Solutions to Interest Theory Sample Questions

Transcription

1 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: e e l(1.336) 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 j) 1 5 (1 j) (1 j) 4 j The accumulated amout at the ed of 40 years is: X 100s , 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, Solutio is 6, ActuarialBrew.com 015 Page 1

2 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 i 1 i 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, , The accumulated value of the deposits is: s , After repayig the loa, the balace is: 1, ,000, The BA-II Plus ca be used to aswer this questio: 10 [N] 14 [I/Y] [PMT] [CPT] [FV] [+/-] [] 10,000 [=] Solutio is, 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 The fud exposure is the average amout i the fud: Fud exposure (Net deposit)(time deposit is i the fud) Page ActuarialBrew.com 015

3 The simple iterest approximatio for the dollar-weighted rate of retur is the icome divided by the fud exposure: Icome 10 i Fud exposure Solutio 6 C Chapter 8, Varyig Auities The equatio of value at the ed of 1 year ca be used to solve for a : ( Ia) v i i a v v a a The BA-II Plus ca be used to solve for : 10.5 [I/Y] [PV] 1 [PMT] [CPT] [N] Result is Solutio 7 C Chapter 8, Varyig Auities The accumulated value is: 9 8 1,000(0.06) (0.06) (0.06) Usig the PI method, we have: P1 160 I 6 10 The preset value is: 10 I I 6 1 (1.09) PV0 P1 a v 160 (1.09) i i The accumulated value at the ed of 10 years is: , 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

4 PV = [PMT] [CPT] [FV] Aswer =, 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: ( ) 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, Xa X 0.10 X 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 Solutio is Solutio 10 B Chapter 15, Bods The book value at the ed of 6 years is: 10,000 BV ,000 a , , The iterest (which is also kow as the ivestmet icome) portio of the 7 th paymet is: IvIc7 BV6 y 10, 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, [] 0.06 [=] Page 4 ActuarialBrew.com 015

5 Result is Aswer is 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: X v 1.08v 1.08 v 1.08 v X X X 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: (1.03) 0(1.03) d 1 4 d Solutio 13 E Chapter 5, Varyig Force of Iterest At time 3, before the deposit of X, the value of the fud is: t t dt e 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 e300 X e 1 X e X e 1 X X X 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 : v 10v 10v 10v 10v 10 (1 K) v (1 K) v (1 Kv ) 0 10a (1 Kv ) 1 (1 K) (1 Kv ) (1 Kv ) (1 Kv ) (1 Kv ) v K The questio referred to K% istead of K, so we multiply the value of K foud above by 100: Solutio 15 B Chapter 1, Loas The amout of the equal aual paymets uder optio (i) is:,000,000, a Page 6 ActuarialBrew.com 015

7 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 The sum of the paymets uder optio (i) is: , 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: i(,000 1,800 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:, ,000 11,000i i 11,000 i Solutio 16 B Chapter 11, Geometric Progressio Auities There are 60 mothly paymets. We use oe moth as the uit of time: (1) i v The 60 paymets are described below: Time Paymet 1 1, 000 1, , , , , ActuarialBrew.com 015 Page 7

8 The outstadig balace after the 40 th paymet is the preset value of the paymets after the 40 th paymet: PV 1, v 1, v 1, v v 0.98 v 1,000 6, v Solutio 17 C Chapter 6, Level Auities The equatio of value at the ed of 3 years ca be used to fid i: 8, s (1 i) 196s 8,000i 98 (1 i) 1 (1 i) 196 (1 i) 1 8,000i 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/ Usig the PI method, we have: P1 I 60 i The preset value is: I I PV0 P1 a v i i 60 1 ( ) 60 ( ) , ,79.1 Usig the BA II Plus, we have: 0.09 [] 4 [] 1 [=] [y x ] 3 [1/x] [=] [] 1 [=] [STO] [N] [RCL] 1 [] 100 [=] [I/Y] [+] [] [RCL] 1 [=] [PMT] [] 60 [] [RCL] 1 [=] [+/-] [FV] [CPT] [PV] Result is PV =, Aswer is,79.1 Page 8 ActuarialBrew.com 015

9 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) X 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 X X 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 or We use the positive value of 15%. Solutio 0 A Chapter 3, Preset Value The equatio of value at the outset is: v 300v 600v v v i ActuarialBrew.com 015 Page 9

10 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: 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, (8 ) ,000 18kdt 0 0,000 k(180 0) k Solutio D Chapter 15, Bods The price of the bod is: 1 v P Coupa Rv 1,000r i i 1, (1 v ) ,031.5( ) , 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, , We ca set this equal to the ext preset value of Project Q: 4,000 X 1, , X 5,460 Page 10 ActuarialBrew.com 015

