SOCIETY OF ACTUARIES FINANCIAL MATHEMATICS EXAM FM SAMPLE SOLUTIONS

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1 SOCIETY OF ACTUARIES EXAM FM FINANCIAL MATHEMATICS EXAM FM SAMPLE SOLUTIONS Ths page dcaes chages made o Sudy Noe FM Aprl 8, 04: Queso ad soluo 6 added. Jauary 4, 04: Quesos ad soluos were added. Copyrgh 03 by he Socey of Acuares. Some of he quesos hs sudy oe are ake from pas SOA/CAS examaos. FM PRINTED IN U.S.A.

2 The followg model soluos are preseed for educaoal purposes. Alerae mehods of soluo are, of course, accepable.. Soluo: C Gve he same prcpal vesed for he same perod of me yelds he same accumulaed value, he wo () measures of eres () δ ad δ mus be equvale, whch meas: ( + ) = e over oe eres measureme perod (a year hs case)..04 Thus, Soluo: E δ ( + ) = e or δ ( +.0) = e ad = l(.0) = l(.0) = δ or 3.96%. Accumulaed value ed of 40 years = 00 [(+) 4 + (+) 8 +..(+) 40 ]= 00 ((+) 4 )[-((+) 4 ) 0 ]/[ - (+) 4 ] ( Sum of fe geomerc progresso = s erm mes [ (commo rao) rased o he umber of erms] dvded by [ commo rao] ) ad accumulaed value ed of 0 years = 00 [(+) 4 + (+) 8 +..(+) 0 ]=00 ((+) 4 )[-((+) 4 ) 5 ]/[ - (+) 4 ] Bu accumulaed value ed of 40 years = 5 mes accumulaed value ed of 0 years Thus, 00 ((+) 4 )[-((+) 4 ) 0 ]/[ - (+) 4 ] = 5 {00 ((+) 4 )[-((+) 4 ) 5 ]/[ - (+) 4 ]} Or, for > 0, -((+) 40 = 5 [-((+) 0 ] or [-((+) 40 ]/[-((+) 0 ] = 5 Bu x - y = [x-y] [x+y], so [-((+) 40 ]/[-((+) 0 ]= [+((+) 0 ] Thus, [+((+) 0 ] = 5 or (+) 0 = 4. So X = Accumulaed value a ed of 40 years = 00 ((+) 4 )[-((+) 4 ) 0 ]/[ - (+) 4 ] =00 (4 /5 )[-((4 /5 ) 0 ]/[ 4 /5 ] = Alerae soluo usg auy symbols: Ed of year 40, accumulaed value = 00 ( s / a ), ad ed of year accumulaed value = 00 ( s / a ). Gve he rao of he values equals 5, he = ( s / s ) = [( + ) ]/[( + ) ] = [( + ) ]. Thus, (+) 0 = 4 ad he accumulaed value a he ed of 40 years s 00( s / a ) = 00[( + ) ]/[ ( + ) ] = 00[6 ]/[ 4 / ] = 40 4 Noe: f = 0 he codos of he queso are o sasfed because he he accumulaed value a he ed of 40 years = 40 (00) = 4000, ad he accumulaed value a he ed of 0 years = 0 (00) = 000 ad hus accumulaed value a he ed of 40 years s o 5 mes he accumulaed value a he ed of 0 years. 59

3 3. Soluo: C Erc s eres (compoud eres), las 6 mohs of he 8 h 5 year: 00( + ) ( ) Mke s eres (smple eres), las 6 mohs of he 8 h year: 00( 5 ). Thus, 00( + ) ( ) = 00( ) or ( + ) 5 =, whch meas / = or = = 9.46% Soluo: A The payme usg he amorzao mehod s The perodc eres s.0(0000) = 000. Thus, deposs o he skg fud are = The, he amou skg fud a ed of 0 years s s 0.4 Usg BA II Plus calculaor keysrokes: d FV (o clear regsers) 0 N, 4 I/Y, PMT, CPT FV +/ = yelds 33.8 (Usg BA 35 Solar keysrokes are AC/ON (o clear regsers) 0 N 4 % PMT CPT FV +/ =) Soluo: E Key formulas for esmag dollar-weghed rae of reur: Fud Jauary + deposs durg year whdrawals durg year + eres = Fud December 3. Esmae of dollar weghed rae of reur = amou of eres dvded by he weghed average amou of fud exposed o earg eres oal deposs = 0 oal whdrawals = 45 Ivesme come = = 0 0 Rae of reur = = 0/ = %

