Course FM Practice Exam 1 Solutios Solutio 1 D Sikig fud loa The aual service paymet to the leder is the aual effective iterest rate times the loa balace: SP X 0.075 To determie the aual sikig fud paymet, we first eed to determie the relevat accumulated value factor: 5 (1.06) 1 s 56% 5.63709 0.06 The aual sikig fud paymet is: X X SFP s 56% 5.63709 The total aual paymet from the borrower is the combiatio of the service paymet ad the sikig fud paymet, so the total paymets made by Lace over the 5-year period is just five times this amout. This give us a equatio of value, which we ca solve for the loa amout, X: X 17,036.75 X(0.075)(5) (5) 5.63709 17,036.75 0.375X 0.88698X 17,036.75 1.6198X X 13,500.00 Solutio C Iterest rate swap The otioal amout is ot eeded to aswer this questio. The zero-coupo bod prices are: 1 P(0,1) 1.045 0.956938 P(0,) 1.048 0.910495 3 P(0,3) 1.051 0.861374 ActuarialBrew.com 017 Page 1
The 1-year implied forward rates are: r (0,1) s 0.045 0 0 1.048 r0 (1,) 1 0.051009 1.045 3 1.051 r0 (,3) 1 0.05706 1.048 The fixed swap rate is: P(0,1) r 0(0,1) P(0,) r0(1,) P(0,3) r0(,3) R P(0,1) P(0,) P(0,3) 0.956938(0.045) 0.910495(0.051009) 0.861374(0.05706) 0.956938 0.910495 0.861374 0.13866 0.0508.78807 A quicker way to determie the aswer is: 3 1 P(0,3) 1 1.051 R 0.0508 P (0,1) P (0,) P (0,3).78807 Solutio 3 A Accumulated value of cash flows at aual effective iterest rate The equatio of value for Sheryl s accumulated value at time 10 years is: 10 5 0 39,661.45 5,000(1 i) 10,000(1 i) 15,000(1 i) 5 We let x (1 i), ad we ca solve the equatio of value for x with the quadratic equatio: 0 5,000x 10,000x 4,661.45 x x 10,000 (10,000) 4(5,000)( 4,661.45) (5,000) 10,000 4,356.98 10,000 We discard the egative solutio sice it does t make sese with iterest rates, ad the we solve for i: x 1.43563 5 (1 i) 1.43563 1 i 1.075 i 0.075 ActuarialBrew.com 017 Page
Solutio 4 B Varyig perpetuity-immediate ad level perpetuity-due Bob s preset value is: 50 50(1 i) 50a i/(1 i) i Tom s paymets are a compoud-icreasig perpetuity-immediate. The preset value of which is: a j lim lim 1 1 (1 j) 1 1 a j 1 e j 1 e j Sice j (1 e) /( i e), we have: a j 1 1 1 1 e 1 1e j 1 e i e ie Tom s preset value is: 5 5a j i 0.05 Sice their preset values are equal, we ca set up the equatio of value ad solve for i: 50(1 i) 5 i i 0.05 50(1 i)( i0.05) 5i 50( i0.05 i 0.05 i) 5i 50i 43.75i1.5 0 43.75 43.75 4(50)( 1.5) i (50).76949 i 100 i 0.077 We discarded the egative solutio sice it did t make sese for a iterest rate. ActuarialBrew.com 017 Page 3
Solutio 5 A Loa repaymet The first 15 paymets pay the pricipal dow at a rate that is equal to 100% of the iterest rate. Sice the iterest rate is 5%, the portio of the pricipal that is paid dow by each of the first 15 paymets is: (.00 1.00) 0.05 0.05 After 15 years, the origial pricipal has bee reduced by 5% for 15 years. The equatio of value at the ed of 15 years is: 50,000 0.95 15 Xa 15 0.05 15 11.05 115,8.8075 X 0.05 115,8.8075 10.37966X X 11,158.63 The BA-II Plus ca be used to aswer this questio: 50,000 [] 0.95 [y x ] 15 [=] [PV] 15 [N] 5 [I/Y] [CPT] [PMT] The result is 11,158.63, so X 11,158.63. Solutio 6 A Callable bod price Let s work i semiaual effective periods ad defie the bod variables first: F C 1,000 30 60 i 0.08 / 0.04 g r 0.06 / 0.03 coupo 0.03 1,000 30 Sice g < i, the bod is a discout bod, ad the miimum yield is determied from a call at the latest possible call date. (If the call price chages, the we would eed to check the price at the latest date of each call price chage.) The latest call date i this case is to assume that the bod will mature at time 30 years. The appropriate auity factor is: 60 1 (1.04) a.6349 60 4% 0.04 ActuarialBrew.com 017 Page 4