11 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, 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 , idicatig that the sikig fud paymet is [PV] 0,000 [FV] [CPT] [I/Y] Result is 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, , The iterest paid by Jaice is: 5, ,000 ActuarialBrew.com 015 Page 11

12 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 a , The total amout of iterest paid o all 3 loas is: 3, ,000 1, , 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: (1 i) 50(1 i) 6 (1 i) i 0.15 The iterest eared i Bruce s accout i the 11 th year is: X 100(1 i) i 100(1.15) 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

13 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 /3 The 4-moth effective iterest rate is foud below: m ( m) ( p) i i 1 1 m p (3) 1/3 i (3) 1/3 1/3 i The preset value of the secod perpetuity is: 1 X 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, ,97 Uder Sceario A, the profit is: ,703s 177, , , ,31.97 Uder Sceario B, the profit is: ,703s 177, , , ,3.78 Choice D best describes the isurace compay s profit. 4 4 ActuarialBrew.com 015 Page 13

14 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, , , , , v 5, v 5, v ,000 1, 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, Solutio 33 B Chapter 18, Spot Rates The value of the bod is: , 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 ) [+/-] [PV] 3 [N] 60 [PMT] 1,000 [FV] [CPT] [I/Y] (Result is ) Page 14 ActuarialBrew.com 015

15 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 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) 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: P' y Div ( y g) ModDur P y 1 Div ( y g) y g ActuarialBrew.com 015 Page 15

16 The Macaulay duratio is: MacDur ModDur (1 y) Questios 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) X 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 The iitial loa balace is: Pmt a , The first pricipal paymet is equal to the level paymet mius the iterest o the iitial loa balace: , Solutio 47 B Chapter 15, Bods Page 16 ActuarialBrew.com 015

17 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, (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 [] [=] [x ] [] 1 [=] Solutio is Solutio 48 A Chapter 7, Level Auities The mothly effective iterest rate is: 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, , His cotributios must accumulate to 310,880.83: Xs , X , X 310, 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

18 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: X ,000 1 v v v 17 k 3 X ,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 , 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, Q I ,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, Page 18 ActuarialBrew.com 015

19 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, The combied prices of Bod I ad Bod II are: 1,000 1, , 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, Xa 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/x] [] 1,7.5 [=] Result is 1, Aswer is 1, 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 , 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 ActuarialBrew.com 015 Page 19

20 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, [STO] 1 40 [N] 1,100 [FV] [CPT] [PV] Result is 1, Sice 1, is less tha 1, , the price-to-worst is 1, 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, Xa 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 [1/x] [] 1,01.50 [=] Result is 1, Aswer is 1, 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 Page 0 ActuarialBrew.com 015

21 [] [=] Result is 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 MacDur PV CF 15,000a t 0 0 t Let X be the percetage of the preset value that is ivested i the 5-year bods: X (1 X)10 X The amout ivested i the 5-year bods is: X 35,000 a , , The BA II Plus ca be used to aswer this questio: 15 [N] 6. [I/Y] 1[PMT] [CPT] [PV] [+/-] Result is [STO] 1 [] 1.06 [] 15 [] 1.06 [y x ] 15 [=] [] 0.06 [=] Result is [] [RCL] 1 [=] Result is [] 10 [=] [] 5 [+/-] [=] Result is [] [RCL] 1 [] 35,000 [=] Solutio is 08, ActuarialBrew.com 015 Page 1

22 Solutio 60 A Chapter 11, Geometric Progressio Auities The 16 paymets are described below: Time Paymet 1,000, , , , The preset value of the paymets is:, , L,000v,000v 1.03,000v , v 0.97v 0.97 v 0.97 v v 1.03 v 0.97v 0.97 v,000, v v v, , , , , Solutio 61 E Chapter 5, Varyig Force of Iterest The equatio of value is: t sds e,000 s t 100 exp ds s 150 t 0 3 s 100 s ds l4 To evaluate the itegral, let s use the followig substitutio: 3 s u s du ds 50 Page ActuarialBrew.com 015

23 The itegral is: s 100 s t st st 1 st 3 ds l 0.5l 3 s 0 u du u 0 s s s t 0.5 l t 150 l(3) l 3 We ca ow solve for t: 3 3 t t 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 (1 ) 5 y y 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 Ry Coup, The discout at the time of purchase is: Discout ( Ry Coup) a, a y , , Solutio 63 A Chapter 1, Loas We begi by fidig the level paymet amout: t t1 Pr v Pmt Pmt Pmt ActuarialBrew.com 015 Page 3

