THE UNIT CIRCLEFor the quarter circle and the pi experiment, see the graph belowx y y-1 0 0-0.999 0.04471 -0.04471-0.998 0.063214 -0.06321-0.997 0.077402 -0.0774-0.996 0.089353 -0.08935-0.995 0.099875 -0.09987-0.994 0.10938 -0.10938-0.993 0.118114 -0.11811-0.992 0.126238 -0.12624-0.991 0.133862 -0.13386-0.99 0.141067 -0.14107-0.98 0.198997 -0.199-0.97 0.243105 -0.2431-0.96 0.28 -0.28-0.95 0.31225 -0.31225-0.94 0.341174 -0.34117-0.93 0.36756 -0.36756-0.92 0.391918 -0.39192-0.91 0.414608 -0.41461-0.9 0.43589 -0.43589-0.89 0.455961 -0.45596-0.88 0.474974 -0.47497-0.87 0.493052 -0.49305-0.86 0.510294 -0.51029-0.85 0.526783 -0.52678-0.84 0.542586 -0.54259-0.83 0.557763 -0.55776-0.82 0.572364 -0.57236-0.81 0.58643 -0.58643-0.8 0.6 -0.6-0.79 0.613107 -0.61311-0.78 0.62578 -0.62578-0.77 0.638044 -0.63804-0.76 0.649923 -0.64992-0.75 0.661438 -0.66144-0.74 0.672607 -0.67261-0.73 0.683447 -0.68345-0.72 0.693974 -0.69397-0.71 0.704202 -0.7042-0.7 0.714143 -0.71414-0.69 0.723809 -0.72381-0.68 0.733212 -0.73321-0.67 0.742361 -0.74236-0.66 0.751266 -0.75127-0.65 0.759934 -0.75993-0.64 0.768375 -0.76837-0.63 0.776595 -0.7766Q u a rte r U n it C irc leS h o w in g s in g le ra n d o m p o in t ra n d (),ra n d ()10 .90 .80 .70 .60 .50 .40 .30 .20 .1000 .10 .20 .30 .40 .50 .60 .70 .80 .91-0.62 0.784602 -0.7846-0.61 0.792401 -0.7924-0.6 0.8 -0.8-0.59 0.807403 -0.8074-0.58 0.814616 -0.81462-0.57 0.821645 -0.82164-0.56 0.828493 -0.82849-0.55 0.835165 -0.83516-0.54 0.841665 -0.84167-0.53 0.847998 -0.848-0.52 0.854166 -0.85417-0.51 0.860174 -0.86017-0.5 0.866025 -0.86603-0.49 0.871722 -0.87172-0.48 0.877268 -0.87727-0.47 0.882666 -0.88267-0.46 0.887919 -0.88792-0.45 0.893029 -0.89303-0.44 0.897998 -0.898-0.43 0.902829 -0.90283-0.42 0.907524 -0.90752-0.41 0.912086 -0.91209-0.4 0.916515 -0.91652-0.39 0.920815 -0.92081-0.38 0.924986 -0.92499-0.37 0.929032 -0.92903-0.36 0.932952 -0.93295-0.35 0.93675 -0.93675-0.34 0.940425 -0.94043-0.33 0.943981 -0.94398-0.32 0.947418 -0.94742-0.31 0.950737 -0.95074-0.3 0.953939 -0.95394-0.29 0.957027 -0.95703-0.28 0.96 -0.96-0.27 0.96286 -0.96286-0.26 0.965609 -0.96561-0.25 0.968246 -0.96825-0.24 0.970773 -0.97077-0.23 0.973191 -0.97319-0.22 0.9755 -0.9755-0.21 0.977701 -0.9777-0.2 0.979796 -0.9798-0.19 0.981784 -0.98178-0.18 0.983667 -0.98367-0.17 0.985444 -0.98544-0.16 0.987117 -0.98712-0.15 0.988686 -0.98869-0.14 0.990152 -0.99015-0.13 0.991514 -0.99151-0.12 0.992774 -0.99277-0.11 0.993932 -0.99393-0.1 0.994987 -0.99499-0.09 0.995942 -0.99594-0.08 0.996795 -0.99679-0.07 0.997547 -0.99755-0.06 0.998198 -0.9982-0.05 0.998749 -0.99875-0.04 0.9992 -0.9992-0.03 0.99955 -0.99955-0.02 0.9998 -0.9998-0.01 0.99995 -0.999957.53E-16 1 -10.01 0.99995 -0.999950.02 0.9998 -0.99980.03 0.99955 -0.999550.04 0.9992 -0.99920.05 0.998749 -0.998750.06 0.998198 -0.99820.07 0.997547 -0.997550.08 0.996795 -0.996790.09 0.995942 -0.995940.1 0.994987 -0.994990.11 0.993932 -0.993930.12 0.992774 -0.992770.13 0.991514 -0.991510.14 