The Value of Being Lucky: Option Backdating and Non-diversifiable Risk

Transcription

1 The Value of Being Lucky: Option Backdating and Non-diversifiable Risk Vicky Henderson Jia Sun A. Elizabeth Whalley April 2015 Empirical evidence shows that backdating of executive stock option grants is prevalent, particularly at firms with highly volatile stock prices. Yet extant Black-Scholes style models used for empirical estimates of gains to executives from backdating suggest that this benefit is not only modest but is greatest for low volatility firms. In this paper we take the stance that, ex ante, executives who have the opportunity to backdate should take this into account in their valuation and we quantify the value to a risk averse executive of a lucky option grant with strike chosen to coincide with the lowest stock price of the month. This enables us to reconcile theory with empirics - we show the ex ante gain to risk averse executives is significant and increases with volatility. Our explanation hinges on valuing the embedded lookback option, and key features of risk aversion, inability to diversify and American early exercise. Keywords: Option backdating, opportunistic timing of option grants, executive stock options, employee stock options, executive compensation. JEL Classification Numbers: G13, G34, J33, K42, M52 We thank participants at the 8th Oxford-Princeton workshop (March, 2014), the Banff Workshop at BIRS (May, 2014), the SIAM FM Chicago conference (November, 2014) and in a seminar in the Economics department, City University London. University of Warwick, Coventry, CV4 7AL. UK and Oxford-Man Institute, University of Oxford, Eagle House, Walton Well Road, Oxford. OX2 6ED. UK. vicky.henderson@warwick.ac.uk China Credit Rating Co. Ltd, Ying Tai Mansion. 28 Jin Rong Street, Beijing, Beijing , China. sunjia@chinaratings.com.cn Finance Group, Warwick Business School, University of Warwick, Coventry. CV4 7AL. UK. elizabeth.whalley@wbs.ac.uk 1

2 1 Introduction The practice of executives influencing their option compensation by setting a grant date retrospectively is known as backdating. Since options are usually granted at-the-money, selecting an advantageous grant date to coincide with a low stock price will be valuable to an executive. 1 Backdating of option awards has received significant attention in the financial press 2 and has led to in excess of 140 firms being the subject of investigations by the Securities and Exchange Commission (SEC). 3 There is a substantial body of empirical evidence documenting that backdating of option awards took place in the US, and was particularly prevalent in firms with highly volatile stock prices. 4 These empirical studies raise unanswered questions: Why are the empirical estimates of the gains to the executive from backdating so modest, when backdating was so commonplace? Why was backdating so prevalent in firms with highly volatile stock prices and high tech firms? Existing Black Scholes based estimates have failed to answer these questions. We take up the challenge of providing answers in this paper. In short, there is a gap between the empirics and the extant theoretical studies which we fill. To answer these questions, we adopt the viewpoint that, ex ante, executives who have the opportunity to backdate should take this into account in their valuation of their option grants. Executives anticipate being able to choose the best grant date over a backdating window - consistent with the characterization of lucky grants of Bebchuk, Grinstein and Peyer (2010) as being at-the-money grants at the lowest stock price of the month. Thus, in contrast to the valuation of standard executive stock options, the value of the executives opportunity to backdate involves a lookback feature. 5 1 Prior to August 2002, executives in the U.S. had either up to 45 days after the end of the fiscal year (Form 5) or up to the 10th day of the month following the grant (Form 4) to report to the Securities and Exchange Commission (SEC). This gave significant scope for retrospective granting. Following the introduction of the Sarbanes-Oxley Act (SOX) in August 2002, they have to report within two business days of the grant. 2 Beginning with a Wall Street Journal article Perfect payday on March 18, 2006, which reports that executives of six firms obtained option grants on multiple dates all exactly at a local minimum, and suggests that this cannot be explained by chance (see Forelle and Bandler (2006a)). 3 As listed online by the Wall Street Journal Options Scorecard, final update September See for example, Yermack (1997), Lie (2005), Heron and Lie (2007, 2009), Narayanan, Schipani and Seyhun (2007), Narayanan and Seyhun (2008) and Bebchuk, Grinstein and Peyer (2010). Details are given in the literature review in Section 2. 5 Armstrong and Larcker (2009) first point out that if the decision to backdate is made ex ante, the option can be viewed as a lookback option and conjecture that this additional option on the strike price is valuable. We confirm their intuition is correct but take it several steps further - we quantify the value of backdating and show that risk aversion is key in enabling us to obtain values which are significant and which increase with stock price volatility, thus explaining the empirical observations. 2