The price of the bod that guaratees a miimum yield of at least a 8% omial aual rate covertible semiaually is: P 60 a 60 4% 30 1,000(1.04) 773.7651 Let s check this aswer usig the BA-35 calculator by pressig [d][cmr], 1,000 [FV], 30 [PMT], 4 [%i], 60 [N], ad [CPT][PV], ad the result is 773.7651. Usig the BA II Plus, we press [d][clr TVM], 1,000 [FV], 30 [PMT], 4 [I/Y], 60 [N], ad [CPT][PV], ad the result is the same. We ca verify that this is the price to guaratee the required miimum retur by calculatig the prices if the bod had bee called at other dates. If the prices at other possible call dates are higher, the the retur would be lower. The table below illustrates the prices at other chose call dates: N Price 40 80.073 41 800.0695 58 775.7043 59 774.7157 Sice all the other prices are higher tha $773.7651, they would provide a lower retur tha the required retur if the bod is ot called early. Solutio 7 C Nomial iterest rate ad force of iterest The accumulated value of Peter s accout after 6.5 years is: i X 1 4 (4) 46.5 The accumulated value i David s accout after 6.5 years is: X e 0.076.5 ActuarialBrew.com 017 Page 5
Sice their accumulated values are equal at this time, we set up the equatio of value ad solve for the aual omial rate of iterest covertible quarterly: 6 (4) i 0.455 X 1 Xe 4 6 (4) i 1 1.576173 4 (4) i 1 1.01765 4 (4) i 0.0706 Alteratively, it is quicker if we ote that both X ad 6.5 years are ot ecessary to work this problem. We the have: 4 (4) i 0.07 1 e 4 (4) i 0.0706 Solutio 8 E Bod coupo Let s work i semiaual effective periods ad defie the bod variables first: F 1,000 C 1,050 30 60 i 0.08 / 0.04 60 1 (1.04) a.6349 60 4% 0.04 We set up the equatio of value for the price of the bod ad we ca solve it for X, the coupo amout: 60 948.19 Xa 1,050(1.04) 60 4% 948.19.6349X 99.8134 X 37.50 The semiaual effective coupo rate is 37.50 /1,000 3.75%, ad the aual coupo rate covertible semiaually is twice this amout: 3.75% 7.5%. ActuarialBrew.com 017 Page 6
Alteratively, we ca use a fiacial calculator to help us determie the aswer. Usig the BA 35, we press [d][cmr], 948.19 [PV], 4 [%I], 1,050 [FV], 60 [N], [CPT][PMT], ad the result is 37.50. Usig the BA II Plus, we press [d][clr TVM], 948.19 [PV], 4 [I/Y], 1,050 [FV], 60 [N], [CPT][PMT], ad we get the same result. Solutio 9 C Macaulay duratio of stock with level divideds The formula for Macaulay duratio is: mt y tcft 1 m MacD mt y CFt 1 m The deomiator i this case is just the preset value of a level perpetuity-immediate, which is give by: 3 1v v 1/(1 i) 1i 1 1v1v 1v v1vv v 1v 1v 11/(1 i) i(1 i) i The umerator is the preset value of a icreasig perpetuity-immediate: a v a 0 1/ 1 ( ) lim ( ) lim d i Ia Ia i i i i We the have: (1 i) / i 1 i MacD 1/ i i We are give tha the Macaulay duratio is 1.0, so we ca solve the equatio for i: 1 i 1.0 i 1.0i 1 i 0.0i 1 i 0.05 Solutio 10 C Loa drop paymet If we accumulate the iitial loa balace to time 1, the we ca treat the loa as a loa with the first paymet occurrig oe year later: 1,000,000 1,041,666.667 1 (1 0.04) ActuarialBrew.com 017 Page 7
The aual effective iterest rate is: 0.04 1 0.04 0.04167 We ca solve the time-0 equatio of value for : 1,041,666.667 100,000a 1 1.04167 1,041,666.667 100,000 0.04167 0.43403 1 1.04167 0.56597 1.04167 l 0.56597l1.04167 l 0.56597 l1.04167 13.9437 Therefore, there are 13 paymets of 100,000 ad a fial drop paymet at time 14: 1,041,666.667 100,000a DropPmt 1 0.04 13 13 1 1.04167 1,041,666.667 100,000 DropPmt 0.96 0.04167 1,041,666.667 100,0009.88317 DropPmt 14 0.96 DropPmt 94,479.31 We ca use the BA II Plus to aswer this questio: 1,000,000 [] 0.96 [=] [+/-] [PV] 0.04 [] 0.96 [] 100 [=] [I/Y] 100,000 [PMT] [CPT] [N] The result is 13.9437. 13 [N] [CPT] [FV] [] 0.96 [=] The aswer is 94,479.31. 14 14 ActuarialBrew.com 017 Page 8
