to Iterest Theory Sample Questios Solutio 1 C Chapter 4, Iterest Rate Coversio After 7.5 years, the value of each accout is the same: 7.5 7.5 0.04 1001 100e 1.336 e l(1.336) 7.5 0.0396 7.5 Solutio E Chapter 7, Level Auities Let j be the effective iterest rate for a iterval of 4 years: 4 j (1 i) 1 There are 10 4-year itervals i 40 years, ad there are 5 4-year itervals i 0 years. Therefore, the accumulated value at the ed of 10 itervals is equal to 5 times the accumulated value at the ed of 5 itervals: 100s 5 100s 10 j 5 j 10 5 (1 j) 1 (1 j) 1 5 j /( j 1) j /( j 1) 10 (1 j) 1 5 5 (1 j) 1 5 (1 j) 1 5 5 (1 j) 4 j 0.3195 The accumulated amout at the ed of 40 years is: 10 1.3195 1 X 100s 100 1.3195 6,194.7 10 0.3195 0.3195 The BA-II Plus ca be used as follows: 4 [y x ] 0. [=] [] 1 [=] [] 100 [=] [I/Y] [ d ] [BGN] [ d ] [SET] [ d ] [QUIT] 10 [N] 100 [PMT] [CPT] [FV] Result is 6,194.7. Solutio is 6,194.7. ActuarialBrew.com 015 Page 1
Solutio 3 C Chapter 3, Simple Iterest Eric ad Mike ear the same amout of iterest durig the last 6 moths of the 8 th year: 15 i i i 1001 00 15 i 1 i 0.946 Solutio 4 A Chapter 1, Sikig Fud The deposits to the sikig fud are equal to 1,67.45 mius the iterest o the loa: 1,67.45 10,000 0.10 67.45 The accumulated value of the deposits is: 10 1.14 1 67.45 s 67.45 67.45 19.3373 1,133.1858 10 0.14 0.14 After repayig the loa, the balace is: 1,133.1858 10,000,133.19 The BA-II Plus ca be used to aswer this questio: 10 [N] 14 [I/Y] 67.45 [PMT] [CPT] [FV] [+/-] [] 10,000 [=] Solutio is,133.19. Solutio 5 E Chapter 13, Dollar-weighted Rate of Retur The icome is the withdrawals mius the deposits, treatig the iitial balace as a deposit ad the fial balace as a withdrawal: Icome Withdrawals Deposits 60 5 5 80 35 75 10 1 10 The fud exposure is the average amout i the fud: Fud exposure (Net deposit)(time deposit is i the fud) 11 10 1 0 10 6.5 75 10 5 5 80 35 1 1 1 1 1 1 1 1 11 1 470 75 10 90.8333 1 1 Page ActuarialBrew.com 015
The simple iterest approximatio for the dollar-weighted rate of retur is the icome divided by the fud exposure: Icome 10 i 0.1101 Fud exposure 90.8333 Solutio 6 C Chapter 8, Varyig Auities The equatio of value at the ed of 1 year ca be used to solve for a : 77.1 1.105 ( Ia) v i 77.1 1.105 i a v v a 77.1 0.105 1.105 a 8.0955 The BA-II Plus ca be used to solve for : 10.5 [I/Y] 8.0955 [PV] 1 [PMT] [CPT] [N] Result is 19.00. Solutio 7 C Chapter 8, Varyig Auities The accumulated value is: 9 8 1,000(0.06) 1001.09 900(0.06) 1001.09 100(0.06) 100 9 8 160 1.09 154 1.09 106 Usig the PI method, we have: P1 160 I 6 10 The preset value is: 10 I I 6 1 (1.09) 610 10 PV0 P1 a v 160 (1.09) i i 0.09 0.09 0.09 93.3333 6.4177 81.607 880.5886 The accumulated value at the ed of 10 years is: 10 880.5886 1.09,084.67 Usig the BA II Plus, we have: 10 [N] 9 [I/Y] 160 [] 6 [] 0.09 [=] [PMT] 6 [] 10 [] 0.09 [=] [FV] [CPT] [PV] ActuarialBrew.com 015 Page 3
PV = 880.5886 0 [PMT] [CPT] [FV] Aswer =,084.67 Questio 8 has bee deleted from the set of sample questios. Solutio 9 D Chapter 1, Loas The first 10 paymets pay the pricipal dow at a rate that is equal to 50% of the iterest rate. Sice the iterest rate is 10%, the portio of the pricipal that is paid dow by each of the first 10 paymets is: (1.50 1.00) 0.10 0.05 After 10 years, the origial pricipal has bee reduced by 5% for 10 years. The equatio of value at the ed of 10 years is: 1, 000 0.95 10 Xa 10 0.10 10 1 1.10 598.7469 X 0.10 X 97.44 The BA-II Plus ca be used to aswer this questio: 1,000 [] 0.95 [y x ] 10 [=] [PV] 10 [N] 10 [I/Y] [CPT] [PMT] Result is 97.4417. Solutio is 97.44. Solutio 10 B Chapter 15, Bods The book value at the ed of 6 years is: 10,000 BV6 0.08 10,000 a 800 3.4651 7,90.9366 40.06 4 1.06 10,693.011 The iterest (which is also kow as the ivestmet icome) portio of the 7 th paymet is: IvIc7 BV6 y 10,693.011 0.06 641.58 The BA-II Plus ca be used to aswer this questio: 4 [N] 6 [I/Y] 0.08 [] 10,000 [=] [PMT] 10,000 [FV] [CPT] [PV] Result is 10,693.011. [] 0.06 [=] Page 4 ActuarialBrew.com 015
Result is 641.5813. Aswer is 641.58. Solutio 11 A Chapter 11, Geometric Varyig Auities At the ed of 5 years, the value of the perpetuity s remaiig paymets is equal to the value of the 5-year auity-immediate: 100 3 4 5 X v 1.08v 1.08 v 1.08 v 0.08 3 5 100 1 1 1 4 1 X 1.08 1.08 1.08 0.08 1.08 1.08 1.08 1.08 100 5 X 0.08 1.08 100 1.08 X 0.08 5 X 54 Solutio 1 C Chapter 5, Accumulated Value The $10 iitial deposit accumulates over the first 10 years (40 quarters) at a omial discout rate of d compouded quarterly ad the over the ext 0 years (40 half years) at a omial iterest rate of 6% compouded semiaually. The $0 deposit at time 15 years accumulates for 15 years (30 half years) at a omial iterest rate of 6% compouded semiaually. The equatio of value at the ed of 30 years ca be solved for d: 10 40 30 (1.03) 0(1.03) 100 40 d 1 4 d 0.0453 Solutio 13 E Chapter 5, Varyig Force of Iterest At time 3, before the deposit of X, the value of the fud is: 3 3 3 t t dt 300 7 0 100 0 300 100e 100e 100e A deposit of X is made at time 3, ad the iterest eared from time 3 to time 6 is equal to X: ActuarialBrew.com 015 Page 5
6 3 3 3 7 300 300 100e300 X e 1 X 7 300 0.63 100e X e 1 X 96.059 0.14X X 784.59 Solutio 14 A Chapter 11, Geometric Progressio Auities Problems ivolvig geometric auities ca be solved usig the formula for a geometric series: 1 a(1 r ) S a ar ar ar 1 r First term Term that would come ext 1 Ratio The preset value of the perpetuity-immediate is 167.50: 3 4 5 6 7 10v 10v 10v 10v 10v 10 (1 K) v (1 K) v 167.50 6 (1 Kv ) 0 10a 10 167.50 5 1 (1 Kv ) 1 (1 K) 5 6 1 1.09 (1 Kv ) 10 10 167.50 0.09 1 (1 Kv ) (1 Kv ) 10 3.8696 10 167.50 1 (1 Kv ) 1 1.8407 v K 0.0400 6 The questio referred to K% istead of K, so we multiply the value of K foud above by 100: 0.0400 100 4.00 Solutio 15 B Chapter 1, Loas The amout of the equal aual paymets uder optio (i) is:,000,000,990.0073 a 6.6889 10 0.0807 Page 6 ActuarialBrew.com 015
