TO: Users of the ACTEX Review Seminar on DVD for SOA Exam FM/CAS Exam 2

Transcription

1 TO: Users of the ACTEX Review Semiar o DVD for SOA Exam FM/CAS Exam FROM: Richard L. (Dick) Lodo, FSA Dear Studets, Thak you for purchasig the DVD recordig of the ACTEX Review Semiar for SOA Exam FM (CAS Exam ). This versio is iteded for the exam offered i May 7 ad thereafter. The purpose of this memo is to provide you with a orietatio to the Iterest Theory ad elemetary fiace compoets of this semiar. A two-page summary of the topics covered i the semiar is attached to this cover memo. Although the semiar is orgaized idepedetly of ay particular textbook, the summary of topics shows where each topic is covered i the textbook Mathematics of Ivestmet ad Credit (Third Editio), by Samuel A. Broverma. Iterest Theory is a topic that is best taught, ad uderstood, i a coceptual (rather tha mathematical) framework. The amout of mathematics ivolved i the topic is quite light, beig little more tha the idea of summig a geometric series plus some very elemetary calculus applicatios i coectio with the force of iterest ad cotiuous auities. However, although it is ot very mathematically sophisticated, it is still a challege to fully uderstad. Throughout my review semiar that you are ow viewig o DVD, you will hear me state that "There are o formulas, there are oly cocepts." Accordigly, the thrust of my review is to get the studets to grasp these cocepts. Oce they are uderstood, problems ca be solved with little difficulty because the mathematics is ot complicated. Actuarial exam review semiars always cosist of a appropriate combiatio of review of theory ad the workig of practice problems. For topics that ivolve complex mathematics i the solutio of problems, more practice problems should be worked by the istructor. For a topic like Iterest Theory, however, with its elemetary mathematics level, my educatioal philosophy is that our time together ca be better spet by gettig to uderstad the topic tha by practicig our algebra. Accordigly, you will fid that a relatively small umber of sample problems (about 35) are worked out for the group i this DVD. A copy of these questios, with detailed solutios, is attached to this cover memo for your referece. Of course you still eed to do may, may, may practice problems as you prepare for your exam, ad it is assumed that you have purchased oe (or more) of the several study guides that have bee prepared for that purpose. With respect to the Fiace Topics (see Sectio H of the Summary of Topics), please uderstad that these topics were added to the Iterest Theory segmet of Exam FM for the first time as of the May 5 exam. Sice a umber of studets might ot have had previous course work i these Fiace Topics, I have tried to give a more thorough presetatio of the backgroud theory, alog with a few illustrative questios. Good luck to you o your exam!

2 ACTEX PUBLICATIONS, INC. DVD SEMINAR FOR EXAM FM-CI SUMMARY OF TOPICS A. Iterest Measuremet 1. Accumulatio Fuctio (1.1.1). Amout Fuctio (1.1.3) 3. Effective Rate of Iterest (1.1.1) 4. Effective Rate of Discout (1.4.1) 5. Simple Iterest (1.1.) 6. Compoud Iterest (1.1.1) 7. Preset Value (1..1) 8. Nomial Rate Notatio (1.3; 1.4.) 9. Equivalece of Rates. Force of Iterest (1.5) B. Basic Cocepts 1. Equatio of Value (1..). Ukow Time (1..) 3. Ukow Iterest Rate (1..) 4. Basic Calculator Usage C. Level Paymet Auities 1. Auity-Immediate ( ). Auity-Due (.1.3) 3. Perpetuity ( ) 4. Ukow Time (Balloo ad Drop) (..3) 5. Ukow Iterest Rate (Calculator Usage) (..4) D. Geeral Auities 1. Geeral Approach for ICP PP (..1). m thly Auities (..1) 3. Cotiuous Auities (..) 4. Varyig Auities a. Geometic (.3.1) b. Arithmetric (.3.) c. Cotiuous Varyig (.3.)

