Stock Based Compensation: Firm-specific risk,
Efficiency and Incentives
Vicky Henderson
Nomura Centre for Quantitative Finance
University of Oxford
henders1@maths.ox.ac.uk
1
http://www.maths.ox.ac.uk/~henders1
Stock Based Compensation: Some statistics ...
In the US, executive stock options represented $893bn worth of shares
in June 2000; Nohel and Todd (2001).
Hall and Murphy (2000) cite that the grant date value of stock options accounted for 40% of total pay for S&P500 CEO’s in 1998
Conyon and Murphy (2000) find US CEO’s earn on average 30%
more cash pay and 500% more total pay than UK CEO’s (1997)
2
However, in UK, proportion of total pay that is performance related
rose by 40% between 1995 and 2002
56% of UK ESO’s are currently OTM (”underwater”) but grants of
options continue to increase, despite market falls
In terms of stock grants (rather than options), UK companies tend
to use more LTIP’s (vest if performance objectives met) whereas US
use stock with no performance criteria
Bryan, Nash and Patel (2002) find there is more equity in the compensation mix for countries with equity-orientated capital markets,
and countries where shareholder rights are strong
3
Stock compensation less significant part of total pay in Japan and
Germany, perhaps as capital markets debt-based.
Stock options were banned in Japan before 1997: flat executive pay
regime and ”job for life”
By 2000, some 342 listed companies in Japan including Sony, NEC
and Sega offered options
However, not on same scale as US plans
4
Purpose of Stock based Compensation
• Link between executive wealth and stock price provides incentives,
and serves to align interests and mitigate agency problems between
shareholders and management
• Used to retain employees
• Pay
5
Accounting Treatment of Stock Compensation
Until 2002:
Little guidance on treatment of stock compensation in most countries
In US, still using GAAP issued in 1972 (pre Black Scholes!) which treats intrinsic
value at grant date as expense → ATM options escape expensing → companies
base design of plans around accounting treatment, most options issued ATM
2002:
IASB issued exposure draft ED2 on share based payments, proposed effective date
Jan 04
Will cover EU plus range of other countries
Main change is to consider ALL stock and options as expense and record their fair
value on grant date in accounts
Following corporate accounting scandals (Enron, World Com etc), US under pressure to follow, many US companies have voluntarily said they will expense
6
FASB reconsidering expensing, but political pressures in US may prevent convergence with IASB
Accounting Treatment of Stock Compensation
Japan is one of seven national accounting standard setters to have
an IASB member in their jurisdiction
ASBJ is considering the proposals of ED2 and actual conditions of
stock options in Japan
Summary issues paper Dec 02 and meeting Feb 03
7
Impact on Performance Indicators
Estimated reduction in operating income when stock compensation
is expensed: For S&P 500 companies:
5% in 1999
8% in 2000
12% in 2001.
Estimated impact on diluted EPS from 2000-2001:
US: 41 %
UK: 40 %
Japan: 5 %
Germany: 13 %
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Company cost vs Value to Executive
Early models used Black Scholes both for company cost and value to the executive
However, executive faces trading restrictions and often cannot trade the company
stock (or sell the options). Usual structure:
More recently, researchers recognise that that the cost of the compensation to the
company overstates the value that executives place on it, due to restrictions faced
by executives in trading the stock → asymmetry
[Lambert, Larcker and Verrecchia (1991), Kulatilaka and Marcus (1994), Huddart
(1994), Detemple and Sundaresan (1999), Hall and Murphy (2000) : utility models,
no short sales and non-option wealth in riskless asset only]
9
Difference called ”deadweight cost” by Meulbroek (2000), opportunity cost to company
Questions ?
• What is the value to the executive of these options and how much
lower is it than the cost to the company itself ?
• What are the incentive benefits of ESO’s to the executive ?
• How should companies design their stock based compensation plans,
in light of the recent accounting changes ? ie. no longer an accounting driven bias to give ATM options.....
10
Approach
Propose continuous time utility model, no trading in company stock
Executive is allowed to trade in the market portfolio but is subject to firm-specific
risk for incentive purposes
Exposed to total risk, can diversify away market risk, but left with firm-specific
risk (less than fully diversified). Since firm-specific risk is not rewarded, leads to
executive placing less value on options than cost to company
By separating market and firm-specific risks, it is clear that it is the firm-specific
risk that is driving the ”deadweight cost”
With the exception of Meulbroek (2001), Jenter (2001) and Jin (2000), most research treats systematic and non-systematic risks together.