Alteratively, the amout of the equal aual paymets uder optio (i) ca be foud usig the BA-II Plus calculator: 10 [N] 8.07 [I/Y],000 [+/-][PV] [CPT] [PMT] Result is 99.0007. The sum of the paymets uder optio (i) is: 99.0007 10,990.0073 Sice the paymets of 00 uder optio (ii) are over ad above the paymet of the iterest, the balace of the loa decreases by 00 year. Therefore, the iterest paymets declie each year. The sum of the paymets uder optio (ii) is: 00 10 i(,000 1,800 00),000 00 i(10 9 1) 10 11,000 00i,000 11,000i Settig the sum of the paymets uder optio (i) equal to the sum of the paymets uder optio (ii) allows us to solve for i:,990.0073,000 11,000i 990.0073 i 11,000 i 0.0900 Solutio 16 B Chapter 11, Geometric Progressio Auities There are 60 mothly paymets. We use oe moth as the uit of time: (1) i 0.09 0.0075 1 1 1 v 1.0075 0.9956 The 60 paymets are described below: Time Paymet 1 1, 000 1, 000 0.98 3 1, 000 0.98 40 39 1, 000 0.98 41 40 1, 000 0.98 60 59 1,000 0.98 ActuarialBrew.com 015 Page 7
The outstadig balace after the 40 th paymet is the preset value of the paymets after the 40 th paymet: 40 40 41 59 0 PV 1,000 0.98 v 1,000 0.98 v 1,000 0.98 v 40 60 1 0.98 v 0.98 v 1,000 6, 889.11 1 0.98v Solutio 17 C Chapter 6, Level Auities The equatio of value at the ed of 3 years ca be used to fid i: 8,000 98 s (1 i) 196s 8,000i 98 (1 i) 1 (1 i) 196 (1 i) 1 8,000i 98 14 196 4 1 i 0.15 Solutio 18 B Chapter 8, Varyig Auities We use oe moth as the uit of time. The mothly effective iterest rate is: 1/3 0.09 1 1 0.00744 4 Usig the PI method, we have: P1 I 60 i 0.00744 The preset value is: I I PV0 P1 a v i i 60 1 (1.00744) 60 (1.00744) 0.00744 0.00744 0.00744 70.6568 48.485 10,39.5813,79.1 Usig the BA II Plus, we have: 0.09 [] 4 [] 1 [=] [y x ] 3 [1/x] [=] [] 1 [=] [STO] 1 60 60 [N] [RCL] 1 [] 100 [=] [I/Y] [+] [] [RCL] 1 [=] [PMT] [] 60 [] [RCL] 1 [=] [+/-] [FV] [CPT] [PV] Result is PV =,78.11. Aswer is,79.1 Page 8 ActuarialBrew.com 015
Solutio 19 C Chapter 13, Dollar-Weighted ad Time-Weighted Retur We ca use the simple iterest approximatio to fid the dollar-weighted rate of retur for accout K. The icome ad fud exposure are: Icome Withdrawals Deposits 15 X 100 X 5 X Fud exposure (Net deposit)(time deposit is i the fud) 100 0.5X 0.5( X) 100 The dollar-weighted rate of retur is approximately: Icome 5 X i Fud exposure 100 We set the dollar-weighted retur for accout K equal to the time-weighted retur for accout L: 5 X 15 105.8 1 100 100 15 X 15 105.8 5 X 100 15 X 13,5 15 X 15 X (15 X) 13, 5 15 X 115 X 10 or X 40 There are two possibilities for the value of i: 5 X 5 10 5 40 15 0.15 or.15 100 100 100 100 We use the positive value of 15%. Solutio 0 A Chapter 3, Preset Value The equatio of value at the outset is: 10 10 100 00v 300v 600v 10 100 00 0.76 300 0.76 600v 0.7088 v i 0.0350 ActuarialBrew.com 015 Page 9
Solutio 1 A Chapter 10, Cotiuously Payable Auities The equatio of value at time 10 is: b b a 10 1 s t 10 (8 ) ds 0,000 k(8 t) e dt 0 t b t s rds AV Pmt e dt We begi by evaluatig the itegral i the expoet: 10 10 t t 1 18 (8 s) ds l(8 s) l(18) l(8 t) l 8 t The equatio of value ca ow be used to solve for k: 18 k t dt t 0,000 10 (8 ) 0 8 10 0,000 18kdt 0 0,000 k(180 0) k 111.11 Solutio D Chapter 15, Bods The price of the bod is: 1 v P Coupa Rv 1,000r 381.50 i i 1,000 1.0315(1 v ) 381.50 1,031.5(1 0.5889 ) 381.50 1,055.11 Solutio 3 D Chapter 13, Net Preset Value The et preset value of Project P is:,000 4,000 NPV0 PV0(Cash Iflows) PV0(Cash Outflows) 4, 000 1.10 1.10 1,13.9669 We ca set this equal to the ext preset value of Project Q: 4,000 X 1,13.9669,000 1.10 1.10 X 5,460 Page 10 ActuarialBrew.com 015
Solutio 4 E Chapter 1, Sikig Fud We ca use the BA-II Plus calculator to fid the amout of the aual paymet: 0 [N] 6.5 [I/Y] 0,000 [PV] [CPT] [PMT] Result is 1,815.179. We reduce the paymet amout by the iterest that is paid to the leder i order to obtai the sikig fud paymet that must accumulate to 0,000 at the ed of 0 years. We ca use the calculator to fid the iterest rate that the sikig fud must ear. Cotiuig from the calculatio above, the keystrokes are: [+] 0.08 [] 0,000 [=] [PMT] Result is 15.179, idicatig that the sikig fud paymet is 15.179. 0 [PV] 0,000 [FV] [CPT] [I/Y] Result is 14.179. The solutio is 14.18%. Solutio 5 D Chapter 6, Perpetuities Bria s share of the preset value of the perpetuity is 40%: X Xa 0.4 i 1v 0.4 v 0.6 The charity s share of the preset value of the perpetuity is K: X X v K i i 0.6 K K 0.36 Solutio 6 D Chapter 1, Loas The iterest paid by Seth is: 10 5, 000 1.06 1 3, 954.385 The iterest paid by Jaice is: 5,000 10 0.06 3,000 ActuarialBrew.com 015 Page 11
The iterest paid by Lori is the sum of the 10 paymets mius the total pricipal paid. The total pricipal paid is equal to the iitial pricipal of 5,000: 5,000 5,000 10 5,000 10 5,000 679.3398 10 5,000 a 7.3601 10 0.06 1,793.3979 The total amout of iterest paid o all 3 loas is: 3,954.385 3,000 1,793.3979 8,747.64 Solutio 7 E Chapter 3, Accumulated Value Sice the amout of iterest eared i Bruce s accout durig the 11 th year is equal to the amout of iterest eared i Robbie s accout durig the 17 th year, the amout i Bruce s accout at the ed of the 10 th year must be equal to the amout i Robbie s accout at the ed of the 16 th year: 10 16 100(1 i) 50(1 i) 6 (1 i) i 0.15 The iterest eared i Bruce s accout i the 11 th year is: 10 10 X 100(1 i) i 100(1.15) 0.15 38.88 Solutio 8 D Chapter 1, Loas The outstadig pricipal at the ed of ( t 1) years is: a a ( t1) t1 The outstadig pricipal at the ed of t years is: a t The sum of the iterest paid i year t ad the pricipal paid i year ( t 1) is: X a i (1 a i) 1 v 1 (1 v ) 1 v v t1 t t t 1 v ( v 1) 1v d The aswer is Choice D. t1 t t1 t Page 1 ActuarialBrew.com 015
Solutio 9 B Chapter 7, Perpetuities The preset value of the first auity ca be used to fid the effective 3-year iterest rate: 10 3 (1/3) i 1/3 (1/3) i 0.315 1/3 The 4-moth effective iterest rate is foud below: m ( m) ( p) i i 1 1 m p (3) 1/3 i 1.315 1 3 (3) 1/3 1/3 i 3 1.315 1 0.03068 The preset value of the secod perpetuity is: 1 X 3.60 0.03068 p 3 Solutio 30 D Chapter 17, Asset-Liability Matchig O 1/31/017, the compay will receive the par value of 8,703, leavig the followig et liability: 1,000,000 8,703 177,97 Uder Sceario A, the profit is: 1.045 1 0.05 8,703s 177,97 0.05 8,703 177,97 40.045 0.045 1,31.97 Uder Sceario B, the profit is: 1.055 1 0.05 8,703s 177,97 0.05 8,703 177,97 40.055 0.055 1,3.78 Choice D best describes the isurace compay s profit. 4 4 ActuarialBrew.com 015 Page 13
