VALUING SURRENDER OPTIONS IN KOREAN INTEREST INDEXED ANNUITIES Changi Kim* * Dr. Changi Kim is Lcturr at Actuarial Studis Faculty of Commrc & Economics Th Univrsity of Nw South Wals Sydny NSW 2052 Australia. Tl: +6 2 9385 2647 Email: c.im@unsw.du.au
2. INTRODUCTION Surrndr Options : Insuranc policy holdrs can withdraw thir account valus from th insuranc companis at any tim. It is a right givn to th policy holdrs. Surrndr Bhaviors and thir Impacts : High mart intrst rats driv insuranc policy holdrs to surrndr thir insuranc contracts and find high yild altrnativs in th mart. Whn intrst rats incras surrndr rats also incras du to th incrasd gap btwn th mart intrst rats and th insuranc crditing rats. Intrst rat movmnts affct th cash flows of assts and liabilitis of insuranc companis.
3 Surrndr Rat Modls Ndd : Surrndr rat is on of th main factors influncing th futur liability cash flows. Modling appropriat surrndr rats is ssntial in managing assts and liabilitis of insuranc companis. Factors affcting surrndr rats : Diffrnc btwn rfrnc mart rat and policy crditing rat sasonal ffct ag and gndr of clints conomy growth rat forign xchang rat inflation rat policy ag sinc issu dat and unmploymnt rat tc
4 2. THE STRUCTURE OF INTEREST INDEXED ANNUITIES Many insuranc companis ar slling singl prmium dfrrd annuitis (SPDA). Intrst-indxd annuitis (IIA) ar on of th most popular SPDA products in Kora. Th distinctiv faturs of IIA ar th surrndr options and annuitization options. Crditing Intrst Rats of IIA Almost all contracts guarant a minimum intrst rat Th crditing intrst rats ar announcd vry month basd on currnt mart rats currnt invstmnt gain rats and th xpctd futur portfolio incom gain rats. Surrndr Chargs of IIA Many contracts crdit th full prmium to th account valu and assss surrndr chargs whn th policy holdr surrndrs. Th amount of surrndr chargs ar usually from 7% dcrasd by % annually to zro ovr a 7 yar priod.
5 Dath Bnfits in IIA Usually th dath bnfit is th account valu. A fw variations of dath bnfits ar considrd according to th companis for xampl th account valu plus 0% of prmium and anothr 0% of prmium in th cas of accidntal dath. Annuitization and Annuity Options in IIA Th policy holdr can choos th initial annuitization dat Th typical typs of annuity options ar : (a) th lump sum withdrawal of th account valu at th dat of annuitization (b) crtain priods btwn 5 and 20 yars (c) lif incom with a crtain priod of 0 yars (d) inhritanc annuity.
6 3. MODELING SURRENDER RATES FOR INTEREST INDEXED ANNUITIES Th variabls considrd ar (a) th diffrnc btwn rfrnc nw mony rats and product crditing rats (b) th policy ag sinc th contract was issud (c) unmploymnt rats (d) conomy growth rats () sasonal ffcts. Cascad Structur for th Surrndr Rat Modling Trm Structur Short Rats Announcd Rats Crditing Rats Surrndr Chargs Rfrnc Mart Rats Economy Growth Rats Economy Cycl Policy Ag Unmploymnt Rats Sasonal Effcts Surrndr Rats
7 Short Rats For a givn trm structur th st of short rats { i (t)} for t = 0 and for =02 t for ach t should b simulatd satisfying th following quation P (0t) = P ˆ(0 t) whr P (0t) = tim 0 pric of a zro coupon bond paying at maturity t and P ˆ(0 t) = tim 0 pric of a zro coupon bond paying at maturity t from th mart. For discrt tim intrst rat modl w may us an intrst rat modl such as Blac-Drman-Toy(BDT) or Ho-L(HL) modl to gnrat short rats { i t t=0 T-}. In valuing a stram of path-dpndnt cash flows w nd to pic a subst of all intrst rat paths with a computationally rasonabl numbr K of sampl paths. Th st of all intrst rat paths is Ω Th numbr of lmnts of Ω is Ω. Thn Ω = 2 20 for 20 months. W dnot th subst of sampl paths as Ωˆ with Ω ˆ = K Ωˆ = { ω2 ω K }. ω...