4 6. Soluo: C + v Cos of he perpeuy = v ( Ia) + a + v v = v a v v = + a = Gve = 0.5%, a a = = 77.0 a = , a 0.5% 0.05 = 9 Tps: Helpful aalyss ools for varyg aues: draw pcure, defy layers of level paymes, ad add values of level layers. I hs queso, frs layer gves a value of / (=PV of level perpeuy of = sum of a fe geomerc progresso wh commo rao v, whch reduces o /) a, or v (/) a 0 d layer gves a value of / a, or v (/) a 0. h layer gves a value of / a, or v (/) a 0 Thus 77. = PV = (/) (v + v +. v ) = (/.05) a. 05 ca be easly solved for usg BA II Plus or BA 35 Solar calculaor 6

5 7. Soluo: C 6 Ds + 00 s ( ) (.09) s ( ) Helpful geeral resul for obag PV or Accumulaed Value (AV) of arhmecally varyg sequece of paymes wh eres coverso perod (ICP) equal o payme perod (PP): Gve: Ial payme P a ed of s PP; crease per PP = Q (could be egave); umber of paymes = ; effecve rae per PP = ( decmal form). The a a PV = P. + Q [(. v )/] (f frs payme s a begg of frs PP, jus mulply hs resul by (+)) To effcely use specal calculaor keys, smplfy o: (P + Q/) a. Q v / = (P + Q/) a. (Q/) v. The for BA II Plus: selec d FV, eer value of selec N, eer value of 00 selec I/Y, eer value of (P+(Q/)) selec PMT, eer value of ( (Q/)) selec FV, CPT PV +/- For accumulaed value: selec d FV, eer value of selec N, eer value of 00 selec I/Y, eer value of (P+(Q/)), selec PMT, CPT FV selec +/- selec eer value of ( (Q/)) = For hs queso: Ial payme o Fud Y s 60, crease per PP = - 6 BA II Plus: d FV, 0 N, 9 I/Y, (60 (6/.09)) PMT, CPT FV +/- + (60/.09) = yelds (For BA 35 Solar: AC/ON, 0 N, 9 %, (6/.09 = +/ =) PMT, CPT FV +/- STO, 60/.09 + RCL (MEM) =) Soluo: D P = 000(.095)(.095)(.096) = 34.3 Q = 000(.0835)(.086)(.0885) = 80.8 R = = Thus, R > P> Q. ( )( )( ) 6

6 9. Soluo: D For he frs 0 years, each payme equals 50% of eres due. The leder charges 0%, herefore 5% of he prcpal ousadg wll be used o reduce he prcpal. A he ed of 0 years, he amou ousadg s ( ) = Thus, he equao of value for he las 0 years usg a comparso dae of he ed of year 0 s = X a. So X = = % a0 0% Aleravely, derve aswer from basc prcples raher ha uo. Equao of value a me 0: 000 =.5(.)(000) (v +.95 v +.95 v v 0 ) + X v 0 a. 0. Thus X = [000 - {.5(.)(000) (v +.95 v +.95 v v 0 )}]/ (v 0 a ) 0. = {000 [50 v ( (.95 v) 0 )/(-.95 v)]}/ (v 0 a )= Soluo: B = 6% 4 BV = 0, 000v + 800a = = 0, I7 = BV6 = , 693 = Soluo: A Value of al perpeuy mmedaely afer he 5 h payme (or ay oher me) = 00 (/) = 00/.08 = 50. Exchage for 5-year auy-mmedae payg X a he ed of he frs year, wh each subseque payme creasg by 8%, mples 50 (value of he perpeuy) mus = X (v +.08 v +.08 v v 5 ) (value of 5-year auy-mmedae) = X ( (.08) (.08) (.08) -5 ) (because he aual effecve rae of eres s 8%) = X ( ) = X [5(.08 - )]. So, 50 (.08) = 5 X or X = 54 63

