The Valuation of M&A Targets by Relative
Indifference Prices
Carolin Mauch
∗
Stefan Rostek
†
University of Tuebingen
May 10, 2012
Abstract
In this paper, we apply the technique of indifference pricing to the valuation of
investment projects or businesses. We suggest that indifference pricing is the intuitive approach to overcome the conceptual drawbacks of classical real options
theory when facing market incompleteness.
Though assuming identical information about future states of nature, we show that
the sell side on the one hand and any potential buyer on the other hand may derive
different values. It may come as no surprise that these values differ by the investor’s
degree of risk aversion. However, beyond that, our approach reveals another source
of investor specific prices: The relative indifference price depends on the investor’s
current level of engagement in the very business field.
∗
University of Tuebingen, Department of Economics and Business Administration, Chair of managerial Accounting, Nauklerstr.47, 72074
Tübingen, Germany, carolin.mauch@uni-tuebingen.de
†
University of Tuebingen, Department of Economics and Business Administration, Chair of Corporate Finance, Sigwartstr.18, 72076
Tübingen, Germany, stefan.rostek@uni-tuebingen.de
1
I
Introduction
Valuation of investment projects and whole businesses by means of real option approaches
had been very popular in the 90s of the last century. When Stewart C. Myers introduced
the term ’real option’ in 1977 to the economics literature, he described this type of option as ’opportunities to purchase real assets on possibly favorable terms’ (Myers(1977),
p. 163). He stressed that DCF methods would systematically undervalue projects and
businesses due to limited consideration of strategic options. A multitude of real option
approaches as well as different valuation methods were suggested afterwards. Early approaches as Brennan and Schwartz (1985) for the valuation of natural resources or Kemna
(1993) for the valuation of growth and abandonment options conceptually refer to the
seminal work of Black and Scholes (1973) and Merton (1973) for the valuation of financial
options using a ’twin security’ for replication purposes. From today’s perspective where
seemingly even financial markets only sometimes happen to be complete, extending this
analogy to project valuation is known to be heroic.
In order to value real options in the presence of incomplete markets, further approaches
such as superhedging, minimal distance measures, coherent and convex risk measures as
well as indifference pricing, have been proposed that are more or less tractable(see for example Rouge/El Karoui (2000), Goll/Rüschendorf (2001), Musiela et al.(2008), Artzner
et al. (1999) or Föllmer/Schied (2002)). However, all of these techniques are rather
driven from a mathematical viewpoint and therefore seem to be not economically intuitive enough to convince practitioners to apply them.
The concept of indifference pricing fills that gap by explicitly readdressing to the funda-
2
mental principle of utility maximization. The approach was introduced by Hodges and
Neuberger (1989). The basic idea is to find an option price which makes the investor, in
consideration of his initial wealth and his utility function, indifferent between holding and
not holding the option. As a very general result, the pricing functional is convex, which
leads to the fact that the option’s price does not increase linearly by scale and a seller’s
price may deviate from a buyer’s price even though they share the same information.
Notably, both characteristics seem to be tailor-made to describe price features of M&A
targets.
Despite this intuitive and economic sound idea, until now, the scientific debate mainly
took place within the field of mathematical finance. Consequently, most of the literature
related to indifference pricing deals with continuous models (see e.g. Rouge/El Karoui
(2000), Henderson (2002), Musiela/Zariphopoulou (2003), Stoikov/Zariphopoulou (2005),
and Davis (2006)), whereas approaches of discrete indifference pricing are rather scarce.
As a commendable exception, Smith and McCardle (1998) as well as Musiela and Zariphopoulou (2004) develop discrete indifference pricing models.
Musiela and Zariphopoulou (2002) were also the first ones to intensely discuss the nonadditive character of indifference prices, which leads to the concept of relative indifference
prices. Most easily, the idea behind is that a buyer may not be willing to pay double the
price for a project of double extent as risk may increase in a non-linear way.
By means of this paper, we want to draw the attention to practical applications of relative indifference pricing within the field of project or business valuation. We promote
the technique of relative indifference pricing for at least three reasons: its easy-to-grasp
3
and economic intuitive concept, its ability to deal with incompleteness as a fundamental
inherence of M&A target valuation, and last but not least its ease of tractability.
The rest of the paper is structured as follows: In the following section, we shortly introduce the concept of indifference pricing in a quite general way. In section III, we specify
the target model as well as investor’s preferences and derive the indifference prices in a
similar way as Musiela and Zariphopoulou (2005). We then motivate the use of relative
indifference prices in section IV. We apply the results to an easy example where relative indifference pricing is used to derive potential sell prices and buy prices for an oil
platform.