Solutio 11 D Pthly auity-due preset value Liz s auity paymets occur today, ad at times years, 4 years, 6 years, ad so o, util at time 18 years. There are 10 paymets i all. Sice these paymets occur every other year, let s work i two-year effective periods. If we let i deote the two-year effective iterest rate, the equatio of value is: 10 1 (1 i) 6,787.98 10,000a 10,000 10 i/(1 i) We could use trial ad error to determie the value of i that satisfies the above equatio. It is a valid approach durig the exam if it saves time, but we ca also use a calculator to quickly determie the aswer. Usig the BA 35, we press [d][cmr], [d][bgn], 10,000 [PMT], 10 [N], 68,75.84 [PV], [CPT][%i], ad the result is 9.50%. Usig the BA II Plus, we press [d][clr TVM], [d][bgn] [d][set] [d][quit], 10,000 [PMT], 10 [N], 68,75.84 [PV], [CPT][I/Y] ad we get the same result. We eed to be careful, sice this rate of 9.5% is the two-year effective iterest rate. We eed to covert this rate to the aual omial iterest rate covertible semiaually: 1/ () (1/) i i 0.5 1 1 (1.095) 1 0.095 1/ () i 0.04590 Solutio 1 E Forward iterest rate The aswer ca be quickly determied by the followig relatioship: 4 4 (1 s4 ) 1.09 f3 1 1 0.1055 3 3 (1 s3 ) 1.08 Alteratively, sice forward rates ca be a little cofusig, it may help to draw a timelie. The timelie below shows the various spot ad forward rates ad where they are placed o the timelie. ActuarialBrew.com 017 Page 9
f0 f1 f f3 0 1 3 4 s1 = 6.0% s = 7.0% s3 = 8.0% s4 = 9.0% The oe-year forward rate coverig the spa of the fourth year from time 3 years to time 4 years is f 3. To determie this forward rate, we eed to calculate f 1 ad f first: f0 s1 6.0% 1.07 f1 1 0.08009 1.06 3 1.08 f 1 0.1008 (1.06)(1.008009) 4 1.09 f3 1 0.1055 (1.06)(1.08009)(1.1008) Solutio 13 B Dollar-weighted iterest rate The fud balace of $1,050 at the start of the year is accumulated with 1 moths of iterest. The withdrawal o May 1 is accumulated with 8 moths of iterest ad the withdrawal o Jue 15 is accumulated with 6.5 moths of iterest. There are deposits at the ed of every moth from Jauary 31 to December 31, ad each of these deposits are accumulated to the ed of the year. The fud value at the ed of the year is $1,160. We let i be the aual effective iterest rate, ad the equatio of value is: 1/1 11/1 10/1 1/1 0/1 1,050(1 i) 90(1 i) 90(1 i) 90(1 i) 90(1 i) 8 /1 6.5 /1 600(1 i) 400(1 i) 1,160 Sice the fud activity occurs durig a 1-moth period, we ca use the simple iterest approximatio to simplify the above equatio: 1 11 10 1 0 1,050(1 i) [90(1 i) 90(1 i) 90(1 i) 90(1 i)] 1 1 1 1 1 8 6.5 600(1 i) 400(1 i) 1,160 1 1 ActuarialBrew.com 017 Page 10
Recogizig that 1 ( )( 1)/, the part i the brackets above becomes: 90 90 90 1 i(11 10 1) 90 1 i(66) 1,080 495i 1 1 We ca ow solve the full equatio for i, the aual effective iterest rate: 1,050 1,050i1,080 495i600 400i400 16.6667i 1,160 98.3333i 30 i 0.033 Lastly, we solve for the omial aual rate compouded mothly: 1 (1.033) 1 0.0318 (1) 1 /1 i Solutio 14 A Loa: amout of pricipal i a paymet The loa paymet P is the sum of the iterest compoet ad the pricipal compoet: P,058.08 97.79,355.87 We ca determie the balace at time 1 years from the iterest compoet of the 13th aual paymet: I13 B1 i,058.08 B1,867.55556 0.09 The balace at time 1 years is also equal to the preset value of the remaiig future loa paymets. There are 1 remaiig loa paymets, so aother equatio for the loa balace at time 1 years is: B 1 a 1 9%,355.87,867.55556 Usig the BA 35 calculator, we press [d][cmr],,867.55556 [PV], 9 [%i],,355.87 [PMT], [CPT][N], ad the result is 4.00014. Usig the BA II Plus, we press [d][clr TVM],,867.55556 [PV], 9 [I/Y],,355.87 [PMT], [CPT][N] ad we get the same result. Sice 1 4, we kow 36, i.e., the loa iitially had 36 aual paymets. To calculate the amout of pricipal i the 3rd paymet, we eed to kow the balace at time years. At time years, there are 36 14 paymets remaiig, so the appropriate auity factor is: 14 1 (1.09) a 7.78615 14 9% 0.09 The balace at time years is the: B,355.87a 18,343.1581 14 9% ActuarialBrew.com 017 Page 11
