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RealOptionsSpreadhSheetforHSM4301.xls

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Real Options Exercises
Harvard Business School Case 9-208-045
Courseware 9-208-703

Tables A & B

Real Options Exercises
Problem 2
Table A
Round 1
Year0123456789
After-tax cash flow10.012.015.015.515.916.416.917.417.9
Investment-145.0
Net cash flow-145.010.012.015.015.515.916.416.917.417.9
Round 2
Year0123456789
After-tax cash flow20.024.030.030.931.832.8
Investment-250.0
Net cash flow000-250.020.024.030.030.931.832.8
Round 3
Year0123456789
After-tax cash flow40.048.060.0
Investment-450.0
Net cash flow000000-450.040.048.060.0
Table B
Rounds 1-3
Year0123456789
After-tax cash flow0.010.012.015.035.539.946.487.897.2110.7
Investment-145.0-250.0-450.0
Net cash flow-145.010.012.0-235.035.539.9-403.687.897.2110.7

Appendix

Black-Scholes-Merton Value of a European Call Option, Expressed as a Percentage of Underlying Asset Value
NPVq = (Underlying asset value) / PV(Exercise price)
0.300.350.400.450.500.550.600.650.700.750.800.820.840.860.880.900.920.940.960.981.001.021.041.061.081.101.121.141.161.181.201.251.301.351.401.451.501.752.002.50
0.050.00.00.00.00.00.00.00.00.00.00.00.00.00.00.00.00.10.30.61.22.03.14.56.07.59.110.712.313.815.316.720.023.125.928.631.033.342.950.060.00.05
0.100.00.00.00.00.00.00.00.00.00.00.00.10.20.30.50.81.21.72.33.14.05.06.17.38.610.011.312.714.115.416.820.023.125.928.631.033.342.950.060.00.10
0.150.00.00.00.00.00.00.00.00.10.20.50.71.01.31.72.22.83.54.25.16.07.08.09.110.211.412.613.815.016.217.420.423.326.028.631.133.342.950.060.00.15
0.200.00.00.00.00.00.00.00.10.40.81.51.92.32.83.44.04.75.46.27.18.08.99.910.911.913.014.115.216.317.418.521.223.926.428.931.233.542.950.060.00.20
0.250.00.00.00.00.00.10.20.51.01.82.83.33.94.55.25.96.67.48.29.19.910.911.812.813.714.715.716.717.718.719.822.324.727.129.431.733.842.950.060.00.25
0.300.00.00.00.10.10.30.71.22.03.14.45.05.76.37.07.88.69.410.211.111.912.813.714.615.616.517.418.419.320.321.223.525.828.130.232.334.343.150.160.00.30
0.350.00.00.10.20.40.81.42.33.34.66.26.87.58.29.09.810.611.412.213.013.914.815.616.517.418.319.220.121.021.922.724.927.129.231.233.235.143.550.260.00.35
0.400.00.10.20.50.91.62.43.54.86.38.08.79.410.211.011.712.513.414.215.015.916.717.518.419.220.120.921.822.623.524.326.428.430.432.334.236.044.050.560.10.40
0.450.10.20.51.01.72.63.75.06.58.19.910.611.412.212.913.714.515.316.217.017.818.619.420.321.121.922.723.524.325.125.927.929.831.733.535.337.044.650.860.20.45
0.500.20.51.01.72.63.75.16.68.210.011.812.613.414.214.915.716.517.318.118.919.720.521.322.122.923.724.525.326.126.827.629.531.333.134.836.438.145.351.360.40.50
0.550.51.01.72.63.85.16.68.310.011.913.814.615.416.116.917.718.519.320.120.921.722.423.224.024.825.526.327.027.828.529.231.032.834.536.137.739.246.151.960.70.55
0.600.91.62.53.75.16.68.310.111.913.815.816.617.418.118.919.720.521.322.022.823.624.325.125.826.627.328.128.829.530.230.932.634.335.937.539.040.447.052.561.00.60
