Answer Key

712 ANSWER KEY 2.15 n= c a b ... Simple interest Compound interest a(t) = 1 + it (1 + i)t ... 12.13 4.53% 12.14 X( 17 + 15 + 12) + Y...

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Answer Key Section 1 1.1 2.94% 1.2 (a) $780 (b) 6.5% 1.3 $1,090 1.4 1 + 0.01t 1.5 9.2% 1.6 3.3 years 1.7 $11,576.25 1.8 $4,971.77 1.9 6.96% 1.10 55.48 years 1.11 No 1.12 P (1 + i1 )(1 + i2 ) · · · (1 + in ) 1.13 $56.74 1.14 (a) $1,004.01 (b) $4.01 (c) 0.401%

Section 2 2.1 (a) A step function (b) A straight line 2.2 $300 2 2.3 (a) t3 + 23 t + 1 (b) 2n + 1 2.4 2n+1 − 2t+1 2.5 (a) $6 (b) $3 2.6 25% 2.7 $3,000 2.8 $666.67 2.9 A(t) is decreasing for t > 10. 1 2.10 125 [225 − (t − 10)2 ] 2.11 $10,250 711

712

ANSWER KEY

2.15 n = c − a − b 2.20 0.09709 2.22 $6,589.19

Section 3

0 2x 2n (c) x2 −x+1 < 0 for x ≥ 1. 3.1 (a) 200% (b) n2 −n+1 3.2 i1 = 3%, i2 = 1.21% 3.3 (a) 4.2% (b) 3.45% 3.4 4.5% 3.5 i1 = 4% and i2 = 5.77% 3.6 4% 3.7 (a) 2,700 (b) 1.5 3.8 (a) 967.74 (b) 14.33 3.9 Either alternative is equivalent as net interest rate is the same at 12.50% 3.10 5.33% 3.11 4.84% 3.12 (a) 12.04% (b) −15.97% 3.13 (a) $2.12 (b) 12.013% 3.14 (a) $100 (b) $125 (c) $5 3.15 $16,817.93 3.16 22.5171 3.17 Bank B offers higher effective interest rate than bank A 3.18 4.06% 3.19 161.051 3.20 146.41 3.22 133.06

Section 4 4.1 $145 (b) $9 4.2 9.2% 4.3 3.3 4.4 $238.10 4.5 0.8% 4.6 $582.50 4.7 17.4 years 4.8 4.8387% 4.9 10.6667% 4.10 3.75%

713 4.11 4.13 4.14 4.15 4.16 4.17 4.18 4.19 4.20 4.21 4.22 4.23

$15,748.03 $10,800 16 0.47393365% If i is a simple interest rate, then i5 = i10 implies i = 0. (a) 107.35% (b) 67.59% (c) 451 days 5% 6  1+it k 1+is (a) $2,700 (b) $1.50 (a) $967.74 (b) 14.33 $2,420

Section 5 5.1 (a) $37.26 (b) $36.89 (c) $37.78 5.2 (a) $100.27 (b) $100 (c) $101.67 t t t 5.3 360 > 365 > 366 t1 t2 5.4 (a) 360 ≥ 360 since Banker’s rule counts the exact number of days. The inequality is reversed for the month of February. (b) Count the number of days using each method between January 31 and February 28. 5.5 (a) $71.58 (b) $71.18 (c) $72.57. 5.6 (a) $ 86.30 (b) $ 86.07 (c) $ 87.50 (d) $ 87.50. 5.7 Amount in Fund A < Amount in Fund C < Amount in Fund B. 5.8 $ 61.70 5.9 52.14% 5.10 (a) 1,086.03 (b) 1,086.11 (c) 1,087.22 72 5.11 73 5.12 $40.02 5.14 5.0% 5.15 12.371% 5.16 12.245%

Section 6 6.1 6.2 6.3 6.4 6.5

$7,163.39 17.67 years $402.63 4.49% 23.79 years

714

ANSWER KEY

6.6 1.523 i−j 6.7 r = 1+j 6.8 $1,184.05 6.9 5.40 years 6.10 38.88 6.11 $6,442.83 6.12 $10,937.50 6.13 21.74% 6.14 $161.05 6.15 7,569.06 6.16 979.93 6.17 2.325 years 6.18 a(t) = a(0) = a(1) = Period during which a(t) is greater in = 6.19 6.20 6.21 6.22 6.23 6.25 6.26 6.27

Simple interest Compound interest 1 + it (1 + i)t 1 1 1+i 1+i 0
t>1

i 1+i(n−1)

i

4.29711% (a) 1.0124 (b) 0.9938 $42.51 $7,693.49 $1,994.38 (a) 10% (b) Compound interest (c) $14.64 11 $24.05

Section 7 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 7.9

5.5332 years $2,540.15 (a) $15,748.03 (b) $15,386.99 $3281.25 √ 1 (3 + 5) 2 $12,830.49 1,414.21 $ 68,725.29 $1,500

715 7.10 7.11 7.12 7.13 7.14 7.15 7.16 7.18 7.19 7.20

$666.67 $43,372.84 (E) i = 1−ν . ν (a) $8.40 (b) $8.47. $98.62 (a) $1,111.11 (b) $1,069.48 i = 6.38% and ν = 94% $693.11 $231.92

Section 8

8.1 $2800 8.2 (a) 753.57 (b) 9.89% 8.3 $4500 8.5 $868.78 8.6 19263.17 8.7 $1,856.61 8.8 $8,500.00 8.9 $5,150.21 8.10 20% 8.11 9.52% 8.12 $6446.53 8.13 38.88 8.14 0.1071 8.15 0.2308 8.16 $2,925.06 8.17 4% 8.18 0.0905 8.19 $700 8.20 (a) $4552.88 (b) $5491.03 (c) 13.18% 8.21 245 days 8.23 $3876.05 8.24 120 121

