Bilgin Demir

ABSTRACT

This research paper aims to briefly explain employee stock options, their functionalities, tax consequences, and their value using the Binomial Lattice Model. Firms measure and record option expenses on their Financial Statements. Options are valued mostly with Black-Scholes Model using the assumptions obtained from the historical data. “Expected life of the option” is one of these assumptions. Firms do not disclose the details of employee stock options to the public, such as exercise behavior, age, income level, etc. Due to a lack of necessary information on all employees’ exercise behavior, this paper will only analyze the exercise behavior of key employees. Key employees must report their stock transactions to SEC; therefore, that data will be used to calculate the value of the options. The main goal is to compare our results with actual financial statement components to seek accuracy, and for this purpose, Federal Express Inc. will be our case study for this research.

1. Employee Stock Option Basics

Employee stock option plans are contracts between a company and its employees that give the right to buy a specific number of the firm’s shares at a fixed price within a certain period. But why do firms grant employee stock options as compensation for any reason? One of the reasons is to minimize the firm’s compensation costs and to conserve cash compensation. Employees granted stock options hope to profit by exercising their options at a higher price than when granted. Firms grant employee stock options as compensation for a number of reasons: to minimize the firm’s compensation costs, conserve cash, and avoid the limits on the tax deductibility of cash compensation.

Although the length of options varies across companies, a typical option has a 10-year life. The typical option is granted at the money, meaning that the strike price at which the option can be exercised is equal to the stock price at the time of the grant. Therefore, if the stock price remains below the strike price at the grant date through its life, the option will expire valueless. However, if the stock price increases, the employee can exercise the option and receive the shares at the strike price specified in the option contract. Typically, an option has a vesting period and schedule, such as 25 percent per year at the end of each of the first four years of the option’s life. Most of the time, there is a waiting period of 3 years, and options cannot be exercised.

When employees exercise their options, they receive shares in exchange for paying the strike price. They can then keep the shares or sell them on the market for the current share price to have the “intrinsic value” of the option. There are three different types of option exercises. Cashless exercise, often facilitated by the employer or a broker, in which the employee never purchases the stock but receives the option’s intrinsic value. Cashless hold is the same structure as cashless sell, but the employee receives the share amount instead of cash. A cash purchase is when an employee must pay the option’s cost, taxes, fees, and commissions.

Figure 1.1 Vesting Schedule for typical Employee Stock Options

u – unvested, v – vested

10-year vesting Schedule

Figure 1.1 shows the vesting schedule of an option. U represents the unvested period, and v represents the vested period when employees can exercise their options if the option is in the money.

One can visualize the option exercise as follows. Assume a company grants 100 options to employees over 10 year period at $10 the grant price. After 3 years as a quiet period, employees can vest 1/3 of the options and exercise them anytime. Assume that the stock price goes above $10 to $20 and that employees exercise the option as a cashless sell. The intrinsic value is the spread between the $20 market price and the strike price of $10.

2. Cost of the Employee Stock Options

When valuing a firm’s equity, existing options must be included in the forecasted operating cash flow stream, as the cost of the options exercised can be part of the cash flow stream. As it comes to option accounting, two facts are clear. First, an option has an expected value (and cost to existing shareholders) even if the options are not in the money. If the firm’s stock price increases, the option payoff; if it does not, the employee receives nothing. But the employee can never lose, so the expected value is always positive.

Based on the discussions that options represent value given to the employee as compensation effort, the Financial Standard Accounting Board has finalized that options should represent part of compensation on a firm’s income statement.

2.1. How to Measure and Record Option Expense

The main question about this is timing. When should firms measure and record the expense? There are at least two potential measurement dates that exist. The first one is to wait until options are exercised, mostly years after the grant date, and measure an expense as the option’s value at the time of the exercise.

