Fixed Deferred Annuities with Appendix for Equity-Indexed Annuities
More Annuities Problems : Varying -Interest rate . Payment · More Annuities Problems : Varying...
Transcript of More Annuities Problems : Varying -Interest rate . Payment · More Annuities Problems : Varying...
MoreAnnuitiesProblems:Varying- Interestrate.Payment
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ChangingInterestRates
• Intheprevioussection,therateofcompoundinterestremainunchangedthroughouttheperiod.
• Inpractice,interestrateofinterestvarywithconsiderablefrequency.
Forinstance:Banksvarytheirdepositrateswithchangesinmarketconditions.
(compoundinterest)
Example(changeofinterestrate- compoundinterest)
Find thepresentvalueofRM10000due in10years’timeif𝑗" = 8% forthefirst3years,𝑗& = 10% forthenext5yearsand 𝑗) = 9% forthelast2years,
RM10,000
Time 0 3 8 10𝑗" = 8%𝑚 = 4, 𝑡 = 3n=12
è𝑖 = 34" = 0.02
𝑗& = 10%𝑚 = 2, 𝑡 = 5n=10
è𝑖 = 37& = 0.05
𝑗) = 9%𝑚 = 1, 𝑡 = 2n=2
è𝑖 = 38) = 0.09
ie thepresentvalueofRM10K:
ValueofRM10Kin3years’time= 8416.80 1+ 0.05 ;)<
= 𝑅𝑀5167.19
ValueofRM10KNow= 5167.19 1+ 0.02 ;)&
= 𝑅𝑀4074.29
ValueofRM10Kin8years’time= 10000 1+ 0.09 ;&
= 𝑅𝑀8416.80
= 1000 1 + 0.12 ;)& 1+ 0.05 ;)< 1+ 0.09 ;& = 𝑅𝑀4074.29
Previously,inevaluatingthePV(orFV)whentheinterestratevariesè thegeneralstrategyisto evaluatewheneverthereisachangeintherateofinterest
MoreAnnuitiesProblems• Findingtheperiodicpayment• Findingtheinterestrate• Findingthetermofannuity• Changesintheinterestrate• Annuitieswherepaymentsvary
Similartotheevaluationofvalueswheninterestratesvaries.
è Itisnecessarytovalueannuitieswheninterestrateschangeduringtheannuitiesterm
è Similartothepreviousmethod,• Eachsuccessiveintermediatevalueshouldhaveafocaldatethatiscloserto
thefinalfocaldate.• Theaimistomovetowardsthefinalfocaldate,gatheringuptheannuity
paymentsaswego.• Eachstepcomprisesthevaluationoftheannuitypaymentswithinthe
appropriateperiodtogetherwiththevalueofthepreviousintermediatevalueatthenewrateofinterest
Findingperiodicpayment
Example1Findthevalueat1Jan2000ofanannuityofRM1000papayablefrom1Jan1991to1Jan2000inclusive,iftheinterestrateis9%papriorto1Jan1997and8%pathereafter
R=1000
FV?
0 1
RM1K
2
RM1K
3
RM1K
1Jan1991
1Jan1992
1Jan1993
4
RM1K
5
RM1K
6
RM1K
1Jan1994
1Jan1995
1Jan1996
7
RM1K
1Jan1997
8
RM1K
9
RM1K
10
RM1K
1Jan1998
1Jan1999
1Jan2000
𝑖 = 0.09n=7
𝑖 = 0.08n=3
𝐹𝑉 == 𝑅1 + 𝑖 𝑛− 1
𝑖𝑛𝑖𝑠𝑅
At1Jan1997
= 10001+ 0.09 7 − 1
0.09
70.09𝑠= 1000
= 𝑅𝑀9200.43
𝐹𝑉 = 𝑛𝑖𝑠+𝑅
At1Jan2000
= 11.589.89 + 10001 + 0.08 3 − 1
0.08
30.08𝑠+1000
= 𝑅𝑀14836.29
BringforwardRM9200.43
= 𝑅𝑀9200.43(1 + 0.08)J
= 11.589.89 + 3246.40
Example2(a)CalculatethePVofa10-yearannuitywithhalf-yearlypaymentsofRM50each,assumingj2=8%foryears1,2,and3;j2=7%foryears4and5;j2=6%foryears6,7,8,9,and10.