Section 9

9.1 $2039.89 9.2 0.15378 9.3 (a) 3.9% (b) 3.83% (c) $3.98  (d) 3.98% (e) 3.91% 1   − (2) 3 9.4 i(6) = 6 1 − d 2 −1 9.5 8.003343431 9.6 20 9.7 ν = r−4 9.8 7.83% 9.9 9.4588% 9.10 18.6069% 9.11 8% and 8.2432% 9.12 14.0579%, $ 1072.38, $72.38

716 9.13 9.14 9.15 9.16 9.17 9.18 9.19 9.20 9.21 9.22 9.23 9.24 9.25 9.26 9.27 9.28 9.29 9.30

ANSWER KEY 13.5% 5.926% (a) 16% (b) 17.74% 13.578% $711.78 0.494% 0.517% 0.51% $604.62 $358.98 12% 13.17% (a) $4225.27 (b) $4225.46 5.48% $1,540.34 559.45 9.08% The first option

Section 10

10.1 $674.93 10.2 6.184%  b 10.3 tt21 ea(t2 −t1 ) 10.4 $142.26 10.5 2.28% 10.6 3.96% 10.7 0.23% 10.8 102.07 10.9 1.58 10.10 (a) a(t) = e0.09t (b) 0.09417 (c) $2619.93 10.11 784.59 10.12 (a) 10.0418% (b) 10.1260% (c) 10.5171% 10.13 $1174.62 δ δ δ d (d(p) ) = e− p (e p − 1 − pδ ) > 0 10.14 d(p) = p(1 − e− p ) and dp 10.15 $1952.75 10.16 11.82% 10.17 $11,639.93 10.18 9.966% 10.19 6.184% 10.20 5 10.21 6.83% 10.22 8.46% 10.23 $367.88 10.24 4.138% 10.25 0.501% 10.26 5.00% 10.27 2.00% 10.28 8.97% 10.29 4.04%

717 10.30 10.31 10.33 10.34 10.35 10.36 10.37 10.39

1,625.45 d t = ln δ−ln δ (I) and (III) 27% −ν 1.2 0.047 δt = dtd (ln [A(t)]) = ln 2 + 2t ln 3 + 2t ln 5 ln 2

Section 11

11.1 7.531% 11.2 1 + n 11.3 1.03873123 11.4 1.005% 11.5 21.8864 11.6 $606.77 11.7 $598.46 11.8 $120 11.9 $1,276.30 11.10 0.81669 11.11 1.35 11.12 9.479% 11.13 (a)7% (b) 7.71% (c) 7.53% 11.14 $1,046.03 11.15 $535.26 11.16 0.0413796 11.17 $4,333.33 11.18 $2,475.09 11.19 n2 11.20 200,000 11.21 24,498.78 11.22 0.08 11.23 5.12% 11.24 6.78% 11.25 8.0% 11.26 17936.13 11.27 7.77% 11.28 29.52% 11.29 1.25

Section 12

12.1 $5,159.24 12.2 $917.76 12.3 $483.11 12.4 $568.15 12.5 690.30 12.6 $2,440.68 12.7 $2,273.79 12.8 $2,859.41 12.9 $5,737.65 12.10 −$374.91

718 12.11 12.12 12.13 12.14 12.15 12.16 12.17 12.18 12.19 12.20 12.21 12.22 12.23

ANSWER KEY $3,635.67 $1,555.79 4.53% X(ν 17 + ν 15 + ν 12 ) + Y (ν 20 + ν 18 + ν 15 ) 0.08 657.84 1500 5% 400 1062.93 2504.12 12/85 406.92

Section 13

13.1 7.46% 13.2 11.96% 13.3 4% 13.4 3.5% 13.5 10% 13.6 2.81% 13.7 9.46% 13.8 12.91% 13.9 3.389% 13.10 14.8% 13.11 $17936.13 13.12 n2 13.13 1.5% 13.14 0.0139 13.15 1.054092553 13.16 2.4% 13.17 (a) 7% (b)7.71% (c) 7.53% 13.18 0.02524 13.19 700% 13.20 0.0443 13.21 5.49% 13.22 4.53%

Section 14

14.1 4.8 years 14.2 6.25267 years 114 14.3 100i 14.4 2.933 years 14.5 36.39 years 14.6 14.26 years 14.7 (a) 7.174 years (b) 6.917 years 14.8 n(2n+1) 3 14.9 1.433 years 14.10 3 years 14.11 6 years 14.12 33.67 years

719 14.13 14.14 14.15 14.17 14.18 14.19 14.20 14.21 14.22

14 years 6 years 6.91 years One year 26 13.436 15 3.03 1.8453

Section 15

15.1 $216.31 15.2 $270,833.34 15.3 8.73316 15.4 2945 15.5 x22x−y +2x−y 15.6 (a) 8.3% (b) 29.52 15.8 $8,316.61 15.9 $11,731.39 15.10 $6,808.51 15.11 (a) $1629.90 (b) $1350 (c) $834.81 15.12 $385.18 15.13 $162,067.01 15.14 $652 n  n 15.15 n2 1 − n+1 15.16 $1,489.36 15.17 Option (a) 15.18 $860.66 15.19 $400.40 15.20 $8747.64 15.21 $1,268.96 15.22 $1,600 15.23 3.095 15.24 (e) 15.25 12.25% 15.26 10.01% 15.27 $2,792.38 15.28 1.079.68 15.29 19,788.47 1000 15.30 k = a +ν 5 a +ν 10 a 15 20 15 10 +ν a5 15.31 (D) 15.32 0.0018 15.33 i−1 [s41 − s15 − 26] 15.34 1310.25 5 1−(1−ki) 15.35 (1−ki) −0.5 −1 15.36 12527.18 15.37 3140 15.38 10%

