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Estimating Expected Term
What is Expected Term?
Employee stock options have contractual terms — often 10 years — as measured from the grant date to the final maturity date of the option. Between grant and maturity, depending on the anticipated distribution of option exercise and forfeiture, there is an expected term that summarizes uncertainty about the term of the option into a single "average" number. The option term is uncertain because of the inability of employees to market or transfer options. Marketability and transferability constraints cause voluntary early suboptimal exercise by continuing employees or involuntary exercise or forfeiture of vested options by employees departing from the company.
Under FAS 123(R), expected term is measured conditional on vesting. It is the term expected of an option that will vest for certain. Recall that FASB standards call for modified grant date accounting, which treats forfeiture during the vesting period as a grant level calculation rather than a factor in the valuation of each option. It is therefore important to discriminate between published option valuations that are low because of a discount for pre-vest forfeiture, or even non-transferability relating to the vesting period, and valuations that are compliant with FAS 123(R). We can think of the definition of expected term consistent with FASB standards as:
The anticipated average amount of time that an option is outstanding, assuming it will vest. The amount of time that an option is outstanding is measured from the grant date to the date of option expiration, regardless of whether expiration is the result of early exercise, post-vest forfeiture or cancellation (usually related to departure from the company), or option maturity at the end of the contractual term.
The original 1995 FASB Statement No. 123 claimed that fair value should be based on the expected term as an assumption or input to an option valuation model — typically Black-Scholes — rather than the contractual term input. The revised statement, FAS 123(R) retains the position that fair value should be based on expected term (see paragraphs A3 and A18 of the revised statement), but with a revised perspective. For a closed-form model expected term is still an assumption or input; but in the case of a binomial (or trinomial) lattice or a Monte Carlo Simulation the expected term becomes a model output. This revised perspective suggests new challenges to measuring expected term, a required disclosure under FAS 123(R), and comparing it across companies.
How is Expected Term Estimated?
There are several ways of estimating expected term. First there is an average term estimated using a company's own history. For off-book option value disclosure in financial statement footnotes companies have been relying on this estimate as a Black-Scholes input. Under FAS 123(R), the use of historical averages as indicative of prospective behavior requires more rigorous support. Expected term should not be confused with the average historical time to exercise. Average historical time to exercise ignores the term of options in three categories:
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options that remain outstanding
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options that expire worthless at the end of the contractual term
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options that are forfeited out-of-the-money by departing employees
Ignoring options in the third category usually results in small error depending on the extent of employee departure post-vesting. Ignoring options in the first and second categories can cause significant downward bias in the estimate of historical term. Historical time to exercise is not a supportable estimator of historical term, let alone expected term.
A second estimate of expected term is the probability-weighted average term implied by a lattice or simulation model. Here, employee exercise and forfeiture behavior is estimated from data and input to the model, which outputs expected term as well as option value. A lattice or simulation model weights the magnitude of each positive or zero cash flow — discounted to the grant date — by the probability of its occurrence. The option value is the probability-weighted average of the discounted cash flows across all possible outcomes. The model can similarly track the probability-weighted time from grant date to cash flow occurrence and average these times across all possible outcomes.
The model implied measure of expected term just described can be used to meet the disclosure requirement of FAS 123(R); however, it will be a risk-neutral measure of expected term. Using the same lattice or simulation model, one can compute a third estimate of expected term, which is risk-adjusted and will typically be shorter than the risk-neutral estimate. Generally, this third estimate of expected term is likely to be the most theoretically sound; that is, assuming that information from the data is properly imported into the lattice or simulation model.
A fourth estimate of expected term is value-implied rather than model-implied. This method calls for calculating fair value using a lattice or simulation model, and then calculating the implied expected term from Black-Scholes. That is, the value-implied term is the term that results in the same fair value as the numerical model, assuming all other inputs to Black-Scholes are consistent with those used for the numerical valuation. This method becomes more ambiguous if term structures rather than constant parameter inputs were used for volatility and/or interest rates in the numerical valuation.
The second, third and fourth methods for estimating expected term involve importing information on employee behavior into a model of future stock prices, whether it be a lattice or simulation. The information might be imported as one or more barriers corresponding to suboptimal exercise factors. A better approach is to estimate a probability surface of early exercise using non-linear regression with a fractional dependent variable; this is a standard econometrics technique. In general, we consider models that incorporate an exercise probability of the form:
λe = G(St,t,x) + ε
where λe is the exercise probability predicted by a function G of independent factors including the stock price and time. The conditional expectation of the error is zero: E(ε|St,t,x)=0. Here x can include past stock prices prior to the current time t, as well as personal factors, such as employee wealth, age, length of service, or home jurisdiction (domestic or foreign). The range of G is all numbers between and including 0 and 1. To estimate the parameters, we specify a functional form for G involving the predicting factors St,t,x and a set of coefficients. Then a data sample consisting of observations on the dependent variable λe, and the predicting factors, St,t,x are used to estimate the coefficients of G.
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