3 We value this opportunity to backdate in a setting which takes into account both the executive s risk aversion and exposure to non-diversifiable risk and the American early exercise feature of option grants. 6 We demonstrate that not only does this approach show that the opportunity to backdate is very valuable - much more so than previously thought - but that its value increases when the volatility of the firm s stock price is higher - thus enabling us to reconcile the theoretical and empirical findings. We now explain our contributions in more detail. The extant literature has provided a number of estimates of the ex post gains to executives and thus the economic significance of backdating. Typically the gains are estimated using the Modified Black Scholes formula. Narayanan, Schipani and Seyhum (2007) calculate an average benefit of 1.25% to 3.66% of the value of the options the executives received during the pre-sox period. Narayanan and Seyhun (2008) demonstrate that by backdating for one month during their sample, an executive could have increased the value of their option grant by around 8%. 7 The natural question to ask upon seeing these numbers is: why did executives backdate options if their estimated gains are relatively small? We show, in contrast, that the magnitude of gains the risk averse executive could expect ex ante from the opportunity to backdate are significant and therefore could have provided a motivation to backdate. For our base level of risk aversion, percentage gains given a one month backdating window range between 7.2% to as much as 25.5%. We also compute corresponding ex ante gains to an otherwise equivalent risk neutral executive under Black Scholes and find they are much smaller at between 2.2% and 8.2%, underscoring the need for risk aversion. Strikingly, under our base risk aversion, and for the same parameters as Narayan and Seyhun (2008), our model estimates a gain of 13.5% due to the backdating opportunity over one month. Just as it is well accepted that the Black Scholes model does not provide a good estimate of the value of options to executives (Hall and Murphy (2000,2002), Lambert, Larcker and Verrecchia (1991), and recently Carpenter, Stanton and Wallace (2010)), we show it also does not provide a good estimate of the value of backdating to executives. Whilst risk averse executives place a lower value on their options when their inability to diversify is taken into account, they place a higher value on the opportunity to engage in backdating. The intution is that non-diversifiable risk has a greater negative impact on 6 It is well understood that executive s who are forced to hold an undiversified portfolio will value standard option grants at much less than their cost to well-diversified shareholders (Lambert, Larcker and Verrecchia (1991), Carpenter (1998), Hall and Murphy (2000, 2002)) and exercise much earlier than if they were not restricted (Carpenter (1998), Bettis, Bizjak and Lemmon (2005)). However, backdating has not yet been considered in this context and we fill this gap. 7 An alternative measure is used by Bebchuk, Grinstein and Peyer (2010). 3

4 American at-the-money (or non-backdated) options than those that are backdated or in-the-money, see Section 4.1 for details. Furthermore, the additional cost of the grant imposed on shareholders when risk averse executives can backdate is also significant, although less so than the magnitude of the gains to executives themselves. In the absence of backdating, non-diversifiable risk exposure lowers the subjective value of options to the executive relative to their cost to well-diversified shareholders. We show that the additional costs to shareholders in the presence of backdating, however, are lower in magnitude than the size of the gain that the executive can expect from backdating. Percentage rises in costs to the firm due to backdating by risk averse executives range from 4.6% to over 10% in our model. We demonstrate that the additional cost to the firm is greater if executives are risk averse, rather than risk neutral, or equivalently, greater when options are valued in a utility-based rather than under a Black Scholes model. We answer the second question by showing that incorporating non-diversifiable risk and early exercise can explain the greater frequency of backdating in high volatility firms. There is substantial evidence across the empirical literature (Bebchuk, Grinstein and Peyer (2010), Heron and Lie (2009), Bizjak, Lemmon and Whitby (2009)) that firms with more volatile stock prices (often high tech firms) were more likely to engage in backdating. Intuition would suggest that executives at firms with high volatility potentially have more to gain from backdating as they have a greater chance of obtaining a date with a lower strike price. We will call this the expected strike discount effect. Bebchuk, Grinstein and Peyer (2010) investigate this further and find that indeed, grant events were more likely to be classified as lucky in months when the difference between the lowest and median stock price was greatest. However, this intuition is incomplete, and Black Scholes option pricing arguments (see Hull (2012), McDonald (2012)) inform us that at-the-money call options are more sensitive to volatility than in-the-money calls. 8 Thus for a given increase in volatility, a non-backdated option will have a larger increase in its Black Scholes value than an otherwise equivalent backdated option. Computations using Modified Black Scholes by Walker (2007) and Dierker and Hemmer (2007) demonstrate that this moneyness on vega effect is dominant. These authors obtain the opposite prediction to that documented empirically. We resolve this puzzle, and by doing so, demonstrate the importance of taking into account both the executive s exposure to non-diversifiable risk and their ability to exercise 8 Phrased differently, there is considerable variation with volatility in option sensitivity with respect to strike. For high volatility, the Black Scholes call value is less sensitive to strike, because for high volatility, there is a greater likelihood of the option expiring out-of-the-money. 4

5 options early when assessing the benefit of backdating to executives. In our model, for our base level of risk aversion in Panel B of Table 2, we see that the benefit of backdating is always increasing with stock price volatility, consistent with the evidence in the empirical literature. Why is this the case? When the executive is risk averse, the difference between the sensitivities of in and at-the-money options to volatility is much smaller than under Black Scholes and thus the moneyness on vega effect is reduced, and is typically dominated by the impact of the expected strike discount. The reasoning is as follows. It is well known that subjective option values and vegas are are reduced by non-diversifiable risk exposure, and vega can even be negative for sufficiently far in-the-money options. However, American early exercise places a lower bound on negative vega, and hence vegas vary within a tighter band than under the Black Scholes model. Much of our discussion has been applicable to the pre-sox era. What can we say about the period since 2002 when executives only had two days to report? Since Narayanan and Seyhun (2008) find that since 2002, 10% of executives reported more than one month late, our analysis is still very relevant post-sox. However, is the opportunity to backdate still valuable for executives who stay within the rules? The surprising answer is yes. We consider a two day window and find that there is still a 5% increase in ex ante value to the executive from the backdating opportunity, for our base case parameters. So, despite the much tighter post-sox reporting rules, somewhat surprisingly, the opportunity for executives to benefit from backdating is still present. In addition to its implications for the backdating literature, our work brings to the forefront some interesting findings for a broader corporate finance audience. Risk aversion is usually associated with a reduction in the value of risky contingent claims (eg. Lambert, Larcker and Verrechia (1991), Detemple and Sundaresan (1999), Miao and Wang (2007)). Our finding in this paper that it instead increases the value of the option to backdate is in sharp constrast to this usual thinking. In the executive compensation literature, risk aversion has been useful in explaining why we see undiversified executives exercise their options early (Bettis, Bizjak and Lemmon (2005), Carpenter, Stanton and Wallace (2010, 2013)), and risk-adjusted pay may explain much of the apparent higher pay for U.S. versus U.K. and international CEOs (Conyon, Core and Guay (2011) and Fernandes, Ferreira, Matos and Murphy (2013)). In the wider corporate finance literature, Cai and Vijh (2007) have shown risk aversion is important in explaining mergers and acqusitions data - acquisitions enable target CEOs to remove liquidity restrictions on stock and option holdings and diminish the illiquidity discount. Similarly, Sorensen, Wang and Yang (2014) show a risk averse limited partner requires the general partner to produce substantial alpha to cover their 2/20 compensation, with break-even values of PE 5

