0.02t if 0 t 3 δ t = if 3 < t


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1 1 Exam FM questons 1. (# 12, May 2001). Bruce and Robbe each open up new bank accounts at tme 0. Bruce deposts 100 nto hs bank account, and Robbe deposts 50 nto hs. Each account earns an annual effectve dscount rate of d. The amount of nterest earned n Bruce s account durng the 11th year s equal to X. The amount of nterest earned n Robbe s account durng the 17th year s also equal to X. Calculate X. (A) 28.0 (B) 31.3 (C) 34.6 (D) 36.7 (E) (# 12, May 2003). Erc deposts X nto a savngs account at tme 0, whch pays nterest at a nomnal rate of, compounded semannually. Mke deposts 2X nto a dfferent savngs account at tme 0, whch pays smple nterest at an annual rate of. Erc and Mke earn the same amount of nterest durng the last 6 months of the 8th year. Calculate. (A) 9.06% (B) 9.26% (C) 9.46% (D) 9.66% (E) 9.86% 3. (# 50, May 2003). Jeff deposts 10 nto a fund today and 20 ffteen years later. Interest s credted at a nomnal dscount rate of d compounded quarterly for the frst 10 years, and at a nomnal nterest rate of 6% compounded semannually thereafter. The accumulated balance n the fund at the end of 30 years s 100. Calculate d. (A) 4.33% (B) 4.43% (C) 4.53% (D) 4.63% (E) (#25 Sample Test). Bran and Jennfer each take out a loan of X. Jennfer wll repay her loan by makng one payment of 800 at the end of year 10. Bran wll repay hs loan by makng one payment of 1120 at the end of year 10. The nomnal semannual rate beng charged to Jennfer s exactly one half the nomnal sem annual rate beng charged to Bran. Calculate X. A. 562 B. 565 C. 568 D. 571 E (#1 May 2003). Bruce deposts 100 nto a bank account. Hs account s credted nterest at a nomnal rate of nterest convertble semannually. At the same tme, Peter deposts 100 nto a separate account. Peter s account s credted nterest at a force of nterest of δ. After 7.25 years, the value of each account s 200. Calculate ( δ). (A) 0.12% (B) 0.23% (C) 0.31% (D) 0.39% (E) 0.47% 6. (#23, Sample Test). At tme 0, deposts of 10, 000 are made nto each of Fund X and Fund Y. Fund X accumulates at an annual effectve nterest rate of 5 %. Fund Y accumulates at a smple nterest rate of 8 %. At tme t, the forces of nterest on the two funds are equal. At tme t, the accumulated value of Fund Y s greater than the accumulated value of Fund X by Z. Determne Z. A B C D E
2 7. (#24, Sample Test). At a force of nterest δ t = 2 k+2t. () a depost of 75 at tme t = 0 wll accumulate to X at tme t = 3; and () the present value at tme t = 3 of a depost of 150 at tme t = 5 s also equal to X. Calculate X. A. 105 B. 110 C. 115 D. 120 E (# 37, May 2000). A customer s offered an nvestment where nterest s calculated accordng to the followng force of nterest: 0.02t f 0 t 3 δ t = f 3 < t The customer nvests 1000 at tme t = 0. What nomnal rate of nterest, compounded quarterly, s earned over the frst four year perod? (A) 3.4% (B) 3.7% (C) 4.0% (D) 4.2% (E) 4.5% 9. (# 53, November 2000). At tme 0, K s deposted nto Fund X, whch accumulates at a force of nterest δ t = 0.006t 2. At tme m, 2K s deposted nto Fund Y, whch accumulates at an annual effectve nterest rate of 10%. At tme n, where n > m, the accumulated value of each fund s 4K. Determne m. (A) 1.6 (B) 2.4 (C) 3.8 (D) 5.0 (E) (# 45, May 2001). At tme t = 0, 1 s deposted nto each of Fund X and Fund Y. Fund X accumulates at a force of nterest δ t = t2. Fund Y accumulates at a nomnal rate of k dscount of 8% per annum convertble semannually. At tme t = 5, the accumulated value of Fund X equals the accumulated value of Fund Y. Determne k. (A) 100 (B) 102 (C) 104 (D) 106 (E) (# 49, May 2001). Tawny makes a depost nto a bank account whch credts nterest at a nomnal nterest rate of 10% per annum, convertble semannually. At the same tme, Fabo deposts 1000 nto a dfferent bank account, whch s credted wth smple nterest. At the end of 5 years, the forces of nterest on the two accounts are equal, and Fabo s account has accumulated to Z. Determne Z. (A) 1792 (B) 1953 (C) 2092 (D) 2153 (E) (# 1, May 2000). Joe deposts 10 today and another 30 n fve years nto a fund payng smple nterest of 11% per year. Tna wll make the same two deposts, but the 10 wll be deposted n years from today and the 30 wll be deposted 2n years from today. Tna s deposts earn an annual effectve rate of 9.15%. At the end of 10 years, the accumulated 2
