MACHINE REPLACEMENTYear 0 1 2CF of single machine -550 100 200Discount rate for machine cash flows 25% - This is the risk-adjusted discount rate.NPV of single machine -24.56 - =NPV(B5,C3:G3)+B3Therefore you would not recommend buying just onemachine (part a of question).Number of machines bought next year 9Option value of single machine purchased half a year from now 45.43 - =B22NPV of total project 384.30 - =B6+B8*B9Black-Scholes Option Pricing FormulaS 525.44 PV of machine CFsX 550.00 Exercise price = Machine costr 10.00% Risk-free rate of interestT 0.5 Time to maturity of option (in years)Sigma 30% - Volatilityd1 0.1264 - (LN(S/X)+(r+0.5*sigma2)*T)/(sigma*SQRT(T)d2 -0.0857 - d1 - sigma*SQRT(T)N(d1) 0.5503 - Uses formula NormSDist(d1)N(d2) 0.4658 - Uses formula NormSDist(d2)Option value = BS call price 45.43 - S*N(d1)-X*exp(-r*T)*N(d2)A B C D123456789101112131415161718192021223 4300 400MACHINE REPLACEMENT- This is the risk-adjusted discount rate.- =NPV(B5,C3:G3)+B3Therefore you would not recommend buying just onemachine (part a of question).Exercise price = Machine costRisk-free rate of interestTime to maturity of option (in years)- (LN(S/X)+(r+0.5*sigma2)*T)/(sigma*SQRT(T)- d1 - sigma*SQRT(T)- Uses formula NormSDist(d1)- Uses formula NormSDist(d2)- S*N(d1)-X*exp(-r*T)*N(d2)E F G12345678910111213141516171819202122机器替换年 0 1 2每天机器的现金流量 -550 100 200机器现金流量的折现率 25% - 这是风险调整的折现率每台机器的NPV -24.56 - =NPV(B5,C3:G3)+B3因此你不会推荐只购买一台机器(问题的一部分).下一年购买机器的数量 9半年前购买的每台机器的期权价值 45.43 - =B22整个项目的NPV 384.30 - =B6+B8*B9布莱克-斯科尔斯期权定价公式S 525.44 机器现金流量的现值X 550.00 执行价格 = 机器成本r 10.00% 无风险利率T 0.5 期权到期时间 (年)Sigma 30% - 波动性d1 0.1264 - (LN(S/X)+(r+0.5*sigma2)*T)/(sigma*SQRT(T)d2 -0.0857 - d1 - sigma*SQRT(T)N(d1) 0.5503 - 使用公式 NormSDist(d1)N(d2) 0.4658 - 使用公式 NormSDist(d2)期权价值 = 布莱克-斯科尔斯价格 45.43 - S*N(d1)-X*exp(-r*T)*N(d2)A B C D123456789101112131415161718192021223 4300 400机器替换- 这是风险调整的折现率- =NPV(B5,C3:G3)+B3因此你不会推荐只购买一台机器(问题的一部分).- (LN(S/X)+(r+0.5*sigma2)*T)/(sigma*SQRT(T)- d1 - sigma*SQRT(T)- 使用公式 NormSDist(d1)- 使用公式 NormSDist(d2)- S*N(d1)-X*exp(-r*T)*N(d2)E F G12345678910111213141516171819202122Cost 50,000Present value of future cash flows 45,000NPV -5,000Cost of course 1,500Sigma 30%Risk-free interest rate 7%Time to maturity 0.5Call option value 2,513 - =bscall(B2,B1,B8,B7,B6)Since the course costs less than the value of the option, its worth doing!成本 50,000未来现金流量的现值 45,000NPV -5,000课程成本 1,500Sigma 30%无风险利率 7%到期时间 0.5看涨期权价值 2,513 - =bscall(B2,B1,B8,B7,B6)由于课程成本少于期权价值, 值得这么做!Project cash flows169 State prices130 qU 0.300091 qD 0.5000-1009170-90Project valuation15.213913.65-10013.6535-22.5NPV -5.99 Answer to part aAnswer to part bProject cash flows169 state prices130 qU 0.300091 qD 0.5000-10091700Project valuation15.213913.65-10013.65350NPV 16.51 - =SUM(A37:E43) Answer to part bAnswer to part cA B C D E F G H1234567891011121314151617181920212223242526272829303132333435363738394041424344454647484950Initial project cash flows169 State prices130 qU 0.300091 qD 0.5000-1009170-90Cash flows accounting for abandonment at 100Salvage value, any time 100169100.7100-100100100100NPV 569.7 - =SUM(A65:E71) Answer to part bA B C D E F G