3 ACTEX PUBLICATIONS, INC. DVD SEMINAR FOR EXAM FM-CI 3 E. Yield Rate Determiatio 1. Geeral Yield Rate Equatio (5.1.1). Uiqueess (5.1.) 3. Project Evaluatio (5.1.3) 4. Book Value vs. Market Value (.4.4) 5. Effect of Reivestmet (.4.1) 6. Sikig Fud Valuatio (.4.5) 7. Dollar-Weighted Yield (5..1) 8. Time-Weighted Yield (5..) 9. Ivestmet Year / Portfolio Rates (5.3.1). Cotiuous Model (5.3.3) F. Loa Repaymet 1. Accumulatio Method (Zero-Coupo-Bod Model). Iterest-Oly Method (Par Bod Model or Sikig Fud Model) 3. Amortizatio Method (3.1.1) 4. Outstadig Balace ( ) 5. Amortizatio Schedules (3.1.) 6. Other Properties (3.1.5; 3.) a. No-Level Iterest b. Varyig Paymets 7. Sikig Fuds (3.3) G. Bods 1. Termiology ad Notatio (4.1). Bod Price (4.1.1) a. Basic Formula b. Alterate Formula (Premium Formula) c. Makeham's Formula 3. Premium ad Discout (4.1.1) 4. Amortizatio ad Accrual (4.) 5. Off-Coupo-Date Values (4.1.) a. Price b. Price-plus-accrued 6. Yield Rate Determiatio (Calculator Usage) 7. Callable Bods (4.3.1)

4 4 ACTEX PUBLICATIONS, INC. DVD SEMINAR FOR EXAM FM-CI H. Fiace Topics 1. Theoretical Stock Prices (8..1). Survey of Moder Fiacial Istrumets a. Mutual Fuds (8..4) b. CD s (8.3.1) c. Moey Market Fuds (8.3.) d. Mortgage Backed Securities (8.3.3) 3. Effects of Iflatio (1.6) 4. Yield Curves (Term Structure of Rates) a. Spot Rates (6.1) b. Forward Rates (6.3) 5. Duratio (Macaulay Duratio) (7.1) 6. Volatility (Modified Duratio) (7.1) 7. Covexity (7.) 8. Immuizatio (7.) 9. Asset/Liability Matchig (7.)

5 ACTEX PUBLICATIONS, INC. DVD SEMINAR FOR EXAM FM-CI 5 ILLUSTRATIVE QUESTIONS 1. Jeifer deposits $ ito a bak accout. The bak credits iterest at omial aual rate j, covertible semiaually, for the first seve years ad omial aual rate j, covertible quarterly, for all years thereafter. The accumulated value of the accout at the ed of 5 years is X. The accumulated value of the accout at the ed of.5 years is $198. Fid X. (ANSWER: ). A deposit of $ is made ito a fud at time t =. The fud pays iterest at a omial rate of discout d (4), compouded quarterly, for the first two years. Begiig at time t =, iterest is 1 credited at a force of iterest give by δ t = ( t+ 1). At time t = 5, the accumulated value of the fud is $6. Fid d (4). (ANSWER:.196) 3. A deposit of X is made ito a fud that pays a aual effective rate of iterest of 6% for years. At the same time, X is deposited ito aother fud that pays a aual effective rate of discout of d for years. The amouts of iterest eared over the years are equal for both fuds. Fid d. (ANSWER:.949) 4. I Fud X moey accumulates at force of iterest δ t =.1 t +., for < t <. I Fud Y moey accumulates at aual effective rate i. A amout of $1 is ivested i each fud, ad the accumulated values are the same at the ed of years. Fid the value i Fud Y at the ed of 1.5 years. (ANSWER: 1.35) 5. If A() t = Kt + Lt + M, for < t <, with A() =, A(1) = 1, ad A () = 136, fid the value of the force of iterest at time t =.5. (ANSWER:.97) 6. Fud A accumulates at omial rate 1%, covertible mothly. Fud B accumulates at force of iterest δ t = t 6, for all t. At time t =, $1 is deposited i each fud. The accumulated values of the two fuds are equal at time >, where is measured i years. Fid. (ANSWER: 1.433) 7. O Jauary 1, 1997, $ is ivested i a fud for which the force of iterest at time t is give by δ t =.( t 1), where t is the umber of years sice Jauary 1, Fid the accumulated value of the fud o Jauary 1, (ANSWER: 68.94) 1 8. At δ t = (1 + t), paymets of $3 at t = 3 ad $6 at t = 6 have the same preset value as paymets of $ at t = ad X at t = 5. Fid X. (ANSWER: 315.8) 9. Jeff ad Joh have equal amouts of moey to ivest. Joh purchases a -year auity-due with aual paymets of $5 each. Jeff ivests his moey i a savigs accout earig 9% effective aual iterest for two years. At the ed of the two years, he purchases a 15-year immediate auity with aual paymets of Z. Both auities are valued usig a effective aual rate of 8%. Fid the value of Z. (ANSWER: )