3 amount of Tna s deposts equals the accumulated amount of Joe s deposts. Calculate n. (A) 2.0 (B) 2.3 (C) 2.6 (D) 2.9 (E) (# 1, November 2001 ). Erne makes deposts of 100 at tme 0, and X at tme 3. The fund grows at a force of nterest δ t = t2, t > 0. The amount of nterest earned from tme 3 to 100 tme 6 s X. Calculate X. (A) 385 (B) 485 (C) 585 (D) 685 (E) (# 24, November 2001). Davd can receve one of the followng two payment streams: () 100 at tme 0, 200 at tme n, and 300 at tme 2n () 600 at tme 10 At an annual effectve nterest rate of, the present values of the two streams are equal. Gven ν n = , determne. (A) 3.5% (B) 4.0% (C) 4.5% (D) 5.0% (E) 5.5% 15. (# 17, May 2003). An assocaton had a fund balance of 75 on January 1 and 60 on December 31. At the end of every month durng the year, the assocaton deposted 10 from membershp fees. There were wthdrawals of 5 on February 28, 25 on June 30, 80 on October 15, and 35 on October 31. Calculate the dollar weghted rate of return for the year. (A) 9.0% (B) 9.5% (C) 10.0% (D) 10.5% (E) 11.0% 16. (#32, Sample Test). 100 s deposted nto an nvestment account on January 1, You are gven the followng nformaton on nvestment actvty that takes place durng the year: Aprl 19, 1998 October 30, 1998 Value mmedately pror to depost Depost 2X X The amount n the account on January 1, 1999 s 115. Durng 1998, the dollar weghted return s 0% and the tmeweghted return s y. Calculate y. (A) 1.5% (B) 0.7% (C) 0.0% (D) 0.7% (E) 1.5% 17. (# 27, November 2000). An nvestor deposts 50 n an nvestment account on January 1. The followng summarzes the actvty n the account durng the year: Date Value Immedately Before Depost Depost March June October
4 On June 30, the value of the account s On December 31, the value of the account s X. Usng the tme weghted method, the equvalent annual effectve yeld durng the frst 6 months s equal to the (tmeweghted) annual effectve yeld durng the entre 1year perod. Calculate X. (A) (B) (C) (D) (E) (#31, May 2001). You are gven the followng nformaton about an nvestment account: Date Value Immedately Before Depost Depost January 1 10 July 1 12 X December 31 X Over the year, the tme weghted return s 0%, and the dollarweghted return s Y. Calculate Y. (A) 25% (B) 10% (C) 0% (D) 10% (E) 25% 19. (#16, May 2000 ). On January 1, 1997, an nvestment account s worth 100, 000. On Aprl 1, 1997, the value has ncreased to 103,000 and 8, 000 s wthdrawn. On January 1, 1999, the account s worth 103, 992. Assumng a dollar weghted method for 1997 and a tme weghted method for 1998, the annual effectve nterest rate was equal to x for both 1997 and Calculate x. (A) 6.00% (B) 6.25% (C) 6.50% (D) 6.75% (E) 7.00% 20. (# 28, November 2001). Payments are made to an account at a contnuous rate of (8k+tk), where 0 t 10. Interest s credted at a force of nterest δ t = 1. After 10 years, the 8+t account s worth 20, 000. Calculate k. (A) 111 (B) 116 (C) 121 (D) 126 (E) (# 2, November, 2000) The followng table shows the annual effectve nterest rates beng credted by an nvestment account, by calendar year of nvestment. The nvestment year method s applcable for the frst 3 years, after whch a portfolo rate s used: Calendar year calendar of orgnal year of Portfolo nvestment Portfolo rate Rate % 10% t% % % 5% 10% 1994 t 1% % t 2% 12% % % 11% 6% % % 7% 10% % 4
5 An nvestment of 100 s made at the begnnng of years 1990, 1991, and The total amount of nterest credted by the fund durng the year 1993 s equal to Calculate t. (A) 7.00 (B) 7.25 (C) 7.50 (D) 7.75 (E) (# 51, November, 2000) An nvestor deposts 1000 on January 1 of year x and deposts another 1000 on January 1 of year x+2 nto a fund that matures on January 1 of year x+4. The nterest rate on the fund dffers every year and s equal to the annual effectve rate of growth of the gross domestc product (GDP) durng the 4 th quarter of the prevous year. The followng are the relevant GDP values for the past 4 years: Year III Quarter IV Quarter Year Quarter III Quarter III x x x x What s the nternal rate of return earned by the nvestor over the 4 year perod? (A) 1.66% (B) 5.10% (C) 6.15% (D) 6.60% (E) 6.78% 23. (#26, Sample Test). Carol and John shared equally n an nhertance. Usng hs nhertance, John mmedately bought a 10year annutydue wth an annual payment of 2500 each. Carol put her nhertance n an nvestment fund earnng an annual effectve nterest rate of 9%. Two years later, Carol bought a 15year annutymmedate wth annual payment of Z. The present value of both annutes was determned usng an annual effectve nterest rate of 8%. Calculate Z. A B C D E (#27, Sample Test). Susan and Jeff each make deposts of 100 at the end of each year for 40 years. Startng at the end of the 41st year, Susan makes annual wthdrawals of X for 15 years and Jeff makes annual wthdrawals of Y for 15 years. Both funds have a balance of 0 after the last wthdrawal. Susan s fund earns an annual effectve nterest rate of 8 %. Jeff s fund earns an annual effectve nterest rate of 10 %. Calculate Y X. A B C D E (# 22, November 2000). Jerry wll make deposts of 450 at the end of each quarter for 10 years. At the end of 15 years, Jerry wll use the fund to make annual payments of Y at the begnnng of each year for 4 years, after whch the fund s exhausted. The annual effectve rate of nterest s 7%. Determne Y. (A) 9573 (B) 9673 (C) 9773 (D) 9873 (E)
