PART Three A Sample Case Study Applying FAS 123 ccc_mun_pt3_131-132.qxd 8/20/04 9:26 AM Page 131 ccc_mun_pt3_131-132.qxd 8/20/04 9:26 AM Page 132 CHAPTER 10 A Sample Case Study T his chapter provides an example case study with detailed empirical justi- fications for the input assumptions used in the ESO valuation. These in- puts were obtained based on the 2004 proposed FAS 123 revision requirements and recommendations, and are used in the customized bino- mial lattice model. The customized binomial lattice used is a proprietary al- gorithm that incorporates the traditional BSM inputs (stock price, strike price, time to maturity, risk-free rate, dividend, and volatility) plus addi- tional inputs including time to vesting, changing forfeiture rates, changing suboptimal exercise behavior multiples, blackout dates, changing risk-free rates, changing dividends, and changing volatilities over time. 1 This propri- etary algorithm can be run to accommodate hundreds to thousands of lat- tice steps as well as incorporate Monte Carlo simulation of uncertain inputs whenever necessary. The following sections describe how each of the inputs was derived in the valuation analysis. The analysis is an excerpt from several real-life FAS 123 consulting projects. The numbers and as- sumptions have been changed to maintain client confidentiality but the re- sults and conclusions are still equally valid. The case study here goes through in selecting and justifying each input parameter in the customized binomial lattice model, and showcases some of the results generated in the analysis. Some of the more analytically intensive but equally important as- pects have been omitted for the sake of brevity. STOCK PRICE AND STRIKE PRICE The first two inputs into the customized binomial lattice are the stock price and strike price. For the ESOs issued, the strike price is always set at the stock price at grant date. This means obtaining the stock price will also yield the strike price. Table 10.1 lists the stock prices estimated by the firm’s investor relations department. Conservative and aggressive closing 133 ccc_mun_ch10_133-166.qxd 8/20/04 9:26 AM Page 133 stock prices were provided for a period of 24 months, generated using growth curve estimations. For instance, the closing stock price for Decem- ber 2004 is estimated to be between $45.17 and $50.70. In order to per- form due diligence on the stock price forecast at grant date, several other approaches were used. Twelve analyst expectations were obtained and their results were averaged. In addition, econometric modeling with Monte Carlo simulation was used to forecast the stock price. Using a path-depen- dent stochastic simulation model (Figure 10.1), 2 the average stock price was forecast to be $47.22 (Figure 10.2), consistent with the investor rela- tions stock price. The valuation analysis will use all three stock prices, and the final result used will be the average of these three stock price forecasts. 