The other method is measuring the option’s value at the time of the grant date based on the estimate of its value. Financial Account Standard Board favors this method under the argument that the fair value of the option is established at the granting date.
The Congressional Budget Office’s analysis of this account issue comes to the following conclusions:

• If firms do not recognize as an expense the fair value of employee stock options, and measure when the options are granted, the firms’ reported net income would be overstated.
• Changes in the value of employee stock options after they are granted and exercised are irrelevant to a firm’s income statement because they affect shares directly, not the firm itself. Specifically, they transfer wealth from existing shareholders to holders of the employee stock options. In 1993, Financial Accounting Standard Board recommended a change in the accounting treatment of employee stock options. It proposed that firms recognize the fair value of the options (measured at the time of the grant date) as an expense on their income statements. However, this proposal encountered opposition, mostly from the managers of firms granting such options. Those managers preferred to continue to use intrinsic value. This preference came from the fact that when options are granted, the intrinsic value is almost always less than the fair value, and thus, a smaller amount is subtracted from firms’ earnings.

3. Tax Implications to Employees

When examining the tax issues, one must consider the two major classes of stock options – incentive Stock Options (ISOs) and nonqualified stock options (NQOs). Incentive Stock Options provide no tax deduction to the issuing company and no taxable income to the employee during the exercise. But when the stock is sold, the employee must pay tax on the difference between the selling price and purchase price, usually the capital gains rate. Conversely, employees must pay taxes at the time of the exercise for nonqualified stock options at ordinary income rates.

4. Option Valuation

Firms mainly value options using the Black-Scholes model, and all existing option-pricing models use assumptions that are generally not designed for employee stock options.

The simple Black-Scholes model for a non-dividend paying stock can be written as:

C =SN(D1) – Ke-rt N(D2),

Where,
C= the value of the option
S = the current market price of the stock
K= the strike price t
t = the time remaining before option is exercised N(D)=the cumulative standard normal density function r= the risk –free interest rate
σ = the standard deviation of the return on the stock

D1= (log(S/K) + (r+σ2/2)t ) / σ/t

D2= (log(S/K) + (r-σ2/2)t) / σ/t

Firms use historical data for model assumptions. These assumptions are:

• The expected life of the option
• Stock Price Volatility
• Risk-free interest rate
• Dividend yield
• Forfeiture rate
• Exercise Price

This research paper will use the same key assumptions with different inputs using historical data. Firms that have been contacted declined to participate in this research but encouraged me to use any public information for SEC filings. Therefore, stock transactions initiated by key employees are used to create our assumptions. First, the Binomial Lattice Tree Model will calculate the option’s value. As mentioned, FEDEX Inc. will be our case study, and key employees’ past 10-year employee stock option transactions will be used for our input. We are interested in valuing FEDEX employee stock options with a Grant Date of 06/09/2014 with a $143.54 grant price. The option has 10 years term. Our model has four steps. As a first step, all available public data is retrieved from company disclosure. Public data includes 10K Annual Statements, 10Q Quarterly Statements, and Form 4. The second step, form Form-4 is used to determine key employees’ transaction behaviors and allows us to calculate the average life of the option. The third step is historical stock return and volatility over 10 years calculated. Also, a risk-free rate is obtained from the Treasury website from the grant date over the option’s lifetime. In the fourth step, we will use our assumptions to construct the Binomial Lattice Model. Using the available public data following information was obtained. Key employees exercised their options with an average of 4.64-year timeframe. We calculate the volatility as 27.37% from the historical 10-year stock movement using 30-day average standard deviations.

Historical Volatility is chosen over Implied Volatility because it reflects the stock movement better than implied volatility because of its long time horizon. Implied Volatility may be a better indicator for short-term exchange-traded options, but historical volatility would be a better input for employee stock options. The risk-free rate is listed as 1.69% for the date of 06/09/2014 for 5-year maturity. The following table compares the values obtained with FEDEX assumptions listed on the 2014-10K Financial Statement.

Figure 2.

As seen in Figure 2 above, only the dividend rate is used as a common indicator.

Next, we will construct our 40-step Binomial Lattice Tree with these assumptions. First, we calculate the up factor using the following formula for FedEx assumptions.

u= eσ√Δt

where Δt=6.20/40 = 0.16

then u can be calculated as e ^ (0.35√0.16) =1.15

Next, we calculate d as follows,

d=1/u

d=1/1.5=0.87

Now we can start constructing our Binomial Tree.

Figure 3.

Binomial Tree

Above construction goes up to step 40, and we will use the end values to calculate the ending option prices.

