[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-seo-398754-105":59,"doc-detail-398754-en":130},{"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":123,"head_meta":125,"extra_data":127,"updated_unix":129},105,"en","interest-theory-formulas-exam-fm2","Interest Theory Formulas - Exam FM/2","","Interest theory formula sheet for SOA Exam FM/2 and CAS Exam 2, providing a free study reference for students. Covers core time value of money relationships (FV, PV, accumulation functions, discounting), effective interest and discount rates with nominal rates and compounding/discounting frequencies, and equivalent rate measures. Includes annuity, perpetuity, and continuous/increasing/decreasing/deferred forms, plus geometric progression payments and key identities. Also addresses varying interest and practical loan amortization notation and setup.",{"@graph":69,"@context":122},[70,84,105],{"@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/interest-theory-formulas-exam-fm2/398754/",{"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/interest-theory-formulas-exam-fm2/398754.png","ImageObject",300,407,{"name":92,"@type":93},"A glass of water","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-10-28","2026-09-27",true,{"@type":102,"interactionType":103,"userInteractionCount":52},"InteractionCounter",{"@type":104},"ViewAction",{"@type":106,"mainEntity":107},"FAQPage",[108,114,118],{"name":109,"@type":110,"acceptedAnswer":111},"What are the fundamental relationships between FV, PV, and interest rate i?","Question",{"text":112,"@type":113},"The sheet states FV and PV relations using (1 + i) raised to n, along with discounting through v = (1 + i)^{-1}. It also provides accumulation and discount identities tied to i, d, and δ.","Answer",{"name":115,"@type":110,"acceptedAnswer":116},"How are nominal interest and discount rates converted to effective rates?",{"text":117,"@type":113},"It gives equivalence formulas showing how a nominal rate convertible m-thly yields an effective interest rate, and similarly how a nominal discount rate convertible p-thly yields an effective discount rate. It also links effective annual rates to accumulation/discount factors.",{"name":119,"@type":110,"acceptedAnswer":120},"Which annuity types are included and what is the difference between immediate and due?",{"text":121,"@type":113},"The document covers annuity immediate (payments at the end of the period) and annuity due (payments at the beginning). It also includes formulas and identities for both, plus continuous annuities and increasing/decreasing variants.","https://schema.org",{"og:url":83,"og:type":124,"og:title":65,"og:site_name":95,"og:description":67},"article",{"robots":126,"canonical":83},"index,follow",{"doc_id":128,"site_id":62},398754,1790551762,{"code":4,"msg":5,"data":131},{"doc_id":128,"user_id":132,"nickname":92,"user_avatar":133,"doc_module":4,"category_id":19,"category_name":20,"doc_title":65,"doc_description":67,"doc_content":134,"file_id":135,"file_url":136,"file_type":137,"file_size":138,"view_count":52,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":139,"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},962090760832,"https://ap-avatar.wpscdn.com/davatar_155a257f0dc6eb9ab79c44ca47cae57d","Exam FM/2 Interest Theory Formulas  \nby (/iropracy  \nThis is a collaboration of formulas for the interest theory section of the SOA Exam FM / CAS Exam 2. This study sheet is a free non-copyrighted document for students taking Exam FM/2 .  \nThe author of this study sheet is using some notation that is unique so that no designation will repeat. Each designation has only one meaning throughout the sheet.  \nFundamentals of Interest Theory and Time Value of Money  \nFV = PV (1 + i) n PV = ~~ (~~1Vi~~)~~n  \n\n| id =  (1 + i)\u003Cbr>1 v =  (1 + i) | d = 1 − v\u003Cbr>v = 1 − d | i − d = id\u003Cbr>d = iv |\n| --- | --- | --- |\n| a (t ) ≡ The amount an initial investment of 1 grows to by time t\u003Cbr>A(t ) ≡ The amount an initial investment of A(0) grows to by time ta (t ) = (1 + i)t = et⋅ln (1+i) A(t ) = A(0) (1 + i)t = A(0) et⋅ln(1+i)δ = ln(1 + i) a (t ) = eδ⋅t\u003Cbr>v n = (1 + i)−n = e −δn\u003Cbr>δ (t ) = ~~ ~~aa′~~(~~~~)~~) e∫t0δ (u ) du = a (t ) A(0) e ∫t0δ (u ) du = A(t ) |  |  |\n| Effective interest rate with nominal rate i (m ) convertible m-thly\u003Cbr>⎛ i (m ) ⎞ m\u003Cbr>i = ⎜⎝ 1 + ~~ ~~m~~ ~~⎠⎟ −1 Effective discount rate with nominal rate d (p ) convertible p-thly\u003Cbr>⎛ d (p ) ⎞p\u003Cbr>1 − d = ⎜1 − ~~ ~~ ⎟\u003Cbr>⎝ p ⎠ Nominal Rate Equivalence\u003Cbr>1 + i = eδ = 1v = 1d = ⎜1 + ~~ ~~im(m~~ ~~)~~ ~~⎟m = ⎜1 − ~~ ~~dp(p~~ ~~)~~ ~~⎟ − p |  |  |\n\nEffective annual rate it during the t-th year is given by:  \n amount earned   a (t ) − a(t −1)  A(t ) − A(t −1) it = beginning amount = a (t −1) = A(t −1)  \nNote that the t-th year is given by the time period [t −1, t ] Therefore, the interest earned during the t-th year is given by:  \nA(t −1) ⋅i = A(t ) − A(t −1)  \n\n| For equivalent measures of interest we have the following relationship:\u003Cbr>d \u003C d(2) \u003C d(3) \u003C \" \u003C δ \u003C \" \u003C i(3) \u003C i(2) \u003C i |  |\n| --- | --- |\n| Annuities\u003Cbr>Annuity Immediate— payments are made at the end of the period Annuity Due— payments are made at the beginning of the period |  |\n| Annuity Immediate\u003Cbr>2 n  1 − v n a ~~ ~~ = v + v + \" + v =\u003Cbr>n|i i | s   = (1 + i)n−1 + (1 + i)n−2 +\"+1 =  (1 + i)n −1\u003Cbr>n|i i |\n| n\u003Cbr>a ~~ ~~ = v ⋅ s ~~ ~~\u003Cbr>n|i n|i | n\u003Cbr>sn~~ ~~|i = (1 + i) ⋅ an~~ ~~|i |\n| Annuity Due\u003Cbr>􀀅 􀀅 n−1 1 − v n\u003Cbr>a   = 1 + v + \" + v = ~~ ~~\u003Cbr>n|i d | 􀀅   = (1+ i)n + (1+ i)n−1 +\"+ (1+ i) =  (1+ i) n −1 n| i d |\n| 􀀅 􀀅 n 􀀅 􀀅\u003Cbr>a ~~ ~~ = v ⋅ s ~~ ~~\u003Cbr>n|i n|i | 􀀅n~~ ~~|i = (1 + i )n ⋅ a􀀅 􀀅 n~~ ~~|i |\n| Identities for Annuity Immediate and Annuity Due\u003Cbr>a􀀅 􀀅 n~~ ~~| = ~~ ~~id~~ ~~ an~~ ~~| = (1 + i) an~~ ~~| 􀀅n~~ ~~| = id~~ ~~ sn~~ ~~| = (1 + i) sn~~ ~~| a􀀅 􀀅 n~~ ~~| = 1 + an−1| 􀀅n~~ ~~| = sn+1| −1 |  |\n\nPerpetuity  \na∞~~ ~~|i =  an~~ ~~|i = v + v 2 + v 3 + \" = 1i a􀀅 􀀅 ∞~~ ~~|i =  a􀀅 􀀅 n~~ ~~|i = 1d  \n\n| Continuous Annuities\u003Cbr>  1 − v n ian |i =  δ  = δ  an |i\u003Cbr>PV =  e − ∫t0δ (u ) du p (t ) dt 0 |   (1 + i)n −1 isn |i =  δ  = δ  sn |i |  |  | nan | i = ∫vt dt\u003Cbr>0\u003Cbr>where p (t ) = payment function |\n| --- | --- | --- | --- | --- |\n|  | n\u003Cbr>FV = ∫ 0 | e∫tn δ (u ) du p ( t ) dt |  |  |\n| Increasing Annuities— Payments are 1, 2,…, n\u003Cbr>(Ia )n~~ ~~|i =  a􀀅 􀀅 n~~ ~~| − nv n  (Ia􀀅􀀅)n~~ ~~|i =  i  (Ia )n~~ ~~|i = (1 + i)(Ia )n~~ ~~|i =  a􀀅 􀀅 n~~ ~~| − nv n \u003Cbr>i d d\u003Cbr>n~~ ~~|i = 1 + n ~~ ~~n~~ ~~| i = 􀀅n~~ ~~| − n  (I􀀅)n~~ ~~|i =  i  (Is )n~~ ~~|i = (1 + i)n(Ia􀀅􀀅)n~~ ~~|i = 􀀅n~~ ~~| − n\u003Cbr>(Is ) ( i ) (Ia )\u003Cbr>i d d\u003Cbr>(Ia )∞~~ ~~|i = (Ia )n~~ ~~|i = ~~ ~~1di = 1i + ~~ ~~i12~~ ~~ (Ia􀀅􀀅)∞~~ ~~|i = (Ia􀀅􀀅)n~~ ~~|i = d12~~ ~~ |  |  |  |  |\n| Decreasing Annuities— Payments are n, n-1,…, 2, 1 |  |  |  |  |\n| (Da )~~ ~~ =  n − an |i \u003Cbr>n| i\u003Cbr>i\u003Cbr>( )  ( ) ( )   n(1 + i)n − s ~~ ~~ \u003Cbr>n n | i\u003Cbr>Ds n | i = 1 + i Da n | i =\u003Cbr>i |  |  | (Da􀀅􀀅)~~ ~~ =  i  (Da )~~ ~~ = (1 + i)(Da )~~ ~~ =  n − an | i \u003Cbr>n|i n|i n|i\u003Cbr>d d\u003Cbr>(D􀀅)n~~ ~~|i = (1 + i)n(Da􀀅􀀅)n~~ ~~|i |  |\n\nPresent Value of the annuity with terms X, X + Y, X + 2Y,…, X + (n −1)Y  \n⎛ a􀀅 􀀅   − nv n ⎞  \nX ⋅ a~~ |~~ + Y⎜ n |  ⎟ n i ⎜⎝ i ⎠⎟  \nPresent Value of the perpetuity with terms X, X + Y, X + 2Y,…  \nX Y  \n~~ ~~ + ~~ ~~  \ni i 2  \nAnnuities with Terms","cbCaiu2EDj3HueVQ","https://ap.wps.com/l/cbCaiu2EDj3HueVQ","pdf",153380,15,"English","# Fundamentals of Interest Theory and Time Value of Money\n## Accumulation, discounting, and key identities\n## Effective/nominal interest and discount rate equivalence\n# Annual Effective Rate and Interest Earned in a Year\n# Annuities\n## Annuity Immediate and Annuity Due\n## Identities for Annuity Immediate and Annuity Due\n## Continuous Annuities\n## Increasing and decreasing annuities\n## Annuities with terms in geometric progression\n# Perpetuity\n## Perpetuity and related formulas\n# Useful Identities and special relationships\n# Variable interest and changing payment/compounding frequency\n# Loan Repayment—Amortization","[{\"question\":\"What are the fundamental relationships between FV, PV, and interest rate i?\",\"answer\":\"The sheet states FV and PV relations using (1 + i) raised to n, along with discounting through v = (1 + i)^{-1}. It also provides accumulation and discount identities tied to i, d, and δ.\"},{\"question\":\"How are nominal interest and discount rates converted to effective rates?\",\"answer\":\"It gives equivalence formulas showing how a nominal rate convertible m-thly yields an effective interest rate, and similarly how a nominal discount rate convertible p-thly yields an effective discount rate. It also links effective annual rates to accumulation/discount factors.\"},{\"question\":\"Which annuity types are included and what is the difference between immediate and due?\",\"answer\":\"The document covers annuity immediate (payments at the end of the period) and annuity due (payments at the beginning). It also includes formulas and identities for both, plus continuous annuities and increasing/decreasing variants.\"}]","Interest Theory Formulas - Exam FM/2 | PDF",1790488684,38]