THRESHOLD THEORY – MODELLING RISK ATTITUDE
by
Tomasz Kasprowicz
tkasprow@matbud.com
Affiliation: Gemini (non academic)
Bobrzyńskiego 23/5, 30-348 Kraków
+48 602 25 25 00
Andrzej Bednorz
abednorz@gmail.com
(no affiliation)
Powstańców 6a, 44-362 Bluszczów
+48 697 52 22 97
68
CHAPTER 1
DEVELOPMENT OF THRESHOLD THEORY
Introduction
Behavior under uncertainty
One of the basic points of interest to economists is the behavior of people
under uncertainty. Analysis of the problem dates back to Daniel Bernoulli (1738),
but was developed in the state it is most widely known today by Von Neumann
and Morgenstern (1947). The basic concept is that each individual has a utility
function that translates uncertain outcomes into expected utility. In the classical
approach, the premise is that an agent behaves rationally to maximize his
expected utility. Assumptions about the form of the utility function were based on
four main observations: (1) people always prefer more wealth to less wealth,
which suggests the first derivative of the utility function should be positive. (2)
The St. Petersburg paradox (Bernoulli, 1738) suggests a negative second
derivative of the function, which is equivalent to risk aversion and decreasing
marginal utility of wealth. (3) The Super St. Petersburg paradox (Menger, 1934)
suggests that the function should be bounded (4) The Inverse Super St.
Petersburg paradox (Musumeci, working paper). Interestingly enough, despite
these observations, the utility functions used most commonly to explain decisions
under uncertainty have been logarithmic, exponential and quadratic functions,
which do not satisfy the third condition. In particular, Markowitz’s (1959) notions
of diversification, as well as the CAPM model that was derived from it (Sharpe,
69
1964) are based on the assumption that either agents have quadratic utility
functions or the assets’ returns follow a normal distribution.
The standard theory does not account for some anomalies (Kahneman &
Tversky, 1979) and gambling, which are persistent and widespread. These
anomalies were confirmed by laboratory experiments (Kahneman & Tversky,
1979; Grether & Plott, 1979). Some considered these anomalies a result of
violation of the rational agent assumption or experiment setup. Others tried to
force fit the properties of the function to explain given phenomena (Alais &
Hagen, 1979) or include subjective probabilities (Friedman and Savage, 1948).
There were also attempts to explain the anomalies by investigating potential
market incompleteness (Heaton and Lucas, 1996) or assuming habit formation in
consumption patterns (Campbell and Cochrane, 1999).
The Threshold Theory developed in this paper uses another route to
model human behavior without relying on utility functions. As the theory is based
on a real options approach, we assume that the market for traded assets
satisfies a no-arbitrage condition and that people are not satiated. The
development of a theory seems difficult under such lax assumptions. However,
we manage to derive an agent's attitude towards risk based on construction of
the agent's portfolio. We define the agent’s portfolio as the set of assets and
contingent securities that may affect the agent’s attitude toward risk, in any given
decision he is facing. The assets in one’s portfolio may be in the form of shares,
currency, real estate, a job, etc. The payoff of contingent claims depends on the
70
value of some underlying asset. It usually takes a monetary form; although in
some cases may take another form (pleasure, satisfaction etc.).
The contribution of the theory developed in this paper is twofold. First, on
a theoretical level, it provides a framework in which the assumption of risk
aversion is replaced with a more versatile and general modeling scheme. In
particular, Threshold Theory can be useful in all cases involving the decisions of
a single individual who cannot diversify his risk, including cases of entrepreneurs
and financing of small business. In addition, the modeling scheme provided may
be helpful in branches of corporate finance, such as agency theory and signaling
theory.
Real options
It is widely agreed that the real options approach to economic phenomena
gives excellent insights. Probably the oldest and the best known successful
generalized option modeling1 is Black and Scholes’ (1973) representation of
equity as a call option on a company. Further applications of real options
followed shortly after that.2 Unfortunately, applications of the real options
approach seem to be limited relative to its apparent potential. The main reason
for this is the fact that volatility and other estimates for many underlying assets
1
By financial options, we understand financial contracts to buy or sell an underlying asset at the strike
price stipulated in the contract in time period also stipulated in the contract. Real options are usually
considered assets of corporations originating in the properties of projects a company holds: the option to
terminate a project, the option to delay a project, the option to suspend a project etc. Here we are dealing
with a much broader view of real options, including also other than the corporate point of view. We include
all situations that arise in the case assets held by an individual that have option-like payoffs. We include
both monetary and non-monetary payoffs. Therefore, we will use term “generalized real options” instead.
2
These standard real options include: option to abandon, option to expand, option to switch, option to
delay.
71
are not available due to infrequent trading or the absence of a market. The
literature for practitioners suggests using estimates of the volatility of actively
traded assets that are closely related to underlying assets of interest (Amram,
Kulatilaka, 1999). However, in most cases basis risk3 seems to be high and this
solution seems to be too risky.
Threshold Theory avoids this common problem of the real options
application because no valuation is made within the model. Because of this, the
model may be applied to a wide range of assets including these that are not
actively traded. For the same reason, Threshold Theory has very limited
application in the field most penetrated by real options – capital budgeting. On
the other hand, the theory introduces a generalized real options approach to
other fields of finance, in particular corporate governance, signaling and agency
theories, and the behavior of undiversified investors. It seems that it is able to
give a theoretical explanation for many phenomena observed in the business
world, hence providing the discipline with a flexible modeling tool designed to
provide predictions and not merely fit the data.
The Model
Options – review
Before we delve into construction of a model we review certain basic but
crucial facts about options. Threshold Theory most often uses the Cash-or-
3
Basis risk arises from imperfect correlation of the actual underlying asset of an option and asset used for
hedging.
72
Nothing (CoN) binary option, sometimes in combination with the standard call
option.4
Following the seminal work of Black and Scholes (1972), subsequent
researchers have developed formulae for pricing various types of contingent
claims. Others have come up with formulae for cases in which the assumptions
of Black and Scholes (henceforth BS) are not met (for review of various
variations on BS model see Smith, 1996). A much more flexible approach was
presented by Cox, Ross, and Rubinstein (1979). The tremendous theoretical and
practical success of this approach suggests that oversimplification in the BS
model does not severely affect its accuracy.
A Cash-or-Nothing option is a binary option that pays a constant amount A
at maturity only if the price of the underlying asset exceeds the exercise price.
This option’s price is just the discounted expected payoff. The closed formula for
the value of such an option in BS’s world at time t=0 is as follows (Ingersoll,
2000):
(2) CoN  Ae  rT N (d 2) ,
Where A is the amount paid in case the option finishes ‘in- the-money’, N(d2)
is the probability of an option finishing in-the-money (in risk neutral terms). This is
exactly the same interpretation as in the case of a vanilla call.5
An Asset-or-Nothing option is an option that pays the value of an asset if
the price of an asset exceeds the strike price at maturity and nothing otherwise.
4
Combination of Cash-or-Nothing and standard call option is known as Asset-or-Nothing option (AoN)
This type of option has no call and put variety. As this option is a pure jump, we deal only with long and
short positions in it. In different words, buying a put is equivalent to writing a call. One may insist that if A
is positive then an investor has long position in option, while if A is positive then an investor has short
position in option. In such a case one can distinguish between put and call type.
5
73
The closed formula in BS world for the value of such call option at time t=0 is as
follows (Ingersoll, 2000):
(3) AoN  SN (d1) ,6
The relationship between various parameters and the value of an option
can be found by taking derivatives of the value function with respect to the
parameter of interest. It is known that for vanilla options, the value of an option
rises as volatility increases.7 This is not necessarily true for a CoN option though,
as the relationship changes with S. The reason for such dependence is that if
S>X then there is no reward for finishing more “in the money”, however there is a
loss of A if the value of the underlying asset falls below X. Therefore, in such a
case additional volatility brings no value from up-side potential and increases the
risk of losses.
This observation is crucial for Threshold Theory. Figure 1 and 2 illustrate
the impact of the change of volatility on the value of a vanilla call and CoN with
respect to S for parameters: X=11, T=1, r=2%, A=10.
6
Here the reader can see that the regular call option is just a composition of a long call AoN option and
short CoN, both with the same strike X. The amount paid off by the CoN option is X. Then the value of
AoN is AoN  SN (d1) , the value of the CoN is CoN  Ae rT N (d 2) but as A=X then
 rT
CoN  Xe rT N (d 2) and hence the value of the portfolio is C  SN (d1)  Xe N (d 2) which is
exactly the formula for the value of the standard call.
7
This result is not theoretically valid as BS assumes constant volatility. Instead of derivatives with respekt
to volatility we should talk about comparing options of parameters identical except for volatility. However
the conclusions are identical hence we proceed with less correct but easier to comprehend ‘derivatives’/
74
Impact of change in volatility on cash or nothing option
2.5
2
C
1.5
Volatility = 5%
1
Volatility = 10%
0.5
0
9
10
11
12
13
-0.5
S
Figure 1 The impact of volatility on the value of a standard call option
Impact of change in volatility on cash or nothing option
12
10
8
6
C
volatility = 5%
4
voaltility = 10%
2
0
9
9.5
10
10.5
11
11.5
12
12.5
13
-2
S
Figure 2 The impact of change in volatility on the value of cash or nothing option
75
Ideas underlying the model
Normally, the parameters used in option pricing are exogenously
determined. Consider the alternative situation in which we own an option but we
have influence on some parameters that determine the value of the option. The
influence is not absolute but is limited and can be exercised only at some points
in time. In fact, these decisions are the points of interest for Threshold Theory.
Which parameters could be interesting from the point of view of model
building? An agent may decide to choose different maturities, switch maturities of
the options or pick a time of the exercise in case of an American option; however
these cases are not taken into consideration in this paper. The risk free rate is a
macroeconomic variable determined by the market as a whole and we do not
take into consideration this variable either. Exercise price seems to be a good
candidate, but that would change the nature of the option contract, creating
serious problems in any further development of the model. The two remaining
parameters are those that we focus on: the value of the underlying asset and its
volatility. If we use our power to change the value of an asset we call it a pure
wealth transfer, while a change of volatility we call pure risk change.
Do examples exist in which such power is feasible? The answer to this
question is “yes” and we present two examples in a subsequent section.
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Some examples
Introduction. Threshold Theory applies option theory to situations that are
not normally associated with this type of modeling. Consequently an illustrative
example is very helpful in understanding the theory. This is why we start with a
simple example and transform it, step by step, into more interesting cases.
The case of financial options. We will start with the case of financial
options. This familiar ground will allow us to proceed towards examples that are
more exotic later.
Consider an agent holding in his portfolio two securities: a cash or nothing
option written on IBM stock (paying $10 if IBM’s stock price at maturity exceeds
$11) and a share of stock in IBM. Note that the underlying asset of the option
held is also a part of the portfolio. Figure 3 presents the payoff at maturity to the
agent with respect to the value of the underlying asset at maturity.
C+S
77
S
Figure 3 Payoff at maturity from the portfolio consisting of the cash-ornothing option and its underlying asset (C+S) versus value of option’s underlying
asset (S).
The following figure presents the value of the portfolio before
maturity (assumed parameters: volatility 5%, risk free rate 2%, time to maturity 1
year).
78
C+S
C
C+
S
S
Figure 4 The value of a portfolio consisting of a cash-or-nothing option and
its underlying asset (C+S) versus value of the underlying asset (S)
Now let us investigate what would be the reaction of the agent to
changes in volatility and in the value of the underlying asset. This can be done by
finding the derivatives of the value of the portfolio in relation to these two
variables. The value of the portfolio is:
(4) W= S + CoN = S + Ae rT N (d 2)
d2 
ln( S / X )  (r 
 T
2
2
)T
Let us start by differentiating with respect to S
79
(5)
d
A
W 1
e rT e
dS
S T 2

