SOA Exam FM / CAS Exam 2

Transcription

1 SOA Exam FM / CAS Exam May 005 Exam Questios Questio 1 Which of the followig expressios does NOT represet a defiitio for + (1 i) 1 A v i 1 v B i C v+ v + + v 1 v D v 1 v s E 1 (1 + i) a? Questio Lori borrows $10,000 for 10 years at a aual effective iterest rate of 9%. At the ed of each year, she pays the iterest o the loa ad deposits the level amout ecessary to repay the pricipal to a sikig fud earig a aual effective iterest rate of 8%. The total paymets made by Lori over the 10-year period is X. Calculate X. A 15,803 B 15,853 C 15,903 D 15,593 E 16,003 Questio 3 A bod will pay a coupo of $100 at the ed of each of the ext three years ad will pay the face value of $1,000 at the ed of the three-year period. The bod s duratio (Macaulay duratio) whe valued usig a aual effective iterest rate of 0% is X. Calculate X. A.61 B.70 C.77 D.89 E

2 Exam FM May 005 Exam - Questios Questio 4 A estate provides a perpetuity with paymets of X at the ed of each year. Seth, Susa, ad Lori share the perpetuity such that Seth receives the paymet of X for the first years ad Susa receives the paymets of X for the ext m years, after which Lori receives all the remaiig paymets of X. Which of the followig represets the differece betwee the preset value of Seth s ad Susa s paymets usig a costat rate of iterest? A X a v a m B X a v a m + C X 1 a v a m D X 1 a v a m + E X 1 va v a m Questio 5 Susa ca buy a zero-coupo bod that will pay $1,000 at the ed of 1 years ad is curretly sellig for $ Istead, she purchases a 6% bod with coupos payable semi-aually that will pay $1,000 at the ed of 10 years. If she pays X, she will ear the same aual effective iterest rate as the zero-coupo bod. Calculate X. A 1,164 B 1,167 C 1,170 D 1,173 E 1,176 Questio 6 Joh purchased three bods to form a portfolio as follows: Bod A has semi-aual coupos at 4%, a duratio of 1.46 years, ad was purchased for $980. Bod B is a 15-year bod with a duratio of 1.35 years ad was purchased for $1,015. Bod C has a duratio of years ad was purchased for $1,000. Calculate the duratio of the portfolio at the time of purchase. A B C D E 16.6 years years 16.7 years years 16.8 years

3 May 005 Exam - Questios Exam FM Questio 7 Mike receives cash flows of $100 today, $00 i oe year, ad $100 i two years. The preset value of these cash flows is $ at a aual effective rate of iterest i. Calculate i. A 10% B 11% C 1% D 13% E 14% Questio 8 A loa is beig repaid with 5 aual paymets of $300 each. With the 10 th paymet, the borrower pays a extra $1,000, ad the repays the balace over 10 years with a revised aual paymet. The effective rate of iterest is 8%. Calculate the amout of the revised aual paymet. A 157 B 183 C 34 D 57 E 383 Questio 9 The preset value of a series of 50 paymets startig at $100 at the ed of the first year ad icreasig by $1 each year thereafter is equal to X. The aual effective rate of iterest is 9%. Calculate X. A 1,165 B 1,180 C 1,195 D 1,10 E 1,5 Questio 10 Yield rates to maturity for zero-coupo bods are curretly quoted at 8.5% for oe-year maturity, 9.5% for two-year maturity, ad 10.5% for three-year maturity. Let i be the oe-year forward rate for year two implied by curret yields of these bods. Calculate i. A 8.5% B 9.5% C 10.5% D 11.5% E 1.5% 3

