ebookg 38-6

6 7 8

6-6 A 6 B

130 6.1 6.1.1 10 W p 0. 6 W 1 15 1 p 0. 4 W 2 8 W 1 0 p 0. 6 1 p 0. 4 [ 1 ] W 1 1 5 W 2 8 E(W) E(W) p W 1 + ( 1 p)w 2 ( 0. 6 150 000)+(0.4 80 000) 122 000 10 2. 2 122 000 100 000 2 2 p[w 1 E(W) ] 2 ( 1 p) [W 2 E(W) ] 2 0.6(150 000 122 000) 2 0.4(80 000 122 000) 2 1 176 000 000 34 292.86 2. 2 5 10 5 000 p 0. 6 50 000 A. 10 1 p 0. 4 20 000 B. 5 000 2. 2 22 000 5 000 17 000 17 000 risk premium 1 [1] 6 8 A

6 131 6.1.2-1 1. 70 100 1. 70 10 1 1 1. 70 2 1 1. 70 p 0. 5 p 0. 5 p 0. 5 / / 2 6.1.3 risk averse 6 B

132 u t i l i t y - E(r) 2 U E(r) 0. 005A 2 ( 6-1 ) U A 0. 005 6-1 6-1 A A 6-1 R - I - S - K Gary Schatsky T. Rowe Price Associates Inc. Zurich Group Inc. s Scudder Kemper Investments Inc. 7 ~ 10 B e r n s t e i n

6 133 Ron Meier 10 % ~ 15 % The Wall Street Journal. 1. 60 20 % a. b. c. 2. 20 % 2A. 5 a. b. c. 2B. 15 a. b. c. 2C. 30 a. b. c. 3. 25 % a. b. c. 4. 15 a. b. c. 5 10

134 5. a.2 000 b. 50% 5 000 c. 20% 15 000 6. a. b. c. 7. 10 a. b. c. a 1 b 2 c 3 9 ~ 1 4 1 5 ~ 2 1 2 2 ~ 2 7 The Wall Street Journal. 6-1 22 3 4 5 17 A 3 6-1 22 ( 0. 005 3 3 4 2 ) 4. 66 0. 005 3 3 4 2 1 7. 34 A 2 11. 56 10. 44 3 20 20 7 A 4 A 8 certainty equivalent rate

6 135 r i s k - n e u t r a l risk lover 6-1 E(r P P P P mean-variance (M-V) Criterion E(r A ) E(r B ) A B 6-1 A B P P P

136 P P Q P i n d i fference curve 6-2 E(r) 6-1 E(r) 0. 005A 2 1 0 2 0. 0 1 0 0. 0 0 5 4 4 0 0 2 1 5 2 5. 5 1 5 0. 0 0 5 4 6 5 0 2 2 0 3 0. 0 2 0 0. 0 0 5 4 9 0 0 2 2 5 3 3. 9 2 5 0. 0 0 5 4 1 150 2 6-1 A 4 4 a. 6-2 b. P 6.2 6.2.1 h e d g i n g

6 137 d i v e r s i f i c a t i o n 6.2.2 H u m a n e x Best Candy 50 [ 1 ] 0. 5 0. 3 0. 2 ( ) 25 10 2 5 1 expected return P r (s) s r(s) E(r) E(r) = E(r ) ( 0. 5 2 5 ) ( 0. 3 1 0 ) 0. 2 ( 2 5 ) 1 0. 5 ( 6-2 ) 2 variance 2 = Pr(s)[r(s) E (r)] s Pr(s)r(s) S 2 0. 5 ( 25 1 0. 5 ) 2 0. 3 ( 10 1 0. 5 ) 2 0. 2 ( 2 5 1 0. 5 ) 2 3 57. 25 ( 6-3 ) standard deviation 2 [1]

