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) 38.9 2. (# 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 8-th 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) 4.73 4. (#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 sem-annual 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. 574 5. (#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. 1625 B. 1687 C. 1697 D. 1711 E. 1721 1
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. 125 8. (# 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 = 0.045 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) 6.2 10. (# 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) 108 11. (# 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) 2392 12. (# 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
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) 3.2 13. (# 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) 785 14. (# 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 = 0.75941, 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, 1998. 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 95 105 Depost 2X X The amount n the account on January 1, 1999 s 115. Durng 1998, the dollar weghted return s 0% and the tme-weghted 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 15 40 20 June 1 80 80 October 1 175 75 3
On June 30, the value of the account s 157.50. 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 (tme-weghted) annual effectve yeld durng the entre 1-year perod. Calculate X. (A) 234.75 (B) 235.50 (C) 236.25 (D) 237.00 (E) 237.75 18. (#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 dollar-weghted 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 1998. 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) 131 21. (# 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 1 2 3 Portfolo rate Rate 1990 10% 10% t% 1993 8% 1991 12% 5% 10% 1994 t 1% 1991 8% t 2% 12% 1995 6% 1993 9% 11% 6% 1996 9% 1994 7% 7% 10% 1997 10% 4
An nvestment of 100 s made at the begnnng of years 1990, 1991, and 1992. The total amount of nterest credted by the fund durng the year 1993 s equal to 28.40. Calculate t. (A) 7.00 (B) 7.25 (C) 7.50 (D) 7.75 (E) 8.00 22. (# 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 1 800.0 808.0 x 850.0 858.5 x + 1 900.0 918.0 x + 2 930.0 948.6 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 10-year annuty-due 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 15-year annuty-mmedate wth annual payment of Z. The present value of both annutes was determned usng an annual effectve nterest rate of 8%. Calculate Z. A. 2330 B. 2470 C. 2515 D. 2565 E. 2715 24. (#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. 2792 B. 2824 C. 2859 D. 2893 E. 2925 25. (# 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) 9973 5
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) 324.73 (B) 326.89 (C) 328.12 (D) 355.45 (E) 450.65 27. (# 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, 1985. 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, 184 28. (# 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) 9.4 30. (# 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) 6195 31. (# 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
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 4-month perod s X. Calculate X. (A) 38.8 (B) 39.8 (C) 40.8 (D) 41.8 (E) 42.8 33. (# 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 annuty-mmedate payng 1 each year, calculated at an annual effectve nterest rate of j% () n year annuty-mmedate 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) 42 35. (# 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) 3.9 36. (#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 7276.35. Gven k > 5, calculate k. A. 8.06 B. 8.21 C. 8.36 D. 8.51 E. 8.68 37. (# 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 167.50. Calculate K, gven K < 9.2. (A) 4.0 (B) 4.2 (C) 4.4 (D) 4.6 (E) 4.8 7
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 12-month 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) 563 39. (# 45, May 2003). A perpetuty-mmedate 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 perpetuty-mmedate payng Y per year. The annual effectve rate of nterest s 8%. Calculate Y. (A) 110 (B) 120 (C) 130 (D) 140 (E) 150 40. (# 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) 329 41. (# 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) 21 42. (# 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) 2880 43. (# 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
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 sem-annually, on ts nvestment. What prce does Bank X receve? (A) 9,792 (B) 10,823 (C) 10,857 (D) 11,671 (E) 11,709 44. (# 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) 40.4 45. (# 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 perpetuty-mmedate 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. 19.4 B. 19.9 C. 20.4 D. 20.9 E. 21.4 47. (# 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