24 The total amout of iterest paid o the loa is equal to the total amout of the paymets mius the iitial loa balace: a 6, , , , ,39.13 The BA II Plus ca be used to aswer this questio: [] [y x ] 4 [=] Result is [PMT] 8 [N] 4.75 [I/Y] [CPT] [PV] Result is 5, [+] 8 [] [RCL] [PMT] [=] Aswer is 1, Solutio 64 D Chapter 7, Level Auities The mothly iterest rate for the first 18 moths is: The accumulated value of the loa after 18 moths mius the accumulated value of the paymets is: ,000(1.007) s 4, , , The ew iterest rate is to refiace the loa is: The equatio of value for the refiaced loa after 18 moths is: 16, Xa , X , X X We ca use the BA II Plus to aswer this questio: 18 [N] 0.7 [I/Y],000 [PV] [+/-] [PMT] [CPT] [FV] (Result is 16, ) [PV] 4 [N] 0.4 [I/Y] 0 [FV] [CPT] [PMT] 18 Page 4 ActuarialBrew.com 015

25 Aswer is Solutio 65 C Chapter 16, Duratio Sice the bod is priced at par, its Macaulay duratio is: 14 ( m) () MacDur a a Solutio 66 A Chapter 16, Duratio The modified duratio is: MacDur ModDur ( m) y m The estimated percetage chage i the price is: ( ) % P ModDur y m ( ) The estimate for the ew price is: 1,000(1 % P) 1,000( ) 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) ActuarialBrew.com 015 Page 5

26 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 ,000 d MacDur w MacDur 0 d The ratio is: 1 Port j j 50 5, ,000 j1 d d d d k 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 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

27 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, B 3,000 B 3, The absolute value of the differece is: AB, , 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, b B 5, b 1.03 The duratio of the assets is equal to the duratio of the liability: 5 5,000 (8 bb ) 8 1, b ,000 96,000 (8 b) 5, b b.5076 We ca ow solve for B: B 5, B 7, The ratio is: B 7, 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

28 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: PV 1.04 A PV L A B PVB 1.04 PV 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, , a 0 i/ a 0 i/ 1 i 1 i 5, , a 5, 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 Aswer is Page 8 ActuarialBrew.com 015

29 Solutio 75 D Chapter 1, Loas The iitial loa paymet is: 400, , ,000 Pmt 4, a 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, a (4, ) a j /1 4, , a 144 j /1 356, , 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, [N] [CPT] [PV] Result is 356, [RCL] [PMT] [] [=] [PMT] [CPT] [I/Y] [] 1 [=] Result is Aswer is 6.90%. Solutio 76 D Chapter 15, Bods The equatio of value that equates the prices of the two bods is: 1, a 5 a j / 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, [FV] [CPT] [I/Y] [] [=] Result is Aswer is 4.40%. ActuarialBrew.com 015 Page 9

30 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 tl(1.03) l() 6tl(1.005) t The smallest iteger that satisfies the equatio above is t = 48. The umber of moths i 48 6-moth itervals is: Solutio 78 B Chapter 13, Time-Weighted Rate of Retur The balace at the ed of the year is: 0.5 5, , , The time-weighted rate of retur is: 5,00 8, ,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: l( K 0.5 t) exp dt exp exp 4 l( K 1) 4 l( K) 0 K 0.5t K 1 K 1 exp 4 l K K Page 30 ActuarialBrew.com 015

31 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: K K 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, i 5 (1 0.8 i) 1,000i 1, i 5 (1 0.8 i) i i 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: i L0 i,500 a v L 5 5,500(1 v ) iv L ,500,500v iv L5,500 v,500 il5 5,500 Xv The fial expressio above matches Choice D. ActuarialBrew.com 015 Page 31

32 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 1 i 100,000 10,000 30, ,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: ,000s 1, , The effective 6-moth iterest rate is: Let be the umber of 6-moth periods that the fud ca support withdrawals of 3,000: 3,000 a 50, 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, (1.04) 3,000s , , , , , Page 3 ActuarialBrew.com 015

33 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, ) [PV] 4 [I/Y] 3,000 [PMT] 0 [FV] [CPT] [N] (Result is ) 6 [N] [CPT] [FV] Aswer is 1, Solutio 85 D Chapter 7, Perpetuities We ca use the price of the first perpetuity to fid the value of i: (1 i) 1 i The aual effective iterest rate used to value the secod auity is: i The secod perpetuity begis at the ed of 1 year, so it ca be valued as a perpetuity immediate accumulated for two years: R R 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, , , , ActuarialBrew.com 015 Page 33