Solutio 31 D Chapter 11, Geometric Progressio Auities The 0 paymets are described below: The preset value of the paymets is: Time Paymet 1 5,000 1.07 5,000 1.07 3 3 5,000 1.07 0 0 5,000 1.07 0 0 5,000 1.07v 5,000 1.07 v 5,000 1.07 v 1 1.07 1.07 1.05 1.05 5,000 1,633.60 1.07 1 1.05 Solutio 3 C Chapter 13, Net Preset Value The 60,000 received at the ed of 3 years is let for 1 year at 4%. The et preset value is: 60,000(1.04) 60,000 NPV 100,000 698.78 4 1.05 Solutio 33 B Chapter 18, Spot Rates The value of the bod is: 60 60 1,060 96.03 1.07 3 1.08 1.09 Solutio 34 E Chapter 18, Spot Rates We ca use the BA-II Plus calculator to aswer this questio: 60 [] 1.07 [+] 60 [] 1.08 [x ] [+] 1,060 [] 1.09 [y x ] 3 [=] (Result is 96.096) [+/-] [PV] 3 [N] 60 [PMT] 1,000 [FV] [CPT] [I/Y] (Result is 8.9180) Page 14 ActuarialBrew.com 015
Aswer is 8.9%. Solutio 35 C Chapter 16, Duratio The modified duratio ca be expressed i terms of the derivative of the bod s price or i terms of the Macaulay duratio: 700 MacDur 100 1 0.08 MacDur 7.56 ( m) P' y ModDur ( m) P y MacDur ( m) y 1 m Solutio 36 C Chapter 16, Duratio The price of the stock ad the derivative of its price are: Div 1 Py ( ) Divy y P'( y) Div y The modified duratio is: P' y Div y 1 1 ModDur 10 P y 1 Div y y 0.10 The Macaulay duratio is: MacDur ModDur (1 y) 10 1.10 11 Solutio 37 B Chapter 16, Duratio The price of the stock ad the derivative of its price are: Div Py ( ) Div( y g) y g P'( y) Div ( y g) The modified duratio is: 33.3333 1 P' y Div ( y g) 1 1 1 ModDur P y 1 Div ( y g) y g 0.05 0.0 0.03 ActuarialBrew.com 015 Page 15
The Macaulay duratio is: MacDur ModDur (1 y) 33.3333 1.05 35 Questios 38-44 have bee deleted from the set of sample questios. Solutio 45 A Chapter 13, Project Appraisal We ca use the fact that the time-weighted rate of retur is zero to solve for X: 1 X (1 0) 10 1 X 10 X 1 1 X 10 10X 1X X 60 The ivestmet icome ad the exposure of the fud to iterest are: Icome Withdrawals Deposits X 10 X 10 Fud exposure (Net deposit)(time deposit is i the fud) 10 0.5X 10 0.5 60 40 The simple iterest approximatio for the dollar-weighted rate of retur is: Icome 10 i 0.5 Fud exposure 40 Solutio 46 A Chapter 1, Loas The fial paymet is the loa balace at the ed of year 3 accrued with iterest: Pmt 559.1 1.08 603.8496 The iitial loa balace is: 4 1 1.08 Pmt a 603.8496,000.065 4 0.08 The first pricipal paymet is equal to the level paymet mius the iterest o the iitial loa balace: 603.8496,000.065 0.08 443.8475 Solutio 47 B Chapter 15, Bods Page 16 ActuarialBrew.com 015
Sice the yield is equal to the coupo rate, the price of the bod is 1,000. Sice the ivestmet i the bod results i a yield of 7%, we have the followig equatio of value: 10 1,000(1.07) 30s 1,000 0.5 0 (1 i) 1 We ca use the BA-II Plus calculator to aswer this questio: 1,000 [] 1.07 [y x ] 10 [] 1,000 [=] [FV] 0 [N] 30 [+/-] [PMT] [CPT] [I/Y] Result is 4.7597. [] 100 + 1 [=] [x ] [] 1 [=] Solutio is 0.09746. Solutio 48 A Chapter 7, Level Auities The mothly effective iterest rate is: 0.08 0.00667 1 To have 3,000 of mothly icome begiig o his 65 th birthday, the ma eeds the followig lump sum o his 65 th birthday: 3,000 1,000 310,880.83 9.65 His cotributios must accumulate to 310,880.83: Xs 51 0.00667 300 310,880.83 1.00667 1 X 1.00667 310,880.83 0.00667 957.3666X 310,880.83 X 34.7 We ca use the BA II Plus to aswer this questio: [ d ] [BGN] [ d ] [SET] [ d ] [QUIT] 3,000 [] 9.65 [] 1,000 [=] [FV] 5 [] 1 [=] [N] 8 [] 1 [=] [I/Y] [CPT] [PMT] Aswer is 34.7 ActuarialBrew.com 015 Page 17
Solutio 49 D Chapter 6, Level Auities The parets make 17 cotributios of X. O the daughter s 18 th birthday, the equatio of value is: 17 16 3 X 1.05 1.05 1.05 50,000 1 v v v 17 k 3 X 1.05 50,000 1 v v v k1 Oly Choices A ad D show 17 cotributios, so the correct aswer must be Choice A or Choice D. The right side of the equatio i Choice A shows the value of the withdrawals at time 17, but the left side of Choice A shows the value of the cotributios at time 0, so the equatio of value is ot correct. The right side of Choice D shows the value of the withdrawals at time 18, ad the left side of Choice D shows the value of the cotributios at time 18, so the equatio of value is valid, ad Choice D is the correct aswer. Questio 50 has bee deleted from the set of sample questios. Solutio 51 D Chapter 17, Dedicatio The quatity of Bod II to purchase is the quatity that produces a cash flow of $1,000 at time 1: 1,000 QII 0.97561 1, 05 The quatity of Bod I to purchase is the quatity that produces the et liability remaiig after the paymet from Bod II is received: 1,000 5 0.97561 Q I 0.93809 1,040 Choice D is the correct aswer. Solutio 5 C Chapter 17, Dedicatio The liability of 1,000 due i two years is met by arragig a paymet of 1,000 i two years from Mortgage II. Sice Mortgage II makes a equal-sized paymet at time 1, it also pays 1,000 at time 1. That leaves 1,000 of et liability at time 1 to be met by Mortgage I. The amout let is: 1,000 1,000 1,000 X Y,751.41 1.06 1.07 1.07 Page 18 ActuarialBrew.com 015
Solutio 53 A Chapter 18, Forward Rates Bod II must produce a cash flow at time that is sufficiet to grow to,000 at a 6.5% iterest rate:,000 1,877.9343 1.065 The combied prices of Bod I ad Bod II are: 1,000 1,877.9343,583.66 1.06 1.07 Solutio 54 C Chapter 15, Callable Bods Sice the coupo rate is greater tha the yield-to-worst, the bod is a premium bod. The yield-to-worst ca therefore be foud by idetifyig each iterval defied by level redemptio prices ad cosiderig the possibility that the bod is called at the begiig of each iterval. I this case, there is oly oe such iterval, ad it rus from time 15 years util maturity. The yield-to-worst is based o the begiig of the iterval, which occurs at the time 15 years. The equatio of value is: X 1,7.5 0.04Xa 30 0.03 30 1.03 We ca use the BA II Plus to aswer this questio: 30 [N] 3 [I/Y] 0.04 [PMT] 1 [FV] [CPT] [PV] Result is 1.1960. [1/x] [] 1,7.5 [=] Result is 1,440.0030. Aswer is 1,440.00. Solutio 55 B Chapter 15, Callable Bods We observe that the coupo of 40 is greater tha the yield-to-worst times the fial redemptio value: YTW R 0.03 1,100 33 Therefore, the bod is a premium bod, ad the earliest possible redemptio withi each iterval of level redemptio prices should be cosidered. The priceto-worst is the miimum of the two resultig prices: 1,00 1,100 Mi 40 a, 40 a 30 0.03 30 40 0.03 40 1.03 1.03 ActuarialBrew.com 015 Page 19
