保險精算課程設(shè)計

臉加端心SouthwestUniversityofScienceandTechnology保險精算課程設(shè)計學(xué)院****學(xué)院專業(yè)***************班級*****班姓名*****學(xué)號*********指導(dǎo)教師*****二零一*年*月摘要生命表是用一個小表來表達諸多相關(guān)概率的方法，表中各項內(nèi)容均為年齡的函數(shù)。生命表可以反映出任意年齡的人在任何期限內(nèi)的生存概率、死亡概率等信息。生命表主要是由以下數(shù)據(jù)構(gòu)成：生存率、生存人數(shù)、死亡人數(shù)等等。壽險精算是精算學(xué)的核心內(nèi)容，揭示了對未來的不確定的財務(wù)事件提供數(shù)量化意見的精算方法。它以概率統(tǒng)計為基礎(chǔ)的生命模型研究人的死亡和疾病的不確定性，以復(fù)利函數(shù)研究資產(chǎn)的時間價值對未來事件進行量化，并將生命模型和復(fù)利函數(shù)結(jié)合，形成了一套全面量化未來不確定的財務(wù)事件的方法。它不僅在保險、金融等領(lǐng)域發(fā)揮著巨大的作用，對于可以通過類似方法描述不確定性和時間價值函數(shù)的事務(wù)，也是一個重要工具，如可以參照死亡保險的量化模型分析大型設(shè)備壽命等。關(guān)鍵詞：生命表；壽險精算；目錄第1節(jié)緒論研究背景意義第2節(jié)主要內(nèi)容生命表建立的理論依據(jù)生存分布生命表常見保險產(chǎn)品保費的厘定案例簡介計算第3節(jié)結(jié)論參考文獻附錄生命表1第一節(jié)緒論研究背景隨著世界經(jīng)濟金融化和金融自由化進程的加快，金融創(chuàng)新加速了金融保險的替代性融合，推動了金融保險資源的全球性流動與市場整合，加快了世界保險業(yè)以結(jié)構(gòu)優(yōu)化升級為核心的一體化趨勢，金融混業(yè)經(jīng)營已成趨勢。金融保險資源的跨國流動及其形成的世界保險關(guān)系更加復(fù)雜，對投資精算師、理財精算師、保險精算師等人才需求缺口更加巨大，而培養(yǎng)這方面專門人才的學(xué)科一一保險精算便應(yīng)運而生了。生命表是壽險定價的重要工具，生命表上所記載的死亡率、生存率是決定人壽保險費的重要依據(jù)。生命表的建立可追溯到公元1661年，英國就有了歷史上最早的死亡機率統(tǒng)計表。到1693年，世界上第一張生命表是英國天文學(xué)家哈萊制定了《哈萊死亡表》，它奠定了近代人壽保險費計算的基礎(chǔ)，到1700年，英國又建立了"均衡保費法"，使投保人每年繳費是同一金額。我國在1929-1931間，金陵大學(xué)的肖富德編制了中國第一張生命表，稱為“農(nóng)民生命表"。1982年第2次全國人口普查得到了完整的生命表資料，直到1995年末才制定出了中國人壽保險業(yè)第一張經(jīng)驗生命表。意義生命表是壽險定價的重要工具，生命表上所記載的死亡率、生存率是決定人壽保險費的重要依據(jù)。生命表以年歲為綱，全面、

完整地反映了某一國家或地區(qū)一定人群從誕生直至全部死亡的生死規(guī)律。生命表的編制為經(jīng)營人壽保險業(yè)務(wù)奠定了科學(xué)的數(shù)理基礎(chǔ)，是計算人身保險的保險費、責(zé)任準備金、退保金的主要依據(jù)。第二節(jié)主要內(nèi)容生命表建立的理論依據(jù)生存分布一、生存函數(shù)定義：s⑺=Pr(XN)概率意義：新生兒能活到工的概率與分布函數(shù)的關(guān)系：與密度函數(shù)的關(guān)系：，⑴二一式(工)二、剩余壽命1、定義：已經(jīng)活到x歲的人(簡記⑴)，還能繼續(xù)存活的時間，稱為剩余壽命，記作T(x)。2、剩余壽命的分布函數(shù)它的概率意義為：⑶將在未來的￡年內(nèi)去世的概率，簡記用&-&-<X<彳+1X>x)=式萬-s(x+￡)3、剩余壽命的生存函數(shù)：學(xué)口)=「皿幻＞。"叱＞工+2它的概率意義為：⑶能活過工十￡歲的概率，簡記1%s(x+f)

