TABLE OF CONTENTS. 1. Interest Rates Level Annuities Different Periods Varying Annuities Yield Rates

Transcription

1 TABLE OF CONTENTS 1. Interest Rates 1 2. Level Annuities Different Periods Varying Annuities Yield Rates Loans Financial Instruments Term Structure Duration McDonald McDonald McDonald McDonald McDonald McDonald 8 391

2 NOTES Questions and parts of some solutions have been taken from material copyrighted by the Casualty Actuarial Society and the Society of Actuaries. They are reproduced in this study manual with the permission of the CAS and SoA solely to aid students studying for the actuarial exams. Some editing of questions has been done. Students may also request past exams directly from both societies. I am very grateful to these organizations for their cooperation and permission to use this material. They are, of course, in no way responsible for the structure or accuracy of the manual. Exam questions are identified by numbers in parentheses at the end of each question. CAS questions have four numbers separated by hyphens: the year of the exam, the number of the exam, the number of the question, and the points assigned. SoA or joint exam questions usually lack the number for points assigned. W indicates a written answer question; for questions of this type, the number of points assigned are also given. A indicates a question from the afternoon part of an exam. MC indicates that a multiple choice question has been converted into a true/false question. Page numbers (p.) with solutions refer to the reading to which the question has been assigned unless otherwise noted. Page references refer to Samuel A. Broverman Mathematics of Investment and Credit (2008); James W. Daniel and Leslie Jane Federer Vaaler, Mathematical Interest Theory (2009); Stephen G. Kellison, The Theory of Interest (2008); Robert L. McDonald, Derivatives Markets, (2006); and Chris Ruckman and Joe Francis, Financial Mathematics (2005). Although I have made a conscientious effort to eliminate mistakes and incorrect answers, I am certain some remain. I am very grateful to students who discovered errors in the past and encourage those of you who find others to bring them to my attention. Please check our web site for corrections subsequent to publication. I would also like to thank Graham Lord for checking some of the solutions in the manual. Hanover, NH 11/1/10 PJM

3 Financial Instruments 263 B. Price of a Bond: Calculating Purchase Prices B1. A bond with a $100 par value has 5.25% annual coupons and is due to mature at the end of 16 years. The bond will be redeemed at maturity for an amount equal to its par value less a service charge. The service charge is equal to 25% of the excess (if any) of the par value over the purchase price. A prospective purchaser offers a price that will produce a yield equivalent to a 7% annual effective interest rate, taking into account the deduction of the service charge. It is noted that (1.07) 16 = 3. In which of the following ranges does this price lie? A. < $65 B. $65 but < $71 C. $71 but < $77 D. $77 but < $83 E. $83 (80S 4 12) B2. A $1,000 bond with quarterly coupons of $25 each will be redeemed in 3.5 years for $1,010. It is purchased to yield a nominal annual rate of 8% compounded quarterly. In which of the following ranges is the purchase price of this bond? A. < $1,000 B. $1,000 but < $1,025 C. $1,025 but < $1,050 D. $1,050 but < $1,075 E. $1,075 (81S 4 13) B3. Two fifteen-year bonds with $100 redemption values are each purchased to yield an effective annual interest rate of 4%. The first bond bears annual g% coupons and is purchased at a premium of $ The second bond bears annual (g + 2)% coupons. Which of the following is closest to the purchase price of the second bond? A. $125 B. $127 C. $129 D. $131 E. $133 ( ) B4. A 9% bond with a 1,000 par value and coupons payable semiannually is redeemable at maturity for 1,100. At a purchase price of P, the bond yields a nominal annual interest rate of 8%, compounded semiannually, and the present value of the redemption amount is 190. Determine P. A. 1,050 B. 1,085 C. 1,120 D. 1,165 E. 1,215 (84S 4 11) B5. A bond of amount 1 sells for (1 + p) at a certain fixed yield rate. If the bond's coupon rate were halved, the price would be (1 + q). What would be the price if the coupon rate were doubled? Throughout, assume the bond is unchanged in all other respects. A p 2q B. (1 + p)2/(1 + q) C. 1 + p + 2q D p q E p 4q ( ) B6. A $1,000 par value bond with 9% coupons payable semiannually is purchased for $1,300. The yield to the purchaser is 6%, convertible semiannually. If the same bond were redeemable at 120% of par, what price would have been paid to obtain the same yield? (Answer to nearest $10.) A. $1,260 B. $1,320 C. $1,380 D. $1,440 E. $1,500 (84F 4 10) B7. A $1,000 bond bearing coupons at an annual rate of 5.5% payable semiannually and redeemable at $1,100 is bought to yield a nominal annual rate of 4% convertible semiannually. If the present value of the redemption value at this yield is $140, what is the purchase price? A. < $1,310 B. $1,310 but < $1,330 C. $1,330 but < $1,350 D. $1,350 but < $1,370 E. $1,370 ( )

