ASC Topic 718 Accounting Valuation Report Company ABC, Inc. Monte-Carlo Simulation Valuation of Several Proposed Relative Total Shareholder Return TSR Component Rank Grants And Index Outperform Grants As of July 1, 2015 This ASC Topic 718 Valuation Report Has been prepared by 718Valuations.com 855.718.VALU
Table of Contents Management Summary 1. Critical Dates 4. Monte Carlo Simulation How it Works 6. Required Valuation Assumptions 8. Valuation Assumptions Summary 16. Fair Value Summary 17. Valuation Certification 18. Appendix A Peer Group Stock Prices, Volatilities, and Correlations Appendix B Correlated Stock Price Simulation Process Description Appendix C Antithetic Variance Reduction Technique
Management Summary This report details the ASC 718 Grant Date Fair Values for several proposed equity plan designs as requested by Company ABC. The types of awards modeled consist of five Component Rank and five Index Outperform Relative TSR plans. The calculations have been performed as of July 1, 2015. The awards analyzed are deemed to have a Market Condition 1 under ASC 718 and therefore the value of all possible future payouts must be factored into the grant date Fair Value. To model the awards, a Monte Carlo simulation based valuation model has been used. Plan Design Plan Type Fair Value Percentage of Grant Price Design 1 Component Rank $44.05 145.97% Design 2 Component Rank $38.03 126.00% Design 3 Component Rank $36.55 121.09% Design 4 Component Rank $35.88 100.00% Design 5 Component Rank $31.43 104.14% Design 6 Index Outperform $34.58 114.56% Design 7 Index Outperform $33.29 110.31% Design 8 Index Outperform $37.11 122.96% Design 9 Index Outperform $37.01 122.63% Design 10 Index Outperform $33.20 110.07% A Component Rank Relative TSR grant is an award that varies the number of shares earned based upon the TSR of Company ABC as compared to the TSRs of the components of the peer group. An Index Outperform Relative TSR grant is an award that varies the number of shares earned based upon the TSR of Company ABC as compared to the TSRs of the selected Index. 1 If a grant s payout is based upon the company s stock price performance then the award is deemed to have a Market Condition under ASC 718. Company ABC, Inc. Relative Total Shareholder Return Plan Analysis 1
Plan Designs Component Rank The vesting percentage of the award will be based upon the Percentile Rank of Company ABC s TSR as compared to the TSRs of the components of the S&P 500 Index, the peer group. The vesting timeframe for the awards is 3.00-years. The five Component Rank plan designs requested by Company ABC are summarized in the charts below. Plan Design 1 Plan Design 2 60 th 200.00% 75 th 200.00% 40 th 100.00% 50 th 100.00% 20 th 50.00% 30 th 50.00% < 20 th 0.00% < 30 th 0.00% Negative TSR Cap None Negative TSR Cap None Overall Payout Cap None Overall Payout Value Cap None Plan Design 3 Plan Design 4 75 th 200.00% 75 th 200.00% 50 th 100.00% 50 th 100.00% 30 th 50.00% 30 th 50.00% < 30 th 0.00% < 30 th 0.00% Negative TSR Cap 100% Negative TSR Cap 100% Overall Payout Value Cap None Payout Value Cap 4x Grant Price Plan Design 5 90 th 200.00% 60 th 100.00% 40 th 50.00% < 40 th 0.00% Negative TSR Cap None Overall Payout Value Cap None Company ABC, Inc. Relative Total Shareholder Return Plan Analysis 2
Plan Designs Index Outperform The vesting percentage for these award designs will be based upon the outperformance or underperformance of Company ABC as compared to the Competitor Index as a whole, the S&P 500 Index. As with the Component Rank plan designs, the timeframe for vesting is 3-years. The vesting percentage schedules are detailed in the chart below. Plan Design 1 Plan Design 2 Plan Design 3 Plan Design 4 Plan Design 3 Relative Performance Company ABC s vs. S&P 500 Change in Shares Earned Change in Shares Earned Change in Shares Earned Change in Shares Earned Change in Shares Earned Max Payout 200.00% 200.00% 200.00% 200.00% 200.00% For each 1% increase in Excess Return +3.00% + 3.00% + 2.00% + 2.00% + 1.00% TSR Company ABC - TSR S&P 500 = 0.00% 100.00% 100.00% 100.00% 100.00% 100.00% For each 1% decrease in Excess Return - 3.00% - 3.00% - 1.00% - 1.00% - 1.00% Underperformance Payout Threshold -33.33% -33.00% -100.00% -50.00% -50.00% Payout Below Threshold 0.00% 0.00% 0.00% 0.00% 0.00% Negative TSR Cap None None None 100% None Overall Payout Value Cap None 3X Grant Price None None None For the Components Rank and the Index Outperform plans the TSR for each company (Components Rank Plan) and for the Index as a whole (Index Outperform Plan) is calculated as follows: TSR = Ending Average Price Beginning Average Price Beginning Average Price Company ABC, Inc. Relative Total Shareholder Return Plan Analysis 3