8 W dnot th -th path of Ωˆ as ω. For a path ω w can dfin a function ω from tim to stat such that ω (t) is th stat s at tim t on path ω ω (t) = s whr = 2 K t = 0 T- and s =0 t. For notational simplicity lt us dnot (t ω ) to b th nod at tim t on path ω Nw Mony Rats{ i m (t ω ) t= 0 T- and =2 K}. W us th maximum of th 3 yar yild rats(undr th incrasing trm structur) and th short rats(undr th dcrasing trm structur) as th rfrnc nw mony rats. At nod (t) to calculat th 3 yar yild rats Y(tt+3yar) for th nxt tn yars w us intrst rat modls such as Blac-Drman- Toy(BDT) or Ho-L(HL) modls with th following formula P(tu) = A( t u ) u = 0 { + Y u)} = u t
9 whr P(tu) is th pric at nod (t) of th zro-coupon bond maturing at tim u>t and A(tu) is th Arrow-Dbru pric. Economy Growth Rats { i EG (t ω ) t= 0 T- and =2 K}. 2π t i EG (t) = 0.00767-0.095883* i m (t) 0.00565*sin( ) 30 2π t + 0.03263* cos( ) + ξ t 30 Unmploymnt Rats { i UE (t ω ) t= 0 T- and =2 K}. i UE (t) = i UE (t-) * { + 0.840 4.360i EG (t) 0.440 DV 3 0.20997 DV 4 0.6229 DV 5 0.2605 DV 6 0.0758 DV 7 0.0894 DV 8 0.545 DV 9 0.09962 DV 0 } + ε t Announcd Rats { i a (t ω ) t= 0 T- and =2 K}. i a (t) = 0.00538 + 0.002527 * i m ( t) + ε t Crditing Rats of IIA { i c (t ω ) t= 0 T- and =2 K}. i c (t ω ) = max{ i a (t ω ) i g }
0 whr th announcd rat i a (t ω ) is givn and th guarantd annual intrst rat i g is i g = 3% annually. In practic considring th surrndr bhaviors of th policy holdrs th crditing rat i c (u ω ) at tim u on path ω is dpndnt on th latst surrndr tim bfor tim u * i c (u ω ) = i c (u ω u ) whr * u = 0 if thr is no surrndr bfor tim u = th latst surrndr tim bfor tim u. For computational purposs w us th formula for th announcd rats * i a (u ω ) = i a (u ω u ) = 0.00538 + 0.002527 0.2i m ( u * ω ) + 0.8i m ( u ω ) + ε u Surrndr Chargs Sc (t) Th surrndr chargs during th yar t Sc (t) ar Sc (t) = 7% 0 t <
Sc (t) = 6% t < 2 Sc (t) = 5% 2 t < 3 Sc (t) = 4% 3 t < 4 Sc (t) = 3% 4 t < 5 Sc (t) = 2% 5 t < 6 Sc (t) = % 6 t < 7 Sc (t) = 0% 7 t < 0. Surrndr Rats { q t ω ) t= 0 T- and =2 K}. s ( W us th Logit Modl for th IIA surrndr rats { q t ω ) t=02 T and =2 K} s ( whr q t ω ) = s ( + xp( α ) α = β + β * i ω ) + β * i ω ) 0 UE UE EG EG + β *{ i ( t ω ) i ( t ω )} + β DV. m c = 0246802 = * month_
2 Paramtr Estimats with Logit Modl(IIA) Analysis of Maximum Lilihood Estimats Standard Paramtr DF Estimat Error Chi-Squar Pr > ChiSq Intrcpt -6.032 0.0067 950275.502 <.000 DIFFLAG0 9.3465 0.0563 2755.98 <.000 DIFFLAG2 0.9728 0.042 557.6077 <.000 DIFFLAG4-6.2020 0.0438 2003.9722 <.000 DIFFLAG6-2.7553 0.0399 4776.8774 <.000 DIFFLAG8.4655 0.0390 40.2 <.000 DIFFLAG0 0.5252 0.0389 82.560 <.000 DIFFLAG2 -.8470 0.0468 557.807 <.000 Unmploy 50.6348 0.640 9534.7985 <.000 Eco GROWTH -5.3360 0.723 959.5427 <.000 MONTH -0.2 0.00409 2662.3227 <.000 MONTH2-0.499 0.00446 8867.322 <.000 MONTH3-0.3629 0.00446 6633.620 <.000 MONTH4 0.2 0.0045 728.9672 <.000 MONTH5 0.2443 0.00408 3589.787 <.000 MONTH6 0.296 0.00424 4879.207 <.000 MONTH7 0.2 0.00429 242.8388 <.000 MONTH8 0.2082 0.00458 2065.2003 <.000 MONTH9 0.4040 0.00452 7970.0766 <.000 MONTH0 0.499 0.00469 024.0567 <.000 MONTH 0.3720 0.00447 693.5047 <.000
3 Ral and Prdictd Surrndr Rats of IIA Expctd Ral 0.00% 9.00% 8.00% 7.00% 6.00% 5.00% 4.00% 3.00% 2.00%.00% 0.00% 980 9803 9805 9807 9809 98 990 9903 9905 9907 9909 99 000 0003 0005 0007 0009 00 00 003 005 007 009 0 020 Tim