4 264 Financial Instruments Solutions are based on Broverman, pp ; Daniel, pp ; Kellison, pp ; Ruckman, pp B1. To earn 7% over 16 years, the AV of the coupons and the net redemption value must equal 3P. 3P = 5.25s 16 + [100 (.25)(100 P)] = [5.25][(1.07)16 1] P = (5.25)(2/.07) + 75 = P B2. P = 25a 14 + (1,010)(1.02) -14 = [25][1 (1.02)-14 ].02 + (1,010)(.75788) = 1, B = ga 15 + (100)(1.04) -15 = [g][1 (1.04)-15 ].04 + (100)(.55526) = g g = g + 2 = P = ga 15 + (100)(1.04) -15 = ( )( ) = 133 Answer: E B4. g = Fr/C = (1,000)(.045)/1,100 = P = K + (g/i)(c K) = (.04091/.04)(1, ) = 1, B p = Fr a_ n + K 1 + q = (Fr/2)a_ n + K K = (2)(1 + q) (1 + p) = 1 + 2q p Fr a_ n = (2)(p q) 2Fr a_ n + K = (2)(2)(p q) + (1 + 2q p) = 1 + 3p 2q Answer: A B6. 1,300 = P = K + (g/i)(c K) = K + (.045/.03)(1,000 K) = 1,500.5K.5K = 200 K = 400 P' = P +.2K = 1,300 + (.2)(400) = 1,380 B7. Solve for n and use to calculate the purchase price. 140 = Cv n = (1,100)(1.02) -n ln (140/1,100) n = ln 1.02 P = 27.5 a K = [27.5][1 (1.02) ].02 = = = (27.5)( ) = 1,340.01

5 Financial Instruments 265 B8. You are given the following information on a bond: i) Its par value and redemption value both equal 1,000. ii) Its coupon rate is 12% convertible semiannually. iii) It is priced to yield 10%, convertible semiannually. The bond has a term of n years. If the term of the bond is doubled, the price will increase by 50. Calculate the price of the n-year bond. A. 1,050 B. 1,100 C. 1,150 D. 1,200 E. 1,250 (85F 4 6) B9. A $1,000 par value 20-year bond has coupons at 5% convertible semiannually. It has a redemption value of $1,100. If the yield rate is 6% convertible semiannually, what is the bond's price? A. < $900 B. $900 but < $910 C. $910 but < $920 D. $920 but < $930 E. $930 ( ) B10. A bond with coupons of 40 sells for P. A second bond with the same maturity value and term has coupons equal to 30 and sells for Q. A third bond with the same maturity value and term has coupons equal to 80. All prices are based on the same yield rate, and all coupons are paid at the same frequency. Determine the price of the third bond. A. 4P 4Q B. 4P + 4Q C. 4Q 3P D. 5P 4Q E. 5Q 4P (87S ) B11. A 1,000 bond with coupon rate c convertible semiannually will be redeemed at par in n years. The purchase price to yield 5% convertible semiannually is P. If the coupon rate were (c.02), then the price of the bond would be (P 300). Another 1,000 bond is redeemable at par at the end of 2n years. It has a coupon rate of 7% convertible semiannually and the yield rate is 5% convertible semiannually. Calculate the price of this second bond. A. 1,300 B. 1,375 C. 1,475 D. 2,100 E. 2,675 (88F ) B12. A 100 par value 6% bond with semiannual coupons if purchased at 110 to yield a nominal rate of 4% convertible semiannually. A similar 3% bond with semiannual coupons is purchased at P to provide the buyer with the same yield. Calculate P. A. 90 B. 95 C. 100 D. 105 E. 110 (88F ) B13. You are given two n-year par value 1,000 bonds. Bond X has 14% semiannual coupons and a price of 1, to yield i, compounded semiannually. Bond Y has 12% semiannual coupons and a price of 1, to yield the same rate i compounded semiannually. Calculate the price of bond X to yield (i 1%). A. 1,500 B. 1,550 C. 1,600 D. 1,650 E. 1,700 (89S )

