ACTUARIAL FUNCTIONS 1_05

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1 ACTUARIAL FUNCTIONS _05 User Guide for MS Office 2007 or laer

2 CONTENT Inroducion Insallaion procedure Demo Version and Acivaion Using formulas and synax Using he help... 6 Noaion Technical documenaion Formulas for person Life Table Formulas for 2 persons - Life Table Formulas for Generaion Life Table Lieraure Acuarial funcion for Windows User Guide Sepember 203 Version _05 Page 2

3 INTRODUCTION The applicaion called Acuarial funcion.0 is specifically designed o help acuaries in heir life calculaions creaed in MS Excel environmen hrough a library of basic acuarial funcions in he form of add-in o Microsof Excel. The purpose of Acuarial funcion.0 is o cover basic acuarial funcions (see chaper 6) and o suppor he user in a highly qualified work wih validaed and auomaed calculaions wihou any addiional ime demands. The poenial benefis include: Enhanced produciviy Decreased ime o final resul Sandardized and opimized work Acuarial funcion.0 is a small and lighweigh applicaion wih minimal memory consumpion. This user s guide will cover he acuarial funcion asks associaed wih Microsof Excel. I is assumed ha he reader is familiar wih he Microsof Excel environmen. 2 INSTALLATION PROCEDURE To run he applicaion, you need o download he.xla file and save i on your disc. NOTE:.xla files do no work wihou opening hrough MS Excel. I is necessary o open he MS Excel firs and hrough he File > Open find he.xla file. If you wan o add Acuarial funcion o MS Excel and have he funcion available whenever your MS Excel is opened, click on File -> Opion and choose Add-Ins. Click on he Go buon a he boom of he form shown below. Acuarial funcion for Windows User Guide Sepember 203 Version _05 Page 3

4 The window wih a lis of available Add-Ins appears. Click on he Browse buon and selec he.xla file on your disc and click on OK. Then he Acuarial funcion is available in he lis of he Add-Ins. Check he box and click on OK. The acuarial funcion is available in he User defined funcion (for more informaion see chaper 4). Acuarial funcion for Windows User Guide Sepember 203 Version _05 Page 4

5 3 DEMO VERSION AND ACTIVATION Afer opening he file, you will be informed abou he demo version running. In he demo version, you are limied o he maximum of 30 calculaions. Once you reach he limi, all furher resuls will be se o zero. Click OK o coninue. Acuarial funcion for Windows User Guide Sepember 203 Version _05 Page 5

6 Now, he Welcome screen appears and you have he opion o acivae he full version of he Acuarial funcion. If you wan o run he demo, click on he Demo buon. Once you wan o acivae he full version, click on Acivae. As you already have he regisraion key, ener i in he Acivaion key cell and click on he Acivae produc buon. In case you do no have he Acivaion key, click on he Purchase now from Tools4F.com buon and inser he Regisraion key (he Copy o clipboard buon makes he process smooher) in he purchase form. Once he license is ordered and paid, he Acivaion key is sen o you and you can acivae he full version. NOTE: The Acivaion key is generaed by means of he Regisraion key provided by his applicaion and boh of he keys are unique o one compuer only. You will no be able o use neiher he Regisraion key nor he Acivaion key on any oher compuer. Please make sure you are using he correc Regisraion-Acivaion key pair, as, once acivaed on one compuer, i will no be possible o move and run he acivaed applicaion on anoher compuer. Acuarial funcion for Windows User Guide Sepember 203 Version _05 Page 6

7 4 USING FORMULAS AND SYNTAX Acuarial funcions.0 includes a se of mahemaical predefined formulas which are used o simplify he process of insering funcions ino he workshees and eliminae errors. NOTE: Once you work wih Acuarial funcion, all he daa you use mus be in one workbook. I is no possible o use he daa which are in oher workbooks han hose wih he calculaions. Before you sar working wih he funcion, i is necessary o se/define he inpu daa lxnumber of he living according o heir age. The daa able mus have he following srucure: index age (ineger) in he firs column (column A) and value lx (real)in he second column (column B) as shown in he following picure. Acuarial funcion for Windows User Guide Sepember 203 Version _05 Page 7