7 . Soluo: C Equao of value a ed of 30 years: ( d ) (.03) + 0(.03) = ( d ) = d = d = Soluo: E 3 d = So accumulaed value a me 3 of depos of 00 a me 0 s: 3 / e = The amou of eres eared from me 3 o me 6 equals he accumulaed value a me 6 mus he accumulaed value a me 3. Thus 3 /300 3 ( ) ( ) ( )( ) X e + X = X + X X = X = X X = Soluo: A = Prese value = ( + k) 0a + 0v [ ] =.09 + k = v 5 9. ( ) because he summao s a fe geomerc progresso, whch smplfes.09 + k.09 o (/(-commo rao)) as log as he absolue value of he commo rao s less ha (.e. hs case commo rao s (+k)/.09 ad so k mus be less ha.09) ( )( ) So = k or 8.80 = + k ) or 0 = (+k)/(0.09-k) 0.09 k 0.09 k ad hus 0.84 = k or k = Aswer s

8 5. Soluo: B Opo : 000 = Pa P = 99 Toal paymes = 990 Opo : Ieres eeds o be = [ ] = [, 000] = 0.09 Tp: For a arhmec progresso, he sum equals he average of he frs ad las erms mes he umber of erms. Thus hs case, = (/) ( ) 0 = 000. Of course, wh oly 0 erms, s farly quck o jus add hem o he calculaor! Soluo: B The po of hs queso s o es wheher a sude ca deerme he ousadg balace of a loa whe he paymes are o level. Mohly payme a me = 000(0.98) Sce he acual amou of he loa s o gve, he ousadg balace mus be calculaed prospecvely, OB 40 = prese value of paymes a me 4 o me 60 = 000(0.98) 40 (.0075) + 000(0.98) 4 (.0075) (0.98) 59 (.0075) 0 Ths s he sum of a fe geomerc seres, wh frs erm, a = 000(0.98) 40 (.0075) commo rao, r = (0.98)(.0075) umber of erms, = 0 Thus, he sum = a ( r )/( r) = 000(0.98) 40 (.0075) [ (0.98/.0075) 0 ]/[ (0.98/.0075)] =

9 7. Soluo: C The paymes ca be separaed o wo layers of 98 ad he equao of value a 3 s 98 S + 98S = ( + ) ( + ) + = 8.63 ( + ) = = = 8.63 =.5% Soluo: B Cover 9% coverble quarerly o a effecve rae per moh, he payme perod. Tha s, solve for j such 3.09 ha ( + j ) = ( + ) or j = or.744% 4 The.. 60 a v ( Ia ) = [ ] = Aleravely, use resul lsed soluo o queso 7 above wh P = Q =, = ad = 60. The (P + Q/) = ( + /.00744) = ad Q/ = Usg BA II Plus calculaor: selec d FV, eer 60 selec N, eer.744 selec I/Y, eer selec PMT, eer selec FV, CPT PV +/- yelds

