Deeper Understanding Fall 10 Manual for FM

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Deeper Understanding, Faster Calculation --Exam FM Insights & Shortcuts Part I: Theories of Interest 12th Edition by Yufeng Guo Fall 2010 http://actuary88.com This electronic book is intended for individual buyer use for the sole purpose of preparing for Exam FM. This book may NOT be resold or otherwise redistributed to others. No part of this publication may be reproduced for resale or multiple copy distribution without the express written permission of the author. 2010, 2011 By Yufeng Guo 1 Contents Part One: Theories of Interest Chapt e r 1 Exam-t aking and s t udy s t rat e gy ............................ 5 A tale of two Exam FM takers, Mr. Busy and Mr. Lazy................................................ 5 Truths about Exam FM................................................................................................... 9 How to study hard yet fail the exam miserably ............................................................ 11 Recommended study method........................................................................................ 13 How to build a 3 minute solution script........................................................................ 14 How to eliminate errors ................................................................................................ 30 Chapt e r 2 Ge t t ing s t art e d ............................................................. 48 Chapt e r 3 FM Fundament al.......................................................... 49 Time value of money .................................................................................................... 49 Principal ........................................................................................................................ 49 Interest rate.................................................................................................................... 49 Simple interest rate ....................................................................................................... 50 Compound interest rate................................................................................................. 50 Force of interest ............................................................................................................ 52 Nominal interest rate..................................................................................................... 57 APR............................................................................................................................... 58 Annual effective interest rate........................................................................................ 59 Continuous compounding............................................................................................. 60 Effective annual rate of discount .................................................................................. 60 Simple annual rate of discount...................................................................................... 63 Nominal annual rate of discount ................................................................................... 63 Relationship between i , , d , ( ) mi , ( ) md ....................................................................... 64 Future value .................................................................................................................. 75 Present value ................................................................................................................. 75 Convert interest rate to discount rate or vice versa....................................................... 77 PV of a stream of cash flows ........................................................................................ 82 Net Present Value ......................................................................................................... 82 Internal rate of return (IRR).......................................................................................... 82 Asset and its price ......................................................................................................... 84 Convert a cash flow from one point of time to another point of time .......................... 85 Collapse multiple cash flows into a single cash flow................................................... 86 Annuity collapsing n parallel cash flows into a single cash flow.............................. 89 Avoid the common pitfall ............................................................................................. 90 Perpetuity.................................................................................................................... 104 Annuity payable m-thly in advance ......................................................................... 