2

ln( X )( r  )T
ln(S )
2


 T
 T


2


 /2



The derivative is always positive. This means that the agent will always
benefit from an increase in the price of an underlying asset.
Now let us consider the influence of changes in volatility. Let us assume
that a change in volatility has no impact on the value of an underlying asset.8
Then we can consider the derivative in terms of volatility:
2
 ln(S / X ) rT  T 
 /2

2 
 T
d
A rT ln( S / X )  rT
T 
(6)
W 
e (

)e
d
2
2
2 T
The middle part of the equation 
ln( S / X )  rT 1
T

determines
2

2
T
the sign of the derivative.
We are interested in finding the values of the underlying asset for which
the derivative is positive or negative; solving
W
 0 will tell us where the sign

changes.
ln( S / X )  rT 1
T
 2T
0

  ln( S )  ln( X )  rT 
2
2
2
T
(7) S  Xe rT e

ln( S )  ln( X )  rT 
 T
2
2
Therefore whenever the value of an underlying asset is below
Xe
 rT
e

 2T
2
the agent benefits from an increase in volatility, and is harmed
otherwise.
8
This assumption may seem to be restrictive. In fact, in this particular example this assumption is met.
 2T
2
80
Generalized real options cases. The marginal analysis performed above is
interesting but seems to have limited importance. So far, we have been analyzing
an example (presented at the beginning of this chapter) in which the agent is the
recipient of market conditions. Therefore, the analysis reflects only the impact of
some exogenous changes on the value of his portfolio. However, in the case of
an agent influencing either the volatility or the value of the underlying asset, then
such an analysis would be able to predict how the influence would be used. In
other words, we could predict whether the influence would be used to decrease
or increase value/volatility of the underlying asset.
In the example given, such an influence would be clearly used to increase
the value of underlying asset in all cases (positive derivative with respect to S);
increase volatility in all cases when the value of the underlying asset is below
Xe
 rT
e

 2T
2
and decrease volatility in cases where the value of the underlying
asset is above Xe
 rT
e

 2T
2
. Cases in which the investor has an influence on IBM’s
stock returns (held by the agent in the example here presented) are exceedingly
rare so the predictions seem to be irrelevant. Therefore, analysis is meaningful if:
1) the payoff is of the same nature as in the case presented; 2) the agent exerts
some influence over the underlying asset. Now we will attempt to construct such
an example, which will serve two purposes: 1) an illustration of how the model
may work; 2) setting the foundation for a comparison of Threshold Theory and
the Expected Utility Framework.
81
Consider an agent who plans to retire at time T, say in 40 years. However,
he may invest towards retirement only today and currently has a fixed amount S 0,
say $100,000. No more money can be invested later on and no money can be
withdrawn for 40 years. However, the agent has full control over the composition
of his investment at each point in time, meaning he can trade securities within the
portfolio at any moment. The agent wishes to retire in London, but as prices are
much higher there he needs to have at least $X, say $500,000, at retirement in
order to be able to afford to do so. The ability to retire in his preferred
surroundings gives him pleasure worth $A.
Now let us analyze the portfolio held by the agent. He holds:

At t=0 an investment worth S0=$100,000

An Asset-or-Nothing option that pays $A, if the value of the underlying asset
exceeds X=$500,000 at maturity in T=40 years. The underlying asset is the
investment held by the agent.
In such a case the agent faces a payoff exactly as in the case of the
financial options presented earlier. Also, the agent has some influence on the
underlying asset, which is just his portfolio. The agent can change the
composition of the portfolio and hence influence its volatility. This violates the BS
assumptions of constant volatility. The option described is clearly more valuable
than a regular option due to extra flexibility. Hence, the ‘exact’ values developed
earlier are not exact and must be treated only as approximations.9
9
More accurate values may be developed via Monte Carlo simulations and this approach should be used if
exact values are needed.
82
Now we may apply marginal analysis to predict the agent’s behavior. A
positive derivative with respect to the value of the underlying asset means that
the agent would always act to increase the value of his investment. This cannot
be done as no additional transfers into the account are allowed. The only
influence the agent has on the underlying asset concerns its volatility. Trading
securities (assuming 0 transaction costs) has no impact on the value of the
underlying asset (investment) but changes its volatility. Hence, the agent will
trade securities in a way that increases volatility of the portfolio whenever the
value of the underlying asset is below the strike of the option (X), discounted by
appropriate amounts.10 We call this point the Threshold Point. The opposite is
true when the value of the underlying asset is above the Threshold Point.
This behavior is in perfect accordance with common sense. We will
illustrate it with a somewhat more extreme example. Imagine a person who
needs to have $200 by the end of the week to buy insulin, otherwise she dies. If
her wealth is lower than $200 (we ignore discounting), then this person will gladly
enter risky investments (even with negative expected value if necessary) if this
gives her a chance to end up with more than $200 by the end of the week.
Hence, she is willing to pay for risk (or accept investments with a negative
expected return). On the other hand if her wealth is higher than $200 then she
will prefer to hold her asset in low risk or risk free investments. Any risky
investment would have to be rewarded with a positive expected return.
10
In general, the point of change of volatility preference is not exactly discounted at risk free rate strike of
the binary option. The adjustment depends on assumptions concerning the stochastic process governing the
behavior of the underlying asset. In the BS case, the adjustment is just the expected value of the lognormal
distribution.
83
Equivalent analysis may be applied to the base example. The agent will prefer
higher volatility (more risky) assets when the value of the investment (the
underlying asset) is below the Threshold Point and assets of lower volatility (less
risky) when the value of the investment is above the Threshold Point.
Another real options example. We will develop another example in the
realm of real options. This provides further illustration of the model and also
prepares the base for empirical application of the model.
Consider an agent who is the CEO of company XYZ. Her salary depends
on the stock price of the company on January 1st each year. If the stock price is
lower than X=$10 then she is fired and suffers loss of reputation. This means that
the present value of her wages in her next job will be lower by $A compared to
current conditions. Otherwise she is paid $1 for each dollar that the stock price S
exceeds $10.
Her portfolio consists of two assets:

A cash or nothing option that pays $A in the event of the value of the
underlying asset (stock of XYZ) exceeding X=$10 11

Call option on stock XYZ with a strike X=$10
Figure 5 presents the agent’s payoff at maturity and the value of the portfolio
before maturity.
11
This in fact is a compound option that pays off another CoN and call with one year to maturity in
addition to regular payoff.
C+CoN
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Payoff
CoN+C
S
Figure 5 Payoff and the value of asset or nothing option
The influence of the agent on the value of the underlying asset is pretty
obvious as exercising this influence is in fact her job. By accepting various
projects she influences both the value and the volatility of the underlying asset.
Now, using again marginal analysis we will try to predict the behavior of the
agent. We assume that A=$10 so these two options blend into an asset or
nothing option.
The derivative of the value of the portfolio function (W) with respect to the
value of the underlying asset is
dW
(8)

dS
1
2
ln(S )

 T
ln( X )( r 
 T
e

2
2
 x2 / 2
)T
1
dx 
e
 2T
ln(S )
(

 T
ln( X )( r 
 T
2
2
)T
)2 / 2
The entire derivative is always positive. Therefore the CEO will accept all
positive NPV projects providing they do not alter the volatility of the company.
85
The derivative of the value function with respect to volatility is:
 ln(S / X )  rT
dW
S
ln( S / X )  rT
T  
(9)

(

)e
d
2
2
2 T

and
 T
T
2

2

 /2


ln( S / X )  rT
T

determines the sign of the derivative.
2
2
 T
ln( S / X )  rT 1
T
 2T
0

 ln( S )  ln( X )  rT 
2
2
2
T
ln( S )  ln( X )  rT 
 2T
2
 T
2
(10) S  Xe rT e
2
Therefore, whenever the stock price is below Xe
 rT
 2T
e
2
then the
CEO will gladly accept NPV=0 projects that increase the volatility of the firm and
also some negative NPV projects, providing volatility increases sufficiently. On
the other hand, if the stock price is above Xe
 rT
 2T
e
2
then the CEO will accept 0
NPV projects if they lead to a decrease in volatility and if a project leads to an
increase of volatility it must be rewarded with sufficient NPV.12
Development of the model
We will assume that:

an agent is endowed with a contingent security which value depends
on the value of the underlying assets (ST)

12
the agent is non-satiated
Note this somewhat disturbing result; it seems that not all positive NPV projects will be undertaken and
also some negative NPV projects will be undertaken.
86

the agent has some influence on the underlying asset S, in particular
the agent can:
o change the value of the underlying asset, providing an
opportunity exists
o change the volatility of the asset, providing an opportunity exists

markets are arbitrage free
The influence is known by the market and this information is included in the
value of the underlying asset. The nature of the influence as well as timing of the
opportunities depend on the situation being modeled.
Let us review the notation used in this paper

T is time to maturity

S is the value of underlying assets

C is the value of the contingent claim

X is the exercise price of the contingent claim

A is the amount of money paid out by the CoN option in the event of
it finishing in the money.
Let us also define:

The Threshold Point is the point at which the preference towards
volatility changes 13

Positive risk attitude – all situations in which an agent tends to
increase the volatility of his portfolio
13
Note that the Threshold Point converges towards the exercise price of the CoN option as expiration of the
option approaches.
87

Negative risk attitude - all situations in which an agent tends to
decrease the volatility of his portfolio

Neutral risk attitude – all situations in which an agent is indifferent to
either increasing or decreasing the volatility of his portfolio
The agent will act to maximize the value of his portfolio at each point in time.
This follows from the assumption of the investor’s non-satiation. Hence, we are
able to predict the behavior of an agent when he faces a decision about the
influence on the underlying asset. All we need to do is to perform marginal
analysis, as described before. A positive derivative with respect to volatility
means that the agent will exert his power over the asset in order to increase
volatility. A positive derivative with respect to S means that the agent will exert
his power over the asset so that the value of the asset increases. Interactions of
these derivatives add a certain structure to the model. If the agent’s portfolio
consists of multiple assets we may find the derivative of each part separately and
add them up. In that way we are able to make predictions in more complicated
cases. This is a very general framework that allows us to analyze many types of
situations. Here we concentrate on those that include a jump in the payoff
function as they allow for the Threshold Point.
It is worth noting that there is no need to calculate the exact value of the
option and hence that there is no need for exact estimates of the parameters to
infer the agent’s decisions. Therefore, the main problem of applications of real
options is alleviated. All we need to do is predict the changes in parameters
88
(volatility, value of underlying asset) given the decision, which is considerably
easier than finding the values of the parameters.
The Threshold Theory and Expected Utility Framework
The Threshold Theory is mostly used to predict an agent’s behavior. This
is also true in the case of the Expected Utility Framework and sometimes these
two theories may be applied in describing similar phenomena. Threshold Theory
was constructed to avoid assumption of risk aversion that is in most cases
bounded to the Expected Utility approach.
The Expected Utility framework is very popular and some researchers
may not find it comfortable to detach from it completely. Fortunately, it is even
possible to use Threshold Theory within the Expected Utility framework but then
analysis must be qualitative, due to interference of utility functions with Threshold
Theory. In some special cases a very detailed quantitative description is possible
even within the Expected Utility Framework.14 These special cases are:

The underlying asset is not a part of the agent’s portfolio and its price is
uncorrelated with the agent’s wealth, with the exception of influencing the
value of the option held by the agent

Underlying assets forms a synthetic option in agent’s portfolio (as in the
case of the rather unfortunate diabetic described earlier in this paper).
14
Detailed quantitative analysis is presented for illustrative purposes only and is not required in most
applications.
89

Values of assets are already adjusted for risk and therefore risk aversion
(or seeking) is already accounted for (e.g. prices are expressed in terms of
utiles)

Agents are assumed to be risk neutral
In the mentioned cases the agent’s decisions are independent of his risk
aversion or risk seeking (as defined by the Expected Utility framework). We
encourage to use the Threshold Theory on its own and now we present its
properties in comparison to standard model.
In the case of Expected Utility, the behavior of a person is determined
based on his or her preference towards symmetric bets with an expected value =
0. If a person prefers riskier bets then we describe him or her as risk seeking,
otherwise we characterize them as risk averters. In the case of indifference we
describe them as risk neutral.
It is worth noting the important difference between Threshold Theory and
the Expected Utility framework: in the latter, we have to give an agent the choice
between two fair bets of different variances in order to decide whether he is a risk
averter or a risk seeker. Then this information would be an underlying
assumption for the model used to predict his further behavior. Threshold Theory
predicts the agent’s attitude towards risk and does not assume it. Agent’s risk
attitude follows from the theory and changes accordingly. Therefore, in some
cases Threshold Theory may be more flexible and accurate than the traditional
framework allowing for functional changes in risk attitude.
90
The decision of an agent in the case of Threshold Theory is easy to
predict regardless of the properties of the bet. One just needs to find the
expected value of the portfolio given the bet and pick the highest one. To simplify
the picture we will present a case with symmetric zero expected value bets only.
In such a case we are able to develop many concepts related to the expected
utility framework.
Now we proceed with a simple analysis of the case of the CoN option.
(Parameters: A=10, X=11, S=10, T-t=1, volatility=5% and r=0.5%.) Below is an
illustration of the first and second derivatives of the value of the option.15
1
0.8
0.6
0.4
First derivative
Second derivative
0.2
0
1 12 23 34 45 56 67 78 89 100111122133144
-0.2
-0.4
Figure 6 First and second derivative of CoN value function with respect to S
The first derivative is positive, as we found in an earlier section. The
second derivative changes sign close to the threshold. Before the threshold it is
15
Note that this result differs somewhat from the findings in the previous section. This is due to simplifying
the assumptions of Arrow’s and Pratt’s measures of risk aversion ARA and RRA. The approach presented
here is conducted for comparison only. For modeling, one should use the results presented earlier.
91
positive (positive risk attitude – preference for risk, negative price of risk – the
equivalence of risk seeking), after the threshold second derivative is negative
(negative risk attitude, positive price of risk, equivalence of risk aversion). The
 rT
change occurs exactly at S0 = X e
e
 2T  1
 2T
2
e
.
It is important to note that within Threshold Theory risk attitude is an effect
of portfolio construction. At the same time the expected utility framework
assumes one’s risk preferences. This fundamental difference should be kept in
mind when comparing frameworks. This type of set-up gives Threshold Theory
extra predicative power that may be useful in modeling certain phenomena.
Some inferences arising from the model
Equity
Black and Scholes (1974) noticed that equity in a leveraged company is
similar to a call option on a company as a whole. This observation is very
insightful, but also somewhat confusing. The value of a call option increases with
volatility, and therefore company owners should exhibit a positive RA regardless
of the company’s value (and become risk neutral only when the company’s value
goes to infinity). This implies that owners should accept some negative NPV
projects providing they are sufficiently volatile. Furthermore, it would be beneficial
for shareholders if companies just made fair bets among themselves, since this
would increase volatility without any loss of value. Managers should be rewarded
by the stock market if they engage in a series of fair bets. In other words,
92
engagement of a company in an infinite series of independent, fair coin flips of
any bet size should significantly drive up the stock price.
This problem, a variation of which is also known as the “asset substitution
problem”, has been recognized before, but only in terms of imminent default.
However, when the option model is treated literally, the asset substitution
problem can be applicable to all companies. This implies that a Fortune 500
company would gamble all of its wealth on a bet with negative expected return
(providing that the variance of returns is big enough); such an implication is very
disturbing and does not find support in real life.
Moreover, if such a model is correct then dominating capital budgeting
schemes are incomplete. Neither NPV nor real options approaches take into
consideration the possibilities mentioned above, which implies that practitioners
are taught techniques that do not maximize investors’ wealth. Furthermore, this
framework shows that the investment policy of a company depends on leverage,
violating one of MM’s assumptions.16
Threshold Theory offers a solution to the problem illustrated in the “Fortune
500” example. Let’s assume that S is the total value of a company, X the face
value of the debt, A the bankruptcy costs, and T the maturity of the debt. In such
a situation, negative risk attitude is exhibited even before the value of the
company reaches the threshold X and the problem vanishes. A positive RA,
before the threshold is reached, is consistent with our standard understanding of
an asset substitution problem.
16
This fact cannot be solved within the new framework and actually opens up interesting possibilities for
further research
93
Bond holders pay most of the bankruptcy costs as they are the ones that
receive the assets of the company net of these costs. Therefore it is not obvious
why such costs would influence equity holders. One answer is related to
diversification of investors. In such a case equity holders will bear the costs of
bankruptcy. As Haugen and Senbet (1978) noted in such a case, in line with
Coase’s (1960) theory, the parties should renegotiate the contract in order to
avoid bankruptcy costs. However, if there are multiple parties involved then
negotiation costs may be higher than bankruptcy costs.
The “underinvestment” (Myers, 1977) problem is also easy to comprehend
within this framework. The problem is: in the case of high leverage not all positive
NPV projects are accepted as most benefits accrue to debt holders. Threshold
Theory stipulates even stronger implications. If the value of the company is much
smaller than the Threshold, then many positive NPV projects will not be accepted
because they are not risky enough. The reason is that in such cases there is
negative price of risk before threshold is crossed because, before the threshold is
reached, there is a negative price on risk for the owners. If no threshold is
included in the model, underinvestment would exist for any value of the company
which again is inconsistent with empirics.
Agency theory and signaling
Agency theory is one of the basic ideas in modern finance and the
assumption of self-interest is appealing to a rational decision making paradigm.
Threshold theory presents a modeling tool for this type of study. For the sake of
94
illustration, let’s assume that the CEO of a company would be fired at the Annual
Meeting of Shareholders if the stock price is less than a pre-specified value
(critical value). Furthermore, if the CEO is fired, the present value of income for
the CEO drops (due to damaged reputation). In such a case, we are dealing with
a cash–or-nothing option on stock S with the threshold at critical value X of the
stock price, and the value of the decrease in the present value of salary equal to
A. If we further assume that there exists some pay-performance sensitivity, we
can construct such a portfolio wherein the contingent security becomes an assetor-nothing option. This is very convenient, as closed valuation formulas for both
types of options exist and hence risk attitude in various situations may be
assessed.
In this framework, many other phenomena are easy to explain. For
example, the hypothesis of managerialism states that managers derive some
kind of utility from managing bigger assets. From a Threshold Theory
perspective, however, this behavior is more clear, as accumulation of assets
(especially of cash or risk free assets) while the CEO has negative RA moves the
stock price further from the threshold and decreases company-specific volatility if