Now we ca determie the amout of iterest ad pricipal i the 3rd paymet: I3 ib 0.09 18,343.1581 1,650.8843 P3 P I3,355.87 1,650.8843 704.98577 Solutio 15 A Auity-immediate preset value With iformatio from auity, we ca determie the aual effective iterest rate. Usig the BA 35, we press [d][cmr], 475.54 [FV], 5 [N], 81.06 [PMT], [CPT][%i], ad the result is 7.9993, or 8.0%. Usig the BA II Plus or the BA II Plus Professioal, we press [d][clr TVM], 475.54 [FV], 5 [N], 81.06 [PMT], [CPT][I/Y] ad we get the same result. Sice the accumulated value of auity at time 5 years is $475.54, the preset value at time 0 of auity is: PV 5 (1.08) (475.54) 33.64453 Auity 1 has five aual paymets from time 6 years to time 10 years. If we use a auity-immediate preset value factor to value these paymets, its value is determied at time 5 years (1 year before the first paymet), so we eed to discout the time 5 preset value back 5 years to determie the time 0 preset value. Sice the preset value of auity 1 is twice the preset value of auity, we set up the equatio of value for auity 1 ad solve for the ukow paymet X: 5 (33.64453) X(1.08) a 58% 5 5 1 (1.08) 647.8907 X(1.08) 0.08.71737X 647.8907 X 38.0 ActuarialBrew.com 017 Page 1
Solutio 16 B Redigto immuizatio We do t eed to kow the amout of the liability paymet to aswer this questio. The duratio of the asset portfolio must be equal to duratio of the liability. Let w be the percetage of the asset portfolio that is ivested i the asset that pays at time 4: 4w+ 8(1 - w) = 6-4w = - 1 w = Sice the preset value of the asset portfolio is equal to the preset value of the liability, the preset values of the asset cash flows at time 0 are: PV PV X Y 1 = PV 1 = PV L L The amouts of the cash flows are foud by accumulatig their preset values: 1 4 4 X PV 1.03 L 1.03 X PV 1 = = = = 0.89 Y 8 1 8 4 PVY 1.03 PV 1.03 1.03 L Alteratively, we ca aswer this questio as follows. The preset values of the assets ad liability are: 4 8 PA X(1 i) Y(1 i) 6 PL 50,000(1 i) The derivatives of the preset value of the assets ad liability are: ' 5 9 PA 4 X(1 i) 8 Y(1 i) ' 7 PL 6(50,000)(1 i) Valuig the above equatios whe the aual effective iterest rate is 3%, we have: PA 0.88849X 0.78941Y PL 41,874.18 ' PA 3.45044X 6.13133Y ' PL 43,97.4534 ActuarialBrew.com 017 Page 13
We set the preset values equal to each other ad the derivatives equal to each other ad we have two equatios with two ukow variables: 0.88849X 0.78941Y 41,874.18 3.45044X 6.13133Y 43,97.4534 We multiply the secod equatio by the factor 0.88849/3.45044: 0.88849X 1.5788Y 6,811.3193 Subtractig this result from the first equatio, we have: 0X 0.78941Y 0,937.1064 Y 6,5.50 With this value for Y, we ca ow solve for X: 41,874.18 0.78941(6,5.50) X 3,564.90 0.88849 The ratio is the: X 3,564.90 0.89 Y 6,5.50 Solutio 17 E Bod yield: reivestmet of coupo paymets Let s work i semiaual effective periods ad defie the bod variables first: F 1,000 C 1,050 15 30 i 0.08 / 0.04 r 0.07 / 0.035 coupo 0.035 1,000 35 a 30 4% 1 (1.04) 0.04 The price of the bod is: 30 4% 30 17.903 30 P 35a 1,050(1.04) 98.95577 Alteratively, usig the BA 35, we press [d][cmr], 1,050 [FV], 30 [N], 35 [PMT], 4 [%i], [CPT][PV], ad the result is 98.95577. Usig the BA II Plus, we press [d][clr TVM], 1,050 [FV], 30 [N], 35 [PMT], 4 [I/Y], [CPT][PV], ad we get the same result. ActuarialBrew.com 017 Page 14
George reivests each of the 30 coupos at a semiaual effective iterest rate of 0.09 / 0.045. The accumulated value of these coupos at time 15 years is: 30 (1.045) 1 35s 35,135.4744 30 4.5% 0.045 To determie his aual effective yield, we recogize that George paid $98.95577 at time 0, ad it grew to $1,050 + $,135.4744 = 3,185.4744 at time 15 years. We set up this equatio of value ad solve for i, the aual effective iterest rate: 15 98.95577(1 i) 3,185.4744 15 (1 i) 3.4885 1 i 1.0856 i 0.0856 Solutio 18 B Loa iterest rates The equatio of value is: 10 10,000 X a v a 10 4% 5 6% Determiig the required values, we have: 10 11.04 a 8.11090 10 4% 0.04 10 10 v 1.04 0.67556 5 11.06 a 4.136 56% 0.06 Solvig for X, we have: 10,000 X 8.11090 0.67556 4.136 10,000 10.9566 91.69041 ActuarialBrew.com 017 Page 15