11
Model itself draws heavily on recent research into options on non-traded assets,
see Davis (2000), Henderson and Hobson (2002 RISK, 2002 JEDC) and Henderson
(2002, MF)
The Model (1)
Executive receives European call options on company stock S, expiry
T , strike K
Cannot trade the stock, but can trade in the market portfolio M
dM/M = µdt + σdB
dS
= νdt + η(ρdB + 1 − ρ2dW )
S
where W is a Brownian motion independent of the Brownian motion
B driving the market.
Total variance of S is η 2, split into
• market or systematic risk ρ2η 2, and
• firm-specific or non-systematic risk (1 − ρ2)η 2
q
12
For |ρ| < 1, INCOMPLETE
The Model (2)
Wealth X generated by executive holding cash amount θ in M and
remainder earns riskless rate r:
dX = θdM/M + r(X − θ)dt
Aim of executive to maximise expected utility of wealth, where receives λ units of the call option
V (t, Xt, St; λ) = sup Et[U (XT + λ(ST − K)+)].
(θu )u≥t
13
−γx , γ > 0, which
We use exponential utility function U (x) = − 1
γe
has CARA. Bliss and Panigirtzoglou (2002) study measures of risk
aversion implied in options prices and their results suggest CRRA
may be too restrictive and CARA is more consistent with the data.
Under exponential utility, U (x) = − 1γ e−γx , γ > 0,
V (t, Xt , St ; λ) =
sup Et [U (XT + λ(ST − K)+ )]
(θu )u≥t
1
r(T −t)
= − e−γXt e
g(T − t, logSt )
γ
+
where g(0, logs) = eλγ(S−K) and g solves NON-LINEAR pde
1 2 0
1 2 00 (σρηg 0 + (µ − r)g)2
−ġ + (ν − η )g + η g −
=0
2
2
2σ 2 g
Following Henderson (2002 MF) and Henderson and Hobson (2002) to reduce to
LINEAR pde
2
Set g(τ, z) = eατ G(τ, z + (β − η 2 /2)τ )b, and take b = (1 − ρ2 )−1 , α = − (µ−r)
, and
2σ 2
β = r + η( ν−r
−
η
ρ(ν−r)
)
σ
we get the heat equation
Ġ =
z
subject to G(0, z) = e−γλ(e −K)
+
1 2 00
η G
2
/b .
14
2
2
2
+
2 −1
g(T − t, logSt ) = e−(µ−r) (T −t)/2σ [E0 e−λγ(1−ρ )(ST −K) ](1−ρ )
(1)
Utility Indifference Value
Value of options to executive is amount of cash he would forego to receive them, calculate by solving for p in V (0, x − p, S; λ) = V (0, x, S; 0)
...Approach of Hodges and Neuberger (1989).
Value of λ calls is given by
2 )(S −K)+
e−r(T −t)
e
0
−λγ(1−ρ
T
log E [e
].
p =−
2
γ(1 − ρ )
Can expand value pe as:
λ
pe = e−r(T −t)λ E0(ST − K)+ − γ(1 − ρ2)Var[(ST − K)+] + ...
2
µ
¶
15
When ρ = 1, drift δ under P0 becomes the riskless rate, so P0 is risk
neutral measure, and pe is the Black Scholes price.
Note the expectation here is no longer with respect to the original
probability measure. Under this new measure P0, the rate of return on
the market is r, but return on innovations orthogonal to the market
are unchanged:
dM
= rdt + σdB 0
M
0
where B 0 = B + µ−r
σ t, a P Brownian motion and
ν − r ρ(µ − r)
dS
= [r + η(
−
)]dt + ηρdB 0 + η 1 − ρ2dW
S
η
σ
q
ρ(µ−r)
(µ−r)ηρ
−
)
=
ν
−
.
where we define the drift δ = r + η( ν−r
η
σ
σ
(2)
16
Observe also pe solves a pde:
1 2 2 e
η S (p ss − (pe s )2 (γ(1 − ρ2 )er(T −t) )) + δspe s = rpe ;
2
Again, |ρ| = 1 gives the Black Scholes pde.
p˙e +
pe (T, s) = λ(s − K)+
Take ρ̃2 > ρ̂2 , fix δ, and consider two option values f˜ and fˆ calculated from the
correlations. Then L̃f˜ = 0, f˜(T, s) = λ(s − K)+ where L̃g = −rg + ġ + 12 η 2 s2 (g 00 −
γ(1 − ρ̃2 )er(T −t) (g 0)2 ) + δSg 0. Similarly L̂fˆ = 0, where L̂ defined with ρ̂.