Jance pays nterest at the end of every sx-month 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 sx-month perod. Calculate the total amount of nterest pad on all three loans. (A) 8718 (B) 8728 (C) 8738 (D) 8748 (E) 8758 48. (#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. 672 49. (# 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) 765 50. (# 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 3700. Calculate t. A. 198 B. 204 C. 210 D. 216 E. 228 51. (# 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 153.86. () The total prncpal repad as of tme (n 1) s 6009.12. () The prncpal repad n the frst payment s Y. Calculate Y. (A) 470 (B) 480 (C) 490 (D) 500 (E) 510 52. (# 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 start-up 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) 465 10
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 356.54 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) 900 54. (# 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 5483.36 at the end of each quarter. After 12 payments, the nterest rate dropped to 6% compounded quarterly. The new quarterly payment dropped to 5134.62. 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 4265.73, payable for fve more years. Determne X. (A) 11, 047 (B) 13, 369 (C) 16, 691 (D) 20, 152 (E) 23, 614 55. (# 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 1076.82 and at the end of the thrd year s 559.12. Calculate the prncpal repad n the frst payment. (A) 444 (B) 454 (C) 464 (D) 474 (E) 484 56. (# 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) 7344 58. (# 36, Sample Test). Don takes out a 10-year 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. 4070 B. 4120 C. 4170 D. 4220 E. 4270 11
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) 2.13 60. (# 26, May 2003). 1000 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) 2431 61. (# 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. 5394 B. 5465 C. 5551 D. 5600 E. 5685 63. (#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 0.75. 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 fve-year perod turned out to be 7.45%, compounded sem-annually. What nomnal rate of nterest, compounded monthly, dd Sally 12
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) 4485 66. (# 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. 162 68. (# 35, Sample Test). Jason and Margaret each take out a 17-year 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. 4840 B. 4940 C. 5040 D. 5140 E. 5240 69. (# 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
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) 411 70. (# 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) 237 72. (# 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) 1115 73. (# 41, May 2001). Bll buys a 10-year 1000 par value 6% bond wth sem-annual coupons. The prce assumes a nomnal yeld of 6%, compounded sem-annually. 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 n-year bond wth sem-annual 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
Par value s 1000. r The rato of the sem-annual bond rate to the desred sem-annual yeld rate, 1.03125. s The present value of the redempton value s 381.50. Bond Y Redempton value s the same as the redempton value of bond X. Prce to yeld s 647.80. What s the prce of bond X? (A) 1019 (B) 1029 (C) 1050 (D) 1055 (E) 1072 75. (# 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) 955 76. (#42, May 2003). A 10,000 par value 10-year 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) 667 77. (#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 6-th coupon payment. (A) 2.00 (B) 2.08 (C) 2.15 (D) 2.25 (E) 2.34 78. (#38, Sample Test). Laura buys two bonds at tme 0. Bond X s a 1000 par value 14-year bond wth 10% annual coupons. It s bought at a prce to yeld an annual effectve rate of 8%. Bond Y s a 14-year 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. 1415 B. 1425 C. 1435 D. 1445 E. 1455 15
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. 1180 B. 1200 C. 1220 D. 1240 E. 1260 80. (# 40, November, 2000). Among a company s assets and accountng records, an actuary fnds a 15-year bond that was purchased at a premum. From the records, the actuary has determned the followng: () The bond pays sem-annual nterest. () The amount for amortzaton of the premum n the 2nd coupon payment was 977.19. () The amount for amortzaton of the premum n the 4th coupon payment was 1046.79. What s the value of the premum? (A) 17,365 (B) 24,784 (C) 26,549 (D) 48,739 (E) 50,445 81. (#48, May 2000). A corporaton buys a new machne for 2000. 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 1000. () 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 33.125% of the book value n the begnnng of the year. Calculate X. (A) 4250 (B) 4500 (C) 4750 (D) 5000 (E) 5250 83. (#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, 250 84. (#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 sum-of-the-years-dgts method. At 16
