Valuing Employee Stock Options Part 10 pdf

The customized binomial lattice used is a proprietary al-gorithm that incorporates the traditional BSM inputs stock price, strikeprice, time to maturity, risk-free rate, dividend, and vo

PART Three

A Sample Case Study

Applying FAS 123

CHAPTER 10

A Sample Case Study

justi-fications for the input assumptions used in the ESO valuation These puts were obtained based on the 2004 proposed FAS 123 revisionrequirements 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, strikeprice, time to maturity, risk-free rate, dividend, and volatility) plus addi-tional inputs including time to vesting, changing forfeiture rates, changingsuboptimal exercise behavior multiples, blackout dates, changing risk-free

propri-etary algorithm can be run to accommodate hundreds to thousands of tice steps as well as incorporate Monte Carlo simulation of uncertaininputs whenever necessary The following sections describe how each of theinputs was derived in the valuation analysis The analysis is an excerptfrom 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 goesthrough in selecting and justifying each input parameter in the customizedbinomial lattice model, and showcases some of the results generated in theanalysis Some of the more analytically intensive but equally important as-pects have been omitted for the sake of brevity

lat-STOCK PRICE AND STRIKE PRICE

The first two inputs into the customized binomial lattice are the stock priceand strike price For the ESOs issued, the strike price is always set at thestock price at grant date This means obtaining the stock price will alsoyield the strike price Table 10.1 lists the stock prices estimated by thefirm’s investor relations department Conservative and aggressive closing

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stock prices were provided for a period of 24 months, generated usinggrowth 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 otherapproaches were used Twelve analyst expectations were obtained andtheir results were averaged In addition, econometric modeling with MonteCarlo simulation was used to forecast the stock price Using a path-depen-

was forecast to be $47.22 (Figure 10.2), consistent with the investor tions stock price The valuation analysis will use all three stock prices, andthe final result used will be the average of these three stock price forecasts

Estimate of Stock Price per Investor Relations

Per Share Stock Price

The next input is the option’s maturity date The contractual maturitydate 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

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

(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 0.66 –1.14 0.87 0.49 1.33 0.28 –0.86 0.38 0.36 –0.62 0.04 –0.23 1.10 –1.07 –0.15

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 inTable 10.2 Using the spot yield curve, the spot rates were bootstrapped toobtain 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 rateapplies 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 twofuture periods For instance, a one-year forward rate three years from nowapplies 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%,

DIVIDENDS

The firm’s stocks pay no dividends, and this parameter will always be set tozero In other cases, if dividend yields exist, these yields are entered into themodel, including any expected changes to dividend policy over the life ofthe option

VOLATILITY

Volatility is the next input assumption in the customized binomial latticemodel There are several ways volatility can be measured, and in the in-terest of full disclosure and due diligence, all methods are used in thisstudy Table 10.4 shows the first method used to estimate the changingvolatility of the firm’s stock prices using the Generalized AutoregressiveConditional Heteroskedasticity (GARCH) model The inputs to themodel are all available historical stock prices since going public The re-sults indicate that the standard GARCH (1,1) model is inadequate to

and bad Akaike and Schwarz criterion statistics As such, GARCH sis is found to be unsuitable for forecasting the volatility for valuing thefirm’s ESOs and its results are abandoned Only GARCH (1,1) is shown

analy-in this example In reality, multiple other model specifications were runand analyzed

TABLE 10.3

Two additional approaches are used to estimate volatility The first is

to use historical stock prices for the last quarter, last one year, last twoyears, and last four years (equivalent to the vesting period) These closingprices are then converted to natural logarithmic returns and their samplestandard deviations are then annualized to obtain the annualized volatili-

In addition, Long-term Equity Anticipation Securities (LEAPS) can beused to estimate the underlying stock’s volatility LEAPS are long-termstock options, and when time passes such that there are six months or soremaining, LEAPS revert to regular stock options However, due to lack oftrading, the bid-ask spread on LEAPS tends to be larger than for regularlytraded equities Table 10.5 lists the two LEAPS closest to the stock priceforecast at grant date Implied volatilities on both bid and ask are listed inTable 10.5

After performing due diligence on the estimation of volatilities, it is foundthat a GARCH econometric model was insufficiently specified to be of statis-tical validity Hence, we reverted back to using the implied volatilities of long-

Convergence achieved after 30 iterations

term options or LEAPS, and compared them with historical volatilities Thebest 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 thislarge spread, Monte Carlo simulation was applied by running a simulation onthese volatility rates; thus, every volatility calculated here will be used in theanalysis 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 andsix 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 employeeleaves within the first six months, the entire option grant is forfeited Afterthe six months, each additional month vests 1/42 additional portions of theoptions 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-