8 This able has o be placed on he named workshee, he name of which is LT_M in our example as shown in he picure below. All he Formulas in his Acuarial Funcion add-in sar wih he prefix AF and all heir argumens can be specified by value or by reference. To display he funcions click on he cell you wan o apply he funcion, click he Inser Funcion buon (fx) o he lef of he formula bar and click on he funcion caegory User defined. Once you have seleced a funcion, click OK. Once you have chosen he funcion, Excel will display a synax window o help you wih consrucing he funcion. For furher explanaion of each argumen see chaper 5. Le us calculae a simple example here. The funcion AF_Axn(LT; X; N; I) (single neo premium for pure deah cover (erm)) has four argumens. The firs one is called LT. As described in chaper 5, LT sands for Life Table. I is he name (sring) of he shee wih he lx able, in our case i is LT_M. As i is a sring variable, i has o be se wihin he quoaion marks. If he name of he workshee wih he lx able is LT_M and is placed in cell E2, you can use synax =AF_Axn(E2;23;4;0,02). Acuarial funcion for Windows User Guide Sepember 203 Version _05 Page 8

9 The oher argumen is X - Age which is he enry age of he insured person in years (ineger). The hird argumen is N Duraion. Duraion is a period in number of years (ineger) (ypically he policy period). The fourh argumen is I Ineres. Ineres is he annual ineres rae (real) used o discoun he paymens. There are wo possibiliies how o wrie his argumen. One opion is o wrie a decimal number (e.g. 0.02), he oher one is o wrie percenage (2 %). When his is finished, click OK in he synax window o inser he funcion ino your workshee. Of course, even here, when you use cell references in he Excel formulas, he formulas will auomaically updae whenever he relevan daa in he spreadshee change. Acuarial funcion for Windows User Guide Sepember 203 Version _05 Page 9

10 Oher examples of synax Funcion AF_ann_xn(LT; X; N; ; I; PT) (presen value of life annuiy for a limied number of years) has six argumens. The argumens LT Life able, X age, N duraion and I Ineres are explained above. Paymen frequency is a number of paymens wihin one year ( - annually, 2 semi-annually, 4 - quarerly, - monhly). As i is a number, here is no need o pu i in he quoaion marks. PT Paymen ype akes values - for paymens a he beginning of he period or 0 - for paymens a he end of he period. =AF_ann_xn("LT_F";30;0;2;0,04;0) Le us calculae funcion AF_I_Axn(LT; X; N; I; INF; IT; IR) (single neo premium for pure deah cover (erm) wih increasing sum assured), which has seven argumens. The argumens LT Life able, X age, N duraion and I Ineres are explained above. INF Increase frequency is an increase frequency during one year (ineger). This argumen acceps he values - annually, 2 - semi-annually, 4 quarerly or - monhly. IT Increase ype is he ype of paymen increasing. Possible values can be for geomeric or 2 for linear/arihmeic. IR Increase rae is he annual nominal increase rae (real) used for paymen increase. For INF =, IR can be 00 % or less. If he increase frequency (INF) seleced is higher han annual (=), he increase relaed o he sub-annual period is IR*freq, e.g. for monhly frequency (=) he monhly increase is IR*. Firs we consider INF =, IR = 00 %. =AF_I_Axn("LT_M";25;5;0,03;;2;00%) = 0, Now we consider he same ineres rae, bu increase frequency during one year is semi-annually. Tha means INF = 2, IR = 200 %. =AF_I_Axn("LT_M";25;5;0,03;2;2;200%) = 0, Acuarial funcion for Windows User Guide Sepember 203 Version _05 Page 0

11 Oher possibiliies Besides working wih he Life Table, you can also work wih Generaion Life Tables (GT). GT mus have he following srucure: birh year (ineger) in he firs column (column B), age (ineger) in he second column (column C) and lx (real) in he hird column (column D) This Acuarial funcions applicaion also includes he calculaion for 2 persons. All funcions can be seen in he Formula bar or in Help as shown in he picure bellow. Acuarial funcion for Windows User Guide Sepember 203 Version _05 Page

