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Posted on Jul 30, 2024

Question 6 Consider 3 assets. Asset 1 is a risk-free bond which has normalized price of unity at time t=0. Assume the risk-free rate of

Question 6

Consider 3 assets. Asset 1 is a risk-free bond which has normalized price of unity at time t=0. Assume the risk-free rate of interest is 5% per year. The price of Asset 2 is $50 at time t=0. In addition, it is estimated that, if the economy is in an upturn, the price of Asset 2 would be $55, while it would be $45 if the economy is in a downturn. The price of Asset 3 is unknown at time t=0 and it is $3 if the economy is in an upturn, while it is $2 if the economy is in a downturn. Use the non-arbitrage theorem to derive the equations that the price of Asset 3 at time t=0 has to satisfy in order to be an arbitrage-free price. (Note: you do not have to solve the equations.)

Table for N(x) When x 50 XI 00 .01 02 03 .041 OS 06 .071 08 091 10.00.50000,49600.49200 48800 48400.450110.4761|0.47210.468110.4641 0.10.46020.45620.45220.4483 0.444510.44040.43640.4325 0.42860.4247 0.210.420710.4168 0.4129 0.4090 0.40520.40130.39740.39360.3897 0.3859 0.3 0.38210.3783 0.3745 0.37070.36690.36320.3594 0.355710.35200.3483 0.40.34460.3409 0.33720.33300 33000 32640.32280.31920.31560.3121 0 510.3085|0.3050 0.3015|0.298110 29460 29120.28770.28430.28100.2776 0.60.27430.2709 0.26760.26430.26110.2578 0.25460.25140.24830.2453 0.70.24200.23890.23580.2327|0.22960.22660.22360.22060.21770.2148 0.80.21190.2090 0.2061 0.2033 0.20050.19770.1999 0.19220.1894 0.1867 0.910.184110.18140.1788 0.1762 0.1736 0.17110.168|0.16600.1635 0.1611 1.00.158710.15620.1539|0.15150,14920 1-4690.14-4610.14230.140110.1379 1.10.13570.13350.13140.1292 0.1271 0.1251 0.12300.1210 0.11900.1170 1.210.11510.113101112 0.10930. 1075 0.1056/0.10380.10200.10030.0985 1.30.09680,09510.09340.09180.0901.688510.08690,08530.08380.0823 1.40.0808 0.07930.0778 0.0764 0.07490.0735020721 0.0708 0.06960.0681 1.510.06680,0655 0.06430,06300,06180.0606 0.0594 0.05820.0571 0.0559 1.60.0548|0.053710.052610,05160.0505 0,049510.048510.047510,046310.0435 1.70.0446 0.0436 0.04270.04180,0409 0.04010.0392|0,0384|0,037510.0367 1.870.03590.0351 0.03440.03360.03290.032210_03140.050710.0301 0.0294 190.028710.0281 0.02740.0268 0.026210.025 0.025 0.0244 0.0239 0,0233 2.00.0228|0.02220.021710.02120.0207 0.02020.0197 0.01920.018810.0183 2.10.01790.01740.01 700.01660.01620.01580_01540.01 500.014610.0143 220.01390.01360.0132 0.01290.0125.01220.01190.0116 0.0113|0.0110 2.310.010710.0104 0.0102|0,0099 0.0096 0.00940.009110.00890.00870.0084 2.40.00820.0080 0.00780.00750.00730.0071 0.00690.0068 0.0066 0.0064 2.5 0.006210.00600.00590.005710,0055 0.00540.00520.00510.0049|0.0048 2.60.00470.00450.00440.00430,004110.00400.00390.00380,00370,0036 270.00350.00340.00330.0032 0.003110.00300.00290.00280.00270.0026 2.8|0.00260.0025 0.002410.0023 0.0023 0.0022 0.00210.00210.00200.00191 2.990.00190.00180.0018|0.00170,0016 0.0016 0.00150.00150.00140.0014 3.00.00130.0013 0.00130.0012 0.0012 0.0012 0.0011 0.00110.0010 0.0010 3.110.0010 0.0009 0.0009 0.00090.0008 0.0008 0.00080.0008 0.0007 0.0007 3.210.00070.000710.0006 0.0006 0.0006 0.0006 0.000 0.000 0.0005 0.0005 3.310.000510.0005 0.000510.00040.0004 0.0004 0.0004 0.00040.0004 0.0003 3.410.