6 6 ACTEX PUBLICATIONS, INC. DVD SEMINAR FOR EXAM FM-CI. Bill deposits moey ito a bak accout at the ed of each year. The deposit for year t is equal to t, for t = 1,,3,. The bak credits iterest at effective aual rate i. $5 of iterest is eared i the accout durig the 11 th year. Fid the value of i. (ANSWER: 7.6%) 11. A perpetuity pays at the ed of the 4 th year, 4 at the ed of the 6 th year, 6 at the ed of the 8 th year, ad so o. Fid the preset value (at time ) of the perpetuity usig a effective aual iterest rate of %. (ANSWER: 45.35) 1. Give that at dt =, fid the value of a i terms of ad δ. (ANSWER: δ ) 13. A cotiuously icreasig auity, with a term of years, makes paymet at aual rate t at time t. The force of iterest is equal to 1. Express the preset value of the auity as a fuctio of. (ANSWER:.644 ) 14. A immediate auity makes mothly paymets for years. Durig the first year, each mothly paymet is $. Each year the mothly paymets are 5% larger tha i the previous year. Fid the preset value of the auity at omial iterest rate i (1) =.6. (ANSWER: 419,55) 15. A ma deposits $ ito Fud A, which pays iterest at the ed of each six-moth period at a certai omial aual rate, covertible semiaually. These iterest paymets are take from Fud A ad immediately reivested i Fud B, which ears iterest at effective aual rate 6%. At the ed of 15 years, the ma has eared a effective aual yield rate of 7.56%. What is the omial rate of iterest beig eared i Fud A? (ANSWER: 8.4%) 16. A ivestor pays P for a auity that provides paymets of $ at the begiig of each moth for years. These paymets, whe received, are immediately ivested i Fud A, payig (1) i =.1. The mothly iterest paymets from Fud A are reivested i Fud B, earig (1) i =.6. The ivestor s effective aual yield rate over the -year period is 8%. Fid the value of P. (ANSWER: 97) 17. You are give the followig cash flow activity to a ivestmet fud ad the values of the fud o the dates show: Date Fud Value Deposit Withdrawal 1/1 1, 3/ /1 1,4 7/ /1 1, / /1 1,6 1/31 1,8 Fid the absolute value of the differece betwee the dollar-weighted ad time-weighted rates of retur. (ANSWER:.5939)