12 Le us calculae a simple example. The funcion AF_Axyn(LT_X; LT_Y; X; Y; N; I)(Single neo premium for pure deah cover (erm) for 2 lives) has six argumens. The firs wo argumens are Life Tables for he persons. In his example, we consider man (LT_M) and woman (LT_F). X is he age of he firs person, Y is he age of he second person. N is he Duraion. Duraion is a period in number of years (ineger) of he calculaion (ypically he policy period, ec.). You need o roll down o see he las argumen. I Ineres is he annual ineres rae (real) used o discoun he paymens. The decimal number or percenage can be inpu. When his is finished, click OK in he synax window o inser he funcion ino your workshee. 5 USING THE HELP If you need exra help in creaing your funcion, click on he Help on his funcion reference in he boom lef corner of he Inser funcion dialog box. The Help will hen be displayed (as in he following picure). Acuarial funcion for Windows User Guide Sepember 203 Version _05 Page

13 Acuarial funcion for Windows User Guide Sepember 203 Version _05 Page 3

14 On he lef side, he lis of funcions is displayed. To learn more abou oher funcions, click on he name of he required funcion and he Help will be displayed on he righ side accordingly. The Help can be opened direcly from he folder i is saved. NOTE: The file *.chm and *.xla mus be saved in he same folder o work wih he Help properly. If you have any rouble in finding he Help for a paricular opic or he Help provided is no sufficien, send an o us ) and our suppor eam will conac you. Your feedback helps us o improve our documenaion, so we welcome your inpu. 6 NOTATION LT Life Table LT is he name of he shee (sring) wih moraliy able, he srucure of he able is: index age (ineger) in he firs column, value lx (real) in he second column. X Age Age is he enry age of he main insured person in years (ineger) N Duraion Duraion is he period in number of years (ineger) of he calculaion (ypically he policy period, ec.) K Deferred period Deferred period is he number of years (ineger) of deferred paymens Paymen Paymen frequency is he number of paymens wihin one year(ineger) ( frequency - annually, 2 - semi-annually, 4 - quarerly, - monhly) I Ineres Ineres is he annual ineres rae (real) used o discoun he paymens INF Increase Increase frequency is he increase frequency during one year (ineger) ( - frequency annually, 2 - semi-annually, 4 - quarerly, - monhly) IT Increase Increase ype is he ype of paymen increasing ( - geomeric, 2 - ype linear/arihmeic) IR Increase rae Increase rae is he annual rae (real) used for paymen increase PT Paymen - for paymens a he beginning of he period, 0 - for paymens a he end ype of he period ω Max age in LT v v = + I Acuarial funcion for Windows User Guide Sepember 203 Version _05 Page 4

15 7 TECHNICAL DOCUMENTATION 7. FORMULAS FOR PERSON LIFE TABLE AF_lx (LT; X) lx is in LT AF_qx (LT; X) q x = l x l x+ l x (.) AF_npx (LT; X; N) n years survival probabiliy np x = l x+n l x (.2) AF_Dx (LT; X; I) Commuaion figure Dx D x = l x ν x (.3) AF_Cx (LT; X; I) Commuaion figure Cx C x = d x ν x+ = (l x l x+ )ν x+ (.4) AF_Nx (LT; X; I) Acuarial funcion for Windows User Guide Sepember 203 Version _05 Page 5

16 Commuaion figure Nx ω x N x = j=0 D x+j (.5) AF_Mx (LT; X; I) Commuaion figure Mx ω x M x = j=0 C x+j (.6) AF_Sx (LT; X; I) Commuaion figure Sx ω x S x = j=0 N x+j (.7) AF_Rx (LT; X; I) Commuaion figure Rx ω x R x = j=0 M x+j (.8) AF_Axn (LT; X; N; I) Single neo premium for deah and survival cover A xn = n E x + A xn (.9) AF_Exn (LT; X; N; I) Single neo premium for pure endowmen cover ne x = D x+n D x (.0) Acuarial funcion for Windows User Guide Sepember 203 Version _05 Page 6