10 9. Soluo: C Key formulas for esmag dollar-weghed rae of reur: Fud Jauary + deposs durg year whdrawals durg year + eres = Fud December 3. Esmae of dollar weghed rae of reur = amou of eres dvded by he weghed average amou of fud exposed o earg eres The for Accou K, dollar-weghed reur: Amou of eres I = 5 00 x + x = 5 x 5 x = = (5 x)/00; or ( + ) K = (5 x)/00 00 x + x 4 Key coceps for me-weghed rae of reur: Dvde he me perod o subervals for each me here s a depos or whdrawal For each suberval, calculae he rao of he amou he fud a he ed of he suberval (before he depos or whdrawal a he ed of he suberval) o he amou he fud a he begg of he suberval (afer he depos or whdrawal) Mulply he raos ogeher o cover he desred me perod The for Accou L me-weghed reur: ( + ) = 5/ /(5 x) = 3.5/(5 x) Bu ( + ) = ( + ) for Accou K. So 3.5/(5 x) = (5 x)/00 or (5 x) = 3,5 x = 0 ad = (5 x)/00 = 5% Soluo: A Equae prese values: v v = 600 v 0 v = = Thus, v 0 = 45.8/600 = = 3.5% Soluo: A Use equao of value a ed of 0 years: d l( 8+ ) 8 8+ ( + ) = e = e = ( 8 + ) , 000 = ( 8k + k ) ( + ) d = k ( 8 ) d , 000 = 8k = 80k k = = Soluo: D Prce for ay bod s he prese value a he yeld rae of he coupos plus he prese value a he yeld rae of he redempo value. Gve r = sem-aual coupo rae ad = he sem-aual yeld rae. Le C = redempo value. The Prce for bod X = P X = 000 r = 000 r ( v ) because gve as a + C v (usg a sem-aual yeld rae hroughou) a = v ad he prese value of he redempo value, C v, s We are also gve r =.035 so 000 r = Thus, P X = 03.5 ( v ) Now oly eed v. Gve v = , v = (0.5889). Thus P X = 03.5 ( (0.5889) ) =

11 Soluo: D Equae e prese values: v+ 4000v = v xv x 000 = x = Soluo: E For he amorzao mehod, payme P s deermed by 0000 = X a, whch yelds (usg calculaor) X = For he skg fud mehod, eres s.08 (000) = 600 ad oal payme s gve as X, he same as for he amorzao mehod. Thus he skg fud depos = X 600 = = 5.3. The skg fud, a rae j, mus accumulae o years. Thus, 5.3 s 0 j = whch yelds (usg calculaor) j =

12 5. Soluo: D The prese value of he perpeuy = X/. Thus, he gve formao yelds: X B= X a = 0.4 C = v Xa X J = v 0.4 a = v = 0.6 X J = 0.36 Tha s, Jeff s share s 36% of he perpeuy s prese value Soluo: D The gve formao yelds he followg amous of eres pad: 0 0. Seh = = = Jace = 5000 ( 0.06)( 0) = Lor = P(0) 5000 = where P= = a 0 6% The sum s Soluo: E X = Bruce s eres s mes he accumulaed value a he ed of 0 years = 00 (+) 0. X = Robbe s eres s mes he accumulaed value a he ed of 6 years = 50 (+) 6 Because boh amous equal X, akg he rao yelds: X/X = v 6 or v 6 = /. Thus, (+) 6 = ad = /6 =.46. So X =.46 [00 (.46) 0 ]= Soluo: D Year ( + ) prcpal repad = v Year eres repad = a + = v + Toal = v + + v = v (v ) = v ( ( v)) = + v (d)

13 9. Soluo: B 3 s gve as PV of perpeuy payg 0 a ed of each 3-year perod, wh frs payme a he ed of 3 years. Thus, 3 = 0 (v 3 + v 6 +,,,,,,, ) = 0 v 3 (/- v 3 ) (fe geomerc progresso), ad v 3 = 3/4 or (+) 3 = 4/3. Thus, = X s gve as he PV, a he same eres rae, of a perpeuy payg a he ed of each 4 mohs, wh he frs payme a he ed of 4 mohs. Thus, X = (v /3 + v /3 +,,,,,,,) = v /3 (/(- v /3 )) = Soluo: D The prese value of he lably a 5% s $8,70.48 ($,000,000/ (.05^4)). The fuure value of he bod, cludg coupos revesed a 5%, s $,000,000. If eres raes drop by ½%, he coupos wll be revesed a a eres rae 4.5%. Aual coupo paymes = 8,703 x.05 = 4,35. Accumulaed value a /3/007 wll be 4,35 + [4,35 x (.045)] + [4,35 x (.045^)] + [4,35 x (.045^3)] + 8,703 = $998,687. The amou of he lably payme a /3/007 s $,000,000, so he shorfall = 998,687,000,000 = -,33 (loss) If eres raes crease, he coupos could be revesed a a eres rae of 5.5%, leadg o a accumulao of more ha he $,000,000 eeded o fud he lably. Accumulaed value a /3/007 wll be 4,35 + [4,35 x (.055)] + [4,35 x (.055^)] + [4,35 x (.055^3)] + 8,703 = $,00,33. The amou of he lably s $,000,000, so he surplus or prof =,00,33,000,000 = +,33 prof Soluo: D. Prese value = 5000 (.07v +.07 v v v v 0 ) 0 (.07v) = v ( ) (.07v) smplfyg o: 5,000 (.07) [ -(.07/.05) 0 ] / ( ) =,