4 Exam FM May 005 Exam - Questios Questio 11 A $1,000 par value bod pays aual coupos of $80. The bod is redeemable at par i 30 years, but is callable ay time from the ed of the 10 th year at $1,050. Based o her desired yield rate, a ivestor calculates the followig potetial purchase prices, P: Assumig the bod is called at the ed of the 10 th year, P = $957 Assumig the bod is held util maturity, P = $897 The ivestor buys the bod at the highest price that guaratees she will receive at least her desired yield rate regardless of whe the bod is called. The ivestor holds the bod for 0 years, after which time the bod is called. Calculate the aual yield rate the ivestor ears. A 8.56% B 9.00% C 9.4% D 9.53% E 9.99% Questio 1 Which of the followig are characteristics of all perpetuities? A B C D E I. The preset value is equal to the first paymet divided by the aual effective iterest rate. II. Paymets cotiue forever. III. Each paymet is equal to the iterest eared o the pricipal. Questio 13 I oly II oly III oly I, II, ad III The correct aswer is ot give by (A), (B), (C), or (D). At a omial iterest rate of i covertible semi-aually, a ivestmet of $1,000 immediately ad $1,500 at the ed of the first year will accumulate to $,600 at the ed of the secod year. Calculate i. A.75% B.77% C.79% D.81% E.83% 4

5 May 005 Exam - Questios Exam FM Questio 14 A auity-immediate pays $0 per year for 10 years, the decreases by $1 per year for 19 years. At a aual effective iterest rate of 6%, the preset value is equal to X. Calculate X. A 00 B 05 C 10 D 15 E 0 Questio 15 A isurace compay accepts a obligatio to pay $10,000 at the ed of each year for years. The isurace compay purchases a combiatio of the followig two bods at a total cost of X i order to exactly match its obligatio: (i) 1-year 4% aual coupo bod with a yield rate of 5% (ii) -year 6% aual coupo bod with a yield rate of 5%. Calculate X. A 18,564 B 18,574 C 18,584 D 18,594 E 18,604 Questio 16 At the begiig of the year, a ivestmet fud was established with a iitial deposit of $1,000. A ew deposit of $1,000 was made at the ed of 4 moths. Withdrawals of $00 ad $500 were made at the ed of 6 moths ad 8 moths, respectively. The amout i the fud at the ed of the year is $1,560. Calculate the dollar-weighted (moey-weighted) yield rate eared by the fud durig the year. A 18.57% B 0.00% C.61% D 6.00% E 8.89% Questio 17 At a aual effective iterest rate of i, the preset value of a perpetuity-immediate startig with a paymet of $00 i the first year ad icreasig by $50 each year thereafter is $46,530. Calculate i. A 3.5% B 3.50% C 3.75% D 4.00% E 4.5% 5

6 Exam FM May 005 Exam - Questios Questio 18 A store is ruig a promotio durig which customers have two optios for paymet. Optio oe is to pay 90% of the purchase price two moths after the date of sale. Optio two is to deduct X% off the purchase price ad pay cash o the date of sale. A customer wishes to determie X such that he is idifferet betwee the two optios whe valuig them usig a effective aual iterest rate of 8%. Which of the followig equatios of value would the customer eed to solve? X 0.08 A 1+ = X 0.08 B = X 16 C ( 1.08) = X 1.08 D = X 16 E 1 ( 1.08) = Questio 19 Calculate the omial rate of discout covertible mothly that is equivalet to a omial rate of iterest of 18.9% per year covertible mothly. A 18.0% B 18.3% C 18.6% D 18.9% E 19.% Questio 0 A ivestor wishes to accumulate $10,000 at the ed of 10 years by makig level deposits at the begiig of each year. The deposits ear 1% aual effective rate of iterest paid at the ed of each year. The iterest is immediately reivested at a aual effective iterest rate of 8%. Calculate the level deposit. A 541 B 57 C 598 D 615 E 61 6

7 May 005 Exam - Questios Exam FM Questio 1 A discout electroics store advertises the followig fiacig arragemet: We do t offer you cofusig iterest rates. We ll just divide your total cost by 10 ad you ca pay us that amout each moth for a year. The first paymet is due o the date of sale ad the remaiig eleve paymets at mothly itervals thereafter. Calculate the effective aual iterest rate the store s customers are payig o their loas. A 35.1% B 41.3% C 4.0% D 51.% E 54.9% Questio O Jauary 1, 004, Kare sold stock A short for $50 with a margi requiremet of 80%. O December 31, 004, the stock paid a divided of $, ad a iterest amout of $4 was credited to the margi accout. O Jauary 1, 005, Kare covered the short sale at a price of X, earig a 0% retur. Calculate X. A 40 B 44 C 48 D 5 E 56 Questio 3 The stock of Compay X sells for $75 per share assumig a aual effective iterest rate of i. Aual divideds will be paid at the ed of each year forever. The first divided is $6, with each subsequet divided 3% greater tha the previous year s divided. Calculate i. A 8% B 9% C 10% D 11% E 1% 7