Solutio 19 D Bod ivestmet icome The book value at time 1 is the preset value of the bod s future coupo paymets: 01 BV1 1,000v 0.04 1,000a 01 5% 8 8 11.05 1,0001.05 40 0.05 676.83936 406.4631 935.36787 The ivestmet icome eared durig the 13 th year, i.e., the iterest portio of the 13 th coupo is: 935.367870.05 46.76839 The BA II Plus ca be used to fid the iterest portio of the 13 th coupo: 0 [N] 5 [I/Y] 40 [PMT] 1,000 [FV] [CPT][PV] The result is 875.37790, so the price of the bod at time 0 is 875.38. To determie the ivestmet icome: [ND][AMORT] 13 [ENTER] 13 [ENTER] By cotiuig to hit the dow arrow key, we observe these values: BAL = 94.1367, so PRN = 6.76839, so INT = 46.76839, so BV 13 94.14 DA 13 6.77 IvIc 13 46.77 Solutio 0 B Auity-due accumulated value If Joh had made all of the aual deposits from 1/1/78 to 1/31/07, he would have made 007 1978 1 30 deposits. The deposits were supposed to occur o Jauary 1 of each year from 1/1/78 to 1/1/07. Joh missed deposits 16 through 1. The 16th deposit was supposed to occur o 1/1/93, sice the first deposit occurred o 1/1/78. Likewise, the 1st deposit was supposed to occur o 1/1/98. Thus, Joh made the first 15 deposits (from 1/1/78 to 1/1/9), missed the ext 6 deposits (from 1/1/93 to 1/1/98), ad made the last 9 deposits (from 1/1/99 to 1/1/07). The accumulated value of the first 15 deposits at time 16 years (1/1/93) is: 15 (1.045) 1 10,000 s 10,000 17,193.3673 15 4.5% 0.045 /1.045 ActuarialBrew.com 017 Page 16
Sice we eed the accumulated value as of 1/31/07 (or 1/1/08), we eed to accumulate this accumulated value for aother 008 1993 15 years. The accumulated value of the first 15 deposits at time 1/31/07 is: 15 17,193.3673(1.045) 40,330.5105 The accumulated value of the last 9 deposits at 1/31/07 (i.e., 9 years after 1/1/99) is: 9 (1.045) 1 10,000 s 10,000 11,88.0937 94.5% 0.045 /1.045 The total accumulated value of these two pieces at time 1/1/08 is: 40,330.5106 11,88.0937 533,1.6043 Alteratively, we ca approach this problem from aother agle. We ca assume that Joh made all 30 paymets. The accumulated value of the 30 aual paymets at 1/31/07 is: 30 (1.045) 1 10,000 s 10,000 637,53.8779 30 4.5% 0.045 /1.045 The accumulated value of the 6 deposits that were missed, valued at 1/1/99 is: 6 (1.045) 1 10,000 s 10,000 70,191.5179 64.5% 0.045 /1.045 We eed to accumulate the 1/1/99 accumulated value of the missig deposits to 1/31/07. There are 007 1999 1 9 years from 1/1/99 to 1/31/07, so the 1/31/07 accumulated value of the missed deposits is: 9 70,191.5179(1.045) 104,311.736 Now we ca subtract the accumulated value at 1/31/07 of the missig deposits from the accumulated value at 1/31/07 of the deposits assumig that oe were missed. The resultig accumulated value is: 637,53.8779 104,311.736 533,1.6043 Solutio 1 C Macaulay duratio of bod The formula for Macaulay duratio is: MacD mt y tcft 1 m mt y CFt 1 m ActuarialBrew.com 017 Page 17
For this bod with coupo paymets of $X at the ed of each year, the Macaulay duratio is: 1 3 10 10 X(1)(1.09) X()(1.09) X(3)(1.09) X(10)(1.09) 400(10)(1.09) 1 3 10 10 X(1.09) X(1.09) X(1.09) X(1.09) 400(1.09) This ca be writte as: 7.466 10 X( Ia) 400(10)(1.09) 10 9% 10 Xa 400(1.09) 10 9% Determiig the required values, we have: 10 1 (1.09) a 6.41766 10 9% 0.09 a (1.09)(6.41766) 6.9955 10 9% 10 6.9955 10(1.09) ( Ia) 30.79043 10 9% 0.09 Pluggig these values ito the above equatio, we ca solve for X: X(30.79043) 1,689.6433 7.466 X(6.41766) 168.9643 47.9143X 1,61.48763 30.79043X 1,689.6433 17.1380X 48.15559 X 5.00 Solutio B Compoud decreasig ad icreasig auity The preset value of a compoud icreasig auity-immediate is: 1 1 1 (1 j) ie a where j 1 e j 1e j 1e Sice we have mothly paymets ad icreases that occur mothly, let s work i mothly periods. The mothly effective iterest rate is: (1) i 1/1 (1.15) 1 0.009864 1 Let s split the auity ito two parts: the compoud decreasig part durig the first two years ad the compoud icreasig part durig the last two years. The mothly paymets for the compoud decreasig part start at time 1 moth. The first paymet is $5,000. There are 4 paymets for the decreasig part ad the mothly rate of decrease is 1%. ActuarialBrew.com 017 Page 18