7 ACTEX PUBLICATIONS, INC. DVD SEMINAR FOR EXAM FM-CI You are give the followig cash flow activity to a ivestmet fud, ad the values of the fud just before the cash flows occur: Date Fud Value Deposit Withdrawal 1/1 1,, 7/1 1,3, 5, X 1,5,, 1/31 1,15, Give that the time-weighted ad dollar-weighted rates of retur are equal, fid the date X. (ANSWER: November 15) 19. O Jauary 1, $9 is deposited ito a ivestmet accout. O April 1, whe the value i the fud has grow to X, a withdrawal of W is made. No other deposits or withdrawals are made throughout the year. O December 31, the value of the accout is $85. The dollar-weighted rate of retur over the year is % ad the time-weighted rate of retur is 16%. Fid the value of X. (ANSWER: 7.6) The followig table, which is used for both Questios ad 1, gives the patter of ivestmet year ad portfolio iterest rates over a three-year period, where m = is the time after which the portfolio method is applicable. Caledar Year of Origial Ivestmet, y Ivestmet Year Rate y i1 Ivestmet Year Rate y i Portfolio Rates y i + Caledar Year of Portfolio Rate, y + Z 9% % 11% Z + Z + 1 7% 8% Z + 5%. Frak ivests $ at the begiig of each of caledar years Z, Z + 1 ad Z +. Fid the amout of iterest credited to Frak s accout for caledar year Z +. (ANSWER: 67.49) 1. A ivestmet of $ is made at the begiig of each of caledar years Z, Z + 1 ad Z +. What is the average aual effective time-weighted rate of retur for the three-year period? (ANSWER: 8.58%). A loa charges iterest at omial aual rate i =.4. The loa is repaid with equal paymets made at the ed of each moth for moths. The th paymet is equally divided betwee iterest ad pricipal repaid. Fid the value of. (ANSWER: 34) (1) 3. A loa of X is repaid by the sikig fud method over years. The sikig fud ears aual effective rate 8%. The amout of iterest eared i the sikig fud i the third year is $ Fid the value of X. (ANSWER: )

8 8 ACTEX PUBLICATIONS, INC. DVD SEMINAR FOR EXAM FM-CI 4. A loa of X, chargig effective aual rate 8%, is to repaid by the ed of years. If the loa is repaid with a sigle paymet at time t =, the total amout of iterest paid is $468.5 more tha it would be if the loa had bee amortized by makig te equal aual paymets. Fid the value of X. (ANSWER: 7) 5. A loa, chargig effective aual rate 8%, is repaid over years with uequal aual paymets. The first paymet is X, ad each subsequet paymet is.16% greater tha the oe before it. The amout of iterest cotaied i the first paymet is $89.. Fid the value of X. (ANSWER: 1) 6. Tia buys a -year bod, of face (ad redemptio) amout $, with % aual coupos at a price to yield % effective aual rate. The coupos, whe received, are immediately reivested at 8% effective aual rate. Immediately after receivig (ad reivestig) the 4 th coupo, Tia sells the bod to Joe for a price that will yield effective aual rate i to the buyer. The yield rate that Tia actually ears o her ivestmet is 8% effective aual rate. Fid the value of i. (ANSWER: 1.%) 7. Joh purchases a -year bod of face amout $. The bod pays coupos at aual rate 8%, payable semiaually. The redemptio value is R. The purchase price is $8, ad the preset value of the redemptio value is $ Fid R. (ANSWER: 8) 8. A -year bod has face value $, but will be redeemed for $1. It pays semiaual coupos, ad is purchased for $1135 to yield 1%, covertible semiaually. The first coupo is X, ad each subsequet coupo is 4% greater tha the precedig coupo. Fid the value of X. (ANSWER: 5) 9. The rate of iterest is 8% ad the rate of iflatio is 5%. A sigle deposit is ivested for years. Let A deote the value of the ivestmet at the ed of years, measured i time dollars, ad let B deote the value of the ivestmet at the ed of years computed at the real rate of iterest. Fid the ratio of B A. (ANSWER: 1.) 3. Rework Questio 9 assumig equal level deposits at the begiig of each year durig the -year period istead of a sigle deposit. (ANSWER:.81995) 31. Cosider the followig table of spot rates: Term of Ivestmet Spot Rate 1 7.% Fid each of (a) the 1-year deferred, -year forward rate, ad (b) the -year deferred, 3-year forward rate. (ANSWERS: 9.64%;.51%)