the end of year 4, the machne has a book value of 2218. At that tme, the deprecaton method s changed to the straght-lne method for the remanng years. Determne the new deprecaton charge for year 8. (A) 200 (B) 222 (C) 286 (D) 370 (E) 464 85. (#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) 1.14 86. (# 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 55,216.36. (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 11,749.22. (v) Annual mantenance cost of the asset s 3000. What was the orgnal purchase prce of the asset? (A) 72,172 (B) 107,922 (C) 112,666 (D) 121,040 (E) 143,610 87. (# 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 1-year perod, Chrs s return on the short sale s, whch s twce 17
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)(1 + 0.1224) 10 = 38.85. 2. Erc s nterest s ( x 1 + 2) 16 ( x 1 + 2) 15 ( = x 1 + ) 15 2 2. Mke s nterest s 2x 1 2 = x. So, (1 + 2 )15 = 2 and = 9.45988%. 3. The followng seres of payments has present value zero: Contrbutons 10 20 100 Tme 0 15 30 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 20 100 Tme 15 30 The present value at tme 10 of the payments at tme 15 and at tme 30 s 20(1.03) 10 + 100(1.03) 40 = 14.8818 + 30.6557 = 15.7738. Ths equals the present value at tme 10 of the ntal depost,.e. 10 ( 1 d 4) 40 = 15.7738. So, d = 4.5318%. 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 = 1120. 2 ( 1 + (2) 1 + (2) 2 ) 20 = 1120 800 = 1.4. So, (2) = (1.4)1/20 1 1 (1.4)1/20 2 = 0.034518 and x = 1120 (1+0.034518) 20 = 568.14. 18
5. Snce s a nomnal rate of nterest convertble semannually, 100(1 + 2 )(2)(7.25) = 200 and = 9.7928%. To fnd the force of nterest, we solve for δ, 100e δ(7.25) = 200, and get δ = 0.09560. We have that δ = 0.097928 0.09560 = 0.0023 = 0.23%. 6. For Fund X, A(t) = 10000(1 + 0.05) 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 1+0.08t ln(1.05) = 0.08. The soluton of ths equaton s t = 1 1 = 8. For Fund X, 1+0.08t ln(1.05) 0.08 A(8) = 10000(1 + 0.05) 8 = 14774.55. For Fund Y, A(8) = 10000(1 + (0.08)8) = 16400. So, Z = 16400 14774.55 = 1625.45. 7. 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 + 10. Dvdng these two equatons, 1 = k+10 2k and k = 10. So, x = 75(k+6) k = 75 16 10 = 120. 8. Frst, we fnd a(4) = e R 4 0 δs ds : 4 0 δ s ds = 3 0 (0.02)s ds + 4 3 (0.045) ds = (0.02)s2 2 3 0 4 +(0.045)s = 0.135. 3 and a(4) = e 0.135. 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) = 0.033893 = 3.3893%. 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 = 8.8499. For Fund Y, a(t) = (1 + 0.10) t. So, 4K = A(n) = a(n) A(m) = (1 + a(m) 0.10)n m 2K. So, n m = ln 2 = 7.2725. Hence, m = 8.8499 7.2725 = 1.5775. ln 1.1 10. 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) = (1 0.08 2 ) (2) 5 = (0.96) 10. From the equaton, e 125/(3k) = (0.96) 10, we get that k = 125 (3)(10)(ln 0.96) = 102.0691. 0 19
11. For Tawny s bank account, a(t) = ( ) 1 + 0.1 2t = (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, = = 0.1905. So, 1+5 1 10 ln(1.05) 2 Z = A(5) = A(0)(1 + t) = 1000 (1 + 5 (0.1905)) = 1952.75.. 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)) = 21 + 46.5 = 67.5. The value of Tna s deposts at tme 10 s 10(1 + 0.0915) 10 n + 30(1 + 0.0915) 10 2n. Call x = (1.0915) n. Then, 10(1 + 0.0915) 10 x + 30(1 + 0.0915) 10 x 2 = 67.5. So, 3x 2 + x = (67.5) 10 1 (1.0915) 10 = 2.8123. So, (1.0915) n = x = 1± 1 2 +(4)(3)(2.8123) = 0.81579 (2)(3) and n = 2.3226. 13. We have that a(t) = e R t 0 δs ds = e R t s 2 0 100 ds = e t3 300. 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 300 33 e 300 63 = (100)(1.094174)(2.054433 1.094174) = 784.0595. (2)(1.094174) 2.054433 300 (e 300 63 e 300 33 ) 14. Snce the equatons of value of the two streams of payments at tme zero are equal, Usng that ν n = 0.75941, we have that So, = 3.51%. 100 + 200ν n + 300ν 2n = 600ν 10. 600(1 + ) 10 = 600ν 10 = 100 + 200(0.75941) + 300(0.75941) 2 = 424.89. 20
15. We have that V 0 = 75 and F V = 60. We have deposts of 10 at tmes 1, 2,..., 12, and 12 12 12 wthdrawals of 5, 25, 80, 35, at respectve tmes 2, 1, 9.5, 10. We have that 12 2 12 12 n I = F V V 0 C j = 60 75 (10)(12) + 5 + 25 + 80 + 35 = 10 and So, j=1 V 0 t + n j=1 (t t j)c j = (75)(1) + (10) 12 j=1 = 75 + 10 (11)(12) 5 10 25 1 12 2 12 2 = 12 j 12 5 ( 1 1 12 ) 25 ( 1 1 2 80 2.5 12 35 2 12 = 90.8347. ) 80 ( 1 9.5 12 I V 0 t + n j=1 (t t = 10 j)c j 90.8347 = 11%. ) ( ) 35 1 10 12 16. Snce the dollar weghted return s 0%, 100 + 2x + x = 115, whch gves that x = 5. The tme weghted return y satsfes So, y = 0.006819 = 0.6819%. 1 + y = 95 105 115 100 105 110 = 0.993181. 