The mothly iterest rate for the first two years is: 0.009864 ( 0.01) j 0.00064 1 ( 0.01) The preset value at time 0 of the decreasig part is: 4 1 1 (1.00064) 5,000 95,455.4166 1 ( 0.01) 0.00064 The mothly paymets for the compoud icreasig part start at time 5 moths. The 3 first paymet for the icreasig part is 5,000(0.99) (1.005). There are also 4 paymets for the icreasig part ad the mothly rate of icrease is 0.5%. The mothly iterest rate for the secod two years is: 0.009864 0.005 j 0.004839 1.005 The preset value at time 4 moths of the icreasig part is: 4 3 1 1 (1.004839) 5,000(0.99) (1.005) 89,706.74436 1 (0.005) 0.004839 The preset value of the decreasig part must be accumulated for five years ad the preset value of the icreasig part must be accumulated for three years to determie the accumulated value of the etire series of paymets at time five years: 5 3 95,455.4166(1.15) 89,706.74436(1.15) 99,740.7459 Solutio 3 E Pricig a bod usig spot rates The bod pays aual coupos of 0.08 1,000 80. The preset value of the bod is: 80 80 80 1,080 PV 973.9560 1.06 3 4 1.07 1.08 1.09 The aual effective yield ca be quickly determied usig a fiacial calculator. Usig the BA 35, we press [d][cmr], 1,000 [FV], 4 [N], 80 [PMT], 973.956 [PV], [CPT][%i], ad the result is 8.80048. Usig the BA II Plus, we press [d][clr TVM], 1,000 [FV], 4 [N], 80 [PMT], 973.956 [PV], [CPT][%i], ad we get the same result of 8.8%. ActuarialBrew.com 017 Page 19
Solutio 4 D Decreasig auity-immediate preset value The paymets start at $1,050 at time 1 year ad decrease by $15 each year. There are 30 paymets, so the last paymet of $615 occurs at time 30 years. The decreases are $15 ad there are 30 paymets, so let s subtract 15 30 450 from the first paymet of $1,050, ad we re left with $600. If we subtract $600 from each of the 30 paymets, we are left with a paymet stream that starts at $450 at time 1 ad decreases by $15 each year to $15 at time 30. The preset value of a level series of 30 paymets of $600, we have: 30 1 1.065 600a 600 600(13.05868) 7,835.0554 30 6.5% 0.065 The preset value of the decreasig paymet stream is: 30 a 30 6.5% 30 13.05868 15( Da) 15 15 3,909.53633 30 6.5% 0.065 0.065 Puttig the two preset values back together, we have: PV 7,835.0554 3,909.53633 11,744.7419 Solutio 5 A Loa balace We are give that the outstadig loa balace at the ed of the 9th year is $1,355.. So the fial loa paymet must be this balace times the quatity of oe plus the effective iterest rate, sice the fial paymet must pay off this balace plus the iterest from the last period o this balace. The level paymet is the: P 1,355. 1.075 1,456.8615 The balace at the ed of the third year is the preset value of the remaiig seve paymets: 7 1 (1.075) B3 1,456.8615a 1,456.8615 7,716.41 77.5% 0.075 Alteratively, sice this is a 10-year loa, the fial loa paymet must cotai exactly this amout as the pricipal amout to pay off the loa, so: P10 1,355. ActuarialBrew.com 017 Page 0
We ca develop a relatioship betwee successive pricipal amouts. We recall that the outstadig loa balace at ay time t is the preset value of the remaiig loa paymets: t 1 v Bt Pa P ti i The amout of iterest i loa paymet t is the periodic effective iterest rate times the prior loa balace, ad the amout of pricipal i loa paymet t is the loa paymet mius the amout of iterest i loa paymet t: t 1 1 v t1 It ibt1 ip P 1 v i t1 t1 Pt P It P P 1 v Pv So if we divide successive loa paymets, we see that a pricipal amout i a loa paymet is (1 + i) times the pricipal amout i the prior loa paymet: Pt 1 Pt t Pv 1 1 i t 1 Pv v The iitial loa amout L is the sum of the pricipal amouts i each loa paymet. I this case, we have: L P1 P P10 We have just determied a iterative relatioship betwee the pricipal amouts i each loa paymet, so we ca restate the above relatioship as: 9 8 L 1,355.v 1,355.v 1,355. 9 1,355.[1 vv v ] 10 1 v 1,355. 1 v 10 1 (1.075) 1,355. 1 (1.075) 1 10,000.0153 So the loa balace at the ed of the third year is the iitial loa amout less the pricipal amouts i the first three loa paymets: B3 LP1 P P3 9 8 7 10,000.0153 1,355.(1.075) 1,355.(1.075) 1,355.(1.075) 7,716.41 ActuarialBrew.com 017 Page 1