17 AF_Axn (LT; X; N; I) Single neo premium for pure deah cover (erm) A xn = M x M x+n (.) D x AF_ann_xn (LT; X; N; ; I; PT) Presen value of life annuiy for limied number of years For paymens a he beginning of he period: n ä xn = ( pay l l x x+j + v j= N ), (.) j + N = {b ; b 0, ; b Z}. (.3) pay = { εz, 0 else. } (.4) For paymens a he end of he period: n a xn = ( pay l l x x+j + v j= N ), (.5) j + N = {b ; b, ; b Z}, (.6) pay = { εz, 0 else. } (.7) Noe: Linear inerpolaion is made beween l x and l x+. AF_kAxn (LT; X; N; K; I) Single neo premium for deah and survival cover wih deferred period years of deferred period k A xn = k n E x + k A xn (.8) Acuarial funcion for Windows User Guide Sepember 203 Version _05 Page 7

18 AF_kExn (LT; X; N; K; I) Single neo premium for pure endowmen cover wih deferred period year of deferred period k ne x = D x+k+n D x (.9) AF_kAxn (LT; X; N; K; I) Single neo premium for pure deah cover (erm) wih deferred period year of deferred period k A xn = M x+k M x+k+n (.20) D x AF_kann_xn (LT; X; N; K; ; I; PT) Presen value of life annuiy for limied number of years wih deferred period years of deferred period For paymens a he beginning of he period: n k ä xn = ( pay l l x x+k+j + v j= N ), (.2) k+j + N = {b ; b 0, ; b Z}. (.22) pay = { εz, 0 else. } (.23) For paymens a he end of he period: n k a xn = ( pay l l x x+k+j + v j= N ), (.24) k+j + N = {b ; b, ; b Z}, (.25) Acuarial funcion for Windows User Guide Sepember 203 Version _05 Page 8

19 pay = { εz, 0 else. } (.26) Noe: Linear inerpolaion is made beween l x and l x+. AF_I_Axn (LT; X; N; I; INF; IT; IR) Single neo premium for deah and survival cover wih increasing sum assured (IA) xn = n (IE) x + (IA) xn (.27) AF_I_Exn (LT; X; N; I; INF; IT; IR) Single neo premium for pure endowmen cover wih increasing sum assured Increaseype is he ype of increasing of paymens: linear/arihmeic geomeric n(ie) x = [ + (n INF ) IR INF ] D x+n D x (.28) n(ie) x = [ + IR INF ]n INF D x+n D x (.29) AF_I_Axn (LT; X; N; I; INF; IT; IR) Single neo premium for pure deah cover (erm) wih increasing sum assured (IA) xn = ( inc l d x x+j + v j= M ), (.30) Increaseype is he ype of increasing of paymens: n j + M = {b ; b, INF ; b Z}. (.3) INF linear/arihmeic Acuarial funcion for Windows User Guide Sepember 203 Version _05 Page 9

20 geomeric inc = inc + IR INF, inc =, (.32) inc = inc ( + IR INF ), inc =. (.33) Noe: Linear inerpolaion is made beween d x and d x+. AF_I_ann_xn (LT; X; N; ; I; INF; IT; IR; PT) Presen value of life annuiy for limied number of years wih increasing paymen For paymens a he beginning of he period: n (Iä) xn = ( pay l inc l x x+j + v j= N ), (.34) j + N = {b ; b 0, freq ; b Z}, freq = max(inf, ), (.35) freq M = {b ; b 0, INF ; b Z}. (.36) INF pay = { εz, 0 else. } (.37) For paymens a he end of he period: n (Ia) xn = ( pay l inc l x x+j + v j= N ), (.38) j + N = {b ; b, freq ; b Z}, freq = max(inf, ), (.39) freq M = {b ; b, INF ; b Z} (.40) INF pay = { εz, 0 else. } (.4) Increaseype is he ype of increasing of paymens: Acuarial funcion for Windows User Guide Sepember 203 Version _05 Page 20

21 linear/arihmeic inc εm = inc + IR INF, inc =, inc M = inc, (.42) geomeric inc εm = inc ( + IR INF ), inc =, inc M = inc. (.43) Noe: Linear inerpolaion is made beween l x and l x+. AF_I_kAxn (LT; X; N; K; I; INF; IT; IR) Single neo premium for deah and survival cover wih increasing sum assured wih deferred period years of deferred period k (IA) xn = k n (IE) x + k (IA) xn (.44) AF_I_kExn (LT; X; N; K; I; INF; IT; IR) Single neo premium for pure endowmen cover wih increasing sum assured wih deferred period years of deferred period Increaseype is he ype of increasing of paymens: linear/arihmeic geomeric k n(ie) x = [ + (n INF ) IR INF ] D x+k+n D x (.45) k n(ie) x = [ + IR INF ]n INF D x+k+n D x (.46) AF_I_kAxn (LT; X; N; K; I; INF; IT; IR) Single neo premium for pure deah cover (erm) wih increasing sum assured wih deferred period years of deferred period n (IA) xn = ( inc l d x x+k+j + v j= M ), (.47) k+j + Acuarial funcion for Windows User Guide Sepember 203 Version _05 Page 2