138 357.25 = 18.9% 50 5 3 3 0. 5 E(r ) 0. 5E(r ) 0. 5r ( 0. 5 1 0. 5 ) ( 0. 5 5 ) 7. 75 4 4 50 5 0. 5 0. 5 1 8. 9 9. 45 1 0. 5 7. 75 5. 5 2. 75 5. 5 2. 75 S a l l y S u g a r k a n e s 0. 5 0. 3 0. 2 1 5 3 5 6 14. 73 3 0. 5 0. 3 0. 2 ( ) 13. 0 2. 5 5. 0

6 139 8. 25 4. 83 10. 50 18. 90 7. 5 7 5 9. 4 5 8. 2 5 4. 8 3 5 20 15 1 8 17 ( c o v a r i a n c e ) [r E(r )][r E(r )] C o v (r r ) Pr( s) [r (s) E(r )][r (s) E(r )] ( 6-4 ) s E(r ) 1 0. 5 E(r ) 6 6-4 C o v (r r ) 0. 5 ( 25 1 0. 5 )( 1 6 ) 0. 3 ( 10 1 0. 5 )( 5 6 ) 0. 2 ( 2 5 1 0. 5 )( 35 6 ) 2 40. 5 0. 5 0. 3 0. 2 2 5 1 0 2 5 1 5 3 5 correlation coeff i c i e n t

140 1 1 ( ) [ C o v (r r )]/( ) 2 40. 5 /( 18. 9 1 4. 73 ) 0. 86 1 5 1 2 22 w 1 w 2 P 2 P 2 w 1 2 1 2 +w 2 2 2 2 + 2w 1 w 2 C o v (r 1 r 2 ) w 1 w 2 0. 5 1 8. 9 1 4. 73 C o v (r r ) 2 40. 5 P 2 ( 0. 5 2 1 8. 9 2 )+ (0.5 2 1 4. 7 3 2 ) + [ 2 0. 5 0. 5 ( 2 4 0. 5 ) ] 2 3. 3 P ( 23. 3 ) 1 / 2 4. 83 5 6 7 5 2 0 ( ) a. b. c. 5 P d. 1. 2. 3. 4.

6 141 5. A r r o w, Kenneth. Essays in the Theory of Risk Bearing. Amsterdam: North Holland, 1971. L e v y, Haim; and Moshe Ben-Horim. S t a t i s t i c s: Decisions and Applications in Business and Economics. New York: Random House, 1984. Wonnacott, Thomas H.; and Ronald J. Wonnacott. I n t ro d u c t o ry Statistics for Business and Economics. New York: Wi l e y, 1984. 1. 70 000 200 0. 5 6 a. 8 b. ( a ) c. 12 d. ( a ) ( c ) 2. 12 18 7 3. - 5 3 ( 5 2 5 - ) 4. 4 A 4 3 4 5. 5 6. A 5 7 8 9 E(r) 1 1 2 3 0 2 1 5 5 0 3 2 1 1 6 4 2 4 2 1

142 U E(r) 0. 005A 2 A 4 7. A 4 a. 1 b. 2 c. 3 d. 4 8. a. 1 b. 2 c. 3 d. 4 9. ( A ) a. b. c. d. 4 1 500 7 8. 5 500 20 / 5 10 12 10. W W 500 0 1. 0 11. 10 A 3 0. 2 0. 8 0. 4 0. 6 0. 6 0. 4 12. A 5 11 0. 8 0. 2 1. 0 0 13 15 0. 5 0. 3 0. 2 ( ) 10 5 20 13. 14. 15. 5 13 1. 22 000 /100 000 0. 22 2 2 5 22 5 1 7 2. 5 1. 50 1. 1 5 1. 05 1. 60 1. 05 1. 05 1. 60 1. 68 1 +r 1. 6 / 1. 50 1. 12 r 1 2 3. A 4 U 2 0 ( 0. 0 0 5 4 2 0 2 ) 1 2