6 performance measures that are reasonably close to empirical averages. 9 This paper presents a new setting - option backdating - where taking into account risk aversion enables the theory to match empirics. There are remarkably few theoretical models of option backdating and surrounding issues. Stannard and Guthrie (2013) develop a model to formalize the managerial power view of option backdating. In contrast, Gao and Mahmudi (2014) (also Dierker and Hemmer (2007)) suggest that firms may have used backdating as an optimal response to distortions in the institutional environment. Whilst Gao and Mahmudi (2014) argue via intuition gleaned from Hall and Murphy (2000) that in-the-money options are optimal for risk averse managers, they do not, as we do, value the lookback option to backdate, consider early exercise, or provide estimates of the magnitude of the benefit or cost of backdating. Finally, Eikseth and Lindset (2011), as we do, recognize the ex ante value of the option to backdate. However, their Black Scholes valuation does not take into account non-diversifiable risk exposure or early exercise, and thus cannot speak to the questions we address in this paper. Indeed, as we show later, an ex ante Black Scholes model gives the opposite relationship with volatility than is documented in the empirical literature. The paper is organized as follows. We review both the empirical evidence on backdating found in the literature and extant executive stock option valuation models in the next section. Section 3 develops the theoretical model to value the option grant with (and in the absence of) the opportunity to backdate and the corresponding costs to shareholders. We quantify the magnitude of the gains to the executive from backdating and the costs to the firm in Section 4. In Section 5 we demonstrate how our model explains the observed link between firms with high volatility stocks and backdating. We close by discussing broader implications of our model for the backdating literature as well as for practice. 2 Literature Review (i) Empirical evidence of backdating There is substantial evidence that backdating of option awards took place in the US, particularly prior to the tightening of the rules on the reporting of grants to the SEC in Before August 29, 9 Furthermore, Chen, Miao, and Wang (2010) study the effects of non-diversifiable risk on entrepreneurial finance and show that more risk-averse entrepreneurs borrow more in order to lower their business risk exposure. Miao and Wang (2007) show that non-diversifiable risk significantly alters option exercising strategies. 10 Cicero (2009) provides empirical evidence of opportunistic timing of option exercise rather than grant date backdat- 6

7 2002 (pre-sox), executives could either report their option grant to the SEC no later than 45 days after the fiscal year end (Form 5) or report no later than the 10th day of the month following the grant (Form 4). After SOX, executives have to report no later than two business days after the grant. Early work by Yermack (1997) identified that option grants were followed by abnormally positive returns. 11 Lie (2005) documents a V-shaped abnormal stock return pattern around option grants and suggests executives may be timing awards to their advantage. Heron and Lie (2007) and Narayanan and Seyhun (2008) 12 exploit the change in reporting requirements due to the implementation of SOX to conduct refined tests of the backdating hypothesis. Heron and Lie (2007) demonstrate that the abnormal stock return pattern around option grants was greatly diminished after 2002 (except when the grants are not reported to the SEC in a timely fashion), consistent with a backdating explanation. However, contrary to the belief that backdating would completely disappear in the post-sox era, Narayanan and Seyhun (2008) find more than 20% of executives report their options late (after the two day rule) with 10% reporting more than a month late. Consistent with backdating, both Heron and Lie (2007) and Narayanan and Seyhun (2008) find that post-grant stock price and stock return reversals around the grant date are positively related to reporting lags. Heron and Lie (2009) and Bebchuk, Grinstein and Peyer (2010) document that backdating practices were pervasive during the period 1996 and Heron and Lie (2009) estimate that 13.6% of option grants were backdated. 13 They find that 29.2% of firms in their sample at some point engaged in manipulation of grants to top executives. Bebchuk, Grinstein and Peyer (2010) develop a new test for backdating using price-ranks and categorize lucky grants as those which are awarded at the lowest stock price of the grant month. In their sample, and pre-sox adoption, about 15% of grants to CEOs and 11% of grants to directors were lucky. (ii) Valuation Models for executive stock options in the absence of backdating The importance of the executive s risk aversion and inability to diversify due to hedging constraints has long been recognized in both the theoretical and empirical literature on valuation of executive compensation. It is well understood that executive s who are forced to hold an undiversified portfolio ing. 11 Related work of Aboody and Kasznik (2000) and Chauvin and Shenoy (2001) found evidence of both the timing of unscheduled option grant dates around the scheduled release of corporate information, and the timing of the release of information around scheduled grant dates. 12 See also Collins, Gong and Li (2005). 13 The fraction is highest for unscheduled at-the-money grants, those that are in the tech sector (32% of firms in the tech sector versus 20.1% of non-tech firms), and those with high stock price volatility (29% of firms with high volatility versus 13.6% with low volatility). 7

