A SWITCH CRITERION FOR DEFINED CONTRIBUTION PENSION SCHEMES

Transcription

1 A SWTCH CTEON O DENED CONTBUTON PENSON HEMES Bas Ars CP Via al Collgio 3 14 Moncaliri (TO, aly Tl ax bas_ars@yahoo.com Elna Vigna Univrsià di Torino Diparimno di Saisica Mamaica Applicaa piazza Arbarllo 8 11 Torino, aly Tl ax lna.vigna@unio.i Absrac n his papr w ry o find an invsmn sragy which givs a high probabiliy of raching a givn arg amoun a rirmn. This sragy dpnds on wo momns: h firs is h appropria im o sop invsing h conribuions ino quiis and h scond is h opimal momn o convr h fund of hs conribuions ino bonds (noing ha h lar may nvr occur. W dfin h sragy and compar i wih ohr invsmn sragis for Dfind Conribuion Pnsion Schms. inally w do no only analys h accumulaion phas bu also ak ino considraion wha happns during h disribuion phas. ésumé Dans c éud, nous allons ssayr d définir un sraégi d'invsissmn qui donn un grand probabilié d'aindr un somm prédérminé au momn d la rrai. C sraégi dépndra d dux momns: la prmièr s la périod approprié pour invsir ls appors dans ds acions la scond s l momn opimal pour convrir ls fonds ds conribuions n irs d'mprun (n rnan qu c drnièr phas pu n jamais s produir. La sraégi sra défini comparé aux aurs sraégis d'invsissmn ds Schémas d rai à Coisaions Définis. En définiiv, nous n'analysrons pas sulmn la phas d'accumulaion mais nous prndrons aussi n comp c qui s produi lors d la phas d disribuion. Kywords: Dfind Conribuion Pnsion Schm, Accumulaion and Disribuion Phas, ncom Drawdown, srv.

2 A Swich Cririon for Dfind Conribuion Pnsion Schms 1 1. nroducion n h liraur on dfind conribuion pnsion schms hr ar wo diffrn approachs o sudy h financial risk born by h mmbr, h horical and h mpirical approach. Th horical approach ris o opimis risk and/or rurn using dynamic programming chniqus (among ohrs, Blak al., a,. Boulir al.,, Dlsra al.,, Habrman &Vigna,, Vigna & Habrman, 1. Th mpirical approach xamins h currnly usd invsmn sragis and ris o propos br alrnaivs (among ohrs, Blak al., b, Booh & Yakoubov,, Knox, 1993, Ludvik, n his papr w adop h scond approach. A widsprad mpirical invsmn policy in dfind conribuion schms is h so-calld lifsyl sragy (among ohrs, Exly al., 1998, Knox, 1993 whrby h individual gradually convrs hr/his quiy porfolio ino a bond porfolio in h las yars bfor rirmn. n h firs yars h invsmns ar mad ino quiis in ordr o g a high xpcd yild, and in h las yars h invsmns ar mad ino bonds which hav a lowr xpcd yild, bu also a lowr volailiy, in ordr o prvn unncssary risk clos o rirmn. Th ida in his papr is inspird by h fac ha, sinc quiis ouprform bonds in h long run (s among ohrs, Bodi, 1995, h mmbr of a dfind conribuion pnsion schm nds in gnral a sufficinly long priod of invsmn in quiis bfor swiching ino bonds. This priod is shorr if rurns on quiis ar high and longr if rurns ar low. Thrfor i dpnds on quiy prformanc during h accumulaion phas and i is no appropria o drmin i in advanc using a fixd rul (lik in lifsyl sragy and saic porfolio allocaions. n his papr w propos a cririon for swiching h porfolio from quiis ino bonds which is parially dynamic, i. aks ino accoun acual ralisaions of rurns on asss. W invsiga an invsmn sragy in which h individual maks wo diffrn kinds of swichs, insad of gradually swiching h porfolio in bonds. Afr invsing h conribuions in quiis for a crain priod, h individual will build up a lss risky fund wih conribuions invsd in bonds. rom his poin on (firs swich, hr ar wo funds, h firs on wih h nw conribuions invsd ino bonds, which will always b invsd ino bonds, and h scond on, wih h old conribuions invsd in quiis, which will b invsd ino quiis unil h swich of h fund occurs (scond swich and ino bonds afrwards. Th priod in which h scond swich can ak plac is from h firs swich unil rirmn bu can b xndd o h yars afr rirmn as wll, if h rir aks h incom drawdown opion. W wan o drmin h bs im for h wo diffrn swichs rying o maximis h probabiliy of raching h arg fund a rirmn, i.. h bs im o sar invsing h conribuions ino bonds insad of quiis and h bs im o swich h quiy fund ino bonds. n wha follows, w propos a mhod for drmining hs wo momns, invsiga and s our sragy by mans of Mon Carlo simulaions and compar i wih ohr invsmn sragis. W will considr no only h pr-rirmn priod (or accumulaion phas, bu also h posrirmn priod (or dcumulaion phas, wih diffrn assumpions rgarding h choic annuiizaion / drawdown opion, dpnding on whhr or no h scond swich has occurrd W ar vry graful o Angla van Hrwaardn (Univrsiy of Amsrdam and Giovanna Nicodano (Univrsiy of Turin for hir commns and suppor.

3 A Swich Cririon for Dfind Conribuion Pnsion Schms in h pr-rirmn priod. Up o our knowldg, i is h firs im ha h wo phass ar considrd oghr. Th papr is organisd as follows. Scion dscribs h basic sragy. n scion 3 w dscrib h adjusmns w mak o h basic sragy. Scion 4 dscribs h dcumulaion phas and scion 5 concluds.. Th basic sragy.1 S up and assumpions n ordr o dscrib h invsmn sragy ha w wan o invsiga, w drmin 6 imporan momns in h lif of h individual: Th momn whn h individual joins h schm, from his poin on all h conribuions ar invsd in quiis. Swich of h conribuions. Th momn whn h individual sops invsing conribuions ino quiis and sars invsing hm ino bonds. rom his poin on hr ar wo funds: h quiy fund ( f which consiss of all h conribuions up o im invsd in quiis and h bond fund ( f which consiss of all h conribuions from im on invsd ino bonds. Th im of rirmn. A Th im whn annuiizaion is compulsory, in cas h individual aks h ncom Drawdown opion (for xampl whn h individual rachs h ag of 75, which w will discuss lar. S Swich of h quiy fund. Th im a which h quiy fund ( f is swichd ino bonds. rom his poin on boh porfolios ar invsd ino bonds. This swich-momn will dpnd on a swich-cririon which will b spcifid lar. W assum ha if his swich occurs bfor rirmn, hn h individual buys a fixd ral annuiy a im (in his cas momn A dos no play any rol. f his swich has no occurrd y a rirmn, h individual will ak h incom drawdown 1 opion, so h swich may occur a any im bwn and A. n his cas whn h swich cririon is saisfid, h individual insad of convring h quiy fund ino bonds, buys a fixd ral annuiy wih boh funds. f h swich has no occurrd y a im A, hn h rmaining funds in quiis and bonds ar usd o buy a fixd ral annuiy. D Th im of dah of h individual. Timlin 1 Th fund rmains invsd afr rirmn wih a flxibl incom gnrad by wihdrawals from h fund. ncom allowd is broadly bwn 35% (minimum incom ha mus b drawn and 1% of h quivaln guarand lvl singl lif annuiy.