II
The concept of indifference pricing in a discrete
setting
The approach of indifference pricing introduced by Hodges and Neuberger (1989), aims
to find the optimal hedging strategy which maximizes expected utility. By comparing
the indirect utility for the case of holding and the case of not holding the option, the
indifference price can be computed.
Assume a one-period setting, i.e. t ∈ {0, T }, with a traded asset St , a non-traded asset
Yt as well as a riskless asset Bt . Without any options traded a rational investor with
utility function U (x) maximizes his expected utility with respect to the physical measure
P. He does so by optimally splitting his initial wealth x into a number of α shares in
St and β shares in Bt . Consequently, at the two relevant points in time, the investor’s
4
wealth is given by:
X0 = αS0 + βB0 = x
XT = αST + βBT .
The investor’s indirect utility function or value function V without option trading is then:
V 0 (x) = max E (U (XT ))
α
Furthermore, we assume that there is an option contract C which is written at time
t = 0 and generates payments C at the end of the planning horizon. These payments
may depend both on the traded asset ST and on a non-traded asset YT . An investor
who is invested in St and Bt and additionally is short this option has to solve a shifted
optimization problem which is expressed by the seller’s value function VSC :
VSC (x) = max E [U (XT − CT )]
α
while the same investor being long the option will have the buyer’s value function VBC :
VBC (x) = max E [U (XT + C)]
α
In order to calculate the indifference price, the value functions V C for holding and V 0
for not holding the option are compared. The seller’s indifference price υS (C) is the
additional amount of initial wealth an investor requests for taking a short position in the
option, i.e.
VSC (x + υS (C)) = V 0 (x) , ∀x ∈ R
(1)
On the other hand, the buyer’s indifference price υB (C) is the amount of initial wealth
an investor is accepting to waive for having a long position in the option, i.e.
VBC (x − υB (C)) = V 0 (x) , ∀x ∈ R
5
(2)
III
Specifying asset evolution and risk preferences
Our valuation model is based on the discrete time indifference pricing model introduced
by Musiela and Zariphopoulou (2005). The latter specifies the riskless asset to have zero
interest rate as well as the two risky assets (traded and non-traded) to be of random
walk type which may be correlated. We slightly extend that framework assuming that
the binomial process of the traded risky asset will be mean-reverting. We do so having in
mind the practical application of the valuation of an oil field, where the dynamics of the
related traded commodity, i.e. the oil price, is usually assumed to be of mean-reverting
type.
The value of the traded asset S(t) is given by
ST = S0 + δ (S ∗ − S0 ) + ε,
S0 > 0
where S ∗ is the mean reversion level or the long run equilibrium price to which the asset
will converge to, δ the mean reversion rate, and ε = εu , εd the random shock to price
from t = 0 to t = T . To avoid riskless arbitrage opportunities the condition
−1 <
δ (S ∗ − S0 ) + εd
δ (S ∗ − S0 ) + εu
<0<
S0
S0
must hold.1
The value of the non-traded asset Y is also binomially distributed, i.e.
YT ∈ YTu , Ytd
These values of the non-traded asset do not need to be measured in monetary units (see
example in section V) and may be correlated to the evolution of the traded asset.
1 In
line with Cox/Ross/Rubinstein (1979), p.232.
6
The probability space is denoted by (Ω, F, P), where Ω = {ω1 , ω2 , ω3 , ω4 } defines the four
possible states in t = T and P is the real (historical) probability measure on the σ-algebra
Ω allocating the probabilities pi = P (ωi ).
Note that in a one-period setting the following four possible states may occur:
I
STu = S0 + δ (S ∗ − S0 ) + εu
YTu
II
STu = S0 + δ (S ∗ − S0 ) + εu
YTd
III
STd = S0 + δ (S ∗ − S0 ) + εd
YTu
IV
STd = S0 + δ (S ∗ − S0 ) + εd
YTd
The option value at maturity is a random variable defined on Ω with values C (ωi ) = ci ∈
R , for i = 1, ..., 4.
We complete our setting by specifying the utility functions of the market participants.
Both the seller and a potential buyer are assumed to have exponential utility functions
that only differ with respect to the risk aversion parameter:
and
US (x) = −e−γS x
for the seller
UB (x) = −e−γB x
for the buyer.