0.651.42.43.64.96.58.210.011.913.815.817.818.619.320.120.921.722.523.224.024.725.526.227.027.728.429.129.830.531.231.932.634.235.837.438.940.341.748.053.361.40.65
0.702.13.34.76.38.19.911.913.815.817.819.820.621.322.122.923.624.425.225.926.627.428.128.829.530.230.931.632.332.933.634.235.837.338.840.341.643.049.054.061.90.70
0.753.04.46.17.99.811.713.715.817.819.821.822.523.324.124.825.626.327.127.828.529.229.930.631.332.032.733.334.034.635.335.937.438.940.341.743.044.350.054.962.40.75
0.804.05.77.59.511.513.615.717.719.821.823.724.525.326.026.827.528.329.029.730.431.131.832.433.133.834.435.135.736.336.937.539.040.441.843.144.445.651.155.863.00.80
0.855.17.19.111.213.315.517.619.721.823.825.726.527.228.028.729.430.230.931.632.232.933.634.234.935.536.236.837.438.038.639.240.641.943.344.545.846.952.256.763.60.85
0.906.48.510.713.015.217.419.621.723.825.827.728.429.229.930.631.332.032.733.434.134.735.436.036.637.337.938.539.139.640.240.842.143.544.746.047.148.353.357.664.30.90
0.957.810.112.514.817.119.421.623.725.727.729.630.431.131.832.533.233.934.635.235.936.537.237.838.439.039.640.140.741.341.842.443.745.046.247.448.549.654.558.665.00.95
1.009.311.814.316.719.121.423.625.727.729.731.632.333.033.734.435.135.736.437.037.738.338.939.540.140.741.241.842.442.943.444.045.246.547.648.849.950.955.659.565.71.00
1.0510.913.616.118.621.023.325.627.729.731.633.534.234.935.636.236.937.638.238.839.440.040.641.241.842.442.943.544.044.545.045.546.848.049.150.251.252.256.760.566.51.05
1.1012.615.418.020.623.025.327.529.631.633.535.436.136.737.438.138.739.340.040.641.241.842.342.943.544.044.545.145.646.146.647.148.349.450.551.652.653.557.961.567.21.10
1.1514.417.220.022.525.027.329.531.633.635.437.237.938.639.239.940.541.141.742.342.943.544.044.645.145.646.246.747.247.748.248.649.850.951.952.953.954.959.062.568.01.15
1.2016.219.121.924.527.029.331.533.635.537.339.139.740.441.041.742.342.943.544.044.645.145.746.246.747.347.848.348.749.249.750.151.352.353.354.355.256.160.263.568.81.20
1.2518.121.123.926.529.031.333.535.537.439.240.941.542.242.843.444.044.645.245.746.346.847.347.848.448.849.349.850.350.751.251.652.753.754.755.756.657.461.364.569.61.25
1.3020.023.025.928.531.033.335.437.439.341.042.743.343.944.545.145.746.346.847.447.948.448.949.449.950.450.951.351.852.252.753.154.155.156.157.057.958.762.465.570.41.30
1.3521.925.027.930.533.035.237.339.341.142.844.445.145.746.346.847.447.948.549.049.550.050.551.051.552.052.452.953.353.754.154.655.656.557.458.359.159.963.566.571.11.35
1.4023.927.029.932.534.937.139.241.142.944.646.246.847.447.948.549.049.650.150.651.151.652.152.653.053.553.954.354.855.255.656.056.957.958.759.660.461.264.667.571.91.40
1.4525.829.031.934.536.939.141.143.044.746.447.948.549.049.650.150.751.251.752.252.753.253.654.154.555.055.455.856.256.657.057.458.359.260.060.961.662.465.768.472.71.45
1.5027.831.033.836.438.840.942.944.846.548.149.650.150.751.251.852.352.853.353.754.254.755.155.656.056.456.857.257.658.058.458.859.760.561.362.162.963.666.869.473.51.50
