Prepared for. The Silicon Valley Chapter of Financial Executives International. Prepared by

Transcription

1 Valuation of Employee Stock Options and Other Equity-Based Instruments: Short-Term and Long-Term Strategies for Complying with FAS 123R and for Optimizing the Performance of Equity-Based Compensation Programs under the New Standard Prepared for The Silicon Valley Chapter of Financial Executives International Prepared by Ronald D. Rudkin, Ph.D. Vice President Analysis Group, Inc. Two Embarcadero Center, Suite 1750 San Francisco, California (415) April, 2006

2 Abstract This report discusses strategic actions firms can take both now and in the future to comply with FAS 123R and to optimize the performance of their equity-based compensation programs (EBCPs) under the new standard. It discusses challenges that firms are expected to face when attempting to comply with the new FAS 123R input requirements and recommends strategies for dealing with them. In the report we evaluate the key differences between the two most often used valuation models -- the modified Black-Scholes (MBS) and the binomial lattice models and show that latticebased models produces more accurate and generally lower fair value estimates than MBS-based models for both traditional employee stock options (ESOs) and for the nontraditional instruments (e.g., premium, capped, indexed and performance-based options) analyzed in the report. It discusses changes firms can make to the features of traditional ESOs that can enable them to better accomplish their HR objectives and to lower compensation expense. Finally, the report discusses how nontraditional instruments can enable firms to better accomplish their HR goals and to reduce compensation expense compared to traditional ESOs. The nontraditional options analyzed are shown to produce lower fair value estimates than traditional ESOs.

3 Acknowledgements I wish to thank the members of the Silicon Valley Chapter of the Financial Executives International (FEI) who participated in this project. Special thanks goes to Randy Bambrough, Gary Bohe-thackwell, Richard Brounstein, Todd Lowenstein, Michael Shahbazian and Dave Wittrock for their support and encouragement during the project and for permitting me to present a summary of this paper during one of their dinner meetings. I wish to especially thank Todd Lowenstein for engaging me in numerous stimulating conversations and exchanges throughout the course of the project.

4 Contents Executive Summary...1 I. Introduction...8 II. Selection of an Appropriate Valuation Model...11 III. Key Issues Concerning the Estimation of Model Inputs and Measures of Employees Exercise and Post-Vesting Employment Termination Behavior...19 IV. Strategies for Optimizing the Performance of Equity-Based Compensation Programs Under FAS 123R...22 V. Conclusions...33 References...34 Appendix A...35 Appendix B...39

5 Executive Summary This report discusses and evaluates strategic actions that firms can take both now and in the future to comply with FAS 123R and to optimize the performance of their equity-based compensation programs (EBCPs) under the new standard. The new standard provides an opportunity for firms to take strategic actions that have the potential to reduce compensation expense and improve the effectiveness of their EBCPs. These actions include determining the most appropriate: Valuation models to use; Data and methods for estimating model inputs and measures of employees exercise and post-vesting termination behavior (calibration measures); and Instruments firms can use to accomplish their human resources goals (attract and retain employees, align employee and shareholder interests and reduce dilution) and to reduce compensation expense. 1 The report is not intended to provide a comprehensive assessment of all aspects of FAS 123R. Rather, it is intended to focus on the key implementation challenges firm s are expected to face and those aspects of FAS 123R that are expected to have the greatest impact on the cost and effectiveness of EBCPs. 1. Report s Main Findings The report s main findings are: Initially, most firms will be able to use fairly simple methods to estimate the inputs required for the modified Black-Scholes (MBS) model. However, the SEC s simple method for estimating Expected Term (ET) will sunset at the end of Also, this method and methods that produce unbiased estimates based on a firm s transaction data are expected to produce larger estimates of both ET and fair value than those produced by the common practice of taking the weighted average of observed settlement times. The report discusses challenges that firms are expected to face when attempting to comply with the FAS 123R input requirements and recommends strategies for dealing with them. Lattice-based models are shown to produce more accurate and generally lower fair value estimates than the MBS model for both traditional employee stock options (ESOs) and the nontraditional instruments analyzed in this report. The report discusses situations where the MBS model either can t be used or should not be used (e.g., to determine the fair value of a Maximum Value Options or to determine the impact on fair value of changing an instrument s features) to perform valuations. 1 The nontraditional instruments include premium, capped, indexed and performance-based options with either performance or market conditions as well as both standard and performance-based versions of restricted stock, restricted stock units and stock appreciation rights. The features include an instrument s contractual term, type and length of the vesting schedule, vesting frequency (e.g., annual or monthly) and strike price. 1

