Table Of ContentSpringerBriefs in Statistics
Jana Špirková · Igor Kollár · Gábor Szűcs ·
Pavel Zimmermann
Selected Payout
Products of the Old-Age
Pension Saving Scheme
SpringerBriefs in Statistics
SpringerBriefs present concise summaries of cutting-edge research and practical
applications across a wide spectrum of fields. Featuring compact volumes of 50
to 125 pages, the series covers a range of content from professionalto academic.
Typicaltopicsmightinclude:
• Atimelyreportofstate-of-theartanalyticaltechniques
• A bridge between new research results, as published in journal articles, and a
contextualliteraturereview
• Asnapshotofahotoremergingtopic
• Anin-depthcasestudyorclinicalexample
• A presentationofcoreconceptsthatstudentsmustunderstandinordertomake
independentcontributions
SpringerBriefs in Statistics showcase emerging theory, empirical research, and
practicalapplicationinStatisticsfromaglobalauthorcommunity.
SpringerBriefs are characterized by fast, global electronic dissemination, stan-
dard publishing contracts, standardized manuscript preparation and formatting
guidelines,andexpeditedproductionschedules.
Jana Špirková • Igor Kollár • Gábor Szu˝cs •
Pavel Zimmermann
Selected Payout Products of
the Old-Age Pension Saving
Scheme
JanaŠpirková IgorKollár
FacultyofEconomics FacultyofEconomics
MatejBelUniversityinBanskáBystrica MatejBelUniversityinBanskáBystrica
BanskáBystrica,Slovakia BanskáBystrica,Slovakia
GáborSzu˝cs PavelZimmermann
FacultyofMathematics,Physicsand FacultyofInformaticsandStatistics
Informatics PragueUniversityofEconomicsand
ComeniusUniversityinBratislava Business
Bratislava,Slovakia Prague,CzechRepublic
ISSN2191-544X ISSN2191-5458 (electronic)
SpringerBriefsinStatistics
ISBN978-3-031-23848-2 ISBN978-3-031-23849-9 (eBook)
https://doi.org/10.1007/978-3-031-23849-9
CopyrightforFigure4.2:©EuropeanCentralBank,FrankfurtamMain,Germany.Thefiguremaybe
obtainedfreeofchargethroughthewebsiteoftheEuropeanCentralBank.
©TheAuthor(s),underexclusivelicensetoSpringerNatureSwitzerlandAG2023
Thisworkissubjecttocopyright.AllrightsaresolelyandexclusivelylicensedbythePublisher,whether
thewhole orpart ofthematerial isconcerned, specifically therights oftranslation, reprinting, reuse
ofillustrations, recitation, broadcasting, reproductiononmicrofilmsorinanyotherphysicalway,and
transmissionorinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilar
ordissimilarmethodologynowknownorhereafterdeveloped.
Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication
doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant
protectivelawsandregulationsandthereforefreeforgeneraluse.
Thepublisher,theauthors,andtheeditorsaresafetoassumethattheadviceandinformationinthisbook
arebelievedtobetrueandaccurateatthedateofpublication.Neitherthepublishernortheauthorsor
theeditorsgiveawarranty,expressedorimplied,withrespecttothematerialcontainedhereinorforany
errorsoromissionsthatmayhavebeenmade.Thepublisherremainsneutralwithregardtojurisdictional
claimsinpublishedmapsandinstitutionalaffiliations.
ThisSpringerimprintispublishedbytheregisteredcompanySpringerNatureSwitzerlandAG
Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland
To Peter, Veronikaand Michal.
J.Š.
To Alena, Martinand Tomáš.
I.K.
To Zuzana andBeáta.
G.Sz.
To Markéta, KláraandViktor.
P.Z.
Preface
Thismonographisintendedforanyoneinterestedintheamountofpensionofold
age that can be expected from pension savings of old age or voluntary pension
savings.Itisprimarilyintendedforstudentsofactuarialandfinancialmathematics
andfutureeconomists.
Itdealswithandanalysestheso-calledpayoutphaseofold-agepensionsavings.
This means that it examines the amount of the monthly lifetime pension that the
beneficiarycanexpectfromtheamountsaved.Thissavedamountrepresentsasingle
premiumforlifetimepensioninsurance.
The monograph analyses five selected pension savings products. In addition
todeterminingandanalysingthebasiclifetimemonthlypensionthatthebeneficiary
willreceiveaslongashe/sheisalive,italsooffersanalysesofotherpensions,which
includebenefitsforsurvivorsorauthorizedpersons,intheeventthatthebeneficiary
dies. For example,selected productsincorporatethe risk for insurance companies
thatifabeneficiarydiesduringthefirstsevenyearsofthebeneficiary’spensionplan,
alumpsumofnotyetpaidmonthlypensionsthatwouldbepaidtothebeneficiary
forsevenyearsifhe/shewerealivewillbepaidtosurvivorsorauthorizedpersons.