8 Exam FM May 005 Exam - Questios Questio 4 A auity pays $1 at the ed of each year for years. Usig a aual effective iterest rate of i, the accumulated value of the auity at time ( + 1) is $ It is also kow that (1 + i ) =.476. Calculate. A 4 B 5 C 6 D 7 E 8 Questio 5 A bak customer takes out a loa of $500 with a 16% omial iterest rate covertible quarterly. The customer makes paymets of $0 at the ed of each quarter. Calculate the amout of pricipal i the fourth paymet. A 0.0 B 0.9 C.7 D 5. E There is ot eough iformatio to calculate the amout of pricipal. 8

9 SOA Exam FM / CAS Exam May 005 Exam Solutios E Solutio 1 Aswer choice (A) is a expressio for the auity-immediate preset value factor ad it is correct sice: a = v s Aswer choices (B) ad (C) are also correct expressios. Aswer choice (D) is also correct sice the part i the brackets is aother way to express the preset value of a auity-due: We the have: a = va 1 1 v 1 v 1 + v+ v + + v = = = a 1 v d Aswer choice (E) is ot a correct expressio for the auity-immediate preset value factor. It should be: a s = (1 + i) C Solutio The total paymet for a sikig fud loa for ay period is comprised of the service paymet (SP) ad the sikig fud paymet (SFP). Sice Lori pays the iterest due o the loa each year, the service paymet equals the iterest due o the loa: SP = 10, 000(0.09) = $ The accumulated value of the sikig fud paymets at the sikig fud iterest rate must equal the loa value at the maturity of the loa. The auity-immediate accumulated value factor is: ( ) s10 8% = = The aual sikig fud paymet is the: 10, 000 SFP = = $ s 10 8% The total aual paymet o the loa is: SP + SFP = = $1, The total paymets made over the 10-year period is: X = 1, = $15,

10 Exam FM May 005 Exam Solutios B Solutio 3 Macaulay duratio ca be calculated directly: 100(1) 100() 1,100(3) , MacD = = = , A Solutio 4 Seth receives aual paymets of $X from time 1 year to time years. The preset value of Seth s auity-immediate at time 0 is just: Xa Susa receives m aual paymets of $X from time + 1 years to time + m years. Sice the auity-immediate preset value factor is valued oe period before the first cash flow, the preset value at time 0 of Susa s paymets is: Xv a m The differece betwee the preset value of Seth s ad Susa s paymets is: X a v a m B Solutio 5 We determie the aual effective iterest rate from the price of the zero-coupo bod: 1,000 P = = (1 + i) 1 1 1,000 (1 + i) = = i = The aual effective iterest rate is 4.0%. The semi-aual effective iterest rate is the: () i 1/ = (1.04) 1 = The 6% coupo bod pays semi-aual coupos of: ,000 = $30 The auity-immediate preset value factor for 0 semi-aual periods at 1.980% semi-aual effective iterest is: 0 1 ( ) a % = = The price of the 6% coupo bod is: 0 P = 30a % + 1, 000( ) = 30(16.384) + 1, 000( ) = 1,

11 May 005 Exam Solutios Exam FM D Solutio 6 This is just a straightforward weighted average questio. The total value of the portfolio is: , , 000 =, 995 The duratio of the portfolio is the: 980 1,015 1, = 16.77,995,995,995 A Solutio 7 The equatio of value for the preset value of the cash flows is: = v v + 00v = 0 We ca solve this equatio for v usig the quadratic equatio: v 00 ± ( 64.46) v = 100 = or We discard the egative solutio sice it does t make sese whe determiig a iterest rate. We use v to solve for i: 1 v= (1 + i) = i = 1.10 i = 0.10 C Solutio 8 The iitial loa paymet is determied by the equatio: L P = a i The auity-immediate preset value factor is: 5 1 (1.08) a5 8% = = The iitial loa amout is: L 300 = L = 3, The outstadig loa balace immediately after the 10th loa paymet by the retrospective method is: 10 B10 = 3, (1.08) 300s10 8% 10 (1.08) 1 = 3, (.15893) = 6, , =,