We ca use the BA II Plus to aswer this questio: 30 [N] 3 [I/Y] 40 [PMT] 1,00 [FV] [CPT] [PV] Result is 1,78.4018. [STO] 1 40 [N] 1,100 [FV] [CPT] [PV] Result is 1,61.8034. Sice 1,61.8034 is less tha 1,78.4018, the price-to-worst is 1,61.8034. Solutio 56 E Chapter 15, Callable Bods The coupo is less tha the product of the yield-to-worst ad the fial redemptio value, so the bod is a discout bod: 0.0 X 0.03 X Coup YTW R tk Therefore, the yield-to-worst is based o the latest possible redemptio, which occurs at the ed of 10 years. The equatio of value is: X 1,01.50 0.0Xa 0 0.03 0 1.03 We ca use the BA II Plus to aswer this questio: 0 [N] 3 [I/Y] 0.0 [PMT] 1 [FV] [CPT] [PV] Result is 0.851. [1/x] [] 1,01.50 [=] Result is 1,00.0349. Aswer is 1,00.03. Solutio 57 B Chapter 15, Callable Bods Sice the price is less tha the redemptio value of 1,100, the bod is a discout bod. Therefore, the yield-to-worst is based o the latest possible redemptio, which occurs at the ed of 10 years. The equatio of value is: 1,100 1,01.50 (0.0 1,100) a 0y 0 (1 y) We ca use the BA II Plus to aswer this questio: 0 [N] 1,01.50 [+/-] [PV] 0.0 [] 1,100 [=] [PMT] 1,100 [FV] [CPT] [I/Y] Result is.4559. Page 0 ActuarialBrew.com 015
[] [=] Result is 4.9117. Aswer is 4.91%. Questio 58 has bee deleted from the set of sample questios. Solutio 59 C Chapter 17, Asset-Liability Maagemet The Macaulay duratio of the liability is: t PV0 CF 15 t a v 15 15,000( Ia) t 0 15 0.06 66.071 MacDur PV CF 15,000a 9.5866 9.5866 t 0 0 t 15 6.891 Let X be the percetage of the preset value that is ivested i the 5-year bods: 6.891 5 X (1 X)10 X 0.616 The amout ivested i the 5-year bods is: X 35,000 a 0.616 35,000 9.5866 08,556.1 15 The BA II Plus ca be used to aswer this questio: 15 [N] 6. [I/Y] 1[PMT] [CPT] [PV] [+/-] Result is 9.5865. [STO] 1 [] 1.06 [] 15 [] 1.06 [y x ] 15 [=] [] 0.06 [=] Result is 66.071. [] [RCL] 1 [=] Result is 6.891. [] 10 [=] [] 5 [+/-] [=] Result is 0.616. [] [RCL] 1 [] 35,000 [=] Solutio is 08,556.1. 15 ActuarialBrew.com 015 Page 1
Solutio 60 A Chapter 11, Geometric Progressio Auities The 16 paymets are described below: Time Paymet 1,000,000 1.03 3,000 1.03 8 7,000 1.03 9 7,000 1.03 0.97 10 7 The preset value of the paymets is:,000 1.03 0.97 16 7 8,000 1.03 0.97 8 7 L,000v,000v 1.03,000v 1.03 7 8 8 8,000 1.03 v 0.97v 0.97 v 0.97 v v 1.03 v 0.97v 0.97 v,000,000 1.03 v 1 1.03v 1 0.97v,000 6.568 1, 431.5956 5.754 13,136.4140 7,55.193 0,688. 63 8 9 9 9 7 8 Solutio 61 E Chapter 5, Varyig Force of Iterest The equatio of value is: t 500 0 sds e,000 s t 100 exp ds 4 0 3 3 s 150 t 0 3 s 100 s 3 150 ds l4 To evaluate the itegral, let s use the followig substitutio: 3 s u 3 150 s du ds 50 Page ActuarialBrew.com 015
The itegral is: s 100 s 3 150 t st st 1 st 3 ds 0.5 0.5l 0.5l 3 s 0 u du u 0 s 0 150 3 s s0 3 3 3 t 0.5 l 150 3 t 150 l(3) l 3 We ca ow solve for t: 3 3 t 150 3 t 18.9 4 Solutio 6 E Chapter 15, Bods The rate of growth of the accumulatio of discout is equal to the yield: DAt k k (1 y) DAt 306.69 (1 ) 5 y 194.8 y 0.09500 We use the discout accumulated i the 15 th year to fid the differece betwee the yield times the redemptio value ad the coupo: t1 DAt ( Ry Coup) v Ry Coup 194.8 1.09500 40151 Ry Coup,06.401 The discout at the time of purchase is: Discout ( Ry Coup) a,06.401 a y 40 0.09500,06.401 10.475 1,134.59 Solutio 63 A Chapter 1, Loas We begi by fidig the level paymet amout: t t1 Pr v Pmt Pmt 699.68 1.0475 Pmt 84.3946 851 ActuarialBrew.com 015 Page 3
The total amout of iterest paid o the loa is equal to the total amout of the paymets mius the iitial loa balace: 1 1.0475 8 84.3946 84.3946 a 6,739.1570 84.3946 8 0.0475 6,739.1570 84.3946 6.590 6,739.1570 5,500.051 1,39.13 The BA II Plus ca be used to aswer this questio: 699.68 [] 1.0475 [y x ] 4 [=] Result is 84.3946. [PMT] 8 [N] 4.75 [I/Y] [CPT] [PV] Result is 5,500.051. [+] 8 [] [RCL] [PMT] [=] Aswer is 1,39.13. 8 Solutio 64 D Chapter 7, Level Auities The mothly iterest rate for the first 18 moths is: 0.084 0.007 1 The accumulated value of the loa after 18 moths mius the accumulated value of the paymets is: 18 1.007 1,000(1.007) 450.30s 4,943.564 450.30 18 0.007 0.007 4,943.564 450.30 19.111 16,337.0983 The ew iterest rate is to refiace the loa is: 0.048 0.004 1 The equatio of value for the refiaced loa after 18 moths is: 16,337.0983 Xa 4 0.004 1 1.004 16,337.0983 X 0.004 16,337.0983.8405X X 715.7 4 We ca use the BA II Plus to aswer this questio: 18 [N] 0.7 [I/Y],000 [PV] 450.30 [+/-] [PMT] [CPT] [FV] (Result is 16,337.0983) [PV] 4 [N] 0.4 [I/Y] 0 [FV] [CPT] [PMT] 18 Page 4 ActuarialBrew.com 015
Aswer is 715.7. Solutio 65 C Chapter 16, Duratio Sice the bod is priced at par, its Macaulay duratio is: 14 ( m) () 1 1.038 MacDur a a 1.038 5.56 7 0.076 Solutio 66 A Chapter 16, Duratio The modified duratio is: MacDur 7.959 ModDur 7.444 ( m) y 1.07 1 m The estimated percetage chage i the price is: ( ) % P ModDur y m 7.444 (0.08 0.07) 0.05940 The estimate for the ew price is: 1,000(1 % P) 1,000(1 0.05940) 940.60 Solutio 67 E Chapter 18, Forward Rates The wordig of this questio is a little ambiguous, as it seems that we are beig asked for either: f, the forward rate that applies from time to time 3, or f 3, the forward rate that applies from time 3 to time 4 Sice we are ot give eough iformatio to fid f 3, we coclude that we are beig asked for f : 1 f 1 f f t (1 st ) t 1 t 1 (1 st 1) 1 0.8589 1 0.90703 0.05601 ActuarialBrew.com 015 Page 5
Solutio 68 C Chapter 16, Duratio Sice we are ot told otherwise, we assume that the yield remais costat at 5%. After the first coupo is paid, the bod is still priced at par, so the remaiig 7- year bod has a price of 5,000. The duratio of a immediate paymet of 50 is 0, ad the duratio of the 7-year bod is d. A portfolio cosistig of a immediate paymet of 50 ad the 7-year bod has a duratio of d 1. 50 5,000 d MacDur w MacDur 0 d The ratio is: 1 Port j j 50 5,000 50 5,000 j1 d d 0.954 d 0.954 d k 1 0.954 d Solutio 69 A Chapter 17, Dedicatio The quatity of Bod C that is purchased is: 100 QC 105 The quatity of Bod A that is purchased is the amout eeded to cover the remaiig liability after the cash flow from Bod C is received: Q A 100 99 5 105 0.8807 107 Solutio 70 B Chapter 17, Redigto Immuizatio A is true, because uless the portfolio is cash-flow-matched, the duratio of the assets ca chage at a differet rate from the duratio of the liabilities. B is false because Redigto immuizatio requires frequet rebalacig. C is true. Full immuizatio protects agaist large chages i the iterest rate, but Redigto immuizatio oly protects agaist small chages. D is true, because Redigto immuizatio is based o a assumptio that the yield curve is flat. E is true, because Redigto immuizatio is based o a assumptio that ay shifts to the yield curve are parallel shifts. Page 6 ActuarialBrew.com 015