8 will value options at significantly less than their cost to well-diversified shareholders (Lambert, Larcker and Verrechia (1991), Carpenter (1998), Hall and Murphy (2000, 2002)). Furthermore, undiversified executives will exercise options far earlier than if they were well-diversified, consistent with the early exercise observed in practice (Carpenter (1998), Bettis, Bizjak and Lemmon (2005) amongst others). 14 Models allowing dynamic optimal investment of non-option wealth for risk averse executives have been studied by Detemple and Sundaresan (1999), Cai and Vijh (2005), Henderson (2005), Grasselli and Henderson (2009), Leung and Sircar (2009) with related contributions by Kahl, Liu and Longstaff (2003) and Ingersoll (2006). Most recently, Carpenter, Stanton and Wallace (2010) consider a riskaverse employee who maximizes expected utility by choosing both an exercise policy for his options in addition to a dynamic trading strategy in the market and riskless asset for his outside wealth. These models provide a benchmark for modeling optimal exercise, option valuation and option cost to shareholders. A general message of all of these papers is that risk averse executives with nondiversified portfolios will have much lower option valuations than if they were not constrained. That is, risk aversion decreases option values. Here, in contrast, risk aversion increases the gain from backdating to executives. In practice, most firms use an approximate valuation method - the Modified Black Scholes method - permitted by FAS 123R, in which the option expiration date is set equal to the options expected life (or a fixed term in the absence of sufficient data). As demonstrated in Carpenter, Stanton and Wallace (2010, 2013), this approximation entails significant errors which vary with firm, option and option holder characteristics, underscoring the importance of accounting for the dependence of exercise on characteristics such as risk aversion. 3 Model A company will have a board meeting at the end of the month, T 0, at which options will be granted to senior executives. Given the vast majority of options are granted at-the-money 15 in the absence of backdating, our benchmark is that at-the-money call options are granted on date T 0. We will assume 14 A number of recent empirical studies investigate the determinants of executive exercise (Armstrong, Jagolinzer and Larcker (2007), Klein and Maug (2010) and Carpenter, Stanton and Wallace (2013)) and find a role for factors arising from both portfolio theory and utility arguments as well as behavioral factors. Bettis, Bizjak and Lemmon (2005) and Carpenter, Stanton and Wallace (2010) find a significant relationship between measures of stock price volatility or option risk, and early exercise, consistent with utility-based arguments. Furthermore, Carpenter, Stanton and Wallace (2013) note the propensity of top employees to exercise faster may stem from greater exposure to non-diversifiable firm risk. 15 In the sample used by Heron and Lie (2007), 86% of options have a strike price within 2% of the stock price. 8

9 options have a life of T years 16 and are American style, and hence are exerciseable until maturity date T 0 + T. 17 Executives know they have the opportunity to backdate the option award and instead grant atthe-money options retrospectively, on an earlier date with a more favorable strike price. Following Bebchuk, Grinstein and Peyer (2010), where lucky grants are awarded at the lowest strike price of the month, we consider a backdating window or lookback period of one month. Executives can select the best date within the one month window on which to grant options. 18 Denote the best strike by J = min 0 u T0 S u, where S is the stock price, and the date on which the minimum occurred by 0 t min T 0. Ex post, executives who decide to backdate report that they received at-the-money (strike J) options at t min. In fact, at the board meeting date, they have options which are necessarily in-the-money, since J S T0. Ex ante, executives who have the opportunity to backdate should take this into account in their valuation. Our aim is to compare the ex ante value of the option grant if backdating can occur, to the benchmark value of the grant if backdating cannot occur. We compare the benefit of backdating to the executive for a fixed size of option grant. 19 Executives are constrained in their ability to trade the stock S which underlies their options. 20. This means they cannot fully hedge risk and face some non-diversifiable risk whilst they continue to hold unexercised options. Although the executive cannot trade the firm s stock, S, in common with Carpenter, Stanton and Wallace (2010) amongst others, we allow them to partially hedge their exposure by taking a position in the market portfolio, M and riskless bond with constant riskless rate 16 Option maturities have been documented to be surprisingly homogeneous. Bettis, Bizjak and Lemmon (2005) document most options in their sample have a ten year maturity whilst Narayanan and Seyhun (2008) report an average maturity of 9.5 years. Most recently, Carpenter, Stanton and Wallace (2013) study over 1 million option grants across 102 firms and find that the term is uniformly ten years. 17 We do not consider a vesting period during which options are not exerciseable. Vesting would not change our key findings. It would slightly reduce all option values (and costs) but should not significantly alter relative benefits (and costs) of backdating or their relationship with Black Scholes estimates. 18 Microsoft were found to have been offering executives the chance to select their own grant date for at-the-money options within a one month window, see Forelle and Bandler (2006b). 19 This assumes that the grant size does not change with the decision to backdate. Bebchuk, Grinstein and Peyer (2010) find no evidence that firms providing opportunistically timed CEO grants reduced CEO s compensation from other sources. 20 Insiders cannot short sell stock as they are prohibited by Section 16-c of the Securities and Exchange Act. Evidence of early exercise across all ranks of employees suggests they are constrained as to the hedging they can carry out (Bettis, Bijak and Lemmon (2005), Carpenter, Stanton and Wallace (2013)). 9