Now for any g, (L̃−L̂)g = 12 η 2 s2 γer(T −t) (ρ̃2 −ρ̂2 )g 02 . Defining h = f˜−fˆ then h(T, s) = 0
and
1
L̂h = −( η 2 S 2 γer(T −t) (ρ̃2 − ρ̂2 ))(f˜0 )2 .
2
Feynman-Kac gives
·Z T µ
¶
¸
1 2 2 r(T −t) 2
h(t, s) = Ê
(ρ̃ − ρ̂2 )(f˜0 )2 du|St = s ≥ 0.
η s γe
2
t
Value increasing in ρ and as ρ tends to 1, pe tends to the Black Scholes value
17
Value decreasing in γ → executive places less value on options if more risk averse
Parameters λ and γ only appear together in the per-unit value, and we set q = λγ.
Choose value for q for the numerics based on equating local absolute risk aversion with power utility: γ = R
, where R is the coefficient of relative risk aversion in
x
the power utility and x is private wealth. R is well studied, previous authors use 3-4.
Eg: Say S0 = 100 cents.
Scenario 1: Receive $25 000 worth of shares so λ = 25000. If this represents
only small fraction of wealth (1/40) then x = $1000000, γ = 4/108 and λγ =
4x25000/108 = 1/1000.
Scenario 2: Shares represent 1/4 of wealth, so x = $100000, γ = 4/107 and
λγ = 4x25000/107 = 1/100.
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T =3
T =5
T = 10
ρ
q1
q2
q3
q4
q1
q2
q3
q4
q1
q2
q3
q4
0.25 0.92 0.87 0.81 0.55 0.85 0.79 0.71 0.43 0.68 0.6
0.5
0.26
0.5
0.93 0.90 0.84 0.60 0.88 0.82 0.75 0.48 0.72 0.64 0.55 0.30
0.75 0.96 0.93 0.90 0.71 0.92 0.88 0.83 0.60 0.80 0.74 0.65 0.39
0.90 0.98 0.97 0.95 0.85 0.96 0.94 0.91 0.76 0.89 0.85 0.78 0.56
0.95 0.99 0.98 0.97 0.91 0.98 0.97 0.95 0.85 0.93 0.90 0.86 0.68
1
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
Table 1 - Ratio of executive value to company cost for ATM option compensation.
We vary expiry (in years), risk aversion, correlation between stock and market (and
thus varying β). Parameters: q1 = 4/4000, q2 = 4/2500, q3 = 4/1500, q4 = 4/400.
We take volatilities and risk premium from the CRSP stock price data (1998). Take
σ = 0.22 the volatility of the NYSE index, η = 0.45 the average volatility of NYSE
firms, and risk premium µ − r = 0.075.
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T =3
T =5
T = 10
ρ
q1
q2
q3
q4
q1
q2
q3
q4
q1
q2
q3
q4
0.25 0.96 0.94 0.9
0.76 0.92 0.88 0.83 0.63 0.78 0.72 0.64 0.42
0.5
0.97 0.95 0.92 0.79 0.93 0.90 0.85 0.67 0.81 0.75 0.67 0.46
0.75 0.98 0.97 0.95 0.85 0.96 0.94 0.90 0.75 0.86 0.81 0.75 0.55
0.90 0.99 0.99 0.98 0.92 0.98 0.97 0.95 0.86 0.92 0.89 0.85 0.68
0.95 0.99 0.99 0.99 0.96 0.99 0.98 0.97 0.91 0.95 0.94 0.90 0.77
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
Table 2: Ratio of executive value to company cost for stock compensation. Parameters as for Table 1, except K = 0.