Solutio 71 D Chapter 17, Full Immuizatio Sice the liability is equidistat from the asset cash flows, the weights (or market values) of the asset cash flows must be equal. This implies that the value of A at time 4 is 3,000 ad the value of B is also 3,000: A(1.05) 3,000 A,71.0884 B 3,000 B 3,307.5000 1.05 The absolute value of the differece is: AB,71.0884 3,307.5000 586.41 Solutio 7 A Chapter 17, Full Immuizatio The preset value of the assets is equal to the preset value of the liability: 5,000 B 1,000 5 8b 8 1.03 1.03 1.03 B 5,159.8669 8b 1.03 The duratio of the assets is equal to the duratio of the liability: 5 5,000 (8 bb ) 8 1,000 5 8b 8 1.03 1.03 1.03 5,000 96,000 (8 b) 5,159.8669 5 8 1.03 1.03 8 b 10.5076 b.5076 We ca ow solve for B: B 5,159.8669 10.5076 1.03 B 7,039.687 The ratio is: B 7,039.687 b.5076,807.1 Solutio 73 D Chapter 17, Immuizatio The first setece of the questio could be more clearly stated as, Trevor has asset cash flows at time of A ad at time 9 of B. ActuarialBrew.com 015 Page 7
The duratio of the asset portfolio must be equal to duratio of the liability. Let X be the percetage of the asset portfolio that is ivested i the asset that pays at time : X 9(1 X) 5 7X 4 4 X 7 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: 4 PVA PVL 7 3 PVB PVL 7 The amouts of the cash flows are foud by accumulatig their preset values: 4 1.04 PV 1.04 A PV L A 7 4 1.013 B 9 3 9 7 PVB 1.04 PV 1.04 3 1.04 L 7 Solutio 74 D Chapter 15, Bods The price of Bod A exceeds the price of Bod B by 5,341.1: i 10,000 i 10,000 10,000 0.0 10,000 0.0 a 0 i/ a 0 i/ 1 i 1 i 5,341.1 10,000 0.04 a 5,341.1 0 i / We ca use the BA II Plus to aswer this questio: 5,341.1 [] 10,000 [] 0.04 [=] [PV] 0 [N] 1 [+/-] [PMT] [CPT] [I/Y] [] [=] Result is 8.4000. Aswer is 0.0840. Page 8 ActuarialBrew.com 015
Solutio 75 D Chapter 1, Loas The iitial loa paymet is: 400,000 400,000 400,000 Pmt 4,057.0663 a 180 98.5934 151 0.0075 1 1.0075 0.0075 The balace after the 36 th paymet ca be foud usig the prospective method. At the ew iterest rate, the smaller paymets pay off the balace i 1 years: 4,057.0663 a (4,057.0663 409.88) a 144 0.0075 144 j /1 4,057.0663 87.8711 3,647.1863 a 144 j /1 356,498.8491 3,647.1863 a 144 j /1 The easiest way to fid j is to use the BA II Plus calculator. Let s use the calculator from the begiig of this questio: 180 [N] 9 [] 1 [=] [I/Y] 400,000 [+/-] [PV] [CPT] [PMT] Result is 4,057.0663. 144 [N] [CPT] [PV] Result is 356,498.8491. [RCL] [PMT] [] 409.88 [=] [PMT] [CPT] [I/Y] [] 1 [=] Result is 6.9000. Aswer is 6.90%. Solutio 76 D Chapter 15, Bods The equatio of value that equates the prices of the two bods is: 1,00 800 5 a 5 a 60 0.05 60 60 j / 60 1.05 j 1 The easiest way to fid j is to use the BA II Plus calculator: 60 [N].5 [I/Y] 5 [PMT] 1,00 [FV] [CPT] [PV] Result is 1,045.4567. 800 [FV] [CPT] [I/Y] [] [=] Result is 4.3985. Aswer is 4.40%. ActuarialBrew.com 015 Page 9
Solutio 77 E Chapter 3, Iterest Rate Coversios Iterest is credited oly at the ed of each iterest coversio period, so Lucas receives iterest oly every 6 moths. Therefore, the momet at which Lucas s accout is at least double the amout i Daielle s accout will occur o some multiple of 6 moths. Let t be the umber of 6-moth periods util Lucas s accout is at least double the amout i Daielle s accout. We eed to fid the miimum iteger value of t that satisfies the followig: t 6t 0.06 0.03 1 1 1 tl(1.03) l() 6tl(1.005) t 47.5490 The smallest iteger that satisfies the equatio above is t = 48. The umber of moths i 48 6-moth itervals is: 48 6 88 Solutio 78 B Chapter 13, Time-Weighted Rate of Retur The balace at the ed of the year is: 0.5 5,000 1.09,600 1.09 8,164.4797 The time-weighted rate of retur is: 5,00 8,164.4797 1 0.08860 5,000 5,00,600 Solutio 79 A Chapter 5, Varyig Rates The iterest accumulatio factors must be the same for Bill ad Joe over the course of 4 years. The iterest accumulatio factor for Joe is: 4 4 1 l( K 0.5 t) exp dt exp exp 4 l( K 1) 4 l( K) 0 K 0.5t 0.5 0 4 K 1 K 1 exp 4 l K K Page 30 ActuarialBrew.com 015
The iterest accumulatio factor for Joe is equal to the iterest accumulatio factor for Bill: 4 4 K 1 K 1 K 5 K 1 K 1 K 5 K K 1 K 5 K 1 5 K 5 Joe s accumulated value at the ed of 4 years is: 4 4 10 K 1 10 6 K 5 0.736 Solutio 80 C Chapter 1, Sikig Fuds Sice the studet pays the iterest o the loa each year, the amout eeded to pay off the loa at the ed of 5 years is the origial amout of 1,000. Therefore, the sikig fud paymets must accumulate to 1,000. 1,000 i s 1,000 50.8i 5 (1 0.8 i) 1,000i 1,000 0.8i 5 (1 0.8 i) 1 0.4 0.8i 0.06961 i 0.08701 Solutio 81 D Chapter 1, Loas The pricipal repaid i year 6 is the paymet of,500 mius the iterest o the outstadig balace at the ed of 5 years: X,500 il 5 The iterest paid i the first year is the iterest rate times the iitial loa balace: 5 5 5 i L0 i,500 a v L 5 5,500(1 v ) iv L5 5 5 5,500,500v iv L5,500 v,500 il5 5,500 Xv The fial expressio above matches Choice D. ActuarialBrew.com 015 Page 31