720

ANSWER KEY

15.39 15.40 15.41 15.43 15.44 15.47 15.48 15.50

4956.06 678.60 2022.29 5.5348 $10,571.40 (1+i)k −1 − k = 2!1 k(k − 1)i + 3!1 k(k − 1)(k − 2)i2 + · · · i (a) (1 + i)k − 1 (a) 4.5194 (b) 4.5230 (c) 4.5160

Section 16

16.1 $147.35 16.2 $ 99.74 16.3 $633.59 16.4 $431.21 16.5 $20,497.75 16.6 $331.08 16.7 16.1% 16.8 (a) $4,055.45 (b) $ 4,217.67 (c) $6,003.05 (d) $6,243.18 16.9 $258.28 16.10 $8.92 16.11 (i) and (iii) 16.12 $2,255.41 16.13 $8,101.65 16.14 s¨n − a ¨n 16.16 $900 16.17 $3,096.92 16.18 479.17 16.19 $53,839.83 16.20 (c) 16.21 $2,514.76 16.22 9,873.21 16.23 log2 (n + 1) 16.24 1600 16.25 Let f (i) = sn · a ¨n . Then f (0) = n2 and f 0 (i) > 0. 16.26 324.73 16.27 75.6 16.28 58016.73 16.29 5100

Section 17 17.1 17.2 17.3 17.4 17.5 17.6 17.7 17.8 17.9

$1,670.73 (v) $3608.32 $14,271.17 $1,272.04 $1,272.04 1.8 (a) $3,256.88 (b) $5,403.15 (c) $6,959.37 $16,178

721 17.10 17.11 17.12 17.13 17.15 17.16 17.17 17.18 17.19 17.20 17.21 17.22 17.23 17.24

$311.77 (i) and (ii) $124.72 7% x = z = 4 and y = 7 a45 6,000 574.725 108.8793 439 135.6 5 $6,230,292.14 $406.21

Section 18

18.1 $462.50 18.2 All three 18.3 500 18.4 $1,614.46 18.5 36% of the original perpetuity ) 18.6 1 − ln (iP δ 18.7 1000(1 + i)10 [(1 + i)20 − 1] 18.8 4 18.9 30 49 18.10 12.8% 18.11 1,000 18.12 8238.52 18.13 Allen’s share < Chris’s share < Julie’s share 18.14 20% 18.15 3.5265% 18.16 (a) 18.17 (II) 18.18 3.709875 18.19 $249.43 18.20 0.04 18.21 151.94 i+3 18.22 i(i+2) 18.23 955.03

Section 19

19.1 17 19.2 (a) 19 (b) $ 41.97 19.3 30.2 years 19.4 $1,124.39 19.5 $1,665.35 19.6 21 and $146.07 19.7 9 and $32.41 19.8 9 years 19.9 14 19.10 29

722

ANSWER KEY

19.11 19.12 19.13 19.14 19.15 19.16 19.17 19.18

8 January 1, 2016. 150.87 149.54 213.48 439.08 26 April 30, 2013

Section 20

20.1 (a) 9.97% (b) 4.23% 20.2 12.25% 20.3 10.016% 20.4 8.91% 20.5 25% 20.6 11.11% 20.7 25% 20.8 7.5% 20.9 12% 20.10 4.10% 20.11 3.984% 20.12 12.82% 20.13 8.688% 20.14 8.333% 20.15 0.7143 20.16 6,194.72 20.17 4.53% 20.18 32.6 20.19 6.67% 20.21 i = 2.2853% 20.22 0.0620 20.23 7.38% 20.24 0.858 20.25 51.2%

Section 21

21.1 $12,514.23 21.2 $1567.72 21.3 $12,203.74 21.4 $15963.86 21.5 $13.30 21.6 (a) $740.12 in Fund A and $776.40 in Fund B (b) Heather will have more money and will have about $87.58 more than Lisa. 1

21.7

P (1+i) 2 1+2a4 +2a5 (1+i)−4 i

j

21.8 8.145 21.9 8.080 21.10 14.5 21.11 $3216.95 21.12 $3520.87

Section 22

22.1 $45,582.96

723 22.2 $1,835.43 22.3 $15,914.19 22.4 $41,470.22 22.5 $6,716.79 22.6 $103,777.91 22.7 $2,998.86 22.8 $8,230.18 22.9 $3,5824 22.10 $11,466 22.11 1% 22.12 2,359.39 22.13 2,300.27 22.14 2,305.90 22.15 $54498.57 22.16 $907,009.42 22.17 $25,360.97 22.18 $610.16

Section 23

23.1 167,460.79 −a a 23.2 (a) 200 · 176s 32 (b) 200 · 4

120

a −a 180 36 a 4

23.3 300 1−(0.9925) 1−(0.9925)6 23.4 11,451.58 23.5 $285.83 a a 23.6 (a) (1 + i)−4 s3r (b) (1 + i)−7 a3r 3 3 23.7 1196.33 23.8 10.15 months 23.9 $908.12 23.10 18 payments with final payment of $110.09 h i a36 a15 23.12 X = Y a a − 1 45 h9 i s 1 v 40 23.13 R = 37, 500 s + a · a2 . 2  40 60 P 2 10 23.14 a4 = 4k=1 10+k . 23.15 8% 23.16 652.23 23.17 2.42 23.18 5 23.19 $8230.18