10 r. Prices for the stock S and market M follow 21 ds S = (ν q)dt + ηdb (1) dm M = µdt + σdz (2) where standard Brownian motions B and Z are defined on a probability space (Ω, F, F u, P) where F u is the augmented σ-algebra generated by {B u, Z u ; 0 u t} and their instantaneous correlation is ρ ( 1, 1). The volatility of stock returns, η, expected return on the stock, ν, proportional dividend yield, q 0, and expected return, µ and volatility of market returns, σ, are all constants. The mean stock return, ν is equal to the CAPM return for the stock, given its correlation with the market, ν = r + β(µ r); β = ρη/σ. Denote by θ t the cash investment held in the market M at time t. The executive s non-option wealth follows dw u = (rw u + θ u (µ r))du + θ u σdz u ; W t = w given initial wealth w. Remaining non-diversifiable risk that cannot be hedged is represented by idiosyncratic volatility (1 ρ 2 )η 2, and is greater, the lower the (absolute value of) correlation between the stock and market. The executive aims to maximize the expected utility of total wealth at the option maturity, T 0 + T, over choice of exercise time τ and outside investment in the market θ u (satisfying the usual integrability condition E T 0 +T t θ 2 udu) and riskless bonds. We assume the executive has constant absolute risk aversion (CARA) denoted by U(x) = e γx ; γ > As such, when the option is exercised, it is optimal to sell shares immediately. 23 The cash proceeds are added to outside wealth and will continue to be invested optimally in the market and riskless bond until the maturity date. 3.1 Valuing the Option grant with the opportunity to backdate We now value the option grant under this framework. Over the period [T 0, T 0 + T ], the option grant with the opportunity to backdate is N American calls with known strike J. We denote the value to 21 In common with the ESO valuation literature, we abstract from leverage and dilution considerations, and assume the firm s stock price is unaffected by the executive. 22 For tractability, we use CARA. As described, the valuation involves numerical solution of a three-dimensional problem. Using alternative wealth dependent utilities such as CRRA would add a further dimension. Solving under alternative utilities would not alter our main findings. 23 The value of continuing to hold the stock is lower than the stock s market value due to non-diversifiable risk. In the absence of minimum holding constraints, immediate sale upon exercise is optimal. Empirically, many executives sell shares immediately upon exercise (Ofek and Yermack (2000)). 10

11 the risk averse executive (at time u [T 0, T 0 + T ] with current wealth W u, current stock price S u ) of the option grant with the opportunity to backdate by Y b (u, W u, S u ; J) and set out how to value this grant in the next subsection. Over the backdating window [0, T 0 ], the option grant cannot be exercised and has features in common with a lookback option (see Goldman, Sosin and Gatto (1979), Conze and Viswanathan (1991), and, for a textbook treatment, Hull (2012)). A (floating strike) European lookback call with maturity T 0 allows the holder to buy the stock at the minimum stock price over the option s life, J, and so has a payoff equal to the difference between the terminal stock price, S T0 and the strike, J. Our backdated option grant instead has a payoff at T 0 equal to Y b (T 0, W T0, S T0 ; J), the value of the N American calls, and hence has a payoff which is non-linear in J. Valuation after the board meeting date, [T 0, T 0 + T ] We work backwards in time from the maturity date T 0 +T, and first consider times between T 0 and T 0 + T when the options are exercisable by the executive. By definition, the strike J = min 0 u T0 S u is known. The value to the executive of having N exercisable strike J options, optimal choice over outside investments θ t and optimal choice over the exercise time of options, 24 Y b (u, W u, S u ; J) solves the variational inequality: (for T 0 u T 0 + T ) where the differential operator L is defined by L = η2 s 2 2 Y b (u, W u, S u ; J) M(u, W u + N(S u J) +, T 0 + T ) (3) Y b t + sup {LY b } 0 (4) θ 2 s 2 + (ν q)s s + ρθσηs 2 w s + θ2 σ [θ(µ r) + rw] w2 w. (5) The inequality (3) says that the options stay alive whilst the value of continuing is greater than the exercise value, where the exercise value M(u, W u + N(S u J) +, T 0 + T ) is the value derived from optimally investing the exercise proceeds and any cash wealth until T 0 + T, given by (Merton (1971)): M(t, w, T ) = sup EU(W T W t = w) = e γwer( T t) e (µ r)2 σ 2 ( T t). (6) {θ s} t s T 24 We effectively assume the option grant is exercised as a block, consistent with most models of ESO valuation (Carpenter, Stanton and Wallace (2010), amongst many others. Henderson, Sun and Whalley (2013) study the impact of having a portfolio of ESOs with differing strikes and maturities on exercise thresholds, values and costs, but do not treat backdating.). By doing this, we obtain a numerically tractable problem and we can compare the value of backdated versus non-backdated options. Non-block exercise will be of second-order importance here as even large changes in exercise thresholds have a relatively small effect on option values. 11