9 ACTEX PUBLICATIONS, INC. DVD SEMINAR FOR EXAM FM-CI 9 3. Fid the duratio of a commo stock that pays divideds at the ed of each year, if it is assumed that each divided is 4% greater tha the prior divided ad the effective rate of iterest is 8%. (ANSWER: 7) 33. Fid the covexity of the commo stock described i Questio 3. (ANSWER: 15) 34. A fiacial istitutio accepts a $85, deposit from a customer o which it guaratees to pay 8% compouded aually for te years. The oly ivestmets available to the istitutio are five-year zero-coupo bods ad preferred stocks, both yieldig 8%. The istitutio develops a ivestmet policy usig the followig reasoig: (a) The duratio of the five-year zero-coupo bod is 5. (b) The duratio of the preferred stock is (c) The duratio of the obligatio to the customer is. (d) Takig the weighted-average of the duratios, the amout ivested i five-year zerocoupo bods is chose to be ( ) 13.5 (85,) = $35, ad the amout ivested i preferred stock is chose to be ( ) 5 (85,) = $5, Verify that this ivestmet strategy is optimal uder immuizatio theory, assumig the customer leaves the fuds o deposit for the full te-year period. 35. A isurace compay accepts a obligatio to pay, at the ed of each year for two years. The compay purchases a combiatio of the followig two bods at a total cost of X i order to exactly match its obligatio: (i) 1-year 4% aual coupo bod with a yield rate of 5%. (ii) -year 6% aual coupo bod with a yield rate of 5%. (a) Fid the value of X. (ANSWER: 18,594) (b) Fid the face amouts of the 4% ad 6% bods. (ANSWER: 971.1: )

10 ACTEX PUBLICATIONS, INC. DVD SEMINAR FOR EXAM FM-CI SOLUTIONS TO ILLUSTRATIVE QUESTIONS 1. We are give that or so The j j ( ) ( ) = 198, 4 j j ( ) 8 1+ = 1.98, 1/ 8 = (1.98) 1 =.47. X = (1.47) = We are give 8 5 (4) 1/ 1 1 d t+ dt e = 6. 4 The itegral evaluates to The which solves for d e 5 l( t+ 1) l(6 / 3) 8 = e =. (4) 1 d = 6 = 1.3, 4 ()() = 1 (1.3) (4) =.196. (4) 1/8 3. We are give AV = X (1.6), so the amout of iterest is X (1.6) 1 =.7985 X. We are also give AV = X (1 d ), so the amout of iterest is X sice the iterest amouts are the same. The (1 d) 1 =.7985 X, d (1 ) 1 = , which solves for d 1/ = 1 (.5817) =.949.

11 ACTEX PUBLICATIONS, INC. DVD SEMINAR FOR EXAM FM-CI I Fud X, I Fud Y, ( ) (.1 t+.) dt.5 t +.t AV = e = e = AV = (1 + i) = , so i =.14. The 1.5 AV 1.5 = (1.14) = The first coditio implies A() = M =, so M =. The secod coditio implies A(1) = K + L+ = 1, so K+ L=. The third coditio implies A() = 4K + L+ = 136, so K+ L= 18. These equatios solve for K = 8 ad L =, so At ( ) = 8t + t+. The ad δ δ t d () dt At 16t + At () 8t + t+ = = 16(.5) + = = (.5) + (.5) +.5, 6. We are give 1 / 6 (1.1). A t dt B = = = AV e AV The 1 /1 (1.1) = e. Takig atural logs, we have 1 l(1.1) = (.995)(1 ) =, 1 or = 1.433, so = ( t 1) dt. 3 AV = e = exp ( t 1) 3. / 3[ 1 ( 1) ]. / 3 = e = e =

12 1 ACTEX PUBLICATIONS, INC. DVD SEMINAR FOR EXAM FM-CI X (1 ) (1 ) (1 ) (1 ) + t dt + t dt + t dt + t dt 3e + 6e = e + X e I geeral, The we have or 1 (1 + t) dt = l(1 + t) = l(1 + ). 3(1+ 3) + 6(1+ 6) = (1+ ) + X (1+ 5), = + X, which solves for X = The amout each has to ivest ca be foud as the preset value of Joh s auity: PV = 5 a = 18, The Jeff ivests a equal amout for two years at 9%, so he has 18,117.18(1.9) = 1,55.3 at time t =. This amout is the the preset value of the immediate auity, so 1,55.3 = Z a 15.8 so Z 1,55.3 = = a The balace i the accout at time t = is ( Is ) i, so the iterest eared i the 11 th year is ( ) 5 = i ( Is) i = s, from which we fid s = 15. We use a BA-35 calculator to directly fid i = 7.6%. i