Proposition 1 Let Q be the probability measure of the following form:
qi = Q (ωi ) = q
pi
p1 + p2
qi = Q (ωi ) = (1 − q)
∗
pi
p3 + p4
, i = 1, 2
, i = 3, 4
d
0 )+ε
where q = − δ(S ε−S
. Then, the seller’s indifference price of an investor with risk
( u −εd )
aversion parameter γS is given by the conditional certainty equivalent
υS (C) = EQ
γ C
1
S T
ln EP e
| ST .
γS
7
(3)
Furthermore, the buyer’s indifference price of an investor with risk aversion parameter
γB is given by the conditional certainty equivalent
υB (C) = −EQ
−γ C
1
ln EP e B T | ST .
γB
(4)
Proof We first derive the seller’s indifference price. Substituting the possible values for
ST and C into the value function we obtain
∗
u
VSC (x) = e−γS x max EP −e−γS α(δ(S −S0 )+ε ) (p1 eγS c1 + p2 eγS c2 )
α
−e−γS α(δ(S
∗ −S )+εd
0
i
) (p eγS c3 + p eγS c4 )
3
4
Maximizing over α yields the optimal number of shares
1
(δ (S ∗ − S0 ) + εu ) (p1 eγS c1 + p2 eγS c2 )
ln −
=
γS (εu − εd )
(δ (S ∗ − S0 ) + εd ) (p3 eγS c3 + p4 eγS c4 )
1
1 − q p1 eγS c1 + p2 eγS c2
=
ln
·
γS (εu − εd )
q
p3 eγS c3 + p4 eγS c4
αS∗
∗
d
0 )+ε
. As a result the value function takes the form
with q = − δ(S ε−S
u −εd
VSC
(x) = e
−γS x
p1 eγS c1 + p2 eγS c2
q
q p3 eγS c3 + p4 eγS c4
1−q
1−q
When the investor is not holding the option the value function VS0 is obtained by setting
c1 = c2 = c3 = c4 = 0. Using the defining equation (1) the indifference sell price results
in
υS (C) = q
1
p1 eγS c1 + p2 eγS c2
1
p3 eγS c3 + p4 eγS c4
+ (1 − q)
ln
ln
γS
p1 + p2
γS
p3 + p4
Based on the evolution of the traded asset, we define the following events:
A = {ω1 , ω2 } = {ω : S1 (ω) = S0 + δ (S ∗ − S0 ) + εu0 }
Ac = {ω3 , ω4 } = ω : S1 (ω) = S0 + δ (S ∗ − S0 ) + εd0 ,
8
which allows us to write
1
1
ln EP eγS CT | A + (1 − q)
ln EP eγS CT | Ac
γS
γS
1
= EQ
ln EP eγS CT | ST .
γS
υS (C) = q
On the other hand, using the defining equation (2) one can show in a perfectly analogous
way that the buyer’s indifference price is given by
υB (C) = −EQ
−γ C
1
B T
ln EP e
| ST
γB
which completes the proof.
Proposition 1 shows that calculating the indifference price is a two-step process. Each of
the steps can be associated with one of the two different types of risks that are present
in the incomplete market.
As a first step, for each possible value of the traded asset the conditional expectation
1
γS
ln EP eγS CT | ST is calculated. It is easy to show that this expression represents the
conditional certainty equivalent (see Musiela and Zariphopoulou (2005)). Most evidently,
investor’s individual preferences influence this part of the price. Moreover, this part of
the pricing functional apparently is non-linear.
As a second step, the remaining hedgeable risk is valued through making use of arbitrage
free pricing characteristics. This is possible as with respect to the tradeable asset St , the
market is assumed to be complete. The value q hence represents the unique risk-neutral
probability for an upward move of St and is solely based on market information instead
of individual preferences.
As an immediate consequence of proposition 1, we receive the following corrolary.
9
Corrolary 1 For identical risk aversion parameters γS = γB we have the following relationship:
υB (C) = −υS (−C) .
IV
Relative indifference prices
The basic setup implicitly assumes that both the seller and the buyer initially are not
invested in the project, i.e. after the deal the seller of the option C will be short in the
project and the buyer long. When applying this to real projects, it seems to be reasonable
to assume that an investment project cannot be sold without possessing it in advance.
From now on, we will therefore assume that before the deal the seller has a long position
of aS C with a ≥ 1, i.e. he sells a fraction of 1/a of the project.