12 The option grant is exercised when the value from continuing to hold it is sufficiently low that it equals the value from exercising and investing the payoff N(S u J) optimally, and thus the optimal exercise time τ is characterized by: τ = inf{t 0 u T 0 + T : Y b (u, W u, S u ; J) = M(u, W u + N(S u J) +, T 0 + T )}. (7) Due to the choice of CARA, we can scale out dependence on wealth. As for standard American options, we can describe the optimal exercise time τ as the first time the stock price reaches an exercise threshold τ = inf{t 0 u T 0 + T : S u = Sb (u)}. (8) We define the certainty equivalent or subjective value of the backdated option grant to the executive, V b (u, S u ; J); u [T 0, T 0 + T ], with strike J = min 0 u T0 S u, to be the cash equivalent which, when invested optimally, would give the same expected utility as is achieved by the option grant: Y b (u, W u, S u ; J) = M(u, W u + V b (u, S u ; J), T 0 + T ). We now solve the free boundary problem, together with appropriate boundary conditions 25 numerically using standard finite difference methods. Although the strike J = min 0 u T0 S u is known during [T 0, T 0 + T ], we will need the values at T 0 for each possible value of J, and hence we need a threedimensional grid (t, S, J). Valuation during the backdating window, [0, T 0 ] To value the option grant prior to T 0, we use conditioning to reduce the valuation to that of a European derivative with payoff at T 0 of V b (T 0, S T0 ; J). We have Y b (u, W u, S u ; J) = sup {θ u} 0 u T0 EM(T 0, W T0 + V b (T 0, S T0 ; J), T 0 + T ); 0 u T 0 During the backdating window u [0, T 0 ], Y b solves the partial differential equation (pde) given in (4) and (5), together with a boundary condition at T 0. Again, using a standard transformation to remove wealth dependence, we solve the pde, together with appropriate boundary conditions 26 numerically using standard finite difference methods. 25 Appropriate boundary conditions are a maturity condition: Y b (T 0+T, W T0 +T, S T0 +T ; J) = M(T 0+T, W T0 +N(S T0 J) +, T 0 + T ), and a condition at S = 0, Y b (u, W u, 0; J) = M(u, W u, T 0 + T ). 26 The boundary conditions are: an upper bound for large S, a lower bound for S = J, and an implicit boundary condition for J = 0. 12

13 3.2 Valuing the option grant without backdating: A benchmark In the absence of backdating, the executive receives at-the-money American call options at T 0 with a T year maturity. Prior to receiving the option grant at T 0, the executive has N forward-starting at-the-money American calls 27 with unknown stochastic strike S T0. At T 0, the option grant are simply American call options with a given strike because at T 0, the strike is known. We again work backwards in time, first valuing after the board meeting date. Valuation after the board meeting date, [T 0, T 0 + T ] During the time period u [T 0, T 0 + T ], the grant is valued using the above framework with the restriction J = S T0, giving the certainty equivalent option value V nb (u, S u, S T0 ). As described earlier in (8), we can give the optimal exercise time as the first time the stock price reaches an exercise threshold, denoted by Snb (u) in the absence of backdating. Valuation during the backdating window, [0, T 0 ] To value the option grant prior to T 0, we use conditioning to reduce the valuation to that of a European derivative with payoff at T 0 of V nb (T 0, S T0, S T0 ). We have Y nb (u, W u, S u, S T0 ) = sup θ u;0 u T 0 EM(T 0, W T0 + V nb (T 0, S T0, S T0 ), T 0 + T ); 0 u T 0. We can now use pde transformation techniques and then re-express in terms of the certainty equivalent value of the option grant to give: 1 V nb (u, S u, S T0 ) = γ(1 ρ 2 )e r(t 0+T u) log )V nb (T 0,S T0,S T0 )e r((t0+t ) T0) E[e γ(1 ρ2 ]; 0 u T 0. We can use numerical integration to compute the certainty equivalent or subjective value of the non-backdated option grant V nb (u, S u, S T0 ) for u [0, T 0 ]. 3.3 The Cost of the option grant to the company We now turn to consider the cost or objective value of an option grant to the company. Shareholders are not restricted in their ability to diversify, and thus, the cost of options to them can be computed under the assumption that they are well-diversified and able to hedge perfectly. The cost or objective value of an option grant is the value of an equivalent tradeable American option, but with the exercise decision controlled by the executive (see Carpenter (1998), Bettis, Bizjak and Lemmon (2005) and Carpenter, Stanton and Wallace (2010) all in the absence of backdating). Tradable American option values can 27 The standard terminology forward-starting refers to the option starting at some point in the future, see for example Hull (2012) for a textbook treatment. 13