Time 0 6𝑗& = 8%𝑚 = 2,𝑡 = 3n=6
è𝑖 = 37&= 0.04
𝑗& = 7%𝑚 = 2,𝑡 = 2n=4
è𝑖 = 37&= 0.035
RM50 RM50 RM50RM50 RM50
10
RM50RM50
RM50 RM50 RM50
20𝑗& = 6%𝑚 = 2,𝑡 = 5n=10
è𝑖 = 37&= 0.03
𝑛𝑖𝑃𝑉 = 𝑅𝑎 = 𝑅
1− 1+ 𝑖 ;M
𝑖AtYear 5
= 50𝑎100.03
= 𝑅𝑀426.51
𝑃𝑉 =𝑛𝑖𝑎+𝑅
AtYear 3
= 𝑅𝑀555.33
Bringbward RM426.51
= 𝑅𝑀426.51(1+ 0.035);"
= 371.68+ 183.6540.035𝑎+50
𝑃𝑉 =𝑛𝑖𝑎+𝑅
Now
= 𝑹𝑴𝟕𝟎𝟎. 𝟗𝟗
Bringbward RM555.33
= 𝑅𝑀555.33(1+ 0.04);S
= 438.89+ 262.1060.04𝑎+50
PV?
RM50 RM50RM50
RM50RM50
RM50 RM50 RM50 RM50 RM50
Example2(cont’d)(b)From(a),whatunchangedrateofinterestthroughoutthe10yearswouldprovidethesamePV?
PV=RM700.99;R=50;m=2;t=10è n=20
𝑛𝑖𝑃𝑉 = 𝑅𝑎 = 𝑅
1− 1+ 𝑖 ;M
𝑖
700.99 = 50𝑎20𝑖
= 𝑅𝑀14.019810𝑖𝑎
200.03𝑎 = 14.877
Fromtable
200.04𝑎 = 13.590
20𝑖𝑎 = 14.0198
Thus,linearinterpolationformulais
𝑖 − 0.030.04− 0.03 =
14.0198− 14.87713.590− 14.877
𝑖 − 0.030.01 =
−0.8572−1.287
𝑖 = 0.0367ie.3.67%
𝑗2 = 𝑖𝑚 = 7.34%
Exercise1. Aworkeraged25investsRM150everyyearfor40years.Howmuchwillshe
retirewithiftheinvestmentearns:• 10%painyears1-5• 8%painyears6-15• 9%painyears16-30• 3%painyears31-40Whatlevelofinterestratethroughout thetermwouldprovide thesameretirementbenefits?
(𝑅𝑀27953.68)
MoreAnnuitiesProblems• Findingtheperiodicpayment• Findingtheinterestrate• Findingthetermofannuity• Changesintheinterestrate• Annuitieswherepaymentsvary
Annuitieswherepaymentsvary
• Inevaluatingannuities,sizeofpaymentswithinthetermofannuityneedtobeconsidered
• Previously,weonlyconsidereduniformsizeofpayments.
• Therearetwomethodsusedwhenthepaymentsvaryintermsofaconstantratio.
Example3Mr AdamswantstobuyanannuityofRM1000ayearfor10yearsthatisprotectedagainstinflation.TheXYZtrustCompanyofferstosellhimanannuitywherepaymentsincreaseeachyearbyexactly10%.Inparticular, thepaymentswillbeRM1100attheendofyear1,RM1210attheendofyear2,RM1331attheendofyear3,andsoonfor10years.Find thecostofthisannuityiftherateofinterestis13%pa.
0 101
1000(1+0.1)1=1100
2 3 9PVie cost
1000(1+0.1)2=1210
1000(1+0.1)3=1331
1000(1+0.1)9 1000(1+0.1)10
Note:Thisdiagramissimilar tothecompound interesttimeline.WeevaluatePVbasedonthecompound interestevaluationie.ateachperiodtothefocaltimeandaccumulateitsamount.
