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Summary of Equity Valuation Web Models
This summary is intended to provide further technical background for Radford’s collection of web-based valuation models. Note that these valuation results are all intended only to be used as reasonable, “rule of thumb” estimates. In summary, the models are:
Stock Options / SARS:o Time Basedo Capped Optionso Indexed Options
Fixed Appreciation Relative to an Index
o Vesting Contingent Stock Price Hurdles
Absolute Performance Shareso Absolute Stock Price Targets
Hurdle Checked at the End of the Performance Period Hurdle Checked at Any Time of the Performance Period
o Market Stock Units Standard Custom
Relative Performance Shareso Composite Indexo Component Rank
Each of the models above is discussed in the text that follows. For any further questions, please contact us.
Terry AdamsonRadford 1650 Market Street Philadelphia, PA 19103 (215) 255-1802 (Office)(215) 908-4036 (Cell)Link to [email protected]
Jon W. Burg, FSARadford199 Fremont Street, Floor 15 San Francisco, CA 94105 (415) 486-7137 (Office)(415) 205-9534 (Cell)Link to [email protected]
Jim LecherRadford1650 Market Street Philadelphia, PA 19103 (215) 255-1859 (Office)(215) 359-8154 (Cell)Link to [email protected]
Time Based Option/SAR Models
The fair value is determined based upon a standard Black-Scholes option pricing model.
The Hull-White model is based upon a standard Hull-White Trinomial Model.
The standard binomial fair value provided is based upon a standard European binomial model, not the Cox, Ross, & Rubinstein (CRR) Model. Note that the CRR binomial model creates higher fair values for dividend paying stocks as it includes a probability that employees will early exercise their options to capture the value of dividends of the underlying share.
Capped Option Valuation Model
The fair values are simulated 50,000 times based upon a risk-neutral stock price model with 12 nodes per year. To ensure reasonable model convergence, an antithetic variance reduction technique is used.
If the user enters a cliff vesting schedule, all of the options are assumed to become vested at the user entered vesting time. If the user enters a graded vesting schedule, an equal number of options are assumed to vest annually over the user entered vesting time.
The model checks at every post vesting node, prior to the user entered expected life, if the appreciation cap is achieved. If the appreciation cap is achieved, the model assumes that the options will be exercised at that point in time. If the appreciation cap is not achieved prior to the user entered expected life, exercise is assumed to be exercised at the expected life.
The appreciation cap (maximum stock price) is calculated by multiplying the stock price by the appreciation cap. For example, if the current stock price is $50.00 and the appreciation cap is 200%, the stock price cap is equal to $100, yielding a gain of $50.00.
Indexed Options - Fixed Appreciation
This model calculates the fair value of an option that has a strike price that appreciates based upon a user-entered annual appreciation rate. The strike price that is in effect as of the user-entered expected life is calculated via the following formula:
Initial Strike Price×(1+Annual Appreciation Rate )Expected Term
The fair value is calculated based upon a modified closed-form Black-Scholes formula.
Indexed Options - Relative to an Index
The fair values are simulated 25,000 times based upon a risk-neutral stock price model based upon a log-normal assumption of future stock prices. Two stock price paths are simulated, one for the Client company and one for the Index. The stock price paths are based upon a volatility assumption for the Index, and a volatility and dividend yield assumptions for the Client company. The stock prices are simulated from the grant date to the user-entered expected life with 12 measurement periods per year. To ensure reasonable model convergence, an antithetic variance reduction technique is used. Please note that the number of simulations and measurement periods were chosen to reduce run time. Therefore, some variability will occur between two identical runs of the model.
The strike price of the option is set based upon the simulated appreciation (or depreciation) percentage of the index over the timeframe from valuation date to user-entered expected life. The appreciation or depreciation percentage is multiplied by the user-entered strike price to determine the strike price that is in effect at the time of assumed option exercise.
Vesting Contingent on Stock Price Hurdles
The model calculates the fair value of an option that vests if the Company’s stock price exceeds the user-entered vesting hurdle.
There are three choices for when the stock price hurdle may be achieved. They are:
1) At Any Time – The stock price based vesting hurdle can be achieved at any time from the date of grant until to the full contractual term of the award.
2) Prior to the Service condition Being Satisfied -The stock price based vesting hurdle can only be satisfied from the date of grant until to the service vesting requirement.
3) After the Service Condition is Satisfied -The stock price based vesting hurdle can only be satisfied from the service vesting requirement to the full contractual term of the award.
If an option becomes vested it is assumed that the option will be exercised at the midpoint of the time in which the option becomes vested and the full contractual term of the award.
The fair values are simulated 25,000 times based upon a risk-neutral stock price model based upon a log-normal assumption of future stock prices. Monthly measurement periods are used. To ensure reasonable model convergence, an antithetic variance reduction technique is used. Please note that the number of simulations and measurement periods were chosen to reduce run time. Therefore, some variability will occur between two identical runs of the model.