Solutio 6 B Net preset value The et preset values for projects X ad Y are: 5 10 NPV 1,000 700(1 i) 700(1 i) X 5 10 NPV 800(1 i) 1,016.31(1 i) Y Sice their et preset values are equal, we set up the equatio of value. We let 5 i x (1 ) ad solve for x usig the quadratic equatio: 5 10 5 10 1,000 700(1 i) 700(1 i) 800(1 i) 1,016.31(1 i) 316.31x 1,500x 1,000 0 1,500 ( 1,500) 4(316.31)(1,000) x (316.31) x 0.8046 or 3.93973 We ca ow solve for the ukow aual effective iterest rate: 5 (1 i) 0.8046 or 5 (1 i) 3.93973 1 i 1.0450 1 i 0.76016 i 0.045 i 0.3984 Sice the egative iterest rate does t make sese, the aswer is 4.5%. Solutio 7 A Surplus The preset value of the liability is: 5,000,000 PVL,086,35.304 15 1.06 The compay eeds to buy a bod that will mature for the liability amout of $5,000,000 at time 15 years, assumig the iterest rate does ot chage. Sice the coupo rate of the 15-year bod is equal to its yield, the 15-year bod is priced at par. The bod s face amout should therefore also be $,086,35.304. To verify this, we have: bod F,086,35.304 aual coupo 0.06,086,35.304 15,179.518 ActuarialBrew.com 017 Page
This bod will exactly match the liability i 15 years, sice the bod s face amout plus the reivested coupos will add to $5,000,000 at time 15 years, assumig the iterest rate does ot chage:,086,35.304 15,179.5185s,086,35.304 15,179.518 15 6% 5,000,000.00 15 1.06 1 The iterest rate chages o 1/31/10, which is exactly 3 years after 1/31/07. The ew iterest rate of 5.5% remais i effect for the remaiig 1 years util 1/31/. The coupos are reivested for 3 years at 6.0%, ad the they are reivested for the remaiig 1 years at 5.5%. The accumulated value of the coupos is the: 0.06 1 AV (coupos) 15 15,179.518 s (1.055) 15,179.518s 36% 15.5% 3 1 1.06 1 1 1.055 1 15,179.518 (1.055) 15,179.518 0.06 0.055,808,81.43 Combied with the face amout, the bod s value at 1/31/ is the:,808,81.43,086,35.30 4,895,137.73 The isurace compay s liability at 1/31/ is still $5,000,000, so the isurace compay s profit at this time is: 4,895,137.73 5,000,000 104,86.7 Solutio 8 B Level auity-due accumulated value factor Expressio A is true sice: (1 i) 1 (1 i) 1 s sice d iv d iv Expressio B is false sice: 1 s (1 i) a (1 i) (1 i) a (1 i) a Expressio C is true sice: 1v 1v s (1 i) a (1 i) (1 i) sice d 1 v d 1 v ActuarialBrew.com 017 Page 3
Expressio D is true sice: 1 1 s (1 i) (1 i) (1 i) (1 i) Expressio E is true sice: s s 1 1 1 (1 i) 1 (1 i) 1 s (1 i) (1 i) so s s 1 Solutio 9 C Callable bod price Workig with what we have, we ca quickly determie the bod s aual coupo paymet $C. The we calculate the price that results i a aual effective yield of 8.34%. There is o eed to determie P. To determie $C, we assume the bod is called at the ed of the 15th year. Usig the BA 35, we press [d][cmr], 1,05 [FV], 15 [N], 8 [%i], 793.89 [PV], [CPT][PMT], ad the result is 55.00. Usig the BA II Plus, we press [d][clr TVM], 1,05 [FV], 15 [N], 8 [I/Y], 793.89 [PV], [CPT][PMT], ad we get the same result. The ivestor actually held the bod for 0 years, whe it was called for $1,05, ad the ivestor s actual aual effective yield was 8.34%. The price that results i a aual effective yield of 8.34% ca ow be determied. Usig the BA 35, we press [d][cmr], 1,05 [FV], 0 [N], 55 [PMT], 8.34 [%i], [CPT][PV], ad the result is 733.1175. Usig the BA II Plus, we press [d][clr TVM], 1,05 [FV], 0 [N], 55 [PMT], 8.34 [I/Y], [CPT][PV], ad we get the same result. Thus, we coclude that the ivestor paid $733.1. Solutio 30 D Mothly auity-immediate preset value Sice Tia s retiremet paymets occur mothly, let s work i mothly periods. We set up the equatio of value for the preset value of these beefits, ad we will be able to determie the aual effective iterest rate. Tia will receive 40 mothly paymets at the ed of each moth for 0 years. 4,000a 587,938.54 40 ActuarialBrew.com 017 Page 4