4 A Swich Cririon for Dfind Conribuion Pnsion Schms 3 conribuion quiis conribuion bonds Pnsion P A D W ak h iniial im o b and h rirmn im o b 4 (if h invsmn sragy dos no work a a long horizon, looking a a shorr horizon has vn lss us, h conribuion (c is a fixd prcnag of h ral salary, and h ral salary rmains consan ovr im. Assumpions on h asss Ass rurns ar assumd o b lognormally disribud. n paricular, h rurn on h bond is, wih ~ N (, σ and h rurn on quiy λ, wih λ ~ N ( λ, σ λ. λ and ar assumd o b uncorrlad. W now dfin a arg ral rurn (r*, which will b usd lar in drmining and in dfining h swich cririon. As h arg rurn w hav chosn h Chisini avrag of and λ rlaiv o h xpcd rurn ovr on yar of a porfolio invsd qually in h wo + λ E r * asss. Thrfor r* is h soluion of ( = E( and his mans ha * 1 1 r = ( + λ + ( σ 8 λ + σ, sinc λ and ar uncorrlad.. Th priod in which h conribuions ar invsd ino quiis n ordr o sima h priod in which h conribuions should b invsd ino quiis, h firs hing an individual has o considr a saring im, is ihr hr/his rquird avrag ral rurn on invsmns or his/hr Targ und ( a rirmn. This rquird rurn dpnds on h risk avrsion of h individual and is supposd o b + 1 σ bwn h xpcd rurn on bonds ( and h xpcd rurn on quiis (. r* Wih his rquird rurn ( w can now calcula h swich yar of h conribuions. W dfin as h soluion of h following quaion: = + TA whr:, TA 1 λ+ σ λ TA is h fund a im obaind by invsing all h conribuions a h ra r*. W call i h iniial projcd Targ und a rirmn; is h fund a im obaind by invsing a rn of conribuions in an ass whos rurn is E( λ and hn invsing his fund for (--yars in an ass whos ral rurn is E (. W call i h iniial Projcd und of conribuions invsd in quiis; [1]

5 A Swich Cririon for Dfind Conribuion Pnsion Schms 4 Or is h fund a im obaind by invsing a rn of (- conribuions in an ass whos rurn is E (. W call i h iniial projcd und of conribuions invsd in bonds. ( 1 j= c ( r* ( j = ( 1 which is quivaln o: c ( λ+ 1/ σ λ ( i ( + 1/ σ ( + 1 c ( + 1/ σ ( i, [] ( 1 j= c ( r* ( j = ( 1 c ( E( λ i ( i ( E( ( + 1 c ( E( [3] n ohr words, is h poin in im such ha invsing conribuions for h whol priod r* (from o a h rquird rurn is quivaln o invsing conribuions from o in an ass wih consan rurn qual o E( λ (xpcd rurn on quiis and hn invsing h fund and h conribuions from o in an ass wih consan rurn qual o E ( (xpcd rurn on bonds. W no ha is drminisic: onc h paramrs of h ass rurns ar givn, is calculad in a drminisic way, using xpcd valus. ( i. Exampl: L ~ N ( 4%,(5% and λ ~ N (6%,(15%. Givn h paramrs of h ass rurns, and givn ha h corrlaion bwn quiis and bonds is zro, r * = 5,315%. Wih h Targ urn (r* 5,315%, h yar in which h conribuions ar swichd from quiis o bonds is 3, his mans ha h firs annual conribuions ar invsd in quiis and afr ha h conribuions ar invsd in bonds. Considring diffrn ras for h rquird ral rurn r*, sill boundd bwn ( +,5σ and ( λ +,5σ λ (i would b unrasonabl o rquir a rurn ousid his rang, w would find h following valus for as rpord in figur 1. n our xampl h rang is [4,15%; 7,15%], his is [ ( +,5 ; ( λ +,5 ]. σ σ λ

6 A Swich Cririon for Dfind Conribuion Pnsion Schms 5 igur 1. Th for h diffrn arg rurns r* 8,% 7,% Targ urn r* 6,% 5,% 4,% 3,%,% 1,%,% Commns is clar ha if h rquird rurn incrass h incrass as wll, his mans also ha h im priod in which h S (swich of h quiy fund can ak plac (bwn and will b shorr. This is consisn wih h fac ha h individual who rquirs a high ra of rurn is lss risk avrs han h individual who rquirs a low ra of rurn, and hrfor invss for a longr priod ino riskir asss. H/sh will also hav lss im bfor rirmn o mak h swich of h quiy fund. TA Th projcd funds (,, a im wr calculad wih h xpcd rurns on h asss, h acual funds a im will obviously dpnd on acual ralisaions of h ass rurns ovr im. W considr h following fund valus (for = : f = ( ( 1 c j= ( i+ 1 λ j, h valu of h fund wih h conribuions invsd ino quiis a im ; f =, h valu of h fund wih h conribuions invsd ino bonds a im (which is zro bcaus w ar bfor im ; f = f + f, h oal of h wo funds buil-up;.3 Th swich cririon for h quiy fund or our invsmn sragy h swich of h quiy fund can occur from im on and h swich cririon will b sd yarly from im. Th swich of h quiy fund occurs a im if h following holds:

7 A Swich Cririon for Dfind Conribuion Pnsion Schms 6 TA + ( 1 j= c ( r* ( j ( ( 1 c j= i+ 1 ( λ ( 1 ( +.5σ ( + 1/ σ + c ( j ( i, which is qual o: ( 1 j= c ( r* ( j f ( 1 ( + 1/ σ ( + 1/ σ + c ( ( i, [4a] which is qual o: ( 1 j= c ( r* ( j f ( E( ( + 1 c ( E( i ( i. [4b] [4c] W noic ha a im h rurns on h conribuions invsd ino quiis ar known. Thrfor in h formula abov w us h acual ralisaions ( λ insad of h xpcaion (whras for h rurn on bonds from o w ak h xpcaion. TA,, ar h projcd funds a im calculad a im. W obsrv ha TA and ar qual o and, whil is diffrn from rurns ar usd insad of xpcaions. n simplr words, h swich of h quiy fund occurs if TA bcaus ralizd ( 1 c j= ( i+ 1 λ j ( 1 c ( E( λ ( i or quivalnly (, [5] ha is, h swich occurs if h invsmn rurns on quiis bhavd on avrag lik or br han hir xpcaions. f his is no h cas, and a im w hav ( 1 c j= ( i+ 1 λ j < ( 1 c ( E( λ i ( i, [6] hn h swich dos no occur and h quiy fund will rmain invsd ino quiis for h nx yar. Afr on yar h swich cririon will b sd again and h swich will occur (so S=+1 if h following holds: ( 1 c ( r* ( i ( ( 1 c j= ( i+ 1 λ j λ ( + 1 ( E( ( ( c ( + 1 ( E( ( ( ( 1 ( + 1 c E( ( i. [7]

8 A Swich Cririon for Dfind Conribuion Pnsion Schms 7 W noic ha a im (+1 also h rurn on bonds in yar (, +1 is known (, and w us i in h capialisaion of h conribuion paid a im. Th swich cririon will b sd ach yar up o h firs im a which h cririon is saisfid. n gnral, h swich occurs a h firs im whn h following holds: TA,, whr: +, = f ( + 1/ σ <. Wih ( f, h Projcd und of conribuions invsd ino quiis for = ( ( 1 c j= ( i+ 1 λ j λ i [8], h valu of h fund wih conribuions ino quiis, if h swich ino bonds has no y akn plac;, +, =, h adjusd projcd final valu of h porfolio on bonds yar <<; 1 j + 1/ σ = ( c ( (, j= i+ 1 alrady invsd ino bonds from up o im ; TA, h iniial Targ und a rirmn; ( 1 + 1/ σ ( i, = c ( in bonds. in, h projcd final valu of h conribuions, h projcd valu of h fuur conribuions o b invsd n ohr words, h lf hand sid of h quaion is h projcd final valu of h conribuions alrady invsd and h righ hand sid of h quaion is h inal Targ und minus h projcd fuur conribuions. f h swich has occurrd bfor im, h valu of h quiy fund a im will b: f = ( ( 1 c j= ( i+ 1 λ j S ( + 1 λ i ( S + 1 i. [9] Th rs of h valus rmain h sam. f h swich dos no ak plac bfor rirmn h individual chooss h incom drawdown -opion, and h swich cririon afr rirmn will chang. W will discuss his lar. Tsing h sragy: a numrical xampl & simulaions n ordr o s how h sragy works w carry ou som simulaions. Wih ~ N ( 4%,(5%, λ ~ (6%,(15% TA N, ρ = and c=1, h Targ und (

9 A Swich Cririon for Dfind Conribuion Pnsion Schms 8 a im will b 14,5. f = 3, h xpcd quiy amoun a rirmn ( will b 115,94 and h xpcd bond amoun ( will b 6,56. W mak 1. Mon Carlo simulaions of invsmn rurns of h asss, applying ach yar h invsmn sragy prviously dscribd. Thrfor w inroduc h final fund buil up a rirmn = +, whr = f and = c j= ( i+ 1 j. Thn w compar h rsuls w g wih h rsuls obaind by invsing h whol porfolio ino quiis for 4 yars. rom now on w will call h invsmn sragy basd on h swich cririon prviously dscribd h swich sragy. Th rsuls w prsn ar: h man and h sandard dviaion of h final fund; h downsid dviaion 3 of h final fund and h man of h shorfall 4 (whr by shorfall w man h diffrnc of h fund from h arg, providd ha his diffrnc is < ; h probabiliy of failing h arg fund; h probabiliy of failing h arg fund givn ha h swich of h quiy fund occurrd bfor rirmn. This probabiliy is vry imporan in his papr bcaus h fund a rirmn is lowr han h arg fund a rirmn and is fully invsd ino bonds. This mans no incom drawdown, w will spcify his lar; h valu a risk a 95% lvl of h fund and h valu a risk a 75% lvl of h fund. Tabl 1. Th 4 yars 1% quiis vrsus h swich sragy (=3 4 yars 1% quiis Swich sragy (=3 Man 36,4 158,1 Sandard Dviaion,6 66,7 Downsid Dviaion 54,3 44,5 Man shorfall from h 46,3 35,9 TA TA P ( < 39,4% 43,3% TA P( < S < 41 Dos no apply 5,5% Va 95% 58,3 67,8 Va 75% 11,1 117,8 Commns W s ha h avrag valu of h final fund is much highr invsing 4 yars fully in quiis han h avrag final valu of h swich sragy, whil h k 3 j TA = (1/ k ( = j 1 k 1 4 = ( k j= 1 TA j, whr ( j TA < for j= 1,,,k. j TA, whr ( > for j= 1,,,k.

10 A Swich Cririon for Dfind Conribuion Pnsion Schms 9 swich sragy has an vn highr probabiliy of failing h arg. All h ohr risk masurs indica ha h swich sragy is lss risky. inally w noic ha h probabiliy of failing h arg afr h S is nar 5,5%: his mans ha h swich cririon is no sufficin in mor han 1 ou of 4 cass, ha is vry high. Up o now h swich only maks sur ha, from whn i is saisfid, h quiy risk is liminad, bu h risk of h rurn on bonds sill rmains. 3. Adjusmns o h basic modl 3.1 Yarly args and a flxibl n ordr o monior h growh of h fund ovr im w inroduc yarly args, whos dfiniion is diffrn in h priods bfor and afr. or h yarly arg is: YT = 1 c ( ( λ+.5σ λ ( i = ( 1 c E( and from < h yarly arg is: λ ( i ( 1 ( 1 ( λ+.5σ ( i ( +.5σ ( ( +.5σ λ YT = c ( ( + c ( W also nd f ( i., h oal of h wo funds buil-up a im, whr f = f + f. [1] [11] igur. Th yarly arg buil up by h wo funds yarly arg Equiy und Bond und yar Th disribuion of S ovr h 1. simulaions is plod in h figur 3. or h momn w assum ha if h swich has no occurrd a rirmn (=41 i will occur h yar afr (41. Thos popl would ak h incom drawdown opion which w will considr lar.