Section 24 24.1 24.2 24.3 24.4 24.5

$10 $2,050 $32,494.69 $1,144.82 $14,621.67

724 24.6 $18,616.53 24.7 All three 24.8 3.33% 1 2 ]) 24.9 ln (20[1−(1−d) ln (1−d) 24.11 $11,466 24.13 $3481.91 24.14 $907,009.42 24.15 127 months

Section 25

25.1 (a) $6,866.52 (b) $6,846.27 25.2 6.93% 25.3 61 25.4 27.0367 years 25.5 $61.85 25.6 0.995839 25.7 3.96% 25.8 40 25.9 $433.75 25.12 1 − 1δ ln δi 25.13 $679.67 25.14 $1419.76 25.15 8291.60 25.16 1224.47 25.17 11.65 25.18 n − 100δ 25.19 27 25.20 (I) 25.21 12,818.13

Section 26

26.1 $14,784.96 26.2 50a20 + 5(Ia)20 26.3 5(Is)20 26.4 $2,100.00 26.5 $8,394.60 26.6 (a) $4,621.75 (b) $5,053.90 26.7 $6,000 26.8 $139,734.76 26.9 $1,504.63 26.10 66 26.11 4.762% q 2q 26.12 (a) p−q (b) p−q 26.13 $906.71 26.14 $10,669.98 26.15 $38.25 a 26.16 dn 26.17 an · a ¨n

ANSWER KEY

725 26.18 26.19 26.20 26.21 26.22 26.23 26.24 26.25 26.26 26.27 26.28 26.29 26.30 26.31 26.32 26.33 26.34 26.35 26.36 26.37 26.38 26.39 26.40 26.41 26.42 26.43 26.44 26.45 26.46 26.47 26.48 26.49 26.50 26.51 26.52 26.53 26.54 26.55 26.56 26.57 26.58 26.59 26.60 26.61 26.62 26.64

30 $2,729.26 6250 − 325A 19 54 129.54 4.0 8.7% 116 15,922.96 1210.49 220.18 3.5% 122,633.60 10.17% 39.45 584.29 (Ia)10 + 10ν 10 a15 ν 4 (Da)10 $16,606.71 $7121.26 6761.90 11.1% 5504.76 20 78.41 4.88% n an − nν 3.586 2084.67 a 3 a2

14.20 a20 (1 + ν 2 ) 4.76% 44 1079.68 46 348.64 20,000 15% 38.86 $0.12 (b) $119.44 200,094.77 119.58 290.72 (Ia)10 + a10

Section 27

27.1 1825.596 27.2 18333.33 27.3The geoemtric sum is divergent. 27.4 1050 27.5 $8,586.96 27.6 785.40

726

ANSWER KEY

27.7 23.32 27.8 $2,851.99 27.9 1.91 27.10 (a) $8831.22 (b) $22,788.32 27.11 $77,284.41 27.12 8.36 27.13 2601.90 27.14 1308.53 27.15 186.0105 27.16 $117.18 27.17 398.43

Section 28

28.1 (a) 3 (b) 25 12 28.2 i(m)1·d(m) = m(i(m)1−d(m) ) 28.3 $45561.82 28.4 (a) $45,094.57 (b) $44,518.37 28.5 $2252.56 28.6 $13,309.47 28.8 48 28.9 112.59 28.10 $40,052 28.11 251.97 28.12 $5,549.27 28.14 34,687.05 28.15 3484.07 28.16 15,141.02 28.17 8279.61 28.18 147929 28.19 374367.44 ¨20 0.04 28.20 2000a12 j a

Section 29

29.1 $216.74 29.2 $160.84 29.3 $156.25 1 29.4 δi −δ k 29.5 $16,020 29.6  84.5  a3 2a − b + (b − a) ln 29.7 (2a−b) 2 29.8 $60 29.9 $217.69 29.10 $62.13 29.11 33.53 29.12 3 29.13 111.11 29.15 (1 + n)2 29.16 48,688.70 29.17 e5 − 1

Section 30

b−a a



727 30.1 −$49, 667.53 30.2 3 30.3 −$57.85 30.4 (a) −$7214.08 (b) 13.52% 30.5 0% 30.6 −3.17288% 30.7 12.4% 30.8 $43,734.71 30.9 7.05622% 30.10 N P V (0.08) = −2699.67 < 0 so reject project. 30.11 $4,896.52 30.12 1.1101% 30.13 Since N P V (13%) > 0, the project is a worthwile investment. 30.14 $20,205.63 30.15 −$13, 761.14 30.16 $6,960.65 30.17 12.96% 30.18 (a) Cannot tell unless the rate for the second 6 years of project A is known (b) 6.04% 30.19 5,460 30.20 8881.52 30.21 −39.71%

Section 31

31.1 5 31.2 2% 31.3 0 < X < 100 31.4 IRR is imaginary 31.5 8.7% 31.6 3 31.7 3.3038% 31.8 Follows from Theorem 31.1(i) with k = 1 31.9 4% or 5% 31.10 17.87% 31.11 213.35% or 27.65% 31.12 27.65% < i < 213.35% 31.1 $3125.96

Section 32

32.1 $140,785.64 32.2 $172,432.81 32.3 $21,374.10 32.4 $2924.73 32.5 $3,416.80 32.6 $18,774.42 32.7 $6200.52 32.8 10.02% 32.10 6.1619905% 32.11 $4,448.42