𝑃𝑉 = 𝑆(1 + 𝑖);M
𝑃𝑉 = 𝑆(1 + 𝑖);) + 𝑆(1 + 𝑖);& + 𝑆(1 + 𝑖);J +…+𝑆(1 + 𝑖);(M;)) + +𝑆(1+ 𝑖);M
= [1000 1.1 ](1+ 0.13);)+[1000(1.1)2](1+ 0.13);& +⋯+ [ 1000 1.1 Z] 1+ 0.13 ;Z
+ [ 1000 1.1 )<] 1+ 0.13 ;)<
Example3- Method1𝑃𝑉 = 𝑆(1 + 𝑖);) + 𝑆(1 + 𝑖);& + 𝑆(1 + 𝑖);J +…+𝑆(1 + 𝑖);(M;)) + +𝑆(1 + 𝑖);M
Note:RHsideoftheequationisthesumofgeometricprogression
𝑎1 − 𝑟M
1 − 𝑟Thus,
𝑃𝑉 = 𝑆(1 + 𝑖);) + 𝑆(1 + 𝑖);& + 𝑆(1 + 𝑖);J +…+𝑆(1 + 𝑖);(M;)) + +𝑆(1 + 𝑖);M
= [1000 1.1 ](1+ 0.13);)+[1000(1.1)2](1 + 0.13);& +⋯+ [ 1000 1.1 )<] 1+ 0.13 ;)<
𝑎 = 100 1.1 (1 + 0.13);)
𝑟 = 1.1 (1 + 0.13);)𝑛 = 10
𝑃𝑉 = 100 1.1 (1 + 0.13);)1− 1.1 (1 + 0.13);) )<
1− 1.1 (1 + 0.13);)= 𝑅𝑀8650.17
Example3- Method2
Noticewecansimplify theaboveas:
Revisit
𝑃𝑉 = 𝑆(1 + 𝑖);) + 𝑆(1 + 𝑖);& + 𝑆(1 + 𝑖);J +…+𝑆(1 + 𝑖);(M;)) + +𝑆(1 + 𝑖);M
= [1000 1.1 ](1+ 0.13);)+[1000(1.1)2](1 + 0.13);& +⋯+ [ 1000 1.1 )<] 1+ 0.13 ;)<
𝑃𝑉 = 10001.11.13
+1.11.13
&+
1.11.13
J+ ⋯+
1.11.13
)<
Rewritingwehave
𝑃𝑉 = 10001.11.13
+1.11.13
&+
1.11.13
J+ ⋯+
1.11.13
)<
𝑖\Letbethenewrateofinterest
1 + 𝑖\ =1.11.13
;)
=1.131.1
𝑖\=0.027
= 1000 1+ 𝑖 \ ;) + 1+ 𝑖 \ ;& + 1+ 𝑖 \ ;J + ⋯+ 1+ 𝑖 \ ;)<
Solveusingordinaryannuity:
𝑖
𝑃𝑉 = 𝑅𝑛𝑖 ’𝑎 𝑃𝑉 = 1000 = 𝑹𝑴𝟖𝟔𝟓𝟎.𝟏𝟕
10𝑖 ’𝑎ie
Example4Find thePVofpaymentsmadeattheendofeachyearfor10yearsusing12%paifthefirstpaymentisRM100,thesecondisRM200,thethirdRM300andsoon
0 101
100
2 3 9PV
200 300 900 1000
Note:Thisdiagramissimilar tothecompound interesttimeline.WeevaluatePVbasedonthecompound interestevaluationie.ateachperiodtothefocaltimeandaccumulateitsamount.
𝑃𝑉 = 𝑆(1 + 𝑖);M
𝑃𝑉 = 𝑆(1 + 𝑖);)+ 𝑆(1 + 𝑖);&+…. +𝑆(1 + 𝑖);(M;))+ +𝑆(1+ 𝑖);M
= [100(1+ 0.12);)+200(1+ 0.12);& + ⋯+ 900 1+ 0.12 ;Z + 1000 1+ 0.12 ;)<
Multiply bothsides by(1+i)i.e 1.12
1.12𝑃𝑉 = 100 +200(1+ 0.12);) +⋯+900 1+ 0.12 ;a + 1000 1+ 0.12 ;)<
(1)
(2)
Multiply bothsides by(1+i)i.e 1.12
Example4(cont’d)𝑃𝑉 = 𝑆(1 + 𝑖);)+ 𝑆(1 + 𝑖);&+…. +𝑆(1 + 𝑖);(M;))+ +𝑆(1+ 𝑖);M
𝑃𝑉 = [100(1+ 0.12);)+200(1+ 0.12);& + ⋯+ 900 1+ 0.12 ;Z + 1000 1+ 0.12 ;)<
Multiply bothsides by(1+i)i.e 1.12
1.12𝑃𝑉 = 100 +200(1+ 0.12);) +⋯+900 1+ 0.12 ;a + 1000 1+ 0.12 ;)< (2)
Subract (2)from(1)
0.12𝑃𝑉 = 100 +100(1+ 0.12);) +⋯+100 1+ 0.12 ;Z−1000 1+ 0.12 ;)<
(1)
NowontheRhside wehavetheformofa10yearannuitydue, ie.
0.12𝑃𝑉 = 100 +100(1+ 0.12);) +⋯+100 1+ 0.12 ;Z+ 100 1+ 0.12 ;)<
𝑃𝑉 = 𝑅𝑛𝑖𝑎
0.12𝑃𝑉 = 100100.12𝑎
0.12𝑃𝑉 = 100(1+ 0.12)<+ 100(1+ 0.12);) + ⋯+ 100 1+ 0.12 ;Z + 100 1+ 0.12 ;)<
−1000 1+ 0.12 ;)<
𝑷𝑽 = 𝑹𝑴𝟐𝟓𝟗𝟎.𝟒𝟑
ExerciseAcourtwastryingtodetermine thePVofthefuture incomeofamanparalysed inacaraccident.Atthetimeoftheaccident,themanwasearningRM25000ayearandanticipatedreceivingan8%riseeachyear.Heis30yearsawayfromretirement.Ifmoney isworth10%pa,whatisthePVofhisfuture income.
(𝑅𝑀571486.98)