H5152535455565758596061626364656667686970717273项目现金流量169 状态价格130 qU 0.300091 qD 0.5000-1009170-90项目估价15.213913.65-10013.6535-22.5NPV -5.99 a部分的答案b部分的答案项目现金流量169 状态价格130 qU 0.300091 qD 0.5000-10091700项目估价15.213913.65-10013.65350NPV 16.51 - =SUM(A37:E43) b部分的答案c部分的答案A B C D E F G H1234567891011121314151617181920212223242526272829303132333435363738394041424344454647484950初始项目现金流量169 状态价格130 qU 0.300091 qD 0.5000-1009170-90在100时放弃期权的现金流量任意时刻的残值 100169100.7100-100100100100NPV 569.7 - =SUM(A65:E71) b部分的答案A B C D E F G H5152535455565758596061626364656667686970717273Market data state pricesExpected market return 15% qu 0.2549 - =(1+B5-B9)/(1+B5)*(B8-B9)Sigma of market return 20% qd 0.6711 - =(B8-1-B5)/(1+B5)*(B8-B9)Risk-free rate 8%One-period up and down of marketup 1.419068 - =EXP(B3+B4)down 0.951229 - =EXP(B3-B4)qu qdData table: Effect 0.2549 0.6711 - =E4 , data table headerof risk-free rate on 0% 0.1042 0.8958state prices 2% 0.1441 0.83634% 0.1824 0.77916% 0.2193 0.72418% 0.2549 0.671110% 0.2891 0.620012% 0.3221 0.570814% 0.3539 0.523216% 0.3847 0.477418% 0.4144 0.4331Data table: Effect qu qdof sigma on 0.2549 0.6711 - =E4 , data table headerstate prices 5% -0.2005 1.126410% 0.1143 0.811615% 0.2117 0.714220% 0.2549 0.671125% 0.2763 0.649630% 0.2870 0.639035% 0.2915 0.634540% 0.2922 0.633745% 0.2904 0.635550% 0.2870 0.638955% 0.2824 0.643660% 0.2769 0.649065% 0.2708 0.655170% 0.2642 0.6617市场信息 状态价格市场收益率的期望值 15% qu 0.2549 - =(1+B5-B9)/(1+B5)*(B8-B9)市场收益率的标准差 20% qd 0.6711 - =(B8-1-B5)/(1+B5)*(B8-B9)无风险利率 8%一期市场的上涨和下跌状态上涨 1.419068 - =EXP(B3+B4)下跌 0.951229 - =EXP(B3-B4)qu qd模拟运算表: 有效 0.2549 0.6711 - =E4 , 模拟运算表头无风险利率上的 0% 0.1042 0.8958状态价格 2% 0.1441 0.83634% 0.1824 0.77916% 0.2193 0.72418% 0.2549 0.671110% 0.2891 0.620012% 0.3221 0.570814% 0.3539 0.523216% 0.3847 0.477418% 0.4144 0.4331模拟运算表: 有效 qu qd标准差上的 0.2549 0.6711 - =E4 , 模拟运算表头状态价格 5% -0.2005 1.126410% 0.1143 0.811615% 0.2117 0.714220% 0.2549 0.671125% 0.2763 0.649630% 0.2870 0.639035% 0.2915 0.634540% 0.2922 0.633745% 0.2904 0.635550% 0.2870 0.638955% 0.2824 0.643660% 0.2769 0.649065% 0.2708 0.655170% 0.2642 0.6617Matching state prices to the cost of capitalProject cost of capital 30% - This is the discount rate for the project if it has no optionsRisk-free rate 5%Project cash flows18013090-5060-50-100Project expected cash flows: Assumes equal state probabilitiesYear 0 1 2Expected CF -50 40 57.5Project NPV 14.79State pricesqu 0.4197qd 0.5327 - NOTE: this was a case where Goal Seek worked better than Solver!Project state-by-state discounting31.706554.560920.1209-5013.4140-26.6341-28.3750State-by-state NPV 14.79Target cell 0.00A B C D E F G H1234567891011121314151617181920212223242526272829303132333435363738- NOTE: this was a case where Goal Seek worked better than Solver!I1234567891011121314151617181920212223242526272829303132333435363738状态价格与资本成本匹配项目资本成本价值 30% - 这是如果该项目没有期权时的折现率无风险利率 5%项目现金流18013090-5060-50-100项目预期现金流量: 假设每个状态概率相等年 0 1 2预期CF -50 40 57.5项目的NPV 14.79状态价格qu 0.4197qd 0.5327 - 注意: 该情况下单变量求解好于规划求解!项目每期状态的折现31.706554.560920.1209-5013.4140-26.6341-28.3750状态的NPV 14.79目标单元格 0.00A B C D E F1234567891011121314151617181920212223242526272829303132333435363738