6 26. (# 27, November 2001). A man turns 40 today and wshes to provde supplemental retrement ncome of 3000 at the begnnng of each month startng on hs 65th brthday. Startng today, he makes monthly contrbutons of X to a fund for 25 years. The fund earns a nomnal rate of 8% compounded monthly. Each 1000 wll provde for 9.65 of ncome at the begnnng of each month startng on hs 65th brthday untl the end of hs lfe. Calculate X. (A) (B) (C) (D) (E) (# 47, May 2000). Jm began savng money for hs retrement by makng monthly deposts of 200 nto a fund earnng 6% nterest compounded monthly. The frst depost occurred on January 1, Jm became unemployed and mssed makng deposts 60 through 72. He then contnued makng monthly deposts of 200. How much dd Jm accumulate n hs fund on December 31, 1999? (A) 53, 572 (B) 53, 715 (C) 53, 840 (D) 53, 966 (E) 54, (# 12, November 2001). To accumulate 8000 at the end of 3n years, deposts of 98 are made at the end of each of the frst n years and 196 at the end of each of the next 2n years. The annual effectve rate of nterest s. You are gven (l + ) n = 2.0. Determne. (A) 11.25% (B) 11.75% (C) 12.25% (D) 12.75% (E) 13.25% 29. (# 34, November 2000). Chuck needs to purchase an tem n 10 years. The tem costs 200 today, but ts prce nflates 4% per year. To fnance the purchase, Chuck deposts 20 nto an account at the begnnng of each year for 6 years. He deposts an addtonal X at the begnnng of years 4, 5, and 6 to meet hs goal. The annual effectve nterest rate s 10%. Calculate X. (A) 7.4 (B) 7.9 (C) 8.4 (D) 8.9 (E) (# 8, May 2003). Kathryn deposts 100 nto an account at the begnnng of each 4 year perod for 40 years. The account credts nterest at an annual effectve nterest rate of. The accumulated amount n the account at the end of 40 years s X, whch s 5 tmes the accumulated amount n the account at the end of 20 years. Calculate X. (A) 4695 (B) 5070 (C) 5445 (D) 5820 (E) (# 33, May 2003). At an annual effectve nterest rate of, > 0, both of the followng annutes have a present value of X: () a 20 year annuty mmedate wth annual payments of 55 () a 30 year annuty mmedate wth annual payments that pays 30 per year for the frst 10 years, 60 per year for the second 10 years, and 90 per year for the fnal 10 years Calculate X. (A) 575 (B) 585 (C) 595 (D) 605 (E) 615 6
7 32. (# 17, May 2001). At an annual effectve nterest rate of, > 0%, the present value of a perpetuty payng 10 at the end of each 3 year perod, wth the frst payment at the end of year 6, s 32. At the same annual effectve rate of, the present value of a perpetuty mmedate payng 1 at the end of each 4month perod s X. Calculate X. (A) 38.8 (B) 39.8 (C) 40.8 (D) 41.8 (E) (# 5, May 2001). A perpetuty mmedate pays X per year. Bran receves the frst n payments, Colleen receves the next n payments, and Jeff receves the remanng payments. Bran s share of the present value of the orgnal perpetuty s 40%, and Jeff s share s K. Calculate K. (A) 24% (B) 28% (C) 32% (D) 36% (E) 40% 34. (# 50, May 2001). The present values of the followng three annutes are equal: () perpetuty mmedate payng 1 each year, calculated at an annual effectve nterest rate of 7.25% () 50 year annutymmedate payng 1 each year, calculated at an annual effectve nterest rate of j% () n year annutymmedate payng 1 each year, calculated at an annual effectve nterest rate of j 1% Calculate n. (A) 30 (B) 33 (C) 36 (D) 39 (E) (# 14, May 2000). A perpetuty payng 1 at the begnnng of each 6 month perod has a present value of 20. A second perpetuty pays X at the begnnng of every 2 years. Assumng the same annual effectve nterest rate, the two present values are equal. Determne X. (A) 3.5 (B) 3.6 (C) 3.7 (D) 3.8 (E) (#29, Sample Test). Chrs makes annual deposts nto a bank account at the begnnng of each year for 20 years. Chrs ntal depost s equal to 100, wth each subsequent depost k% greater than the prevous year s depost. The bank credts nterest at an annual effectve rate of 5%. At the end of 20 years, the accumulated amount n Chrs account s equal to Gven k > 5, calculate k. A B C D E (# 5, November 2001). Mke buys a perpetuty mmedate wth varyng annual payments. Durng the frst 5 years, the payment s constant and equal to 10. Begnnng n year 6, the payments start to ncrease. For year 6 and all future years, the current year s payment s K% larger than the prevous year s payment. At an annual effectve nterest rate of 9.2%, the perpetuty has a present value of Calculate K, gven K < 9.2. (A) 4.0 (B) 4.2 (C) 4.4 (D) 4.6 (E) 4.8 7
8 38. (# 9, May 2000). A senor executve s offered a buyout package by hs company that wll pay hm a monthly beneft for the next 20 years. Monthly benefts wll reman constant wthn each of the 20 years. At the end of each 12month perod, the monthly benefts wll be adjusted upwards to reflect the percentage ncrease n the CPI. You are gven: () The frst monthly beneft s R and wll be pad one month from today. () The CPI ncreases 3.2% per year forever. At an annual effectve nterest rate of 6%, the buyout package has a value of 100, 000. Calculate R. (A) 517 (B) 538 (C) 540 (D) 548 (E) (# 45, May 2003). A perpetutymmedate pays 100 per year. Immedately after the ffth payment, the perpetuty s exchanged for a 25 year annuty mmedate that wll pay X at the end of the frst year. Each subsequent annual payment wll be 8% greater than the precedng payment. Immedately after the 10th payment