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.00030.0003 0.0002 3.50.0002 0.0002 0.0002 0.0002 0.0002 0.0002 0.0002 0.0002 0.0002 0.0002 3.60.0002 0.0002 0.0001 0.0001 0.0001 0.0001 0.0001 0.00010.0001 0.0001 3.70.000 10.0001 0.000 10.0001 0.000100001 0.0001 0.0001 0.00010.0001 3.810.0001|0.0001 0.000 10.0001 0.0001 0.0001 0.00010.000 10.0001 0.0001 3.990.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.000 0.0000 0.0000 0.0000 14.00.0000 0.0000 0.0000 0.00000000000000 0.000 0.0000 0.0000 0.0000 Appendix Table for N(x) When x 20 .00 01 02 .031 04 OS 06 .07 .08 0 0.0 0.5000 0.5040 0.50800.51200.51600.51990.52390.52790.5319 0.5359 0.1 0.53980.54380.54780.5517|0.55570.559680.56360.5675057140.5753 0.210.5793 0.58320.587110.59100.59480.59870.6026 0.6064 0.610306141 10.310.61790.6217|0.6255|0.6293|0.6331|0.6368|0.6406|0.64430,6480 0.6517 10.410.6554|0.6591|0.6628|0.66640.6700|0.6736 0.677210.6808|0.68440.6879 0.5 0.691510.6950|0.69850.70190.70540.7088|0.71230.71570.71900.7224 10.60.7257|0.72910.7324|0.73570.73890.7422 0.74540.74860751707549 10.700.750 0.7611|0.764210.76730,77040.77340.77640.7794|0.782307852 10.80.788110.79100.7939|0.7967|0.799510.80230.80510.80780.8106|0.8133 0.90.81590.81860.821210.8238|0.82640.82890.8315|0.8340 0.83650,8389 1.00.84130.8438|0.8461|0.848510.85080.85310.83540.8577/0.85990.8620 110.86430.8665|0.86860.87080.8729 0.87490.8770|0.8790 0.88100.8830 120.88490.88690.88880 8907|0.89250.89440.89620.89800.89970.9015 1.30.90320.90490.90660.908210.90990-91150.913110.9147/0.916210.9177 1.40.91920.9207 0.92220.9236 0.9251 0.9265|0.927910.92920.93060.9319 1.510.93320.93450.9357 0.93700.93820.93940.94060.94180.94290.9441 1.60.943210 94630.9474094840.949510 950510.951 510.952510.953510.9545 1.70.95540.95640.957310 95820.95910 9599|0.96080.96160.962510.9633 1.80.9641|0.9649|0.96560.96640.9670096780.96860.96930.96990.9706 1.90.97130.97190.97260.97320.97380.97440.9750097560.97610.9767 2.00.97720.97780.97830.97880.97930.97980.98030.98080.98120.9817 2.110.98210.98260.98300.9834|0.9838|0.98420.98460.9850|0.98540.9857 2.210.9861|0.98640.9868|0.98710.98750.9878|0.98810.98840.9887.9890 2.30.98930.98960.98980.9901|0.99040.990610.990910.99110.9913|0.9916 2.4 0.991810 99200.99220.992510.99270.9929|0 99310.99320 99340 9936 2 $10.993870.99400.99410.99430.994510.99460.9948|0.99490 99510.9952 2.60.99530.9955099560.99570.99590.9960099610.9962099630-9964 2.70.996510,99660.9967|0.99680.99690.9970/0.997110.9972|0.99730 9974 2.80 99740.997510.99760.997710.99770-99780 9979|0.9979|0.998009981 2.90.998 10.99820.99820.99830.9984|0 99840.998|0.998510.998609986 3.00.99870.99870.99870.99880.99880.9989099890.9989099900 