6 Using a lattice model that allows inputs, especially stock price volatility, to vary with time is shown to produce more accurate and, under today s conditions, substantially lower fair value estimates than would result if the inputs were held constant. Changing the features of traditional ESOs can enable firms to better accomplish their HR objectives and, for both types of models, to produce lower fair value estimates than are were produced under the base case. Nontraditional instruments have the potential to enable firms to better accomplish their HR goals and to reduce compensation expense compared to traditional ESOs. For example, stock settled stock appreciation rights are preferred by employees, reduce dilution and have the same fair value as traditional ESOs. Lastly, for both types of models the nontraditional options analyzed were shown to produce lower fair value estimates than traditional ESOs. The following pages provide added detail about the report s main findings. 2. Selection of an Appropriate Valuation Model The report discusses the key differences between, as well as, the pros and cons of the two most often used types of valuation models the modified MBS and the binomial lattice (lattice) models. 2 It also compares the fair value estimates produced by both types of models for a traditional employee stock option (ESO). 3 The report shows (see Figure 1) that the fair values produced by the MBS model are 20% to 40% (average 27%) greater than those produced by the lattice model for a three-year graded vested option. 4 For comparability the Expected Term (ET) output by the lattice model is used in the MBS model The MBS model is the traditional Black-Scholes model with the instrument s expected term substituted for its contractual term. All of the valuations shown in this report are based on MBS-based and lattice-based models that are specifically designed to reflect the features of the traditional employee stock options and nontraditional instruments being valued. The valuations are based on model inputs that are similar to those used in Illustration 4 of FAS 123R. The calibration measures, such as expected term, that are output by the lattice model are based on what are known as risk-neutral probabilities (i.e., the probability of the stock price increasing depends on the risk-free rate). For completeness we also provide estimates of expected term and other calibration measures based on risk-adjusted probabilities (i.e., the probability of a stock price increase depends on the risk-adjusted return). Using risk-adjusted probabilities to estimate expected term, the fair values produced by the MBS model are roughly 15% to 30% (average 22%) greater than those produced by the lattice model. 2

7 Figure 1. Fair Values for Three-Year Graded Vested Options Lattice Model vs. Modified Black-Scholes Model $16.00 Lattice Modified Black-Scholes $14.00 $12.00 $ % $ % $11.07 $ % $12.46 $14.72 $8.00 $9.21 $6.00 $4.00 $2.00 $0.00 Tranche 1 Tranche 2 Tranche 3 In FAS 123R firms are asked to consider the effect on fair value of expected changes in model inputs. By allowing the risk-free rate and volatility to vary with time, we show that the fair value estimates are more accurate and 13% lower than those that result from holding these inputs constant. The fair value produced by the MBS model (with inputs based on the averages of the time-varying values for the risk-free rate and volatility) are 40% greater than those produced by the lattice model when these two inputs are allowed to vary with time. 3. Estimation of Model Inputs and Measures of Employees Exercise and Post-Vesting Employment Termination Behavior Following are the key challenges facing firms with respect to the FAS 123R input requirements: Inputs are to be forward-looking and reflect expected changes during the instrument's contractual term (lattice model) or ET (MBS model); When estimating volatility, firms are to consider mean reversion 6 and the implied volatility method; As discussed in Footnote 58 of FAS 123R, mean reversion is the tendency of volatility to converge to some long-term equilibrium value. With this method, volatility is inferred from the market price of traded instruments. 3

8 Firms are required to estimate the number of options expected to vest. Under the old standard, firms could assume that all of the options would vest and were then trued up based on actual experience; When estimating ET, firms are required to aggregate awards into relatively homogeneous groups with respect to exercise and post-vesting termination behavior for all types of valuation models; and When estimating ET, firms are to consider the effect of factors such as length of the vesting period, volatility, blackout dates, path of the firm s stock price and employee characteristics in order to reflect differences between historical and expected future conditions. In Staff Accounting Bulletin 107 (SAB 107), the SEC staff provided additional guidance that has the potential to simplify the estimation of some of the key model inputs, at least initially. 8 SAB 107 provides a simplified method that firms can use to compute ET for plain vanilla options. Also, SAB 107 provides guidance concerning when firms are allowed to rely exclusively on either the implied or historical methods for estimating volatility. The implications of the FAS 123R standard and additional guidance given by SAB 107 are discussed below. a. SAB 107 simplified method for estimating ET The estimation of expected term will be greatly simplified for firms that are able to use this method. However, the report recommends that firms carefully assess the pros and cons of using this method compared to methods that utilize a firm s transaction data. b. Estimating volatility The process of estimating volatility will be greatly simplified for firms that are able to place exclusive reliance on either the implied method or the historical method. In the event that a firm is either unable, chooses not to place exclusive reliance on either of these methods, or there is a considerable difference between the estimates of short-term and long-term volatility, then we recommend that the firm combine the estimates produced by these two methods. The report recommends the use of a statistical method for estimating the convergence or mean reversion from short-term volatility (based on implied volatility) to longterm volatility (based on the historical method). c. Estimating pre- and post-vesting terminations Under the new standard, firms are required to estimate the number of options expected to vest. A key input to this calculation is the pre-vesting departure rate or turnover rate. The report discusses both simple and complex methods that can be used to estimate the departure rate. The more complex methods are able to reflect the influence of factors, such as the level of the firm s stock price on a firm s departure rate. The termination rate is also used to reflect the effect of post-vesting termination behavior on fair value. 8 SEC Staff, Staff Accounting Bulletin No. 107, dated March 29,