6 143 U 7 ( 0. 0 0 5 4 0 ) 7 ( ) A 8 U 2 0 ( 0. 005 8 2 0 2 ) 4 7 4. 5. 6. a. 0. 5 0. 3 0. 2 2 5 1 0 2 5 7 5 2 0 5 5 5 E(r ) ( 0. 5 7 ) + 0. 3 ( 5 ) ( 0. 2 2 0 ) 6 [ 0. 5 ( 7 6 ) 2 + 0. 3 ( 5 6 ) 2 0. 2 ( 20 6 ) 2 ] 1 / 2 8. 72 C o v ( ) 0. 5 ( 7 6 )( 25 1 0. 5 ) 0. 3 ( 5 6 )( 10 1 0. 5 ) + 0. 2 ( 20 6 )( 2 5 1 0. 5 ) 90. 5 ( ) [ C o v ( )]/ 90. 5 /( 8. 72 1 8. 90 ) 0. 55 0. 55 0. 86 50 5 0. 5 0. 3 0. 2 1 6 2. 5 2. 5 E(r ) ( 0. 5 1 6 ) + ( 0. 3 2. 5 ) + 0. 2 ( 2. 5 ) 8. 25 [ 0. 5 ( 16 8. 25 ) 2 + 0. 3 ( 2. 5 8. 25 ) 2 + 0. 2 ( 2. 5 8. 25 ) 2 ] 1 / 2 7. 94

144 b. E(r) 7. 75 9. 45 7. 94 c, d. 5 2 ( 0. 5 2 2 ) + ( 0. 5 2 2 ) + [ 2 0. 5 0. 5 C o v ( )] ( 0. 5 2 1 8. 9 2 ) + ( 0. 5 2 8. 72 2 ) + [ 2 0. 5 0. 5 ( 9 0. 5 )] 6 3. 06 7. 94 6A 6A.1 1 E(r) = n s=1 Pr( s)r(s) s 1... n r(s) s P r (s) - mean absolute deviation, MAD

6 145 n s=1 Pr(s) [r(s) E(r) ] n 2 = Pr(s) [r(s) E(r)] 2 s=1 6 A - 1 a) b) 6A-1 6 A - 1 a 6 A - 1 b b a a b a b a b a b a b M 4 M 3 M 5 U E(r) b 0 n M 3 = Pr(s)[r(s) E(r)] 3 s =1 2 b 1 M 3 b 2 M 4 b 3 M 5

146 [1] 1) 2) - 1987 25 - - - 6 A - 1 6A-1 N 1 N 8 N 3 2 N 1 2 8 7 1. 1 N A 1 2. 4 N A 6. 5 N A 16. 4 N A 5 1 4. 4 3 9. 2 8. 1 4. 6 17. 4 16. 7 22. 7 22. 6 2 0 0. 5 6. 3 16. 3 16. 1 22. 2 22. 3 25. 3 25. 3 5 0 19. 6 28. 2 26. 4 28. 2 27. 8 28. 2 28. 1 28. 2 7 0 38. 7 49. 7 33. 8 35. 7 31. 6 32. 9 30. 0 30. 0 9 5 96. 3 95. 6 54. 3 51. 8 40. 9 39. 9 34. 1 33. 8 442. 6 N A 136. 7 N A 73. 7 N A 43. 1 N A 28. 2 28. 2 28. 2 28. 2 28. 2 28. 2 28. 2 28. 2 41. 0 41. 0 14. 4 14. 4 7. 1 7. 1 3. 4 3. 4 (M 3 ) 255. 4 0. 0 88. 7 0. 0 44. 5 0. 0 17. 7 0. 0 1 227 131 072 32 768 16 384 Lawrence Fisher and James H. Lorie, Some Studies of Variability of Returns on Investments in Common Stocks, Journal of Business 43(April 1970). [1] Paul A. Samuelson, The Fundamental Approximation Theorem of Portfolio Analysis in Terms of Means, Variances, and Higher Moments, Review of Economic Studies 37 (1970).