Absolute Performance Shares
Absolute Stock Price Targets – Hurdle Checked at the End of the Performance Period
The fair value is based upon a modified Black-Scholes model. Specifically, upon the first term of the Black-Sholes model with the strike price being set equal to the to the vesting hurdle price. There are 4 distinct calculations depending on hurdle type (stock price or TSR) and if the holder of the award does or does not receive dividends that are paid during the performance period. If the hurdle is set as a required return it is assumed that dividends are reinvested for purposes of calculating the return. Within the Radford calculators, TSR hurdles are converted to stock prices hurdles in Scenarios 1 and 2.
Scenario 1 – Hurdle is based upon TSR and the holder receives dividend equivalents.
The fair value is calculated via the following formula:
Performance Share = S× NormSDist (d1)
where:
d1=log S
H+(r+ σ2
2 )×tσ √ t
SH
=Ratio of Share Price to Hurdle
r=Risk-Free Rate
σ=Volatility
t=Duration of time between Grant Date and End of Performance Period
Scenario 2 – Hurdle is based upon TSR and the holder does not receive dividend equivalents
Performance Share = S×e−qt×NormSDist (d1 )
Where:
d1=log S
H+(r+ σ2
2 )×tσ √ t
q=Annual Dividend Yield
Scenario 3 – Hurdle is based upon a stock price and the holder receives dividend equivalents
Performance Share = S× NormSDist (d1)
Where:
d1=log S
H+(r−q+ σ2
2 )×tσ √ t
Scenario 4 – Hurdle is based upon a stock price and the holder does not receive dividend equivalents
Performance Share = S×e−qt×NormSDist (d1 )
Where:
d1=log S
H+(r−q+ σ2
2 )×tσ √ t
Absolute Stock Price Targets – The hurdle can be achieved and the shares will vest any time during the performance period
The fair values are simulated 25,000 times based upon a risk-neutral stock price model based upon a log-normal assumption of future stock prices. To ensure reasonable model convergence, an antithetic variance reduction technique is used. In order for the model to run in a reasonable amount of time it is assumed that there are 12 measurement periods per year. If TSR is entered as a vesting hurdle dividends are assumed to be reinvested. If stock price is entered as a vesting hurdle no credit is given with respect to the vesting hurdle for dividends paid over the performance period. Please note that the number of simulations and measurement periods were chosen to reduce run time. Therefore, some variability will occur between two identical runs of the model.
At every node the simulated price is checked to see if the hurdle is achieved. If the hurdle is achieved the option is assumed to be exercised, uniformly, from the point of vesting (the greater of the node at which the hurdle is achieved and the time vesting requirement) to the contractual term of the award.
Market Stock Units
Standard MSU – A standard MSU represents a restricted stock unit in which the number of shares earned is based upon the targeted shares granted and the TSR over the performance period. The targeted number of shares is multiplied by (1+ TSR) to determine the number of awards earned. For purposes of calculating TSR, dividends are assumed to be reinvested. The model allows for the input of a “Maximum”. All TSRs over the maximum will result in the payout being capped at the “Maximum” payout. If a “Floor” is entered for payout type, the “Floor” entered will be the minimum number of shares earned regardless of TSR over the performance period. If a “Threshold” is entered for payout type, any simulated TSR that is less than the “Threshold” will result in no stock units being earned.
The fair values are simulated 25,000 times based upon a risk-neutral stock price model based upon a log-normal assumption of future stock prices. To ensure reasonable model convergence, an antithetic variance reduction technique is used. The stock prices are simulated out to the user entered performance period. The simulated payout at time t is calculated based upon the TSR and stock price. The future payout is then discounted to the grant date with the risk-free rate of return. Please note that the number of simulations was chosen to reduce run time. Therefore, some variability will occur between two identical runs of the model.
Custom MSU – The Custom MSU model was developed to add more flexibility to the payout pattern of the award. The modeling and basic inputs are identical to the Standard MSU model with the exception of the vesting schedule which allows for user defined payout patterns. A standard MSU determines the payout based solely on the TSR of the company’s stock over the performance period. Please note that the payouts are linearly interpolated between the vesting bend-points which are entered.
Relative Performance Shares
Composite Index Valuation Model – The Composite Index Model determines payout based upon the Client companies’ TSR as compared to the TSR of the Index that is used as a comparator. The model simulates two stock price paths, one for the Client company and one for the Index. Performance is measured using the user-input Payout Curve data. The simulated stock prices are based upon the risk-free rate of return, Client and Index volatility, and the correlation assumed between the Client Company and the Index.
Component Rank Valuation Model – The Component Rank Model determines payout based upon the Client companies’ TSR as compared to the TSR of each component within the Index. The model simulates a stock price path for the Client company and for each combination of correlation and volatility (90 in total). Performance is measured using the user-input Payout Curve data with payout being interpolated between bend points by default. The simulated stock prices are based upon the risk-free rate of return, Client and component volatility, and the correlation assumed between the Client Company and the Index.
Company Return = e(r−σ2
2 )t+σ√ t ( ρ τ Index+√1−ρ2 τCompany )
where
ρ=Correlation between Company and Index
τ Index=Inverse Normal of Index's Random Uniform
τCompany=Inverse Normal of Company's Random Uniform