of the 25 year annuty, the annuty wll be exchanged for a perpetutymmedate payng Y per year. The annual effectve rate of nterest s 8%. Calculate Y. (A) 110 (B) 120 (C) 130 (D) 140 (E) (# 51, May 2000). Seth deposts X n an account today n order to fund hs retrement. He would lke to receve payments of 50 per year, n real terms, at the end of each year for a total of 12 years, wth the frst payment occurrng seven years from now. The nflaton rate wll be 0.0% for the next sx years and 1.2% per annum thereafter. The annual effectve rate of return s 6.3%. Calculate X. (A) 303 (B) 306 (C) 316 (D) 327 (E) (# 22, May 2003). A perpetuty costs 77.1 and makes annual payments at the end of the year. The perpetuty pays 1 at the end of year 2, 2 at the end of year 3,... n at the end of year (n + 1). After year (n + 1), the payments reman constant at n. The annual effectve nterest rate s 10.5%. Calculate n. (A) 17 (B) 18 (C) 19 (D) 20 (E) (# 16, November 2001). Olga buys a 5 year ncreasng annuty for X. Olga wll receve 2 at the end of the frst month, 4 at the end of the second month, and for each month thereafter the payment ncreases by 2. The nomnal nterest rate s 9% convertble quarterly. Calculate X. (A) 2680 (B) 2730 (C) 2780 (D) 2830 (E) (# 26, May 2000). Betty borrows 19,800 from Bank X. Betty repays the loan by makng 36 equal payments of prncpal at the end of each month. She also pays nterest on the unpad 8
9 balance each month at a nomnal rate of 12%, compounded monthly. Immedately after the 16th payment s made, Bank X sells the rghts to future payments to Bank Y. Bank Y wshes to yeld a nomnal rate of 14%, compounded semannually, on ts nvestment. What prce does Bank X receve? (A) 9,792 (B) 10,823 (C) 10,857 (D) 11,671 (E) 11, (# 44, November 2000). Joe can purchase one of two annutes: Annuty 1: A 10 year decreasng annuty mmedate, wth annual payments of 10, 9, 8,..., 1. Annuty 2: A perpetuty mmedate wth annual payments. The perpetuty pays 1 n year 1, 2 n year 2, 3 n year 3,..., and 11 n year 11. After year 11, the payments reman constant at 11. At an annual effectve nterest rate of, the present value of Annuty 2 s twce the present value of Annuty 1. Calculate the value of Annuty 1. (A) 36.4 (B) 37.4 (C) 38.4 (D) 39.4 (E) (# 20, November 2000). Sandy purchases a perpetuty mmedate that makes annual payments. The frst payment s 100, and each payment thereafter ncreases by 10. Danny purchases a perpetuty due whch makes annual payments of 180. Usng the same annual effectve nterest rate, > 0, the present value of both perpetutes are equal. Calculate. (A) 9.2% (B) 9.7% (C) 10.2% (D) 10.7% (E) 11.2% 46. (# 30, Sample Test). Scott deposts: 1 at the begnnng of each quarter n year 1; 2 at the begnnng of each quarter n year 2; 8 at the begnnng of each quarter n year 8. One quarter after the last depost, Scott wthdraws the accumulated value of the fund and uses t to buy a perpetutymmedate wth level payments of X at the end of each year. All calculatons assume a nomnal nterest rate of 10% per annum compounded quarterly. Calculate X. A B C D E (# 7, May 2001). Seth, Jance, and Lor each borrow 5000 for fve years at a nomnal nterest rate of 12%, compounded sem annually. Seth has nterest accumulated over the fve years and pays all the nterest and prncpal n a lump sum at the end of fve years. 9
10 Jance pays nterest at the end of every sxmonth perod as t accrues and the prncpal at the end of fve years. Lor repays her loan wth 10 level payments at the end of every sxmonth perod. Calculate the total amount of nterest pad on all three loans. (A) 8718 (B) 8728 (C) 8738 (D) 8748 (E) (#28, Sample Test). A loan of 10, 000 s to be amortzed n 10 annual payments begnnng 6 months after the date of the loan. The frst payment, X, s half as large as the other payments. Interest s calculated at an annual effectve rate of 5% for the frst 4.5 years and 3% thereafter. Determne X. A. 640 B. 648 C. 656 D. 664 E (# 12, November 2002). Kevn takes out a 10 year loan of L, whch he repays by the amortzaton method at an annual effectve nterest rate of. Kevn makes payments of 1000 at the end of each year. The total amount of nterest repad durng the lfe of the loan s also equal to L. Calculate the amount of nterest repad durng the frst year of the loan. (A) 725 (B) 750 (C) 755 (D) 760 (E) (# 37, Sample Test). A loan s beng amortzed by means of level monthly payments at an annual effectve nterest rate of 8%. The amount of prncpal repad n the 12 th payment s 1000 and the amount of prncpal repad n the t th payment s Calculate t. A. 198 B. 204 C. 210 D. 216 E (# 10, May 2000). A bank customer borrows X at an annual effectve rate of 12.5% and makes level payments at the end of each year for n years. () The nterest porton of the fnal payment s () The total prncpal repad as of tme (n 1) s () The prncpal repad n the frst payment s Y. Calculate Y. (A) 470 (B) 480 (C) 490 (D) 500 (E) (# 24, May 2000). A small busness takes out a loan of 12, 000 at a nomnal rate of 12%, compounded quarterly, to help fnance ts startup costs. Payments of 750 are made at the end of every 6 months for as long as s necessary to pay back the loan. Three months before the 9th payment s due, the company refnances the loan at a nomnal rate of 9%, compounded monthly. Under the refnanced loan, payments of R are to be made monthly, wth the frst monthly payment to be made at the same tme that the 9th payment under the old loan was to be made. A total of 30 monthly payments wll completely pay off the loan. Determne R. (A) 448 (B) 452 (C) 456 (D) 461 (E)