9 d. Estimating ET The report discusses reasons why developing unbiased estimates of ET, based on a firm s transaction data, is expected to be difficult and proposes methods for overcoming these difficulties. Fairly detailed data and sophisticated estimation methods are expected to be required to 1) aggregate data into relatively homogeneous groups, 2) deal with the potential bias due to censoring (incomplete life cycle data as of the evaluation date), and 3 control for differences between the historical values of key explanatory variables (e.g., length of the vesting period, volatility, blackout dates, path of the firm s stock price and employee characteristics) affecting ET and the values of these variables that are used for valuation purposes. e. Estimating measures of employees exercise behavior required by lattice models In the following report, we also discuss methods that can be used to estimate measures of employees exercise behavior (e.g., expected time-to-exercise, probability of exercise and ratio of the stock price at exercise to the strike price) that are required to calibrate lattice models. For the most part, these methods are similar to those that are recommended for estimating ET. 4. Strategies for Optimizing Performance under FAS 123R a. Evaluation of impact of changing ESO features By changing the features of an ESO (e.g., contractual term, strike price, vesting schedule and attribution method), the report discusses ways that firms may be able to better accomplish their HR objectives and reduce compensation expense. Using a lattice model, we show that reducing the contractual term by three years, reducing the length of the graded vesting schedule by one year, increasing the strike price by $3.50 and increasing the vesting frequency from annual to monthly reduces the fair value produced by the lattice model by roughly 3%, 7%, 9% and 13%, respectively. The MBS model shows directionally similar, but generally smaller reductions than the lattice model (See Figure 2). Reducing the option s contractual term is the only exception. The MBS model shows 7.7% reduction in fair value compared to a 2.5% reduction in fair value for the lattice model. This occurs because the reduction in contractual term is predicted by the lattice model to produce a much greater reduction in ET (16%) than in fair value (2.5%). The greater reduction in ET results in a greater reduction in fair value predicted by the MBS than that predicted by the lattice model. 5

10 Figure 2. Effect on Fair Values of Changing the Features of a Traditional Employee Stock Option Lattice Model vs. Modified Black-Scholes Model $14.00 $12.00 $10.00 $8.00 $ % $10.64 Lattice Model -7.20% -8.80% $10.14 $ % Modified Black-Scholes Model $6.00 $4.00 $2.00 $0.00 Base Reduce Contractual Term Reduce Vesting Schedule Length Increase Strike Price Increase Vesting Frequency Base Reduce Contractual Term Reduce Vesting Schedule Length Increase Strike Price Increase Vesting Frequency $9.53 $ % -4.40% -7.70% $12.17 $12.63 $ % $12.28 b. Evaluation of potential benefits of using nontraditional instruments Firms are expected to make greater use of nontraditional instruments because all instruments will be treated the same under the new standard and nontraditional instruments have the potential to allow firms to better accomplish their EBCP goals and reduce compensation expense. In this report, we discuss the objectives, strengths and weaknesses and provide fair value estimates for the types of nontraditional instruments expected to be most widely used by firms. We also show that for both types of valuation models, fair values are lower for the nontraditional options analyzed than for ESOs (see Figure 3). As shown in Figure 4, the MBS-based models generally produce higher fair value estimates than lattice-based models. For example, for premium, purchased and indexed options, the MBS-based models produce fair value estimates that are 27%, 31% and 47% greater than those produced by the lattice-based models. The sole exception is the Maximum Value Option (MVO), where the MBS-based model produces a lower fair value estimate than the lattice-based model. This understatement occurs because the MBS-based model understates the cash flows and overstates the discounting associated with an MVO. 6

11 Figure 3. Fair Values for Various Non-Traditional Instruments Compared to a Traditional ESO Lattice Model vs. Modified Black-Scholes Model $14.00 $12.00 $10.00 $8.00 $6.00 $ % $9.95 Lattice Model -9.44% % $9.88 $ % % $13.19 Modified Black-Scholes Model -1.59% -4.40% $4.00 $2.00 $0.00 Traditional ESO Premium Option Purchased Option Maximum Value Option Indexed Option Market-Based Option Traditional ESO Premium Option Purchased Option Maximum Value Option Indexed Option Market-Based Option $12.61 $ % $ % $9.01 $6.15 $5.89 NA 7

12 Figure 4. Fair Values for Various Non-Traditional Instruments Lattice Model vs. Modified Black-Scholes Model $14.00 $ % $ % $ % $12.98 Lattice Modified Black-Scholes $10.00 $8.00 $10.91 $9.95 $ % $ % $9.01 $6.00 $7.60 $6.15 $5.89 $4.00 $2.00 $0.00 Traditional ESO Premium Option Maximum Value Option Purchased Option Indexed Market-Based Option This report shows how a lattice model can be used to determine exchange ratios that will make employees indifferent between a traditional ESO and a particular nontraditional instrument. It provides an example that demonstrates how to compute the exchange ratio that will make employees indifferent between restricted stock and a traditional ESO. The example demonstrates that even though the fair value of the restricted stock is greater than that of a traditional ESO, total compensation expense is less, because fewer shares of stock are required to make employees indifferent toward the two instruments. 8