14 be calculated under Black Scholes model assumptions. In our model, the cost of a backdated option grant is its equivalent tradeable value, given the backdating decision, exercise decision and optimal partial hedging are optimized by the executive. Cost after the board meeting date, [T 0, T 0 + T ] The cost to shareholders after T 0 of the N American calls with strike J, satisfies the standard Black Scholes pde for stock prices below the executive s exercise threshold, s S b (t) C b t + (r q)s C b s + η2 s 2 2 C b 2 s 2 rc b = 0 (9) with boundary conditions C b (u, 0, J) = 0, C b (T 0 + T, S T0 +T, J) = N(S T0 +T J) + and C b (u, S b, J) = N(S b (u) J)+. Cost during the backdating window, [0, T 0 ] During [0, T 0 ], C b (u, S u ; J) solves the Black Scholes pde in (9), together with a boundary condition at T 0. Again, we solve the pde numerically together with boundary conditions. 28 Finally, the cost of the option grant in the absence of backdating, C nb, is computed using the same methods as above, taking into account the corresponding no-backdating exercise threshold S nb (u); T 0 u T 0 + T. 4 The Value and Cost of Backdating: Results In this section we present results from numerically solving the models outlined in Section 3. Parameters are chosen as follows. We take our base case value of absolute risk aversion γ = 0.2 in line with those in the literature (Leung and Sircar (2009), Miao and Wang (2007) and Chen, Miao and Wang (2010)). They are also comparable to the levels of relative risk aversion and outside wealth used by Carpenter, Stanton and Wallace (2010). Consider an executive with a grant of one million options with strike $10, on stock worth $10. If she has relative risk aversion of 4 and outside wealth of $20 million, her absolute risk aversion is γ = 0.2. We also consider a very low value of absolute risk aversion γ = 0.01, to show how values and costs vary significantly from those under a Black Scholes model, even for executives who are only slightly risk averse. Our range of parameters for the riskfree rate, dividends, volatility and the market asset are broadly consistent with those used in the backdating and ESO valuation literature. 28 The boundary conditions are: an upper bound for large S, a lower bound for S = J, and an implicit boundary condition for J = 0. 14

15 Before continuing, we briefly summarize notation. We denote by V b = V b (t, S t, J) the certainty equivalent value of the option grant with the opportunity to backdate, with strike J = min u T0 S u ; V nb = V nb (t, S t, S T0 ) the certainty equivalent value of the benchmark non-backdated option grant with strike S T0 ; C b = C b (t, S t, J) the cost of the backdated option grant to well-diversified shareholders (under optimal exercise by executive); and C nb = C nb (t, S t, S T0 ) the cost of the benchmark nonbackdated option grant to well-diversified shareholders (under optimal exercise by executive). Table 2 reports these certainty equivalent values and costs for a range of values of riskfree rate, dividend yield, volatility and risk aversion. For comparison purposes with the extant literature we are also interested in the Black Scholes equivalents of the above. We denote by V BS b = Vb BS (t, S t, J) the Black Scholes value of the option grant with the opportunity to backdate, with strike J = min u T0 S u, and V BS nb = V BS nb (t, S t, S T0 ) the Black Scholes value of non-backdated option grant with strike S T0. Table 1 reports these Black Scholes values for the same set of parameters as used in Table 2. Under Black Scholes, all risk is assumed hedgeble, and so option values to the unconstrained (risk neutral) executive and option costs to well-diversified shareholders are the same. 4.1 How large are the ex ante gains to the executive from the opportunity to backdate? The magnitude of the gains that could be expected ex ante by the executive from having the opportunity to backdate the option grant are significant. For an executive with our base level of risk aversion, and with stock parameters of r = 0.05, q = 0 and volatility 60%, the value of the grant in the absence of backdating is $ million. This increases by 17% to a value of $ 2.48 million when the executive has the opportunity to backdate over a one month lookback window. For an executive with our base risk aversion level, Table 2 shows the percentage gains from the opportunity to backdate vary between 7.2% and 25.5% across our range of other parameters. Percentage gains or the relative gain from backdating are calculated by %G = (V b V nb )/V nb. Furthermore, unreported simulations show that the value of the option with backdating opportunity increases when the length of the lookback window increases, keeping all else fixed. How do these gains compare to those calculated in the absence of executive risk aversion? We compute the relative gain from backdating under the Black Scholes model, given by %G BS = (V BS b V BS nb )/V BS. Table 1 displays these comparable values for the benefit of backdating to executives under nb a Black Scholes framework. For the same parameters as in Table 2, we see the percentage gains under 15

16 Black Scholes are much smaller and vary between 2.2% and 8.2%. Why are the gains to executives so much larger when non-diversifiable risk is taken into account? In fact, non-diversifiable risk reduces option values of both backdated and non-backdated options relative to their Black Scholes equivalents. We return to our earlier example to illustrate this. A risk averse executive with the base level of risk aversion and r = 0.05, q = 0 and volatility 60% placed an ex ante value of $ 2.48 million on the grant if she could backdate over a one month window. However, an otherwise identical but risk neutral executive would have valued this opportunity at $ 7.63 million. 29 The fact that an executive with exposure to non-diversifiable risk places a much lower value on an option grant (in the absence of backdating) is well established (Lambert, Larcker and Verrecchia (1991) and Hall and Murphy (2000, 2002) for example). The presence of non-diversifiable risk represents a cost to the executive which remains until the options are exercised. This cost encourages the executive to exercise earlier, at a lower moneyness, than in a Black Scholes world (see Carpenter, Stanton and Wallace (2010)). This intution carries over to the situation here where the executive has the opportunity to backdate. Although risk aversion reduces the option values of both backdated and non-backdated options relative to their Black Scholes equivalents, there is a larger reduction in value for at-the-money options. Thus there is a larger difference between the certainty equivalent or utility value and the Black Scholes value of non-backdated options than for the backdated options. And this means that the ratio V b /V nb is larger than the equivalent Black Scholes ratio V BS b /Vnb BS. Why is this the case? Call option time-value is maximized close-to-the-money so that far in-the-money call options have little time-value. Risk aversion impacts on option values primarily through its effect on option time-value, since it only affects values whilst the executive continues to hold the exercisable option. Thus risk aversion has greater impact on at-the-money than in-the-money American options. We also see from Table 2 that backdating is most valuable to an executive with higher risk aversion, and when a stock has a higher dividend yield and volatility. Each of these magnifies the impact of non-diversifiable risk on at-the-money relative to in-the-money American exercise options. Up to this point, our analysis has focussed on a one-month backdating window, and is most applicable to the pre-sox era when backdating was prevalent. What can we say about the period since 2002 when executives only had two days to report? In fact, since Narayanan and Seyhun (2008) find that 10% of executives reported more than one month late, our analysis is still relevant for the post-sox period. However, if we consider those executives who stay within the reporting rules, is the 29 Similarly, the risk averse executive values the option grant in the absence of backdating at $ 2.12 million whereas the risk neutral executive places a much higher value of $ 7.4 million on this grant of options, see Tables 1 and 2. 16