Solutio 8 A Chapter 13, Dollar-Weighted Weight of Retur The formula describes the simple iterest approximatio for the iteral rate of retur. The smaller the et deposits (i.e., the cash flows) betwee time 0 ad time 1, relative to the iitial deposit, the more exact the estimate becomes. This matches Choice A. Solutio 83 E Chapter 13, Time-Weighted Weight of Retur The time-weighted rate of retur is based o the product of the time itervals correspodig accumulatio factors: 10,000 130,000 100,000 1 i 100,000 10,000 30,000 130,000 50,000 i 0.30 Solutio 84 C Chapter 6, Level Auities At the ed of 0 years, the value i the fud is: 0 1.0816 1 1,000s 1,000 1.0816 50,38.1558 0 0.0816 0.0816 The effective 6-moth iterest rate is: 0.5 1.0816 1 0.04 Let be the umber of 6-moth periods that the fud ca support withdrawals of 3,000: 3,000 a 50,38.1558 0.04 1 1.04 1.04 16.7941 0.04 6.4719 The 6 th paymet of 3,000 is made at the begiig of the 6 th 6-moth period, which is the same as the ed of the 5 th period, so the 6 th paymet is made at the ed of 1.5 years. Six moths after the 6 th paymet of 3,000 is made, the balace i the fud is: 6 50,38.1558(1.04) 3,000s 6 0.04 6 1.04 1 139,683.0046 3,000 1.04 0.04 139,683.0046 3,000 46.084 1,430.36 Page 3 ActuarialBrew.com 015
We ca use the BA II Plus to aswer this questio: [ d ] [BGN] [ d ] [SET] [ d ] [QUIT] 0 [N] 8.16 [I/Y] 1,000 [PMT] [CPT] [FV] (Result is 50,38.1558) [PV] 4 [I/Y] 3,000 [PMT] 0 [FV] [CPT] [N] (Result is 6.4719) 6 [N] [CPT] [FV] Aswer is 1,430.36. Solutio 85 D Chapter 7, Perpetuities We ca use the price of the first perpetuity to fid the value of i: 1 7.1 1 (1 i) 1 i 0.0775 The aual effective iterest rate used to value the secod auity is: i 0.01 0.0775 0.01 0.0875 The secod perpetuity begis at the ed of 1 year, so it ca be valued as a perpetuity immediate accumulated for two years: R 7.1 1.0875 3 1.0875 1 R 1.7446 Solutio 86 E Chapter 8, Varyig Auities At the ed of 5 years, the balace of the loa is: 5 s 5 10,000(1.05) 100( Is) 1,76.8 100 5 5 0.05 5.8019 5 1,76.8 100 1,76.8 100 16.0383 0.05 11,158.99 ActuarialBrew.com 015 Page 33
The loa is paid off with 15 level paymets of X: Xa 15 11,158.99 15 1 v X 11,158.99 0.05 X 1,075.08 We ca use the BA II Plus to aswer this questio: 5 [N] 5 [I/Y] 1 [PMT] [CPT] [FV] [+/-] [] 1.05 [] 5 [=] [] 0.05 [=] [] 100 [+/-] [=] [+] 10,000 [] 1.05 [y x ] 5 [=] [PV] [15] [N] 0 [FV] [CPT] [PMT] Result is 1,075.08. Aswer is 1,075.08. Solutio 87 C Chapter 6, Level Auities The balace of the fud at the ed of 10 years is: 5,000 3,917.6308 5 1.05 The 10 level paymets of X accumulate to the balace at the ed of 10 years: Xs 10 0.06 10 3,917.6308 1.06 1 X 3,917.6308 0.06 X 97. We ca use the BA II Plus to aswer this questio: 5,000 [] 1.05 [y x ] 5 [=] [FV] 10 [N] 6 [I/Y] [CPT] [PMT] Result is 97.. Aswer is 97.. Solutio 88 E Chapter 1, Loas The mothly effective rate at which the loa is origially made is: 0.08 0.006667 1 The origial paymet amout is: 65,000 61.1739 a 180 0.006667 Page 34 ActuarialBrew.com 015
After the 1 th paymet, the remaiig balace ca be foud usig the prospective method: L 61.1739 a 6,661.3994 1 168 0.006667 The ew paymet amout is the amout eeded to pay off the remaiig balace at the ew iterest rate of 6% compouded mothly: 6,661.3994 6,661.3994 55.19 a a 1801 0.06 1 168 0.005 Alteratively, we ca use the BA II Plus to aswer this questio: 180 [N] 8 [] 1 [=] [I/Y] 65,000 [+/-] [PV] [CPT] [PMT] Result is 61.1739. 168 [N] [CPT] [PV] Result is 6,661.3994. 6 [] 1 [=] [I/Y] [CPT] [PMT] Aswer is 55.19. Solutio 89 E Chapter 6, Level Auities The first tuitio paymet is due at the begiig of the 18 th year, which is at the ed of 17 years. The paymet of X is made at the ed of 18 years. The equatio of value at the ed of 17 years is: 17 18 750s Xv 6,000 1.05 1.05 v 18 17 18 750s Xv 6,000 1.05 1.05 v 18 18 1.07 1 X 750 6,000 4.541 0.07 1.07 X 750 33.9990 7,47.1710 1.07 X 5, 499.744 7,47.1710 1.07 X 1,870.5 We ca use the BA II Plus to aswer this questio: 18 [N] 7 [I/Y] 750 [PMT] [CPT] [FV] 1.05 [y x ] 17 [+] 1.05 [y x ] 18 [] 1.07 [=] [] 6,000 [+] [RCL] [FV] [=] [] 1.07 [=] Aswer is 1,870.5. ActuarialBrew.com 015 Page 35
Solutio 90 B Chapter 15, Bods The price of the bod is: 1,00 P Coupa Rv 45 a 45 17.1591 170.4548 y 40 0.05 40 1.05 94.61 We ca use the BA II Plus to aswer this questio: 40 [N] 5 [I/Y] 45 [PMT] 1,00 [FV] [CPT] [PV] Result is 94.6137. Aswer is 94.61. Solutio 91 A Chapter 15, Callable Bods The bod is a discout bod, because the coupo is less tha the yield-to-worst times the redemptio value: 0.05 1,000 0.06 1,000 Coup YTW R tk Sice the bod is a discout bod, its price-to-worst is calculated based o the latest possible redemptio: 0 P 50 a 1,000v 50 11.4699 311.8047 885.30 0 0.06 We ca use the BA II Plus to aswer this questio: 0 [N] 6 [I/Y] 50 [PMT] 1,000 [FV] [CPT] [PV] Result is 885.3008. Aswer is 885.30. Solutio 9 C Chapter 18, Forward Rates The questio is askig for the rate that applies from time 4 to time 5. 1 f 4 t (1 st ) t 1 t 1 (1 st 1) 5 1.095 1 f4 4 1.09 f 0.115 Solutio 93 D Chapter 11, Geometric Progressio Auities The mothly effective iterest rate is: 0.06 0.005 1 Page 36 ActuarialBrew.com 015
The aual effective iterest rate is: 1 1.005 1 0.06168 The preset value of the first year s paymets is: 1 1 1.005,000a,000 3,37.8641 1 0.005 0.005 The paymets i the 4 subsequet years icrease by % per year: 5 1.0 1 4 1.06168 3,37.86411 1.0 (1.0 ) 3,37.8641 1.0 1 1.06168 3,37.8641 16.1135 374, 443.38 P v v The differece betwee the lump sum ad P is: 374,500 P 374,500 374, 443.38 56.6 Choice D is the correct aswer. Solutio 94 A Chapter 3, Accumulated Value The mothly effective iterest rate is: 0.07 0.006 1 The equatio of value that equates the value of the deposits with $6,500 five years from today is: 60 60 1,700 1.006 3, 400 1.006 6,500 60 1,700 1.006 3, 400 1.006 6,500 1.006 3, 400X 1,700X 4,539.7769 0 where: X 1.006 We ca use the quadratic formula to solve for X: 1,700 1,700 4(3, 400)( 4,539.7769) X 3, 400 X 1.433 or X 0.936 Usig the positive value of X, we have the maximum possible value of : X 1.006 0.933 1.006 11.764 Sice must be less tha or equal to 11.764, the maximum itegral value of is 11. ActuarialBrew.com 015 Page 37
Solutio 95 C Chapter 3, Accumulated Value We ca use the ratio of S to T to fid d: S T 39 38 d 4 d 4 d d 4 4 1,000 1 39 4 38 1,000 1 1 39 1 38 39 19.5d 38 9.5d 1 10d d 0.1 4 The value of d covertible semiaually is equivalet to a aual effective iterest rate of i: d 1 1 i 0.95 1 i i 0.1080 Solutio 96 C Chapter 7, Level Auity The mothly effective iterest rate is: 0.04 0.0035 1 May 1, of the year (y + 10) is 14 moths after Jauary 1, of the year y. Durig these 14 moths, there are 41 quarterly paymets of 100. The equatio of value at time 0 is: 100 100 100 1.9X X 1.0035 1.0035 1.0035 1.0035 41 100 1.9X X 3k 14 1.0035 1.0035 k1 The aswer is Choice C. 3 6 13 14 Page 38 ActuarialBrew.com 015