13 I. Introduction This report discusses and evaluates strategic actions that firms may wish to take both to comply with FAS 123R and to optimize the performance of their equity-based compensation programs (EBCPs) under the new standard. The new standard provides an opportunity for firms to take strategic actions that have the potential to reduce compensation expense and improve the effectiveness of their EBCPs. These actions include determining the most appropriate: Valuation models to use; Data and methods for estimating model inputs and measures of employees exercise and post-vesting termination behavior (calibration measures); and Instruments and features firms can use to accomplish their human resources goals (attraction, retention and alignment of employee and shareholder interests) and to reduce compensation expense. The instruments include traditional, capped, indexed and performance-based options with either performance or market conditions as well as both standard and performance-based versions of restricted stock, restricted stock units and stock appreciation rights. The features include the instrument s contractual term, type and length of the vesting schedule, vesting frequency (e.g., annual or monthly) and strike price. The report is not intended to provide a comprehensive assessment of all aspects of FAS 123R. Rather, it is intended to focus on the key implementation challenges firm s are expected to face and those aspects of FAS 123R that are expected to have the greatest impact on the cost and effectiveness of EBCPs. Section II discusses key issues related to the selection of an appropriate valuation model. The discussion focuses on the key differences between as well as and the pros and cons of the two most often used types of valuation models the Modified Black-Scholes (MBS) model and binomial lattice (lattice) model. Section II and Section IV also provide fair value estimates for both traditional employee stock options and nontraditional instruments based on MBS-based and lattice-based models. As required in FAS 123R these models are specifically designed to reflect the substantive features of the instruments being valued. 9 Using assumptions that are similar to those used in FAS 123R, the report shows that latticebased models generally produce more accurate and lower fair value estimates than the MBS model for both traditional ESO and nontraditional instruments. Section III discusses the pros and cons of both simple and complex methods that firms can employ to estimate model inputs and measures of employees exercise and post-vesting termination behavior required for both types of models. Finally, Section IV discusses and evaluates changes to the features of ESOs and discusses and evaluates the pro and cons of various nontraditional instruments. It shows that the nontraditional instruments are expected to enable firms to better accomplish their HR goals and to reduce compensation expense. This section also discusses method for estimating the exchange ratios that are designed to make employees indifferent between a particular nontraditional ESO and traditional ESOs. 9 The valuations discussed in this report are based on both lattice- and MBS-based models that were specifically designed to value both the traditional and nontraditional instruments discussed above. The lattice model and the methods used to estimate its inputs recently passed an audit by a Big Four accounting firm under the new FAS 123R standard. As part of the audit process, we prepared comparison of the results produced by our model compared to the best known models in the literature. These results, which are available upon request, show that our lattice model produces valuations and measures of employees exercise and post-vesting termination behavior that are either identical to or are close to those produced by the models in the literature. 9

14 The paper has two appendices. The first appendix discusses the changes to the traditional lattice model that are required to comply with the new standard. It also discusses the pros and cons of the various lattice models that have been developed to comply with the new standard. 10 The second appendix discusses the lattice and Black-Scholes-based models that were used in the report. 10 Analysis Group, Inc. has developed and evaluated the three types of lattice models discussed in Appendix A of this report. We have concluded that the first model (Generalized Version of the Traditional Lattice Model) provides the most flexible and accurate framework for valuing both traditional and nontraditional instruments. 10

15 II. Selection of an Appropriate Valuation Model One of the major challenges confronting firms under FAS 123R is the selection of appropriate valuation models. A. Requirements for Selecting a Valuation Model In the absence of market-based instruments, firms are responsible for selecting valuation models that comply with the FAS 123R requirements. Under FAS 123R, firms are allowed to use different types of models for different types of instruments. However, the model(s) that are selected must comply with the measurement objectives listed in Paragraph 8 of FAS 123R. This paragraph requires that the valuation model: Is applied in a manner that is consistent with FASB s fair value measurement objectives (which require firms to estimate, as of the grant date, the fair value of the equity instruments that the entity is required to issue when the employees have rendered the requisite service and satisfied any other conditions required to benefit from the instrument); Is based on generally accepted economic and financial theory; and Reflects the substantive characteristics of the instrument being valued. In addition to meeting these requirements, firms are required to use valuation models that, at a minimum, incorporate the following inputs: Exercise price; Expected term of the instrument; Grant date stock price; Expected volatility of the firm s underlying stock price; Expected dividends on the underlying shares; and The risk-free rate for either the expected term of the award (closed-form model) or contractual term of the award (lattice model). It should be noted that under FAS 123R firms are allowed to change the type of valuation model they initially select if a different technique is likely to result in a better estimate of fair value. Although neither FAS 123R nor SAB 107 state a preference for any particular type of model, they do discuss advantages that they believe lattice-based models have over the MBS model. For example, when comparing the two types of models, Paragraph 15 of FAS 123R states: A lattice model can be designed to accommodate dynamic assumptions of expected volatility and dividends over the option s contractual term, and estimates of expected option exercise patterns during the option s contractual term, including blackout periods. Therefore, the design of a lattice model more fully reflects the substantive characteristics of a particular employee share option or similar instrument. Also, SAB 107 states that for certain instruments, the MBS model may not be able to satisfy the third FAS 123R measurement requirement, because it is not designed to reflect certain 11