6 266 Financial Instruments B8. Calculate the PV of the redemption value and use to calculate the price of the bond: P = K + (g/i)(c K) = (1,000)(1.05) -2n + [.06/.05][1,000][1 (1.05) -2n ] P = 1,200 (200)(1.05) -2n P' = P + 50 = 1,250 (200)(1.05) -2n P' = (1,000)(1.05) -4n + [.06/.05][1,000][1 (1.05) -4n ] = 1,200 (200)(1.05) -4n 1,250 (200)(1.05) -2n = 1,200 (200)(1.05) -4n.25 = [1.05] -2n [1 (1.05) -2n ] (1.05) -2n =.5 P = 1,200 (200)(.5) = 1,100 Answer: B B9. P = 25a 40 + (1,100)(1.03) -40 = [25][1 (1.03)-40 ].03 P = (25)( ) = (1,100)(.30656) B10. P = K + 40a_ n Q = K + 30a_ n 10a_ n = P Q K = 4Q 3P X = K + 80a_ n = (4Q 3P) + (8)(P Q) = 5P 4Q B11. Calculate the value of (1.025) -2n and use to calculate the price of the bond: P = (1,000)(1.025) -2n + (1,000c/2)a 2n P 300 = (1,000)(1.025) -2n + [1,000][c.02)/2]a 2n 300 = [10][1 (1.025)-2n ].025 P' = (1,000)(.25) 2 + [35][1 (.25)2 ].025 Answer: B (1.025) -2n =.25 P' = (1,000)(1.025) -4n + [35][1 (1.025)-4n ].025 = , = 1,375 B = C + (Fr Ci)a 2n = (100)(.03.02)a 2n a 2n = 10 P = C + (Fr Ci)a 2n = (100)( )(10) = 95 Answer: B B13. 1, = 70a 2n + 1,000v -2n 1, = 60a 2n + 1,000v -2n = 10a 2n a 2n = , (70)(13.59) = 1,000v -2n v -2n =.4564 i = 1 v-2n a 2n = =.04 (1.04) -2n =.4564 ln i 1 = (2)(.04) =.07 n = 2 ln 1.04 = (2)(.03922) = 10 P = [70][1 (1.035)-(2)(10) ]] (1,000)(1.035) -(2)(10) = (70)( ) + (1,000)(.50257) = 1, Answer: A

7 Financial Instruments 267 B14. John buys a ten-year 1,000 par value bond with 8% semiannual coupons. The price of the bond to earn a yield of 6% convertible semiannually is 1, The redemption value is more than the par value. Calculate the price John would have to pay for the same bond to yield 10% convertible semiannually. A. 875 B. 913 C. 951 D. 989 E. 1,027 (89F ) B15. On June 1, 1990, an investor buys three fourteen-year bonds, each with a par value 1,000, to yield an effective annual interest rate of i on each bond. Each bond is redeemable at par. You are given: i) The first bond is an accumulation bond priced at ii) The second bond has 9.4% semiannual coupons and is priced at iii) The third bond has 10% annual coupons and is priced at P. Calculate P. A. 825 B. 835 C. 845 D. 855 E. 865 (90S ) B16. A ten-year bond with par value 1,000 and annual coupon rate r is redeemable at 1,100. You are given: i) The price to yield an effective annual interest rate of 4% is P. ii) The price to yield an effective annual interest rate of 5% is (P 81.49). iii) The price to yield an effective annual interest rate of r is X. Calculate X. A. 1,061 B. 1,064 C. 1,068 D. 1,071 E. 1,075 (90S ) B17. Jim buys a ten-year bond with par value of 10,000 and 8% semiannual coupons. The redemption value of the bond at the end of ten years is 10,500. Calculate the purchase price to yield 6% convertible quarterly. A. 11,700 B. 14,100 C. 14,600 D. 15,400 E. 17,700 (90F 140 5) B18. Bart buys a 28-year bond with a par value of 1,200 and annual coupons. The bond is redeemable at par. Bart pays 1,968 for the bond, assuming an annual effective yield rate of i. The coupon rate on the bond is twice the yield rate. At the end of 7 years, Bart sells the bond for P, which produces the same annual effective yield rate of i to the new buyer. Calculate P. A. 1,470 B. 1,620 C. 1,680 D. 1,840 E. 1,880 (90F ) B19. A Treasury bond pays semiannual coupons at 8.8%, has a face value of $1,000, and a yield to maturity of 9.4%. The bond matures in two years. What is the price of the bond? A. < $980 B. $980 but < $990 C. $990 but < $1,000 D. $1,000 but < $1,010 E. $1,010 (91 5B 51 1)