728

ANSWER KEY

32.13 32.14 32.15 32.16 32.17 32.18 32.19 32.20 32.21 32.22 32.23 32.24 32.25 32.26 32.27 32.28 32.29 32.30 32.31 32.32

8,438.71 10% 8.8% 3 $6,338.02 9041.49 698.79 541.47 123.56 10% 2.03 5393.53 9.6% 4485.49 4.6% 7.95% 7316.72 12.78% 4585.09 373.43

Section 33

33.1 (I) and (III) 33.2 9.78% 33.3 $943 33.4 6% 33.5 11% 33.6 8% 33.7 18.57% 33.8 6.175% 33.9 9.52% 33.10 September 1 33.11 September 1 33.12 $6006.36 33.13 (a) 4.87% (b) 4.87% (c) 5.26% 33.14 3.297% 33.15 15.65% 33.16 18.39% 33.17 (a) 4.18% (b) 4.2% 33.18 28.1% 33.19 2.6906% 33.20 28.85% 33.21 1371.4216

Section 34 34.1 34.2 34.3 34.4 34.5 34.6 34.7

(I) and (IV) (a) 6.7% (b) 6.67% (c) 14.54% 0% 18.39% 14.04% −25% $236.25

729 34.8 (a) 10.62% (b) 18.79% 34.9 (a) yes (b) no 34.10 (a) yes (b) The time-weighted method cannot be used here since the balance in the fund is not known on July 1 34.11 (a) 22.88% (b) 29.73% 34.12 93,000 34.13 −3.54% 34.14 (a) 6.524% (b) 9.54% C (b) iD = 2(C−A−D) , iT = B · B+D − 1 (c) 34.15 (a) iD = iT = C−A A 2A+D A 2(C−A−D) B−D C iD = 2A+D , iT = A · B − 1 (d) they are equal (e) the rate in (b) is larger than the rate in (c) 34.16 −0.6818% 34.17 0.022 34.18 0.0622 34.19 107.63 34.20 15% 34.21 T W > IRR > T D

Section 35

35.1 9.0%, 9.5%, 9.7%, 9.8%, 9.7%, and 9.35% 35.2 (a) 112.47 (b) 112.25 (c) 112.78 35.3 $1,976.88 35.4 $1,421.73 35.5 $1,142.24 35.6 $3,188.85 35.7 $2.67, $123.25 35.8 $ 16.92 35.9 1.7%, 2.1%, 2.5%, 5.5% 35.10 3.7%, 3.2%, 2.7%, 2.2% and 1.7% 35.11 5% 35.12 9.47% 35.13 (a) 1,222.98 (b) 3,430.34 35.14 7.749 35.15 6.61 35.16 267.49 35.17 6.0

Section 36

36.1 Rejected in both cases 36.2 (a) N P V (0.09) = 75.05 and N P V (0.10) = −57.85 (b) Reject 36.3 Pay cash 36.4 Net present value method 36.5 No yield rates 36.6 Neither one is ideal for the borrower. However, option (ii) is slightly preferable

Section 37

37.1 $8,863.25

730

ANSWER KEY

37.2 $8,876.56 37.3 $9,409.16 37.4 $87,724.70 37.5 $69,430.10 37.6 $17,653.36 37.7 $63,573.59 37.8 $45,435.32 37.9 $15,000 37.10 $4,917.72 37.11 $16,514.37 37.12 (a) 3000a3 + 2000v 3 a5 (b) L(1 + i)7 − 4000s5 (1 + i)2 − 3000s2 , where L is the original amount 37.13 $10,814.16 37.14 $17,142.86 37.15 $97.44 37.16 $6,889.11 37.17 $37,174.63 37.18 16691.05 37.19 461.13 37.20 635.3157 2 37.21 20,000(1+i) a15 a13 a20 37.23 1510.60 37.24 5736.10

Section 38

38.1 10,000 38.2 6,500 38.3 68.06 38.4 80 38.5 8.1442% 38.6 16,105.10 38.7 $11,820.91 38.8 8.23% 38.9 51st payment 38.10 $20,821.21 38.12 641.86 38.13 an − nν n+1 38.15 13th installment 38.16 724.59 38.17 (a) i[a6 i + νi6 a10 j ] (b) νj6 38.18 1751.81 38.19 443.84 38.20 0 38.21 585.04 38.22 754.92n−t 38.23 1 + ν d 38.24 72 38.25 632.20 38.26 7077.03 38.27 609.74

731 38.28 38.29 38.30 38.31 38.32 38.33 38.34 38.35 38.36 38.37

5.0864% 22.6896 83.61 (a) 7303.89 (b) 5740.78 (c) 1563.11 (d) 661.33 479.74 704.05 341.77 825.23 216 1344.89

Section 39

39.1 (a) 1000 (b) 690.29 (c) 4049.68 39.2 10.89% 39.3 2050 and 3388.80 39.4 (a) 163.30 and 274.18 (b) 9040.94 39.5 58.68 39.6 6.78% 39.7 58,933.79 39.8 10 years 39.9 76,713.64 39.10 676.43 39.11 7610.4798 39.12 2221.41 39.13 229.87 39.14 7% 39.15 4057.43 39.16 8.696% 39.17 330.34 39.18 14.18% 39.19 15902.95 39.20 6.5% 39.21 204.41 39.22 684.30 39.23 35.85 39.24 6.9% 39.25 5238.63 39.26 122.29 39.27 a) 4.04% (b) 5920.36 39.28 12.12% 39.30 14,552.79 39.31 5%

Section 40 40.1 40.2 40.3 40.4 40.5

$32987.69 $6183.70 $1344.89 $14522.80 $963.1919

Section 41

41.1 -$448.07. This means that the outstanding balance after three payments is $10448.07. A negative principal means that the interest is larger than the payment