16 characteristics of the instrument. SAB 107 gives as an example that if exercise depends on a specific increase in the price of the underlying shares, then the MBS model would not generally be appropriate, because it is not designed to take market conditions into account. B. Pros and Cons to Consider When Evaluating MBS and Lattice Models Since most companies currently use the MBS model, their choice is expected to be either to continue using it or to switch to another type of model, such as a lattice model. This section discusses issues that should be considered when deciding to stay with or to switch to another type of model. The key difference between lattice and the MBS model is that lattice models are able to explicitly reflect the features of the instrument (e.g., vesting schedule, contractual term and blackout dates) being valued as well as employee exercise and post-vesting termination behavior. They are also able to accurately assess the effects of dynamic or time-varying inputs on fair value. With the MBS model, the effects of these factors can only be implicitly reflected by changing the instrument s ET. Also, the MBS model can only use fixed or static inputs. As a result, it has limited ability to accurately reflect the effect of time-varying or dynamic inputs. Pros and cons of the MBS include: Pros: 1. Both the firm and its auditor are familiar with this type of model; 2. This type of model can usually be implemented by the firm s own staff; 3. The cost of implementing this type of model is generally less than that of a lattice model; 4. The MBS model uses a single standard formula, which simplifies both valuation and the audit process. This is to be contrasted with the lattice model, where there are at least three different types of lattice models. These models differ primarily with respect to the methods used to reflect employees exercise and post-vesting termination behavior (see Appendix A for a description of these methods as well as the pros and cons of each); and 5. With the MBS model, firms are potentially able to use simple methods for estimating ET and stock price volatility. 11 Cons: 1. As is discussed in more detail below, in the longer term firms using the MBS model are expected to be required to use detailed data and sophisticated methods to estimate ET; 2. This model is generally not appropriate for certain types of instruments (e.g., options with caps or market-based performance conditions); 11 Firms using the MBS model will be able to use the simplified method developed by the SEC staff for estimating ET until December 31, 2007 and may be able to place exclusive reliance on either the implied volatility method or the historical method when estimating stock price volatility. 12

17 3. The MBS model is not able to accurately reflect the effect of time-varying or dynamic inputs; and 4. As shown in this report, the MBS-based model tends to produce less accurate and generally higher fair value estimates than lattice models. Pros and cons of a lattice model include: Pros: 1. Lattice models are viewed by both FASB (in FAS 123R) and the SEC (in SAB 107) as being better able to reflect the important features of the instrument being valued and the impact of dynamic or time-varying inputs on fair value than the MBS model; 2. Lattice models are able to accurately value most, if not all, equity-based instruments, including complex instruments with caps or market conditions; 3. Lattice models are able to accurately value instruments for which model inputs are expected to experience significant changes during the instrument s contractual term; 4. As shown later in this report, using typical inputs, a lattice model is generally more accurate and provides lower estimates of fair value for both traditional ESOs and nontraditional instruments than the MBS model; and 5. Lattice models can be calibrated to measures of exercise and post-vesting termination behavior that are generally easier to estimate (e.g., expected time-to-exercise and probability exercise) than ET (the measure of exercise and termination behavior required for the MBS model). Cons: 1. Initial audits are expected to be more involved for a lattice model than for the MBS model; 2. Lattice models are more costly to implement than the MBS model; and 3. Lattice models generally required a greater level of expertise than the MBS model. As a result, firms may be required to use outside experts. C. Comparative Evaluation of Fair Value Estimates Produced by Lattice and Closed-Form Models This section provides numerical evaluations of the lattice and MBS models. Table 1 shows the inputs and calibration measures used. 13