11 53. (# 55, November 2000). Iggy borrows X for 10 years at an annual effectve rate of 6%. If he pays the prncpal and accumulated nterest n one lump sum at the end of 10 years, he would pay more n nterest than f he repad the loan wth 10 level payments at the end of each year. Calculate X. (A) 800 (B) 825 (C) 850 (D) 875 (E) (# 34, November 2000). An nvestor took out a loan of 150, 000 at 8% compounded quarterly, to be repad over 10 years wth quarterly payments of at the end of each quarter. After 12 payments, the nterest rate dropped to 6% compounded quarterly. The new quarterly payment dropped to After 20 payments n total, the nterest rate on the loan ncreased to 7% compounded quarterly. The nvestor decded to make an addtonal payment of X at the tme of hs 20th payment. After the addtonal payment was made, the new quarterly payment was calculated to be , payable for fve more years. Determne X. (A) 11, 047 (B) 13, 369 (C) 16, 691 (D) 20, 152 (E) 23, (# 37, May 2001). Seth borrows X for four years at an annual effectve nterest rate of 8%, to be repad wth equal payments at the end of each year. The outstandng loan balance at the end of the second year s and at the end of the thrd year s Calculate the prncpal repad n the frst payment. (A) 444 (B) 454 (C) 464 (D) 474 (E) (# 13, May 2001). Ron has a loan wth a present value of a n. The sum of the nterest pad n perod t plus the prncpal repad n perod t + 1 s X. Calculate X. (A) 1 + νn t (B) 1 + νn t d (C) 1 + ν n t (D) 1 + ν n t d (E) 1 + ν n t 57. (# 9, November 2001). A loan s amortzed over fve years wth monthly payments at a nomnal nterest rate of 9% compounded monthly. The frst payment s 1000 and s to be pad one month from the date of the loan. Each succeedng monthly payment wll be 2% lower than the pror payment. Calculate the outstandng loan balance mmedately after the 40th payment s made. (A) 6751 (B) 6889 (C) 6941 (D) 7030 (E) (# 36, Sample Test). Don takes out a 10year loan of L, whch he repays wth annual payments at the end of each year usng the amortzaton method. Interest on the loan s charged at an annual effectve rate of. Don repays the loan wth a decreasng seres of payments. He repays 1000 n year one, 900 n year two, 800 n year three,..., and 100 n year ten. The amount of prncpal repad n year three s equal to 600. Calculate L. A B C D E
12 59. (# 26, May 2001). Susan nvests Z at the end of each year for seven years at an annual effectve nterest rate of 5%. The nterest credted at the end of each year s renvested at an annual effectve rate of 6%. The accumulated value at the end of seven years s X. Lor nvests Z at the end of each year for 14 years at an annual effectve nterest rate of 2.5%. The nterest credted at the end of each year s renvested at an annual effectve rate of 3%. The accumulated value at the end of 14 years s Y. Calculate Y X. (A) 1.93 (B) 1.98 (C) 2.03 (D) 2.08 (E) (# 26, May 2003) s deposted nto Fund X, whch earns an annual effectve rate of 6%. At the end of each year, the nterest earned plus an addtonal 100 s wthdrawn from the fund. At the end of the tenth year, the fund s depleted. The annual wthdrawals of nterest and prncpal are deposted nto Fund Y, whch earns an annual effectve rate of 9%. Determne the accumulated value of Fund Y at the end of year 10. (A) 1519 (B) 1819 (C) 2085 (D) 2273 (E) (# 9, November 2000). Vctor nvests 300 nto a bank account at the begnnng of each year for 20 years. The account pays out nterest at the end of every year at an annual effectve nterest rate of %. The nterest s renvested at an annual effectve rate of ( 2) %. The yeld rate on the entre nvestment over the 20 year perod s 8% annual effectve. Determne. (A) 9% (B) 10% (C) 11% (D) 12% (E) 13% 62. (# 31, Sample Test). Jason deposts 3960 nto a bank account at t = 0. The bank credts nterest at the end of each year at a force of nterest δ t = 1. Interest can be renvested 8+t at an annual effectve rate of 7%. The total accumulated amount at tme t = 3 s equal to X. Calculate X. A B C D E (#33, Sample Test). Erc deposts 12 nto a fund at tme 0 and an addtonal 12 nto the same fund at tme 10. The fund credts nterest at an annual effectve rate of. Interest s payable annually and renvested at an annual effectve rate of At tme 20, the accumulated amount of the renvested nterest payments s equal to 64. Calculate, > 0. A. 8.8% B. 9.0% C. 9.2% D. 9.4% E. 9.6% 64. (# 39, May 2000). Sally lends 10, 000 to Tm. Tm agrees to pay back the loan over 5 years wth monthly payments payable at the end of each month. Sally can renvest the monthly payments from Tm n a savngs account payng nterest at 6%, compounded monthly. The yeld rate earned on Sally s nvestment over the fveyear perod turned out to be 7.45%, compounded semannually. What nomnal rate of nterest, compounded monthly, dd Sally 12