也)4、整值剩余壽命(1)定義：⑶未來存活的完整年數(shù)，簡記K(x)K㈤=上,左式了)心+1#=0。(2)概率函數(shù)：在因艾)=2=Pr(A<TM<t+l)=?鼻—必=rPx-wPt=鈔界%"=咨2.1.2生命表對于具體含義為人的壽命(或未來生命時間長度)的隨機變量而言，想要找到一個簡單的函數(shù)作為其分布函數(shù)(或密度函數(shù))幾乎是不可能的。需要利用其它描述隨機變量的方法，來描述我們所要研究的特定的隨機變量X和T(x)。生命表就是一種行之有效的描述隨機變量X和T(x)近似特征的方法。生命表函數(shù)與生存函數(shù)。1、生命表的定義根據(jù)已往一定時期內(nèi)各種年齡的死亡統(tǒng)計資料編制成的由每個年齡死亡率所組成的匯總表.2、原理在大數(shù)定理的基礎(chǔ)上，用觀察數(shù)據(jù)計算各年齡人群的生存概率。(用頻數(shù)估計頻率)3、常用符號(1)新生生命組個體數(shù)：％(2)年齡：工(3)極限年齡：起%個新生生命能生存到年齡X的期望個數(shù)：1兔)%個新生生命中在年齡X與工+月之間死亡的期望個數(shù):a特別，當"二1時，記作4=L科器=k照&丸二。一乙+i二乙心%個新生生命在年齡［與區(qū)間共存活年數(shù)：工4］砂%個新生生命中能活到年齡X的個體的剩余壽命總數(shù)：7；%=］，力(8)生存率和死亡率分別為Px,qx,有PxSx(1),qx=Fx⑴Px-qx.(9)隨機變量Tx的數(shù)學(xué)期望值叫做x歲的人的完全生命期望0值，又叫平均余命，用ex表示，即0WBW-xex=E［Tx】ytMfx^dt］tPx—"dt.

(10)簡單(或整數(shù)化)生命期望值，記為ex,有e-P[Tx-2bkPxkJk4.(11)生命表中最重要也是最基本的數(shù)據(jù)是lx,它表示l0個個體中在X歲時仍活著的個體數(shù)目，lx=l0、Px,lx=l0、S(X)；(12)Lx表示在這個群體中實際活到了x歲的人；(13)用dx來表示未來一年死亡人數(shù)dx=lxXqx.(14)常用的基數(shù)公式：x1Cx=vdxxDx=vlxoOMxCxkk：0CONx八Dxkk0QORx八Mxkk=0QOSx八Nxkk=0(15)常用基數(shù)的推導(dǎo)：A1-二A1-二Mx:nDx八_Dxn八_DxnMxMAx：nDxxqn''ax:nNx-NxnDx在已有數(shù)據(jù)的基礎(chǔ)上完善生命表（見附件EXCELS）2.2常見保險產(chǎn)品保費的厘定案例簡介30歲男性，投保國壽鴻壽年金保險，保額10萬元保費支出：每年保費8000元，交20年保單利益：養(yǎng)老保障：55歲開始每年領(lǐng)取5000元，至79歲止?jié)M期返還：80歲返還滿期金20萬元身故保障：80歲前身故領(lǐng)取20萬元身故保險金計算計算前提相關(guān)假設(shè)：假定利率i=0.06。貨幣計量單位為人民幣元。因為30歲投保，從55歲起每年領(lǐng)取養(yǎng)老金，到79歲止,共領(lǐng)取25次。這是25年延期25年定期期首付生存年金。折合為30歲時的精算現(xiàn)值為：

5000(N3025-N30.25.255000(N3025-N30.25.25D30_(43031.29-2184.81)5000―16542.86=12345.65如前文的描述這是一款生死和險（養(yǎng)老保險），與定期生存年金的組合，因此還要計算出養(yǎng)老保險的精算現(xiàn)值：_1200000A30面_M30-M3050D80D30_200000(1695.3711-246.3230369.99)―16542.86=21991.82建繳凈保費應(yīng)為：P=12345.65+21991.82=34337.47因為投保人選擇的是20年繳清，期首支付，下面計算均衡凈保費：那么繳納保費的30歲時的折現(xiàn)值是:第三節(jié)結(jié)論壽險精算是精算學(xué)的核心內(nèi)容，它依據(jù)經(jīng)濟學(xué)的基本原理和知識，利用現(xiàn)代數(shù)學(xué)方法，對各種保險經(jīng)濟活動未來的財務(wù)風(fēng)險進行分析、估價和管理的一門綜合性的應(yīng)用科學(xué)。如研究保險事故的出險規(guī)律、保險事故損失額的分布規(guī)律、保險人承擔(dān)風(fēng)險的平均損失及其分布規(guī)律、保險費率和責(zé)任準備金、保險公司償付能力等保險具體問題。是以概率論和數(shù)理統(tǒng)計為基礎(chǔ)，是應(yīng)用數(shù)學(xué)、統(tǒng)計學(xué)、金融理論、保險理論以及人口學(xué)等學(xué)科的知識和原理，去解決商業(yè)保險與各種社會保險業(yè)務(wù)中需要精確計算的項目。從保險精算自身理論看，保險精算除應(yīng)用數(shù)理科學(xué)外，還緊密聯(lián)系金融、保險等知識和理論，從而體現(xiàn)出新型、綜合交叉和邊緣性質(zhì)。而生命表的出現(xiàn)為保險公司進行更加合理的保險產(chǎn)品的價格厘定提供了理論方法，同時方便著人們的生活和社會的完善。參考文獻【1】壽險精算原理；李曉林；中國財政經(jīng)濟出版社?！?】國外保險精算學(xué)科發(fā)展、人才培養(yǎng)及啟示；蒲成毅；重慶工商大學(xué)

