# Displaying risk in mergers: a diagrammatic approach for exchange ratio
determination

Alessandra Mainini  Dipartimento di scienze economiche e sociali, Università
Cattolica del Sacro Cuore, Piacenza, Italia  Enrico Moretto∗, Dipartimento di
economia, metodi quantitativi e strategie di impresa, Università di Milano-
Bicocca, Milano, Italia
∗corresponding author<EMAIL_ADDRESS>Daniela Visetti  Dipartimento
di economia, metodi quantitativi e strategie di impresa, Università di Milano-
Bicocca, Milano, Italia
∗corresponding author<EMAIL_ADDRESS>
###### Abstract

This article extends, in a stochastic setting, previous results in the
determination of feasible exchange ratios for merging companies. A first
outcome is that shareholders of the companies involved in the merging process
face both an upper and a lower bounds for acceptable exchange ratios.
Secondly, in order for the improved ‘bargaining region’ to be intelligibly
displayed, the diagrammatic approach developed by Kulpa is exploited.

Keywords— Mergers and acquisitions, exchange ratio determination, synergy,
risk-adjusted performance, diagrammatic representation

##  1 Introduction, literature review and motivation

Mergers and acquisitions have been, and still are a widely studied topic in
financial literature under both a theoretical and an empirical point of view.
A concise summary of contributions in this field is virtually impossible;
recent attempts are King et al. [1] and Risberg et al. [2].

Companies merge for various reasons, but with a unique goal: the creation of
synergy, that is a, hopefully positive, change in the behavior of the equity
of the resulting company (here denoted by $M$) when compared to those of the
equities of the preexisting ones, namely the acquiring, $A$, and the acquired,
or target, $B$.

Relevant contributions to the financial evaluation of synergies are due to
Gupta and Gerchak [3], Leland [4], and are broadly summarized in Chapter 15 of
Damodaran [5].

This paper focuses on the impact of synergy for shareholders of the merging
companies and extends, in a stochastic environment, known results (Larson and
Gonedes, [6] and Yagil [7]; in what follows, this approach is referred to as
LG-Y).

The attention of this article resides on stock-for-stock agreements, where
shareholders of the target company swap, if the merger is finalized, $r$
stocks of company $M$ for each share of company $B$ they own. Shareholders of
the acquiring company $A$ receive, instead, one stock of $M$ for each share
they own. Stockholders of either company $A$ or $B$ become shareholders of
company $M$.

A topic that has been almost totally neglected in financial research is the
fair determination of the exchange ratio (ER) $r$. In a deterministic
framework, the LG-Y setting is the only known contributions. Here, companies
are evaluated using the the dividend discount model (Gordon and Shapiro, [8]).
The result is the explicit determination of the bargaining region (BR), that
is the range for exchange ratios that do not reduce shareholders’ wealth.

A generalization with a dynamical extension of the LG-Y setting is due to
Cigola and Modesti [9] while a recent contribution, that encompasses a number
of accounting observations, is in Taliento [10].

As noted by Amihud and Lev [11], risk reduction can be a driver pushing
companies to merge. An attempt to introduce risk in the determination of the
exchange ratio has been proposed by Moretto and Rossi [12]. This contribution,
though, does not identify a BR; it provides, instead, a unique risk-corrected
ER.

More recently, Toll and Hering [13] analyze, in a stochastic context, the
effects of a merger for shareholders by means of expected utility theory.

Entering the realm of randomness in financial markets requires a solid theory
for risk measures. The fundamental contribution in this context is provided by
Artzner et al. [14]; here riskiness is measured by means of functions that
translate random variables into a cash amount that, if positive, has to be
paired to the risk in order to make it acceptable.

In this contribution, no specific risk measure is chosen. Our analysis starts
by mimicking results in LG-Y replacing the deterministic result with expected
values of random stock prices and measures changes in risk the merger grants
to stockholders.