11 A Swich Cririon for Dfind Conribuion Pnsion Schms 1 igur 3. Th Disribuion of h S Numbr of S yar f w simula (1. simulaions wih h =3, h cririon abov is saisfid in lss hn 4% of h cass afr 3 yars, so P(S=3<4%. W hink ha his rsul can b xplaind by h fac ha in h lognormal disribuion h mdian (or 5 h prcnil is lowr han h man. Thrfor w would xpc in mor han 5% of h cass a rurn lowr han h xpcd valu. This would lad o h cririon no o b saisfid in mor han 5% of h cass. Considring h xampl inroducd bfor, h xpcaion of h sum of h prmiums 3 λ invsd in quiis a h xpcd swich im (3 yar YT 3 = ( c E ( = i = j = ( i + 1 W wan o analys wha happns if w chang h from 3 o rspcivly 4, 5,..., 31, ha mans invsing xra yars in quiis, and look a h probabiliy ha h prviously dfind yarly arg fund ( YT,..., YT, for which h is fixd a yar 3, in ha yar will 4 31 b rachd. Tabl. suls of invsing h conribuion xra yars in quiis Yar Avrag f = 55,4 6,6 65,8 71,8 79, 85,6 9,9 1,3 18,7 YT a 54,9 58, 61,7 65,3 69,1 73,1 77, 81,5 86, P ( S = = 38,8% 41,6% 44,% 45% 46,7% 47,7% 5,% 5,9% 54,7% P ( S = 3 38,8% 44,9% 49,% 5,5% 55,5% 57,5% 59,9% 63,% 65,6% TA P( < 5,5% 3,7% 1,6% 1,% 18,6% 18,% 17,% 15,9% 14,% = Commns f you invs h conribuions for som xra yars in quiis h diffrnc bwn h oal fund a im and h yarly arg a im is on avrag incrasing. P ( S = = is smallr han P ( S = 3 bcaus in h firs cas h swich cririon is sd only a im =, in h scond cas h swich sragy is sd a any im bwn =3 and im (i is h

12 A Swich Cririon for Dfind Conribuion Pnsion Schms 11 accumulaion of all h swichs from =3 unil. Th probabiliy of failing h arg if h swich is mad afr invsing som xra yars in quiis is dcrasing. 3. Th nd for a buffr or rsrv W now wan o giv an sima of h volailiy of h final fund ( undr h assumpion ha h yarly arg YT is xacly rachd afr rspcivly and 3 yars, and is no rachd bfor, and is 3 (his mans you invs h conribuions ino quiis for yars and convr h whol fund f from quiis ino bonds a yar 3 and yar 31. Suppos YT is xacly sufficin a h bginning of yar 3, which mans ha TA =, hn h final fund a rirmn will b: i ( YT3 + c j= ( i+ 1 j. [1] W carry ou again 1. simulaions for invsmn rurns and w look a h probabiliy of failing h arg, h man shorfall and h downsid dviaion of h final fund. Th probabiliy of failing h arg is 53,4% wih a man shorfall of h final fund a of,4 and a downsid dviaion of h final fund of 4,5. f YT is xacly sufficin a yar 31, h final fund a rirmn will b: 4 ( YT31 + c = = 3 j= ( + 1 j. [13] W us h sam 1. simulaions as bfor (considring h invsmn rurns simulad in h las n yars. Th probabiliy of failing h final arg is 5,3% wih a man shorfall a rirmn ( of 16,3 and a downsid dviaion of h final fund of 19,6. Tabl 3 shows rsuls whn YT is xacly rachd afr 3, 5,, 39 yars. W dfin: SMS= man shorfall from SDD= Downsid dviaion of h TA, w call i simulad man shorfall. TA, w call i simulad downsid dviaion. Tabl 3. h rsuls whn f = YT a yar 3, 5,..., 39 S SMS,4 19,1 17,9 16,8 16,3 14,1 1,5 9,5 5,7 SDD 4,5 3, 1,7,4 19,6 17,1 15,3 11,7 7,1 TA P < f = YT 53,4% 5,% 5,7% 51,9% 5,3% 51,8% 5,7% 5,4% 5,7% ( W obsrv wo hings, h man shorfall and h downsid dviaion of h final fund ar dcrasing as h S is incrasing and in boh cass h probabiliy of failing h arg wih

13 A Swich Cririon for Dfind Conribuion Pnsion Schms 1 h cririon w us is vry high. This is a rasonabl rsul bcaus, whn S is high, rirmn is closr and risk on bond rurn affcs o a lssr xn h final fund rsul. Givn his high probabiliy of failing h final arg, w hink ha h yarly arg a S is no nough, i dos no includ a rsrv or buffr in ordr o hdg agains poor prformanc of bonds. How can a rsrv b crad? Th cririon has o b changd. An asy way of incrasing h xpcaion of h valu of h fund a rirmn is simply o invs h conribuions som xra yars in quiis, as w saw bfor. W assum, for xampl, ha =31 (which mans invsing all h annual conribuions h firs 3 yars in quiis, wihou a buffr, carry ou 1. simulaions and compar i wih h lifsyl sragy 5 of annually swiching 1 % from quiis ino bonds vry yar during h las n yars bfor rirmn and wih h swich sragy =3. Tabl 4. Th lifsyl sragy vrsus h swich sragy for boh =3 and =31 Lifsyl sragy Swich sragy (=3 =31 Man 1,8 158,1 184,8 Sandard dviaion 136,4 66,7 14,7 Downsid dviaion 48,7 44,5 5, TA Man shorfall from 4, 35,9 4,7 TA P( < 4,5% 43,3% 36,% TA P( < S < 41 Dos no apply 5,5% 14,% Valu a isk 95% 69,4 67,8 6,9 Valu a isk 75% 111,6 119,6 13,7 Commns Changing h from 3 o 31 (wihou including a rsrv givs a highr avrag final fund whil h wo probabiliis of failing h arg dcras subsanially. Th ohr risk masurs of h swich sragy =3 ar lowr, bu if w ak h 36 lows valus of h swich sragy (=3 (considring ha h probabiliy of failing h arg in =31 is 36%, h downsid dviaion of h final fund is 49,1 and h man shorfall is 4,5, no much br han in h =31 sragy (undrlining h fac ha somims you mus b carful in drawing conclusions whn comparing h sam risk masur on diffrn sragis. Th lifsyl sragy has a highr avrag han h wo swich sragis, h probabiliy of failing h arg is bwn h wo. inally, if w look a h disribuion of h fund w s ha in h xrm cass h valu a risk 95% h lifsyl sragy has a highr valu han h swich sragis whil a lowr valu rgarding h valu a risk 75%. Considring h possibiliy of building up a rsrv, on should sima how much h individual nds as a rsrv a ach im. To do his, w look a h downsid dviaion or h man shorfall of h final fund. W hav sn bfor boh ha h downsid dviaion and h man shorfall ar dcrasing as h S incrass. 5 Th valu of h fund of h lifsyl sragy a im, (, is: LS LS λ f = ( f ( 1 + c (1 + ω EQ + (1 ω EQ,wih ω EQ = h proporion of fund invsd in quiis a h bginning of yar. LS f