Critical Dates The dates shown in the table below have been used to determine the fair value estimates of the proposed awards. Since the fair value can be very sensitive to the dates used to model the award, it is critical that all dates shown are accurate and in alignment with the grant design specifications. Date Definition Date Grant Date July 1, 2015 Performance Period Begin Date July 1, 2015 Performance Period End Date June 30, 2018 Beginning Average Price - Date Range July 1, 2015 to July 29, 2015 Ending Average Price - Date Range June 4, 2018 to June 30, 2018 Service Vesting Date June 30, 2018 A brief description of the dates detailed above are provided below: Grant Date (Valuation Date) The Grant Date (Valuation Date) is the date on which the award is issued. All valuation assumption used in the modeling process must be calculated as of this date. Performance Period Begin Date This is the first date in the performance period. The stock price performance that will ultimately be used to determine the plan payout is based upon the stock prices starting on this date. Performance Period End Date This is the last date in the performance period. The stock prices that will ultimately be used to determine the plan payout are based upon the stock prices ending on this date. Since the beginning average price date range is short for the grants analyzed is grant, the ending average prices will have a minimal impact on the grant date fair value. Company ABC, Inc. Relative Total Shareholder Return Plan Analysis 4
Critical Dates Beginning Average Price Date Range This is the date range over which the beginning average prices for the company and the components of the S&P 500 Index. Since the beginning average prices are based upon the 20 trading-day period including and following the grant date, the beginning average prices will be simulated in the valuation model. Ending Average Price Date Range This is the date range over which the ending average prices for the company and the components of the S&P 500 Index are calculated in the simulation model. Since the ending average price date range is short for this grant, the ending average prices will have a minimal impact on the grant date fair value. Service Vesting Date The shares earned will vest as of this date for all holders of the awards that are employed by Company ABC as of this date. At this time the number of units earned and any dividend equivalents due, based upon the award agreement, will be issued to and fully owned by holders of the awards. The values and dates detailed on the previous page are critical to the proper valuation of the award and are based upon 718 Valuations, LLC interpretations of the award designs requested by Company ABC. Company ABC must confirm that the values and dates shown are in alignment with their interpretation of the designs requested. If the values are not in alignment with the requested designs, the fair values calculated and detailed in this report may vary substantially from the correct grant date fair values and will be non-compliant with fair value measurement standards defined under ASC 718. Company ABC, Inc. Relative Total Shareholder Return Plan Analysis 5
Monte Carlo Simulation How It Works Monte-Carlo simulation is a commonly used financial modeling technique. It is extremely useful when modeling complex instruments due to the technique s inherent flexibility and ability to take into account ALL features of a financial instrument, regardless of the complexity of the features. The following 4-step process details how the Monte-Carlo simulation model calculates the fair value of the award. This 4-step process is repeated over and over again based upon the number of simulations performed. The result for each simulation is recorded. After the simulations have been completed, the results are averaged and the fair value of the award is determined. Monte Carlo Simulation Process: Step 1 Determine Stock Prices The first step in the valuation process requires the simulation of stock prices for Company ABC and the members of the Peer Groups for the Component Rank Plan. For the Index Outperform Plan only Company ABC and the S&P 500 Index as a whole