12 Exam FM May 005 Exam Solutios A extra $1,000 is paid at this time, so the outstadig balace becomes:, ,000 = 1, This ew balace is repaid over 10 years with a revised aual paymet. The ew paymet is: 1, , P = = = a % 1 (1.08) 0.08 D Solutio 9 The first paymet is $100 at time 1, ad each subsequet aual paymet icreases by $1. There are 50 paymets, so the last paymet of $149 occurs at time 50 years. This series of paymets ca be split ito two parts cosistig of a level auity-immediate ad a icreasig auityimmediate. The first part is a level auity-immediate i which paymets of $99 are made at the ed of each year for 50 years. The secod part is a 50-year icreasig auity-immediate, where the first paymet is $1 at time 1, ad the each subsequet aual paymet icreases by $1. The preset value of both parts of this paymet series is: a + Ia % ( ) 50 9% Determiig the required values, we have: 50 1 (1.09) a50 9% = = a50 9% = (1.09)( ) = (1.09) ( Ia) 50 9% = = So the preset value of the paymet series is: X = 99( ) = 1, Solutio 10 C or E The SOA origially scored aswer choice (C) as the correct aswer, but the it also scored aswer choice (E) as correct due to ambiguity regardig which forward rate is to be determied. Aswer choice (C): The oe-year forward rate for year two ca be iterpreted as f 1, the oe-year forward rate i effect from time 1 year to time years, sice the secod year is from time 1 year to time years. Forward rates ca be determied from the yields of the zero-coupo bods. I this case, we use the yield from the two-year zero-coupo bod sice the yield o a two-year zerocoupo bod is the same as the two-year spot rate. We have: (1 + s) = (1 + f0)(1 + f1) = (1.085)(1 + f1) f1 = f = 10.51% 1 4

13 May 005 Exam Solutios Exam FM Aswer choice (E): The oe-year forward rate for year two ca also be iterpreted as the oeyear forward rate that starts at time two years. This forward rate is f, the oe-year forward rate i effect from time years to time 3 years. I this case, we use the yield from the 3-year zerocoupo bod sice the yield o the 3-year zero-coupo bod is the same as the 3-year spot rate. Usig the above result for f 1, we have: 3 (1 + s3) = (1 + f0)(1 + f1)(1 + f) = (1.0850)(1.1051)(1 + f ) f = (1.0850)(1.1051) f = 1.53% C Solutio 11 We are give two potetial bod purchase prices, either oe of which ca be used to determie the desired yield rate. Assumig the bod is called after 10 years for the call price of $1,050, the equatio of value for the bod price is: , 050(1 ) 10 i P = = a + + i Usig the TI BA-35 calculator, we ca easily determie the yield. Press 957 [PV], 80 [PMT], 1,050 [FV], 10 [N], ad the [CPT] [%i]. The desired yield is 9.0%. Assumig the bod is held util maturity, the equatio of value for the bod price is: , 000(1 ) 30 i P = = a + + i Usig the TI BA-35, press 897 [PV], 80 [PMT], 1,000 [FV], 30 [N], ad the [CPT] [%i]. The desired yield is agai 9.0%. This ivestor buys the bod at the highest price that guaratees she will receive at least her desired yield of 9.0% regardless of whe the bod is called. The bod is called after 0 years. The ivestor either purchased this bod for a price of $957 or $897. The price that provides a higher yield is the lower price, so she purchased the bod for $897. Assumig she paid $897 for the bod, we ca determie her actual yield. The equatio of value assumig the bod is called after 0 years for the call price of $1,050 is: , 050(1 ) 0 i P = = a + + i Usig the TI BA-35, press 897 [PV], 80 [PMT], 1,050 [FV], 0 [N], ad the [CPT] [%i]. Her actual yield is 9.4%. If this ivestor had paid $957 for the bod, her yield would have bee 8.56%, which is lower tha her desired yield. Sice she bought the bod at a price to guaratee at least a 9% yield, she must have paid $987 for the bod. B Solutio 1 Statemet I is false. Statemet I is ot true for level perpetuities-due or variable perpetuities. Statemet II is true by defiitio. Statemet III is false. Statemet III is ot true for variable perpetuities. 5