22 Increase ype is he ype of increasing of paymens: M = {b ; b, INF ; b Z}, (.48) INF linear/arihmeic geomeric inc = inc + IR INF, inc =, (.49) inc = inc ( + IR INF ), inc =. (.50) Noe: Linear inerpolaion is made beween d x and d x+. AF_I_kann_xn (LT; X; N; K; ; I; INF; IT; IR; PT) Presen value of life annuiy for limied number of years wih increasing paymen For paymens a he beginning of he period: n k (Iä) xn = ( pay l inc l x x+k+j + v j= N ), (.5) k+j + N = {b ; b 0, freq ; b Z}, freq = max(inf, ), (.52) freq M = {b ; b 0, INF ; b Z}. (.53) INF pay = { εz, 0 else. } (.54) For paymens a he end of he period: n k (Ia) xn = ( pay l inc l x x+k+j + v j= N ), (.55) k+j + N = {b ; b, freq ; b Z}, freq = max(inf, ), (.56) freq M = {b ; b, INF ; b Z}, (.57) INF Acuarial funcion for Windows User Guide Sepember 203 Version _05 Page 22

23 pay = { εz, 0 else. } (.58) Increase ype is he ype of increasing of paymens: linear/arihmeic inc εm = inc + IR INF, inc =, inc M = inc, (.59) geomeric inc εm = inc ( + IR INF ), inc =, inc M = inc. (.60) Noe: Linear inerpolaion is made beween l x and l x FORMULAS FOR 2 PERSONS - LIFE TABLE AF_lxy (LT_X; LT_Y; X; Y) lx and ly are in LT l xy = l x l y (2.) AF_qxy (LT_X; LT_Y; X; Y) q xy = p xy = p x p y = l x+ l x l y+ l y (2.2) AF_npxy (LT_X; LT_Y; X; Y; N) n years survival probabiliy for 2 lives np xy = l x+n l x l y+n l y (2.3) AF_Dxy (LT_X; LT_Y; X; Y; I) Commuaion figure Dxy Acuarial funcion for Windows User Guide Sepember 203 Version _05 Page 23

24 D xy = l x l y ν x+y 2 (2.4) AF_Cxy (LT_X; LT_Y; X; Y; I) Commuaion figure Cxy C xy = d xy ν x+y 2 + = (l x l y l x+ l y+ )ν x+y 2 + (2.5) AF_Nxy (LT_X; LT_Y; X; Y; I) Commuaion figure Nxy ω max (x,y) N xy = j=0 D x+j,y+j (2.6) AF_Mxy (LT_X; LT_Y; X; Y; I) Commuaion figure Mxy ω max (x,y) M xy = j=0 C x+j,y+j (2.7) AF_Sxy (LT_X; LT_Y; X; Y; I) Commuaion figure Sxy ω max (x,y) S xy = j=0 N x+j,y+j (2.8) AF_Rxy (LT_X; LT_Y; X; Y; I) Commuaion figure Rxy ω max (x,y) R xy = j=0 M x+j,y+j (2.9) AF_Axyn (LT_X; LT_Y; X; Y; N; I) Acuarial funcion for Windows User Guide Sepember 203 Version _05 Page 24