must be simulated. The prices are simulated to determine the Beginning Average Prices and the Ending Average Prices in order to determine the TSR for every company/index being simulated. We have described the process used to simulate stock prices in Appendix B. Since stocks tend to act in a correlated manner, we have modeled the stock prices based upon our best estimate of the correlation coefficients between Company ABC, the components of the S&P 500 Index and the S&P 500 Index as a whole. Step 2 Determine the Number of Awards Earned The plan designs analyzed specifies that the number of awards earned as of the service vesting date will be based upon the Relative Performance of the TSRs of Company ABC and the Peer Group for the Component Rank Plan and the outperformance/underperformance of the S&P 500 Index as a whole for the Index Outperform plan. Please see pages 2 and 3 for further detail of the design payout schedules. Step 3 Determine the Value of the Award If it is determined, based upon the simulated stock prices, that awards will vest, the value of the payout is then determined. The value is Company ABC, Inc. Relative Total Shareholder Return Plan Analysis 6
Monte Carlo Simulation How It Works determined by multiplying the number of awards earned by Company ABC s simulated stock price on the date of vesting as shown below: Value of Award = Vesting Percentage Simulated Company Price on Date of Vesting Step 4 Determine the Preset Value of the Award Now that the future value of the award has been determined, the present value of the award as of the valuation date, July 1, 2015 must be calculated. This is accomplished by calculating a present value factor based upon the risk-free rate of return and the service vesting requirement. The calculation of the present value of the award is shown below. ( RFR x Service Vesting Requirement ) Present Value = Future Value e The 4-step process described above is for one simulation in the procedure. In order to determine the Fair Value of the awards, the process must be repeated many times. The number of times that the process is repeated is the number of simulations performed. The number of simulations that need to be performed in order for the Fair Value to be properly calculated is dependent on many factors. However, one of the more critical factors is the stock price volatilities used in the modeling process. We have performed 100,000 simulations to ensure proper convergence. Company ABC, Inc. Relative Total Shareholder Return Plan Analysis 7
Required Valuation Assumptions In order to model future stock prices several economic assumptions must be determined. These assumptions are used in the modeling process and will result in the stock prices being modeled in a manner consistent with all publicly known information as of the valuation date. Please note that the Monte Carlo model, like the famed Black-Scholes option pricing model, does not reflect any market sentiment as to the expectations that the stock price will more likely than not increase or decrease over time but is based upon a risk-neural framework. The assumptions used in the Monte Carlo model are detailed as follows along with a description of how the assumptions impact the resulting fair value. The required valuation assumptions, which are summarized on the following pages, are summarized below: Closing Stock Prices For Company ABC, Peer Group companies and the S&P 500 Index as a whole. Volatilities For Company ABC, the components of the S&P 500 Index and the S&P 500 Index a whole. Correlations o Component Rank Plan Company ABC and Peer Group Members vs. the S&P 500 Index as a whole. o Index Outperform Plan Company ABC vs. the S&P 500 Index as a whole. Risk-Free Rate Dividend Yield Timeframe Until Vesting Assumed Trading Days per Year Company ABC, Inc. Relative Total Shareholder Return Plan Analysis 8
Required Valuation Assumptions Closing Stock Prices Since the Beginning Average Prices ( BAPs ) are based upon the 20-trading days including and following the grant date, we have simulated the BAPs as well as the Ending Average Prices ( EAPs ). Due to fact that the BAPs do not include stock prices for any dates prior to the Grant Date, the BAPs are not needed as an input for the valuation model. The only required stock price inputs are the closing stock prices as of the Grant Date. Company ABC s Closing Stock Price as of 7/1/2015 = $30.18 A summary of the Closing Stock Prices for the components of the Peer Group are summarized in Appendix A. Company ABC, Inc. Relative Total Shareholder Return Plan Analysis 9