14 Exam FM May 005 Exam Solutios D Solutio 13 The $1,000 deposit accumulates from time 0 to time years ad the $1,500 deposit accumulates from time 1 year to time years. Sice the omial iterest rate i is covertible semiaually, let s work i semiaual periods. So the first deposit accumulates for 4 semiaual periods ad the secod deposit accumulates for semiaual periods. The equatio of value is: 4 i i 1, , =,600 i Let x = 1 +, ad we ca rewrite the equatio of value so that we ca solve for x usig the quadratic equatio: 1, 000x + 1, 500x, 600 = 0 1,500 ± 1, ,000 (,600) x = 1, 000 x = or.5834 We discard the egative solutio sice it does t make sese with iterest rates. We ca ow solve for i: i x = 1 + = i 1 + = i =.81% E Solutio 14 Paymets of $0 occur at the ed of each year from time 1 year to time 10 years, ad the the paymets decrease by $1 each year so that a paymet of $19 occurs at time 11 years, $18 at time 1 years, ad so o, dow to $1 at time 9 years. If we split the paymets ito two parts, it will be easier to fit them to stadard auity factors. The first part is a 9-year auity-immediate of $0, ad the secod part is a 0-year decreasig auity-immediate i which the first paymet occurs at time 10 years. The preset value factor of the decreasig auity-immediate is valued oe year before the first cash flow, which is at time 9 years i this case, so this preset value eeds to be discouted for 9 years to brig it to time 0. The preset value of both parts of the paymet series is the: 6 0 a + v 9 ( Da ) 9 0 Calculatig the required values, we have: 9 1 (1.06) a9 = = (1.06) a0 = = ( Da) 0 = = The preset value of the paymet series is: 9 0( ) + (1.06) ( ) = 0.18

15 May 005 Exam Solutios Exam FM D Solutio 15 With the dedicatio ivestmet strategy, we start matchig liability ad asset cash flows at the ed ad work backwards to the begiig. At time years, the liability obligatio is $10,000. This amout eeds to be matched at time with $10,000 of asset cash flows. Assumig a par value of $100, the -year bod pays $6 at time 1 year ad $106 at time years. To match the liability cash flow at time, we eed to buy uits of the -year bod: 10, 000 = Sice we purchase uits of the -year bod, it pays = at time 1. So the et liability cash flow that eeds to be matched at time 1 by the 1-year bod is 10, = 9, Assumig a par value of $100, the 1-year bod pays $104 at time 1 year. To match this et liability cash flow at time 1, we eed to buy uits of the 1-year bod: 9, = The combiatio of uits of the -year bod ad uits of the 1-year bod exactly matches the liability cash flows at times 1 ad. The prices of the 1-year ad -year bods are: 104 P1 yr = = P yr = + = The cost to buy uits of the -year bod ad uits of the 1-year bod is: = 18, There is a shortcut to workig this problem if we recogize that the yield o both bods is 5%, so all asset cash flows are discouted at 5%. To match the liability cash flows, the asset portfolio must have cash flows of $10,000 at time 1 ad $10,000 at time. The price of this portfolio is: 10, , = 18,