107 Annuity payable m-thly in arrears ........................................................................... 109 Increasing annuity....................................................................................................... 121 Continuously increasing annuity................................................................................. 121 Decreasing annuity...................................................................................................... 122 2 Chapt e r 4 Calc ulat or t ips ............................................................ 137 Best calculators for Exam FM .................................................................................... 137 New features added in BA II Plus Professional.......................................................... 137 How to reset calculators to their best conditions for FM............................................ 137 Compound interest ...................................................................................................... 140 Annuity ....................................................................................................................... 144 Loan/bond amortization.............................................................................................. 153 Compare Cash Flow Worksheet with TVM Worksheet ............................................. 159 Increasing annuity....................................................................................................... 159 Comprehensive calculator exercise............................................................................. 167 Chapt e r 5 Ge ome t ric ally inc re as ing annuit y ..................... 175 Chapt e r 6 Re al vs . nominal int e re s t rat e ............................. 204 Chapt e r 7 Loan repayment and amort izat ion.................... 210 Chapt e r 8 Sinking fund ................................................................ 240 Chapt e r 9 Callable and non-c allable bonds ......................... 247 Chapt e r 1 0 Valuat ion of s t oc ks ................................................... 272 Chapt e r 1 1 Pric e of a bond s old bet we en t wo coupon payment s 275 Chapt e r 1 2 Time we ight ed re t urn and dollar we ight ed re t urn 288 Chapt e r 1 3 Inve s t ment year & port folio met hod ............... 303 Chapt e r 1 4 Short Sale s ................................................................... 305 Chapt e r 1 5 Te rm s t ruct ure of int e re s t rat e , s pot rat e, forward rat e , and arbit rage ............................................................. 316 Chapt e r 1 6 Macaulay durat ion, modified durat ion, c onvexit y 341 Chapt e r 1 7 Immunizat ion.............................................................. 376 Chapt e r 1 8 Cas h flow mat ching .................................................. 394 Va lu e of t h is PDF s t u dy ma nu a l...................................................... 401 About t he aut hor................................................................................. 402 3 4 http://actuary88.com Chapt e r 1 Exam-t aking and s t udy s t rat e gy Re ad t his c hapt e r be fore ope ning your t e xt books ! Th is ch a pt er r equ ir es s ome kn owledge a bou t t h e t ime va lu e of mon ey, a n n u it y, a n d loa n a mor t iza t ion . If you don t kn ow t h es e con cept s , don t wor r y. J u s t s kip t h e det a iled ma t h ca lcu la t ion s a n d focu s on t h e ma in idea s in t h is ch a pt er . La t er on , a ft er you u n der s t a n d t h e t ime va lu e of mon ey, a n n u it y, a n d loa n a mor t iza t ion , come ba ck t o t h is ch a pt er a n d go t h r ou gh t h e ma t h . It s cr it ica l t h a t you u n der s t a n d t h e es s en ce of t h is ch a pt er befor e you r u s h t o r ea d t h e t ext books . A t ale of t wo Exam FM t ake rs , Mr. Bus y and Mr. Lazy It wa s t h e bes t of t he t imes , it wa s t h e wor s t of t imes , it wa s t h e a ge of bein g la zy, it wa s t h e a ge of bein g bu s y, it wa s t h e epoch of pa s s in g Exa m FM, it wa s t h e epoch of fa ilin g Exa m FM, it wa s t h e h ope of get t in g ASA, it wa s t h e des pa ir of goin g n owh er e. Two a ct u a r ia l s t u den t s , Mr . Bu s y a n d Mr . La zy, a r e bot h pr epa r in g for Exa m FM. Th ey h a ve t h e s a me h eigh t a n d weigh t . Th ey h a ve t h e s a me level of in t elligen ce. As a ma t t er of fa ct , t h ey a r e s imila r a bou t a lmos t ever yt h in g except t h a t Mr . Bu s y is ver y bu s y a n d Mr . La zy is ver y la zy. Mr . Bu s y a n d Mr . La zy bot h wou ld h a ve a wor r y-fr ee life if t h ey don t n eed t o a mor t ize a loa n wit h geomet r ica lly in cr ea s in g pa ymen t s . Challe nge -- loan amort izat ion wit h ge ome t ric ally inc