附錄生命表1年齡(x)死亡率qx生存人數(shù)lX死亡人數(shù)dx生存人年數(shù)LxTx期望壽命ex00.00072210000007229996397670992576.7110.0006039992786039989777571028675.7620.0004999986754989984267471130974.8130.0004169981774159979697371288373.8540.0003589977623579975837271491372.8850.0003239974053229972447171733071.9060.0003099970823089969287072008670.9370.0003029967743019966246972315869.9580.0003019964733009963236872653468.9790.0003039961733029960236773021167.99100.0003059958723049957206673418867.01110.0003089955683079954156573846866.03120.0003139952613129951056474305465.05130.0003249949503229947896374794864.07140.0003439946273419944576275316063.09150.0003729942863709941016175870362.11160.0004129939164099937126076460261.14170.0004599935074569932795977089060.16180.0005129930515089927975877761159.19190.0005649925425609922625778481558.22200.0006129919836079916795679255257.25210.0006549913756489910515580087356.29220.0006899907276839903865480982255.32230.0007169900447099896905381943654.36240.0007399893367319889705282974653.40250.0007579886047489882305184077652.44260.0007759878567669874735085254651.48270.0007949870917849866994986507350.52280.0008189863078079859034887837449.56290.0008499855008379850824789247148.60300.0008899846638759842264690738947.64310.0009379837889229833274592316346.68320.0009939828669769823784493983645.72330.00105598189010369813724395745844.77340.00112198085411009803044297608643.81350.00119497975511709791704199578142.86360.00127597858512489779614101661241.91370.00136797733713369766694003865140.97380.00147297600114379752833906198140.02

390.00158997456415499737903808669939.08400.00171597301616699721823711290838.14410.001845971347179297045136140727137.21420.00197896955519189685963517027636.27430.002113967637204596661534201680|35.35440.002255965593217796450433235065|34.42450.00241396341523259622533227056133.50460.002595961091249495984431308308132.58470.002805958596268995725230348464P31.66480.00304295590829089544542939121230.75490.00329995300031449514282843675929.84500.00357949856339194816027485331「28.94510.003847946465364194464426537171128.04520.00413294282438969408762559252627.14530.00443493892841639368462465165026.26540.004778934765446693253223714804125.37550.00520393029948409278782278227224.49560.00574492545853169228002185439423.61570.006427920142591491718520931594122.75580.0072691422966379109102001440821.89590.008229907591746990385719103498121.05600.009313900123838389593118199641120.22610.01064489174094928869941730371019.40620.0118058822481041587704116416716118.61630.01309287183311414866126155396751—17.82640.014519860419124928541731467354917.05650.0161847927136528411011381937616.30660.017898834275149328268091297827515.56670.0198958193431630181119312151466114.83680.022113803042177587941641134027314.12690.0245757852851929877563610546109P13.43700.027307765986209177555289770474112.76710.0303474507022605733767901494612.10720.033703722464243497102908281179r11.46730.037434698115261336850487570889110.84740.04146367198227862658051688584110.25750.04591864411929577629331622779019.67760.05083961454331243598921559845919.11770.0562745833003282556688849995388.57780.0622735504753428053333544326508.05790.0688915161963556149841538993157.55800.0761874806343661846232534009007.08810.0842244440163739742531829385746.62820.0930714066193784438769725132566.18830.10283687753791034982021255595.76840.1134893308653755031209017757395.37850.1252212933153672927495114636494.99860.138082565863542923887111886984.63870.152157221157336512043319498274.29880.167543187506314151717997454953.98890.184333156091287731417045736973.68900.202621127318257971144194319923.39910.222510152122588902273175733.13920.2440597893219264693002273462.88930.2673835966815954516911580462.65940.2925444371412788373201063552.43950.31960430926988425984690352.23960.34860621042733517374430512.05970.37957213706520311105256771.87980.412495850435086750145721.71990.44733449962235387978221.571000.4840127611336209339431.431010.5223971425744105318511.301020.5623176803834897981.171030.6035392981802083091.041040.6457711876801010.851051424221210.50年齡(x)生存人數(shù)lxDxNxCxMxRxSx010000001000000.0034417381.49704.39160551.6711263066.35949326920.4911999278974905.3733417381.49573.53159847.2811102514.68914909539.002998675950553.6632442476.12462.76159273.7510942667.40881492157.523998177926906.6631491922.46376.19158810.9910783393.65849049681.404997762903923.0030565015.80315.71158434.8110624582.66817557758.935997405881560.3829661092.81277.80158119.0910466147.85786992743.136997082859781.1128779532.42259.19157841.2910308028.76757331650.337996774838551.6527919751.31247.07157582.1010150187.46728552117.908996473817852.1027081199.67240.17157335.049992605.36700632366.599996173797664.3226263347.57235.80157094.879835270.32673551166.92110995872777973.3025465683.25231.49156859.079678175.46647287819.35

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