9990 3.10.99900.9991 0.99910.9991 0.99920999210.9992099920.99930-999 3.210.99930.99930-99940.99940.99940 99940.99940.999809995|0.9995 3.30.9995|0.999$10.99950 99960.99900 99960 99960.99900 99900 9997 3:40.9997/0.9997 0.9997 0.9997 0.9997 0.9997 0.9997/0.9997 0.999709998 350.9998|0.9998 0.9990 99980.9998 0.99980999809998 0.999809998 3.6 0.9998 0.99980.9999|0.9999 0.9999|0.99900 99999999|0.9999|0.9999 3.70.9999|0.9999 0.9999 0.9999 99990.99990.9999999999999999 380.999999999999 0.9999 0.99990999999999999|0.9999|0 9999 3.9 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.0000 1.0000 4.0 1.0000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.0000 Table for N(x) When x 50 10 SO 190 80 00 02 03 041 07 09 0.00450000/4960/0,49200,48800.4840 0.4801 0.47610.4721 0.46810.4641 10.10.46020.45620.452210.448.30.44430 44040 43640.432510.42860.4247 0.210.420704160-4129 0.40900.40520.40130 3974039360.38970 3859 0.30.38210.3730.374510.370710.36690.36320.35940.35570.35200.3483 0.40.3446014090 33720.33360.33000.32640.32280.3192 0.31560.3121 0.50 308510.30500.301510.298110 29460 29120.287710.28430.28100.2776 0.60.27430.27090.26760.26430.2611 0.25780.2546|0.25140.24830.2451 0.70.24200 23890.23580.23270.22960.22660.22360.22060.21770.2148 0.80.21190.2000 0.20610 2033 0.200510.19770.1449 0.19220.18940-1867 090.18410.18140.1780.176210.17360.17110.1685|0.16600.1635 0.1611 100.1587 0.15620.15390.1515|0.1492 0.146.0.14400.14230.14010.1379 10.1357013350.13140.1292 0.1271 0.125 10.12.30 0.1210 0.11900 1170 1:20.11510113110.11120.10930. 10750 10560.1038/0.10200.10030 0985 1. 30.09680.09510.09340.091400901.10.086 08530.08380.0823 1.40.08080.07930,07780,07640.07490.07350.0721 0,0708 0,06940,0681 1.510.0668|0.065 0.06430.06360.06100606 0.05940.05820.05710.0559 1.60.0548 0.05370_0526.05140.05051004950.048510.0475004650,0455 1.71001446|0.0436 0.04270.041810,040910040110 01920.0384|0.0378 0.0367 1.80.035900351003440.0336 0.03290.03220.031400307 0.0000294 1.910.02870,0281 0.02740,02680_0262 0.025 0.02500.02440,02390.0233 2010.0228 0.02220.02170.02120.0207 0.02020.01970.01920.01880.0183 2010.01790.017410.0170001660.0162.01580.01540.01 500.014600143 2.20.01.39 0.01360.0132 0.01290.0120.01220.01190.0116 0.011300110 2.30.0107 0.01040_0102.0099 0.00% 0.009410.0091 0.00890.00870.0084 240.00820.008 0.007 0.007510.00730.00710.00690.006%0.00660.0064 250.00620.00600.00590.00570,005 0,00540.00520.005 0.00490.0048 260.00470.004450.004410.00-130.00410.0010 0.00390.003 0.00370.0036 20.0035 0.00340.00330_0032 0.00310.00300.00290.0028 0.0002 0.0026 280.0026 0.002 0.0024 0.0023 0.0023 0.0022 0.00210.00210.00200.0019 2.910.00190.001%0.00180,001710.0010000100.001 0.00150.00140.0014 3.00.00130.0012 0.00130.0012 0.0012 0.00110.00 110.0011000100,0010 3.1 0.0010 0.0000 0.00090.0009 0.00080.0008 0.000 0.000 0.00070.0007 3.210.00070.000 0.0006 0.0006 0.06 0.0006 0.00000.0005 0.0005 0.0005 3.30.000 0.000 0.000 0.0004 0.0004 0.0004 0.00040.00040.0004 0.0003 3.40.000.0003 0.0003 