13 ACTEX PUBLICATIONS, INC. DVD SEMINAR FOR EXAM FM-CI The paymets are made every two years, so we first fid the effective iterest rate over a twoyear period. This is j = (1.) 1 =.1. The value of d correspodig to this j is d.1 j = 1.1. The preset value of the perpetuity at time t = is PV = , j d = = j (.1) ( 1.1) ad the preset value at time t = is (1.) = t a 1 1 t 1 t dt v dt dt v dt a. = = = = δ δ δ The a = δ. 13. t δt sice e 1 i. PV = t v dt = t e dt, δ = + I this case δ = 1, so we have PV = t/ t e dt t/ dt e 1/ t/ t/ = t e + e dt t/ 1 e e = + 1/ 1 1 = e e + 1 = (1 e ) =.644

14 14 ACTEX PUBLICATIONS, INC. DVD SEMINAR FOR EXAM FM-CI 14. MP= MP= (1.5) MP= (1.5) 19 MP= (1.5) Note that the paymet is level withi each year, ad the icreases for the ext year. The value of the first year s paymets at time t = 1 is s 1.5 = 4,671.. The value of the secod year s paymets at time t = is therefore 4,671(1.5), ad so o. The the overall preset value is 19 PV = 4,671v + 4,671(1.5) v + + 4,671(1.5) v, where time is measured i years. The effective aual rate is 1 j = (1.5) 1 = The PV 19 4, = 1 j j j j 1.5 ( 1+ j ) 1 4,671 = 1+ j 1.5) 1 ( 1+ j ) 4,671 = ( ) = 418, There will be roudig errors i this problem j j 9 j 3 j The iterest paymets from Fud A are j each, where j is the effective semiaual rate. 1/ They are the ivested i Fud B, which ears k = (1.6) 1 =.9563 effective semiaually. After 15 years ( t = 3), the value i Fud B is j s = 47, 4.1 j, 3 k ad the value i Fud A is still. The the equatio of value to defie the yield rate, kow to be 7.56% effective aual, is 15 (1.756) = + 47, 4.1 j, which solves for j =.4199, ad the omial rate is j =.8399.

15 ACTEX PUBLICATIONS, INC. DVD SEMINAR FOR EXAM FM-CI P A: B: Fud A ears.1 effective per moth, so the iterest is $1 from each ivested i Fud A. Thus the deposits to Fud B are 1,,..., at the ed of each moth. At time t = the balace i Fud A is ()() = 1, ad the balace i Fud B is s.5 ( Is).5 = = 8, The equatio of value to defie the yield rate is the so P P (1.8) = 1, + 8,939.88, = (1, + 8,939.88)(1.8) = [Note that deposits ad withdrawals at the same time poit ca be etted.] Time-weighted: ( )( )( )( ) i T = 1 = [Note that the actual time poit values are ot used.] Dollar-weighted: 3/ 4 5 /1 1/ 6 1.(1 + i).5(1 + i) +.(1 + i) +.(1 + i) = 1.8 i 1. (.5)(.75) + (.) 5 1 ( 1 ) + (.)( 6 ) = i.35 D = = [Note that the itermediate fud balaces are ot used.] The, i i = D T

16 16 ACTEX PUBLICATIONS, INC. DVD SEMINAR FOR EXAM FM-CI 18. This time we are solvig for a ukow date t 1 5 Recall that the time-weighted rate does ot use the dates. The ( )( )( ) i T = 1 = But we are give that it = id. Therefore we are ot solvig for i D, but rather solvig for the ukow time poit t. Let r = 1 t. 1/ (1 + i) 5(1 + i) + (1 + i) = 115 i 1 (5) r + = = The i =.14, r = which solves for r = This is the fractio of year remaiig as of date X. The 365r = 46.5 days remai, so date X is 46.5 days back from the ed of the year, which is about November 15. r X W Both X ad W are ukow. But the dollar-weighted method does ot use the value of X, so we use it to solve for W. 3/4 9(1 + i) W(1 + i) = 85 The i D = W =. 9 3W 4 W 5 = 18.15W W = X 85 ( ) 1+ it = = X 85X = 4.4X 88 X = 7.63.