re as ing payme nt s Mr . Bu s y a n d Mr . La zy bot h love s t a n da r d a n n u it y pr oblems t h a t r equ ir e t h e u s e of memor ized for mu la s s u ch a s i na a n d i na`` . Th ey bot h h a t e geomet r ica lly in cr ea s in g a n n u it y a n d loa n a mor t iza t ion pr oblems . Th ey wou ld gla dly s olve on e h u n dr ed s t a n da r d a n n u it y pr oblems t h a n a mor t ize a mes s y loa n . Sa dly t h ou gh , SOA loves t o t es t t h e pr oblems t h a t Mr . Bu s y a n d Mr . La zy h a t e. 5 Fin a lly t h e exa m da y h a s come. Mr . Bu s y a n d Mr . La zy wa lk in t o t h e exa m r oom. Th e 1s t pr oblem in t h e exa m is a bou t in cr ea s in g a n n u it y a n d loa n a mor t iza t ion . Proble m 1 Da t e of loa n 1/ 1/ 2005 Amou n t of loa n $150,000 Ter m of loa n 25 yea r s Pa ymen t s An n u a l pa ymen t s wit h fir s t pa ymen t du e 12/ 31/ 2005. Ea ch s u bs equ en t pa ymen t is 2% la r ger t h a n t h e pr eviou s pa ymen t . In t er es t r a t e 8.5% a n n u a l effect ive Qu es t ion : Wh a t s t h e t ot a l in t er es t pa id du r in g t h e fir s t 18 pa ymen t s ? Mr. Bus y: Th ou gh n ot fon d of t h is t ype of pr oblems , Mr . Bu s y wa s n ot a bit wor r ied. Wh en pr epa r in g for Exa m FM, Mr . Bu s y bou gh t t h e s olu t ion ma n u a l for Br over ma n s t ext book. He s olved a ll of t h e pr a ct ice pr oblems in t h e s olu t ion ma n u a l. Some of t h e pr oblems in t h e s olu t ion ma n u a ls a r e geomet r ic a n n u it y a n d loa n a mor t iza t ion pr oblems . In a ddit ion , Mr . Bu s y bou gh t a n ot h er ma n u a l wit h t on s of pr a ct ice pr oblems a n d s olved a ll t h e pr a ct ice pr oblems in t h e ma n u a l. Mr . Bu s y felt r ea dy t o t a ckle t h is pr oblem. He s olved over a t h ou s a n d pr a ct ice pr oblems . Su r ely s ome of t h e pr oblems h e s olved wer e a bou t loa n a mor t iza t ion wh er e pa ymen t s wer e geomet r ica lly in cr ea s in g. He s t a r t ed t o men t a lly s ea r ch for h ow h e s olved s u ch pr oblems in t h e pa s t , h opin g t o r eca ll a qu ick s olu t ion . To h is dis ma y, Mr . Bu s y cou ldn t r emember a n y qu ick s olu t ion s t o loa n a mor t iza t ion wit h geomet r ica lly in cr ea s in g pa ymen t s . Th ou gh h e s olved ma n y pr oblems befor e t h e exa m, Mr . Bu s y wa s a lwa ys in a r u s h t o s olve t h e n ext pr a ct ice pr oblem. He n ever h a d t h e t ime t o con den s e h is s olu t ion s t o qu ickly r eca lla ble s olu t ion s r ea dy t o be u s ed in t h e exa m. He didn t even h a ve t h e t ime t o t h or ou gh ly u n der s t a n d t h e ba s ic con cept beh in d loa n a mor t iza t ion a n d beh in d t h e pr es en t va lu e ca lcu la t ion of a geomet r ica lly in cr ea s in g a n n u it y. He wa s a lwa ys in a big h u r r y t o s olve mor e pr a ct ice pr oblems . Time s eemed t o go mu ch fa s t er in t h e exa m r oom. An d t h e pr es s u r e wa s keen . 5 min u t es pa s s ed. Mr . Bu s y wa s goin g n owh er e. Relu ct a n t ly, Mr . Bu s y a ba n don ed t h is pr oblem a n d moved t o t h e n ext on e. 6 Mr. Lazy: Un like Mr . Bu s y, Mr . La zy h a s a la zy a ppr oa ch t o loa n a mor t iza t ion . Mr . La zy r ea lized t h a t loa n a mor t iza t ion a n d geomet r ic a n n u it y pr oblems wer e r epea t edly t es t ed in t h e pa s t . Yea r a ft er yea r , SOA a s ks ca n dida t es t o a mor t ize a loa n . Somet imes t h e loa n t o be a mor t ized h a s level pa ymen t s ; ot h er t imes t h e pa ymen t s a r e a r it h met ica lly or geomet r ica lly in cr ea s in g or decr ea s in g. Loa n a mor t iza t ion a n d geomet r ic a n n u it y pr oblems a r e s o pr edict a ble t h a t Mr . La zy s u s pect ed t h a t SOA wou ld t es t it a ga in t h is yea r wh en h e t a kes t h e exa m. Mr. Lazy s t art s t o s t rat e gize : 1. SOA loves t o t es t loa n a mor t iza t ion . Su ch a pr oblem is doomed t o occu r wh en I t a ke t he exa m. 2. Loa n a mor t iza t ion is n a s t y, es pecia lly wh en t h e pa ymen t s a r e geomet r ica lly in cr ea s in g or decr ea s in g. It s h a r d for me t o figu r e it ou t fr om s cr a t ch in t h e h ea t of t h e exa m. 3. Im la zy. I wa n t t o pas s Exa m FM wit h lea s t effor t . Mr. Lazys c onc lus i on: 1. Befor e t h e exa m, Ill des ign a s t a n da r d cookie-cu t t er s olu t ion t o loa n a mor t iza t ion wit h geomet r ica lly in cr ea s in g pa ymen t s . Th is wa y, I don t h a ve t o in ven t a s olu t ion fr om s cr a t ch in t h e exa m. 2. Ill ma ke my s olu t ion les s t h a n 3 min u t e lon g; 3 min u t es is pr et t y mu ch a ll t h e t ime I h a ve per qu es t ion in t h e exa m. 3. Ill wa lk in t o t h e exam r oom wit h t h e 3 min u t e s olu t ion s cr ipt r ea dy in my h ea d. Ill u s e t h is s cr ipt t o s olv...