8 268 Financial Instruments B14. Calculate the redemption value of the bond and use to calculate the price of the bond to yield 10%. P Fr a 10 C = v n = 1, [40][1 (1.03)-20 ]/.03 (1.03) -20 C = P' = 40a 10 + (1,100)(1.05) -20 = [40][1 (1.05)-10 ].05 P' = (40)( ) = 913 Answer: B + (1,100)(.37689) 1, (40)( ) = 1,100 B15. v 14 (g)(c K) (.047)(1, ) = P/C = /1,000 = i = = =.06 P K (1.06) 2 = P = K + (g/i)(c K) = (.10/.1236)(1, ) = B16. P = (1,000r)a 10 + (1,100)(1.04) -10 = [1,000r][1 (1.04)-10 ].04 + (1,100)(.67556) P = (1,000r)( ) = 8,110.90r P = (1,000r)a 10 + (1,100)(1.05) -10 = [1,000r][1 (1.05)-10 ].05 + (1,100)(.61391) P = (1,000r)( ) = ,721.73r = 8,110.90r r = X = (1,000)(.03513)a 10 + (1,100)( ) -10 = [35.13][1 ( )-10 ] X = (35.13)( ) = 1, B17. (1.015) 2 = P = 400a 20 + (10,500)( ) -20 = [400][1 ( )-20 ] P = (400)( ) + 5, = 11, Answer: A + (10,500)(.55121) B18. 1,968 = (1,200)(2i)a ,200v 28 = (2,400)(1 v 28 ) + 1,200v 28 v 28 =.36 v 27 = P = (1,200)(2i)a ,200v 21 P = (1,200)(2)( ) + (1,200)(.46479) = 1, B19. Assume the redemption amount equals the face amount = P = 44a_ 4 + (1,000)( ) -4 = [44][1 ( )-4 ] P = (44)( ) = (1,000)(.83555)

9 Financial Instruments 269 B20. An n-year zero-coupon bond with par value of 1,000 was purchased for 600. An n-year 1,000 par value bond with semiannual coupons of X was purchased for 850. A 3n-year 1,000 par value bond with semiannual coupons of X was purchased for P. All three bonds have the same yield rate. Calculate P. A. 686 B. 696 C. 706 D. 716 E. 726 (93S ) B21. You are given: i) A 10-year 8% semiannual coupon bond is purchased at a discount of X. ii) A 10-year 9% semiannual coupon bond is purchased at a premium of Y. iii) A 10-year 10% semiannual coupon bond is purchased at a premium of 2X. iv) All bonds were purchased at the same yield rate and have par values of 1,000. Calculate Y. A. X/3 B. 2X/5 C. X/2 D. 2X/3 E. X (93F ) (Sample1 2 10) B22. Two 1,000 par value bonds are purchased. The 2n-year bond costs 250 more than the n-year bond. Each has 13% annual coupons and each is purchased to yield 6.5% annual effective. Calculate the price of the n-year bond. A. 1,200 B. 1,300 C. 1,400 D. 1,500 E. 1,600 (95S ) B23. A 30-year bond has 10% annual coupons and a par value of 1,000. Coupons can be reinvested at a nominal annual rate of 6% convertible semiannually. X is the highest price that an investor can pay for the bond and obtain an effective yield of at least 10%. Calculate X. A. 518 B. 618 C. 718 D. 818 E. 918 (95S ) (Sample1 2 12) B24. In July 1995, you purchase a July 1999 U.S. Treasury bond paying interest annually. Given the following, what is the market value at the time of purchase (as a percentage of the face value)? Face value $5,000 Yield to maturity 9.5% Coupon rate 8% Expected inflation rate 6% A. < 90.0% B. 90.0% but < 94.0% C. 94.0% but < 98.0% D. 98.0% but < 102.0% E % (95F 5B 1 1) B25. On January 1, 1995, you purchased a bond with a December 31, 1999 maturity. At the time of purchase, bonds with the same risk were yielding 7%. The purchased bond has a coupon rate of $80 paid annually and a face value of $1,000. A year later, after you received the first $80 coupon, you sold the bond when bonds with the same risk had a yield of 6.8%. What was the difference between your selling price and buying price in nominal terms? A. The buying price was at least $25 more than the selling price. B. The buying price was at least $15 more than the selling price, but less than $25 more than the selling price. C. The buying price was at least $5 more than the selling price, but less than $15 more than the selling price. D. The buying price was within $5 of the selling price. E. The buying price was less than the selling price by at least $5. (98F 5B 1 1)