In the LG-Y setting, in terms of ER determination shareholders of the
acquiring and acquired companies have clear-cut and opposite interests: the
larger $r$, the larger the stake of company $M$ $B$’s shareholders own, and
the larger the portion of synergy they receive. This is the reason why
stockholders of company $A$ aim at keeping $r$ as small as possible. Our
contribution shows that, when a risk measure is introduced, something
completely different and less straightforward occurs.

As a matter of fact, imposing bounds on expected values and risk measure leads
to more accurate and realistic BRs.

In finance, nothing comes for free. Adding a second dimension embeds into the
stochastic framework a drawback: BRs cannot be easily depicted, unless some
appropriate methodology is applied. Luckily, the ‘diagrammatic’ representation
of intervals developed by Kulpa (see [15], [16], and [17]) permits to
transform intervals into points on a two-dimension Cartesian plane and solve
the issue.

The paper is organized as follows: Section 2 describes the theoretical
framework on which the extension in a stochastic fashion of the LG-Y model
abides. Section 3 introduces Kulpa’s diagrammatic representation, translates
BRs in legible and easy-to-understand plots, and provides some financial
insights. Section 4 concludes

##  2 Arithmetic for merger agreements

###  2.1 The deterministic benchmark: Larson and Gonedes, and Yagil setting

For the reader’s sake, in this subsection we summarize the main findings of
contributions from Larson and Gonedes, and Yagil (LG-Y from now on). Their
approach is deterministic and highlights conditions under which shareholders
of the acquiring ($A$) and acquired ($B$) companies are better off once the
merger agreement is settled, leading to the birth to the resulting company
$M$. The LG-Y setting assumes that, in a stock-for-stock agreement, each stock
of company $A$ translates into one stock of company $M$ while each stock of
company $B$ becomes $r$ stocks of company $M$, where $r\geq 0$ is the exchange
ratio (ER).

Stock-for-stock refers to the exchange of stocks of the resulting with stocks
of the acquired company at a predetermined conversion rate. Companies that pay
for their acquisitions with stock share both the value and the risks of the
transaction with the shareholders of the company they acquire. Nevertheless,
as mentioned above, the LG-Y settingis purely deterministic, and stockholders
share only the value of the transaction.

Let $p_{i}$, $N_{i}$, and $\mu_{i}=p_{i}\cdot N_{i}$ be, respectively, the
stock price, the number of outstanding stocks, and the equity value for
companies $i=A,B,M$. The number of $M$’s outstanding stocks is

$N_{M}=N_{A}+rN_{B}.$

A merger creates some positive synergy when the equity value of $M$
corresponds at least to the sum of equity values of $A$ and $B$:

$\mu_{M}\geq\mu_{A}+\mu_{B}.$

A relevant quantity is ratio

$r_{\ast}=p_{B}/p_{A}\text{.}$ (1)

which represents the relative value of a stock of company $B$ when compared
with the value of one stock of company $A$.

The LG-Y approach provides an explicit expression for the bounds of the
bargaining region (BR), that is the interval containing all ERs for which
shareholders of companies $A$ and $B$ simultaneously experience an increase in
their equity values and, thus, might accept to conclude the merger.

Existence and magnitude of this interval depend on synergy created by the
merger.

Company $A$’s stockholders enjoy a benefit from this agreement when the price
of stocks they own after the merger is equal, at least, to the price of stocks
they previously owned, that is

$p_{M}=\frac{\mu_{M}}{N_{A}+rN_{B}}\geq\frac{\mu_{A}}{N_{A}}=p_{A}\text{.}$

Simple algebra yields an upper bound for $r$:

$r\leq\frac{\mu_{M}-\mu_{A}}{\mu_{A}}\cdot\frac{N_{A}}{N_{B}}=r_{\ast}\cdot\frac{\mu_{M}-\mu_{A}}{\mu_{B}}=\overline{r}_{\mu}\left(\mu_{M}\right).$
(2)

Condition (2) denotes the largest ER company $A$’s shareholders accept.