There are also other risks built into the products, e.g. the risk that a pension will
be paid to survivors or an authorized person for either one or two years after the
beneficiary’sdeath.Italsoanalysesaproductthatincludesageometricallygrowing
pensionannuity.
This work provides a perspective on the so-called effective premium, that
is, that part of the accumulated amount that is intended directly for permanent
monthly pensions. This effective premium is obtained by applying various risk
loadings and other loadings that the actuary must use to prudently determine the
amountofmonthlypension.TheaforementionedloadingsincludeKannistö’smodel
with selection, Kannistö’s model with selection and with 25% shock, market risk
ofaninterestrate,andexpenseloadings.
Theworkalsooffersprofitabilitytestingasamethodthatallowsforthechange
ofindividualinputparameterssothatthemonthlypermanentpensionisdetermined
correctlyandprudently.Italsoanalysesthesensitivityofchangesinindividualinput
vii
viii Preface
parametersto the level of profitability rates, such as the net present value of cash
flows,theprofitmarginandtheinternalrateofreturn.
It answers questions concerning the effectiveness of pension savings and old-
ageinsuranceandtakesapositiononvarioushypotheses.Thebasicquestionbeing:
“Howmanyyearswouldthebeneficiaryhavetoliveinordertobepaidtheamount
he/sheinvestedinsuchinsurance?”
ThefundamentaldataresourcesarethemortalitytablesoftheSlovakRepublic
processed using the Kannistö’s model with selection with respect to the specific
insurancepoolofclientsenteringintosuchaninsurancerelationship.
Itis veryimportantto talk aboutpensions.Notonlyaboutpensionsin general,
butespeciallyabouttheiramounts.Peopleusuallythinkthattheentireamountsaved
willbespentonmonthlypayments.However,financialliteracyshouldclarifythat
insurancecompaniesarefor-profitjointstockcompanies,and,inaddition,actuaries
needtocalculatepensionsprudentlyduetoincreasinglifeexpectancyanduncertain
interestrates,butalsoduetotheinsurancecompany’sinvestmentcostsandrunning
an insurance company. It is essential that future pensioners know what the term
effectivepremiummeans,whatisitsamountinrelationtothemoneyinvested,and
atleastwhatyieldcurveshouldbeusedincalculatingpensions.Thevastmajority
ofpeoplehavealmostnoknowledgeoffinancialandactuarialmathematics,andthis
mustchange.
BanskáBystrica,Slovakia JanaŠpirková
BanskáBystrica,Slovakia IgorKollár
Bratislava,Slovakia GáborSzu˝cs
Prague,CzechRepublic PavelZimmermann
July2022
Acknowledgements
Our thanks go to the Slovak Scientific Grant Agency under projects VEGA no.
1/0150/21, VEGA no. 1/0760/22, and to the Slovak Research and Development
AgencyunderprojectAPVV-20-0311.
We express our great gratitude to Mr. Martin Janecˇek from Tools4F for his
willingnesstodiscussandconsultonourwork.
ix
Contents
PartI IntroductionandPreliminaries
1 Preliminaries .................................................................. 5
1.1 BasicProbabilities ....................................................... 5
1.2 YieldCurves.............................................................. 6
1.3 AdministrativeandCollectionCosts.................................... 7
References...................................................................... 7
2 ModelofLongevity ........................................................... 9
2.1 ModelofAdultAgeLongevity.......................................... 9
2.2 Lee-CarterModelImplementation...................................... 10
2.3 Old-AgeLongevity ...................................................... 11
2.4 ImplementationofKannistö’sModel................................... 13
2.5 SelectionFactor.......................................................... 14
References...................................................................... 15
PartII SelectedPayoutProducts
3 SelectedLifeOld-AgePensionProducts ................................... 19
3.1 SelectedProducts ........................................................ 19
3.2 BasicNotations........................................................... 21
3.3 Product1 ................................................................. 21
3.4 Product2 ................................................................. 23
3.5 Product3 ................................................................. 25
3.6 Product4 ................................................................. 27
3.7 Product5 ................................................................. 29
References...................................................................... 30
4 AnalysisoftheProducts...................................................... 31
4.1 LoadingAssessment..................................................... 31
4.1.1 CalculationofLoadingComponents ........................... 33
4.1.2 EffectivePremium............................................... 34
4.1.3 MonthlyPensionwithRespecttotheRetirementAge......... 36
xi