34 The loa is paid off with 15 level paymets of X: Xa 15 11, v X 11, X 1, 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, Aswer is 1, Solutio 87 C Chapter 6, Level Auities The balace of the fud at the ed of 10 years is: 5,000 3, The 10 level paymets of X accumulate to the balace at the ed of 10 years: Xs , X 3, 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: The origial paymet amout is: 65, a Page 34 ActuarialBrew.com 015

35 After the 1 th paymet, the remaiig balace ca be foud usig the prospective method: L a 6, The ew paymet amout is the amout eeded to pay off the remaiig balace at the ew iterest rate of 6% compouded mothly: 6, , a a 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 [N] [CPT] [PV] Result is 6, [] 1 [=] [I/Y] [CPT] [PMT] Aswer is 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: s Xv 6, v s Xv 6, v X 750 6, X , X 5, , 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, ActuarialBrew.com 015 Page 35

36 Solutio 90 B Chapter 15, Bods The price of the bod is: 1,00 P Coupa Rv 45 a y 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 Aswer is 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: , ,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 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 Aswer is 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) f f Solutio 93 D Chapter 11, Geometric Progressio Auities The mothly effective iterest rate is: Page 36 ActuarialBrew.com 015

37 The aual effective iterest rate is: The preset value of the first year s paymets is: ,000a,000 3, The paymets i the 4 subsequet years icrease by % per year: , (1.0 ) 3, , , P v v The differece betwee the lump sum ad P is: 374,500 P 374, , Choice D is the correct aswer. Solutio 94 A Chapter 3, Accumulated Value The mothly effective iterest rate is: The equatio of value that equates the value of the deposits with $6,500 five years from today is: , , , , , , , 400X 1,700X 4, where: X We ca use the quadratic formula to solve for X: 1,700 1,700 4(3, 400)( 4, ) X 3, 400 X or X Usig the positive value of X, we have the maximum possible value of : X Sice must be less tha or equal to , the maximum itegral value of is 11. ActuarialBrew.com 015 Page 37

38 Solutio 95 C Chapter 3, Accumulated Value We ca use the ratio of S to T to fid d: S T d 4 d 4 d d 4 4 1, , d d 1 10d d The value of d covertible semiaually is equivalet to a aual effective iterest rate of i: d 1 1 i i i Solutio 96 C Chapter 7, Level Auity The mothly effective iterest rate is: 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: X X X X 3k k1 The aswer is Choice C Page 38 ActuarialBrew.com 015

39 Solutio 97 D Chapter 7, Level Auity Let s work i -year periods. For the first six years, the -year effective iterest rate is: The accumulated value at time six years of three $100 paymets made at the begiig of each -year period is: s % / For the last four years, the -year effective iterest rate is: 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% / We use the BA II Plus to obtai the aual effective yield: [ d ] [BGN] [ d ] [SET] [ d ] [QUIT] 5 [N] 100 [PMT] [FV] [CPT] [I/Y] (Result is ) [] 100 [+] 1 [=] [y x ] 0.5 [=] 1 [=] Aswer is Solutio 98 C Chapter 7, Level Auity The mothly effective iterest rate is: 1/ The withdrawals of $5,000 are made at times 15, 16, 17, ad 18. ActuarialBrew.com 015 Page 39

40 The equatio of value at time 0 ca be used to fid X: Xa 5, 000 a X , X , X 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 ) [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 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 ,000 7,89.37 X , ,89.37 X X 17, 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 ) [+/-] [STO] 1 15 [N] 8 [I/Y] 1 [PMT] [CPT] [PV] [] 1.10 [y x ] 10 [=] (Result is ) [+/-] [STO] 11 [N] 10 [I/Y] 15,000 [PMT] [CPT] [PV] [+/-] [+] 15,000 [] 0.08 [] 1.10 [y x ] 10 [=] (Result is 179, ) [] [(] [RCL] 1 [+] [RCL] [)] [=] Aswer is 17, Page 40 ActuarialBrew.com 015

41 Solutio 100 A Chapter 8, Varyig Auity The 6-moth effective iterest rate is: Sice the last coupo paymet was.50, the ext coupo paymet is X.50. The preset value of the bod is 1, The time 0 equatio of value ca be used to solve for X:.50 X.50 X.50 14X 300 1, a X( Ia) 1, a 14(1.03).50 X , (1.03) X , X , X 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, [=] [] [RCL] 1 [=] Aswer is 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 The preset value is: 30 I I (1.05) PV9 P1 a v 1,000 (1.05) i i , , , To fid the preset value at time zero, we discout for 9 years: 99, , ActuarialBrew.com 015 Page 41

