[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-seo-399162-105":59,"doc-detail-399162-en":131},{"code":4,"msg":5,"data":6},0,"success",[7,13,18,23,28,33,38,43,48,51,55],{"id":8,"doc_module":4,"doc_module_name":9,"category_name":10,"show_sort_weight":11,"slug":12},1,"Document","Story & Novel",90,"story-novel",{"id":14,"doc_module":4,"doc_module_name":9,"category_name":15,"show_sort_weight":16,"slug":17},2,"Literature",80,"literature",{"id":19,"doc_module":4,"doc_module_name":9,"category_name":20,"show_sort_weight":21,"slug":22},4,"Exam",70,"exam",{"id":24,"doc_module":4,"doc_module_name":9,"category_name":25,"show_sort_weight":26,"slug":27},5,"Comic",60,"comic",{"id":29,"doc_module":4,"doc_module_name":9,"category_name":30,"show_sort_weight":31,"slug":32},6,"Technology",50,"technology",{"id":34,"doc_module":4,"doc_module_name":9,"category_name":35,"show_sort_weight":36,"slug":37},7,"Healthcare",40,"healthcare",{"id":39,"doc_module":4,"doc_module_name":9,"category_name":40,"show_sort_weight":41,"slug":42},8,"Research & Report",30,"research-report",{"id":44,"doc_module":4,"doc_module_name":9,"category_name":45,"show_sort_weight":46,"slug":47},9,"Religion & Spirituality",20,"religion-spirituality",{"id":46,"doc_module":4,"doc_module_name":9,"category_name":49,"show_sort_weight":46,"slug":50},"World Cup","world-cup",{"id":52,"doc_module":4,"doc_module_name":9,"category_name":53,"show_sort_weight":52,"slug":54},10,"Lifestyle","lifestyle",{"id":56,"doc_module":4,"doc_module_name":9,"category_name":57,"show_sort_weight":24,"slug":58},19,"General","general",{"code":4,"msg":60,"data":61},"ok",{"site_id":62,"language":63,"slug":64,"title":65,"keywords":66,"description":67,"schema_data":68,"social_meta":124,"head_meta":126,"extra_data":128,"updated_unix":130},105,"en","mas200-actuarial-statistics-qmw-mock-exam-paper-model-solutions","MAS200 Actuarial Statistics QMW - MOCK EXAM PAPER: MODEL SOLUTIONS","","MAS200 Actuarial Statistics QMW provides model solutions to a mock exam, covering discount rates, annuity-due valuation, and equations of value for repayment and benefit problems. The solutions apply key actuarial constructs such as nominal and effective forces of discount, conversion between i and v, survival and mortality functions, and De Moivre-style mortality assumptions. Later questions address survival interpolation, curtate expectation of life, moments under distributional assumptions, and present value calculations for annuities and benefit payments.",{"@graph":69,"@context":123},[70,84,106],{"@type":71,"itemListElement":72},"BreadcrumbList",[73,77,79,82],{"item":74,"name":75,"@type":76,"position":8},"https://docshare.wps.com","Home","ListItem",{"item":78,"name":9,"@type":76,"position":14},"https://docshare.wps.com/document/",{"item":80,"name":20,"@type":76,"position":81},"https://docshare.wps.com/document/exam/",3,{"item":83,"name":65,"@type":76,"position":19},"https://docshare.wps.com/document/mas200-actuarial-statistics-qmw-mock-exam-paper-model-solutions/399162/",{"url":83,"name":65,"@type":85,"image":86,"author":91,"headline":65,"publisher":94,"fileFormat":97,"inLanguage":63,"description":67,"dateModified":98,"datePublished":99,"encodingFormat":97,"isAccessibleForFree":100,"interactionStatistic":101},"DigitalDocument",{"url":87,"@type":88,"width":89,"height":90},"https://docshare.wps.com/thumbnails/mas200-actuarial-statistics-qmw-mock-exam-paper-model-solutions/399162.png","ImageObject",300,407,{"name":92,"@type":93},"Patrick","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-10-31","2026-09-27",true,{"@type":102,"interactionType":103,"userInteractionCount":105},"InteractionCounter",{"@type":104},"ViewAction",13,{"@type":107,"mainEntity":108},"FAQPage",[109,115,119],{"name":110,"@type":111,"acceptedAnswer":112},"How