134 A SAMPLE CASE STUDY APPLYING FAS 123 TABLE 10.1 Stock Price Forecast from Investor Relations Estimate of Stock Price per Investor Relations Per Share Stock Price Grant Date Conservative Aggressive Comment 4-Mar-04 $37.51 $37.51 Actual 2-Apr-04 $33.40 $33.40 Actual May-04 $34.87 $35.56 Computed Jun-04 $36.34 $37.72 Computed Jul-04 $37.81 $39.88 Computed Aug-04 $39.28 $42.05 Computed Sep-04 $40.75 $44.21 Computed Oct-04 $42.22 $46.37 Computed Nov-04 $43.69 $48.53 Computed Dec-04 $45.17 $50.70 Per Investor Relations Jan-05 $45.89 $51.52 Computed Feb-05 $46.61 $52.34 Computed Mar-05 $47.34 $53.16 Computed Apr-05 $48.06 $53.98 Computed May-05 $48.78 $54.81 Computed Jun-05 $49.51 $55.63 Computed Jul-05 $50.23 $56.45 Computed Aug-05 $50.95 $57.27 Computed Sep-05 $51.68 $58.09 Computed Oct-05 $52.40 $58.92 Computed Nov-05 $53.13 $59.74 Computed Dec-05 $53.85 $60.56 Per Investor Relations ccc_mun_ch10_133-166.qxd 8/20/04 9:26 AM Page 134 MATURITY The next input is the option’s maturity date. The contractual maturity date is 10 years on each option issue. This is consistent throughout the en- tire ESO plan. Therefore, 10 years is used as the input in the binomial lat- tice model. A Sample Case Study 135 FIGURE 10.1 Stock price forecast using stochastic path-dependent simulation techniques. Starting Value $31.95 Time Asset Value Simulate Annualized Drift 60.00% 0.0000 31.95 Annualized Volatility 80.44% 0.0067 27.64 Forecast Horizon 0.67 0.0133 28.82 Granularity 100.00 0.0200 29.18 Step-Size 0.0067 0.0267 30.56 0.0333 28.40 0.0400 30.14 0.0467 31.72 0.0533 32.87 0.0600 34.16 0.0667 37.27 0.0733 38.10 0.0800 36.11 0.0867 37.16 0.0933 37.98 0.1000 39.03 0.1067 40.64 0.1133 39.14 0.1200 39.40 0.1267 40.12 0.1333 39.67 0.1400 42.70 0.1467 39.88 0.1533 39.64 Brownian Motion with Drift Forecast Values 0.00 10.00 20.00 30.00 40.00 50.00 60.00 70.00 80.00 90.00 100.00 0.0000 0.1000 0.2000 0.3000 0.4000 0.5000 0.6000 0.7000 (c) Johnathan Mun 2003 (Risk Analysis, Wiley 2003) This model illustrates the Brownian Motion stochastic process with a drift rate. An example application includes the simulation of a stock price path. This model requires Crystal Ball to run. Click on Crystal Ball's Single Step button to perform a step wise simulation and see why it is so difficult to predict stock prices. Click on Start Simulation to estimate the distribution of stock prices at certaintime intervals. Enter some values into the input boxes above (default values are $100 for starting value,10% for annualized drift, 45% for annualized volatility, and 1 for forecast horizon). –2.11 0.59 0.13 0.66 –1.14 0.87 0.74 0.49 0.54 1.33 0.28 –0.86 0.38 0.28 0.36 0.57 –0.62 0.04 0.22 –0.23 1.10 –1.07 –0.15 FIGURE 10.2 Results of stock price forecast using Monte Carlo simulation. ccc_mun_ch10_133-166.qxd 8/20/04 9:26 AM Page 135 RISK-FREE RATES The next input parameter is the risk-free rate. A detailed listing of the U.S. Treasury spot yields were downloaded from www.ustreas.gov as seen in Table 10.2. Using the spot yield curve, the spot rates were bootstrapped to obtain the forward yield curve as seen in Table 10.3. Spot rates are the in- terest rates from time zero to some time in the future. For instance, a two- year spot rate applies from year 0 to year 2 while a five-year spot rate applies from year 0 to year 5, and so forth. However, we require the for- ward rates for the options valuation, which we can obtain from bootstrap- ping the spot rates. Forward rates are interest rates that apply between two future periods. For instance, a one-year forward rate three years from now applies to the period from year 3 to year 4. Based on the date of valuation, the highlighted risk-free rates in Table 10.3 are the rates used in the chang- ing risk-free rate binomial lattice model (i.e., 1.21%, 2.19%, 3.21%, 3.85%, and so forth). 