0.000 0.0003 0.0003 0.0003 0.0003 0.000300002 3.5 0.0002 0.0002 0.0002 0.0002 0.0002 0.0002 0.0002 0.0002 0.0002 0.0002 160.0002 0.0002 0.00010000100001 0.0001 0.0001 0.0001100001 0.0001 3.70,0001 0.0001 0.0001 0.0001 0.0001 0.000 0.0001 0.0001 0.0001 0.0001 380.0001 0.0001 0.0001 0.0001 0.0001 0.000100001 0.0001 0.0001 0.0001 390.0000 0.000000000000000-000000000 0.000 0.0000 0.0000 0.0000 400.0000 0.0000 0.0000 0.00000-0000000000-0000000000-000000000 Table for N(x) When x 50 XI 00 .01 02 03 .041 OS 06 .071 08 091 10.00.50000,49600.49200 48800 48400.450110.4761|0.47210.468110.4641 0.10.46020.45620.45220.4483 0.444510.44040.43640.4325 0.42860.4247 0.210.420710.4168 0.4129 0.4090 0.40520.40130.39740.39360.3897 0.3859 0.3 0.38210.3783 0.3745 0.37070.36690.36320.3594 0.355710.35200.3483 0.40.34460.3409 0.33720.33300 33000 32640.32280.31920.31560.3121 0 510.3085|0.3050 0.3015|0.298110 29460 29120.28770.28430.28100.2776 0.60.27430.2709 0.26760.26430.26110.2578 0.25460.25140.24830.2453 0.70.24200.23890.23580.2327|0.22960.22660.22360.22060.21770.2148 0.80.21190.2090 0.2061 0.2033 0.20050.19770.1999 0.19220.1894 0.1867 0.910.184110.18140.1788 0.1762 0.1736 0.17110.168|0.16600.1635 0.1611 1.00.158710.15620.1539|0.15150,14920 1-4690.14-4610.14230.140110.1379 1.10.13570.13350.13140.1292 0.1271 0.1251 0.12300.1210 0.11900.1170 1.210.11510.113101112 0.10930. 1075 0.1056/0.10380.10200.10030.0985 1.30.09680,09510.09340.09180.0901.688510.08690,08530.08380.0823 1.40.0808 0.07930.0778 0.0764 0.07490.0735020721 0.0708 0.06960.0681 1.510.06680,0655 0.06430,06300,06180.0606 0.0594 0.05820.0571 0.0559 1.60.0548|0.053710.052610,05160.0505 0,049510.048510.047510,046310.0435 1.70.0446 0.0436 0.04270.04180,0409 0.04010.0392|0,0384|0,037510.0367 1.870.03590.0351 0.03440.03360.03290.032210_03140.050710.0301 0.0294 190.028710.0281 0.02740.0268 0.026210.025 0.025 0.0244 0.0239 0,0233 2.00.0228|0.02220.021710.02120.0207 0.02020.0197 0.01920.018810.0183 2.10.01790.01740.01 700.01660.01620.01580_01540.01 500.014610.0143 220.01390.01360.0132 0.01290.0125.01220.01190.0116 0.0113|0.0110 2.310.010710.0104 0.0102|0,0099 0.0096 0.00940.009110.00890.00870.0084 2.40.00820.0080 0.00780.00750.00730.0071 0.00690.0068 0.0066 0.0064 2.5 0.006210.00600.00590.005710,0055 0.00540.00520.00510.0049|0.0048 2.60.00470.00450.00440.00430,004110.00400.00390.00380,00370,0036 270.00350.00340.00330.0032 0.003110.00300.00290.00280.00270.0026 2.8|0.00260.0025 0.002410.0023 0.0023 0.0022 0.00210.00210.00200.00191 2.990.00190.00180.0018|0.00170,0016 0.0016 0.00150.00150.00140.0014 3.00.00130.0013 0.00130.0012 0.0012 0.0012 0.0011 0.00110.0010 0.0010 3.110.0010 0.0009 0.0009 0.00090.0008 0.0008 0.00080.0008 0.0007 0.0007 3.210.00070.000710.0006 0.0006 0.0006 0.0006 0.000 0.000 0.0005 0.0005 3.310.000510.0005 0.000510.00040.0004 0.0004 0.0004 0.00040.0004 0.0003 3.410.