17. Accordng to the tme weghted method, the equvalent annual effectve yeld durng the frst 6 months satsfes (1 + ) 1/2 = 40 50 80 60 157.5 160. Accordng to the tme weghted method, the annual effectve yeld satsfes So, and 18. Snce the tme weghted return s 0%, 12 10 y satsfes 1 + = 40 50 80 60 175 160 x 250. ( 40 50 80 60 157.5 ) 2 = 40 160 50 80 60 175 160 x 250 y = x = 40 80 (157.5)2 250 50 60 160 175 x 12+x 10 + 60 60 (10)(1) + (60)(1/2) = 236.5 = 1. So, x = 60. The dollar weghted return = 0.25 = 25%. 19. Aprl 1 s tme 0.25. Let A be the value of the nvestment on January 1, 1998. Usng the dollar weghted method for 1997, 100, 000(1+) 8, 000(1+(0.75)) = 92000+94000 = A. Usng the weghted method for 1998, A(1 + ) = 103, 992. So, we must solve for n the equaton, (92000+94000)(1+) = 1039992, whch s equvalent to 94 2 +186 11.992 = 0. So, = 186± 186 2 (4)(94)( 11.992) = 0.06249905 = 6.249905%. (2)(94) 21
20. We have that a(t) = e R t 0 cashflow s 1 8+s ds = e ln(8+t) ln 8 = 8+t 8. The future value of the contnuous 20000 = 10 0 C(s) a(10) a(s) ds = 10 0 k(8 + s) 8+10 8+s ds = 10 0 18k ds = 180k. So, k = 20000 180 = 111.11. 21. The total amount of nterest earned durng the year 19993 s 28.4 = (100)(1.10)(1.10)(1 + t)(0.08) + (100)(1.12)(1.05)(0.10) + (100)(1.08)(t 0.03)(0.08) = (100)(1.10)(1.10)(0.08) + (100)(1.12)(1.05)(0.10) (100)(1.08)(0.03)(0.08) +t ((100)(1.10)(1.10)(0.08) + (100)(1.08)(0.08)) = 19.28 + 117.68t, and t = 28.4 19.28 117.68 = 7.74983%. 22. The return n the nvestment s ( ) 4 ( ) 4 ( 808 858.5 918.0 100 800 850.0 900.0 ) 4 ( ) 4 948.6 + 100 930.0 ( 918.0 900.0 So, 2440.399 = 1000(1 + ) 4 + 1000(1 + ) 2 and = 6.78218856%. ) 4 ( ) 4 948.6 = 2440.399. 930.0 23. The present value of John s nhertance s 2500ä 10 0.08 = 18117.22. 18117.22 s also the present value at tme 0 of Carol s nhertance. Two years later, the value of Carol s nhertance s 18117.22(1.09) 2 = 21525.07. To fnd z, we solve za 15 8 = 21525.07 to get z = 2514.76. 24. Susan s account at the end of 40 years s worth 100s 40 8% = 25905.63. We fnd x solvng xa 15 8% = 25905.63 and get x = 3026.55. Jeff s account at the end of 40 years s worth 100s 40 10% = 44259.25. We fnd x solvng xa 15 10% = 44259.25 and get y = 5818.83. The Jeff s wthdrawals are of 5818.83. Fnally, 5818.83 3026.55 = 2792.38. 25. We have that (4) = 6.8234%. The future value of Jerry s deposts at tme 10 s 450s 40 6.8234/4 = 25513.22. The future value of Jerry s deposts at tme 15 s 25513.22(1.07) 5 = 35783.61. Then, we fnd y solvng the equaton, 35783.61 = yä 4 7% to get y = 9873.20. 26. To get an ncome of $1, the man needs 1000. So, to get an ncome of $3000, the man needs 9.65 (3000) (1000) = 310880.83. We solve for x n the equaton x s 9.65 (25)(12) 8%/12 = x s 300 0.33333% = 310880.83 and we get x = 324.72. 27. Countng the tme n months, wth January 1, 1985 as tme 1, the cashflow s Contrbutons 200 200 0 0 200 200 0 Tme (n months) 1 59 60 72 73 180 181 22
We have to fnd the future value of ths cashflow at tme 181. The value of the frst 59 payments at tme 60 s 200 s 59 6%/12 = 13754. The value of ths money 121 months later (on December 31, 1999) s 13754 ( ) 1 + 0.06 121 12 = 25149.11. The future value of the last 108 payments on December 31, 1999 s 200 s 108 0.5% = 28690.72. The total value of Jm s retrement account at the end s 25149.11 + 28690.72 = 53839.83. 28. We have that So, = 980 8000 8000 = 98(1 + ) 2n s n + 196s 2n = 98(1 + ) 2n (1+) n 1 = (98)2 2 2 1 + 196 22 1 = 980 = 0.1225 = 12.25%. + 196 (1+)2n 1 29. The future value of the tem at tme 10 s (200)(1.04) 10 = 296.05. The future value of Chuck s nvestments at tme 10 (20) s 6 0.1 (1.1) 4 + x s 3 0.1 (1.1) 4 = 248.52 + 5.3308x. So, 296.05 = 248.52 + 5.3308x and x = 8.92. 30. The cashflow s Contrbutons 100 100 100 100 0 Tme 0 4 8 36 40 The 4 year effectve rate of nterest s (1 + ) 4 1. So, the accumulated amount n the account at the end of 40 years s X = 100 s 10 (1+) 4 1 = 100(1 + )4 ((1 + ) 40 1), (1 + ) 4 1 and the accumulated amount n the account at the end of 20 years s 100 s 5 (1+) 4 1 = 100(1 + )4 ((1 + ) 20 1). (1 + ) 4 1 So, 5 = (1 + ) 20 + 1 and (1 + ) 4 = 1.3195. So, X = 100 (1 + )4 ((1 + ) 40 1) (1 + ) 4 1 = 100 (1.3195)(42 1) 1.3195 1 = 6194.68. 31. The cashflows are: Contrbutons 55 55 55 55 Tme 1 2 3 20 23