4 4. VALUING THE SURRENDER OPTIONS IN INTEREST INDEXED ANNUITIES At tim t and on path ω th probability of surrndr qs ω ) is givn by q t ω ) = s ( + xp( α ) whr α = β + β * i ω ) + β * i ω ) 0 UE UE EG EG + β *{ i ( t ω ) i ( t ω )} + β DV. m c = 0246802 = month_ * Th account valu at tim 0 and on path ω AV (0 ω ) is th initial singl prmium AV (0 ω ) = A. Th account valu at tim t and on path ω is givn by AV (t ω ) = AV (t- ω ) [+ i c (t- ω )] whr i c (t- ω ) is th crditing rat at tim t- and on path ω. Ω (t ω ) = th st of th whol paths from th nod (t ω ) Ωˆ (t ω ) = a subst of sampl paths of Ω (t ω ). Ωˆ = { ω ω 2 ω K }
5 A finit squnc of partitions { P 0 P P t P T } of Ωˆ is dfind satisfying P0 P P t P T. ( t) ( t) (t) v t = P t and P t = { H H 2 H v t }. (t) Th lmnts { H } of P t ar calld to th tim-t historis. Th lmnt of partition P t which contains a path ω is dnotd by (t) H (ω ). Th policy holdr has th option to surrndr at any tim on any path or to p his/hr policy. Dcision Trs surrndr surrndr p p surrndr p
6 Th surrndrd cash valu is accumulatd by th rfrnc rats(nw mony rats). And w us th crditing rats to calculat th accumulatd valu whn th policy holdr dos not surrndr. C(t ω T ω ) = th st of surrndr bhaviors of th policy holdr from (t) tim t to tim T- givn surrndr at tim t on a path ω H ( ω ) C(t ω T ω ) = { ω T ω ) ω T ω ) 2 ω T ω ) 3 ω T ω ) T 2 t } whr ach ω T ω ) l T t l = 2 2 is a surrndr bhavior of th policy holdr from tim t to tim T- givn surrndr at tim t on a path (t) ω H ( ω ). Hr w assum that th policy holdrs always rinvst. Exampl of th st C(T-3 ω T ω ) of surrndr bhaviors givn surrndr at T-3 C(T-3 ω T ω ) = { ( T 3 ω T ω ) ( T 3 ω T ω ) 2 ( T 3 ω T ω ) 3 ( T 3 ω T ω ) 4 } with ach surrndr bhavior is as follows and ( T 3 ω T ω ) ( T 3 ω T ω ) 2 ( T 3 ω T ω ) 3 ( T 3 ω T ω ) 4 = (s s s) = (s s) = (s s ) = (s )
7 whr s dnots surrndring and dnots ping. SV ( ω T ω ) l ) = th accumulatd valu from surrndr at tim T and on a (t) path ω H ( ω ) undr th surrndr bhavior T t 2. ω T ω ) l l = 2 Exampl : If th surrndr bhavior hav ( t ω T ω ) = (ss. ) thn w SV ( whr ( t ω T ω ) ) = AV (t ω ) (-Sc(t)) (+ i m (t ω )) (-Sc()) (+ m T * u = 0 if thr is no surrndr bfor tim u = th latst surrndr tim bfor tim u i (t+ ω )) * ( + i * c ( u ω u )) = + * (t) and i c (u ω u ) is th crditing rat at tim u on path ω H ( ω ). Sinc w assum that surrndr occurs at th nd of priod w apply th surrndr charg Sc() in th abov formula. W also assum that thr is no surrndr charg at tim T. u t 2 SV (t ω T ω ) = th avrag of th accumulatd valu from surrndr (t) givn ω H ( ω )At tim T SV (t ω T ω ) = Q ω T ω ) ( t) E [ SV ( ) H ( ω )]
8 ω T ω ) ω T ω ) = Pr( ) SV ( ) ( t ω T ω ) C ω T ω ) whr Pr( ( T t ω ω ) ( T t ω ω ) C(t ω T ω ). ) is a ris nutral probability of th surrndr bhavior SV (t ω ) = th valu from surrndr at currnt tim t SV (t ω ) = Q E = SV ( t ) ω H ( ω ) whr Pr(ω ) is a ris nutral probability. ( ω ) ω T ω ) ( t) H T ( + i( u ω)) u= t SV ω T ω ) Pr( ω ) T ( + i( u ω)) u= t F(t ω T ω ) = th st of surrndr bhaviors of th policy holdr from (t) tim t to tim T- givn ping at tim t on a path ω H ( ω ) F(t ω T ω ) = { ω T ω ) f f ω T ω ) 2 f ω T ω ) 3 f ω T ω ) T 2 t } whr ach ω T ω ) f l T t l = 2 2 is a surrndr bhavior of th policy holdr from tim t to tim T- givn ping at tim t on a path (t) ω H ( ω ).