A fiacial calculator ca quickly determie the iterest rate. Sice we re workig i moths, the result will be a mothly effective iterest rate, which we ca covert to a aual effective iterest rate. Usig the BA 35, we press [d][cmr], 40 [N], 4,000 [PMT], 587,938.54 [PV], [CPT][%i], ad the result is 0.44717. Usig the BA II Plus, we press [d][clr TVM], 40 [N], 4,000 [PMT], 587,938.54 [PV], [CPT][%i], ad we get the same result. The aual effective iterest rate is: 1 i (1.0047717) 1 0.0550 The preset value of the zero-coupo bod is the: 1,000,000 X 381,465.904 18 1.055 Solutio 31 B Amortizig swap The forward rats are: f f f 0,1 1, 0.05 1.03 1 0.0350 1.05 1.035 1 0.04507 1.03,3 The swap rate is: 1 3 500,000(0.05)(1.05) 850,000(0.0350)(1.03) 1,100,000(0.04507)(1.035) 0.0375 1 3 500,000(1.05) 850,000(1.03) 1,100,000(1.035) Solutio 3 C Immuizatio Statemet I is ot true sice Redigto (i.e., classical) immuizatio oly protects a portfolio agaist small chages i iterest rates. Statemet II is ot true sice the duratio of the assets must be established so that it approximately matches the duratio of the liabilities i order to meet the secod immuizatio coditio. Statemet III is true sice the covexity of the assets must be greater tha the covexity of the liabilities i order to meet the third immuizatio coditio of Redigto immuizatio. ActuarialBrew.com 017 Page 5
Oly statemet III is true, so choice C is the correct aswer. Solutio 33 D Dollar-weighted ad time-weighted iterest rates Let s make a table of the give iformatio, where the fud values are valued immediately before the ext cash flow occurs. t Ft ct 0 100 0 0.5 90 0 0.75 105 40 1.0 X 0 The equatio for the time-weighted iterest rate, i, is: 1 90 105 X (1 i) 100 90 0 105 40 1 i 0.009310X We are give that the time-weighted rate, i, equals the dollar-weighted rate, j, plus 0.097. Substitutig this ito the above equatio, we have: 1 ( j0.097) 0.009310X j 0.009310X 1.097 The equatio for the dollar-weighted iterest rate, j, is: 1 0.5 0.5 X 100(1 j) 0(1 j) 40(1 j) Sice all of the cash flows occur withi a year, we ca use the simple iterest approximatio o the above equatio. We have: X 100(1 1 j) 0(1 0.5 j) 40(1 0.5 j) 100 100 j0 10 j40 10 j 10 100j Substitutig our earlier result for j ito this equatio, we ca solve for X: X 10 100(0.009310X 1.097) 0.068966X 10.3 X 149.35 ActuarialBrew.com 017 Page 6
Solutio 34 A Immuizatio The preset value of the liability is: 1 (1.05) PL 5,000a 5,000 5,0007.7173193,043.373 10 5% 0.05 The asset portfolio has the same preset value. The duratio of the liability is: We have: 10 10 5,000(1) v5,000() v 5,000(10) v 5,000 Ia 193,043.373 193,043.373 a 10 i a 10 10 8.1078 10 1.05 Ia 10 5% (1 ) 1.05 7.7173 8.1078 0.05 The duratio of the liability is the: 5,00039,37378 5.09909 193,043.373 39.37378 10 5% The asset portfolio has the same duratio. The duratio of the 4-year zero-coupo bod is 4 ad the duratio of the 8 year zero-coupo bod is 8. If we let X equal the percetage of the asset portfolio to ivest i the four-year zero-coupo bod, we ca set up the equatio of value ad solve for X: 4X 8(1 X ) 5.09909 4X 8 8X 5.09909 X 0.75 The amout ivested i the four-year bods is: 0.75193,043.373 140,000.6039 Solutio 35 D Quoted T-bill rate The curret price of the Caadia T-bill is: 365 75 Caadia Quoted Rate 90 P 365 75 P 4,94.9784 90 0.06176 ActuarialBrew.com 017 Page 7
The aual effective rate is: 365 90 5,000 i 1 0.063 4,94.9784 ActuarialBrew.com 017 Page 8