Contrbutons 30 30 60 60 90 90 Tme 1 10 11 20 21 30 Snce both annutes have the same present value: 55a 20 = 30a 10 + 60ν 10 a 10 + 90ν 20 a 10. Snce we have that a 20 a 10 = 1 ν 20 1 ν 10 = 1 + ν 10, 55(1 + ν 10 ) = 30 + 60ν 10 + 90ν 20 or 90ν 20 + 5ν 10 25 = 0. Ths s equaton s equvalent to 18ν 20 + ν 10 5 = 0, whose soluton s ν 10 = 1 ± 1 + 5 18 4 = 1 2 18 2. So, (1 + ) 10 = 2, = 7.1774% and x = 55a 20 7.1774% = 574.72. 32. Let j be the rate of nterest correspondng to a perod of 3 years. Then, 1+j = (1+) 3. The present value of the perpetuty at tme 3 years s 10. The present value of the perpetuty j 10 at tme 0 s. So, j(j+1) j2 + j 5 = 0, and j = 1± 1+(5/4) = 1±1.5 = 0.25. Hence, 16 2 2 = (1 + 0.25) 1/3 1 = 7.7217%. Then, (3) = 7.5311%. The present value of a perpetuty mmedate payng 1 at the end of each 4 month perod s x = 3 (3) = 3 0.075311 = 39.83. 33. The total value of the perpetuty mmedate s xa = x. The present value of Bran payments s xa n = x(1 νn ). So, Bran s proporton of the perpetuty s 1 ν n = 0.40. So, ν n = 0.6. The present value of Jeff s payments s ν 2n x 1. So, Jeff s share s ν2n = 0.36 = 36%. 34. The value of the frst perpetuty s a = 1 = 1 = 13.7931. The value of the second 0.0725 annuty s 13.7931 = a 50 j. Hence, j = 7%. From the equaton, a n 6 = 13.7931, we get that n = 30.1680. 35. The present value of the frst perpetuty s 20 = 1+ (2) 2, where (2) s the nomnal rate of (2) 2 nterest convertble semannually. So, (2) = 2 = 10.5263%. Let 19 (0.5) be the nomnal rate of ( nterest convertble twce a year. Then, 1 + (2) 2 The value of the second perpetuty s 20 = x (1+2(0.5) ) 36. The cashflow s ) 2 = ( 1 + 2 (0.5) ) 0.5 and (0.5) = 11.3869%. = 1.2277 2 (0.5) 0.2277 x. So, x = 3.71. 24
Payments 100 100(1 + r) 100(1 + r) 2 100(1 + r) 19 Tme 0 1 2 19 where r = k 100. Snce r >, the present value at tme 1 of ths cashflow s s 100 1+ 20. r 1+ The accumulated value at tme 1 of Chrs cashflow s 7276.35(1 + ) 21. So, s 20 r = 1+ 72.7635(1 + ) 20 = 27.4238. Usng the calculator, we get that r = 0.032 and r = 1+ 0.05 + (1.05)(0.032) = 0.0836. Hence, k = 8.36. 37. Le r = k. The cashflow of the perpetuty s 100 Payments 10 10 10 10 10 10(1 + r) 10(1 + r) 2 10(1 + r) 3 Tme 1 2 3 4 5 6 7 8 The present value at tme 0 of ths perpetuty s 167.5 = 10a 4 + 10ν 4 1 r = 32.2555 + 7.0325 1 0.092 r. So, r = 0.092 7.0325 167.5 32.2555 = 4%. 38. The nomnal rate convertble monthly s (12) = 5.8411%. The value of the payments whch the senor executve accumulates at the end of the frst year s Rs 12 5.8411/12% = 12.3265R. So, the cashflow s equvalent to Payments 12.3265R 12.3265R(1 + r) 12.3265R(1 + r) 2 12.3265(1 + r) 19 Tme 1 2 3 20 where r = 3.2% and = 6%. The present value of the package s 1 100000 = 12.3265R 1 + 0.032 a 20 = 11.9443Ra 0.06 0.032 1.032 20 0.027132 = 182.5063R. So, R = 547.93. 39. The value of the perpetuty mmedately after the ffth payment s 100 = 1250. The payments of the 25 year annuty mmedate are Inflows x x(1.08) x(1.08) 2 x(1.08) 24 Tme 1 2 3 25 Snce = r, the present value at tme 0 of ths annuty s 1250 = 25x(1.08) 1. Hence, x = 50(1.08). The 15 last payments of the annuty are Inflows 50(1.08) 11 x(1.08) 12 x(1.08) 15 Tme 1 2 15 25
Snce = r, the present value at tme 0 of ths annuty s (50)(15)(1.08) 10 = 1619.19. Hence, annual payment of the perpetuty s 1619.19(0.08) = 129.54. 40. The cashflow s Contrbutons 0 0 50(1.012) 50(1.012) 2 50(1.012) 11 tme 0 6 7 8 18 So, X = (1.063) 6 (50)a 12 0.063 0.012 1.012 = 34.6533a 12 5.0395% = 306.4765. 41. The cashflow s Payments 0 1 2 n 1 n n Tme 1 2 3 n n + 1 n + 2 The present value at tme 0 of the perpetuty s 77.1 = ν (Ia) n + ν n+1 n 1 = ν a n (1 + ) nνn + νn+1 n So, 8.0955 = a n 10.5% and n = 19. 42. The cashflow s: Payments 2 4 6 120 Tme (n months) 1 2 3 60 = a n. We have (4) = 9%. So, = 9.3083% and (12) = 8.933%. We use the formula for the ncreasng annutes wth n = 60 and = 8.933%/12 = 0.74444%: 2 (Ia) n = 2 än nνn = 2(10.158740) 0.00744 = 2729.21. 43. Snce Betty pays 19800 nto 36 payments, each monthly payment s 19800 36 = 550. Between the j 1 th and j th month she owes (37 j)550. The nterest n ths quantty s (37 j)5.50. So, the cashflow for Betty s: Payments 550 + (36)(5.5) 550 + (35)(5.5) 550 + (1)(5.5) For bank Y, the cashflow s Tme 1 2 36 Payments 550 + (20)(5.5) 550 + (19)(5.5) 550 + (1)(5.5) Tme 1 2 20 26
For bank Y, we have (2) = 14%. So, = 14.49% and (12) = 13.608312%. Takng = 13.608312%/12 = 1.134026% and n = 20, Bank X receves 550a n + 5.5 (Da) n = 550a 20 1.134026% + 5.5(20 a 20 1.134026% ) 1.134026% = 9792.39 + 5.5(20 17.80343) 0.0113426 = 10857.72 44. The present value of Annuty 1 s The present value of Annuty 2 s (Da) 10 = 10 a 10. (Ia) 10 + ν 10 11a + = (1 + )a 10 10ν10 + 11ν10 = (1 + )a 10 ( where we have used that ν 10 10 a10 ) = 1 a 10. Hence, 2 So, = 9.30%, and the present value of Annuty 1 s 10 a 10 45. Snce the present value of the two perpetutes are equal, we have that 90 + 10(1 + ) 2 = So, 18 2 + 8 1 = 0 and = 4± 16+18 18 = 0.1017. 180(1 + ). + ν10 = 1 + a 10. = 1+a 10 and a 10 = 19/3. = 10 (19/3) 0.0930 = 39.42. 46. The value of the Scott deposts n the frst year are s 4 2.5 = 4.2563. So, the cashflow s equvalent to the followng Payments 4.2563 2(4.2563) 8(4.2563) 1 2 8 We have that (4) = 0.10 and = 0.103813. The value of ths cashflow s (4.2563) (Is) 8 0.103813 = (4.2563) s 8 0.103813 8 0.103813 = (4.2563) 12.7992 8 0.103813 The perpetuty payment s 196.7657 = (196.7657)(0.103813) = 20.4269. = 196.7657. 47. Seth s nterest s 5000((1.06) 10 1) = 3954.23. Jance s nterest s 5000 (10)(5000)(0.06) = 3000. Lor s semannual payment s = 679.34. So, Lor s nterest a 10 0.06 s (10)(679.34) 5000 = 1793.4. The total amount of nterest pad s 3954.23 + 3000 + 1793.4 = 8747.63. 48. The casflow s 27