0.990152 -0.990150.15 0.988686 -0.988690.16 0.987117 -0.987120.17 0.985444 -0.985440.18 0.983667 -0.983670.19 0.981784 -0.981780.2 0.979796 -0.97980.21 0.977701 -0.97770.22 0.9755 -0.97550.23 0.973191 -0.973190.24 0.970773 -0.970770.25 0.968246 -0.968250.26 0.965609 -0.965610.27 0.96286 -0.962860.28 0.96 -0.960.29 0.957027 -0.957030.3 0.953939 -0.953940.31 0.950737 -0.950740.32 0.947418 -0.947420.33 0.943981 -0.943980.34 0.940425 -0.940430.35 0.93675 -0.936750.36 0.932952 -0.932950.37 0.929032 -0.929030.38 0.924986 -0.924990.39 0.920815 -0.920810.4 0.916515 -0.916520.41 0.912086 -0.912090.42 0.907524 -0.907520.43 0.902829 -0.902830.44 0.897998 -0.8980.45 0.893029 -0.893030.46 0.887919 -0.887920.47 0.882666 -0.882670.48 0.877268 -0.877270.49 0.871722 -0.871720.5 0.866025 -0.866030.51 0.860174 -0.860170.52 0.854166 -0.854170.53 0.847998 -0.8480.54 0.841665 -0.841670.55 0.835165 -0.835160.56 0.828493 -0.828490.57 0.821645 -0.821640.58 0.814616 -0.814620.59 0.807403 -0.80740.6 0.8 -0.80.61 0.792401 -0.79240.62 0.784602 -0.78460.63 0.776595 -0.77660.64 0.768375 -0.768370.65 0.759934 -0.759930.66 0.751266 -0.751270.67 0.742361 -0.742360.68 0.733212 -0.733210.69 0.723809 -0.723810.7 0.714143 -0.714140.71 0.704202 -0.70420.72 0.693974 -0.693970.73 0.683447 -0.683450.74 0.672607 -0.672610.75 0.661438 -0.661440.76 0.649923 -0.649920.77 0.638044 -0.638040.78 0.62578 -0.625780.79 0.613107 -0.613110.8 0.6 -0.60.81 0.58643 -0.586430.82 0.572364 -0.572360.83 0.557763 -0.557760.84 0.542586 -0.542590.85 0.526783 -0.526780.86 0.510294 -0.510290.87 0.493052 -0.493050.88 0.474974 -0.474970.89 0.455961 -0.455960.9 0.43589 -0.435890.91 0.414608 -0.414610.92 0.391918 -0.391920.93 0.36756 -0.367560.94 0.341174 -0.341170.95 0.31225 -0.312250.96 0.28 -0.280.97 0.243105 -0.24310.98 0.198997 -0.1990.99 0.141067 -0.141070.992 0.126238 -0.126240.994 0.10938 -0.109380.996 0.089353 -0.089350.998 0.063214 -0.063211 0 00.499829 0.608505 endQ u a rte r U n it C irc leS h o w in g s in g le ra n d o m p o in t ra n d (),ra n d ()10 .90 .80 .70 .60 .50 .40 .30 .20 .1000 .10 .20 .30 .40 .50 .60 .70 .80 .91step 0.01单位圆四分之一圆和pi实验, 见下面的图形x y y-1 0 0-0.999 0.04471 -0.04471-0.998 0.063214 -0.06321-0.997 0.077402 -0.0774-0.996 0.089353 -0.08935-0.995 0.099875 -0.09987-0.994 0.10938 -0.10938-0.993 0.118114 -0.11811-0.992 0.126238 -0.12624-0.991 0.133862 -0.13386-0.99 0.141067 -0.14107-0.98 0.198997 -0.199-0.97 0.243105 -0.2431-0.96 0.28 -0.28-0.95 0.31225 -0.31225-0.94 0.341174 -0.34117-0.93 0.36756 -0.36756-0.92 0.391918 -0.39192-0.91 0.414608 -0.41461-0.9 0.43589 -0.43589-0.89 0.455961 -0.45596-0.88 0.474974 -0.47497-0.87 0.493052 -0.49305-0.86 0.510294 -0.51029-0.85 0.526783 -0.52678-0.84 0.542586 -0.54259-0.83 0.557763 -0.55776-0.82 0.572364 -0.57236-0.81 0.58643 -0.58643-0.8 0.6 -0.6-0.79 0.613107 -0.61311-0.78 0.62578 -0.62578-0.77 0.638044 -0.63804-0.76 0.649923 -0.64992-0.75 0.661438 -0.66144-0.74 0.672607 -0.67261-0.73 0.683447 -0.68345-0.72 0.693974 -0.69397-0.71 0.704202 -0.7042-0.7 0.714143 -0.71414-0.69 0.723809 -0.72381-0.68 0.733212 -0.73321-0.67 0.742361 -0.74236-0.66 0.751266 -0.75127-0.65 0.759934 -0.75993-0.64 0.768375 -0.76837-0.63 0.776595 -0.7766四 分 之 一 单 位 圆显 示 简 单 随 机 点 rand(),rand()10 .90 .80 .70 .60 .50 .40 .30 .20 .1000 .10 .20 .30 .40 .50 .60 .70 .80 .91-0.62 0.784602 -0.7846-0.61 0.792401 -0.7924-0.6 0.8 -0.8-0.59 0.807403 -0.8074-0.58 0.814616 -0.81462-0.57 0.821645 -0.82164-0.56 0.828493 -0.82849-0.55 0.835165 -0.83516-0.54 0.841665 -0.84167-0.53 0.847998 -0.848-0.52 0.854166 -0.85417-0.51 0.860174 -0.86017-0.5 0.866025 -0.86603-0.49 0.871722 -0.87172-0.48 0.877268 -0.87727-0.47 0.882666 -0.88267-0.46 0.887919 -0.88792-0.45 0.893029 -0.89303-0.44 0.897998 -0.898-0.43 0.902829 -0.90283-0.42 0.907524 -0.90752-0.41 0.912086 -0.91209-0.4 0.916515 -0.91652-0.39 0.920815 -0.92081-0.38 0.924986 -0.92499-0.37 0.929032 -0.92903-0.36 0.932952 -0.93295-0.35 0.93675 -0.93675-0.34 0.940425 -0.94043-0.33 0.943981 -0.94398-0.32 0.947418 -0.94742-0.31 0.950737 -0.95074-0.3 0.953939 -0.95394-0.29 0.957027 -0.95703-0.28 0.96 -0.96-0.27 0.96286 -0.96286-0.26 0.965609 -0.96561-0.25 0.968246 -0.96825-0.24 0.970773 -0.97077-0.23 0.973191 -0.97319-0.22 0.9755 -0.9755-0.21 0.977701 -0.9777-0.2 0.979796 -0.9798-0.19 0.981784 -0.98178-0.18 0.983667 -0.98367-0.17 0.985444 -0.98544-0.16 0.987117 -0.98712-0.15 0.988686 -0.98869-0.14 0.990152 -0.99015-0.13 0.991514 -0.99151-0.12 0.992774 -0.99277-0.11 0.993932 -0.99393-0.1 0.994987 -0.99499-0.09 0.995942 -0.99594-0.08 0.996795 -0.99679-0.07 0.997547 -0.99755-0.06 0.998198 -0.9982-0.05 0.998749 -0.99875-0.04 0.9992 -0.9992-0.03 0.99955 -0.99955-0.02 0.9998 -0.9998-0.01 0.99995 -0.999957.53E-16 1 -10.01 0.99995 -0.999950.02 0.9998 -0.99980.03 0.99955 -0.999550.04 0.9992 -0.99920.05 0.998749 -0.998750.06 0.998198 -0.99820.07 0.997547 -0.997550.08 0.996795 -0.996790.09 0.995942 -0.995940.1 0.994987 -0.994990.11 0.993932 -0.993930.12 0.992774 -0.992770.13 0.991514 -0.991510.14 0.990152 -0.990150.15 0.988686 -0.988690.16 0.987117 -0.987120.17 0.985444 -0.985440.18 0.983667 -0.983670.19 0.981784 -0.981780.2 0.979796 -0.97980.21 0.977701 -0.97770.22 0.9755 -0.97550.23 0.973191 -0.973190.24 0.970773 -0.970770.25 0.968246 -0.968250.26 0.965609 -0.965610.27 0.96286 -0.962860.28 0.96 -0.960.29 0.957027 -0.957030.3 0.953939 -0.953940.31 0.950737 -0.950740.32 0.947418 -0.947420.33 0.943981 -0.943980.34 0.940425 -0.940430.35 0.93675 -0.936750.36 0.932952 -0.932950.37 0.929032 -0.929030.38 0.924986 -0.924990.39 0.920815 -0.920810.4 0.916515 -0.916520.41 0.912086 -0.912090.42 0.907524 -0.907520.43 0.902829 -0.902830.44 0.897998 -0.8980.45 0.893029 -0.893030.46 0.887919 -0.887920.47 0.882666 -0.882670.48 0.877268 -0.877270.49 0.871722 -0.871720.5 0.866025 -0.866030.51 0.860174 -0.860170.52 0.854166 -0.854170.53 0.847998 -0.8480.54 0.841665 -0.841670.55 0.835165 -0.835160.56 0.828493 -0.828490.57 0.821645 -0.821640.58 0.814616 -0.814620.59 0.807403 -0.80740.6 0.8 -0.80.61 0.792401 -0.79240.62 0.784602 -0.78460.63 0.776595 -0.77660.64 0.768375 -0.768370.65 0.759934 -0.759930.66 0.751266 -0.751270.67 0.742361 -0.742360.68 0.733212 -0.733210.69 0.723809 -0.723810.7 0.714143 -0.714140.71 0.704202 -0.70420.72 0.693974 -0.693970.73 0.683447 -0.683450.74 0.672607 -0.672610.75 0.661438 -0.661440.76 0.649923 -0.649920.77 0.638044 -0.638040.78 0.62578 -0.625780.79 0.613107 -0.613110.8 0.6 -0.60.81 0.58643 -0.586430.82 0.572364 -0.572360.83 0.557763 -0.557760.84 0.542586 -0.542590.85 0.526783 -0.526780.86 0.510294 -0.510290.87 0.493052 -0.493050.88 0.474974 -0.474970.89 0.455961 -0.455960.9 0.43589 -0.435890.91 0.414608 -0.414610.92 0.391918 -0.391920.93 0.36756 -0.367560.94 0.341174 -0.341170.95 0.31225 -0.312250.96 0.28 -0.280.97 0.243105 -0.24310.98 0.198997 -0.1990.99 0.141067 -0.141070.992 0.126238 -0.126240.994 0.10938 -0.109380.996 0.089353 -0.089350.998 0.063214 -0.063211 0 00.552412 0.277212 end四 分 之 一 单 位 圆显 示 简 单 随 机 点 rand(),rand()10 .90 .80 .70 .60 .50 .40 .30 .20 .1000 .10 .20 .30 .40 .50 .60 .70 .80 .91step 0.01COMPUTING PI USING MONTE CARLO METHODSINITIAL EXPERIMENTNumber of data points 30 3*150,000, else takes end-year wealth/3Current wealth 1,000,000Riskfree rate 6%Parameters of risky investmentExpected annual return 8%Standard deviation of return 20%Proportion invested in risky 70%Safety cushion 3Annual drawdown 150,000YearWealth atbeginningof yearInvested inriskyInvested inbondsRandomnumber,normallydistributed1+return onriskyinvestment1 1,000,000 700,000 300,000 -0.5471 0.97102 848,253 593,777 254,476 0.4716 1.19043 827,072 578,950 248,121 -0.8906 0.90654 638,306 446,814 191,492 0.4212 1.17855 579,907 405,935 173,972 0.9129 1.30036 562,560 393,792 168,768 0.0895 1.10297 463,500 324,450 139,050 -0.3590 1.00828 324,770 227,339 97,431 -0.9809 0.89039 203,906 142,734 61,172 -1.1469 0.861210 125,255 87,678 37,576 -0.2989 1.0204Normally-distributed random numbersgenerated by =NORMSINV(RAND()Wealth at beginning of year =G20-H20 1+return on risky investment=EXP($B$5+$B$6*E21)Investment in risky asset =B21*$B$7A B C D E F12345678910111213141516171819202122232425262728Wealth atend ofyearDrawdown Left at endof 10 years998,253 150,000977,072 150,000788,306 150,000729,907 150,000712,560 150,000613,500 150,000474,770 150,000305,860 101,953187,882 62,627129,369 43,123 86,246=C21*F21+D21*EXP($B$3)Drawdown calculated as=IF(G21$B$8*$B$9,$B$9,G21/$B$8)f o r M S p a g e 1 3PLANNING YOUR RETIREMENTUsing a safety cushion of 3Investor takes 150,000 if end-year wealth 3*150,000, else takes end-year wealth/3G H I J K L M N12345678910111213141516171819202122232425262728计划你的退休金安全系数为3如果年末资金3*150,000,投资者提取$150,000,否则提取年末资金的三分之一当前资金 1,000,000无风险利率 6%风险投资参数年期望收益率 8%收益的标准差 20%风险投资比例 70%安全系数 3年支出 150,000年 年初资金 风险投资 债券投资 正态分布随机数 1+风险投资收益1 1,000,000 700,000 300,000 -0.5471 0.97102 848,253 593,777 254,476 0.4716 1.19043 827,072 578,950 248,121 -0.8906 0.90654 638,306 