EPn = MAX((Sn-K)

Where EPn is the ending option price, Sn is the ending stock price at node 40, and K is our strike price, in other terms, the grant price. Therefore, at expiration, the possible ending option values are shown as follows,

Figure 4.

↓

Figure 4. shows the ending stock prices with ending prices. Because our data is large and it is impossible to put all input on this paper, the arrow indicates more data.

Now we need to calculate the risk-neutral probability using the following formula,

p= e(r-div )(Δt)

p=e(0.0147-0.0056) * (0.16)

p=0.47

Now we will calculate the values from the last step to the current time, step 0. We do this discount the ending option values back through each step using the following formula,

EPn-1 = (p*EPu) + (1-p)*(EPd)*e-rΔt

Figure 5.

Finally, the $49.05 at the valuation date is calculated using FedEx assumptions.

Now we will use the same approach for our research assumptions that we retrieved from public data.

where Δt=4.64/40 = 0.12

Now we can calculate,

u= eσ√Δt
u= e0.27.37√0.12 =1.10

d=1/u

d=1/1.10=0.91

We can construct the Binomial tree as follow for the research assumptions.

Figure 6.

Using the same formula, we can again calculate the ending option values

EPn = MAX((Sn-K)

Figure 7

Further, we need to calculate the risk-neutral probabilities

p= e(r-div )(Δt)

p=e(0.0169-0.0056)(0.12)

p=0.48

As the last step now, we will discount back the ending option value to the current value.

EPn-1 = (p*EPu) + (1-p)*(EPd)*e-rΔt

Figure 8.

Finally, we calculate the option value as $ 35.093 using the research assumptions.

40-Step Binomial Tree using R

# Define input parameters using Fedex Assumptions
2
time <- 6.2 # Time in years
3
volatility <- 0.35 # Volatility
4
risk_free_rate <- 0.0147 # Risk-free rate
5
dividend <- 0.0056 # Dividend yield
6
​
7
# Calculate parameters for the binomial tree
8
delta_t <- time / 40
9
u <- exp(volatility * sqrt(delta_t))
10
d <- 1 / u
11
p <- (exp((risk_free_rate - dividend) * delta_t) - d) / (u - d)
12
​
13
# Initialize the stock price and option value matrices
14
S <- matrix(0, nrow=41, ncol=41)
15
V <- matrix(0, nrow=41, ncol=41)
16
​
17
# Calculate the stock prices at each node of the tree
18
S[1,1] <- 143.54 # Initial stock price
19
for (i in 2:41) {
20
  S[i,1] <- S[i-1,1] * u
21
  for (j in 2:i) {
22
    S[i,j] <- S[i-1,j-1] * d
23
  }
24
}
25
​
26
# Calculate the option values at the final nodes of the tree
27
for (j in 1:41) {
28
  V[41,j] <- max(0, S[41,j] - 143.54)
29
}
30
​
31
# Work backwards through the tree to calculate the option values at earlier nodes
32
for (i in 40:1) {
33
  for (j in 1:i) {
34
    V[i,j] <- exp(-risk_free_rate * delta_t) * (p * V[i+1,j] + (1-p) * V[i+1,j+1])
35
  }
36
}
37
​
38
# Print the option value at the root of the tree
39
cat("Option value:", V[1,1])
Option value: 49.04722

# Define input parameters using Research Assumptions
2
time <- 4.64 # Time in years
3
volatility <- 0.2737 # Volatility
4
risk_free_rate <- 0.0169 # Risk-free rate
5
dividend <- 0.0056 # Dividend yield
6
​
7
# Calculate parameters for the binomial tree
8
delta_t <- time / 40
9
u <- exp(volatility * sqrt(delta_t))
10
d <- 1 / u
11
p <- (exp((risk_free_rate - dividend) * delta_t) - d) / (u - d)
12
​
13
# Initialize the stock price and option value matrices
14
S <- matrix(0, nrow=41, ncol=41)
15
V <- matrix(0, nrow=41, ncol=41)
16
​
17
# Calculate the stock prices at each node of the tree
18
S[1,1] <- 143.54 # Initial stock price
19
for (i in 2:41) {
20
  S[i,1] <- S[i-1,1] * u
21
  for (j in 2:i) {
22
    S[i,j] <- S[i-1,j-1] * d
23
  }
24
}
25
​
26
# Calculate the option values at the final nodes of the tree
27
for (j in 1:41) {
28
  V[41,j] <- max(0, S[41,j] - 143.54)
29
}
30
​
31
# Work backwards through the tree to calculate the option values at earlier nodes
32
for (i in 40:1) {
33
  for (j in 1:i) {
34
    V[i,j] <- exp(-risk_free_rate * delta_t) * (p * V[i+1,j] + (1-p) * V[i+1,j+1])
35
  }
36
}
37
​
38
# Print the option value at the root of the tree
39
cat("Option value:", V[1,1])
Option value: 35.09266