Contrbutons x 2x 2x 2x 2x Tme 0.5 1.5 4.5 5.5 9.5 wth nterest rates changng at tme 4.5. The present value of the seres of payments s So, x = 10000 15.2508 = 655.70. 10000 = x(1.05) 0.5 + 2x(1.05) 0.5 a 4 0.05 + 2x(1.05) 4.5 a 5 0.03 = x(0.9759 + 6.9210 + 7.3539) = 15.2508x. 49. The fnance charge s L = nl L. So, a a n = n,.e. a 2 10 = 5,and = 15.0984%. The n loan s L = 1000a n = 1000a 5 = 5000. The amount of nterest pad durng the frst year of the loan s L = 754.92. 50. Let P be the monthly payment. (12) = 0.077208 and 0.077208/12 = 0.006434. The amount of prncpal repad n the 12 th payment s 1000 = P (1+0.006434) (n+1 12) and the amount of prncpal repad n the t th payment s 3700 = P (1 + 0.006434) (n+1 t). From these two equatons, 3.7 = (1 + 0.006423) t 12 and t = 216. 51. Let P be the nstallment payments. Let L be the loan. The nterest porton of the fnal payment s 153.86 = P (1 ν) = P (1 (1.125) 1 ). So, P = 1384.74. The total prncpal repad as of tme (n 1) s 6009.12 = L P a 1 0.125 = L 1230.88. So, L = 7240. The loan of L = 7240 s pad n annual payments of P = 1384.74 at an nterest of 12.5%. So, 7240 = 1384.74a n 0.125 and n = 9. The prncpal repad n the frst payment s Y = P ν n = 1384.74(1.125) 9 = 479.73. 52. We have that (4) = 12%. So, = 12.550881% and (2) = 12.18%. The balance after the 8 th payment s 12000(1+0.0609) 8 750s 8 0.0609 = 11809.35. The value of ths outstandng balance 3 months later s 11809.35(1.03) = 12163.63. Two months later, the outstandng balance s 12163.63(1 + 0.09 12 )2 = 12346.77. Let P be the new payment of the loan, then 12346.77 = P a 30 0.09/12 and P = 461.12. 53. Assume frst that Iggy makes a payment p Iggy makes at the end of each year. Then, x = pa 10 6% = p7.36. So, p = x 10x an total amount of the payments s 10p =. If he 7.36 7.36 pays the prncpal and accumulated nterest n one lump sum at the end of 10 years, he would pay x(1 + 0.06) 10 = 1.7908x. Hence, the dfference between the two payments s 356.54 = 1.7908x 10x 7.36 = 0.4321x. So, x = 825.13. 54. The outstandng balance of the loan after 20 payments s 5134.62a 20 0.06/4 = 88154.43. The present value at tme 20 months of the future payments of the customer s 4265.73a 20 0.07/4 = 71463.27. So, x = 88154.43 71463.27 = 16691.16. 28
55. Let P be the amount of each payment. The outstandng loan balance at the end of the second year s 1076.82 = P a 2 0.08. So, P = 603.85. The prncpal repad n the frst payment s P ν n = 603.85(1.08) 4 = 443.85. 56. The nterest pad n perod t s 1 ν n+1 t. The prncpal repad n perod t + 1 s ν n t. So, x = 1 ν n+1 t + ν n t = 1 + ν n t (1 ν) = 1 + ν n t d. 57. The nterest per perod s = (9/12)% = 0.75%. Let r = 0.02 and let P = 1000. The outstandng balance mmedately after the 40th payment s B 40 = P (1 + r) k 1 a n k r 1+r = 1000(0.98) 39 a 20 0.0075+0.02 0.98 = 454.7963a 20 0.028061 = 6889.06. 58. We have that 600 = 100a 8. So, = 6.876%. The loan s 100 (Da) n = 100 10 a 10 0.06876 0.06876 = 4269.84. 59. We may assume that Z = 1. Susan s accumulated value conssts of her $7 nvested plus the cashflow for her nterest. The cashflow of her nterest payments are The accumulated value of ths cashflow s Payments 0 0.05 2(0.05) 6(0.05) Tme 1 2 3 7 (0.05) (Is) 6 6% = (0.05) So, Susan s accumulated value s X = 8.1653. ( (1+0.06)s6 6% 6 0.06 ) = 1.1653. Lor s accumulated value conssts of her $14 nvested plus the cashflow for her nterest. The cashflow of her nterest payments are Payments 0 0.025 2(0.025) 13(0.025) Tme 1 2 3 14 The accumulated value of ths cashflow s (0.025) (Is) 13 3% = (0.025) So, Susan s accumulated value s Y = 16.5719. We have that Y X = 2.0296. 60. The balances along tme n the Fund X are: ( (1+0.03)s13 3% 13 0.03 ) = 2.5719. 29
Hence, the deposts at Fund Y are: Balance 1000 900 800 0 Tme 0 1 2 10 Balance 100 + (6) (10) 100 + (6) (9) 100 + (6) (8) 100 + (6) (1) Tme 1 2 3 10 The accumulated value of Fund Y at the end of year 10 s 100s 10 9% + 6 (Ds) 10 9% = 100s 10 9% + 6(10(1+0.09)10 s 10 9% ) = 1519.29 + 6 (8.4807) 0.09 = 2084.67. 61. Snce the yeld rate on the entre nvestment over the 20 year perod s 8% annual effectve, the accumulated value at the end of 20 years s 300 s 20 0.08 = 14826.87. The cahsflow of the nvestments s Payments 300 2(300) 3(300) 20(300) Tme 1 2 3 20 So, the future value at tme 20 of ths cahsflow s So, 300 (Is) n = (300) s 20 /2 20 /2 0.09 = (600)( s 20 /2 20). 14826.87 = (300)(20) + (600)( s 20 /2 20). and 20826.87 = 600 s 20 /2. Hence, /2 = 5% and = 10%. 62. We have that t δ 0 s ds = t 0 1 t ds = ln(8 + s) 8+s 0 = ln(8 + t) ln 8. So, a(t) = e R t 0 δs = e ln(8+t) ln 8 = 8+t a(1) a(0). The nterest earned n the frst year s 3960 = 3960(a(1) 1) = 8 a(0) 3960( 9 1 a(2) a(1) 1) = 3960 = 495. The nterest earned n the second year s 3960 = 8 8 a(1) 3960 10 8 9 8 9 8 3960 11 8 10 8 10 8 = 3960 1 9 = 440. The nterest earned n the thrd year s 3960 a(3) a(2) a(2) = = 3960 1 = 396. Snce the 3 nterest payements are accumulated at rate of 10 7%, the accumulated amount of money at tme t = 3 s 3960+495(1.07) 2 +440(1.07)+396 = 3960 + 566.72 + 470.8 + 396 = 5393.53. 63. The cahsflow of nterest s Payments 12 12 12 12 24 24 24 Tme 1 2 3 10 11 12 20 30
