the relation... D ' x+n Dx+n > D'x Dx...gives the result to pay more living benefit : negative effect for the insurance company. C. COMPONENT 3 There will be more survivor ( or more policyholder, they have to pay premium ) for each insurance year than expected. So, the relation... D'x Dx N'x - N'x+n Nx - Nx+n...causes to collect overpremia also : positive effect for the insurance company. Proof for third component : If the expression below D'x Dx is true, N'x - N'x+n Nx - Nx+n then, Nx - Nx+n N'x - N'x+n < has to be true. ^^«^s^fc, Dx D ` x,, <3i. /......... _ w_. ' <8>» / -f^``* ??,83. Therefore, Nx - Nx+n Dx + Dx+1 + Dx+2 + Dx+3 +... + Dx+n-1 Dx Dx Dx+1 Dx+2 Dx+n-1 1 + + +... + Dx Dx Dx 1 + lx+1 (1+i) ~x * + lx+2 (1+i) ~x * + (l+i)~(x+l) lx (l+i)~(x+2) lx lx+1 1 lx+2 1 = 1 + * + * +... lx (1+i) lx (1+i) `2 px(l) px(2) px(n-l) = 1 + + +... + (1+i) (l+i)~2 (l+i)~(n-l) and, respectively.. N`x - N`x+n p'x(l) p'x(n-l) = 1 + +... + D«x (1+i) (l+i)~(n-l) Furthermore, p'x = 1 - q'x = 1 - k * qx = 1 - k * (1-px) = l-k+k*px k < 1 => p'x > px ( for each age ) J^'f- /,,84. i > O p»x(l) px(l) > (1+i) (1+i) => N'x - N'x+ıı Nx - Nx+n > D'x Dx PRODUCT- REACTION TO THE DEVIATION INTERPRETATION EXAMPLE FOR MATRICES TABLE [ 1.1 ] Technical interest : i = 0,03 Deviation : k = 0,90 ( 10 % mortality decrease ) Index Definition Post-deviational premium rate in %, where the predefined premium rate for each x & n is 100 %...ages at entry... p e r i o d s From table 1.1. we can get the information, that insurance company gains an excess income of 2 % maximal (100 % - 97,99 %), for the technical interest i = 3 % and at mortality decrease rate 10 %. '??/. 85. The tables from Nr. 1.1. to Nr. 3.3. can be used as tools of a sensitivity analysis. A better method to interpret a reaction-matrix is to reduce the matrix to a single value by weighting with the demand profile of the customers. A SAMPLE DEMAND MATRIX FOR DEVIATION-WEIGHTING BASED ON THE DATABASE OF A INSURANCE COMPANY P e r i o d s sum of cells gives 100 The reduction process means multiplying each cell on a reaction-matrix with the correspondent cell on the demand matrix and having the sum of these 16 products. See page 50 for graphical results of this reduction process. In extremum, the difference between pre-defined premium and post-deviational premium is about 3 %. This rate points out the stability of endowment insurance in favor of the insurance company at altering the mortality values, in particular during mortality decrease. On the other hand, this rate is a significant indicator of the protective characteristic of endowment insurance in favor of the policy holder compared with average-age based accumulating life insurance.. 86 APPENDIX CALCULATION MODEL In this chapter, we give a full list of SPREADSHEET model in 123 notation, on which, the numeric analyses of this study are done. To reduce the typing time of formulas, it is useful to skip Turkish explanation fields and calling Copy command to fill the analysis -matrices in electronic worksheet. Mathematical reserves are calculated is this worksheet model by `recurrence` method that minimizes processing / recalculation time. Minimized use of LOOKUP functions is also useful to improve the computing performance. ^ NT5*-'