18 Parameter Table 1. Model Inputs Value Stock price $30 Strike price $30 Option duration Volatility Dividend yield Risk-free rate Annual departure rate Vesting period Expected time-to-exercise 10 years 50 percent 1.0 percent 4.1 percent 3.0 percent 3.0 years 4.1 years The inputs are similar to those used in Illustration 4 in FAS 123R. 12 However, instead of calibrating the lattice model to a sub-optimal exercise factor or Exercise Multiple (EM) of 2.0, we calibrated it to an estimate of Expected Time-To-Exercise (ETTE) of 4.1 years. The 4.1 year value was based on the results of a statistical model that was estimated from company data. The model is designed to provide forward looking estimates of ETTE that reflect the influence of the factors shown in Table 1 (e.g., volatility, departure rate, contractual term and length of the vesting period). We used ETTE rather then EM as a measure of employees exercise behavior, because this measure is generally easier to estimate accurately than EM. Also, we used a slightly different risk-free rate than is used in Illustration 4. The risk-free rate in Table 1 is the average of the time-varying risk-free rates that are used in Section II.E of the report. However, changing the risk-free rate from the average value used in Illustration 4 of 2.9% to the value shown in Table 1 has virtually no impact on the values of the calibration measures. Table 2 shows the fair value estimates and the calibration measures produced by the MBS and lattice models for a three-year graded vested option. For comparability the estimates of ET that are output by the lattice model are used as inputs to the MBS model. The method we used to calculate ET is similar to the method recommended in Paragraph A27 of FAS 123R. We use a three-year graded vested option, because, in essence, it also provides valuations for one-year, two-year and three-year cliff vested options. 12 In his comments, one of the reviewers of a prior draft stated that the inputs were not typical of those generally found in practice. In an attempt to deal with this concern, we have selected inputs that are similar to those used in FAS 123R. It should be noted that changing the inputs did not change our prior conclusions. 14

19 Tranche Lattice Model Table 2. Valuations Based on Risk-Neutral Probabilities Lattice Model vs. Modified Black-Scholes Model Modified Black- Scholes Model Percent Difference Expected Time-To- Exercise Expected Term Exercise Multiple 1 $9.21 $ % $11.07 $ % $12.46 $ % Average $10.91 $ % Table 2 shows that the values produced by the MBS model are 18% to 39% greater than those produced by the lattice model, with an average difference of 28%. Notice that the model is correctly calibrated, because the ETTE for the third tranche equals the required value of 4.10 years. Notice also that increasing the length of the vesting period, increases fair value for both types of models. It occurs for a lattice model because increasing the length of the vesting period prevents risk-averse employees from exercising their options as early as they would like. Delaying exercise will increase the time value of the option (opportunity for the stock price to increase) and thus the option s fair value. For the MBS model, increasing the length of the vesting period increases the option s expected term, which in turn, increases the option s fair value. D. Should the Calibration Measures Be Based on Risk-Neutral or Risk- Adjusted Probabilities? There is an unsettled debate as to the correct probability measure to be used to compute the calibration measures, especially ET. Two types of probability measures have been advocated in the literature: risk-neutral and risk-adjusted. The risk-neutral probability measure is the one typically used to value instruments. This measure assumes that a firm s stock price will increase at the risk-free rate. The risk-adjusted probability measure assumes that a firm s stock price will increase at the risk adjusted rate. The risk-adjusted rate includes a premium above the risk-free rate for the additional risk associated with holding a firm s stock. Most of the academic literature has used risk-neutral probabilities to compute various calibration measures. 13 According to Mark Rubinstein, it is also the method that practitioners use to compute ET. 14 FASB has also advocated the use of this probability measure. In FAS 123R, firms that use lattice models are required to output expected term. Paragraph A27 of FAS 123R discusses a method, based on risk-neutral probabilities, for computing ET. Also Paragraph 282 of FAS 123, FASB discusses a method, which is also based on risk-neutral See for example J. Hull and A. White; M. Rubinstein; J. Ingersoll; S. Huddard; and M. Garman. He states: we can use a binomial tree to calculate the (risk-neutral) expected life of the option, known in the trade as the option fugit. 15

20 probabilities, that firms can use to compute expected option life indirectly (using a lattice model) instead of directly using transaction data. Table 3 provides the same information shown in Table 2 using risk-adjusted probabilities to compute ET and the other calibration measures. Tranche Table 3. Valuations Based on Risk-Adjusted Probabilities Lattice Model vs. Modified Black-Scholes Model Lattice Model MBS Model Percent Difference Expected Time-to Exercise Expected Term Exercise Multiple 1 $9.21 $ % $11.07 $ % $12.46 $ % Average $10.91 $ % A comparison of Table 2 and Table 3 shows that the use of risk-adjusted probabilities does not affect the fair value produced by the lattice model and has only a minor effect on ETTE and EM. However, it does cause ET to decrease. The decrease in ET causes the fair value produced by the Black-Scholes model to also decrease. Table 3 shows that if risk-adjusted probabilities are used, the fair values produced by the MBS model are 15% to 31% greater than those produced by the lattice model, with an average difference of 22% (compared to 18% to 39% for risk-neutral probabilities). To be conservative, the remaining valuations shown in this report are based on risk-adjusted probabilities. 15 It should be noted that for the inputs in Table 1, calibrating the model to an ETTE of 4.1 years is equivalent to calibrating the model to an EM of 1.97, which is virtually identical to the EM of 2.0 used in Illustration However, the fair value estimates produced by the lattice model are not directly comparable with the fair value estimate shown in Illustration 4 of FAS 123R for two reasons. First, the fair value estimate in Illustration 4 does not reflect postvesting employment termination behavior, which, as required in FAS 123R, is reflected in our analysis. 17 Second, the model used in Illustration 4 assumes that, for vested options, exercise occurs whenever the stock price equals or exceeds twice the strike price. In essence, the model One of the reviewers that provided comments on a prior draft contended that using risk-adjusted probabilities would cause the difference between the results produced by the two types of models (MBS and lattice models) to essentially disappear. As can be seen from Table 2 this is not the case. As will be shown later, the difference between the fair value estimates produced by two types of models will increase even further when the model inputs are allowed to change during the instrument s contractual term. It should be noted that the lattice model used to perform the valuations is able compute the joint probability of either exercise or termination occurring at each node. As a result, it is able to output virtually any measure of employee exercise and post-vesting termination behavior, including ET, expected time-to-exercise, pre- and post-vesting cancellation rates and the probabilities of exercise. We were able to match the $14.69 figure shown in Illustration 4 by using a trinomial model that is able to place the layers of the lattice so that one of them equals the barrier where exercise is assumed to occur ($60) and by assuming that the risk free rate and the volatility are the averages of the values shown for each of these inputs in the illustration and the post-vesting termination rate is zero. Using a post-vesting termination rate equal to the forfeiture rate of 3.0%, shown in the illustration, produces a lower fair value estimate than the $14.69 shown in the illustration. 16