Solutio 97 D Chapter 7, Level Auity Let s work i -year periods. For the first six years, the -year effective iterest rate is: 1.04 1 0.0816 The accumulated value at time six years of three $100 paymets made at the begiig of each -year period is: 3 1.0816 1 100s 100 351.6778 3 8.16% 0.0816 / 1.0816 For the last four years, the -year effective iterest rate is: 1.05 1 0.105 The accumulated value at time te years of the time-6 accumulated value ad of the last two $100 paymets made at the begiig of each of the remaiig - year periods is: s 10.5% 351.6778 1.105 100 1.105 1 351.6778 1.155 100 0.105 / 1.105 47.4665 31.8006 659.671 We use the BA II Plus to obtai the aual effective yield: [ d ] [BGN] [ d ] [SET] [ d ] [QUIT] 5 [N] 100 [PMT] 659.671 [FV] [CPT] [I/Y] (Result is 9.3636) [] 100 [+] 1 [=] [y x ] 0.5 [=] 1 [=] Aswer is 0.0458. Solutio 98 C Chapter 7, Level Auity The mothly effective iterest rate is: 1/1 1.08 1 0.006434 The withdrawals of $5,000 are made at times 15, 16, 17, ad 18. ActuarialBrew.com 015 Page 39
The equatio of value at time 0 ca be used to fid X: Xa 5, 000 a 1.08 181 0.006434 15 4 0.08 16 4 1 1.006434 1 1.08 X 1.06434 7, 881.046 1.08 0.06434 0.08 X 117.787 7, 881.046 3.5771 X 40.378 We ca use the BA II Plus to aswer this questio: [ d ] [BGN] [ d ] [SET] [ d ] [QUIT] 1.08 [y x ] 1 [1/x] [=] [] 1 [=] [] 100 [=] [I/Y] 18 [] 1 [=] [N] 1 [PMT] [CPT] [PV] (Result is 117.787) [STO] 1 4 [N] 8 [I/Y] 5,000 [PMT] [CPT] [PV] (Result is 89,47.447) [] 1.08 [y x ] 15 [=] [] [RCL] 1 [=] Aswer is 40.378. Solutio 99 B Chapter 6, Level Auity The equatio of value at time 0 ca be used to fid X: 1 15,000 X 15,000a Xa a 11 0.10 10 0.08 10 0.10 10 15 0.08 1.10 1.10 1 1.10 1 1.10 1 1 1.08 15,000 7,89.37 X 11 10 15 0.10 0.10 10 0.08 1.10 1.10 1.10 1.08 15,000 7.1446 7,89.37 X 6.7590 3.5641 X 17,384. 14 We ca use the BA II Plus to aswer this questio: [ d ] [BGN] [ d ] [SET] [ d ] [QUIT] 10 [N] 10 [I/Y] 1 [PMT] [CPT] [PV] (Result is 6.7590) [+/-] [STO] 1 15 [N] 8 [I/Y] 1 [PMT] [CPT] [PV] [] 1.10 [y x ] 10 [=] (Result is 3.5641) [+/-] [STO] 11 [N] 10 [I/Y] 15,000 [PMT] [CPT] [PV] [+/-] [+] 15,000 [] 0.08 [] 1.10 [y x ] 10 [=] (Result is 179,457.8734) [] [(] [RCL] 1 [+] [RCL] [)] [=] Aswer is 17,384.14. Page 40 ActuarialBrew.com 015
Solutio 100 A Chapter 8, Varyig Auity The 6-moth effective iterest rate is: 0.06 0.03 Sice the last coupo paymet was.50, the ext coupo paymet is X.50. The preset value of the bod is 1,050.50. The time 0 equatio of value ca be used to solve for X:.50 X.50 X.50 14X 300 1,050.50 1.03 14 14 1.03 1.03 1.03 300.50 a X( Ia) 1,050.50 14 0.03 14 0.03 14 1.03 14 14 1 1.03 a 14(1.03).50 X 14 198.3353 1,050.50 0.03 0.03 14 11.6350 14(1.03).50 11.960 X 198.3353 1,050.50 0.03 54.1616 X 79.310 198.3353 1,050.50 X 7.5401 We ca use the BA II Plus to aswer this questio: 14 [N] 3 [I/Y] 1 [PMT] [CPT] [PV] [] 1.03 [=] [+/-] [] 14 [] 1.03 [y x ] 14 [=] [] 0.03 [=] [STO] 1.50 [PMT] 300 [FV] [CPT] [PV] + 1,050.50 [=] [] [RCL] 1 [=] Aswer is 7.54001. Solutio 101 D Chapter 8, Varyig Auity Let s use the PI method fid the value of the auity at the ed of 9 years. We have: P1 1,000 I 500 30 The preset value is: 30 I I 500 1 (1.05) 500 30 PV9 P1 a v 1,000 (1.05) i i 0.05 0.05 0.05 11,000 15.375 69,413.346 99,683.767 To fid the preset value at time zero, we discout for 9 years: 99,683.767 64,57.0 9 1.05 30 ActuarialBrew.com 015 Page 41
Usig the BA II Plus, we have: 30 [N] 5 [I/Y] 1,000 [+] 500 [] 0.05 [=] [PMT] 500 [] 30 [] 0.05 [=] [+/-] [FV] [CPT] [PV] [] 1.05 [y x ] 9 [=] Result is 64,57.0. Aswer is 64,57.0. Solutio 10 C Chapter 11, Geometric Progressio Auity The equatio of value after 30 years ca be used to solve for i: 5,000 (1 i) (1 i) (1.03) 1.03 9 8 9 50,000 1 1.03 v (1.03 v) 30 9 1.03 1.03 (1 i) 1 1 1 5,000 i i 50,000 1.03 1.03 1 1 1 i 1 i 30 30 9 1.03 1.03 (1 i) 10 1 1 i 1 i 30 30 9 1.03 1.03 (1 i) 1 10 1 1 i 1 i 9 (1 i) 10 i 0.0864 the fial deposit is: 30 30 9 1.03 9 1.03 9 30 The accout balace after (1 i) (1.0864) 5,000 1 i 5,000 1.0864 797,836.8 1.03 1.03 1 1 1 i 1.0864 Solutio 103 D Chapter 14, Divided Discout Model Although this does ot refer to the perpetuity as paymets from a share of commo stock, we ca treat the paymets as divideds ad use the divided discout model to fid the preset value of the paymets. The rate of growth of the quarterly paymets is:,010 1 0.005,000 Page 4 ActuarialBrew.com 015
The preset value of the paymets ca be used to solve for the quarterly effective iterest rate: 100,000,000 (4) i 4 0.0551 (4) i 4,010 0.005 The aual effective iterest rate is: 4 (4) i 4 1 1 1.0551 1 0.1060 4 Solutio 104 A Chapter 10, Cotiuously Payable Auity Let s break the perpetuity ito two parts. The first part cosists of the paymets made i the first 10 years. The preset value of the first part is: a 10 10 1 v 1 1.06 7.5787 r l(1.06) The secod part cosists of the paymets made after 10 years. The formula for the preset value of a cotiuously payable auity is: b t rds a s PV Pmt e dt a a t The preset value at time 10 of the paymets made after 10 years is: 10 t 10 10 l(1.06) t ds t10 l(1.06)( t10) PV 1.03 e dt 1.03 e dt 10 10 t10 (10 t) 1.06 1.03 1.03 1.06 dt 1.03 1.06 10 10 10 t 10 1.06 1.03 1 1.06 1.03 0 1.03 1.06 l 1.03 1.06 l 1 34.8309 1.03 1.06 10 10 t dt 10 1.03 1.06 l 1.03 1.06 preset value at time 0 of the paymets made after 10 years is the preset value at time 10, discouted for 10 years: 10 10 1 1 PV0 PV10 34.8309 19.4494 1.06 1.06 1 The ActuarialBrew.com 015 Page 43
The preset value of the perpetuity is equal to the sum of the preset values of the two parts of the perpetuity: 7.5787 19.4494 7.08 Solutio 105 C Chapter 5, Varyig Force of Iterest The preset value at time 5 of the 75,000 paymet is: 5 10 10 10 1 10 rds 5 s ds l( s 1) 5 s1 5 PV AV e 75,000 e 75,000 e l(6) l(11) 6 75,000 e 75,000 11 The equatio of value at time 0 ca be used to solve for X: X 6 1 10,000 75,000 1.06 11 1.06 X 4,498.79 3 5 Solutio 106 D Chapter 1, Drop Paymets If we accumulate the iitial loa balace to time, the we ca treat the loa as a loa with the first paymet occurrig oe year later: 15,000,000 16,796,84.19 (1 0.055) The aual effective iterest rate is: 0.055 0.0580 1 0.055 We ca solve the time-0 equatio of value for : 16,796,84.19 1,00,000a 1 1.0580 16,796,84.19 1,00,000 0.0580 0.1853 1.0580 9.7960 Therefore, there are 9 paymets of 1,00,000 ad a fial drop paymet at time 30: 9 16,796,84.19 1,00,000a DropPmt 0.945 9 1 1.0580 16,796,84.19 1,00,000 DropPmt 0.945 0.0580 DropPmt 960,73.59 30 30 Page 44 ActuarialBrew.com 015