Company’s $B$ shareholders, instead, obtain an increment in their wealth when

$r\cdot
p_{M}=r\cdot\frac{\mu_{M}}{N_{A}+rN_{B}}\geq\frac{\mu_{B}}{N_{B}}=p_{B}\text{.}$

This condition is equivalent to a lower bound for $r$:

$r\geq
r_{\ast}\cdot\frac{\mu_{A}}{\mu_{M}-\mu_{B}}=\underline{r}_{\mu}\left(\mu_{M}\right)\text{.}$
(3)

Bounds (2) and (3) depend on the equity value of $M$. Further, it is immediate
to see that, if $\mu_{M}\geq\mu_{A}+\mu_{B}$, then
$\underline{r}_{\mu}(\mu_{M})\leq\overline{r}_{\mu}(\mu_{M})$. In this case,
the above bounds identify set

$\mathcal{BR}_{\mu}\left(\mu_{M}\right)=[\underline{r}_{\mu}(\mu_{M}),\overline{r}_{\mu}(\mu_{M})].$
(4)

The absence of synergy (i.e. $\mu_{M}=\mu_{A}+\mu_{B}$) shrinks set
$\mathcal{BR}_{\mu}\left(\mu_{M}\right)$

into a singleton:
$\mathcal{BR}_{\mu}\left(\mu_{A}+\mu_{B}\right)=\left\\{r_{\ast}\right\\}$.
This expression highlights the importance of ratio (1).

Shaded region in Figure 1 represents set (4) in terms of $\mu_{M}$.

Figure 1: Graphical representation of a BR (shaded area) in the LG-Y’s setting
- $p_{A}=4$, $p_{B}=2$, $N_{A}=20$, $N_{B}=10$, $r_{\ast}=0.5$,
$\mu_{A}+\mu_{B}=100$, $\mu_{M}\in\left[100,200\right]$.

###  2.2 A stochastic extension

As mentioned above, in a stock-for-stock merger companies not only share the
change in expected equity values but, also, the change in risk created by this
transaction.

In order to expand the LG-Y results in a stochastic environment, we need to
introduce and measure risk.

Randomness is represented by (non-negative) random variables $\tilde{p}_{i}$,
$i=A,B,M$, that describe stock prices. We make no a priori choice on their
distribution, as long as expected values $m_{i}$ assumed to be a measure of
performance and homogeneous measures of risk $\varphi_{i}$, in the sense of
Artzner et al. (see [14]), for $\tilde{p}_{i}$, exist. We also assume that
$\varphi_{i}\geq 0$ so that stocks are considered a non- acceptable risk and
shareholders might be interested in reducing it.

For the sake of notational simplicity, we decide to set $m_{i}=p_{i}$ so that
formulas (2) and (3) are recovered. This claim is easy to justify; the
deterministic approach corresponds to the one presented in this subsection if
expected values only are considered.

Similarly to expression (1), we stress the importance of ratio

$r_{\ast\ast}=\varphi_{B}/\varphi_{A},$ (5)

that expresses the relative risk of a share of company $B$ w.r.t. riskiness of
a stock of company $A$.

Introducing a risk measure for equities allows to define ratio
$\lambda_{i}=p_{i}/\varphi_{i}$, that is the coefficient of variation of
random variable $\tilde{p}_{i}$ as well as a relative value that expresses a
stock’s risk-corrected measure of performance.

Condition $\lambda_{A}>\lambda_{B}$ (respectively $\lambda_{A}<\lambda_{B}$),
that is the risk-corrected performance of stock $A$ is better (respectively
worse) than $B$’s, is equivalent to $r_{\ast\ast}>r_{\ast}$ (respectively
$r_{\ast\ast}<r_{\ast}$).

Let $\rho_{i}=\varphi_{i}\cdot N_{i}$. Shareholders of company $A$ benefit of
a smaller risk after the merger when

$\varphi_{M}=\frac{\rho_{M}}{N_{A}+rN_{B}}\leq\frac{\rho_{A}}{N_{A}}=\varphi_{A}.$
(6)

This inequality yields a lower bound for ERs:

$r\geq
r_{\ast\ast}\cdot\frac{\rho_{M}-\rho_{A}}{\rho_{B}}=\underline{r}_{\rho}(\rho_{M}).$
(7)

Here, to prevent $r$ from becoming negative, condition $\rho_{M}>\rho_{A}$
must hold.