3 DIVIDENDS The firm’s stocks pay no dividends, and this parameter will always be set to zero. In other cases, if dividend yields exist, these yields are entered into the model, including any expected changes to dividend policy over the life of the option. VOLATILITY Volatility is the next input assumption in the customized binomial lattice model. There are several ways volatility can be measured, and in the in- terest of full disclosure and due diligence, all methods are used in this study. Table 10.4 shows the first method used to estimate the changing volatility of the firm’s stock prices using the Generalized Autoregressive Conditional Heteroskedasticity (GARCH) model. The inputs to the model are all available historical stock prices since going public. The re- sults indicate that the standard GARCH (1,1) model is inadequate to forecast the stock’s volatility due to the low R-squared, 4 low F-statistics, 5 and bad Akaike and Schwarz criterion statistics. As such, GARCH analy- sis is found to be unsuitable for forecasting the volatility for valuing the firm’s ESOs and its results are abandoned. Only GARCH (1,1) is shown in this example. In reality, multiple other model specifications were run and analyzed. 136 A SAMPLE CASE STUDY APPLYING FAS 123 ccc_mun_ch10_133-166.qxd 8/20/04 9:26 AM Page 136 TABLE 10.2 U.S. Treasuries Risk-Free Spot Rates Date 1mo 3mo 6mo 1yr 2yr 3yr 5yr 7yr 10yr 20yr 2/2/2004 0.87 0.94 1.03 1.29 1.83 2.36 3.18 3.70 4.18 5.02 2/3/2004 0.93 0.94 1.02 1.27 1.78 2.30 3.12 3.65 4.13 4.98 2/4/2004 0.91 0.94 1.01 1.27 1.80 2.32 3.15 3.67 4.15 5.00 2/5/2004 0.89 0.94 1.02 1.29 1.85 2.40 3.21 3.72 4.20 5.02 2/6/2004 0.89 0.93 1.01 1.26 1.77 2.29 3.12 3.63 4.12 4.95 2/9/2004 0.89 0.94 1.02 1.25 1.76 2.26 3.08 3.60 4.09 4.93 2/10/2004 0.91 0.95 1.02 1.27 1.82 2.33 3.13 3.64 4.13 4.97 2/11/2004 0.89 0.93 1.00 1.23 1.73 2.23 3.03 3.56 4.05 4.90 2/12/2004 0.90 0.93 1.00 1.24 1.75 2.26 3.07 3.58 4.10 4.94 2/13/2004 0.90 0.92 0.98 1.21 1.70 2.19 3.01 3.54 4.05 4.92 2/17/2004 0.90 0.95 1.00 1.21 1.70 2.20 3.02 3.54 4.05 4.91 2/18/2004 0.93 0.94 1.00 1.23 1.72 2.22 3.03 3.55 4.05 4.91 2/19/2004 0.93 0.94 1.00 1.23 1.70 2.20 3.02 3.54 4.05 4.91 2/20/2004 0.93 0.94 1.01 1.26 1.75 2.25 3.08 3.59 4.10 4.96 2/23/2004 0.95 0.97 1.02 1.22 1.69 2.21 3.03 3.55 4.05 4.92 2/24/2004 0.97 0.97 1.02 1.23 1.69 2.20 3.01 3.53 4.04 4.90 2/25/2004 0.96 0.96 1.02 1.23 1.67 2.16 2.98 3.51 4.02 4.89 2/26/2004 0.97 0.96 1.02 1.23 1.69 2.18 3.01 3.54 4.05 4.92 2/27/2004 0.95 0.96 1.01 1.21 1.66 2.13 3.01 3.48 3.99 4.85 Source: http://www.ustreas.gov/offices/domestic-finance/debt-management/interest-rate/yield20040201.html. 137 ccc_mun_ch10_133-166.qxd 8/20/04 9:26 AM Page 137 TABLE 10.3 Forward Risk-Free Rates Resulting from Bootstrap Analysis Annual Forward Curve Years 1 2 3 4 5 678910 2/2/2004 1.29% 2.37% 3.43% 4.01% 4.84% 4.75% 5.27% 4.99% 5.31% 5.63% 2/3/2004 1.27% 2.29% 3.35% 3.95% 4.78% 4.72% 5.25% 4.94% 5.26% 5.58% 2/4/2004 1.27% 2.33% 3.37% 3.99% 4.83% 4.72% 5.24% 