Summary and Conclusion

Our main goal was to value the employee stock options to compare our results with firms’ disclosure to the Securities Exchange Commission. I contacted public companies to obtain the necessary information, but the firms decided not to participate. Because of this and the lack of detailed information, I decided1 to use public information on the FedEx Investor Relations website; therefore, I retrieved the public information of key employees’ transaction history for the past 10 years. Also, FedEx’s 10K Financial Statements from the 2004 and 2014 fiscal years were studied.

As this research concluded, I observed that in 10 years, key employees at FedEx exercised a total of 5,455,899 options which means the company either issued new or repurchased shares from the market. If new shares are issued, the cash flow stream would also be approximately $249 million, negatively affecting the earnings per share portion of the financial statement. Key employees used cash purchase or cashless sell as their exercise transaction type. September is the month with no exercise history as a quiet period. Options are mostly granted by around June 5th. 2008 was the year with the least exercise with 4, and 2012 was the maximum with 25 options exercise.

As stated above, we found the average expected life of the options as 4.64 years compared to the 6.20 FedEx assumption. The risk-free interest rate was obtained as 1.69% from the Treasury website, and the historical sigma was calculated as 27.37%. The only assumption we preserved was the dividend rate of 0.56 %.

Binomial Lattice Tree Model was used to calculate the option values for both FedEx assumptions and our assumptions. FedEx disclosed the option value for 2014 options as $35.79. When the same data is calculated, we find the option value as $49.05. Our input assumptions were calculated as a next step, and the option value was retrieved at $35.093.

My professors, Mr. Ionut Florescu and Mr.Khaldoun Khashanah, encouraged the decision.

Even though FedEx’s assumptions and our inputs are different we observed that the option values are very similar.

When we think about the reason behind that result, FedEx may include the forfeiture rate into the option calculation that offsets and, of course, reduce the average expected life of the option. As a final statement, even though; we observed very similar results between the option values indicated by FedEx and our research, it does not necessarily prove that observing only key employees’ transaction behaviors are sufficient to calculate the fair market value of the employee stock options.

Bilgin Demir December 2014

References:

• Charles Baril, Luis Betancourt, John Briggs – Valuing Employee Stock Option under SFAS 123R using Black-Scholes-Merton and lattice model approaches – Journal of Accounting Education 25 (2007) 88-101
• Mark Rubenstein – On the Accounting Valuation of Employee Stock Options – Journal of Derivatives, Fall 2015
• Anthony Banks – The Binomial Lattice option-pricing model for valuing American type of employee stock options
• Aswath Damodaran, Stern School of Business – Employee Stock Options (ESOPs) and Restricted Stock: Valuation Effects and Consequences, September 2005
• Nalin Kulatilaka, Alan J.Marcus – Valuing Employee Stock Options – Financial Analyst Journal, November-December 1994
• Mark Lang, Kenan-Flagler Business School University of North Carolina – Employee Stock Options and Equity Valuations, ISBN 0- 943205-67-0
• Congress of the United States Congressional Budget Office, April 2004, Accounting for Employee Stock Options. Online Literature
• http://www.sec.gov/answers/empopt.htm
• http://www.nceo.org/articles/employee-stock-options-factsheet
• http://www.american-appraisal.com/US/Library/Articles/The-Basics- of-Stock-Option-Val.htm
• http://knowledge.wharton.upenn.edu/article/how-employees-value- often-incorrectly-their-stock-options

Employee Stock Options and Valuation with R