Required Valuation Assumptions Volatilities ASC 718-55-40 States: An entity shall establish a process for estimating expected volatility and apply that process consistently from period to period. The volatility is the amount by which a stock is expected to fluctuate over the projection period. Generally, the volatility assumption has a significant impact on the fair value of an award. The higher the volatility, the higher the fair value will be, all else being equal. This is due to the fact that the higher the volatility, the greater the chance that the stock price will appreciate substantially over the performance period. Since most companies have long established methodologies for determining a volatility assumption under ASC 718, it is critical that the modeling of the award valued in this report incorporates a volatility assumption that is calculated in a consistent manner with the company s established process. Company ABC has established a process for determining volatility that is based upon Company ABC s daily historical volatility, commensurate with the time to maturity of the award. To be compliant with ASC 718, we have based the calculation of the volatility assumption upon historical daily stock price observations. Since the awards will vest on the third anniversary of the grant, June 30, 2018, we have calculated the volatility assumptions based upon a 3.00 year look-back period. Company ABC Most Recent 3.00 Year Volatility = 22.17% A summary of the volatilities for the components of the Peer Group and the S&P 500 Inde4x as a whole are summarized in Appendix A. Company ABC, Inc. Relative Total Shareholder Return Plan Analysis 10
Required Valuation Assumptions Correlations Since stocks tend to move in a correlated manner, correlation estimates for Company ABC and each company in the peer group must be calculated. We have used the Index Correlation Model 2 in determining the Fair Value of the award since the Peer Group is large and well diversified. This method requires a correlation assumption of Company ABC and each company in the peer group with the S&P 500 Index as a whole. The correlations have been based upon a 3-year look-back period. Company ABC Correlation with S&P 500 = 0.5323 A summary of the correlations for the components of the Peer Groups vs. the S&P 500 Index as a whole are summarized in Appendix A. 2 The Index Correlation Model bases the stock price movements on the stock price correlations between each company and the Index as a whole. Company ABC, Inc. Relative Total Shareholder Return Plan Analysis 11
Required Valuation Assumptions Risk-Free Rate = 1.08% ASC 718 55-12 States: Option-pricing models call for the risk-free interest rate as an assumption to take into account, among other things, the time value of money. A U.S. entity issuing an option on its own shares must use as the risk-free interest rates the implied yields currently available from the U.S. Treasury zero-coupon yield curve over the contractual term of the option if the entity is using a lattice model incorporating the option s contractual term. If the entity is using a closedform model, the risk-free interest rate is the implied yield currently available on U.S. Treasury zero-coupon issues with a remaining term equal to the expected term used as the assumption in the model. The risk-free rate is the amount that the stock price is expected to return, on an annual basis, during the simulation process. The risk-free rate has a fairly minimal impact on the grant date fair value but in general, the fair value will increase as the risk-free rate increases. Below, is a summary of the U.S. Treasury Risk-Free Rates as of July 1, 2015. Since the simulation period is 3.00 years, we have applied the rate associated with a bond with a 3.00 year term. U.S. Treasury Bond Yields as of July 1, 2015 Time to Maturity Yield 0.250 0.01% 0.500 0.01% 1.000 0.13% 2.000 0.28% 3.000 1.08% 5.000 1.70% 7.000 2.14% 10.000 2.43% 1 The rates shown in the chart above can be found at the following link: http://www.treasury.gov/resourcecenter/data-chart-center/interest-rates/pages/textview.aspx?data=yield Company ABC, Inc. Relative Total Shareholder Return Plan Analysis 12