The future value at tme 20 of ths cashflow s 64 = 12s 20 0.75 + 12s 10 0.75 = 12 (1+0.75)20 1 + 12 (1+0.75)10 1 0.75 0.75 = 16((1 + 0.75) 20 1) + 16((1 + 0.75) 10 1) = 16(1 + 0.75) 20 + 16(1 + 0.75) 10 32 So, (1 + 0.75) 20 + (1 + 0.75) 10 6 = 0, (1 + 0.75) 10 = 1± 1+24 2 = 1±5 2 = 2 and = 9.5698%. 64. The value of the end of the 5 years of Sally s nvestment s 10000 ( ) 1 + 0.0745 10 2 = 14415.65. Let P be the amount of the payments Sally receves from Tm. Then, P s 60 0.06/12 = 14415.65. So, P = 206.6167. Let (12) be the nomnal rate of nterest, compounded monthly, whch Sally charged Tm on the loan. Then, 10000 = 206.6167a 60 (12) /12. So, (12) /12 = 0.7333 and (12) = 8.801%. 65. A nomnal rate of 8% convertble quarterly s equvalent to an effectve annual rate of 8.2432%. The nterest earned each year s (2000)(0.082432) = 164.86. A nomnal rate of 9% convertble quarterly s equvalent to an effectve annual rate of 9.3083%. The nterest obtaned from 164.86 nvested at an effectve annual rate s 9.3083% s (164.86)(0.093083) = 15.3457. So, the cashflow gong nto the account payng an annual effectve rate of 7% s Contrbutons 0 0 15.35 2(15.35) 9(15.35) Tme 0 1 2 3 10 The future value of ths cashflow s 15.3457 (Is) 9 0.07 = 15.3457 s 9 0.07 9 0.07 = 15.3457 12.8165 9 0.07 The total value of all funds s 2000 + (10)(164.84) + 836.66 = 4485.06. = 836.66. 66. Let P be the payment under the frst opton Then, 20000 = P a 20 0.065 and P = 1815.12. The annual nterest payment under the second opton s 20000(0.08) = 1600. depost nto the snkng fund s 1815.12 1600 = 215.12. So, 20000 = 215.12s 20 j j = 14.18%. So, the 67. The P be the ntal amount of the payments to the snkng fund. Then P s 10 8% = 10000 and P = 690.29. The present value at tme t = 5 of the frst 5 payments s 690.29s 5 8% = 4049.68. The Q be the amount of the payments to the snkng fund n the last 5 years. Snce the snkng fund has to accumulate to 10000, we have that 4049.68(1.06) 5 + Qs 5 6% = 10000. and Q = 812.59. So, X = 812.59 690.29 = 122.30. 31 and
68. Margaret pays (0.1)L L to the snkng fund each years. The value of snkng fund accumulates to 10000 n 17 years at 4.62%. So, L = (0.1 )Ls 17 0.0462 and 1 0.1 = s 17 0.0462 = 25.0003. So, = 6%. For Jason, L = 500a 17 0.06 = 5238.63. 69. We have that (8000)(1.08) 12 = 800s 12 8% + xs 12 4%. So, 20145.36 = 15181.68 + 15.0258x and x = 330.34. 70. Let P be the amount of equal annual payments made under the frst opton. Then, P a 10 0.0807 = 2000. So, P = 299. The total payments are 2990. Under the second opton, at tme j, the loan s 2000 200j = (200)(10 j). Hence, the payment at tme j s (200)(10 (j 1)) = (200)(11 j). So, the total payments under the second opton are 2990 = 10 j=1 (200 + (200)(11 j)) = 2000 + 200 10 j=1 (11 j) = 2000 + 200 10 j=1 j = = 2000 + 200 10 11 2 = 2000 + 11000. So, = 9%. 71. We fnd P solvng 100 = P a 10 10%, we get that P = 162.75. The deposts to the snkng fund are of 62.75. The value of the snkng fund after the last depost s 62.75s 10 14% = 1213.33. So, the balance n the snkng fund mmedately after repayment of the loan s 213.33. 72. The prce of the bond s the present value of all payments at the annual effectve rate of 8.25%: P = 75 a 1.03 20 + 1050(1.0825) 20 0.0825 0.03 1.03 = 72.8156a 20 0.050970 + 215.10 = 900.02 + 215.10 = 1115.12. 73. Snce Bll got an effectve annual rate of nterest of 7% n hs nvestment, Bll got at the end 1000(1.07) 10 = 1967.15. The coupon payments are 1000(.03) = 30. So, 1967.15 = 30s 20 (2) /2 + 1000 and 967.15 = 30s 20 (2) /2. So, (2) /2 = 4.7597% and = 9.7459%. 74. Let C be the redempton value of the bonds are. Let be the semannual effectve nterest rate. Let ν = (1 + ) 1. Then, 381.50 = Cν 2n and 647.80 = Cν n. So, 381.50 647.80 = νn and (1 + ) n = 1.6980. The present value of the coupons n bond X s F ra 2n = F r 1 ν2n = (1000)(1.03125)(1 (1.6980) 2 ) = 673.59. So, the total present value of the bond X s 381.50 + 673.59 = 1055.09. 75. We have that F = 100 and 2r = 6.75%. So, the coupon payment s F r = (1000) 0.0675 2 = 33.75. Snce the nomnal rate of nterest n the nvestment s 7.4% convertble semannually, we have that the prce of the bond s (33.75)a 20 0.037 + (1000)(1.037) 20 = 954.63. 32