17 ACTEX PUBLICATIONS, INC. DVD SEMINAR FOR EXAM FM-CI 17. Z Z + 1 Z + The ivestmet o 1/1/Z ears 9%, so the fud value o 1/31/Z is 9.. I year Z + 1, the ivestmet from year Z ears % ad the ivestmet from year Z + 1 ears 7%, so the fud value o 1/31/ Z + 1 is (9)(1.) + ()(1.7) = 69.. I year Z +, the ivestmet from year Z ears 11%, the ivestmet from year Z + 1 ears 8%, ad the ivestmet from year Z + ears 5%, so the fud value o 1/31/ Z + is (9)(1.)(1.11) + (7)(1.8) + ()(1.5) = The the time-weighted rate of retur is ( )( )( ) i T = 1 =.818, effective over the three-year period. The aual effective rate is 1/3 i = (1.818) 1 = Z Z + 1 Z + The balace i the accout o 1/31/ Z + 1 is The balace i the accout o 1/31/ Z + is (1.9)(1.) + (1.7) = 69.. (1.9)(1.)(1.11) + (1.7)(1.8) + (1.5) = The the growth durig Z + is = , of which comes from the 1/1/ Z + deposit, so the iterest eared durig caledar year Z + is

18 18 ACTEX PUBLICATIONS, INC. DVD SEMINAR FOR EXAM FM-CI. The effective mothly rate is i =. ad the term of the loa is moths. The ad But PR = I, so we have 1 PR = P v + ( ) I = P 1 v v = v = 1. The l = = l.5 = 35, ad = 34. l v 1 l(1.) 3. The sikig fud deposit is the third year) is X, s.8 so the sikig fud balace after paymets (begiig of SFB = X s s.8.8. The iterest i the third year is i SFB, so we have X = s.8 = (1.8) 1 i SFB X = (1.8) 1 = 85.57, s so.8 4. Uder the accumulatio method, TI = X (1.8) 1. Uder the amortizatio method, TI = P X = X X. a We are give X (1.8) X 1 = +, so a.8.8 X = = (1.8) a The amout of iterest i the first paymet is i L, so we have.8l = 89., so L = 11,15.5. The so ,15.5 = X v+ X(1.16) v + x(1.16) v + + X(1.16) v 9 X = ( 1.8 ) + + ( 1.8 ) X ( 1.8 ) = = X, ,15.5 X = =

19 ACTEX PUBLICATIONS, INC. DVD SEMINAR FOR EXAM FM-CI For the bod, C = F = ad g = r = i=.. Sice g = i, the P= C = is the price that Tia ivests at time t =. At time 4, the reivested coupos have accumulated to s 4.8 = The price Joe pays for the bod is 6 P= + (. ia ) = v + a. 6i i 6i Sice Tia ears 8% the, for her, (1.8) = P, so Solvig by calculator gives i = 1.%. 4 6 i P= = v + a. 6 i 7. Coupos are paid semiaually, ad each coupo is.4() = 4. The price is give, so we have i P = 8 = R v + 4 a. i We are also give that R v i = 31.51, so we have 8 = a, i so a i = = 1.465, which solves for i =.5. The 4 R = [ 8 4(1.465) ](1.5) = There are semiaual coupos. The effective semiaual yield rate is.6. The preset value of the coupos is 3 19 Xv + X (1.4) v + X (1.4) v + + X (1.4) v 1.4 ( ) ( ) ( ) X = X = X. 1.6 = The we have so = X + 1 v, (1.6) X = =