Stockholders of company $B$ obtain a reduction in their equity risk when

$r\cdot\varphi_{M}=r\cdot\frac{\rho_{M}}{N_{A}+rN_{B}}\leq\frac{\rho_{B}}{N_{B}}=\varphi_{B}\text{.}$
(8)

This inequality leads to an upper bound for exchange ratios:

$r\leq
r_{\ast\ast}\cdot\frac{\rho_{A}}{\rho_{M}-\rho_{B}}=\overline{r}_{\rho}(\rho_{M})$
(9)

where imposing $\rho_{M}>\rho_{B}$ forbids $r$ to become negative.

Bounds (7) and (9) yield positive ERs when
$\rho_{M}>\max\left\\{\rho_{A},\,\rho_{B}\right\\}$. This constraint implies
that the reduction in riskiness a merger can achieve reduces the risk of
company $M$ at most up to the largest risk of the merging ones.

Inequalities (7) and (9) identify a second bargaining region that is non empty
whenever $\rho_{M}\leq\rho_{A}+\rho_{B}$ and contains positive ERs if
$\rho_{M}>\max\left\\{\rho_{A},\rho_{B}\right\\}$. These conditions lead to

$\max\left\\{\rho_{A},\rho_{B}\right\\}<\rho_{M}\leq\rho_{A}+\rho_{B}$ (10)

while this second set is

$\mathcal{BR}_{\rho}\left(\rho_{M}\right)=\left[\underline{r}_{\rho}(\rho_{M}),\overline{r}_{\rho}(\rho_{M})\right]$
(11)

for $\max\left\\{\rho_{A},\,\rho_{B}\right\\}<\rho_{M}\leq\rho_{A}+\rho_{B}$.
Interval (11) collapses into a singleton when the merger does not reduce
riskiness, that is $\rho_{M}=\rho_{A}+\rho_{B}$. In fact,
$\mathcal{BR}_{\rho}\left(\rho_{A}+\rho_{B}\right)=\left\\{r_{\ast\ast}\right\\}$.

A comparison between inequalities (2) and (7), that relate to company $A$’s
stockholders, and between inequalities (3) and (9), that identify which
exchange ratios are acceptable for company $B$’s stockholders, leads to a
counterintuitive as well as interesting outcome.

If bound (2) suggests that stockholders of company $A$ want the exchange ratio
to be as small as possible as this would grant them a large stake of expected
synergy, bound (7) indicates that, if the number of $M$’s outstanding stocks
is not large enough, the risk each stock ends up carrying is not less than the
risk of each $A$’s stock.

A similar line of reasoning can be done for company $B$’s stockholders. Bound
(3) implies that shareholders of the acquired company want to fix the exchange
ratio as large as possible because, in this case, they seize a large portion
of expected synergy. Bound (9), though, reveals that acceptable ERs cannot
exceed a given threshold. If this happens, the stocks these stockholders
obtain in exchange of each share they owned carry larger risk111The trivial
arithmetic explanation of this claim is in Appendix A.1..

The existence of exchange ratios upper and lower bounds for both categories of
shareholders permits to introduce two more sets that we call consistency
region (CR). These intervals, when non-empty, contain all ERs that
simultaneously grant both an increase in expected equity values and a
reduction in equity risk for both groups of shareholders.

These sets are defined as

$\mathcal{CR}_{\text{A}}\left(\mu_{M},\rho_{M}\right)=\left[\underline{r}_{\rho}\left(\rho_{M}\right),\overline{r}_{\mu}\left(\mu_{M}\right)\right]$
(12)

with $\mu_{M}\geq\mu_{A}+\mu_{B}$ and
$\max\left\\{\rho_{A},\,\rho_{B}\right\\}<\rho_{M}\leq\rho_{A}+\rho_{B}$ for
shareholders of company $A$, and

$\mathcal{CR}_{\text{B}}\left(\mu_{M},\rho_{M}\right)=\left[\underline{r}_{\mu}\left(\mu_{M}\right),\overline{r}_{\rho}\left(\rho_{M}\right)\right]$
(13)

with $\mu_{M}\geq\mu_{A}+\mu_{B}$ and
$\max\left\\{\rho_{A},\,\rho_{B}\right\\}<\rho_{M}\leq\rho_{A}+\rho_{B}$ for
those of company $B$.