14 3. Soluo: C. NPV = (.05) -4 (60000(.04) ) = (.05) -4 (400) = Tme Cash Ial -00,000 Flow Ivesme Ivesme 60,000 60,000 Reurs Revesme Reurs 60,000*.04 = 400 Toal amou o be dscoued -00, =400 Dscou Facor /(.05)^4 = , , Soluo: B. Usg spo raes, he value of he bod s: 60/(.07) + 60/((.08) ) + 060/((.09) 3 ) = Soluo: E. Usg spo raes, he value of he bod s: 60/(.07) + 60/((.08) ) + 060/((.09) 3 ) = Thus, he aual effecve yeld rae,, for he bod s such ha = 60a + 000v a. Ths ca be 3 easly calculaed usg oe of he calculaors allowed o he acuaral exam. For example, usg he BA II PLUS he keysrokes are: 3 N, PV, 60 +/- PMT, 000 +/- FV, CPT I/Y = ad he resul s 8.9% (rouded o oe decmal place)

15 35. Soluo: C. Durao s defed as = = v v R R, where v s calculaed a 8% hs problem. (Noe: There s a mor bu mpora error o page 8 of he secod edo of Broverma s ex. The referece "The quay brackes Equao (4.) s called he durao of he vesme or cash flow" s o correc because of he mus sg he brackes. There s a erraa ls for he secod edo. Check hp:// f you do o have a copy). The curre prce of he bod sv R, he deomaor of he durao expresso, ad s gve as 00. The = dervave of prce wh respec o he yeld o maury s v = + R = - v mes he umeraor of he durao expresso. Thus, he umeraor of he durao expresso s - (.08) mes he dervave. Bu he dervave s gve as So he umeraor of he durao expresso s 756. Thus, he durao = 756/00 = Soluo: C Durao s defed as = = dvded amou. Thus, he durao = v v R R, where for hs problem v s calculaed a = 0% ad R s a cosa D, he = = v v D D = = = v v. Usg he mahemacs of fe geomerc progressos (or jus rememberg he prese value for a u perpeuy mmedae), he deomaor = v (/(-v)) (frs erm mes dvded by he quay mus he commo rao; coverges as log as he absolue value of he commo rao, v hs case, s less ha ). Ths smplfes o / because - v = d = v. The umeraor may be remembered as he prese value of a creasg perpeuy mmedae begg a u ad creasg by I u each payme perod, whch equals + = +. So durao = S Num /deomaor =((+)/ )/(/) = (+)/ =./. =

16 37. Soluo: B Durao s defed as = = amou, mes (.0) -. Thus, he durao = v v R R, where for hs problem v s calculaed a = 5% ad R s D, he al dvded = = v D(.0) v D(.0) = = = v v (.0) (.0) Usg he mahemacs of fe geomerc progressos (or jus rememberg he prese value for a u geomercally creasg perpeuy mmedae), he deomaor = v, whch smplfes o. I ( v(.0)).0 + ca be show* ha he umeraor smplfes o. So durao = umeraor/deomaor (.0) + + = / =. (.0).0.0 Thus, for =.05, durao = (.05)/.03 = 35. Alerave soluo: A shorer alerave soluo uses he fac ha he defo of durao ca be ca be show o be equvale o (+) P ()/P() where P() = = v R. Thus, hs case P() = = D v (.0) = P () (he dervave of P() wh respec o ) = D ( ). Thus, he durao = (.0). D.0 ad D( ) (.0) ( + ) = D.0 +, yeldg he same resul as above *Noe: The process for obag he value for he umeraor usg he mahemacs of seres smplfcao s: Le S Num deoe he sum he umeraor. The S Num = v + (.0) v + 3 (.0) v (.0) - v +.. ad S Num = (.0)v + (.0) v (-) (.0) - v +.. Thus, (-(.0)v) S Num = v + (.0)v + (.0) v (.0) - v +..= v = ( v(.0)) ad S Num = /( (.0) v) = / = / =. (.0) (.0) (.0) (.0)v 73