20 ACTEX PUBLICATIONS, INC. DVD SEMINAR FOR EXAM FM-CI 9. The accumulated value is (1.8) = But this is measured i dollars valued years after time, ad is ot equal to.1589 time dollars because of iflatio. The value i time dollars is A =.1589 = , (1.5) which is ( ) = (1 + i ) = B, so the ratio B A is I this case each deposit accumulates at 8% from its deposit date, but is the discouted back to time at 5% to express the accumulated value i time dollars. This produces s.8 A = = (1.5) The B= s i = s.857 = , so the ratio is A B = 31. (a) A uit earig the curret 1-year spot rate followed by the 1-year deferred -year forward rate will accumulate to the same result as usig the curret 3-year spot rate. That is, 3 (1.7)(1 + f ) = (1.875), which solves for f =.9636, or 9.64%. (b) Similarly, accumulatio at the curret -year spot rate followed by the -year deferred 3- year forward rate is the same as accumulatio at the curret 5-year spot rate. The 3 5 (1.8) (1 + f ) = (1.95), which solves for f =.51, or.51%. 3. If the divided at the ed of the first year is D, the the preset value of all paymets is The so the volatility is 1 3 Pi ( ) = D (1 + i) + (1.4)(1 + i) + (1.4) (1 + i) + ad the duratio, evaluated at 8%, is 1.4) 1.4 ( ) ( ) 1 = D(1 + i) i 1+ i D(1 i) i = + + = Di (.4). i.4 v P ( i) = Di (.4), P () i = = ( i.4) Pi () 1 1 d = v(1.8) = (.8.4) (1.8) = 7.

21 ACTEX PUBLICATIONS, INC. DVD SEMINAR FOR EXAM FM-CI From Questio 3 we have 1 Pi ( ) = Di (.4). The P ( i) = Di (.4) ad 3 P ( i) = Di (.4), so c P () i = = ( i.4), Pi () which, whe evaluated at i =.8, gives c = The cash flow o the assets (the ivestmet pla) is 35,(1.8) at time 5 (from the bod) ad (.8)(5,) = 4 at each of times 1,, (from the stock). The cash flow o the liability (the deposit) is 85,(1.8). The preset value of the asset cash iflow, as a fuctio of i, is PA( i) = 35,(1.8) (1 + i) + 4 i, ad the preset value of the liability cash outflow, also as a fuctio of i, is The first derivatives are i PL ( i) = 85,(1.8) (1 + ). ad 5 6 PA( i) = (5)(35,)(1.8) (1 + i) 4i 11 i PL ( i) = ()(85,)(1.8) (1 + ), ad the secod derivatives are ad PA( i) = (5)(6)(35,)(1.8) (1 + i) + 8i 1 i PL ( i) = ()(11)(85,)(1.8) (1 + ). At 8% effective aual rate, we fid P (.8) = P (.8) = 85,. Furthermore, A L ad Furthermore, ad v v c A L A c PA (.8) 787,37.4 = = = P (.8) 85,. A PL (.8) 787,37.4 = = = P (.8) 85,. L L PA (.8) 16,675,6 = = = P (.8) 85, A PL (.8) 8,16,118 = = = P (.8) 85, L The immuizatio strategy is successful because (1) PA(.8) = PL(.8), () va = vl, ad (3) c > c, as required. A L

22 ACTEX PUBLICATIONS, INC. DVD SEMINAR FOR EXAM FM-CI 35. (a) To exactly match the obligatios, we eed total bod paymets of, at each of t = 1 ad t =. Sice each bod has a yield rate of 5%, the total preset value (which is the total cost) is X =,a = 18, (b) This is a more meaigful questio tha that posed i part (a). The 4% bod pays a coupo ad its maturity value at t = 1. If the face amout of this bod is F 1, the the total paymet is 1.4 F 1. The 6% bod pays a coupo at t = 1 ad a coupo plus maturity value at t =. If the face amout of this bod is F, the the paymet at t = 1 is.6f ad the paymet at t = is 1.6 F. F 1 ad F must satisfy the equatios ad 1.6F =,.6F + 1.4F =, 1 from which we fid F 1 = ad F =