We ca use the BA II Plus to aswer this questio: 15,000,000 [] 0.945 [x ] [=] [+/-] [PV] 0.055 [] 0.945 [] 100 [=] [I/Y] 1,00,000 [PMT] [CPT] [N] Result is 9.7960. 9 [N] [CPT] [FV] [] 0.945 [=] Aswer is 960,73.59. The closest aswer choice is Choice D. It appears that whe the aswer choices were prepared for this questio, there was a uusually large discrepacy itroduced by iteral roudig. Solutio 107 C Chapter 1, Loas We make use of the followig formula for the iterest portio of a paymet: t1 It (1 v ) Pmt t The formula ca be used to fid the followig expressios: 1 3 1 It (1 v ) Pmt 4 It (1 v ) Pmt It (1 v ) Pmt The iterest portio of the paymet at time ( 1) is equal to 0.550 of the iterest portio of the paymet at time ( 3), which allows us to solve for v: 4 (1 v ) Pmt 0.550(1 v ) Pmt 1 0.550(1 v ) v 0.951 The iterest portio of the paymet at time ( 1) is equal to 0.147 of the iterest portio of the first paymet, which allows us to solve for : (1 v ) Pmt 0.147(1 v ) Pmt 0.0954 0.147(1 v ) 0.336 v.00 ActuarialBrew.com 015 Page 45
Solutio 108 B Chapter 1, Sikig Fuds The sikig fud paymet is: L SFP s 11 0.047 The amout owed to the leder is remais costat at L util the loa is paid off, so the equatio of value at the ed of 7 years is: L L s 6,41 s 7 11 0.047 7 1.047 1 L 1 6,41 11 1.047 1 L 14,749.318 The sikig fud paymet is: L 14,749.318 SFP 1,054.57 s 13.9861 11 0.047 Solutio 109 C Chapter 1, Drop Paymets The level paymets satisfy the followig time-0 equatio of value: 00,000 Pmt a 360 0.005 00,000 Pmt 166.7916 Pmt 1,199.1011 Whe we subtract the preset value of the extra paymets, the ew equatio of value is: 00,000 10,000 a 1,199.1011 a 1 51.005 1 0.005 1 1.005 00,000 10,000 a 1,199.1011 5 0.06168 0.005 1 1.005 00,000 10,000 4.193 1,199.1011 0.005 1 1.005 158,067.9347 1,199.1011 0.005 1 1.005 158,067.9347 1,199.1011 0.005 1.005 0.3409 15.7768 Page 46 ActuarialBrew.com 015
The fial paymet therefore occurs after 16 moths, which is 18 years: 16 18 1 The loa origiated o Jauary 1, 003, ad the fial paymet is made 18 years later. Addig 18 years to Jauary 1, 003 brigs us to Jauary 1, 01. The begiig of Jauary 1, 01 is the same as the ed of December 31, 00. We ca use the BA II Plus to aswer this questio: 00,000 [PV] 360 [N] 0.5 [I/Y] [CPT] [PMT] Result is 1,199.1011. [STO] 1 5 [N] 1.005 [y x ] 1 [] 1 [=] [] 100 [=] [I/Y] 10,000 [PMT] [CPT] [PV] [+] 00,000 [=] Result is 158,067.9347 [PV] 0.5 [I/Y] [RCL] 1 [PMT] [CPT] [N] Result is 15.7768, so the fial paymet occurs after 16 moths. 16 [] 1 [=] Result is 18. [+] 003 [=] Result is 01. The fial paymet is made at the begiig of Jauary 1, 01, which is equivalet to the ed of December 31, 00. Solutio 110 D Chapter 1, Loas The aual effective iterest rate is: 0.08 i 0.08696 1 0.08 We ca solve for the amout of the 5 level paymets: 500,000 Pmt a 5 0.08696 500,000 Pmt 3.906 Pmt 17,53.707 If the first 4 paymets were istead 18,000, the the balace at the ed of 4 years would be: 500,000 18,000 s 500,000 18,000 4.556 4 4 0.08696 4 0.9 0.9 115,0.7473 ActuarialBrew.com 015 Page 47
The fial paymet is the balace at the ed of 4 years, accumulated for oe additioal year of iterest: 115,0.7473 15,0.38 0.9 We ca use the BA II Plus to aswer this questio: 0.08 [] 0.9 [] 100 [=] [I/Y] 5 [N] 500,000 [+/-] [PV] [CPT] [PMT] Result is 17,53.707. 18,000 [PMT] 4 [N] [CPT] [FV] [] 0.9 [=] Aswer is 15,0.38. Solutio 111 B Chapter 18, Spot Rates The price of a zero-coupo bod as a percetage of its redemptio value is equal to the iverse of the accumulatio factor achieved by ivestig i the bod. Therefore ivestig X i the 6-moth bod, for example, results i a accumulated value of: X 0.94 Ivestig X i each of the bods results i a accumulated value of: X X X X X X 0.94 0.95 0.96 0.97 0.98 0.99 1 1 1 1 1 1 X 6.196X 0.94 0.95 0.96 0.97 0.98 0.99 Settig this accumulated value equal to 100,000 allows us to solve for X: 6.196X 100,000 X 16,078.9 Solutio 11 D Chapter 1, Loas We use the followig formulas: t1 Itt (1 v ) Pmt t1 Prt v Pmt Usig the iformatio provided i the questio, we have: 10 It1 (1 v ) Pmt 3, 600 5 Pr6 v Pmt 4,871 Page 48 ActuarialBrew.com 015
Dividig the first equatio by the secod equatio allows us to solve for v: 10 (1 v ) Pmt 3,600 5 v Pmt 4,871 10 3,600 5 1 v v 4,871 10 3,600 5 v v 1 0 4,871 We ca use the quadratic formula to solve for 3,600 3,600 41 ( 1) 5 4,871 4,871 v 5 5 v 0.6966 or v 1.4356 5 v : 5 We use the positive value of v to solve for i: 5 v 0.6966 i 0.07500 Sice the iterest paid i the first year is 3,600, we have: Xi 3,600 X 0.07500 3,600 X 48,000.18 Solutio 113 A Chapter 15, Bods The equatio of value ca be solved for R: P Coupa Rv y R 10,000 R0.035 a 0.5 501.0705 1 5 1.0705 5 1 1.0705 1 10,000 R 0.035 1.0705 0.5 1 1.0705 5 1 10,000 R 0.035 3.6045 1.0705 5 R 9,917.99 We ca use the BA II Plus to aswer this questio: 1.0705 [y x ] 0.5 [] 1 [=] [] 100 [=] [I/Y] 50 [N] 0.035 [PMT] 1 [FV] [CPT] [PV] Result is 1.00869. [1/x] [] 10,000 [=] ActuarialBrew.com 015 Page 49
Result is 9,917.991. Aswer is 9,917.99. Solutio 114 B Chapter 15, Bods At the ed of each moth, the et cash flow to Jeff is the coupo paymet from the bod mius the iterest o the loa: 0.09 0.08 10,000,000 61.6667 1 1 At the ed of 10 years, Jeff receives 10,000 from the bod ad pays back the,000 loa, givig him a et cash flow of: 10,000,000 8,000 Sice the cost of eterig this positio is 8,000, the time-0 equatio of value is: 8,000 61.6667a 10 i (1) 1 8,000 (1) 10 1 i 1 We ca use the BA II Plus to aswer this questio: 10 [N] 8,000 [+/-] [PV] 10,000 [] 0.09 [] 1 [],000 [] 0.08 [] 1 [=] [PMT] 8,000 [FV] [CPT] [I/Y] Result is 0.7708. [] 100 [+] 1 [=] [y x ] 1 [] 1 [=] Aswer is 0.0965. Solutio 115 B Chapter 15, Bods The first equatio of value below sets the value of the first bod equal to the value of the secod bod, ad the secod equatio sets the value of the secod bod equal to the value of the third bod: 0.058 1,000 a 1,000v 0.0440 1,100 a 1,100v 0.0440 1,100 a 1,100v 1,30r a 1,30v The two equatios above ca be simplified as follows: 4.4 a 100v (48.4 1,30 r) a 0v Page 50 ActuarialBrew.com 015