10 270 Financial Instruments B20. v n = P/C = 600/1,000 =.6 K' = 1,000v 3n = (1,000)(.6) 3 = 216 g/i = P K C K = , =.625 P = K' + (g/i)(c K') = (.625)(1, ) = 706 B21. 1,000 X = K + 40a 20 1, X = K + 50a 20 10a 20 = 3X a 20 = 3X/10 K = (2,000 + X 90a 20 )/2 = 1,000 + X/2 (45)(3/10)X = 1,000 13X Y = K + 45a 20 1,000 = (1,000 13X) + (45)(3X/10) =.5X B22. P = P' 130a_ n + (1,000)(1.065) -n = 130a 2n + (1,000)(1.065) -2n [130][1 (1.065) -n ] (1,000)(1.065) -n = [130][1 (1.065)-2n ] (1,000)(1.065) -2n (1,000)(1.065) -2n (1,000)(1.065) -n = 0 (1.065) -n = ( 1,000) ± ( 1,000)2 (4)(1,000)(250) (2)(1,000) =.5 (130)(1.5) P = (1,000)(.5) = 1,500 B23. Equate the AV value of the investment at the specified yield and that of its components and solve for P: (1.03) 2 = P(1.10) 30 = P = 100s ,000 = [100][(1.0609)30 1] , P = (100)( ) + 1,000 = 9, P = Answer: A B24. P = 400a_ 4 + (5,000)(1.095) -4 = [400][1 (1.095)-4 ] (5,000)(.69557) P = (400)( ) + 3, = 4, P/F = 4,758.64/5,000 = 95.2% B25. P = 80a_ 5 + (1,000)(1.07) -5 = [80][1 (1.07)-5 ].07 + (1,000)(.71299) P = (80)( ) = 1, P' = 80a_ 4 + (1,000)(1.068) -4 = [80][1 (1.068)-4 ] (1,000)(.76863) = 1, P' = (80)( ) = 1, P P' = 1, , =.18

11 Financial Instruments 271 B26. For an asset, you are given: i) The value of the asset at the beginning of year 1 is 1,000. ii) The asset returns 10% each year for five years. iii) The asset pays a dividend of 50 at the end of each year for five years. Your opportunity cost of capital is 12%. Calculate the present value of the asset at the beginning of year 1. A. 909 B. 921 C. 930 D. 939 E. 951 (Sample1 2 47) B27. Consider the following bonds: Bond A: a ten-year bond with 5% annual coupons and a face value of 1000 Bond B: a five-year bond with 10% annual coupons and a face value of 700 Bond C: a ten-year bond with 12% annual coupons and a face value of 600 Assuming a discount rate of 10%, rank the following: I. Price of bond A II. Price of bond B III. Price of bond C A. I > II > III B. I > III > II C. II > I > III D. III > I > II E. III > II > I (Sample1 2 49) B28. A firm has proposed the following restructuring for one of its 1,000 par value bonds. The bond presently has ten years remaining until maturity. The coupon rate on the existing bond is 6.75% per annum paid semiannually. The current nominal semiannual yield on the bond is 7.40%. The company proposes suspending coupon payments for four years with the suspended coupon payments being repaid, with accrued interest, when the bond comes due. Accrued interest is calculated using a nominal semiannual rate of 7.40%. Calculate the market value of the restructured bond. A. 755 B. 805 C. 855 D. 905 E. 955 (00S 2 29) B29. You are given the following information about a bond: i) The term-to-maturity is two years. ii) The bond has a 9% annual coupon rate, paid semiannually. iii) The annual bond-equivalent yield-to-maturity is 8%. iv) The par value is $100. Calculate the current price of the bond. ( a.5) B30. A 1,000 par value twenty-year bond with annual coupons and redeemable at maturity at 1,050 is purchased for P to yield an annual effective rate of 8.25%. The first coupon is 75. Each subsequent coupon is 3% greater than the preceding coupon. Determine P. A. 985 B. 1,000 C. 1,050 D. 1,075 E. 1,115 (00F 2 30)