Definition The relative indifference price of a seller with current position aS C is the
amount the seller postulates when reducing his position from aS C to (aS − 1)C) and is
denoted by υSrel (C; aS ):
υSrel (C; aS ) = υS ((1 − aS )C) − υS (−aS C) .
(5)
In the same way, we allow the buyer to be already invested to an extent of aB C, where
aB ≥ 0 defines the range of possible parameters. Following the same idea, we define the
relative indifference price of a buyer with current position aB C:
Definition The relative indifference price of a buyer with current position aB C is the
amount the buyer is willing to pay when enlarging his position from aB C to (aB + 1)C)
10
and is denoted by υBrel (C; aB ):
υBrel (C; aB ) = υB ((aB + 1)C) − υB (aB C)
(6)
Based on these definitions, we can derive the following proposition:
Proposition 2 If the seller and the buyer show the same level of risk aversion, they
agree on the same indifference price if and only if they exactly interchange their level of
investment.
Proof By this, the seller’s relative indifference price is the (positive) difference between
the (negative) amount he would postulate for holding a short position of (1 − a)C and the
(negative) amount he would postulate for holding a short position of (−a)C. The idea
of the definition becomes much clearer when transforming the sell prices with negative
short positions (which basically are long positions) into buy prices:
υSrel (C; aS ) = υS ((1 − aS )C) − υS (−aS C)
= υB (aS C) − υB ((aS − 1)C)
= υBrel (C; aS − 1) .
Consequently, seller and buyer agree on the same price if and only if aB = aS − 1. In this
case the seller’s relative indifference price for reducing his position from aS C to (aS − 1)C
equals the incremental amount the same investor would pay as a buyer for enlarging his
position from (aS − 1)C to aS C, i.e. the parties interchange their positions.
For the practical application in the next section, we provide further characteristics of the
relative indifference prices.
11
Proposition 3 For aS ≥ 1, the seller’s relative indifference price is a decreasing function
both with respect to the risk aversion parameter γS and with respect to the level of current
position aS .
For aB ≥ 0, the buyer’s relative indifference price is a decreasing function both with
respect to the risk aversion parameter γB and with respect to the level of current position
aB .
Proof We recall the characteristics of the function of absolute indifference prices stated
by Musiela and Zariphopoulou (2005): The seller’s indifference price function is continuous, increasing and convex with respect to payoffs, while the buyer’s indifference price
function is continuous, increasing and concave with respect to payoffs. The proposition
then is a straightforward consequence of the definitions given by equations (5) and (6).
V
An Example: Relative indifference prices of an oil
field
In this section, we illustrate the concept of indifference pricing by a simplified example:
We assume that the target is an unconventional oil reserve whose future possible exploitation is contingent on technological progress and/or legal framework.
The future benefit for an investor is determined by two key influences. First, the mere
exploitability plays the role of our non-traded asset and has a typical go/no go character
taking a value of one or zero. Second, in case that exploitation is possible, the main
driver of profits is assumed to be the level of the oil price which represents the traded
12
%
Y0
&
YTu = 1
%
Stu = 110
&
Std = 90
S0 = 105
YTd = 0
Figure 1: Evolution of the traded and the non-traded asset. Parameters of the oil price
evolution are S0 = 105, S ∗ = 90, δ = 13 , εu = +10, εd = −10.
%
{STu , YTu } : ω1 = 0.3
&
{STu , YTd } : ω2 = 0.2
%
{STd , YTu } : ω3 = 0.2
&
{STd , YTd } : ω4 = 0.3
%
{Ω, F, P}
&
Figure 2: Joint distribution of the traded and the non-traded asset.
asset.
Figure 1 shows the evolution of the traded as well as of the non-traded asset. Note that
the traded asset was modeled to be of mean-reverting type. Moreover, we assume that
the joint probability distribution is given by Figure 2: Here we implicitly assumed a slight
positive correlation between oil price and exploitability, as higher oil prices might cause
an enhancement of research efforts. The net profits (in Mio USD) to be earned from
the target are assumed to have the form given by Figure 3 (the profit formation here is
treated as a black box): Let the seller of the target be a company that holds a multitude
of similar oil reserves all dependent on this same type of risks and with potential net
profits linearly related to project C. Consequently, we can assume that the portfolio of
the seller is a multiple of project C and can be modeled to be of the shape aS C. Here,
we assume aS = 3, i.e. the vendor is selling one third of his potential oil reserves. The
13
%
C(ω1 ) = c1 = 50
&
C(ω2 ) = c2 = 0
%
C(ω3 ) = c3 = 20
&
C(ω4 ) = c4 = 0
%
C
&
Figure 3: Distribution of net profits of the target.
risk aversion parameter of the seller’s utility function2 is assumed to be γS = 0.005.