9 KV ( ω T ω ) f l ) = th accumulatd valu at tim T and on a path (t) ω H ( ω ) if th policy holdr invsts th amount of mony AV (t ω ) ω T ω ) until th option maturity T undr th surrndr bhavior l = 2 T t 2. f l KV (t ω T ω ) = th avrag of th accumulatd valu at tim T givn (t) ω H ( ω ) KV (t ω T ω ) = Q ω T ω ) ( t) E [ KV ( f ) H ( ω )] ω T ω ) ω T ω ) = Pr( f ) KV ( f ) ( t ω T ω ) f F ω T ω ) whr Pr( f ( t ω T ω ) ) is a ris nutral probability of th surrndr bhavior f ( t ω T ω ) F(t ω T ω ). KV (t ω ) = th ping valu at currnt tim t KV (t ω ) = Q E = ω H ( t ) KV ( ω ) whr Pr(ω ) is a ris nutral probability. ( ω ) ω T ω ) ( t) H T ( + i( u ω)) u= t KV ω T ω ) Pr( ω ) T ( + i( u ω)) u= t
20 EV (t ω ) = xrcis valu at tim t on a path ω EV (t ω ) = p ω ) * qs ω ) *{ SV (t ω ) - KV (t ω )}. Valu of surrndr option(vso) for th total xrcis valus at tim 0 K T EV ω ) VSO = Pr( ω ) t = t= ( + i( u ω )) u= 0 whr K dnots th numbr of paths at tim 0. Valu of Surrndr Option (VSO) Surrndr Charg VSO(BDT) VSO(HL) Cas 7% - 0% -58.07-56.63 Cas 2 0% 4.77 4.8 Hr th initial singl prmium is 0000.
2 Surrndr chargs rally hav an ffct on th valu of th surrndr option(vso) of IIA. Th two valus of th surrndr option(vso) of IIA with BDT modl and HL modl ar almost th sam but not xactly th sam. So th valus may b dpndnt on th particular choic of intrst rat modl. Th valus of th surrndr option (VSO) with surrndr chargs ar ngativ numbrs. Ths ngativ valus may b som profits to th insuranc companis not to th policy holdrs who hav th option (or th right!). Th surrndr option is a right givn to th policy holdrs and w may xpct that th valu of th surrndr options b positiv. It may not b rally surprising for somon who notic that som insuranc companis gt positiv gains from surrndr.
22 5. FAIR SURENDER CHARGES Th valus of th surrndr option with surrndr chargs ar ngativ numbrs; som profits to th insuranc companis not to th policy holdrs who hav th option W may find fair surrndr chargs not only for th company but also for th policy holdrs Th choic of intrst rat modl is a considration in valuing intrst rat contingnt cash flows. Tabl 3. Finding Fair Surrndr Chargs Surrndr VSO (BDT) VSO (HL) Chargs 7% - 0% -58.07-56.63 5% - 0% -94.79-94.2 % -25.60-25.27 0% 4.77 4.8
23 6. CONCLUSION Conclusions W considr surrndr rat modls for IIA using Logit functions with variabls (a) th diffrnc btwn rfrnc rats(nw mony rats) and product crditing rats with surrndr chargs (b) th policy ag sinc th contract was issud (c) unmploymnt rats (d) conomy growth rats () sasonal ffcts and so on. It is intrsting to not that th valus of th surrndr option (VSO) with surrndr chargs ar ngativ which may b som profits to th insuranc companis not to th policy holdrs who hav th option (or th right!). Finding fair surrndr chargs should b considrd.
24 Futur Rsarch Topics W can invstigat rational and irrational surrndr bhaviors of th policy holdrs. W may also try to calculat th valu of th optimal surrndr options. Thr ar a fw thoris on th pricing of Amrican options with Marov proprtis. But it is still difficult to pric Amrican options with path dpndnt cash flows which do not hav Marov proprtis. W may considr this problm for th futur rsarch topic. THANK YOU VERY MUCH!!!