6 147 6A.2-6 A - 1 1 8 3 2 1 28 n 1 28. 2 41. 0 5 39. 2 14. 4 19. 6 28. 2 128 32 1 00 r r e r e e r 1 e r r e 1 1 00 r e 6 A - 2 6A-2 J. Atchison and J. A. C. Brown, The Lognormal Distribution(New York: Cambridge University Press, 1976).

148 r e (t) t r e (t) e rt 1 rt 40 0. 40 2 0. 16 1 2 ( ) 2 / 1 2 0. 1 6 / 1 2 0. 0 1 3 3 ( 0. 0133 ) 1 / 2 0. 1155 50 8 400 8 450 16 250 2 ( ) ( ) ( 1 / 2 5 0 ) 1 / 2 0. 1 6 / ( 2 5 0 ) 1 / 2 0. 1 0 1 1. 0 1 8 400 8 400 0. 101 8 4. 8 1 50 6 A - 2 6 A - 1 6 A 1. 2. 3. 6 A - 1 1. K L K L 12 / 1 0. 1 0 0 0 2 0. 2 0 0. 2 5 2. 0 0 / 3 0. 4 0 0. 4 0 1 4. 0 0 4 0. 2 5 0. 6 0 2 0. 0 0 5 0. 0 5 0. 8 5 3 0. 0 0 a. b.

6 149 c. 6A1. 6 A 1 6 A 2 K L 6A2. 6 A - 1 6 A - 1 6B 1738 Daniel Bernoulli 1725 1733 n R R(n) 2 n n 0 1 / 2 2 0 1 n 1 1 / 2 1 / 2 2 1 2 n 2 1 / 2 1 / 2 1 / 2 R n 0 1 / 2 1 1 / 2 1 1 / 4 2 1 / 2 2 1 / 8 4 1 / 2 3 1 / 16 8 1 / 2............ n (1/2) n+ 1 2 n 1 / 2 E(R) = n=0 Pr(n)R(n) = 1/ 2 + 1/2+ = 1 [1] [1]

150 l o g R R [ 1 ] 2 l o g 2 0. 693 2 1964 Von Neumann M o rg e n s t e r n 1 000 20 200 001 1 6 B - 1 6B-1 100 000 p 1/2 150 000 1 p 1/2 50 000 6 B - 1 6 B - 2 6 B - 2 5 p 0. 5 10 15 [1] R l o g (R) ( ) V(R) V (R) = Pr(n)log[ R(n)] = ( 1 2 )n+1 log(2 n ) = 0.693 n =0 n =0

6 151 l o g 100 000 11. 51 l o g 150 000 11. 92 G G 11. 92 11. 51 0. 41 p G 0. 5 0. 41 0. 2 6B-2 10 5 L L l o g 100 000 l o g 50 000 11. 51 1 0. 82 0. 69 1 p L 0. 5 0. 69 0. 35 E[U(W) ] p U(W 1 ) + ( 1 p)u(w 2 ) 1/2log(50 000)+1/2log(150 000) 11. 3 7 10 l o g 100 000 11. 51 11. 37 Mary Smith 6 B - 2 11. 37 11. 37 W C E l o g W C E 11. 37 W C E e 11. 3 7 86 681.87 W C E 6 B - 2 Y Y E(W) W C E 100 000 86 681.87 13 318.13 86 681.87 100 000 6 B 1 U(W) W 1 / 2 a. 5 15 b. p 0. 5 c. d.

e. 1926 1996 500 8. 5 6 B 1. 250 000 200 6 0. 001 ( 200 000 ) 2. 1 000 1 a. 1/2 b. c. 1. 5 6B1.a. U (W ) = U(50 000) = W 50 000=223.61 U(150 000) 3 87. 30 b. E(U) ( 0. 5 2 23. 61 ) + ( 0. 5 3 87. 30 ) 3 05. 45 c. 305. 45 W C E W CE = 305.45 W CE = 305.45 2 = 93 301 d. 100 000 e.