4.96% 5.28% 5.60% 2/5/2004 1.29% 2.41% 3.51% 4.03% 4.85% 4.75% 5.26% 5.01% 5.33% 5.65% 2/6/2004 1.26% 2.28% 3.34% 3.96% 4.80% 4.66% 5.17% 4.94% 5.27% 5.60% 2/9/2004 1.25% 2.27% 3.27% 3.91% 4.74% 4.65% 5.17% 4.91% 5.24% 5.57% 2/10/2004 1.27% 2.37% 3.36% 3.94% 4.75% 4.67% 5.18% 4.95% 5.28% 5.61% 2/11/2004 1.23% 2.23% 3.24% 3.84% 4.65% 4.63% 5.16% 4.87% 5.20% 5.53% 2/12/2004 1.24% 2.26% 3.29% 3.89% 4.71% 4.61% 5.12% 4.97% 5.32% 5.67% 2/13/2004 1.21% 2.19% 3.18% 3.84% 4.67% 4.61% 5.14% 4.91% 5.25% 5.59% 2/17/2004 1.21% 2.19% 3.21% 3.85% 4.68% 4.59% 5.11% 4.91% 5.25% 5.59% 2/18/2004 1.23% 2.21% 3.23% 3.85% 4.67% 4.60% 5.12% 4.89% 5.23% 5.56% 2/19/2004 1.23% 2.17% 3.21% 3.85% 4.68% 4.59% 5.11% 4.91% 5.25% 5.59% 2/20/2004 1.26% 2.24% 3.26% 3.92% 4.76% 4.62% 5.13% 4.96% 5.30% 5.64% 2/23/2004 1.22% 2.16% 3.26% 3.86% 4.69% 4.60% 5.12% 4.89% 5.23% 5.56% 2/24/2004 1.23% 2.15% 3.23% 3.83% 4.65% 4.58% 5.10% 4.90% 5.24% 5.58% 2/25/2004 1.23% 2.11% 3.15% 3.81% 4.64% 4.58% 5.11% 4.88% 5.22% 5.56% 2/26/2004 1.23% 2.15% 3.17% 3.85% 4.69% 4.61% 5.14% 4.91% 5.25% 5.59% 2/27/2004 1.21% 2.11% 3.08% 3.90% 4.79% 4.43% 4.90% 4.85% 5.19% 5.53% 138 ccc_mun_ch10_133-166.qxd 8/20/04 9:26 AM Page 138 Two additional approaches are used to estimate volatility. The first is to use historical stock prices for the last quarter, last one year, last two years, and last four years (equivalent to the vesting period). These closing prices are then converted to natural logarithmic returns and their sample standard deviations are then annualized to obtain the annualized volatili- ties seen in Table 10.5. 6 In addition, Long-term Equity Anticipation Securities (LEAPS) can be used to estimate the underlying stock’s volatility. LEAPS are long-term stock options, and when time passes such that there are six months or so remaining, LEAPS revert to regular stock options. However, due to lack of trading, the bid-ask spread on LEAPS tends to be larger than for regularly traded equities. Table 10.5 lists the two LEAPS closest to the stock price forecast at grant date. Implied volatilities on both bid and ask are listed in Table 10.5. After performing due diligence on the estimation of volatilities, it is found that a GARCH econometric model was insufficiently specified to be of statis- tical validity. Hence, we reverted back to using the implied volatilities of long- A Sample Case Study 139 TABLE 10.4 Generalized Autoregressive Conditional Heteroskedasticity for Forecasting Volatility Dependent Variable: LOGRETURNS Method: ML–ARCH Date: 04/10/04 Time: 10:48 Sample: 1901 2603 Included observations: 703 Convergence achieved after 30 iterations Coefficient Std. Error z-Statistic Prob. GARCH –4.36065 1.794356 –2.430222 0.0151 C 0.004958 0.002188 2.266192 0.0234 Variance Equation C 3.10E-07 2.12E-06 0.145964 0.8839 ARCH(1) 0.031233 0.005472 5.707787 0.0000 GARCH(1) 0.971900 0.004750 204.5979 0.0000 R-squared 0.010575 Mean dependent var –0.001054 Adjusted R-squared 0.004905 S.D. dependent var 0.038647 S.E. of regression 0.038552 Akaike info criterion –3.841624 Sum squared resid 1.037432 Schwarz criterion –3.809224 Log likelihood 1355.311 F-statistic 1.865084 Durbin-Watson stat 2.125897 Prob(F-statistic) 0.114774 ccc_mun_ch10_133-166.qxd 8/20/04 9:26 AM Page 139 term options or LEAPS, and