is the effective annual rate of discount computed from a nominal quarterly convertible rate?","Question",{"text":113,"@type":114},"The solution uses the relationship between d and d(4) for quarterly conversion, then derives the effective annual discount rate as 0.0682, reported as 6.82% to two decimals.","Answer",{"name":116,"@type":111,"acceptedAnswer":117},"How is the monthly repayment for the PlayStation loan determined in Question 1(c)?",{"text":118,"@type":114},"It models the payments as an annuity-due over 26 monthly units, sets the equation of value using v and the appropriate per-month discount factor, and solves for P, giving $9.25 to two decimals.",{"name":120,"@type":111,"acceptedAnswer":121},"What is the curtate expectation of life as defined in Question 2(d)?",{"text":122,"@type":114},"Curtate expectation is defined as ex = E(K(x)), where ex is expressed using probabilities of K(x) and shown to simplify to a sum involving kpx terms.","https://schema.org",{"og:url":83,"og:type":125,"og:title":65,"og:site_name":95,"og:description":67},"article",{"robots":127,"canonical":83},"index,follow",{"doc_id":129,"site_id":62},399162,1790554920,{"code":4,"msg":5,"data":132},{"doc_id":129,"user_id":133,"nickname":92,"user_avatar":134,"doc_module":4,"category_id":19,"category_name":20,"doc_title":65,"doc_description":67,"doc_content":135,"file_id":136,"file_url":137,"file_type":138,"file_size":139,"view_count":105,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":29,"language":140,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":141,"faqs":142,"seo_title":143,"seo_description":67,"update_tm":144,"read_time":145,"preview_page_count":4},549758146520,"https://ap-avatar.wpscdn.com/avatar/80002397d8c0411e94?_k=1775819394049821470","MAS200 Actuarial Statistics QMW  \nMOCK EXAM PAPER: MODEL SOLUTIONS  \nQuestion 1 .  \n(a) d (4) = 0:07 (nominal rate of discount convertible quarterly) .  \nd = 1􀀀 􀀔1 􀀀 d(4)4􀀕 4 = 1􀀀 􀀔1 􀀀 0~~ ~~:074􀀕 4 = 0:0682 (e􀀋ective annual rate of discount)  \nAnswer: 6.82%(to 2 decimal places) .  \n(b) The amount to be paid is  \nexp Z01~~1~~2 􀀎 dt! 􀀀 1 = e ~~ ~~1~~􀀎~~2 􀀀 1 = e0:01 􀀀 1 = 0:01 Answer: 0.01 (to 2 decimal places) .  \n(c) The following problem replaces part (c) of Question 1 on the original sample examination paper:  \nA Sony playstation with a cash price of $ 189.99 can be purchased for 26 equal monthly repayments with the 􀀌rst repayment being due at the moment of purchase. Find the amount of each monthly repayment if the APR charged is 26 .7% .  \nSet: (unit of time) = (one month); the interest rate per unit time is then i􀀃 = ~~i ~~(1122) . Correspondingly,  \nv 􀀃 = 1~~ ~~~~ ~~i􀀃 = 1~~ ~~+1~~i ~~(1122) = i)~~  ~~1~~1~~2~~ ~~ ; i being the APR (i=0.267) Let P be the monthly repayment. The loan of $189 .99 is to be repaid by an annuity-due of $P payable at unit time intervals over 26 units of time.  \nHence, the equation of value:  \nP 26 j@v =v 􀀃 = 189:99  \nFrom this,  \n 189:99   189:99  P = 26~~ ~~j@v =v 􀀃 = ~~1~~26v j @v =v 􀀃 = 9:25  \nAnswer: $9.25 (to 2 decimal places) .  \n(d) Using the formulae appended to the end of the examination paper: a􀀁a􀀁􀀁􀀁~~ ~~n(pnjj)~~ ~~ = ~~ 1~~p~~ ~~v􀀔(~~ ~~1111􀀀􀀀􀀀􀀀vvvvn~~1p~~n)~~ ~~􀀕~~ ~~ = \"p(1~~ ~~~~ ~~v~~ ~~1p~~ ~~) \\# 􀀔 ~~ ~~1~~ ~~~~ ~~v 􀀕  \n= ~~ ~~d~~ ~~i(p)~~ ~~ 􀀒 as p(1 􀀀 v ~~1~~p ) = d (p) and ~~ ~~1~~ ~~~~ ~~v = i􀀓  \n(e) Set: (unit of time)=(one quarter); the interest rate per unit time is then i􀀃 = ~~i ~~(44) = ~~0:12~~4 = 0:03 . Correspondingly, v 􀀃 = 1i􀀃 = 1:~~1~~03 .  \nLet P be the initial amount of the quarterly repayment. The loan is to be repaid by a combination of an immediate annuity and a deferred annuity. The immediate annuity pays $P at unit time intervals for 40 time units, the 􀀌rst payment being due at time t = 1 . The deferred annuity pays $100 at unit time intervals for 20 time units, the 􀀌rst payment being due at time t = 21 . Hence the equation of value reads  \nPa40 j@v =v 􀀃 + (v􀀃)20 􀀂 (100a20 j@v =v 􀀃 ) = 10000  \nTherefore  \n10000 􀀀 (v􀀃 )20 100 v 􀀃 [1􀀀(v 􀀃 )20]  \nP = ~~  ~~v~~ ~~􀀃~~ ~~[1􀀀(v~~ ~~􀀃~~ ~~)40]~~ ~~1􀀀v~~ ~~􀀃~~ ~~ = 396:99  \n1􀀀v 􀀃  \nAnswer: $396.99 (to 2 decimal places) .  \nQuestion 2 .  \n(a) 􀀖 (x) = 􀀀 s~~ ~~(~~1~~x) ~~d~~dxs (x), where s (x) is the survival function.  \nThe force of mortality can attain values greater than 1, for example 􀀖(x) = 2 whens (x) = e􀀀2x.  \n(b) t qx is the probability (conditional) that a life aged x at the present time will die within next t years.  \nt qx = s~~ ~~(x)(sx()x+t)  \n􀀖 (x) = !~~ ~~x for De Moivre's Law of Mortality.  \n(c) s (x) = e􀀀􀀖x if X has an exponential distribution. Hence, tpx = s~~ ~~(sx(+x)t) = ~~e ~~􀀀e(x􀀖+xt) = e􀀀􀀖t .  \n(d)  \nP (K(x)=k) = P (k 􀀔T(x)\u003Ck+1)=P(k \u003CT(x)􀀔k+1)  \n= P (T(x) 􀀕k) 􀀀 P(T(x) 􀀕k+1)  \n= kpx 􀀀 k+1px:  \nThe curtate expectation of life is de􀀌ned as ex = E(K(x)) .  \n1 1 1 1 1  \nex = X k P (K(x) = k) =Xk kpx 􀀀X k k+1px =Xk kpx 􀀀X(k 􀀀 1) kpx  \nk=0 k=0 k=0 k=1 k=1  \n1  \n= X kpx:  \nk=1  \n(e) Follows from  \nkpx = s~~ ~~(xs(+x)k) = 􀀔 ~~ ~~s~~ ~~(xs(+x)~~ ~~1) 􀀕 􀀔 ~~ ~~s~~ ~~(x~~ ~~+s(1x~~ ~~~~ ~~k1)􀀀~~ ~~1)􀀕 = (px )(k􀀀1px+1) ; k 􀀕 1 : Namely,  \n1~~ ~~+exex+1 = P1~~1~~exx = px~~ ~~P1~~1~~ex1px = px~~ ~~P1~~0~~exp1x+1 = px  \n(f) The p.d.f. of the random variable Xj (x1 \u003C X \u003C x2 ) is P~~ ~~(xf1X(x\u003C)~~ ~~x~~ ~~2) when x 1 \u003C x \u003C x2 and zero otherwise, fX (x) being the p.d.f of X . Therefore  \nE (Xj (x1 \u003C X \u003C x2 )) = Zxx12 P~~ ~~(x1xXX(x\u003C)~~ ~~x2~~ ~~) dx = Zxx12 sx(x(~~ ~~)ss(x(x)2)) dx  \n= Zxx12 s~~ ~~x(xs1()x􀀀)􀀖s(x(x)2~~ ~~) dx;  \nwhere we have used  \n d   \nfX (x) = 􀀀 dx s (x) = s(x)􀀖 (x) and P (x1 \u003C X \u003C x2 ) = s(x1 ) 􀀀 s(x2 ) :  \nQuestion 3 .  \n(a) 1) 10p50 = ~~l 60~~l~~ ~~50 = ~~78924~~90085 = 0:876 (to 3 s.d.) .  \n2) 5j5q60 = ~~l 65~~l0~~l 70~~ = ~~6849~~7~~0~~8~~5~~2~~4~~4~~806~~ = 0:173 (to 3 s.d.) .  \n3) 1000","cbCaijOnbL7xQjwI","https://ap.wps.com/l/cbCaijOnbL7xQjwI","pdf",136796,"English","# Question 1\n## Discount rate, annuities, and equation of value\n# Question 2\n## Force of mortality, survival probabilities, and life expectancy\n# Question 3\n## Select periods, death interpolation, and commutation functions\n# Question 4\n## Present value of benefits and premiums","[{\"question\":\"How is the effective annual rate of discount computed from a nominal quarterly convertible rate?\",\"answer\":\"The solution uses the relationship between d and d(4) for quarterly conversion, then derives the effective annual discount rate as 0.0682, reported as 6.82% to two decimals.\"},{\"question\":\"How is the monthly repayment for the PlayStation loan determined in Question 1(c)?\",\"answer\":\"It models the payments as an annuity-due over 26 monthly units, sets the equation of value using v and the appropriate per-month discount factor, and solves for P, giving $9.25 to two decimals.\"},{\"question\":\"What is the curtate expectation of life as defined in Question 2(d)?\",\"answer\":\"Curtate expectation is defined as ex = E(K(x)), where ex is expressed using probabilities of K(x) and shown to simplify to a sum involving kpx terms.\"}]","MAS200 Actuarial Statistics QMW - MOCK EXAM PAPER: MODEL SOLUTIONS | PDF",1790494072,15]