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.00030.0003 0.0002 3.50.0002 0.0002 0.0002 0.0002 0.0002 0.0002 0.0002 0.0002 0.0002 0.0002 3.60.0002 0.0002 0.0001 0.0001 0.0001 0.0001 0.0001 0.00010.0001 0.0001 3.70.000 10.0001 0.000 10.0001 0.000100001 0.0001 0.0001 0.00010.0001 3.810.0001|0.0001 0.000 10.0001 0.0001 0.0001 0.00010.000 10.0001 0.0001 3.990.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.000 0.0000 0.0000 0.0000 14.00.0000 0.0000 0.0000 0.00000000000000 0.000 0.0000 0.0000 0.0000 Appendix Table for N(x) When x 20 .00 01 02 .031 04 OS 06 .07 .08 0 0.0 0.5000 0.5040 0.50800.51200.51600.51990.52390.52790.5319 0.5359 0.1 0.53980.54380.54780.5517|0.55570.559680.56360.5675057140.5753 0.210.5793 0.58320.587110.59100.59480.59870.6026 0.6064 0.610306141 10.310.61790.6217|0.6255|0.6293|0.6331|0.6368|0.6406|0.64430,6480 0.6517 10.410.6554|0.6591|0.6628|0.66640.6700|0.6736 0.677210.6808|0.68440.6879 0.5 0.691510.6950|0.69850.70190.70540.7088|0.71230.71570.71900.7224 10.60.7257|0.72910.7324|0.73570.73890.7422 0.74540.74860751707549 10.700.750 0.7611|0.764210.76730,77040.77340.77640.7794|0.782307852 10.80.788110.79100.7939|0.7967|0.799510.80230.80510.80780.8106|0.8133 0.90.81590.81860.821210.8238|0.82640.82890.8315|0.8340 0.83650,8389 1.00.84130.8438|0.8461|0.848510.85080.85310.83540.8577/0.85990.8620 110.86430.8665|0.86860.87080.8729 0.87490.8770|0.8790 0.88100.8830 120.88490.88690.88880 8907|0.89250.89440.89620.89800.89970.9015 1.30.90320.90490.90660.908210.90990-91150.913110.9147/0.916210.9177 1.40.91920.9207 0.92220.9236 0.9251 0.9265|0.927910.92920.93060.9319 1.510.93320.93450.9357 0.93700.93820.93940.94060.94180.94290.9441 1.60.943210 94630.9474094840.949510 950510.951 510.952510.953510.9545 1.70.95540.95640.957310 95820.95910 9599|0.96080.96160.962510.9633 1.80.9641|0.9649|0.96560.96640.9670096780.96860.96930.96990.9706 1.90.97130.97190.97260.97320.97380.97440.9750097560.97610.9767 2.00.97720.97780.97830.97880.97930.97980.98030.98080.98120.9817 2.110.98210.98260.98300.9834|0.9838|0.98420.98460.9850|0.98540.9857 2.210.9861|0.98640.9868|0.98710.98750.9878|0.98810.98840.9887.9890 2.30.98930.98960.98980.9901|0.99040.990610.990910.99110.9913|0.9916 2.4 0.991810 99200.99220.992510.99270.9929|0 99310.99320 99340 9936 2 $10.993870.99400.99410.99430.994510.99460.9948|0.99490 99510.9952 2.60.99530.9955099560.99570.99590.9960099610.9962099630-9964 2.70.996510,99660.9967|0.99680.99690.9970/0.997110.9972|0.99730 9974 2.80 99740.997510.99760.997710.99770-99780 9979|0.9979|0.998009981 2.90.998 10.99820.99820.99830.9984|0 99840.998|0.998510.998609986 3.00.99870.99870.99870.99880.99880.9989099890.9989099900 9990 3.10.99900.9991 0.99910.9991 0.99920999210.9992099920.99930-999 3.210.99930.99930-99940.99940.99940 99940.99940.999809995|0.9995 3.30.9995|0.999$10.99950 99960.99900 99960 99960.99900 99900 9997 3:40.9997/0.9997 0.9997 0.9997 0.9997 0.9997 0.9997/0.9997 0.999709998 350.9998|0.9998 0.9990 99980.9998 0.99980999809998 0.999809998 3.6 0.9998 0.99980.9999|0.9999 0.9999|0.99900 99999999|0.9999|0.9999 3.70.9999|0.9999 0.9999 0.9999 99990.99990.9999999999999999 