Required Valuation Assumptions Dividend Yield The Grant Agreement specifies that the shares will vest based upon future stock prices, inclusive of dividends, paid over the performance period. The agreement also states that holders of the award will receive dividend equivalents, at the time of vesting, for any dividends paid prior to the vesting date. The amount of the divided equivalents will be based upon the number of shares earned at vesting. When a company pays a dividend, its stock price will theoretically drop, as of the opening of trading the day immediately following the Ex-dividend date, by the amount of the dividend paid to shareholders. The reason for this is that when a dividend is paid, it comes directly from cash assets at the firm. Since ownership of the cash assets are transferred directly from the company to shareholders, the value of the company's equity must be reduced to reflect this loss of assets from the point of view of the company. If a plan bases the calculation of the payout percentage inclusive of dividends and pays dividend equivalents to the award holders for all dividends paid over the performance period, the dividend yield will have no impact on the grant date fair value. However, if the award does not base the calculation of the payout percentage inclusive of dividends paid during the performance period, the dividend yield will have an inverse (and possibly substantial) impact on the grant date fair value. This is due to the decrease in stock prices, as a result of dividends being paid, causing the vesting hurdles to be harder to reach. Similarly, if dividend equivalents are not paid to the award holders for dividends paid during the performance period, the grant date fair value of the award would be reduced by the present value of dividends paid between the performance period begin date and the vesting date. To correctly calculate the grant date fair value, one could model the stock price to each projected future Ex-dividend date and then calculate the fraction of shares that could be purchased with the dividend amount paid at that time. The sum of these fractional shares would be used to determine an adjusted stock price that reflects dividend payments. The adjusted Company ABC, Inc. Relative Total Shareholder Return Plan Analysis 13
Required Valuation Assumptions prices would then be compared to the vesting hurdles at every future date over the performance period to determine if a tranche vests. There is however a much simpler way to model dividend reinvestment in this case. If we assume that the company does not pay a dividend, or use a dividend yield assumption of 0.00%, and simulate stock prices over the performance period, the value of the dividends would remain within the equity of the firm. Assuming a dividend yield of 0.00% would result in an identical fair value as if future dividend payments were modeled and tracked and the final value of the payout was based upon stock price appreciation and dividend payments. We have therefore applied a dividend yield of 0.00% within the modeling process to reflect the plan s provisions that the achievement of the vesting hurdles is based upon reinvestment of dividends paid over the performance period and that the holders of the awards will receive dividend equivalents based upon the number of awards earned. Dividend Yield Assumption All companies = 0.00% Company ABC, Inc. Relative Total Shareholder Return Plan Analysis 14
Required Valuation Assumptions Simulation Term = 3.00 Years The time to maturity is the time frame from the valuation date to the end of the performance period. Generally, the longer the time to maturity, the greater the grant date fair value, as there is a longer time frame over which Company ABC s stock price can appreciate. Trading-Days (Per Calendar Year) = 250 We have assumed that there are 250 trading days per calendar year in the simulation model. This is an approximation of the number of trading days in an average calendar year. This assumption has a de minimus impact on the grant date fair value. Company ABC, Inc. Relative Total Shareholder Return Plan Analysis 15
Valuation Assumptions Summary The assumptions summarized below are based upon the information provided on pages 8 through 14 of this report. Company ABC should verify the assumptions to ensure that they are accurate and consistent with the methodologies used to calculate valuation assumptions for all awards that have been issued previously by Company ABC. Valuation Assumption Summary Assumption Company ABC Peer Group Stock Prices as of 7/1/2015 $30.18 See Appendix A Volatility 22.17% See Appendix A Correlation with S&P 500 Index 0.5323 See Appendix A Risk-Free Rate 1.08% 1.03% Applicable Dividend Yield 0.00% 0.00% Simulation Term Years 3.00 3.00 Trading Days per Calendar Year 250 250 Company ABC, Inc. Relative Total Shareholder Return Plan Analysis 16