Snce the company proposes to pay the nterest accordng wth the same the rate of nterest, the value of the restructured bond s the same as the prce of the old bond: 954.63. 76. We have that F = C = 10000, F r = 800, = 6% and n = 10. So, I 7 = C + (F r C)(1 ν n+1 k ) = 600 + (800 600)(1 (1.06) 4 ) = 641.5813. 77. We have that F = 1000, r = 0.04, = 0.0375 and n = 10. So, Hence, the amortzaton of the premum n the 6 th year s (F r C)ν n+1 k = (1000)(0.04 0.0375)(1.0375) 5 = 2.079694. 78. For the bond X, n = 14, C = F = 1000, r = 10%, and = 8%. So, the wrte down n premum s (F r C)ν n+1 k = (1000)(0.10 0.08)(1.08) 9. For the bond Y, n = 14, C = F, r = 6.75%, and = 8%. So, the wrte down n dscount s (C F r)ν n+1 k = F (0.08 0.0675)(1.08) 9. We have that (1000)(0.10 0.08)(1.08) 9 = F (0.08 0.0675)(1.08) 9, F = (1000)(0.10 0.08) = 1600 and (0.08 0.0675) P = (1600)(0.0675)a 14 0.08 + (1600)(1 + 0.08) 14 = 1435.115. 79. We know that F = 10, n = 18, = 5% and 20 = B 9 B 10 = (F r C)ν n+1 k = (1000)(r 0.05)(1.05) 9. So, r = 8.1027%. The book value at the end of year 10 s B 10 = F ra n k + Cν n k = (1000)(0.081027)a 8 0.05 + (1000)(1 + 0.05) 8 = 1200.53. 80. We know that n = 30. The amortzaton n the k th payment s B k 1 B k = (F r C)ν n+1 k. So, (F r C)ν 29 = 977.19 and (F r C)ν 27 = 1046.79. Hence, (1+) 2 = 1046.79 977.19, = 3.5%. and (F r C) = (1 + 0.035) 29 977.19 = 2650. The premum s P C = (F r C)a n = (2650)a 30 3.5% = 48739. 81. The deprecaton charge n the k th year s D k = (A S)(1 + )k 1 s n = 1600(1 + )k 1 s 10. The present value of the deprecaton charges s So, s 10 = 16 and = 8.3946%. 1000 = 10 1600(1+) k 1 k=1 (1 + ) k = 16000 = 16000 s s (1+) 10 10 s. 10 33
82. We know nether X, nor Y, nor n. Snce usng the straght lne method, the annual deprecaton expense s 1000, we have that X Y n method, the deprecaton expense n year 3 s = 1000. Usng the sum of the years dgts 800 = D 3 = (A S)(n 2) S n = 2(n 2)(X Y ). n(n + 1) Usng the declnng balance method, we get that Y = X(1 0.33125) n = X(0.66875) n. We have the equatons 1000 = X Y n, 800 = 2(n 2)(X Y ) n(n + 1) and Y = X(0.66875) n. Dvdng the frst equaton over the second one, we get that 1.25 = 1000 X Y 800 = n 2(n 2)(X Y ) n(n+1) = n + 1 2(n 2) and n = 4. So, we have that X Y = 4000. Usng the thrd equaton, Y = X(0.66875) 4 = X(0.2). From the equatons, X Y = 4000 and Y = X(0.2), we get that X = 5000 and Y = 1000. 83. The deprecaton n the k th year s D k = Ad(1 d) k 1 = 20000(0.2)(0.8) k 1 = 4000(0.8) k 1, where A = 20000 and d = 0.2. We have that the cashflow nto the fund s Contrbutons 4000 4000(0.8) 1 4000(0.8) 2 4000(0.8) 14 Tme 1 2 3 15 So, the accumulated value n the fund at the end of the 15 years s 4000 0.8 a 15 1.06 0.80 1(1 + 0.06)15 = 11982.79a 15 0.325 = 36328.83. 84. Accordng wth the sum-of-the-years-dgts method at the end of year 4, the machne has a book value of 2218. So, 2218 = S + (n k)(n k + 1) (A S) = S 6 7 68 (5000 S) = 1909.09 + n(n + 1) 10 11 110 S. So, S = 110 (2218 1909.09) = 499. At the year 4, the machne has a value of 2218. 68 The salvage value of the machne s 449. So, the new deprecaton charge s D 8 = A S = 6 2218 499 = 286.38 s per year. 6 34
85. For Machne I, S = A(1 d) n gves that X 8 = X(1 d)10. So, 1 d = (1/8) 1/10 = 0.125 0.1. Hence, B 7 = A(1 d) 7 = X(0.125 0.7 ). The total amount of deprecaton n the frst seven years s A B 7 = X X(0.125 0.7 ) = X(1 (0.125) 0.7 ). For Machne II, the total amount of deprecaton n the frst seven years s D 1 + + D 7 = 7 = ( Y X 8 So, X(1 (0.125) 0.7 ) = ( Y X 8 j=1 ( ) Y X 8 2 (10)(11) ) 1 55 (10 + + 4) = ( Y X 8 ) 49 55 and Y X ) 49 55. (11 j) = 0.125 + 55 49 (1 (0.125)0.7 ) = 0.9856. 86. We have that k = 6, = j = 9%, H = 11749.22, M = 3000, B 6 = 55216.36. Hence, the formulas become B k = A (A S)s k j, H = A + A S s n j s n + M 55216.36 = A (A S)s 6 9%, 11749.22 = A(0.09) + A S + 3000. s n 9% s n 9% Solvng for (A S) s n 9% n both equatons A 55216.36 s 6 9% = (A S) s n 9% = 11749.22 A(0.09) 3000 and A = 11749.22 3000 + 55216.36 s 6 9% (0.09) + 1 = 11749.22 3000 + 55216.36 1 (0.09) + s 7.523334565 6 9% 7.523334565 = 72172. 87. For Erc, P = 800 (800 2X) = 2X, M = (800)(0.50) = 400, I = (400)(0.08) = 32 and D = 16. Erc s net proft s P + I D = 2X + 32 16 = 2X + 16. Erc s nterest s = P +I D = 2X+16 = X+8. For Jason, P = 800 (800+X) = X, M = (800)(0.50) = 400, M 400 200 I = (400)(0.08) = 32 and D = 16. Jason s net proft s P +I D = X+32 16 = X+16. Jason s nterest s = P +I D M and = X+8 = 4+8 = 6%. 200 200 = X+16 400. Solvng for X n X+8 200 = 2 X+16 400, we get that X = 4 88. Let x be the prce at whch Jose and Chrs each sell a dfferent stock short. For Jose, P = 760 x, M = x(0.50), I = x(0.50)(0.06) = (0.03)x and D = 32. Jose s yeld s x 760+(0.03)x 32. For Chrs, P = 760 x, M = x(0.50), I = x(0.50)(0.06) = (0.03)x and (0.5)x D = 0. Chrs yeld s = x 760+(0.03)x. We have (0.5)x = x 760 + (0.03)x (0.5)x = 2 x 760 + (0.03)x 32 (0.5)x 35
or 2 1520 x whch mples ( 1520 + 3040 + 128) 1 x The nterest s = 800 760+(0.03)(800) = 16%. (0.5)(800) 3040 128 + 0.06 = 4 + 0.12 x x 1520+3040+128 = 4 + 0.12 2 0.06 and x = = 800. 4+0.12 2 0.06 36