17 opportunity to backdate still valuable for these executives? We re-run our base case numbers taking a 2 day window and find that there is still a 5% increase in value to the executive from backdating. The Black Scholes model would suggest a much smaller benefit of around 1%. So, despite the backdating window shrinking dramatically under the post-sox rules, the benefit to the risk averse executive from backdating is still present. 4.2 How large are the costs to shareholders? In this section, we analyze the cost of backdating to shareholders. We calculate the increased cost of the option grant from the perspective of well-diversified shareholders if executives have the opportunity to backdate as %C = (C b C nb )/C nb. These additional costs are also significant in magnitude. For the firm employing our executive with base level of risk aversion and parameters r = 0.05, q = 0, η = 0.6, the cost to the firm of the grant in the absence of backdating is $ 3.19 million. If the executive can backdate over a one month window, this cost rises to $ 3.42 million, a proportional increase of 7.4%. Table 2 shows the percentage rises in compensation costs to the firm range from 4.67% to 11.8%, assuming our base level of risk aversion. Furthermore, unreported simulations show the rise in cost is larger when the backdating or lookback window is longer, keeping all else fixed. We also see from Table 2 that backdating is most costly to shareholders when executives are more risk averse, and when a stock has a higher dividend yield and volatility. How does the magnitude of the cost increase compare to that of the gains to the executive? Recall, in the absence of backdating, non-diversifiable risk exposure means that the subjective value of options to the executive is less than their objective cost to shareholders. (We see here for example, that the executive values the option grant at $ 2.12 million whilst the cost to shareholders is $ 3.19 million.) However, when we measure the benefit of the opportunity to backdate relative to the increased cost to shareholders, we find the opposite is true. The additional cost to the firm s shareholders from backdating is lower than the magnitude of the gains that are expected by the risk averse executive. For example, our base case executive enjoyed a potential gain of 17% from having the opportunity to backdate, whilst the cost to the firm increased by only 7.4%. Furthermore, the additional cost to the firm is greater when the executive is risk averse, than if the executive were risk neutral. In our example, from Table 2, the increase in cost to shareholders if the executive is risk neutral is only 3.16% (which is the same as the gains to the risk neutral executive). A similar intuition to that used earlier can be used to explain these relationships. First, observe that calculation of the objective cost of options involves using the optimal exercise threshold of the 17

18 executive (see earlier). Thus for in-the-money options, those that are backdated, there is much less of a gap between value and cost, as they are both tied to the intrinsic value at the same threshold, and the threshold is closer for in-the-money than at-the-money options. This results in the ratio V b /V nb exceeding C b /C nb, ie. the benefit of backdating to the executive is greater than the increased cost to shareholders. We now give an explanation as to why the increased cost to shareholders is greater if executives are risk averse than if they are risk neutral. Observe that although shareholders are assumed risk neutral, they use the exercise threshold of the risk averse executive, which is lower than that of a risk neutral executive. This translates into option costs being lowered for both backdated and non-backdated options, relative to Black Scholes costs, but as before, there is a greater reduction in at-the-money costs. Thus there is a larger difference between the objective cost and Black Scholes cost for non-backdated options than for the backdated options, again because of the effect of risk aversion on option time value. 4.3 A Comparison to ex post estimates in the literature A motivating factor for undertaking our study was the observation that ex post empirical estimates found in the extant literature of the gains from backdating options were quite modest. Whilst the media focus was on the large strike price discounts uncovered 30, researchers have typically estimated the gains to executives using the Modified Black Scholes formula. This formula is an approximation which assumes all executives exercise at a single point in time regardless of the stock price path. It estimates the impact of early exercise by making a fixed reduction to option maturity in the European Black Scholes formula. The expected life of an option can potentially be estimated from firm data. However, Carpenter, Stanton and Wallace (2010, 2013) demonstrate that significant valuation errors are introduced using modified Black Scholes to value options when compared to optimal exercise (in the absence of backdating). The most common measure used involves estimating the increase in grant value for a percentage reduction in strike price. We will take estimates from Narayanan and Seyhun (2008) as a case study for comparison. Narayanan and Seyhun (2008) demonstrate by backdating for one month during their sample, that an executive can increase the value of their option grant by around 8%. The details of their calculation are as follows. Narayanan and Seyhun (2008) estimate the gain by calculating the Modified Black Scholes value of the average at-the-money option grant with S = $30, r = 5%, volatility of 30%, T = 9.5 years, and zero dividends. These parameters result in an option value of 30 See, for example, Forelle and Bandler (2006a). 18