25 Single neo premium for deah and survival cover for 2 lives A xyn = n E xy + A xyn (2.0) AF_Exyn (LT_X; LT_Y; X; Y; N; I) Single neo premium for pure endowmen cover for 2 lives ne xy = D x+n,y+n D xy (2.) AF_Axyn (LT_X; LT_Y; X; Y; N; I) Single neo premium for pure deah cover (erm) for 2 lives A xyn = M x,y M x+n,y+n (2.) D xy AF_ann_xyn (LT_X; LT_Y; X; Y; N; ; I; PT) Presen value of life annuiy for limied number of years for 2 lives For paymens a he beginning of he period: ä xyn = n ( pay l l xy x+j + v j= N ), (2.3),y+j + j + N = {b ; b 0, ; b Z}. (2.4) pay = { εz, 0 else. } (2.5) For paymens a he end of he period: a xyn = n ( pay l l xy x+j + v j= N ), (2.6),y+j + j + Acuarial funcion for Windows User Guide Sepember 203 Version _05 Page 25

26 N = {b ; b, ; b Z}, (2.7) pay = { εz, 0 else. } (2.8) Noe: Linear inerpolaion is made beween l x and l x+. AF_kAxyn (LT_X; LT_Y; X; Y; N; K; I) Single neo premium for deah and survival cover for 2 lives wih deferred period years of deferred period k A xyn = k n E xy + k A xyn (2.9) AF_kExyn (LT_X; LT_Y; X; Y; N; K; I) Single neo premium for pure endowmen cover for 2 lives wih deferred period years of deferred period k ne xy = D x+k+n,y+k+n D xy (2.20) AF_kAxyn (LT_X; LT_Y; X; Y; N; K; I) Single neo premium for pure deah cover (erm) for 2 lives wih deferred period years of deferred period k A xyn = M x+k,y+k M x+k+n,y+k+n (2.2) D xy AF_kann_xyn (LT_X; LT_Y; X; Y; N; K; ; I; PT) Presen value of life annuiy for 2 lives for limied number of years wih deferred period years of deferred period For paymens a he beginning of he period: Acuarial funcion for Windows User Guide Sepember 203 Version _05 Page 26

27 k ä xyn = n ( pay l l xy x+k+j + v j= N ), (2.22),y+k+j + k+j + N = {b ; b 0, ; b Z}. (2.23) pay = { εz, 0 else. } (2.24) For paymens a he end of he period: k a xyn = n ( pay l l xy x+k+j + v j= N ), (2.25),y+k+j + k+j + N = {b ; b, ; b Z}, (2.26) pay = { εz, 0 else. } (2.27) Noe: Linear inerpolaion is made beween l x and l x+. AF_I_Axyn (LT_X; LT_Y; X; Y; N; I; INF; IT; IR) Single neo premium for deah and survival cover for 2 lives wih increasing sum assured (IA) xyn = n (IE) xy + (IA) xyn (2.28) AF_I_Exyn (LT_X; LT_Y; X; Y; N; I; INF; IT; IR) Single neo premium for pure endowmen cover for 2 lives wih increasing sum assured Increase ype is he ype of increasing of paymens: linear/arihmeic geomeric n(ie) xy = [ + (n INF ) IR INF ] D x+n,y+n D xy (2.29) Acuarial funcion for Windows User Guide Sepember 203 Version _05 Page 27

28 n(ie) xy = [ + IR INF ]n INF D x+n,y+n D xy (2.30) AF_I_Axyn (LT_X; LT_Y; X; Y; N; I; INF; IT; IR) Single neo premium for pure deah cover (erm) for 2 lives wih increasing sum assured (IA) xyn = n ( inc l d xy x+j + v j= M ), (2.3),y+j + Increase ype is he ype of increasing of paymens: j + M = {b ; b, INF ; b Z}. (2.32) INF linear/arihmeic geomeric inc = inc + IR INF, inc =, (2.33) inc = inc ( + IR INF ), inc =. (2.34) Noe: Linear inerpolaion is made beween d x and d x+. AF_I_ann_xyn (LT_X; LT_Y; X; Y; N; ; I; INF; IT; IR; PT) Presen value of life annuiy for 2 lives for limied number of years wih increasing paymen For paymens a he beginning of he period: (Iä) xyn = n ( pay l j= N inc l xy x+j +,y+j + j + v ), (2.35) N = {b ; b 0, freq ; b Z}, freq = max(inf, ), (2.36) freq M = {b ; b 0, INF ; b Z}. (2.37) INF Acuarial funcion for Windows User Guide Sepember 203 Version _05 Page 28