14 A Swich Cririon for Dfind Conribuion Pnsion Schms 13 Thr ar many ways of dfining a swich cririon which allows for h prsnc of a rsrv. n gnral, h swich of h Equiy und a im will occur if: (,, TA + (1 + Buffr (. or h or 1.. Buffr w can us for xampl: SMS buffr = YT SMS buffr = YT ( ( +.5σ (. [14] [15a] [15b] Th firs dfiniion aks as h buffr h full simad shorfall a im dividd by h yarly arg a im, whil h scond aks h discound simad shorfall a im dividd by h yarly arg a im. W will considr h firs dfiniion. Th buffr w dfin dpnds on h simulaions on invsmns w mad. To mak i mor gnral, w mak a linar rgrssion 6 o hav smoohd rsuls. Tabl 5. Swich sragy (=3 vrsus swich sragy (=3 wih buffr Swich sragy =3 =3 wih buffr Man 158,1 167,8 Sandard dviaion 66,7 7,8 Downsid dviaion 44,5 5,5 TA Man shorfall from h 67,8 45,8 TA P( < 43,5% 34,7% TA P( < S < 41 5,5% 5,% Valu a isk 95% 67,8 66,4 Valu a isk 75% 117,8 114, Commns Th sragy wih buffr has a highr man and a lowr probabiliy of failing h arg. f w look a h 95% and h 75% valu a risk w s ha h valus of h swich sragy wihou buffr ar only slighly highr. Th sragy wih buffr sms o b mor appropria. 6 S appndix for h linar rgrssion w usd.

15 A Swich Cririon for Dfind Conribuion Pnsion Schms 14 Tabl 6. Swich sragy (=31 vrsus swich sragy (=31 wih buffr Swich sragy =31 =31 wih buffr Man 184,8 187,6 Sandard dviaion 14,7 15,1 Downsid dviaion 4, 56,4 TA Man shorfall from h 4,9 49, TA P( < 36,% 3,7% TA P( < S < 41 14% 4,% Valu a isk 95% 6,9 6,9 Valu a isk 75% 1,6 117,7 Commns Th sragy wih buffr has a slighly highr man, bu h wo probabiliis of failing h arg ar significanly lowr whil h valus a risk ar no much lowr. W hav also lookd a an alrnaiv sragy: is variabl, wih a minimum of 3 and a maximum of 31 yars, and h now occurs whn h swich cririon on h quiy fund holds or afr 3 yars (if h cririon is no saisfid bwn 3 and 31. Th maximum of has bn chosn o b 31 bcaus in h lifsyl sragy afr 3 yars h individual sars invsing ino bonds. or his sragy h swich occurs a im, 3 31, if h following holds: ( 1 c ( r* ( i ( ( 1 c j= ( i+ 1 λ j ( E( ( + c ( E( ( + ( 1 c E( [16] Adoping his alrnaiv sragy h cumulaiv disribuion of S compard o h swich sragy (=3 is as displayd in igur 4. igur 4. h cumulaiv disribuion of S ( i. S =3 =3_31 Yar

16 A Swich Cririon for Dfind Conribuion Pnsion Schms 15 Th numbr of S a yar 3 is h sam, bu afrwards h numbr of S a any yar 4 4 wih h variabl is highr. Tabl 7 givs h rsuls of h simulaions wih h flxibl. Tabl 7. Swich sragy (=3_31 vrsus swich sragy (=3_31 wih buffr Swich sragy =3_31 =3_31 wih buffr Man 159,4 169, Sandard dviaion 66,9 71,4 Downsid dviaion 46,5 56,8 TA Man shorfall from h 36,8 48,9 TA P( < 41,4% 3,1% TA P( < S < 41 5,4% 5,8% Valu a isk 95% 63,46 6,9 Valu a isk 75% 1,65 11, Commns Also in his cas h sragy wih buffr sms br and i has h lows probabiliy of failing h arg a rirmn of all h sragis w sd. So w conclud ha in gnral h swich sragis wih a buffr sm o b br han h sragis wihou a buffr. Wha abou h swich sragy compard o h lifsyl sragy and h sragy of invsing fully ino quiis? To do his w ak h swich sragy (=31 wih buffr bcaus i has h highs avrag final fund of all h swich sragis (bu sill lowr han h lifsyl sragy and h full quiy sragy. Tabl 8. Th lifsyl sragy vrsus swich sragy (=31 wih buffr Lifsyl sragy Swich =31 wih buffr Man 1,8 187,6 Sandard dviaion 136,4 15,1 Downsid dviaion 48,7 56,4 TA Man shorfall from h 4, 49, TA P( < 4,5% 3,7% TA P( < S < 41 Dos no apply 4,% Valu a isk 95% 69,4 6,9 Valu a isk 75% 111,6 117,7 Commns Th man of h lifsyl sragy is much highr, bu h probabiliy of failing h arg as wll. Th diffrnc bwn h valu a risk 95% indicas ha in xrm cass h =31 is much lowr.