additional assets are uncorrelated with existing business (cash, most securities,
diversifying mergers). This maximizes the CEO’s option value assuming that the
threshold does not change. Similarly, value-destroying diversifying mergers will
be performed in the event of a drop in S being more than offset by the increase in
the manager’s option value due to a decrease of volatility.17
17
In case we drop the assumption of constant threshold we have to take into consideration how the
threshold changes in relation to company’s assets size.
95
Other problems, such as CEO entrenchment and pay performance
sensitivity, may be explained in natural and simple ways. More complicated
phenomena, including dividend announcements and signaling effects, require
more study and modeling. Several existing papers have used a logic similar to
the one used in the development of Threshold Theory, such as Degeorge, Patel,
Zeckhauser (1999). In this study, the authors found that managers manage
earnings if earnings are not up to the expectations of the market (thresholds were
set at: 0, recent years’ earnings and analysts’ consensus). In this case managers
were trying to influence the value of the underlying asset (earnings) of their
option to remain employed at the cost of increased volatility in the future. The
slight deviation from the approach presented in this paper is that the underlying
asset used for modeling is earnings instead of stock price. However the authors
point to the close relationship between these variables.
96
CHAPTER 2
APPLICATION OF THRESHOLD THEORY TO CORPORATE GOVERNANCE
Introduction
Threshold Theory was designed to explain some anomalies unaccounted
for by the Expected Utility Framework, but can it be used for empirical research?
We believe that this is definitely the case. Here we would like to present one
possible application of the Threshold Theory.18
Description of the model and development of a testable hypothesis
As noted earlier in this paper, a part of the human wealth of a manager
may be modeled by a composition of a cash-or-nothing option and a vanilla call,
each with stock of the managed company as an underlying asset. The strike of
both options is set at the level of the stock price at which the majority of
shareholders would become sufficiently unhappy and fire the manager. The call
option reflects the performance-based part of salary, while the cash-or-nothing
option reflects the reputational loss in case the manager is fired (which translates
in lower future personal earnings).19 According to Threshold Theory, the manager
18
19
I would like to thank Wallace N. Davidson III for noticing this possible application.
For a detailed description of this case please refer to section II.3.4 of Chapter 1
97
should exhibit a positive risk attitude if the stock price is below the threshold
20
and a negative risk attitude if the stock price is higher than the threshold.
However, if the manager decides to retire in the near future, then his
portfolio changes as there is no loss due to damage to his reputation because
reputation is no longer needed. Therefore, the cash or nothing option is no longer
in the manager’s portfolio and all that is left is a vanilla call option. In such a case
the manager should exhibit positive risk attitude at all levels of the stock price.
Hence, managers close to retirement are more likely to undertake actions that
increase risks to the company. This inference is supported by Davidson, Xie, Xu,
Ning (2005) who find that CEOs nearing voluntary retirement tend to engage in
earnings management. This is a long-run volatility-increasing behavior, which
also intends to influence value of the underlying asset. This sort of behavior is in
line with the theory. In this study, we aim to check whether managers exhibit
other types of risky behavior as retirement nears.
The overall riskiness of the company cannot be measured with beta alone
as it omits the idiosyncratic risk. Therefore, as a proxy of total risk we will use the
volatility of the stock price. According to the Threshold Theory, after the decision
concerning retirement is made the manager should tend to increase the volatility
of the company as a whole. Of course, the moment of such a decision is
unobservable but managers usually are not able to adjust the volatility
immediately; therefore, the process of volatility adjustment, as well as coming to
the final decision, should take at least some time. To measure changes in
20
The threshold point is discounted strike further modified depending on type on diffusion process
governing the behavior of the underlying asset. In the case of the Weiner process, the threshold is further
adjusted by the expected value of lognormal distribution (see section II.3)
98
volatility we compare volatility in years preceding the year of voluntary turnover (t
-1 to t-3). Therefore, the testable hypothesis is:
H0: If a company’s CEO is close to retirement then relative to company’s
peers the volatility of the stock price should be increasing over time.
Methodology and data
This study focus on a sample of companies in which the CEO voluntarily
stepped down. The Executive Compensation Database was used to identify CEO
turnovers between 1995 and 2007. The following screening criteria were used:
Executive Compensation Database indicates that the reason for the turnover was
retirement, CEO was 60 or older, CEO had at least 4 years tenure when leaving
the office.21 This screening yielded 98 observations. In order to further assure
that the turnover was planned, or it was the last CEO position held by an
individual, we screen whether the individual held another position after retiring at
any company. Moreover we screened press releases obtained from Lexis –
Nexis concerning the turnover of CEOs aged 60-65 in order to eliminate forced
turnovers disguised as retirements. In this way we seek to assure that there is
little or no jump in the CEO’s payoff function. These additional screens reduce
the sample size to 96 observations.
The study investigates whether the total firm risk, proxied by volatility,
changes over time. The time series nature of the test precludes effective use of
21
This also assures that there was no CEO turnover during the test period.
99
regression analysis in such study. There is no prediction concerning
development of the process over time but only concerning its general direction (in
particular increasing volatility). Therefore, it seems sufficient to compare volatility
at given points in time using a paired t-test.22 We used yearly periods and the set
under investigation consists of years t to t-3 where t is year of retirement. To
have a basis for comparison, the changes in volatility will be compared against
volatility of matched companies.
This comparison calls for a methodology of picking the match as well as
control variables. In this study the inclusion of control variables is somewhat
complicated. There are several types of variables influencing volatility that should
be considered.
1. Variables that represent tools that the CEO may use to change the
company’s volatility.
These include but are not limited to: leverage, R&D spending, earnings
management, etc. Such variables should not be included in the test. This
is because if we remove the influence of such variables then we would be
unable to detect any meaningful relationship between CEO characteristics
and volatility. By removing the influence of the tools that CEO may use to
change volatility, we also remove the link between CEO and volatility.
2. Variables that may determine the level of volatility that are not easily
controlled by the CEO. These are the variables that should be controlled
22
Paired t-test is represented as t n1

dˆ   0
sd / n
, where d is difference in values of variables between pairs,
µ0 is difference tested under null (in this paper always 0), sd is standard deviation of observed differences
and n is number of pairs.
100
in the test as they may vary between companies and obscure the results.
These variables are:

Industry
The industry life-cycle is known to be related to stock price volatility
(Mazzucato & Tancioni, working paper; Mazzucato & Semmler,
2002)

CEO age and compensation
In general, the risk appetite is believed to be related to the agent’s
age (MacCrimmon, Wehrung, 1990); at the same time, the current
level of compensation may determine the potential loss in case of
being fired (A. Rashad Abdel-khalik, working paper). Such person
specific variables should be controlled for as they may influence the
behavior of the CEO. In particular retiring CEOs are typically old
and this fact may bias the results. Furthermore, compensation may
be highly correlated with CEO wealth, which in turn may influence
the risk attitude.
3. Variables that may affect the pace of changes in the volatility need to be
included in the tests. As we are investigating the changes of the volatility
over time we have to take into consideration whether CEOs of the
compared companies are able to manipulate the volatility to a similar
extent. The variables in question are firm specific:
101

It is much easier to manipulate characteristics of a smaller
company. Therefore, when companies are compared they should
be of a similar size.