17 skpped Soluo: A Key coceps for me-weghed rae of reur: Dvde he me perod o subervals for each me here s a depos or whdrawal For each suberval, calculae he rao of he amou he fud a he ed of he suberval (before he depos or whdrawal a he ed of he suberval) o he amou he fud a he begg of he suberval (afer he depos or whdrawal) Mulply he raos ogeher o cover he desred me perod Thus, for hs queso, me-weghed reur = 0% meas: +0 = (/0) (X/(+X) or X = X ad X = 60 Key formulas for esmag dollar-weghed rae of reur: Fud Jauary + deposs durg year whdrawals durg year + eres = Fud December 3. Esmae of dollar weghed rae of reur = amou of eres dvded by he weghed average amou of fud exposed o earg eres Thus, for hs queso, amou of eres I = X X 0 = - 0 ad dollar-weghed rae of reur s gve by Y = [-0/(0 + ½ (60)] = - 0/40 = -.5 = -5% Soluo: A Gve he erm of he loa s 4 years, ad he ousadg balace a ed of hrd year = 559., he amou of prcpal repad he 4 h payme s Bu gve level paymes, he prcpal repad forms a geomerc progresso ad hus he prcpal repad he frs year s v 3 mes he prcpal repad he fourh year = v Ieres o he loa s 8%, hus prcpal repad frs year s (/(.08) 3 )*559. = Soluo: B Prce of bod = 000 because he bod s a par value bod ad he coupo rae equals he yeld rae. A he ed of 0 years, he equao of value o Bll s vesme s he prce of he bod accumulaed a 7% equals he accumulaed value of he vesme of he coupos plus he redempo value of 000. However, he coupos are vesed semaually ad eres s a aual effecve rae. So he equao of value s: 000 (.07) 0 = 30 s 0 j where j s such ha (+j) =+ Rearragg, 30 s 0 j = 000 (.07) = Solvg for j (e.g. usg oe of he approved calculaors) yelds j = %, ad hus = (+j) = Soluo: A 3,000/9.65 = s he umber of housads requred o provde he desred mohly rereme beef because each 000 provdes 9.65 of mohly beef ad he desred mohly rereme beef s Thus, 30,88 s he capal requred a age 65 o provde he desred mohly rereme beef. Usg he BA II Plus calculaor, selec d BGN, selec ND SET ul BGN appears o he scree (mohly corbuos sar oday),selec CE, eer *5 = 300 (he oal umber of mohly corbuos) selec N, eer 8/ = (8% compouded mohly) selec I/Y, eer 30,88 selec +/- selec FV, selec CPT PMT o oba Soluo: D Usg he daugher s age 8 as he comparso dae ad equag he value a age 8 of he corbuos o he value a age 8 of he four 50,000 paymes resuls : X [(.05) + (.05) +...(.05) ] = 50,000[ +... v.05 ] Soluo: D The problem ess he ably o deerme he purchase prce of a bod bewee bod coupo daes. Fd he prce of he bod o he prevous coupo dae of Aprl 5, 005. O ha dae, here are 3 coupos (of $30 each) lef. So he prce o Aprl 5, 005 s: 75

18 P = 000 v a all a j = or P = (30-35) 3 a a j = Thus P = $906.3 The Prce (Jue 8) = 906.3[+(74/83)(0.035)] = $ Soluo: D The followg able summarzes wha s requred by he lables ad wha s provded by oe u of each of Bods I ad II. I 6 mohs I oe year Lables requre: $,000 $,000 Oe u of Bod I provdes: $,040 Oe u of Bod II provdes: $ 5 $,05 Thus, o mach he lably cash flow requred oe year, (/.05) =.9756 us of Bod II are requred us of Bod II provde (.9756*5) = mohs. Thus, ( )/040 = us of Bod I are requred. Noe: Checkg aswer choces s aoher approach bu akes loger! Soluo: B Toal cos = cos of us of Bod I + cos of.9756 us of Bod II =.93809*040 v *(5 v v.035 ) =