732

ANSWER KEY

41.2 10472.28 41.3 37779.22 41.4 8338.85 41.5 97.44 41.6 4.88 41.7 9794.50 41.9 9191 41.12 5 − a5 41.13 4% 41.14 6889.11 41.15 −180.27 41.16 26275.7718 41.17 275.19 41.18 109.17 41.19 1,100 41.20 4269.8376 41.21 10,857.27 41.22 13 41.23 (a) 1287.76 (b) 276.24

Section 42 42.1 42.2 42.3 42.4 42.5 42.6 42.7 42.8 42.9

7.18% 9.38% 9,810.42 (a) 7.91% (b) 8.51% $9797.78 9 years 12,311.48 12,429.62 $1056.62

Section 43

43.1 $919.15 43.2 (a) 8.4% (b) 8% (c) 9.14% (d) 10% 43.3 −7.72% 43.4 $794.83 43.5 $945 43.6 $1200 43.7 $1291.27 43.8 $1100 43.9 $12,464.76 43.10 $9319.06 43.11 All three 43.12 the currebt yield is 5.26% and the nominal annual yield to maturity is 5.88% 43.13 $250 43.14 $106 43.15 $1167.04 43.16 14.16% 43.17 $100.66 43.18 9.19% 43.19 $1055.4592 43.20 9.746%

733 43.21 43.22 43.23 43.24 43.25 43.26 43.27 43.28 43.29 43.30 43.31 43.32 43.33 43.34

$902 9 8.7% 5P − 4Q 1055.09 1,497.43 504,568.85 846.41 1070.89 1842.29 800.1573 $1011.26 50.00 502.40

Section 44

44.3 1 − 0.5p 44.4 $20 44.5 33.98 44.6 573.60 44.8 15.18 44.9 14,877.47 44.10 −7.4518% 44.11 8.71 44.12 13 years 44.13 48,739.29 44.14 641.58 44.15 2.08 44.16 Write-up in the value of 0.77 44.17 819.4081 44.18 1435.11 44.19 1181.8878 44.20 72 44.21 685.87 44.22 7.18% 44.23 9.5% 44.24 2.00% 44.25 7.0018% 44.26 1122.38 44.27 652

Section 45

45.1 12.30% convertible semi-annually 45.2 1 45.3 (a) 94,031.03 (b) 95,902.37 (c) 1,658.44 (d) 94,243.93 (e) B f4 = 95, 911.65, F r 4 = 6

= 94, 244.98 (f) B f4 = 95, 902.37, F r 4 = 1, 666.67, B m = 1, 666, 67, B m 4 4 6 6 6 6 94, 235.70 45.4 (II) only 45.5 200 45.6 13.16 45.7 B f3 = 118.19, F r 3 = 2.50, B m 3 = 115.69 6

6

6

45.9 (a) practical > theoretical = semi-theoretical (b) market price for prac-

6

734

ANSWER KEY

tical and theoretical are larger than the market price for semi-theoretical. No conclusion can be deduced between pratical and theoretical. 45.10 9448.81 45.11 5563.8234 45.12 $6.19 45.13 $208.16 45.14 919.15

Section 46

46.1 8.73% 46.2 8.042% 46.3 9.13% 46.4 8.4% 46.5 4.63% convertible semi-annually 46.6 6.4828% 46.7 4.00% 46.8 9.2% 46.9 7.42% (exact) and 7.37% (salesman’s) 46.10 (a) $913.54 (b) $1074.39 (c) 10.4340%

Section 47

47.1 (a) 1148.77 (b) 846.28 47.2 1085.91 47.3 922.05 47.4 58.65 47.5 (a) 902.88 (b) 1081.11 47.6 (a) 902.88 (b) 1111.18 47.7 1391.99 47.8 1440.01 47.9 3.2% 47.10 1200.07 47.11 4.912% 47.12 25 47.13 9.24% 47.14 (I) and (III) 47.15 (a) 5423.39 (b) 4527.29 (c) 4527.29 (d) 5528.30 47.16 857.11

Section 48

48.1 17.14 48.2 2160 48.3 33.81 48.4 15.71% 48.5 42 48.6 60.44 48.7 75 48.8 (1) 40 (b) 66.67 (c) 200 48.9 1% 48.10 110.81 48.11 38.99 48.12 11%

Section 49

49.1 (a) $550.55 (b) $516.98 (c) $33.57

735 49.2 49.3 49.4 49.5 49.6 49.7 49.8 49.9

$10.35; $11.00; $32.50 $71,428.57 (a) $10,000 (b) $4,400 (c) −6.7% (d) 0% (a) $7,000 (b) $4,400 (c) −56.67% (d) −30% $2250 47.47% No (a) $30,580 (b) $89.36 (c) 58.64% (d) 14.90% (e) 8.27%

Section 50

50.1 38% 50.2 35% 50.3 8% 50.4 45.6% 50.5 987.5 50.6 5% 50.7 465 50.8 90 50.9 10.0% 50.10 20% 50.11 20% 50.12 23,300 50.13 3.375 50.14 6% 50.15 16% 50.16 44 50.17 All three are correct

Section 51

51.1 $504.67 51.2 $10,500 51.3 5.48% 51.4 $112,486.40 51.5 15.65% 51.6 7.54717% 51.7 (a) 12.04% (b) −15.97% 51.8 $87,724.70 51.9 $63,573.59 51.10 $45,435.32 51.11 $37,174.63 2 51.12 20,000(1+i) a15 a a 13 20

Section 52 52.1 52.2 52.3 52.4 52.5 52.6 52.7 52.8

206,225.56 1 2.86% 13,253.93 17887.09 12.9% 122,633.60 X = (1.063)−6 50a12 i0 = 306.47.