42 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) , v (1.03 v) (1 i) ,000 i i 50, i 1 i (1 i) i 1 i (1 i) i 1 i 9 (1 i) 10 i the fial deposit is: The accout balace after (1 i) (1.0864) 5,000 1 i 5, , i 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:, ,000 Page 4 ActuarialBrew.com 015

43 The preset value of the paymets ca be used to solve for the quarterly effective iterest rate: 100,000,000 (4) i (4) i 4, The aual effective iterest rate is: 4 (4) i 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 v 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 l(1.06) t ds t10 l(1.06)( t10) PV 1.03 e dt 1.03 e dt t10 (10 t) dt t l l t dt l preset value at time 0 of the paymets made after 10 years is the preset value at time 10, discouted for 10 years: PV0 PV The ActuarialBrew.com 015 Page 43

44 The preset value of the perpetuity is equal to the sum of the preset values of the two parts of the perpetuity: Solutio 105 C Chapter 5, Varyig Force of Iterest The preset value at time 5 of the 75,000 paymet is: 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, The equatio of value at time 0 ca be used to solve for X: X ,000 75, X 4, 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 ( ) The aual effective iterest rate is: We ca solve the time-0 equatio of value for : 16,796, ,00,000a ,796, ,00, Therefore, there are 9 paymets of 1,00,000 ad a fial drop paymet at time 30: 9 16,796, ,00,000a DropPmt ,796, ,00,000 DropPmt DropPmt 960, Page 44 ActuarialBrew.com 015

45 We ca use the BA II Plus to aswer this questio: 15,000,000 [] [x ] [=] [+/-] [PV] [] [] 100 [=] [I/Y] 1,00,000 [PMT] [CPT] [N] Result is [N] [CPT] [FV] [] [=] Aswer is 960, 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: 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 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 v ) v The iterest portio of the paymet at time ( 1) is equal to of the iterest portio of the first paymet, which allows us to solve for : (1 v ) Pmt 0.147(1 v ) Pmt (1 v ) v.00 ActuarialBrew.com 015 Page 45

46 Solutio 108 B Chapter 1, Sikig Fuds The sikig fud paymet is: L SFP s 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 L 1 6, L 14, The sikig fud paymet is: L 14, SFP 1, s Solutio 109 C Chapter 1, Drop Paymets The level paymets satisfy the followig time-0 equatio of value: 00,000 Pmt a ,000 Pmt Pmt 1, Whe we subtract the preset value of the extra paymets, the ew equatio of value is: 00,000 10,000 a 1, a ,000 10,000 a 1, ,000 10, , , , , , Page 46 ActuarialBrew.com 015

47 The fial paymet therefore occurs after 16 moths, which is 18 years: 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, [STO] 1 5 [N] [y x ] 1 [] 1 [=] [] 100 [=] [I/Y] 10,000 [PMT] [CPT] [PV] [+] 00,000 [=] Result is 158, [PV] 0.5 [I/Y] [RCL] 1 [PMT] [CPT] [N] Result is , 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 We ca solve for the amout of the 5 level paymets: 500,000 Pmt a ,000 Pmt Pmt 17, 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, , ActuarialBrew.com 015 Page 47

48 The fial paymet is the balace at the ed of 4 years, accumulated for oe additioal year of iterest: 115, , 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, ,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 X 6.196X 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, Pr6 v Pmt 4,871 Page 48 ActuarialBrew.com 015

49 Dividig the first equatio by the secod equatio allows us to solve for v: 10 (1 v ) Pmt 3,600 5 v Pmt 4, , v v 4, ,600 5 v v 1 0 4,871 We ca use the quadratic formula to solve for 3,600 3, ( 1) 5 4,871 4,871 v 5 5 v or v v : 5 We use the positive value of v to solve for i: 5 v i Sice the iterest paid i the first year is 3,600, we have: Xi 3,600 X ,600 X 48, 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 ,000 R ,000 R R 9, We ca use the BA II Plus to aswer this questio: [y x ] 0.5 [] 1 [=] [] 100 [=] [I/Y] 50 [N] [PMT] 1 [FV] [CPT] [PV] Result is [1/x] [] 10,000 [=] ActuarialBrew.com 015 Page 49

50 Result is 9, Aswer is 9, 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: ,000, 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, a 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 [] 100 [+] 1 [=] [y x ] 1 [] 1 [=] Aswer is 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: ,000 a 1,000v ,100 a 1,100v ,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