21 assumes that the exercise boundary is horizontal. It is well known that the exercise boundary (see Hall and Murphy, 2002) monotonically declines as one approaches the instrument s expiration date. Consistent with the literature, our lattice model assumes that exercise occurs whenever the stock price equals or exceeds a monotonically declining exercise boundary. The exercise boundary is placed so that the calibration measures output by the model match those estimated from a firm s historical data. E. The Effect of Time-Varying Inputs In FAS 123R firms are asked to consider the effect of expected changes in the key model inputs on fair value. This section evaluates the potential impact of allowing inputs to change during an instrument s contractual term. More specifically, we consider the effect of allowing the risk-free rate and the stock price volatility to vary with time. We forecast the term structure of both of these inputs by using company data and well-known estimation techniques. The risk-free rate is forecast by estimating forward rates using the bootstrap method (see Hull, 2003). When estimating the future path of volatility we estimate short-term volatility using the implied volatility method and long-term volatility using the historical volatility method. These methods are approved by both FASB and the SEC. Short term volatility is estimated to be roughly 35% and long-term volatility is estimated to be slightly in excess of 50%. We then estimate the rate of convergence or mean reversion from short-term to long-term volatility by using the variance targeting technique discussed in Chapter 17 of Hull, The method shows that volatility will converge or mean revert from short- to long-term volatility in about four to five years from the grant date. 19 For comparability, the MBS model is based on the averages of the time-varying values for the risk-free rate and stock price volatility. This is also the approach that is recommended in the literature when using the Black-Scholes model to value instruments with inputs that vary with time (see Wilmott, 1998, p. 121). As shown in Table 1, the resulting average values for the risk-free rate and stock price volatility are 4.1% and 50% respectively. In most cases, allowing the inputs to a lattice model to vary with time is straight forward. The sole exception is volatility. Allowing volatility to vary with time prevents a typical lattice model from recombining. That is, an up move followed by a down will not end up at the same position as a down move followed by an up move. When a lattice fails to recombine, the number of nodes that must be evaluated at each time step increases exponentially, instead of linearly, as in the case of a recombining tree. This prevents typical instruments from being accurately valued in a reasonable period of time. 20 The model used in this report is specially designed to accurately value instruments with time-varying volatility in a reasonable period of time It should be noted that the variance targeting (VT) approach is a variant of the GARH method. However, unlike the GARCH method that estimates long-term volatility, the VT approach merely estimates the rate of convergence or mean reversion from short- to long-term volatility; where short- and long-term volatility can be estimated based on methods that have been approved by both the SEC and FASB. The VT approach we recommend in this report has successfully passed an audit by a Big Four accounting firm. This conclusion contradicts the statement by one of the reviewers of a prior draft of this report that the volatility curve tends to be flat over most of the term of an option, only getting steep towards the end of the term when (a) most employees have probably already exercised and (b) the effect of discounting over the long term of the option both serve to mitigate the effect. A valuation problem will typically have 300 or more time steps (e.g., ten years and 30 steps per year). If this is the case, then the maximum number of nodes that must be valued for a recombining lattice is 301. For a nonrecombining lattice, the maximum number of nodes that must be valued is two multiplied by ten to the 90th power. This number is so large that even the fastest computer can t solve the problem in a reasonable period of time. 17