446,814 191,492 0.4212 1.17855 579,907 405,935 173,972 0.9129 1.30036 562,560 393,792 168,768 0.0895 1.10297 463,500 324,450 139,050 -0.3590 1.00828 324,770 227,339 97,431 -0.9809 0.89039 203,906 142,734 61,172 -1.1469 0.861210 125,255 87,678 37,576 -0.2989 1.0204正态分布随机数产生=NORMSINV(RAND()年初资金 =G20-H20 1+风险投资收益=EXP($B$5+$B$6*E21)风险资产投资=B21*$B$7A B C D E F12345678910111213141516171819202122232425262728年末资金 支出 10年后剩余998,253 150,000977,072 150,000788,306 150,000729,907 150,000712,560 150,000613,500 150,000474,770 150,000305,860 101,953187,882 62,627129,369 43,123 86,246=C21*F21+D21*EXP($B$3)支出计算=IF(G21$B$8*$B$9,$B$9,G21/$B$8)f o r M S p a g e 1 3计划你的退休金安全系数为3如果年末资金3*150,000,投资者提取$150,000,否则提取年末资金的三分之一G H I J K L M N12345678910111213141516171819202122232425262728HOW WELL DO WE DO?PERCENTAGE OF POSITIVE OUTCOMESCurrent wealth 1,000,000Riskfree rate 8%Parameters of risky investmentExpected annual return 10%Standard deviation of return 40%Proportion invested in risky 40%Annual drawdown 100,000Years of investment 10Runs 1,000Successful runs 86.20% $B$22,$B$7*C30*$B$18+(1-$B$7)*C30*$B$17-$B$8,$B$7*C30*$B$19+(1-$B$7)*C30*$B$17-$B$8)Number of year 10 wealth 0 89 0)Percentage of successful simulations 89.00% $B$22,$B$7*C30*$B$18+(1-$B$7)*C30*$B$17-$B$8,$B$7*C30*$B$19+(1-$B$7)*C30*$B$17-$B$8)F G H I J K L M12345678910111213141516171819202122232425262728293031323334353637383940Simulation12Simulation13Simulation14Simulation15Simulation16Simulation17Simulation18Simulation191,000,000 1,000,000 1,000,000 1,000,000 1,000,000 1,000,000 1,000,000 1,000,0001,209,461 846,300 1,209,461 846,300 846,300 846,300 1,209,461 1,209,4611,044,512 700,853 1,044,512 700,853 700,853 700,853 1,483,741 1,483,7411,267,748 563,217 888,421 817,739 563,217 563,217 1,304,064 1,304,0641,099,669 637,510 1,063,353 673,826 637,510 432,972 1,607,620 1,134,035940,616 734,795 1,292,419 782,349 503,276 309,721 1,421,290 1,384,9741,131,700 595,336 1,123,015 640,337 559,020 305,567 1,761,124 1,210,600970,927 463,366 962,709 505,950 429,000 300,129 1,566,551 1,045,5911,171,391 338,483 1,160,630 562,522 305,963 184,011 1,382,426 1,269,1601,433,891 220,306 998,303 636,600 189,532 140,956 1,710,233 1,101,0051,256,890 108,476 844,694 502,415 148,185 84,576 2,139,483 1,341,723N O P Q R S T U12345678910111213141516171819202122232425262728293031323334353637383940Simulation20Simulation21Simulation22Simulation23Simulation24Simulation25Simulation26Simulation271,000,000 1,000,000 1,000,000 1,000,000 1,000,000 1,000,000 1,000,000 1,000,0001,209,461 1,209,461 1,209,461 1,209,461 846,300 846,300 1,209,461 1,209,4611,044,512 1,044,512 1,044,512 1,483,741 700,853 1,008,196 1,483,741 1,044,5121,267,748 1,267,748 888,421 1,304,064 817,739 1,220,193 1,304,064 1,267,7481,560,066 1,560,066 740,713 1,607,620 673,826 1,054,668 1,134,035 1,099,6691,376,290 1,376,290 600,936 2,005,116 537,642 898,032 973,137 940,6161,202,382 1,702,197 468,666 2,525,620 604,020 749,807 820,879 1,131,7001,0