17 A Swich Cririon for Dfind Conribuion Pnsion Schms 16 Tabl 9. 4 yars 1% quiis vrsus h swich sragy (=31 wih buffr : 4 yars 1% quiis Swich =31 wih buffr Man 36,4 187,6 Sandard dviaion,6 15,1 Downsid dviaion 54,3 56,4 TA Man shorfall from h 46,3 49, TA P( < 39,4% 3,7% TA P( < S < 41 Dos no apply 4,% Valu a isk 95% 58,3 6,9 Valu a isk 75% 11,1 117,7 Commns Hr h man of h 4 yars 1% in quiis is much highr han ha of h swich (=31 wih buffr, bu i is also mor risky, which sms qui logical. 4. Th dcumulaion phas 4.1 irmn and byond A his sag h individual has rird and hus nds a pnsion incom. Thr is h choic bwn incom drawdown and a fixd ral annuiy 7. Th assumpion in his papr is ha, if h swich has occurrd bfor rirmn, a fixd ral annuiy is bough a rirmn and if his is no h cas h individual chooss incom drawdown wihou bqus 8 unil h swich cririon is saisfid. Th annuiy opion n cas h annuiy opion is chosn, h S has occurrd bfor and a h annuiy will b bough. Th yarly annuiy pnsion ( P a& will b h oal fund a im, dividd by h annuiy facor 9 : Pa & = (, whr x is h ag of h individual whn joining h schm. a & x+ TA This yarly pnsion will b grar han h arg pnsion if, bu will b lowr if TA. A his fund is 1% invsd in bonds, lik a h nd of h lifsyl sragy. Th diffrnc bwn h swich sragy and h lifsyl sragy is ha wih h swich sragy h porfolio invsd in bonds is usd a o buy annuiis, whras wih h lifsyl sragy, if h incom drawdown opion is akn, h individual has o convr again h porfolio from bonds ino quiis. n fac, during h incom drawdown i sms o b opimal (among ohrs, Blak al., 1, Grrard al.,3, Kapur & Orszag, 1999, 7 Moraliy abl usd: STAT, Annuario saisico ialiano, avol di moralià pr ssso pr éa This mans ha should h individual di, h mony gos o ohr pnsionrs of h schm and no o rlaivs σ Th ra of inrs applid o h fixd ral annuiy a & x + will b akn fixd, v=, wih h sam xpcd ra of rurn as h invsmn ino bonds. This mans ha w do no ak ino accoun h annuiy a x+ = i px+ +.5 risk, aking & σ and iniial ag x = 5 yars and =4.

18 A Swich Cririon for Dfind Conribuion Pnsion Schms 17 Yaari, 1965 invsing h porfolio a las parially ino quiis 1. nvsing fully in bonds during incom drawdown is no suiabl bcaus in absnc of bqus moivs annuiis would prform br han 1% bonds, bcaus of moraliy drag, hdg agains longviy risk, absnc of volailiy c c. Thus, h choic of h incom drawdown opion a rirmn in a schm whr h lifsyl sragy has bn applid producs a disconinuiy in h porfolio composiion. ncom drawdown W assum ha for h incom drawdown opion h pnsion P wihdrawn ach yar is TA P =. This is h Targ und dividd by h facor of h fixd ral annuiy a rirmn, a & x+ w ak his pnsion bcaus w wan o analys h sragy, which givs h highs TA probabiliy of raching. n ral lif hr ar rsricions on h amoun of pnsion o b wihdrawn from h fund, h pnsion ha can b akn in h scnarios in which w dcid o ak incom drawdown will always b lss hn P (bcaus h acual fund is lowr han h, ohrwis h annuiy would b bough, s lar. TA TA Th args in h yars afr rirmn will chang o P a& x + (>. A ag 75 h drawdown opion is no longr allowd, wih h rmaining fund an annuiy will b bough. n our xampl h individual nds a fund of P a& 75 = 46, 45 a ag 75. f h incom drawdown opion is akn hn h pnsion P will b dducd from h fund in bonds and if his fund is no sufficin pnsion will b dducd from h fund in quiis. Th formula for h swich cririon afr rirmn now bcoms: (( f? q?? min( f? P;?? max( f? P;??(1?? p P? a x? q whr (1 + is h bonus facor for pooling (lik in Blak al.,1.6 p TA [17] This mans ha h incom drawdown firs will b akn from h fund invsd in bonds whil i is sufficin (his will happn wih high probabiliy wihin or 3 yars afr rirmn and hn mony will b wihdrawn from h fund in quiis. W compar h invsmn sragis of h accumulaion phas and analys h ffc of h possibiliy o do incom drawdown. As h saring poin for ach sragy w do no mak 1. simulaions, bu w go on wih all h simulaions in which h fund did no rach h arg a (which is h probabiliy of failing h arg a rirmn; his maks i a bi mor complicad o compar h sragis. 1 Khorasan (1996 obsrvs An incom wihdrawal fund mus arn a highr rurn han a whol lif annuiy fund... an incom wihdrawal fund is likly o b invsd wholly or parly in quiis, as quiis ar xpcd o ouprform h govrnmn bonds hld by insurr s annuiy funds.

19 A Swich Cririon for Dfind Conribuion Pnsion Schms 18 Th rsuls w prsn in h abls ar: Abov h hick lin: h probabiliy of failing h Targ und (14,51 a rirmn; h condiional probabiliy of failing h arg a rirmn givn ha h swich of h quiy fund has occurrd; h probabiliy of no aking h incom drawdown opion whil h Targ und a rirmn has no bn rachd. Th S insad has occurrd; h xpcaion of h final fund givn ha h S has occurrd and final arg has no bn rachd. Undr h hick lin (analyss of h drawdown: h probabiliy of saring h incom drawdown, h Targ und has nvr bn rachd; h avrag of h oal fund a h sar of h incom drawdown; h probabiliy of raching h arg afr rirmn; h avrag numbr of yars afr rirmn a which h arg has bn rachd, givn ha i has bn rachd; h probabiliy of no going bankrup bfor +1 bu also no bing abl o mak h swich; TA h avrag rmaining fund a +1 (rcalling ha a +1 h + 1 = 46,45; h probabiliy of going bankrup bfor im +1; h avrag yar in which bankrupcy occurs. Tabl 1. Swich sragy (=3 vrsus swich sragy (=3 wih buffr Swich sragy =3 =3 wih buffr TA P ( < 43,% 34,7% TA P( < S < 41 5,5% 5,% TA P(No Drawdown & < 19,4% 3,6% E( NoDrawdown & TA < 16,4 13,6 P(Drawdown & TA < 3,8% 31,1% E( Drawdown 89,9 9,8 P(Swich bwn and (+1 6,% 9,4% Avrag yar swich afr 4, 4,1 TA P ( < < & ;<<( ,7% 5,6% TA E ( < ,9,6 P (uin 1,9% 16,1% Avrag yar of uin afr 7,9 8, Commns