19 53. Soluo: D Ivesme corbuo = 904; vesme reurs = mohs ad 000 oe year. Thus, he effecve yeld rae per 6 mohs s ha rae of eres j such ha 904 = 000 v j v j = 000 a. Usg BA II Plus calculaor keys: selec d FV; eer 904, selec +/-, selec PV; eer 000, selec PMT; eer, selec N; selec CPT, selec I/Y yelds % forma. Thus, he aual effecve rae = (.03343) = Noe: Eve f s used as PV, he resulg aual effecve eres rae s 6.8% whe rouded o oe decmal po Soluo: C Gve he coupo rae s greaer ha he yeld rae, he bod sells a a premum. Thus, he mmum yeld rae for hs callable bod s calculaed based o a call a he earles possble dae because ha s mos dsadvaageous o he bod holder (earles me a whch a loss occurs). Thus, X, he par value, whch equals he redempo value because he bod s a par value bod, mus sasfy: Prce = 7.5 =.04Xa + Xv or X = 7.5/ (. 04a + v ) = 7.5/.96 = Soluo: A Gve he prce s greaer ha he par value, whch equals he redempo value hs case, he mmum yeld rae for hs callable bod s calculaed based o a call a he earles possble dae because ha s mos dsadvaageous o he bod holder (earles me a whch a loss occurs). Thus, he effecve yeld rae per coupo perod, j, mus sasfy: 30 Prce = 7.5 = 44a + 00v 30 j j or, usg calculaor, j =.608%. Thus, he yeld, expressed as a omal aual rae of eres coverble semaually, s 3.6% Soluo: E Gve he coupo rae s less ha he yeld rae, he bod sells a a dscou. Thus, he mmum yeld rae for hs callable bod s calculaed based o a call a he laes possble dae because ha s mos dsadvaageous o he bod holder (laes me a whch a ga occurs). Thus, X, he par value, whch equals he redempo value because he bod s a par value bod, mus sasfy: 0 0 Prce = 0.50 =.0Xa + Xv or X = 0.50/ (. 0a + v ) = 0.50/.85 = Soluo: B Gve he prce s less ha he par value, whch equals he redempo value hs case, he mmum yeld rae for hs callable bod s calculaed based o a call a he laes possble dae because ha s mos dsadvaageous o he bod holder (laes me a whch a ga occurs). Thus, he effecve yeld rae per coupo perod, j, mus sasfy: 0 Prce = 0.50 = a + 00v 0 j j or, usg calculaor, j =.45587%. Thus, he yeld, expressed as a omal aual rae of eres coverble semaually, s 4.9% j 77

20 58. Soluo: E The rasaco coss are ( for he forward ad for he sock) The prce of he forward s herefore: (50 + ) * (.06) = Soluo: C Frs, he PV of he lably s: PV = 35,000a 5 6.% = 335, The durao of he lably s: d v 35,000v + *35,000v *35,000 = 335, v = v R R 5 =,3,5.95 = , Le X deoe he amou vesed he 5 year bod. X X The, (5) + ( )0 = => X = 08, , , Soluo: A The prese value of he frs egh paymes s: v 000*.03 * v PV = 000v + 000(.03) v (.03) v = = 3, v The prese value of he las egh paymes s: = = PV 000(.03) 0.97v 000(.03) 0.97 v (.03) 0.97 v (.03) 0.97v 000*.03 *0.97 v = = 7, v Therefore, he oal loa amou s L = 0,688.63

21 6. Soluo: E Sce he -year forward prce s hgher ha he -year forward prce, he buyer, relave o he forward prces, overall pays more a he ed of he frs year bu less a he ed of he secod year. So hs meas ha he buyer pays he swap couerpary a he ed of he frs year bu receves moey back from he swap couerpary a he ed of he secod year. So he buyer leds o he swap couerpary a he -year effecve forward eres rae, from he ed of he frs year o he ed of he secod year, amely 6%.

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