Companies with comparably higher/lower volatility at the beginning
of the comparison period may have a hard time to
increase/decrease its volatility even further. Therefore, compared
companies should have comparable volatility to start with.
The screening variables from the third group are the ones best suited for
specifying the match for each company in the original sample. We also include
industry to be one of the variables used for picking a match as this is a
categorical variable and therefore not well suited for regression analysis.
We picked a match in the following way. Each company in the original
sample will be assigned a peer. The first screen will have two basic conditions:
-
the same industry (4-digit SIC code) – to accommodate potential temporal
changes in volatility within industry
-
no CEO turnover in the investigated period
The second screen will include screening variables from the group 3
-
similar size at the beginning of (t – 3) – to ensure that the pace of volatility
change is comparable, assuming that it is harder to change the volatility of
bigger companies. Size will be measured as market value of equity on the
month of turnover as measured by Compustat.
-
similar volatility at the beginning of (t – 3) – to make sure that companies
we are comparing are of similar risk at the start of the period under the
102
investigation and also to eliminate influence of possible mean reversion in
the volatility process. Volatility was measured as the standard deviation of
the daily returns of each company’s market value in a given year.
The similarity of volatility and size will be assessed using normalized
Euclidean distance and Mahalanobis (1936) distance which resulted in identical
matches in all but one case. The peer was selected as the one with the smallest
distance to the data point in the turnover sample out of all companies that were
identified with screen 1.23 Data concerning size were pulled from the Compustat.
Data on the stock performance was obtained from CRSP. Data on executive age
and compensation were obtained from the Executive Compensation Database.
Lack of suitable matches reduced sample size to 65 observations. Table 3
presents the main statistics of the samples.
23
We have also used Euclidean distance on normalized variables, which is equivalent to Mahalanobis
distance with assumed zero covariance between the variables. The matches assigned were the same for all
but one company.
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Table 1 Summary statistics of retirement and matched sample
t-3
t-2
t-1
t
Average retirement sample volatility
2.11%
2.15%
2.06% 2.20%
Average matched sample volatility
2.16%
2.33%
2.14% 2.13%
Median retirement sample CEO age
65
Median matching sample CEO age
59
Average retirement sample market size ($ mln)
4143
Average matched sample market size ($ mln)
2234
Average retirement sample CEO total annual
compensation ($ ths)
3819
4457
4110
4780
2632
2893
3318
4377
Average matched sample CEO total annual
compensation ($ ths)
The data provides two sets of information. Firstly, the matching procedure
provided a close match with respect to average starting volatility; however the
retirement sample companies are almost twice as big as matched companies.
Finally, we observed a much higher median of CEO age in the retirement sample
which also should be expected due to imposed constraints on CEO age, as well
as the fact that retiring individuals are usually elder.
After the match was identified we investigated the influence of the CEO’s
characteristics. To do so we ran a regression of observed volatility on CEO’s age
and compensation.24 The form of the regression is as follows:
Vol   0  1 Age   2 Compensation   3 Re tirement
24
Total compensation is defined as Executive Compensation Database TDC1 variable consisting of
(Salary + Bonus + Other Annual + Restricted Stock Grants + LTIP Payouts + All Other + Value of Option
Grants)
104
Where Age denotes CEO age, Compensation is total compensation
received in a given calendar year and Retirement is a dummy that takes a value
of 1 if the observation comes from the retirement sample. The dummy was
introduced because the event of retirement, which is hypothesized to have
influence on volatility, is highly related to the age of the CEO. The regression
sample consists of all observations from the matched sample (t-3 to t) for which
all necessary data is available and all observations from the retirement sample
for which all necessary data is available except for the year of retirement (t-3 to t1). The results of the regression are presented in Table 4
Table 2 Results of the regression of volatility on age and compensation
Variable
Coefficient
p-value
Age
β1=-0.0001
19%
β2=-8.87E-08
69%
β3=0.0013
27%
Observations =403
p-value of F = 45%
Compensation
Retirement
R2=8.1%
The regression shows that the CEO’s age and compensation are not
significant determinants of firm’s volatility. Therefore further analysis will be
performed on unadjusted variables.
The difference between adjusted volatilities was computed. The evolution
of average distance between volatilities is presented in Figure 7.
105
0.003
0.002
0.001
vol - s.e.
0
t-3
-0.001
t-2
t-1
t0
volatility
vol - s.e.
-0.002
-0.003
-0.004
Figure 7 Evolution of volatility differentia between retirement and matched
sample
We can observe peculiar development of the process. The volatility
differential increases for the last three years before CEO retirement which
supports Threshold theory. No year-to-year changes are statistically significant,
but the difference between the peak at t and the trough at t-2 is significant at the
10% level for a two-sided test (p-value of 6.6%). The result implies that on
average the retirement decision might be made two years prior to the retirement.
The final step is an investigation of changes in systematic risk. Betas were
calculated using S&P500 as a proxy for the market portfolio. The changes in beta
differentials are neither significant on year to year basis nor comparing extreme
values.
106
CHAPTER 3
THRESHOLD THEORY AND SOCCER STRATEGIES
Introduction
Threshold Theory is potentially applicable to behavior well outside of
economics and finance. Here we would like to present an analysis of football
strategies.
Football, also known in the USA as soccer, is a sport played between two
teams. The aim of the game is to place a round ball in the opponent’s goal.
Players, except for the goalkeeper in his own goal area, are not allowed to touch
the ball with their hands. The game takes 90 minutes divided into two 45 minute
periods. The team that scores more goals wins the game. If both teams score the
same number of goals, the game is a draw. In some cases a draw is not
permitted and the game is extended into two extra 15 minute periods. If this extra
time does not bring a resolution the result is determined by penalty shots.
Each team consists of eleven players; one goalkeeper and ten field
players. There are few restrictions on a player’s position, but three main roles
evolved: strikers or forwards are the offensive players, midfielders whose role is
to pass the ball to forwards, and defenders who specialize in preventing their
opponents from scoring. Usually three substitutions per game are allowed.
107
Often matches are a part of larger competitions that determine the best
among a group of teams. There are two main systems used to determine the
best team which are the cup or the league competition. In the cup competition,
the teams are paired and the losing team quits the tournament (draws are not
permitted). The teams that remain in the tournament after the first round are
paired again and the cycle repeats until only one team remains. This is the team
that wins the tournament. In the league competition, each team plays with each
competing team twice (at home and away). Each match won earns the team 3
points while draws earn the team 1 point. The team that gathers the most points
at the end of the season wins the competition. In case there is a draw other
criteria are used, including goal difference and number of goals scored.
It is not as apparent as in previous cases, but here the Threshold Theory
may come in handy in modeling behavior of football strategy setters. Teams face
a non-continuous payoff: scoring just one goal more than the opponent results in
a game won. Each extra goal does not significantly change the general outcome
of the game. The most important thing is who won; by how many goals is of
secondary importance. Therefore, we observe a huge jump in payoff from a
game played at the threshold level of a draw. The Threshold Theory predicts that
teams should use riskier strategies when under the threshold (losing) and safer
strategies when past the threshold (winning), and this is the hypothesis tested in
this section of the paper.
108
Model and tests
In order to evaluate the behavior of soccer strategy makers we
need to collect data; starting composition of teams, time of all of the goals
scored, and the substitutions of players. Such data for the UK Premier League
2006/2007 games was obtained from the ESPN website.
The testing of the hypothesis requires data concerning results and
riskiness of strategies employed. The riskiness of a strategy must be measured
by a proxy. The approach used will be based on indexing the players. Each
player receives a number of points dependent on their role in the field. We
assume that points assigned are: 0 for a goalkeeper, 1 for a defender, 2 for a
midfielder and 3 for a forwarder. We define a strategy employed by a team by the
sum of points of players at a given time. The sum of points indicates how
offensive the strategy is with higher marks indicating a more offensive strategy.
An offensive strategy increases the probability of scoring a goal, but at the
expense of a weakened defense and a larger vulnerability to counterstrikes.
Hence such a strategy should increase the volatility of the game’s score and may
be seen as riskier than defensive strategies. In particular we expect that when
the teams employ riskier strategies we would observe more goals. More goals
per fixed time of the game translate into higher volatility of the result. In order to
verify this conjecture we analyze the relationship between sum of strategies
employed by the teams and number of goals scored per minute of game. Figure
8 presents main findings of the analysis.
109
0.04
y = 0,0012x - 0,016
R² = 0,5444
0.035
0.03
0.025
0.02
0.015
0.01
0.005
0
32
33
34
35
36
37
38
39
40
41
Figure 8 Relationship between sum of strategies employed by both teams
and average number of goals scored per minute
The observed relationship is positive and the sum of strategies
employed is able to explain over 50% of variation in average goals scored per
minute and p-value of the t-statistics of strategy variable is less than 2%. The
correlation between the variables is in excess of 70%. The results reassure that
the use of the riskiness of strategy construct is indeed strongly related to the
volatility of underlying process.
The first step we undertake is to define the measurement of a match result at
a given point of time. The most intuitive one is Differential = goals scored by the
team investigated minus goals scored by the opposing team. It allows us to
group results in more meaningful categories than regular match results for the
following reasons:
110