736

ANSWER KEY

Section 53

53.1 Only (I) 53.2 442.02 53.3 f35 = 8.63% 53.4 324.12 53.5 (a) 43.49 (b) 4.84% 53.6 (a) 45.69 (b) 4.72% 53.7 (a) 0.007 (b) 5.5% 53.8 11.03% 53.9 (a) $926.03 (b) 8.9% 53.10 13,152.50 53.11 4.745% 53.12 10.509% 53.13 (a) 9.64% (b) 10.51% 53.14 (a) 1036.53 (b) 7.953% 53.15 5.9% 53.16 (a) 9% (b) 6.5% 53.17 6.00% 53.18 All except (III) 53.19 1.85%

Section 54

54.1 10.9001 54.2 5 54.3 All except (II) 54.4 All true except (I) 54.5 (a) 5.92 (b) 5.64 54.6 20 54.7 1 54.8 3.584 54.9 2.70 54.10 2.29 and 2.12 54.11 30 and 27.78 54.12 4.56 54.13 35 54.14 11 54.15 7.56 54.16 5.989 54.17 16.773 54.18 208.35 54.19 16 years

Section 55 55.1 55.2 55.3 55.4 55.5 55.6

All except (I) x1 = 11.88 and x6 = 18.17 x = 500 and y = 500 2572.02 in money market and 1714.68 in bonds ν = 1.85 and c = 10.29 (a) 0 (b) 94.31 (c) 312.50

737 55.7 (n+1)(n+2) 3 55.8 1250 55.10 x0 = 201.85 and x6 = 602.06

Section 56 56.1 56.2 56.3 56.4 56.5 56.6 56.7

(a) yes (b) $278,528.37 (c) $19,113.02 2503.47 18594.03 0.93809 of the first bond and 0.97561 of the second bond 1904.27 6.78% 0.758

Section 57

57.1 (II) 57.2 (III) 57.3 See notes 57.4 (I) 57.5 (I) 57.6 (II) 57.7 The value of the reimbursement plan is derived from the grade you earn 57.9 (I) Speculator (II) Hedger (III) Arbitrageur

Section 58

58.2 Diversifiable and non-diversifiable. 58.3 (I) 58.4 (a) and (c) are diversifiable risks; (b) and (d) are non-diversifiable risks 58.5 (a) and (b) are diversifiable risks; (c) and (d) are non-diversifiable risks 58.6 (c) 58.7 non-diversifiable (b),(c),(f),(g),(e); diversifiable: (a) and (d) 58.8 True 58.9 (D) 58.10 (D) 58.11 (B)

Section 59

59.3 (a) A profit of £7,150 (b) A loss of £18,465 59.4 6% 59.5 9.89% 59.6 −10; −5; 0; 5 ; 10 59.7 10; 5; 0; −5 ; −10 59.8 Payoff and profit diagrams coincide. 59.9 $63.50 59.10 (a) $20(long) and −$20(short); (b) −$60(long) and $60(short) 59.11 Buy a foreign exchange forward contract 59.12 A gain of $10 59.13 $25; no profit 59.14 (a) $1000 (b) −$2000 59.15 (D)

738

ANSWER KEY

59.16 (A)

Section 60

60.1 (A) 60.4 (a) $80 (b) buyer walks away 60.5 (a) a loss of $80 (b) $0 60.6 (a) $95.68 (b) $4.32 (c) −$95.68 60.7 A vertical shift by the future value of the premium 60.8 0; 0; 0; 5; 10 60.9 (a) Any date on or before the experition date (b) December 17,2005 (c) not worthless 60.10 $0; (b) $5 60.11 (a) $100, (b) Do not exercise. The option is thus expired without exercise. 60.13 0 60.14 written call option 60.16 $0 60.17 One strategy is to buy 200 shares; a second strategy is to buy 200 call options 60.18 (A) 60.19 (B) 60.20 (c) 60.22 PT > $88 (for exercising the option) and PT > 91.50 for making profit 60.23 (C)

Section 61

61.1 Buy a put option with strike price $25 61.2 (a) $0 (b) $100 61.3 (a) $75.68 (b) −$75.68 (c) $24.32 61.4 (a) $0 (b) $15 61.5 (a) 61.6 Written put option 61.7 Buy 5000 put options with a strike price of $30 and an expiration date in 4 months. 61.8 A long position one-year put option 61.9 Long call option + short put option = long forward. 61.12 (a) At-the-money (b) In-the-money (c) Out-of-the-money (d) Out-ofthe-money (e) In-the-money 61.14 Buy a call option 61.16 If the price of the underlying asset at expiry is above $26 61.17

739

Derivative Position Long Forward Short Forward Long Call Short Call Long Put Short Put

61.18 61.19 61.20 61.21 61.22 61.23 61.24

Maximum Loss −Forward Price Unlimited −FV(premium) unlimited −FV(premium) FV(premium) - Strike Price

Maximum Gain Unlimited Forward Price Unlimited FV(premium) Strike price - FV(premium) FV(premium)

Position with Respect to Underlying Asset Long(buy) Short(sell) Long(buy) Short(sell) Short(sell) Long(buy)

Strategy Guaranteed price Guaranteed price Insures against high price Sells insurance against high price Insures against low price Sells insurance against low price

C D D C A (A) (D)