Assuming a merger yields an improvement in terms of both larger expected
wealth and smaller risk might seem a rather ‘too-good-to-be-true’ requirement.
Still, the key financial notion of diversification is based on better
performance paired with less risk.

Interval (12) is non-empty when

$\frac{\mu_{M}-\mu_{A}}{\rho_{M}-\rho_{A}}\geq\frac{p_{A}}{\varphi_{A}}=\lambda_{A}\text{.}$

To proceed with this analysis, let $s\geq 0$ be the expected synergy and
$v\geq 0$ the reduction in risk, so that $\mu_{M}=\mu_{A}+\mu_{B}+s$ and
$\rho_{M}=\rho_{A}+\rho_{B}-v$. The above condition becomes

$\frac{\mu_{B}+s}{\rho_{B}-v}\geq\lambda_{A}\text{.}$

Consider, at first, the limit case where $s=0$ and $v=0$ (that is, the merger
carries no benefit whatsoever). Interval (12) is non-empty if

$\frac{\mu_{B}}{\rho_{B}}=\lambda_{B}\geq\lambda_{A}\text{,}$

that is when company $B$ is preferable to $A$ in terms of risk-corrected
performance. Shareholders of company $A$ are favourable to complete the merger
regardless of its positive synergy creation and/or risk reduction because
company $M$ has an improved risk-corrected performance when compared $A$’s
one.

The trivial algebra that explains the above claim, and the one regarding
company $B$ below, is relegated in Appendix A.3.

This, of course, holds also when $s>0$ and/or $0<v<\rho_{B}$ as, if
$\lambda_{B}\geq\lambda_{A}$,

$\frac{\mu_{B}+s}{\rho_{B}-v}>\lambda_{B}\geq\lambda_{A}\text{.}$

Further, if $\lambda_{B}\geq\lambda_{A}$ (that is $r_{\ast}\geq r_{\ast\ast}$)

$\mathcal{CR}_{\text{A}}\left(\mu_{A}+\mu_{B},\rho_{A}+\rho_{B}\right)=\left[r_{\ast\ast},r_{\ast}\right]\text{.}$

Consider case $\lambda_{B}<\lambda_{A}$. Here, the no improvement due to
synergy case yields

$\mathcal{CR}_{\text{A}}\left(\mu_{A}+\mu_{B},\rho_{A}+\rho_{B}\right)=\emptyset.$

The resulting company the merger creates has a risk-corrected performance
which is smaller than $\lambda_{A}$. Company A’s shareholders agree on the
merger only if they receive some compensation that offsets their reduction in
risk-corrected performance. This occurs if

$s\geq\lambda_{A}\left(\rho_{B}-v\right)-\mu_{B}=\overline{s}_{A}\text{,}$

that is the condition for non-emptiness of set $\mathcal{CR}_{A}$. Here, for
each $v\in\left[0,\rho_{B}\right)$ there exists a threshold $\overline{s}_{A}$
that linearly decreases w.r.t. $v$. The above inequality indicates that the
larger the reduction in risk, the smaller the minimum expected synergy that
provides company $A$’s stockholders with a range of acceptable ERs.

A similar analysis can be carried out for interval (13). This set is non-empty
when

$\frac{\mu_{M}-\mu_{B}}{\rho_{M}-\rho_{B}}=\frac{\mu_{A}+s}{\rho_{A}-v}\geq\frac{\mu_{B}}{\rho_{B}}=\lambda_{B}\text{.}$

Limit case $s=0$, $v=0$ yields $\lambda_{A}\geq\lambda_{B}$ so that company
$B$’s shareholders agree on the merger even if there is no synergistic effect
if company $A$ has a better risk-adjusted performance.