12 272 Financial Instruments B26. VA 1 = (1,000)(1.1) 50 = 1,050 VA 2 = (1,050)(1.1) 50 = 1,105 VA 3 = (1,105)(1.1) 50 = 1, VA 4 = (1,165.5)(1.1) 50 = 1, VA 5 = (1,232.05)(1.1) 50 = 1, PV = 50a_ 5 + (1,305.26)(1.12) -5 = [50][1 (1.12)-5 ].12 + (1,305.26)(.56743) PV = (50)( ) = 921 Answer: B B27. P A = 50a 10 + (1,000)(1.10) -10 = [50][1 (1.10)-10 ].10 + (1,000)(.38554) P A = (50)( ) = P B = 70a_ 5 + (700)(1.10) -5 = [70][1 (1.10)-5 ].10 + (700)(.62092) P B = (70)( ) = P C = 120a 10 + (600)(1.10) -10 = [120][1 (1.10)-10 ].10 + (600)(.38554) P C = (72)( ) = B28. Since interest is accrued at the yield rate, there is no effect on the bond's market value..0675/2 = /2 =.037 P = 33.75a 20 + (1,000)(1.037) -20 = [33.75][1 (1.037)-20 ] (1,000)(.48353) P = (33.75)( ) = 955 Answer: E B29. P = 4.5a_ 4 + (100)(1.04) -4 = [4.5][1 (1.04)-4 ].04 P = (4.5)( ) = (100)(.85480) B /1.03 = P = (1.03) -1 75a 20 + (1,050)(1.0825) -20 = [ ][1 ( )-20 ] P = ( )( ) = 1,115 + (1,050)(.20485) Answer: E

13 Financial Instruments 273 B31. You have decided to invest in two bonds. Bond X is an n-year bond with semiannual coupons, while bond Y is an accumulation bond redeemable in n/2 years. The desired yield rate is the same for both bonds. You also have the following information: i) Par value of bond X is 1,000. ii) For bond X, the ratio of the semiannual bond rate to the desired semiannual yield rate, (r/i), is iii) The present value of the redemption value of bond X is iv) Redemption value of bond Y is the same as the redemption value of bond X. v) Price to yield of bond Y is What is the price of bond X? A. 1,019 B. 1,029 C. 1,050 D. 1,055 E. 1,072 (01F 2 31) B32. You have decided to invest in bond X, an n-year bond with semiannual coupons, with the following characteristics: i) Par value of bond is 1,000. ii) For bond X, the ratio of the semiannual bond rate to the desired semiannual yield rate, (r/i), is iii) The present value of the redemption value X is Given v n =.5889, what is the price of bond X? A. 1,019 B. 1,029 C. 1,050 D. 1,055 E. 1,072 (Sample FM 22) B33. Susan can buy a zero-coupon bond that will pay 1,000 at the end of twelve years and is currently selling for Instead she purchases a 6% bond with coupons payable semiannually that will pay 1,000 at the end of ten years. If she pays X, she will earn the same annual effective rate as the zero-coupon bond. Calculate X. A. 1,164 B. 1,167 C. 1,170 D. 1,173 E. 1,176 (05S FM 5)

14 274 Financial Instruments B31. Use the PV of the two redemption values to calculate n and the redemption value. Then use the base amount to calculate the price of bond X: K X = = Cv n K Y = = Cv n/2 C = 1,100 v n = G X = F(r/i) = (1,000)( ) = 1, P X = G + (C G)v n = 1, (1,100 1,031.25)(.34682) = 1, B32. See B31. B33. i = (1,000/624.60) 1/12 1 =.04 i' = (1.04).5 = P = 60a 10 + (1,000)(1.0198) -20 = [30][1 (1.0198)-20 ] (1,000)(.67556) P = (30)( ) = 1, Answer: B