We can now start calculating the seller’s indifference price.
υSrel (C; aS ) = υS ((1 − aS )C) − υS (−aS C)
−0.005∗2∗C
1
T
= EQ
ln EP e
| ST
5
−0.005∗3∗C
1
T
ln EP e
| ST
−EQ
5
The values of the conditional certainty equivalents are:
1
1
ln EP e−0.005∗2∗CT | ST = S u =
ln 0.6 ∗ e−0.01∗50 + 0.4 ∗ e−0.01∗0
0.005
0.005
= −53.86
1
1
ln EP e−0.005∗2∗CT | ST = S d =
ln 0.4 ∗ e−0.01∗20 + 0.6 ∗ e−0.01∗0
0.005
0.005
= −15.05
1
1
ln EP e−0.005∗3∗CT | ST = S u =
ln 0.6 ∗ e−0.015∗50 + 0.4 ∗ e−0.015∗0
0.005
0.005
= −76.13
1
1
ln EP e−0.005∗3∗CT | ST = S d =
ln 0.4 ∗ e−0.015∗20 + 0.6 ∗ e−0.015∗0
0.005
0.005
= −21.89
2 Note
that x is measured in Mio USD.
14
∗
d
1
0 )+ε
= −3
With q = − δ(S ε−S
( u −εd )
(90−105)−10
10+10
= 34 , the seller’s relative indifference price equals
υSrel (C; aS ) = 0.75 ∗ (−53.86) + 0.25 ∗ (−15.05) − (0.75 ∗ (−76.13) + 0.25 ∗ (−21.89))
= −44.16 + 62.57
= 18.41 Mio USD
From the theoretical considerations of section IV, we know that a potential buyer with
the same risk aversion parameter γB = 0.005 and a risk exposure aB = 2 would derive
the same indifference price. All potential buyers with the same risk aversion but lower
risk exposure as well as any buyer with same risk exposure but lower risk aversion will
certainly also agree on the seller’s price and may still have room for higher offers in case of
competing bids. However, the combination of higher risk aversion but lower risk exposure
or vice versa are not clear in advance and have to be computed.
For example, the buyer B1 with combination γB1 = 0.008 and aB1 = 1 yields an relative
indifference price of υBrel (C; aB1 ) = 18.66 Mio USD which is slightly above the amount the
seller requires. On the other hand, an investor B2 with lower risk aversion γB2 = 0.002 but
already high exposure aB2 = 8 would only be willing to pay a maximum amount of 16.26
Mio USD. Figure 4 shows the function of buyer’s relative indifference price depending on
risk aversion and risk exposure. For the seller’s relative indifference price of 18.41 Mio
USD given in the example above, it is easy to derive graphically the range of parameter
combinations that can lead to a successful transaction. Figure 5 shows the very plot.
The red area depicts the combinations where buyer’s relative indifference price is above
the one of the seller. The combinations of investors B1 and B2 are marked.
15
Figure 4: Buyer’s relative indifference price depending on risk aversion γB and
risk exposure aB .
VI
Conclusion
We suggested the approach of relative indifference prices to valuation in incomplete markets. We showed that this concept is based on a very intuitive economic idea. By the
concavity of risk-averse utility function, the price function of an option is non-linear as
soon as the option payoff contains an undhedgeable part. The approach is in line with the
existing theory: In the limit case where the market is complete and the option is perfectly
hedgeable, the function will be linear again and of Cox-Ross-Rubinstein type. However,
16
Figure 5: Parameter combinations of potential buyers
the far more general approach of indifference pricing treats this limit case of complete
markets as what it is: an exception that most of the cases will not be appropriate for
modeling the characteristics of investment projects or M&A targets.
Our easy example showed how the technique can be implemented in the context of real
world applications. Such applications certainly have to be modeled by a slightly more
complex structure allowing for multi-period decisions and introducing interest rates. We
decided to abstain from these extensions in order to ease the understanding of the main
concept. However, the mathematical framework for such extensions has already been
17
developed (see e.g. Musiela and Zariphopoulou (2002)) and is basically the same as in
our setting.
A next step to extend this model could be to introduce imperfect information about risk
exposure and risk aversion of the counterparty. A game-theoretic approach may then be
appropriate to model auction type M&A processes in a more realistic way.
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