22 Table 4 shows the average fair value estimates (across the three tranches) for the base case where the inputs are held constant and the cases where the risk-free rate and stock price volatility are allowed to change during the instrument s contractual term. Table 4. Average Fair Value Estimates Based on Time-Varying Inputs Inputs Lattice Model Percent Difference from Constant Inputs Modified Black- Scholes Model Percent Difference from Constant Inputs Constant Inputs $10.91 NA $13.19 NA Risk-Free Rate Varies $ % $ % Risk-Free Rate and Volatility Vary $ % $ % Allowing the risk-free rate to vary with time causes the fair value produced by the lattice model to decline by about 1.0% (from $10.91 to $10.80). Allowing both the risk-free rate and volatility to vary with time causes the fair values produced by the lattice model to decline by 13%. However, the fair value produced by the MBS model does not change, because it is based on the constant inputs, which are derived from the averages of the time-varying values for the risk-free rate and volatility. Lastly, the fair value produced by the MBS model with constant inputs is 40% ((13.19/9.49-1)*100) greater than the fair value produced by the lattice model with time-varying inputs (for both the risk-free rate and volatility). 18

23 III. Key Issues Concerning the Estimation of Model Inputs and Measures of Employees Exercise and Post-Vesting Employment Termination Behavior The following section discusses the key challenges facing firms with respect to the FAS 123R input requirements as well as the additional guidance given in SAB 107. A. FAS 123R Input Requirements Below are key FAS 123R input requirements: Inputs are to be forward-looking and reflect expected changes during the instrument's contractual life (lattice model) or ET (closed-form model); When estimating volatility, firms are to consider mean reversion 21 and the implied volatility method; 22 Firms are required to estimate the number of options expected to vest. Under the old standard, firms could assume that all of the options would vest and then be trued up based on actual experience; When estimating ET, firms are required to aggregate awards into relatively homogeneous groups with respect to exercise and post-vesting termination behavior for all types of models; and Firms are to consider the effect of factors such as length of the vesting period, volatility, blackout dates, path of the firm s stock price and employee characteristics when estimating ET. B. Additional Guidance in SAB The additional guidance in SAB 107 has the potential to simplify the estimation of the key model inputs, at least initially. SAB 107 provides a simplified method that firms can use to compute ET for plain vanilla options. SAB 107 also provides guidance concerning when firms are allowed to place exclusive reliance on either the implied volatility method or the historical method. Firms are allowed to rely exclusively on implied volatility if: The company s valuation model is based on a constant volatility assumption (e.g., Black-Scholes model); The market prices of both the traded options and the underlying stock are measured at similar points in time and the dates are reasonably close to the grant date; The traded options are both near-the-money and close to the exercise price of the ESO; The maturities (new or remaining) of the traded options are at least one year; and The options are actively traded. Firms can place exclusive reliance on the historical method if: 1) there is no reason to assume that volatility in the future will differ from what it has been in the past, and 2) As discussed in Footnote 58 of FAS 123R, mean reversion is the tendency of volatility to converge to some long-term equilibrium value. The footnote states that statistical models have been developed that can reflect mean reversion. With this method, volatility is inferred from the market price of traded instruments. SEC Staff, Staff Accounting Bulletin No. 107, dated March 29,

24 historical data covers a reasonable period of time (at least equal to the expected term of MBS-based models and the contractual term of lattice-based models). C. Potential Implications for Firms 1. SAB 107 simplified method for estimating expected term The estimation of expected term will be greatly simplified for firms that are able to use this method. 24 However, the simplified method can only be used until December 31, Before adopting this method, it is recommended that firms carefully assess its pros and cons compared to approaches that utilize a firm s own transaction data. The SEC method will overstate ET (and thus fair value) for firms using the MBS model if the ET, based on transaction data, is less than the midpoint between the average length of the vesting period and the option s expiration date. 2. Estimating volatility The process of estimating volatility will be greatly simplified for firms that are able to place exclusive reliance on either the implied method or the historical method. 25 In the event that a firm is unable or there is a significant difference between short- and long-term volatility, then we recommend that firms consider combining the estimates produced by the two methods. One way to do this would be to use statistical methods, based on a company s stock price and dividend data, to estimate the rate of convergence or mean reversion from short-term volatility (based on implied volatility) to long-term volatility (based on the historical method). 26 Under today s conditions, where short-term volatility tends to be less than long-term volatility, using a lattice model with time-varying volatility can reduce fair value compared to holding it constant at the average of the values that volatility is expected to take during the instrument s contractual term. 3. Firms will need to estimate the number of options expected to vest Under the new standard, firms are required to estimate the number of options expected to vest. A key input to this calculation is the pre-vesting departure or turnover rate. The turnover rate can be estimated as the ratio of the number of employees departing each period to the number of employees at the beginning of the period. To produce more accurate departure rates rate estimates, firms can use statistical methods that are able to reflect the influence of factors, such as the level of the firm s stock price, health of the industry and employee characteristics. 27 The termination rate is also a key measure of the effect of post-vesting termination behavior on fair value With the SEC method, ET is computed as the midpoint between the average length of the vesting period and the instrument s expiration date. If a firm uses a lattice model and there is significant difference between the implied and historical volatility estimates, then the firm should consider the use of year-by-year or time-varying volatility estimates, possibly along the lines discussed later on in the paragraph. This view is consistent with that of PWC (see Page 4-33 of their document: FAS 123(R), Share-Based Payment-a multidisciplinary approach, May See Hull 2003, Chapter 17. See Green (2003). 20