Fair Value Summary The Fair Value per targeted share issued has been determined to be: Component Rank Plan Value Design 1 Design 2 Design 3 Design 4 Design 5 Simulated Fair Value $44.05 $38.03 $36.55 $35.88 $31.43 Stock Price on Grant Date $30.18 $30.18 $30.18 $30.18 $30.18 Fair Value as a Percentage of Grant Price 145.97% 126.00% 121.09% 118.88% 104.14% Amortization Period - Years 3.00 3.00 3.00 3.00 3.00 Index Outperform Plan Value Design 1 Design 2 Design 3 Design 4 Design 5 Simulated Fair Value $34.58 $33.29 $37.11 $37.01 $33.22 Stock Price on Grant Date $30.18 $30.18 $30.18 $30.18 $30.18 Fair Value as a Percentage of Grant Price 114.56% 110.31% 122.96% 122.63% 110.07% Amortization Period - Years 3.00 3.00 3.00 3.00 3.00 The fair values calculated and summarized above are based upon the vesting tables shown on pages 2 and 3 and the required valuation assumptions detailed in this report. All calculations are as of the Grant Date, July 1, 2015 unless otherwise stated. The valuations provided in this report are based upon 718 Valuations interpretation and experience performing valuations of awards containing Market Conditions. Please feel free to call us if any questions arise for any information provided in this report. Company ABC, Inc. Relative Total Shareholder Return Plan Analysis 17
Valuation Certification Company ABC must review the plan description and assumptions calculated in this report to ensure compliance under ASC 718. We have taken all reasonable steps and tests to confirm that the valuation assumptions have been calculated correctly based upon the stock prices that we have provided you in the file named Company ABC Design Estimates July 1, 2015 Supporting Data.xls has based the valuation assumptions upon the data summarized in the report. has made every effort to avoid any data errors which would impact the valuation assumptions used in the modeling process. Company ABC should review the data used to calculate the valuation assumptions to ensure compliance under ASC 718. The valuation process used to determine the Fair Value of the award detailed in this report is based upon the most current financial and derivative valuation theory. The Monte Carlo simulation techniques used are in congruence with the underlying principles of the Black-Scholes option pricing model with respect to stock price projection and stock price probability distribution. Please feel free to contact us at any time should any questions arise with regards to this report including any questions or concerns that might arise from your auditors. As with most financial analysis the results of the model can be very sensitive to the model inputs. Additional sensitivity analysis is available upon request. James Lecher, CEP Zaneta Chapman, PhD Company ABC, Inc. Relative Total Shareholder Return Plan Analysis 18
Appendix A Closing Stock Prices as of July 1, 2015 for the Components of the S&P 500 Index Peer Group - Summary of Closing Stock Price Min Price Max Price No. of Companies $0.00 $25.00 44 $25.01 $50.00 134 $50.01 $75.00 123 $75.01 $100.00 89 $101.01 $125.00 43 $125.01 $150.00 26 $150.01 $175.00 14 $175.01 $200.00 7 $200.00 23 Total 503 The chart above shows a summary of the Closing Stock Prices for the components of the S&P 500 Index. A complete listing of Closing Stock Prices for all companies has been provided in an Excel file named: Company ABC Design Estimates July 1, 2015 Supporting Data.xls
Appendix A (Continued) Volatilities as of July 1, 2015 for the Components of the S&P 500 Index Peer Group Summary of Volatilities Min Volatility Max Volatility No. of Companies 10.00% 0 10.00% 15.00% 24 15.00% 20.00% 164 20.00% 25.00% 158 25.00% 30.00% 82 30.00% 35.00% 44 35.00% 40.00% 18 40.00% 45.00% 6 45.00% 7 Total 503 The chart above shows a summary of the Volatilities for the components of the S&P 500 Index. A complete listing of the Volatilities for all companies has been provided in an Excel file named: Company ABC Design Estimates July 1, 2015 Supporting Data.xls