The Differential can be measured on an interval scale in contrast to
regular match results that are ordered pairs.25 Differential transforms a
two-dimensional variable into a scalar, which is easier to manipulate.

0 on the scale used will always indicate a single threshold of interest
instead of multiple indications of game results (e.g. 0-0, 1-1, 2-2 etc)
Before we proceed to testing the hypothesis we should first relate the
situation to the Threshold Theory in a more rigorous way so the test is properly
specified. In particular we have to define the nature of the underlying asset and
the type of option. For the purpose of modeling we will assume that the teams
are only interested in winning a match. The value of Differential at the end of the
game (option expiration) is of secondary importance as long as the sign is
preserved. In most league games this should be the case because Differential
over number of matches is important only in rare cases when there is a tie on the
final league table of results.26 Under such assumptions we are dealing with two
cash-or-nothing options. The first one pays off 1 point for the total score table if
Differential is equal or greater than 0. The second option pays off 2 points if
Differential is greater than 0. The payoff of the portfolio and the value of the
portfolio before maturity are illustrated in Figure 9.27
In fact Differential is a set of equivalence classes with respect to relation “~”, defined as (x,y)~(a,b) iff xy=a-b, on the set of results, which preserves ordering.
26
In case of cup tournaments, such assumption is fully met as only winning teams participate in further
stages of tournament.
27
For simulation purposes we assumed BS based valuation of binary options with volatility 5%, risk free
rate 2%, t=1, and Differential=0 at S=10
25
111
3.5
3
2.5
2
Payoff
1.5
Portfol
io
1
0.5
0
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
Figure 9 Payoff and value of the portfolio for soccer example
Clearly, we deal here with two thresholds but they are stacked one on the
other hence their influence should magnify. The point of a draw is the only
situation where thresholds may affect behavior in a different manner and hence
the predictions at this point are unclear. Differential=0 will be treated separately
in the tests.
For each Differential we will find the average strategy used by teams. The
average strategy is defined as time weighted average of all strategies employed
for each observation of the Differential. Assuming that strategies which are more
offensive result in a higher volatility of the Differential, the hypothesis will be
confirmed if the average value of strategy for positive values of Differential is
significantly lower than for negative values of Differential. Such comparison will
112
be performed using a standard t-test. Figure 10 present the relationship between
Differential and average strategy.
Average strategy
19
35000
18.5
30000
18
25000
17.5
20000
17
15000
16.5
10000
16
5000
15.5
Duration…
0
-8
15
-6
-4
-2
0
2
4
6
8
Figure 10 Relationship between Differential and average strategy
employed
We can observe that average strategy employed is higher if a team is
under the threshold, and fluctuates at low levels if a team is at or above the
threshold. This would indicate that a draw is perceived to be as important to
maintain as a wining position. Secondly, we observe strange behavior in case the
Differential is below -4 and above 4. It is important to indicate that these cases
are very rare (in fact they have the duration of about 0.1% of total duration of
games in the sample) and therefore are easily influenced by outliers. Moreover,
as the number of goals scored per match is in general very low (in sample at
hand we observe an average of less than 2.5 goals per game and a median of 2
113
goals per game) then it seems clear that losing by 5 goals takes away any hope
of winning. The flight to safety is possibly a defense against utter shame. On the
other hand, winning by 5 goals is equal to almost sure victory and provides a
possibility of trying more exotic tactics on actual opponent.
The statistical testing of difference of means presents some challenge as it
is hard to decide on the number of observations. On one hand, each minute (or
even second) of the game is a separate observation, however, such observations
are hardly independent. On the other hand, there are 380 separate games
observed so possibly the sample size should be set at this level. However,
averaging data over the game would remove most of the observed variability.
Therefore, we define one observation for the purpose of this statistical test as
each time period in which given Differential was observed. This means that we
effectively divide each game into uninterrupted strips of time during which one
and only one value of Differential was observed. It could be argued that such
partition does not entirely eliminate the problem of independence as two (or
more) stripes of given Differential may occur during one game. However, one
should note that even if such event occurs then those two stripes of time are
divided by at least one other stripe of time (and two goals) so both observations
are spaced in time and hence, to some extent, may be considered independent.
Two goal separation also gives some reassurance. Starting Differential is
always 0 (which is not taken into consideration in the tests) and with a median of
2 goals scored we may safely conclude that at least half of the sample is free of
114
the problem. Proceeding in this manner we have identified 2510 non-0-length
strips,28 of which 726 are under the threshold and 726 are above the threshold.
Table 5 presents p-values of two-sided t-test that compare average
strategy employed by teams below, at, and over the threshold.
Table 3 p-values of t-tests of mean strategies employed by the teams
under the threshold
at the threshold
0.03%
over the threshold
0.03%
at the threshold
74%
The tests confirm the implication of the visual inspection. In particular, the
average strategy employed above the threshold is significantly higher than the
average strategy employed below the threshold. The average strategy employed
when at the threshold is undistinguishable from the average strategy employed
when above threshold, but is significantly different from the average strategy
employed when below the threshold. This supports the hypothesis derived from
the theory.
It may be claimed that more offensive strategies are simply less effective
than defensive ones. Therefore, teams that employ such strategies tend to lose
and that is why we observe more offensive strategies under the threshold. In
other words, one may claim that strategy determines Differential and not the
other way around. In such a case we could observe these results without any
deliberate actions on the side of the teams. There are at least two arguments to
28
Zero length strips arise if a goal was scored in the last minute of the game. Such stripes are removed from
the analysis.
115
refute such approach. Firstly, if offensive strategies were inferior then nobody
would use them in the first place. Secondly, if this is a case then the graph would
have to be symmetric around a Differential of 0. Clearly it is not, which implies
that either losing teams are switching to more risky strategies or winners are
switching to safer strategies (or both do the switch simultaneously).
In order to further address such doubts we will take into consideration
exchanging players with more defensive role with players that will take on more
offensive role and vice versa at various values of Differential. If offensive players
enter the field more often when a team is losing and defensive players enter the
field more often when a team is winning, then the hypothesis is confirmed. The
player entering the field will be considered more offensive if he has more points
assigned to him than the player leaving the field. This approach does not take
into consideration the starting strategies and general preferred set-up of
particular teams hence it is able to further address the reverse causality issue.
Table 6 presents the data concerning player exchanges at various levels
of the threshold, while Table 7 presents net player changes. Change denoted as
-2 is a change of defender for forwarder, -1 is a change of defender for midfielder
or midfielder for forwarder, 0 is change of a player of the same type, 1 is change
of forwarder for midfielder or midfielder for defender, and 2 is a change of
forwarder for defender.
116
Threshold
Table 4 Number of player changes with respect to threshold values
-5
-4
-3
-2
-1
0
1
2
3
4
5
-2
-1
1
1
3
6
17
17
7
2
1
3
3
20
40
94
86
36
21
4
Changes
0
5
36
94
211
358
187
108
41
17
3
1
1
2
11
64
134
109
32
16
10
2
2
2
17
32
8
1
2
Threshold
Table 5 Number of net player changes with respect to threshold values
-5
-4
-3
-2
-1
0
1
2
3
4
5
-2
0
1
0
0
0
9
16
5
2
1
0
Net changes
-1
0
0
0
1
0
0
0
0
0
0
0
0
0
54
0
20
0
11
0
2
0
0
0
1
1
0
8
44
94
15
0
0
0
0
0
2
0
0
1
14
26
0
0
0
0
0
0
Clearly the substitutions of defensive players for offensive players are
much more dominant when the team is winning and the opposite is true when the
team is losing. These conjectures are confirmed by the two sided tests of
proportions with the p-value much lower than 0.1%.29 The behavior of teams at a
29
This is true for both in case of confidence interval consistent version and chi-squared goodness of fit test.
The sample was partitioned into three groups: winning situation, draw, losing situation. The same partition
is kept for later tests.
117
draw is inconclusive in this setting. This refutes the reverse causality argument
and further supports the theory.
So far we have modeled a strategy to be conditioned exclusively on the
Differential. However, it seems reasonable that the strategy of a team depends
also on the strategy employed by the opponent. Inclusion of this factor requires a
somewhat more complicated model. Let us assume that goals scored by a given
team are distributed according to Poisson process with parameter λ. This
parameter depends on how offensive a team’s strategy is (with more offensive
strategies increasing λ) and how defensive the strategy of the opponent is (with
more defensive strategies decreasing λ). Let us define:

f is a function valuing the effectiveness of the offense of a given
strategy (identical for each team) b f(b)≠0

g is a function valuing the effectiveness of the defense of a given
strategy (identical for each team) b g(b)≠0

a is a strategy deployed by team A

b is a strategy deployed by team B
We expect the first derivative of f to be positive and the first derivative of g
to be negative.
We assume that  
f (a)
and that each team A playing with team
g (b)
B will tend to maximize the quotient
 A f (a) g (a)

, which translates into finding
B g (b) f (b)
an optimal point at which the probability of scoring a goal is high but does not
expose their own goal too much. Taking the natural logarithm of both sides, we
118
arrive at the result ln  A  ln B  ln f (a)  ln g (a)  ln f (b)  ln g (b)  F (a)  F (b) ,
with F ( x)  ln f ( x)  ln g ( x) . At the same time team B will try to maximize the
reciprocal of the quotient
B
which translates into maximizing
A
ln B  ln  A  ln f (b)  ln f (a)  ln g (b)  ln g (a)  F (b)  F (a) . We will construct a
function Λ(x, y) = ln  x  ln  y for x, y being possible strategies, basing on λs
obtained from the data.30 The anti-symmetry property derived earlier ensures
that: Λ(x,y)= -Λ(y,x) and Λ(x,x)=0. This indicates that we are dealing with a zerosum game.31
Changes of players indicate a change of strategy and hence we will
examine changes of players again but now from a slightly wider perspective. If a
change of a player results in an increase of the quotient then such a change is a
natural reaction to increase a team’s odds of winning. However, if a change of
strategy results in a decrease of the quotient then the team plays sub-optimally.
In particular, if a new player is more offensive than the replaced one then the
team has a higher probability to score a goal but the probability of losing a goal is
increased proportionally more. This is clearly a shift towards a more risky
strategy. The opposite is true if a more defensive player enters the field. If shifts
towards a more risky strategy are more frequent when a team is losing and shifts
towards a safer strategy are more frequent when a team is winning then the
hypothesis will be confirmed. Tables 8 and 9 present results of this analysis.
In fact, Λ is a matrix that defines game space. The geometry of Λ determines optimal choice of
strategies.
31
If additive relation is used to define λ (λ=f(a)-g(b)) then we arrive at the same conclusions about Λ.
However the geometry of game space may be somewhat altered. For robustness, we have also used additive
approach and, the change of specifications does not materially influence test’s results.
30
119
Table 6 Number of player changes with respect to threshold values, game
theory approach
Threshold
-2
-4
-3
-2
-1
0
1
2
3
4
1
5
7
12
4
1
1
Changes
-1
1
3
12
24
50
52
20
11
3
1
1
3
27
60
41
12
6
6
2
2
1
4
12
3
1
Table 7 Number of net player changes with respect to threshold values,
game theory approach
Threshold
-2
-4
-3
-2
-1
0
1
2
3
4
4
11
4
1
1
Net Changes
-1
1
2
15
36
1
3
7
9
40
14
5
1
Here the picture is even clearer than in the case of the previous test.
Suboptimal, defense enhancing player changes take place much more often
when a team is winning and offense enhancing player changes take place much
more often when a team is losing. This is particularly visible on the net changes
table where the distinction is very clear-cut. Moreover, we can much easier
observe that teams playing at the threshold behave very similar to winning
120
teams. Yet statistical tests reveal at the significance level better than 0.1% that
proportion of changes in each group (winning vs. draw vs. losing) is different.
This suggests, unlike the first test, that at the threshold the risk attitude is less
positive than in the case of winning teams, which is consistent with the theory.
The possible weak point of this test is that it assumes that all teams have
similar offensive and defensive effectiveness with various strategies. However,
some teams may be better with an offensive play while others excel in a
defensive play. The initial strategy may be an indicator of such preferences.
Therefore, we will use starting strategies as a point with Λ(a,a)=0. Both strategies
will be rescaled so their values are equal to the target value a which is equal to
the integer closest to the average of both strategies initially employed. Tables 10
and 11 present results of the tests.
Table 8 Number of player changes with respect to threshold values,
Threshold
modified game theory approach
-4
-3
-2
-1
0
1
2
3
4
-2
1
1
3
5
11
9
5
1
1
Changes
-1
1
1
8
19
45
46
19
15
2
1
1
3
48
79
55
18
9
8
2
2
1
6
8
3
1
121
Table 9 Number of net player changes with respect to threshold values,
Threshold
modified game theory approach
-4
-3
-2
-1
0
1
2
3
4
-2
1
8
8
5
1
1
Net changes
-1
1
2
2
40
60
10
3
3
28
10
7
The results remain unchanged including significance levels of
statistical tests.
122
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