380.999999999999 0.9999 0.99990999999999999|0.9999|0 9999 3.9 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.0000 1.0000 4.0 1.0000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.0000 Table for N(x) When x 50 10 SO 190 80 00 02 03 041 07 09 0.00450000/4960/0,49200,48800.4840 0.4801 0.47610.4721 0.46810.4641 10.10.46020.45620.452210.448.30.44430 44040 43640.432510.42860.4247 0.210.420704160-4129 0.40900.40520.40130 3974039360.38970 3859 0.30.38210.3730.374510.370710.36690.36320.35940.35570.35200.3483 0.40.3446014090 33720.33360.33000.32640.32280.3192 0.31560.3121 0.50 308510.30500.301510.298110 29460 29120.287710.28430.28100.2776 0.60.27430.27090.26760.26430.2611 0.25780.2546|0.25140.24830.2451 0.70.24200 23890.23580.23270.22960.22660.22360.22060.21770.2148 0.80.21190.2000 0.20610 2033 0.200510.19770.1449 0.19220.18940-1867 090.18410.18140.1780.176210.17360.17110.1685|0.16600.1635 0.1611 100.1587 0.15620.15390.1515|0.1492 0.146.0.14400.14230.14010.1379 10.1357013350.13140.1292 0.1271 0.125 10.12.30 0.1210 0.11900 1170 1:20.11510113110.11120.10930. 10750 10560.1038/0.10200.10030 0985 1. 30.09680.09510.09340.091400901.10.086 08530.08380.0823 1.40.08080.07930,07780,07640.07490.07350.0721 0,0708 0,06940,0681 1.510.0668|0.065 0.06430.06360.06100606 0.05940.05820.05710.0559 1.60.0548 0.05370_0526.05140.05051004950.048510.0475004650,0455 1.71001446|0.0436 0.04270.041810,040910040110 01920.0384|0.0378 0.0367 1.80.035900351003440.0336 0.03290.03220.031400307 0.0000294 1.910.02870,0281 0.02740,02680_0262 0.025 0.02500.02440,02390.0233 2010.0228 0.02220.02170.02120.0207 0.02020.01970.01920.01880.0183 2010.01790.017410.0170001660.0162.01580.01540.01 500.014600143 2.20.01.39 0.01360.0132 0.01290.0120.01220.01190.0116 0.011300110 2.30.0107 0.01040_0102.0099 0.00% 0.009410.0091 0.00890.00870.0084 240.00820.008 0.007 0.007510.00730.00710.00690.006%0.00660.0064 250.00620.00600.00590.00570,005 0,00540.00520.005 0.00490.0048 260.00470.004450.004410.00-130.00410.0010 0.00390.003 0.00370.0036 20.0035 0.00340.00330_0032 0.00310.00300.00290.0028 0.0002 0.0026 280.0026 0.002 0.0024 0.0023 0.0023 0.0022 0.00210.00210.00200.0019 2.910.00190.001%0.00180,001710.0010000100.001 0.00150.00140.0014 3.00.00130.0012 0.00130.0012 0.0012 0.00110.00 110.0011000100,0010 3.1 0.0010 0.0000 0.00090.0009 0.00080.0008 0.000 0.000 0.00070.0007 3.210.00070.000 0.0006 0.0006 0.06 0.0006 0.00000.0005 0.0005 0.0005 3.30.000 0.000 0.000 0.0004 0.0004 0.0004 0.00040.00040.0004 0.0003 3.40.000.0003 0.0003 0.000 0.0003 0.0003 0.0003 0.0003 0.000300002 3.5 0.0002 0.0002 0.0002 0.0002 0.0002 0.0002 0.0002 0.0002 0.0002 0.0002 160.0002 0.0002 0.00010000100001 0.0001 0.0001 0.0001100001 0.0001 3.70,0001 0.0001 0.0001 0.0001 0.0001 0.000 0.0001 0.0001 0.0001 0.0001 380.0001 0.0001 0.0001 0.0001 0.0001 0.000100001 0.0001 0.0001 0.0001 390.0000 0.000000000000000-000000000 0.000 0.0000 0.0000 0.0000 400.0000 0.0000 0.0000 0.00000-0000000000-0000000000-000000000