13 charge Tm on the loan? (A) 8.53% (B) 8.59% (C) 8.68% (D) 8.80% (E) 9.16% 65. (# 35, November 2001). At tme t = 0, Sebastan nvests 2000 n a fund earnng 8% convertble quarterly, but payable annually. He renvests each nterest payment n ndvdual separate funds each earnng 9% convertble quarterly, but payable annually. The nterest payments from the separate funds are accumulated n a sde fund that guarantees an annual effectve rate of 7%. Determne the total value of all funds at t = 10. (A) 3649 (B) 3964 (C) 4339 (D) 4395 (E) (# 4, May 2001). A 20 year loan of 20, 000 may be repad under the followng two methods: () amortzaton method wth equal annual payments at an annual effectve rate of 6.5% () snkng fund method n whch the lender receves an annual effectve rate of 8% and the snkng fund earns an annual effectve rate of j Both methods requre a payment of X to be made at the end of each year for 20 years. Calculate j. (A) j 6.5% (B) 6.5% < j 8.0% (C) 8.0% < j 10.0% (D) 10.0% < j 12.0% (E) j > 12.0% 67. (# 34, Sample Test). A 10 year loan of 10, 000 s to be repad wth payments at the end of each year consstng of nterest on the loan and a snkng fund depost. Interest on the loan s charged at a 12% annual effectve rate. The snkng fund s annual effectve nterest rate s 8%. However, begnnng n the sxth year, the annual effectve nterest rate on the snkng fund drops to 6%. As a result, the annual payment to the snkng fund s then ncreased by X. Calculate X. A. 122 B. 132 C. 142 D. 152 E (# 35, Sample Test). Jason and Margaret each take out a 17year loan of L. Jason repays hs loan usng the amortzaton method, at an annual effectve nterest rate of. He makes an annual payment of 500 at the end of each year. Margaret repays her loan usng the snkng fund method. She pays nterest annually, also at an annual effectve nterest rate of. In addton, Margaret makes level annual deposts at the end of each year for 17 years nto a snkng fund. The annual effectve rate on the snkng fund s 4.62%, and she pays off the loan after 17 years. Margaret s total payment each year s equal to 10% of the orgnal loan amount. Calculate L. A B C D E (# 48, November 2000). A 12 year loan of 8000 s to be repad wth payments to the lender of 800 at the end of each year and deposts of X at the end of each year nto a snkng fund. 13
14 Interest on the loan s charged at an 8% annual effectve rate. The snkng fund annual effectve nterest rate s 4%. Calculate X. (A) 298 (B) 330 (C) 361 (D) 385 (E) (# 6, November 2001). A 10 year loan of 2000 s to be repad wth payments at the end of each year. It can be repad under the followng two optons: () Equal annual payments at an annual effectve rate of 8.07%. () Installments of 200 each year plus nterest on the unpad balance at an annual effectve rate of. The sum of the payments under opton () equals the sum of the payments under opton (). Determne. (A) 8.75% (B) 9.00% (C) 9.25% (D) 9.50% (E) 9.75% 71. (# 15, May 2003). John borrows 1000 for 10 years at an annual effectve nterest rate of 10%. He can repay ths loan usng the amortzaton method wth payments of P at the end of each year. Instead, John repays the 1000 usng a snkng fund that pays an annual effectve rate of 14%. The deposts to the snkng fund are equal to P mnus the nterest on the loan and are made at the end of each year for 10 years. Determne the balance n the snkng fund mmedately after repayment of the loan. (A) 213 (B) 218 (C) 223 (D) 230 (E) (# 30, November, 2000). A 1000 par value 20 year bond wth annual coupons and redeemable at maturty at 1050 s purchased for P to yeld an annual effectve rate of 8.25%. The frst coupon s 75. Each subsequent coupon s 3% greater than the precedng coupon. Determne P. (A) 985 (B) 1000 (C) 1050 (D) 1075 (E) (# 41, May 2001). Bll buys a 10year 1000 par value 6% bond wth semannual coupons. The prce assumes a nomnal yeld of 6%, compounded semannually. As Bll receves each coupon payment, he mmedately puts the money nto an account earnng nterest at an annual effectve rate of. At the end of 10 years, mmedately after Bll receves the fnal coupon payment and the redempton value of the bond, Bll has earned an annual effectve yeld of 7% on hs nvestment n the bond. Calculate. (A) 9.50% (B) 9.75% (C) 10.00% (D) 10.25% (E) 10.50% 74. (# 31, November 2001). You have decded to nvest n two bonds. Bond X s an nyear bond wth semannual coupons, whle bond Y s an accumulaton bond redeemable n n 2 years. The desred yeld rate s the same for both bonds. You also have the followng nformaton: Bond X 14