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Effect of Risks on Value
By separating the firm-specific and market risks in model, can examine effect of each on value
(1) Effect of options on risk taking
Vary stock volatility η and fix ρ.
↑ η will increase firm-specific risk, causing a fall in value. But also
value increases via convexity of the call option
Therefore we can expect a mixed effect on value, depending on which
dominates
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Implication : Executives given options generally do NOT have an
interest in increasing stock volatility
(2) Effect of correlation
Vary ρ and keep total volatility η fixed → changes proportion of firmspecific and market risks
As ρ ↑ firm-specific risk falls, and value increases. By varying firm
risk as a proportion of total risk, we obtain a negative relationship
between firm risk and value.
Market risk increases with correlation and thus so does value. Varying
market risk as a proportion of total risk gives a positive relationship
between market risk and value.
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(3) Effect of firm-specific risk
Vary firm-specific risk, keeping market-risk constant
Again, two forces are interacting: increasing firm-specific risk is reducing value, whilst the increase in η will increase value via convexity
This convexity effect should have less of an impact than earlier however as market risk is constant.
The convexity only has an effect on the value for low values of firmspecific risk, for the out-of-the-money option.
The at and in-the-money option values are decreasing in firm-specific
risk over the whole range.
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(4) Effect of market risk
Fix firm-specific risk and vary market risk ηρ
Value increasing over a realistic range of market risk
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Incentives
The idea is that by giving stock based compensation to executives,
they will act like an owner of the company.
Incentives are defined here to be the options’ effect on motivation of a
manager to increase the company’s stock price and are approximated
by the derivative of the utility value with respect to the stock price
rather than BS delta, which overstates incentive effects.
e−r(T −t) log E0 [e−λγ(1−ρ2 )(ST −K)+ ] and set
Recall pe = − γ(1−ρ
2)
2 )(S −K)+
0
−λγ(1−ρ
T
.
W (t, s) = E e
Differentiating gives
e−r(T −t) Ws(t, s)
∂ e
p (t, s) = −
.
2
∂S
γ(1 − ρ ) W (t, s)
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As Ws < 0, the derivative of value with respect to the stock price is
always greater than zero.
Cost vs Incentive Benefits of Options and Stock
How can the firm balance cost and incentive benefits? What type of
option or stock compensation is most efficient for the company ?
Incentives from stock greater than options, but more expensive to
compensate with stock. Same true for ITM vs OTM options. →
Vary number of options/stock granted to give constant ”cost” to
company (BS value plus cash pay)
Efficient contracting theory → when stock compensation given, executive’s base salary reduced (assume by the value to executive of stock
compensation).
26
Eg: Executive starts with $100 000 cash salary, and is offered options
with BS value $10 000 but executive values at $6 000. Company gives
executive $94 000 cash and the options, costing the company $ 104
000. Company now keeps the $ 100 000 to the executive and the $
104 000 cost constant, equivalent to keeping the difference $ 4 000
constant. Now company can:
e
p
∂pe
such that λ[BS K − K ] = c
maxK
∂S
λ
where peK /λ is the value per option to the executive.
Here we choose c to be the cost when λ = 1 and K = 100. Thus can
grant more than 1 OTM option, but less than 1 ITM or stocks.
27
For most values of stock volatility, leads to an optimal strike of K = 0
→ incentives are maximised when compensation is in the form of
STOCK, not options. For low volatilities optimal to grant OTM options.
In agreement with Jenter (2001), Hall and Murphy (2002), we find
options are an inefficient means of creating incentives. These papers
do not get the low volatility effect, as it is a function of the varying
importance of convexity vs firm-specific risk across different strikes.
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Abstracting from the current accounting biases, our results imply
companies should grant stock rather than options.... In light of the
new IASB standard, companies will need to analyse the form of compensation they award, as there is no longer a bias to ATM options.
Conclusions and Further Research
• Difference between company cost and the value to the executive is significant →
deadweight cost
• Model allows us to examine effects of market and firm-specific risks separately on
value (2 asset model)
• Options are often given to encourage executives to take on more risky projects
- however our model shows this is not always achieved due to their risk aversion
towards unhedgeable firm risk
• Economic justification given for compensating with stock rather than options.
Important as option expensing in ED2 and possible expensing in the US eliminates
biases.
• American style exercise ?
29