16 Exam FM May 005 Exam Solutios A Solutio 16 The dollar-weighted yield is just the IRR of the cash flows. I this case, the aswer choices are give as aual effective rates, so let s work i aual periods. The $1,000 deposit at time 0 accumulates for 1 year. The $1,000 deposit at time 4 moths accumulates for (1 4)/1 = 8/1 = /3 years. The $00 withdrawal at time 6 moths accumulates for (1 6)/1 = 6/1 = 1/ years. The $500 withdrawal at time 8 moths accumulates for (1 8)/1 = 4/1 = 1/3 years. Let i be the aual effective iterest rate, ad we have: 1, 000(1 + i) 1 + 1, 000(1 + i) /3 00(1 + i) 1/ 500(1 + i ) 1/3 = 1, 560 There are too may terms i this equatio to solve usig the quadratic equatio. Oe approach to determie i is to use trial ad error at this poit o the left-had side ad see which iterest rate results i a aswer of $1,560. Startig with a guess of.61%, we get $1,615.05, which is a good bit too high, so our ext guess should be a lower iterest rate. If our secod guess is 18.57%, we get $1,558.96, which is very close ad is the correct solutio. With trial ad error, we ca usually aswer the questio after or 3 guesses. Aother approach is to use the simple iterest approximatio sice all of these cash flows are withi a year of each other. Usig the simple iterest approximatio, it is fairly easy to solve the equatio: 1, 000(1 + i) + 1, 000( i) 00( i) 500( i) = 1, 560 1, 400i = 60 i = B Solutio 17 The preset value of the cash flows is: v+ 50v + 300v + 350v + The patter of cash flows does ot exactly match the patter expected by ay of the stadard perpetuity factors, so we eed to split the cash flows util they do. Whe we subtract out a level perpetuity-immediate of $00, we are left with $50 at time years, $100 at time 3 years, $150 at time 4 years, ad so o. The remaiig cash flows fit the patter of a icreasig perpetuity-immediate i which the first cash flow of $50 occurs at time years ad each subsequet aual cash flow icreases by $50. Sice the preset value factor for a perpetuity-immediate is valued oe year before the first cash flow, this value must be discouted for oe more year to brig it to time zero i this case. So the preset value of both parts is: 00a + 50 v( Ia ) = 46, 530 If we do t recall the formula for the preset value factor for a icreasig perpetuity-immediate, it is easy to derive from the formula for the preset value factor for a icreasig auityimmediate: 1 v v a v i/(1 + i) 1 + i ( Ia) = lim ( Ia) = lim = lim = i i i 8

17 May 005 Exam Solutios Exam FM So the preset value of the paymet series becomes: i + = 46,530 i 1 + i i = 46,530 i i Now we ca solve for i usig the quadratic equatio: 46,530i 00i 50 = 0 00 ± ( 00) 4 46, 530 ( 50) i = 46, 530 i = E Solutio 18 Sice the aswers are expressed as accumulated values at time moths, we let the valuatio date be at time moths. The optio to pay 90% of the purchase price two moths after the date of sale is valued at 0.90 times the purchase price at time moths. The optio to deduct X% off the purchase price at the date of sale (time 0) implies that customers pay X 1 times the purchase price at time 0. Sice this amout is paid at time 0, it eeds to 100 be accumulated to time moths so that we ca equate it to the value of the secod optio. The two-moth accumulatio factor usig a aual effective iterest rate of 8% is 1 16 (1.08) = (1.08). Whe we equate these two values at time moths, the purchase price cacels out of each side ad we have: X 16 1 (1.08) = C Solutio 19 We ca relate the omial discout rate covertible mothly to the omial iterest rate covertible mothly ad solve for the omial discout rate covertible mothly: (1) (1) 1 i d 1+ = (1) d 1+ = (1) d = 1 1 (1) d =

18 Exam FM May 005 Exam Solutios A Solutio 0 We are give that the ivestor makes level deposits ito a bak accout at the begiig of each year for 10 years ad that it ears iterest at the ed of every year at a aual effective iterest rate of 1%. The iterest ears iterest at a aual effective rate of 8%. We ll let the amout of each aual level deposit be X. At the ed of the first year, the ivestmet of X has eared 0.1X i iterest, which is reivested at 8% for 9 years to time 10 years. At the ed of the secod year, two paymets of X have eared 0.1( X ) i iterest, which is reivested at 8% for 8 years util time 10 years. At the ed of the third year, three paymets of X have eared 0.1(3 X ) i iterest, which is reivested at 8% for 7 years util time 10 years. That patter cotiues, util at the ed of te years, te paymets of X have eared 0.1(10 X ) i iterest, but sice this occurs at time 10, there is o time for this to be reivested at 8% util time 10 years. I total, there are 10 aual paymets of X ito the fud. The total ivestmet over 10 years is 10X. We ca set up the equatio for this series of ivestmets valued at time 10 years as: ( ) ( ) ( ) ( ) 10X+ 0.1X X X X 1.08 This ca be reduced to: X+ 0.1X ( ) ( ) ( ) ( ) The part i the brackets is the accumulated value of a icreasig auity-immediate, or ( Is ) 10 8%. The accumulated value at time 10 is $10,000, so we ca set up the equatio of value: ( ) 10, 000 = 10X+ 0.1X Is 10 8% Determiig the required values, we have: s 10 8% ( Is ) 10 8% 10 ( ) = = / = = Pluggig these values ito the equatio of value, we have: 10,000 = 10X+ 0.1 X( ) 10, 000 X = X =