15 Par value s r The rato of the semannual bond rate to the desred semannual yeld rate, s The present value of the redempton value s Bond Y Redempton value s the same as the redempton value of bond X. Prce to yeld s What s the prce of bond X? (A) 1019 (B) 1029 (C) 1050 (D) 1055 (E) (# 29, May 2000). A frm has proposed the followng restructurng for one of ts 1000 par value bonds. The bond presently has 10 years remanng untl maturty. The coupon rate on the exstng bond s 6.75% per annum pad semannually. The current nomnal semannual yeld on the bond s 7.40%. The company proposes suspendng coupon payments for four years wth the suspended coupon payments beng repad, wth accrued nterest, when the bond comes due. Accrued nterest s calculated usng a nomnal semannual rate of 7.40%. Calculate the market value of the restructured bond. (A) 755 (B) 805 (C) 855 (D) 905 (E) (#42, May 2003). A 10,000 par value 10year bond wth 8% annual coupons s bought at a premum to yeld an annual effectve rate of 6%. Calculate the nterest porton of the 7th coupon. (A) 632 (B) 642 (C) 651 (D) 660 (E) (#43, May 2000). A 1000 par value 5 year bond wth 8.0% semannual coupons was bought to yeld 7.5% convertble semannually. Determne the amount of premum amortzed n the 6th coupon payment. (A) 2.00 (B) 2.08 (C) 2.15 (D) 2.25 (E) (#38, Sample Test). Laura buys two bonds at tme 0. Bond X s a 1000 par value 14year bond wth 10% annual coupons. It s bought at a prce to yeld an annual effectve rate of 8%. Bond Y s a 14year par value bond wth 6.75% annual coupons and a face amount of F. Laura pays P for the bond to yeld an annual effectve rate of 8%. Durng year 6, the wrtedown n premum (prncpal adjustment) on bond X s equal to the wrteup n dscount (prncpal adjustment) on bond Y. Calculate P. A B C D E
16 79. (#39, Sample Test). A 1000 par value 18 year bond wth annual coupons s bought to yeld an annual effectve rate of 5%. The amount for amortzaton of premum n the 10th year s 20. The book value of the bond at the end of year 10 s X. Calculate X. A B C D E (# 40, November, 2000). Among a company s assets and accountng records, an actuary fnds a 15year bond that was purchased at a premum. From the records, the actuary has determned the followng: () The bond pays semannual nterest. () The amount for amortzaton of the premum n the 2nd coupon payment was () The amount for amortzaton of the premum n the 4th coupon payment was What s the value of the premum? (A) 17,365 (B) 24,784 (C) 26,549 (D) 48,739 (E) 50, (#48, May 2000). A corporaton buys a new machne for It has an expected useful lfe of 10 years, and a salvage value of 400. The machne s deprecated usng the snkng fund method and an annual effectve rate of. Deprecaton s taken at the end of each year for the 10 year perod. The present value of the deprecaton charges over the 10 year perod s 1000 at the annual effectve rate. Calculate. (A) 8.4% (B) 8.8% (C) 9.2% (D) 9.6% (E) 10.0% 82. (#42, November 2001). A coper costs X and wll have a salvage value of Y after n years. () Usng the straght lne method, the annual deprecaton expense s () Usng the sum of the years dgts method, the deprecaton expense n year 3 s 800. () Usng the declnng balance method, the deprecaton expense s % of the book value n the begnnng of the year. Calculate X. (A) 4250 (B) 4500 (C) 4750 (D) 5000 (E) (#45, November 2001). A manufacturer buys a machne for 20, 000. The manufacturer estmates that the machne wll last 15 years. It wll be deprecated usng the constant percentage method wth an annual deprecaton rate of 20%. At the end of each year, the manufacturer deposts an amount nto a fund that pays 6% annually. Each depost s equal to the deprecaton expense for that year. How much money wll the manufacturer have accumulated n the fund at the end of 15 years? (A) 29, 663 (B) 34, 273 (C) 36, 329 (D) 38, 509 (E) 46, (#5, May 2003). A machne s purchased for 5000 and has a salvage value of S at the end of 10 years. The machne s deprecated usng the sumoftheyearsdgts method. At 16
17 the end of year 4, the machne has a book value of At that tme, the deprecaton method s changed to the straghtlne method for the remanng years. Determne the new deprecaton charge for year 8. (A) 200 (B) 222 (C) 286 (D) 370 (E) (#11, May 2001). Machne I costs X, has a salvage value of X/8, and s to be deprecated over 10 years usng the declnng balance method. Machne II costs Y, has a salvage value of X/8, and s to be deprecated over 10 years usng the sum of the years dgts method. The total amount of deprecaton n the frst seven years for Machne I equals the total amount of deprecaton n the frst seven years for Machne II. Calculate Y. X (A) 0.94 (B) 0.99 (C) 1.04 (D) 1.09 (E) (# 8, November 2000) An actuary s tryng to determne the ntal purchase prce of a certan physcal asset. The actuary has been able to determne the followng: () Asset s 6 years old. () Asset s deprecated usng the snkng fund method, wth an annual effectve rate of 9%. () Book value of the asset after 6 years s 55, (v) The loss of nterest on the orgnal purchase prce s at an annual effectve rate of 9%. (v) Annual cost of the asset s 11, (v) Annual mantenance cost of the asset s What was the orgnal purchase prce of the asset? (A) 72,172 (B) 107,922 (C) 112,666 (D) 121,040 (E) 143, (# 