Volatility Inputs: Triangular

distribution with the following

parameters into Monte Carlo

simulation

month vesting option is divided into 48 minigrants and expensed over the

vesting period See Chapter 6 for details on allocating expense schedules

SUBOPTIMAL EXERCISE BEHAVIOR MULTIPLE

The next input is the suboptimal exercise behavior multiple In order to tain this input, data on all options exercised within the past year were col-lected We used the past year, as trading from 2000 to 2002 was highlyvolatile and we believe the high-tech bubble caused extreme events in thestock market to occur that were not representative of our expectations of thefuture In addition, only the past year’s data are available Figure 10.3 illus-trates the calculations performed (The table is truncated to save space.) Thesuboptimal exercise behavior multiple is simply the ratio of the stock pricewhen it was exercised to the contractual strike price of the option Termi-nated employees or employees who left voluntarily were excluded from theanalysis This is because employees who leave the firm have a limited time toexecute the portion of their options that have vested In addition, all un-vested options will expire worthless Finally, employees who decide to leavethe firm would have potentially known this in advance and hence have a dif-ferent exercise behavior than a regular employee Suboptimal exercise be-havior does not play a role under these circumstances The event of anemployee leaving is instead captured in the rate of forfeiture The median be-havior multiple is found to be 1.85, and is the input used in the analysis.The median is used as opposed to the mean value because the distribu-tion is highly skewed (the coefficient of skewness is 39.9), and as means arehighly susceptible to outliers, the median is preferred Figure 10.3’s histogramshows that the median is much more representative of the central tendency ofthe distribution than the average or mean In order to verify that this is thecase, two additional approaches are applied to validate the use of the median:trimmed ranges and statistical hypothesis tests Table 10.6 illustrates atrimmed range where the range of the suboptimal exercise behavior multiplesuch that the option holder will exercise at a stock price exceeding $500 is ig-nored This is justified because given the current stock price it is highly im-

this subjective trimming is 1.84, close to the initial global median of 1.85

In addition, a more objective analysis, the statistical hypothesis test, wasperformed using the single-variable one-tailed t-test, and the 99.99th statisti-cal percentile (alpha of 0.0001) from the t-distribution (the t-distribution wasused to account for the distribution’s skew and kurtosis—its extreme valuesand fat tails) was found to be 3.92 (Table 10.7) The median calculated fromthe suboptimal exercise behavior range between 1.0 and 3.92 yielded 1.76

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144 A SAMPLE CASE STUDY APPLYING FAS 123

Trimmed Ranges (Table Truncated)

We therefore conclude that using the global median of 1.85 is the most

FORFEITURE RATE

The rate of forfeiture is calculated by comparing the number of grants thatwere canceled to the total number of grants This value is calculated on amonthly basis and the results are shown in Table 10.8 The average forfei-ture rate is calculated to be 5.51 percent In addition, the average employeeturnover rate for the past four years was 5.5 percent annually Therefore,5.51 percent is used in the analysis

NUMBER OF STEPS

The higher the number of lattice steps, the higher the precision of the sults Figure 10.4 illustrates the convergence of results obtained using aBSM closed-form model on a European call option without dividends, and

Hypothesis Tests

One-Sample Hypothesis t-Test:

Suboptimal Exercise Behavior

Test of null hypothesis: mean = 3.754

Test of alternate hypothesis: mean < 3.754

Therefore, the 99.99th statistical percentile cut-off is 3.925.

The average for the range between 1.000 and 3.925 is

Therefore, with the three values indicating a suboptimal behavior multiple at around 1.7689, 1.8450 and 1.8531, using the median of all data points provides the best indication as all data are used

The resulting suboptimal behavior multiple used is

comparing its results to the basic binomial lattice Convergence is generallyachieved at 1,000 steps As such, the analysis results will use 1,000 steps

in-stance, a nonrecombining binomial lattice with 1,000 steps has a total of 2

Table 10.9 illustrates the calculation of convergence by using sively higher lattice steps The progression is based on sets of 120 steps (12months per year multiplied by 10 years) The results are tabulated and themedian of the average results is calculated It shows that 4,200 steps is thebest estimate in this customized binomial lattice, and this input is used

RESULTS AND CONCLUSIONS

Using the customized binomial lattice methodology coupled with MonteCarlo simulation, the fair-market values of the options at different grantdates and different forecast stock prices are listed in Table 10.10 For in-stance, the grant date of January 2005 has a conservative stock price fore-cast of $45.17 and its resulting binomial lattice result is $17.39 for theone-month-vesting option, and $17.42 for the six-month-vesting option

In contrast, if we modified the BSM to use the expected life of the option(which was set to the lowest possible value of four years, equivalent to the

Convergence in Binomial Lattice Steps

$17.20 S17.10

higher at $19.55 This is a $2.16 cost reduction compared to using theBSM, or a 12.42 percent reduction in cost for this simple option alone.When all the options are calculated and multiplied by their respective

grants, the total valuation under the traditional BSM is $863,961,092

af-ter accounting for the 5.51 percent forfeiture rate In contrast, the total

valuation for the customized binomial lattice is $813,997,676, a

reduc-tion of $49,963,417 over the period of two years Figure 10.5 and Table10.11 show one sample calculation in detail

Figure 10.5 illustrates a sample result from a 124,900-trial MonteCarlo simulation run on the customized binomial lattice where the error iswithin $0.01 with a 99.9 percent statistical confidence that the fair-marketvalue of the option granted at January 2005 is $17.39 Each grant illus-trated in Table 10.10 will have its own simulation result like the one inFigure 10.5

The example illustrated in Table 10.11 shows a nạve BSM result of

$26.91 versus a binomial lattice result of $17.39 (the BSM using an justed four-year life is $19.55) This $9.52 differential can be explained bycontribution in parts In order to understand this lower option value ascompared to the nạve BSM results, Table 10.12 illustrates the contribution

ad-to options valuation reduction

The difference between the nạve BSM valuation of $26.91 versus a