1.5529.833.035.838.440.742.844.846.648.249.851.251.852.352.853.353.854.354.855.355.756.256.657.057.457.858.258.659.059.459.760.161.061.862.663.364.164.767.870.374.31.55
1.6031.835.037.840.342.644.646.548.349.951.452.853.453.954.454.955.455.956.356.857.257.658.058.558.959.259.660.060.460.761.161.462.363.163.864.565.265.968.871.375.11.60
1.6533.836.939.742.244.446.448.350.051.653.154.454.955.455.956.456.957.357.858.258.659.159.559.960.260.661.061.461.762.162.462.763.564.365.065.766.467.069.972.275.91.65
1.7035.838.941.644.046.248.250.051.753.254.756.056.557.057.557.958.458.859.259.760.160.560.961.261.662.062.362.763.063.463.764.064.865.566.266.967.568.270.973.176.61.70
1.7537.740.843.545.948.050.051.753.454.856.257.558.058.558.959.459.860.260.761.161.561.862.262.662.963.363.664.064.364.664.965.366.066.767.468.068.769.271.974.077.41.75
2.0047.350.152.554.656.558.259.761.162.463.664.665.065.465.866.266.666.967.367.667.968.368.668.969.269.569.870.070.370.670.871.171.772.372.973.473.974.476.578.381.02.00
2.2556.158.660.762.564.165.666.868.069.170.070.971.371.671.972.272.572.873.173.473.773.974.274.474.774.975.275.475.675.876.076.376.877.277.778.178.578.980.682.184.32.25
2.5064.066.167.969.470.872.073.174.074.975.776.476.777.077.277.577.778.078.278.478.778.979.179.379.579.779.980.080.280.480.680.781.181.581.982.282.682.984.385.487.22.50
2.7570.972.774.275.476.677.578.479.279.980.581.181.481.681.882.082.282.482.682.782.983.183.383.483.683.783.984.084.284.384.484.684.985.285.585.886.086.387.488.389.72.75
3.0076.978.379.580.581.482.282.983.584.184.685.185.385.485.685.885.986.186.286.486.586.686.886.987.087.187.387.487.587.687.787.888.188.388.588.889.089.290.090.791.83.00
3.5086.086.987.688.388.889.389.790.190.590.891.191.291.391.491.591.691.691.791.891.992.092.192.192.292.392.492.492.592.692.692.792.893.093.193.393.493.594.094.495.13.50
4.0092.092.592.993.393.693.994.294.494.694.894.995.095.095.195.295.295.395.395.495.495.495.595.595.695.695.795.795.795.895.895.895.996.096.196.296.296.396.696.897.24.00
4.5095.796.096.296.496.696.796.997.097.197.297.397.397.397.497.497.497.597.597.597.597.697.697.697.697.697.797.797.797.797.897.897.897.997.997.998.098.098.298.398.54.50
5.0097.897.998.198.298.398.398.498.598.598.698.698.698.698.798.798.798.798.798.798.798.898.898.898.898.898.898.898.898.898.998.998.998.998.999.099.099.099.199.199.25.00
6.0099.599.599.699.699.699.699.799.799.799.799.799.799.799.799.799.799.799.799.799.799.799.799.799.799.799.799.799.799.799.899.899.899.899.899.899.899.899.899.899.86.00
Note: Values in the table represent percentages of underlying asset values: e.g., 39.3 denotes a call option worth 39.3% of the underlying asset value.
Values in the table were computed from the Black-Scholes option pricing model.
The format of the table was adapted from Brealey and Myers, Principles of Corporate Finance, 4th edition, McGraw-Hill, Inc., New York, 1991, Appendix Table 6, pp. AP12-13.
&C
&C
Cumulative Volatility s√t

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