19 May 005 Exam Solutios Exam FM Solutio 1 D Let the purchase price be X. This purchase price is divided ito 1 mothly paymets of X /10 that begi today. Workig i moths with the mothly effective iterest rate, the equatio of value is: X X = a = a 10 1 i i (1) 1 (1) 1 Usig the TI BA-35 calculator, we ca solve for the mothly effective iterest rate. Press [ d ] [BGN] for begiig of moth paymets, 1 [PV], 0.1 [PMT], 10 [N], ad [CPT] [%i]. The mothly effective iterest rate is %. The aual effective iterest rate is the: i 1 = ( ) 1 = 51.16% B D Solutio Kare sells the stock short for $50 ad receives margi iterest of $4 at the ed of the year. The stock pays a divided of $ at the ed of the year. Her short sale yield is 0%. Pluggig these values ito the short sale yield formula, we ca solve for the buyback price, B: Solutio 3 (50 B) = 50(0.8) 5 B 0.0 = 40 5 B = 8 B = 40 The price of the stock is $75. The first divided oe year from ow is $6, ad each subsequet divided grows by 6% per year. This matches the patter of paymets expected by the costat divided growth stock valuatio formula, so we use this formula to solve for the aual effective iterest rate i: 6 75 = i i 0.03 = 75 i =

20 Exam FM May 005 Exam Solutios Solutio 4 E We use the accumulated value of the auity at time ( + 1) to solve for the aual effective iterest rate: AV + 1 = = s (1 + i) s = 1 + i (1 + i) = i 1 + i = i 1 + i 1.476(1 + i) = i 1.3i = i = 0.1 We use the aual effective iterest rate to solve for : (1 + i) =.476 l(1.1) = l(.476) l(.476) = l(1.1) = 8 A Solutio 5 The amout of iterest i the fourth paymet is the periodic effective iterest rate times the loa balace at time 3 quarters. The amout of pricipal i the fourth paymet is the total paymet mius the amout of iterest i the fourth paymet. i 0.16 The quarterly effective iterest rate is = = (4) Workig i quarters, we calculate the loa balace at time 3 quarters usig the retrospective method as the accumulated value of the loa mius the accumulated value of the paymets: 3 B3 = 500(1.04) 0s34% 3 (1.04) 1 = 500(1.1486) = 500 The loa paymet exactly matches the amout of iterest due o the loa each quarter, so the loa amout stays level. Sice all of the loa paymet is used to pay iterest, there is othig left over to pay dow the pricipal, so the amout of pricipal i the fourth paymet is zero. Mathematically, we ca determie the amout of iterest i the fourth paymet: (4) i I4 = B 3 = = So the amout of pricipal i the fourth paymet is: P4 = P I 4 = = 0.0 1

21 Course FM May 005 Exam Idex May 005 Exam Idex This idex provides details of how the sectios i the BPP Course FM course otes correspod to the questios i the May 005 SOA Exam FM / CAS Exam. Questio & Aswer Correspodig Sectio(s) i the BPP Course FM Course Notes Commets 1: E.1 Level auity-immediate PV factor : C 5.6 Sikig fud loa 3: B 7.3 Macaulay duratio 4: A.1 Level auity-immediate 5: B 6. Bod valuatio 6: D 7.7 Weighted average duratio 7: A.6 Determiig i from PV cash flows 8: C 5.5 Aual paymet loa; refiace loa 9: D 3.1 Icreasig auity-immediate 10: C or E 8.1, 8.3 Forward rates ad yields 11: C 6. Callable bod 1: B.5 Perpetuities 13: D 4., 4.4 No-aual iterest rates 14: E.1, 3.3 Level & decreasig auity-immediate 15: D 7.9 Dedicatio 16: A 5.3 Dollar-weighted iterest rate 17: B 3.1 Icreasig perpetuity-immediate 18: E.6 Equatio of value 19: C 4., 4.3 No-aual iterest & discout rates 0: A 5.1 Iterest reivested at differet rate 1: D 4.5 No-aual auity-due : B 6.4 Short sale 3: D 6.3 Stock valuatio; divided growth 4: E.1 Auity-immediate AV factor 5: A 5.5 Pricipal amout i loa paymet BPP Professioal Educatio: 005 exams Page 1