compared them with historical volatilities. The best single-point estimate of the volatility going forward would be an average of all estimates or 49.91 percent as shown in Table 10.5. However, due to this large spread, Monte Carlo simulation was applied by running a simulation on these volatility rates; thus, every volatility calculated here will be used in the analysis. For the purposes of benchmarking, the Wilshire 5000 and Standard & Poors 500 indices for the same period were found to be 20.7% and 20.5% respectively. The firm’s stock price has a stable beta of 2.3, making the beta- adjusted volatility 47%, which falls within the calculated volatility range. VESTING All ESOs granted by the firm vest in two different tranches: one month and six months. The former are options granted over a period of 48 months, where each month 1/48 of the options vest, until the fourth year when all op- tions are fully vested. The latter is a cliff-vesting grant, where if the employee leaves within the first six months, the entire option grant is forfeited. After the six months, each additional month vests 1/42 additional portions of the options. Consequently, one-month (1/12 years) and six-month (1/2 year) vesting are used as inputs in the analysis. The results of the analysis are sim- ply the valuation of the options. To obtain the actual expenses, each 48- 140 A SAMPLE CASE STUDY APPLYING FAS 123 TABLE 10.5 Volatility Estimates Volatility 4 Years 72.50% 2 Years 58.00% 1 Year 46.25% Quarter 43.55% LEAP: $45 Bid 45.50% LEAP: $45 Ask 47.50% LEAP: $50 Bid 41.50% LEAP: $50 Ask 44.50% Volatility Inputs: Triangular distribution with the following parameters into Monte Carlo simulation Min 41.50% Average 49.91% Max 72.50% ccc_mun_ch10_133-166.qxd 8/20/04 9:26 AM Page 140 [...]... 13,534,530.21 $183,963,675.91 $ 13,917,526.02 $ 14 ,109 ,035.58 $ 25,395,860.42 $ 14,480,685.27 $ 14,666, 510. 11 $ 14,852,311.66 $ 16,541,950.16 $ 16,746,357.48 $ 16,950,764.81 $ 94,488,780.60 $ 17,359,579.47 $235,401,537.28 $ 17,768,368.50 Aggressive $ 10, 151,679.73 $ 10, 315,358.76 $ 10, 479,07.80 $ 10, 642,716.84 $ 10, 806,395.88 $ 12,067,082.40 $ 12,247 ,106 .05 $ 12,427,152.99 $ 64,703,067.40 $ 12,787,246.87... illustrates the contribution to options valuation reduction The difference between the naïve BSM valuation of $26.91 versus a FIGURE 10. 5 Monte Carlo simulation of ESO valuation result 154 A SAMPLE CASE STUDY APPLYING FAS 123 TABLE 10. 11 Options Valuation Results Conservative Stock Price Aggressive Stock Price Average of Two Stock Prices Forfeiture Year 5.51% Risk Free Stock Price Strike Price Maturity... Average $45.17 $45.17 10 1.21% 49.91% 0% 2,400 1.8531 0.08 $12.93 TABLE 10. 9 Convergence of the Customized Binomial Lattice (Table Truncated) $45.17 $45.17 10 1.21% 49.91% 0% 3,000 1.8531 0.08 $12.88 $45.17 $45.17 10 1.21% 49.91% 0% 3,600 1.8531 0.08 $12.91 $45.17 $45.17 10 1.21% 49.91% 0% 4,200 1.8531 0.08 $13.00 $45.17 $45.17 10 1.21% 49.91% 0% 4,800 1.8531 0.08 $13.08 $45.17 $45.17 10 1.21% 49.91% 0%... simply the ratio of the stock price when it was exercised to the contractual strike price of the option Terminated employees or employees who left voluntarily were excluded from the analysis This is because employees who leave the firm have a