Section 62 62.1 $40 62.2 A call option on XYZ stock with a strike price of 25 that expires on the Saturday following the third Friday of April. 62.3 Put options 62.4 (a) Stock price greater than $50 on expiration date (b) below $54 on the expiration date. 62.5 S − D 62.6 Right before the dividend payment date 62.7 $32 62.8 (a) You take your old shares, and give those to the broker. (b) Yes 62.9 C 62.10 B 62.11 D 62.12 (D)

Section 63 63.1 C 63.2 (B) 63.3 (C)

Section 64 64.1 (A) 64.2 (D)

740 64.3 64.4 64.5 64.6 64.7 64.8 64.9

ANSWER KEY (B) (B) (C) B (D) (a) $7500 (b) $1000 (c) $1250 (d) $500 (a) $150 (b) $5.75

Section 65 65.1 (C) 65.2 (A) 65.3 (B) 65.7$480 65.8 $1020 65.9 2% 65.10 Buy a call, sell a put, and buy the zero-coupon bond. 65.11 $2.52 65.12 $43.85 65.13 9.97% 65.14 (E)

Section 66 66.2 (a) $1.85 66.3 (a) −$3.15 66.4 (a) $1.58 (b) −$3.42 66.5 False 66.7 (a) $3.42 (b) −$1.58 66.8 (b) $18.43 (c) $20 66.10 (A)

Section 67 67.1 67.2 67.3 67.4 67.8 67.9

25 (A) (B) (D) Bear spread (D)

741

Section 68 68.1 (A) 68.6 (B) 68.7 (D) 68.8 (B) 68.9 (a) $3.95 (b) −$1.05 (c) 51.05 and 58.95 68.11 $6

Section 69 69.1 5.3846 European call options 69.2 40 cents 69.3 (b) 69.4 $13,850 69.5 (II), (III), and (IV) 69.7 $0 for a call option and the initial principal for an ELCD 69.8 $319,416.80 69.9 A combination of a zero-coupon bond with par value of $1,000,000 and expiration date in 5 years and a 2300 units of European call options with strike price of $500 and delivery date in 5 years

Section 70 70.1 F0,T > S0 Cash Flow t=0 t=T Buy Stock −S0 PT P Sell Prepaid Forward F0,T −PT P Total F0,T − S0 0 F0,T < S0 Cash Flow t=0 t=T Short Stock S0 −PT Buy Prepaid Forward −F0,T PT Total S0 − F0,T 0 70.3 $100 70.4 $1

742

ANSWER KEY

70.5 1.08285 shares 70.6 0.02 70.7 $46.15 70.8 $46.16 70.10 $41.75 70.11 (C)

Section 71 P P = $33.56, F0,1 = = $35, F0,1 = $37.16, 0.05988 (b) F0,1 71.1 (a) F0,1 P $35.64, 0.018 (c) F0,1 = $33.80, F0,1 = $35.90, 0.025 71.2 (a) 1.093 (b) $60.11 71.3 (a) 0.1333 (b) $62.84 71.4

Short one forward Borrow S0 e−δT Total

Cash Flows t=0 t=T 0 F0,T − PT −δT S0 e −S0 e(r−δ)T −δT S0 e −PT

71.5

Short tailed position in Stock Lend S0 e−δT Total

Cash Flows t=0 t=T −δT S0 e −PT −δT −S0 e S0 e(r−δ)T 0 S0 e(r−δ)T − PT

71.6 6.25% 71.7 (a) $1142.02 (b) Cash-and-carry strategy (c) Reverse cash-and-carry strategy 71.8 (a) $1129.26 (b) Cash-and-carry strategy (c) Reverse cash-and-carry strategy 71.9 (a)$7.15 (b) $12.85 71.10 (a) $3.38 (b) $6.62 71.11 1.5% 71.12 (a) Reverse cash-and-carry with profit of $2.72 (b) cash-and-carry with profit of $4.03 71.13 $1.7178 71.15 (a) $56.22 (b) $7800

743 71.16 F − = $841.02 and F + = $845.23 71.17 F − = $837.44 and F + = $848.82

Section 72 72.1 (A) 72.2 (B) 72.3 (C) 72.4 (B) 72.5 (D) 72.6 (B) 72.7 (C) 72.8 (C) 72.9 Marking to market 72.12 $50,000 72.13 After the third day, the margin account has a balance of $2837.50. 72.14 $562.50 which is less than the initial margin. A margin call will be issued. 72.15 (a) $1,200,000 (b) $120,000 72.16 (a) $2,375,000; $237,500 (b) $930.89

Section 73 73.1 (a)-(IV); (b)-(II); (c)-(I); (d)-(III) 73.2 (B) 73.3 (a) $696,493.5565 (b) $ 56.533 73.4 (a) $59.02 (b) $2020 (c) $2831.68 73.5 (a) $774.1535 (b) P1 (1.05)−1 + P2 (1.06)−2 = 750(1.05)−1 + 800(1.06)−2 (c) 100 ounces (d) $24.1535 (e) $25.8465 (f) 7.01% 73.6 $804.1535 73.7 (a) $797.59 (b) $824.04 73.8 Going long two forward contracts. 73.9 Going short two forward contracts.

Section 74 74.1 74.2 74.4 74.5 74.6

(A) (E) Firm ABC pays Firm XYZ the amount $25,000 5.3%; 5.32%; 5.33%; 5.35% 5.36%

744 74.7 $955 74.8 1.66% 74.9 1.66096% 74.11 5.867%

ANSWER KEY

Bibliography [1] S. Kellison , The Theory of Interest, 2nd Edition (1991), McGraw-Hill. [2] R.L. McDonald , Derivatives Markets, 2nd Edition (2006), Pearson. [3] SOA/CAS, Exam FM Sample Questions.

745