Appendix A (Continued) Correlations as of July 1, 2015 for the Components of the S&P 500 Index vs. the S&P 500 Index as a Whole Peer Group - Summary of Correlations Min Correlation Max Correlation No. of Companies 0.3000 0 0.300 0.350 6 0.350 0.400 10 0.400 0.450 32 0.450 0.500 41 0.500 0.550 62 0.550 0.600 81 0.600 0.650 93 0.650 0.700 92 0.700 0.750 52 0.750 0.800 40 0.800 24 Total 503 The chart above shows a summary of the correlations for the components of the S&P 500 Index vs the S&P 500 Index as a whole. A complete listing of the correlations for all companies has been provided in an Excel file named: Company ABC Design Estimates July 1, 2015 Supporting Data.xls
Appendix B Correlated Stock Price Simulation Process Description Stock Price Projection Probability Distribution The underlying assumption for how the stock prices will evolve over time is that stock prices follow a Log-Normal probability distribution. This assumption is identical to the underlying assumption of how stock prices change over time used in the famous Black-Scholes option pricing model. Using a Log-Normal distribution allows the model to project stock prices over long periods of time while not allowing the stock prices to become negative. This is very important as owners of a company's common equity have no liability incurred upon them due to owning shares of the company. The constraint that a company's stock price cannot become negative reflects this limited liability of holders of common equity. The following equation describes how the model projects stock prices over time for the S&P 500 Index: P T= P 0 e [(RFR DY σ2 2 ) T+σ W TI] Now, since we must take into account the correlation coefficient between Company ABC and the S&P 500 Index, the following equation is used to project stock prices over time for Company ABC, This equation ensures that the Company ABC s stock prices are correlated that S&P 500 Index prices, based upon the assumed correlation coefficient. P T= P 0 e [(RFR DY σ2 2 ) T+σ (W TI r + W TC 1 r 2 )]
Appendix B (Continued) Where: P T = The stock price at the end of the performance period. P 0 = The starting stock price as of the grant date. RFR = The annualized risk-free rate of return. DY = The annualized dividend yield. σ = The annualized expected stock price volatility. T = The timeframe over which the stock price is being simulated. r = The correlation coefficient of Company ABC and the S&P 500. W TI W TC = a Wiener process (Brownian Motion) for the S&P 500 Index. = a Wiener process (Brownian Motion) for Company ABC. The modeling of stock prices in this manner is sometimes referred to as modeling stock price changes based upon Geometric Brownian Motion. The principles explained above are identical to the principles used to project stock prices in the Black-Scholes option pricing model. Therefore, if a plain vanilla stock option were valued using a Monte-Carlo simulation model or the Black-Scholes model, the predicted values would be identical.
Appendix C Antithetic Variance Reduction Technique In order to ensure that the fair value will converge within a reasonable number of simulations, we have incorporated a variance reduction technique into the modeling process. The variance reduction technique used is referred to as the Antithetic Variance reduction technique. This technique is based on the equation that describes the variance of two random variables X and Y. The variance of the two random variables X and Y is shown below: Variance (X, Y) = Variance(X) + Variance(X) + 2 Covariance(X, Y) Where, and, ρ = Correlation between X and Y Covariance(X, Y) = ρ σ x σ y σ x = σ y = Standard Deviation of X Standard Deviation of Y
Appendix C (Continued) Now if we can force ρ to be a negative number it will cause the covariance to be a negative number and will reduce the overall variance of the random variables X and Y. Luckily this is quite easy to do. If within the simulation model we make an initial random uniform draw over the interval [0,1] and then covert the random draw to a Cumulative Inverse Normal Draw, we can simulate one stock price path to time T. If we then subtract the initial random uniform from 1, and covert that random draw to a Cumulative Inverse Normal Draw, we can generate a second stock price path. If we then calculate the award payout as described on Page 7 based upon both randomly generated stock prices and average the two fair values, and repeat this process for all simulations performed it will cause the fair value to converge, will less simulations, than if we were to simulate one fair value per simulation and average the fair values over all simulations performed.