limited time to execute the portion of their options that have vested In addition, all unvested options will expire worthless Finally, employees who decide to... still significantly 148 Stock Price Strike Price Maturity Risk-Free Rate Volatility Dividend Lattice Steps Suboptimal Behavior Vesting Binomial Option Value $45.17 $45.17 10 1.21% 49.91% 0% 10 1.8531 0.08 $20.55 $45.17 $45.17 10 1.21% 49.91% 0% 50 1.8531 0.08 $17.82 $45.17 $45.17 10 1.21% 49.91% 0% 100 1.8531 0.08 $17.32 120 600 1,200 1,800 2,400 3,000 3,600 4,200 4,800 5,400 6,000 1 5 10 15 20 25 30 35... lattice (see Figure 10A.3) This section exists only for simple options For more complex options with exotic inputs and changing inputs, the intermediate calculations are not shown The resulting ESO value calculated using the 10- step binomial approach is shown in the Results section as seen in Figure 10A.4 The trinomial lattice calculations have been disabled in the demo version The employee stock purchase... 9,719,311.21 $ 9,872,590.87 $ 10, 025,849.35 $ 10, 179 ,107 .83 $ 11,365,602.93 $ 11,534, 210. 56 $ 11,702,794.88 $ 60,926,665.19 $ 12,039,963.53 $163,625,443.32 $ 12,377,155.48 $ 12,545,739.81 $ 12,709,221.87 $ 12,872,727.23 $ 13,036,232.59 $ 13,199,714.65 $ 14,699,542.02 $ 14,879,372.29 $ 15,059,202.56 $ 83,935,039.46 $ 15,418,914.35 $209,062,661.14 $ 15,778,600.52 Conservative $ 10, 737,306.72 $ 10, 911,406.31 $ 11,085,484.73... 269 202 18 231 259 133 280 217 0 154 29 Years 0.34 Days to Cancellation 147 A Sample Case Study Convergence in Binomial Lattice Steps $17.20 S17 .10 Option Value $17.00 $16.90 Black-Scholes $16.80 $16.70 $16.60 $16.50 1 10 100 100 0 100 00 Lattice Steps FIGURE 10. 4 Convergence of the binomial lattice to closed-form solutions comparing its results to the basic binomial lattice Convergence is generally achieved... Table 10. 10 will have its own simulation result like the one in Figure 10. 5 The example illustrated in Table 10. 11 shows a naïve BSM result of $26.91 versus a binomial lattice result of $17.39 (the BSM using an adjusted four-year life is $19.55) This $9.52 differential can be explained by contribution in parts In order to understand this lower option value as compared to the naïve BSM results, Table 10. 12... was found to be 3.92 (Table 10. 7) The median calculated from the suboptimal exercise behavior range between 1.0 and 3.92 yielded 1.76 142 105 18 89961 95867 4103 8 14289 5025 16831 2200 34637 36058 35882 46869 30738 80763 25998 5817 81744 46899 75145 77678 20244 36156 08/28/2003 08/28/2003 09/03/2003 12/31/2003 12/31/2003 09/29/2003 10/ 28/2003 08/28/2003 07/28/2003 10/ 30/2003 10/ 30/2003 07/28/2003 12/01/2003 . $45.17 $45.17 $45.17 $45.17 $45.17 $45.17 $45.17 $45.17 $45.17 $45.17 Maturity 10 10 10 10 10 10 10 10 10 10 10 10 10 10 Risk-Free Rate 1.21% 1.21% 1.21% 1.21% 1.21% 1.21% 1.21% 1.21% 1.21% 1.21%. Value Lattice Steps Black-Scholes 1 10 100 100 0 100 00 ccc_mun_ch10_133-166.qxd 8/20/04 9:26 AM Page 147 TABLE 10. 9 Convergence of the Customized Binomial Lattice (Table Truncated) Stock Price $45.17 $45.17. 665,500 665,500 November 665,500 8,241, 310 8,906, 810 December 665,500 665,500 149 ccc_mun_ch10_133-166.qxd 8/20/04 9:26 AM Page 149 TABLE 10. 10 (Continued) Options Valuation (Monthly) Date Conservative