29 pay = { εz, 0 else. } (2.38) For paymens a he end of he period: (Ia) xyn = n ( pay l j= N inc l xy x+j +,y+j + j + v ), (2.39) N = {b ; b, freq ; b Z}, freq = max(inf, ), (2.40) freq M = {b ; b, INF ; b Z}, (2.4) INF pay = { εz, 0 else. } (2.42) Increase ype is he ype of increasing of paymens: linear/arihmeic inc εm = inc + IR, inc INF =, inc M = inc, (2.43) geomeric inc εm = inc ( + IR INF ), inc =, inc M = inc. (2.44) Noe: Linear inerpolaion is made beween l x and l x+. AF_I_kAxyn (LT_X; LT_Y; X; Y; N; K; I; INF; IT; IR) Single neo premium for deah and survival cover for 2 lives wih increasing sum assured wih deferred period years of deferred period k (IA) xyn = k n (IE) xy + k (IA) xyn (2.45) AF_I_kExyn (LT_X; LT_Y; X; Y; N; K; I; INF; IT; IR) Single neo premium for pure endowmen cover for 2 lives wih increasing sum assured wih deferred period years of deferred period Acuarial funcion for Windows User Guide Sepember 203 Version _05 Page 29

30 Increase ype is he ype of increasing of paymens: linear/arihmeic k n(ie) xy = [ + (n INF ) IR INF ] D x+k+n,y+k+n D xy (2.46) geomeric k n(ie) xy = [ + IR INF ]n INF D x+k+n,y+k+n D xy (2.47) AF_I_kAxyn (LT_X; LT_Y; X; Y; N; K; I; INF; IT; IR) Single neo premium for pure deah cover (erm) for 2 lives wih increasing sum assured wih deferred period years of deferred period k (IA) xyn = n ( inc l d xy x+k+j + v j= M ), (2.48),y+k+j + Increase ype is he ype of increasing of paymens: k+j + M = {b ; b, INF ; b Z}. (2.49) INF linear/arihmeic geomeric inc = inc + IR INF, inc =, (2.50) inc = inc ( + IR INF ), inc =. (2.5) Noe: Linear inerpolaion is made beween d x and d x+. AF_I_kann_xyn (LT_X; LT_Y; X; Y; N; K; ; I; INF; IT; IR; PT) Presen value of life annuiy for 2 lives for limied number of years wih increasing paymen wih deferred period years of deferred period For paymens a he beginning of he period: k (Iä) xyn = n ( pay l j= N inc l xy x+k+j +,y+k+j + k+j + v ), (2.52) Acuarial funcion for Windows User Guide Sepember 203 Version _05 Page 30

31 N = {b ; b 0, freq ; b Z}, freq = max(inf, ), (2.53) freq M = {b ; b 0, INF ; b Z}. (2.54) INF pay = { εz, 0 else. } (2.55) For paymens a he end of he period: k (Ia) xyn = n ( pay l inc l xy x+k+j + v j= N ), (2.56),y+k+j + k+j + N = {b ; b, freq ; b Z}, freq = max(inf, ), (2.57) freq M = {b ; b, INF ; b Z}, (2.58) INF pay = { εz, 0 else. } (2.59) Increase ype is he ype of increasing of paymens: linear/arihmeic inc εm = inc + IR INF, inc =,inc M = inc, (2.60) geomeric inc εm = inc ( + IR INF ), inc =, inc M = inc. (2.6) Noe: Linear inerpolaion is made beween l x and l x FORMULAS FOR GENERATION LIFE TABLE The Formulas are he same like above, jus firs parameer is changed - GT (Generaion Life Table) is wrien insead of parameer LT (Life able). AF_qx_GT (GT; Year; X) Acuarial funcion for Windows User Guide Sepember 203 Version _05 Page 3

32 AF_lx_GT (GT; Year; X) AF_qxy_GT (GT_X; GT_Y; Year_X; X; Year_Y; Y) AF_lxy_GT (GT_X; GT_Y; Year_X; X; Year_Y; Y)... 8 LITERATURE CIPRA, Tomáš. Finanční a pojisné vzorce. Praha: Grada Publishing, a. s., ISBN X. Acuarial funcion for Windows User Guide Sepember 203 Version _05 Page 32

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