36, May 2003) Erc and Jason each sell a dfferent stock short at the begnnng of the year for a prce of 800. The margn requrement for each nvestor s 50% and each wll earn an annual effectve nterest rate of 8% on hs margn account. Each stock pays a dvdend of 16 at the end of the year. Immedately thereafter, Erc buys back hs stock at a prce of (800 2X) and Jason buys back hs stock at a prce of (800 + X). Erc s annual effectve yeld,, on the short sale s twce Jason s annual effectve yeld. Calculate. (A) 4% (B) 6% (C) 8% (D) 10% (E) 12% 88. (# 27, May 2001) Jose and Chrs each sell a dfferent stock short for the same prce. For each nvestor, the margn requrement s 50% and nterest on the margn debt s pad at an annual effectve rate of 6%. Each nvestor buys back hs stock one year later at a prce of 760. Jose s stock pad a dvdend of 32 at the end of the year whle Chrs s stock pad no dvdends. Durng the 1year perod, Chrs s return on the short sale s, whch s twce 17
18 the return earned by Jose. Calculate. (A) 12% (B) 16% (C) 18% (D) 20% (E) 24% 2 Solutons 1. Snce each account earns an annual effectve dscount rate of d, each account earns the same annual effectve nterest rate. Bruce s nterest n the 11 th year s 100((1+) 11 (1+) 10 ) = 100(1 + ) 10. Robbe s nterest n the 17 th year s 50((1 + ) 17 (1 + ) 16 ) = 50(1 + ) 16. So, 100(1 + ) 10 = 50(1 + ) 16, (1 + ) 6 = 2, = 12.24% and x = 100(1 + ) 10 = 100(0.1224)( ) 10 = Erc s nterest s ( x 1 + 2) 16 ( x 1 + 2) 15 ( = x 1 + ) Mke s nterest s 2x 1 2 = x. So, (1 + 2 )15 = 2 and = %. 3. The followng seres of payments has present value zero: Contrbutons Tme Snce the rate changes at 10, we fnd the present value of the payments at ths tme. We have at tme 10, the followng seres of payments have the same value Contrbutons 10 Tme 0 Contrbutons Tme The present value at tme 10 of the payments at tme 15 and at tme 30 s 20(1.03) (1.03) 40 = = Ths equals the present value at tme 10 of the ntal depost,.e. 10 ( 1 d 4) 40 = So, d = %. 4. Let (2) be the nomnal semannual rate of Jennfer s loan. The future values of the loans are Dvdng these equatons ) 20 x (1 + (2) = 800 and x ( 1 + (2)) 20 = ( 1 + (2) 1 + (2) 2 ) 20 = = 1.4. So, (2) = (1.4)1/ (1.4)1/20 2 = and x = 1120 ( ) 20 =
19 5. Snce s a nomnal rate of nterest convertble semannually, 100(1 + 2 )(2)(7.25) = 200 and = %. To fnd the force of nterest, we solve for δ, 100e δ(7.25) = 200, and get δ = We have that δ = = = 0.23%. 6. For Fund X, A(t) = 10000( ) t and δ t = d ln A(t) = ln(1.05). For Fund Y, A(t) = dt 10000(1 + (0.08)t) and δ t = d 0.08 ln A(t) =. The two forces of nterest are equal, when, dt t ln(1.05) = The soluton of ths equaton s t = 1 1 = 8. For Fund X, t ln(1.05) 0.08 A(8) = 10000( ) 8 = For Fund Y, A(8) = 10000(1 + (0.08)8) = So, Z = = We have that ( a(t) = e R t ) ( ) t t 0 δs ds 2 = exp 0 k + 2s ds = exp ln(k + 2s) The nformaton says that 0 = e ln(k+2t) ln k = k + 2t. k x = 75a(3) = 75(k + 6), x = 150 a(3) k a(5) = 150(k + 6) k Dvdng these two equatons, 1 = k+10 2k and k = 10. So, x = 75(k+6) k = = Frst, we fnd a(4) = e R 4 0 δs ds : 4 0 δ s ds = 3 0 (0.02)s ds (0.045) ds = (0.02)s (0.045)s = and a(4) = e Let (4) be the nomnal rate of nterest compounded quarterly. Then, ( ) 4 4 ( ) 16. a(4) = 1 + (4) 4 = 1 + (4) 4 So, (4) = 4(e 0.135/16 1) = = %. 9. We have that for Fund X, ( ) a(t) = e R t 0 δs ds = e R t t 0 (0.006)s2 ds = exp (0.002)s 3 = e (0.002)t3 = e t3 /500. For Fund X, 4K = A(n) = A(0)a(n) = Ke n3 /500. So, n = (500 ln 4) 1/3 = For Fund Y, a(t) = ( ) t. So, 4K = A(n) = a(n) A(m) = (1 + a(m) 0.10)n m 2K. So, n m = ln 2 = Hence, m = = ln For the Fund X, a(5) = e R 5 0 δs ds = e R 5 s 2 0 k ds = e 125/(3k). For the Fund Y, a(5) = ( ) (2) 5 = (0.96) 10. From the equaton, e 125/(3k) = (0.96) 10, we get that k = 125 (3)(10)(ln 0.96) =
20 11. For Tawny s bank account, a(t) = ( ) t = (1.05) 2t and the force of nterest s δ t = d ln a(t) = 2 ln(1.05). For Fabo s account, a(t) = 1 + t and δ dt t = 2 ln(1.05) = 2 ln(1.05). So, = = So, ln(1.05) 2 Z = A(5) = A(0)(1 + t) = 1000 (1 + 5 (0.1905)) = At tme t = 5, 1+t 12. The value of Joe s deposts at tme 10 s 10 (1 + (0.11)(10 0)) + 30 (1 + (0.11)(10 5)) = = The value of Tna s deposts at tme 10 s 10( ) 10 n + 30( ) 10 2n. Call x = (1.0915) n. Then, 10( ) 10 x + 30( ) 10 x 2 = So, 3x 2 + x = (67.5) 10 1 (1.0915) 10 = So, (1.0915) n = x = 1± 1 2 +(4)(3)(2.8123) = (2)(3) and n = We have that a(t) = e R t 0 δs ds = e R t s ds = e t The accumulaton of the nvestments at tme t = 3 s 100a(3) + x. The accumulaton of the nvestments at tme t = 6 s a(6) (100a(3) + x). The amount of nterest earned from a(3) tme 3 to tme 6 s x = a(6) a(6) a(3) (100a(3) + x) (100a(3) + x) = (100a(3) + x). a(3) a(3) Solvng for x, we get that = 100 e 3 3 2a(3) a(6) x = 100 a(3)(a(6) a(3)) 2e e = (100)( )( ) = (2)( ) (e e ) 14. Snce the equatons of value of the two streams of payments at tme zero are equal, Usng that ν n = , we have that So, = 3.51% ν n + 300ν 2n = 600ν (1 + ) 10 = 600ν 10 = ( ) + 300( ) 2 =
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