20 A Swich Cririon for Dfind Conribuion Pnsion Schms 19 Summing h probabiliy of drawdown and h probabiliy of no drawdown, you g h probabiliy of failing h arg a rirmn. No incom drawdown Th avrag oal fund of h individuals who do no ak h incom drawdown opion is qui high (16,4 and 13,6 wih rspc o h Targ und of 14,51. This is rasonabl bcaus h fund was highr han h yarly arg a las onc (so h swich was mad, bu hn wn blow h arg; in h cas of incom drawdown h yarly arg has nvr bn rachd so h avrag will b lowr. W s ha h probabiliy of no raching h Targ und, givn h S has occurrd, is much highr in h swich sragy wihou a buffr (5,5% vrsus 5,%; w hink his is anohr srong rason o includ a buffr. ncom Drawdown TA Summing h hr prcnags of P ( + 1 < + 1, P (uin and P(Swich TA bwn and +1 givs h P(Drawdown & <. Thus for h =3 wih buffr w hav: 5,6%+9,4%+16,1% = 31,1%. Th oal probabiliy of no raching h arg for h sragy =3 wih buffr is 5,3% and is h sum of: 3,6%, h probabiliy of no aking h incom drawdown, h individual gs a pnsion, bu no as high as wand; 5,6%, h probabiliy ha a im +1 h oal fund lf is no sufficin o buy h annuiy dsird; 16,1%, h probabiliy of going bankrup bfor im +1 (w assum ha h individual rmains aliv during h incom drawdown. Tabl 11. Swich sragy (=3_31 vrsus swich sragy (=3_31 wih buffr Swich sragy =3_31 =3_31 wih buffr TA P ( < 41,4% 3,% TA P( < S < 41 5,3% 5,7% TA P(No Drawdown & < 19,8% 4,1% E( NoDrawdown & TA < 16, 131,3 P(Drawdown & TA < 1,6% 7,9% E( Drawdown 87, 88,4 P(Swich bwn and (+1 5,7% 7,5% Avrag yar swich afr 3,8 3,9 TA P ( < < & ;<<( ,4% 4,4% TA E ( < ,7, P (uin 1,5% 16,% Avrag yar of ruin afr 7,6 7,6 Commns

21 A Swich Cririon for Dfind Conribuion Pnsion Schms n gnral h rsuls ar vry similar o h sragis wih =3. Th oal probabiliy of no raching h arg dcrass in comparison wih h sragis wih =3, bu on avrag ruin occurs arlir and h avrag fund afr 1 yars is a bi smallr. Tabl 1. Swich sragy (=31 vrsus swich sragy (=31 wih buffr Swich sragy =31 =31 wih buffr TA P ( < 36,% 3,7% TA P( < S < 41 14,% 4,% TA P(No Drawdown & < 1,4%,8% E( NoDrawdown & TA < 13, 133,9 P(Drawdown & TA < 5,6% 9,9% E( Drawdown 87,4 89,7 P(Swich bwn and (+1 6,9% 8,7% Avrag yar swich afr 3,9 3, TA P ( < < & ;<<( ,% 4,9% TA E ( < ,8 19,9 P (uin 14,5% 16,3% Avrag yar of ruin afr 7,6 7,7 Commns Th rsuls ar again qui similar o h ohr swich sragis. W finally compar h =31 wih buffr wih 4 yars 1% ino quiis. Tabl yars 1% quiis 11 vrsus swich sragy (=31 wih buffr 4 yars 1% quiis Swich =31 wih buffr TA P ( < 39,4% 3,7% TA P( < S < 41 Dos no apply 4,% TA P(No Drawdown & < Dos no apply,8% E( NoDrawdown & TA < Dos no apply 133,9 P(Drawdown & TA < 39,4% 9,9% 11 n h sragy 4 yars 1% quiis h swich from quiis o bonds nvr occurs, so hr is no S and h opion of No Drawdown dos no occur. Afr h fund of his sragy will rmain invsd fully ino quiis.

22 A Swich Cririon for Dfind Conribuion Pnsion Schms 1 E( Drawdown 96, 87,4 P(Swich bwn and (+1 16% 8,7% Avrag yar swich afr 3,6 3, TA P ( < < & ;<<( ,3% 4,9% TA E ( < ,1 19,9 P (uin 17,1% 16,3% Avrag yar of ruin afr 7,5 7,7 Commns Th oal probabiliy of failing h arg for h sragy 4 yars 1% quiis is 17,1%+6,3%=3,4%, whil h oal probabiliy of failing h arg of h swich sragy (=31 wih buffr is 16,3%+4,9%+,8%=4,%. This mans ha h sragy 4 yars 1% quiis, which is considrd vry risky in h accumulaion phas, acually has h highs probabiliy of raching h arg if incom drawdown is considrd as wll. Of h individuals no raching h arg a (39,4%, many (16% will rach h arg bfor +1. Th swich sragy (=31 wih buffr has h advanag ha on avrag h swich is mad arlir and h,8% which dos no ak h incom drawdown, wihou raching h arg, sill hav on avrag of 133,9 o buy an annuiy, ha mans mor han 9% of h arg. 5. Conclusion and furhr rsarch n his papr h ida has bn o ry o find a suiabl sragy in ordr o rach h Targ und a rirmn. This is don by aking all h quiy risk a h lowr ags, as is common in Dfind Conribuion Schms, invsing h conribuions ino bonds from a crain momn onwards and waiing for h righ momn for swiching h quiy fund from quiis ino bonds (buying bonds wih h conribuions in h manim. sms also imporan o considr boh h priods bfor and afr rirmn, sinc w hav h incom drawdown opporuniy. nvsing fully in quiis sms o b lss risky han usually considrd and h lifsyl sragy lss appropria. Analysing wak and srong poins of his sragy, w hink ha h main waknsss ar: 1. opimaliy. A criic ha can b movd o h swich sragy is ha i is no opimal in h sns of h dynamic programming approach; howvr, also h lifsyl sragy, as wll as many ohr invsmn sragis proposd in h acuarial liraur ar no opimal in ha sns;. man. Th bs sragy ha w found, h =31 wih buffr, producs an avrag final fund ha is sill lowr han h avrag final fund producd by h lifsyl sragy and h 4 yars 1% quiis sragy ; 3. lngh of h sragy. Th sragy is appropria for long priods only. Howvr, mos of h popl work for a long priod bfor rirmn and, in h cas of DC schms, h posiion of a mmbr can b movd from on schm o anohr in cas of urnovr, wihou losing valu; 4. annuiy risk. Th valu of h annuiy a rirmn and afr is akn o b consan, whras i is no, dpnding on h yilds on bonds a h im of purchas. Thus, h annuiy risk is no akn ino considraion;

23 A Swich Cririon for Dfind Conribuion Pnsion Schms 5. salary. Th salary is akn o b consan ovr im. Should h salary incras ovr im, h simad and S would b highr, implying riskir sragis; 6. no corrlaion. Asss ar assumd o b uncorrlad, which is no ralisic. 7. moraliy. Moraliy is no akn ino accoun in h dcumulaion phas. W hink ha h inrsing poins ar h following: 1. h swich cririon is dynamic, i volvs ovr im aking ino accoun acual ralisaions of rurns on asss and bonds (i.. h pas hisory;. w considr boh h accumulaion and h dcumulaion phas oghr. n h liraur, h wo phass ar usually considrd sparaly. A dynamic programming approach ha considrs h wo phass would complica h racabiliy of h modl and his is probably h rason why his has no bn don y (up o our knowldg. This could b an inrsing subjc for furhr rsarch; 3. h ida of spliing up h wo momns of swich of h conribuions from quiis ino bonds and h swich of h quiy fund from quiis ino bonds is nw; 4. h swich cririon is linkd o h achivmn of a crain arg: h swich from quiis ino bonds occurs whn and if hr ar good chancs of raching h arg fund a rirmn; 5. in h papr an indicaion is givn for h rsrv ndd a ach yar, in ordr o compnsa for h fuur risk on bonds. inally w suggs hr lmns for furhr rsarch ha migh improv h invsmn sragy: 1. finding a mor appropria sima for h buffr ndd in ach yar;. changing h swich cririon in such a way ha i aks ino accoun h yild givn on bonds a h momn of swiching from quiis ino bonds; 3. adding h opion of swiching from quiis no only ino bonds, bu also ino dfrrd annuiis. W hink his migh improv h sragy, giv highr rsuls and lad o lowr varianc of h rsuls, wih h disadvanag ha buying dfrrd annuiis implis no bqus in cas of dah bfor rirmn.

24 A Swich Cririon for Dfind Conribuion Pnsion Schms 3 frncs Blak, D., Cairns, A., Dowd, K. (a, Opimal dynamic ass allocaion for dfindconribuion pnsion plans. 1 h A Colloquium, Tromsø, Jun,, pp Blak, D., Cairns, A., Dowd, K. (b, Pnsionmrics : Sochasic pnsion plan dsign and Valu-a-isk during h accumulaion phas. BS-Gamma oundaion, Working Papr Collcion, Numbr 19. Blak, D., Cairns, A., Dowd, K. (1, Pnsionmrics : Sochasic pnsion plan during h disribuion phas. Th Pnsion nsiu, Birkbck Collg, Discussion Papr P-13. Bodi, Z.(1995, On h risk of socks in h long run. inancial Analyss Journal, May/Jun, pp.18-. Bondr, C.G.E., van Hoogdalm, S., Janswijr,.M.A., van Lochm, E. (, nrgnraionl solidarii n individualii in d wd pnsionpijlr: En scnarioanalys. por W114 of h Scinific Council of h Duch Cnral Govrnmn (Duch. Booh, P.M., Yakoubov, Y. (, nvsmn policy for dfind-conribuion pnsion schm mmbrs clos o rirmn: an analysis of h lifsyl concp. Norh Amrican Acuarial Journal, Volum 4, Numbr. Boulir, J-.,Huang, S., Taillard, G. (, Opimal managmn undr sochasic inrs ras: h cas of a procd dfind conribuion pnsion fund. nsuranc: Mahmaics and Economics, Volum 8, pp Dlsra, G., Grasslli, M., Kohl, P-. (, Opimal nvsmn Sragis in a C ramwork. Journal of Applid Probabiliy, Numbr 37, pp Exly, J., Mha, S., Smih, A., Bzooyn, J. (1998, Lif Syl Sragis for Dfind Conribuion Pnsion Schms. GEMS (hp:// Grrard,., Habrman, S., Vigna, E. (3, Opimal nvsmn Choics Pos irmn in a Dfind Conribuion Pnsion Schm. Mimo. Habrman, S., Vigna, E. (, Opimal invsmn sragis and risk masurs in dfind conribuion pnsion schms. nsuranc: Mahmaics and Economics, Numbr 31, pp Kapur, S., Orszag, M. (1999, A Porfolio Approach o invsmn and Annuiizaion during irmn, Procdings of h Third nrnaional Congrss on nsuranc: Mahmaics and Economics, London. Khorasann, M.,Z. (1996, Annuiy choics for pnsionrs, Journal of acuarial pracic vol 4 no. Knox, D.M.. (1993, A criiqu of dfind conribuion using a simulaion approach. Journal of Acuarial Pracic, Numbr 3 (1, pp Ludvik, P. (1994, nvsmn Sragy for Dfind Conribuion Plans. 4 h A nrnaional Colloquium, Volum 3, pp Vigna, E., Habrman, S. (1, Opimal invsmn sragy for dfind conribuion pnsion schms. nsuranc: Mahmaics and Economics, Numbr 8, pp Yaari, M.,E. (1965, Uncrain Lifim, Lif nsuranc, and h Thory of h Consumr. Th viw of Economic Sudis, Vol.3,

25 A Swich Cririon for Dfind Conribuion Pnsion Schms 4 Appndix 1 -Wih swich sragy (=3, hr diffrn cass igur A BO 4% EQ 1% urn on quiis is vry high, so h swich will occur immdialy =3. igur B BO 4% EQ 6% Th rurn on quiis is no oo high, you nd som xra im o swich from quiis o bonds.

26 A Swich Cririon for Dfind Conribuion Pnsion Schms 5 igur C BO 4% EQ % urn on quiis is oo low in his cas, h swich from h quiy fund o bonds will no b mad bfor rirmn.

27 A Swich Cririon for Dfind Conribuion Pnsion Schms 6 Appndix B - Linar rgrssion igur D. Th linar rgrssion for SMS a for h yars 3 o 4.,4,35,3,5,,15,1,5, y = -,181x +,3474 =, igur E. Th linar rgrssion lin for SMS for h yars 31 o 4.,5,,15 y = -,156x +,191 =,9967,1,5,

28 A Swich Cririon for Dfind Conribuion Pnsion Schms 7 Appndix C - Discound Buffr Tabl A. Th discound buffr in comparison wih h original buffr Swich sragy =3 wih discound buffr =3 wih buffr Man 163,77 167,8 Sandard dviaion 68,7 7,8 Downsid dviaion 5,8 5,5 TA Man shorfall from h 43,4 45,8 TA P( < 35,8% 34,7% TA P( < S < 41 1,% 5,% Valu a isk 95% 66,5 66,4 Valu a isk 75% 117,4 114, Tabl B. Swich sragy =31 wih discound buffr =31 wih buffr Man 186,5 187,6 Sandard dviaion 15, 15,1 Downsid dviaion 55,8 56,4 TA Man shorfall from h 48,1 49, TA P( < 33,3% 3,7% TA P( < S < 41 5,7% 4,% Valu a isk 95% 6,9 6,9 Valu a isk 75% 119,3 117,7 Tabl C. Swich sragy =3-31 wih discound buffr =3-31 wih buffr Man 165, 169, Sandard dviaion 69, 71,4 Downsid dviaion 53,6 56,8 TA Man shorfall from h 45, 48,9 TA P( < 34,% 3,1% TA P( < S < 41 1,8% 5,8% Valu a isk 95% 61,9 6,9 Valu a isk 75% 1,7 11,