In this case

$\mathcal{CR}_{\text{B}}\left(\mu_{A}+\mu_{B},\rho_{A}+\rho_{B}\right)=\left[r_{\ast},r_{\ast\ast}\right]\text{.}$

When, instead, $\lambda_{A}<\lambda_{B}$, set $\mathcal{CR}_{\text{B}}$ is
non-empty when expected synergy fulfills condition

$s\geq\lambda_{B}\left(\rho_{A}-v\right)-\mu_{A}=\overline{s}_{B}\text{.}$

The bargaining region in the stochastic setting is the bivariate set

$\mathcal{BR}_{\mu,\rho}\left(\mu_{M},\rho_{M}\right)=\mathcal{BR}_{\mu}\left(\mu_{M}\right)\cap\mathcal{BR}_{\rho}\left(\rho_{M}\right)\text{.}$

Set $\mathcal{BR}_{\mu,\rho}$ can be in one of six different cases, summarized
in Table 1; four of them are non-empty. Each case corresponds to one of the
sets $\mathcal{BR}$ and $\mathcal{CR}$ defined above. A detailed description
of such cases is in Section 3.

| $\overline{r}_{\rho}<\underline{r}_{\mu}<\overline{r}_{\mu}$ | $\underline{r}_{\mu}<\overline{r}_{\rho}<\overline{r}_{\mu}$ | $\underline{r}_{\mu}<\overline{r}_{\mu}<\overline{r}_{\rho}$
---|---|---|---
$\underline{r}_{\rho}<\underline{r}_{\mu}<\overline{r}_{\mu}$ | $\emptyset$ | case 1 - $\left[\underline{r}_{\mu},\overline{r}_{\rho}\right]$ | case 2 - $\left[\underline{r}_{\mu},\overline{r}_{\mu}\right]$
$\mathcal{CR}_{B}(\mu_{M},\rho_{M})$ | $\mathcal{BR}_{\mu}(\mu_{M})$
$\underline{r}_{\mu}<\underline{r}_{\rho}<\overline{r}_{\mu}$ | NA | case 3 - $\left[\underline{r}_{\rho},\overline{r}_{\rho}\right]$ | case 4 - $\left[\underline{r}_{\rho},\overline{r}_{\mu}\right]$
$\mathcal{BR}_{\rho}(\rho_{M})$ | $\mathcal{CR}_{A}(\mu_{M},\rho_{M})$
$\underline{r}_{\mu}<\overline{r}_{\mu}<\underline{r}_{\rho}$ | NA | NA | $\emptyset$
Table 1: Possible cases for
$\mathcal{BR}_{\mu,\rho}\left(\mu_{M},\rho_{M}\right)$. Rows and columns in
this table display all possible juxtapositions of endpoints
$\underline{r}_{\rho}\left(\rho_{M}\right)$ and
$\overline{r}_{\rho}\left(\rho_{M}\right)$ w.r.t. interval
$\left[\underline{r}_{\mu}\left(\mu_{M}\right),\overline{r}_{\mu}\left(\mu_{M}\right)\right]$.
For sake of compactness, in the topmost row and leftmost column, variables
$\mu_{M}$ and $\rho_{M}$ have been dropped. NA denotes non-attainable cases.

Along the lines of [7] and [6], the issue is now the geometric representation
of the bargaining region corresponding to the stochastic setting,
$\mathcal{BR}_{\mu,\rho}\left(\mu_{M},\rho_{M}\right)$, by varying $\mu_{M}$
and $\rho_{M}$. Its representation is a subset of the $\mathbb{R}^{3}$ space.
However, to display this set in a more intelligible way, we decide to
eliminate one dimension, and represent the bargaining region in the
$\mathbb{R}^{2}$ plane.

Fortunately, Kulpa does the trick, as he provides a methodology for
representing intervals as points on the plane. His results are summarized and
applied in the next section.

##  3 Diagrammatic representation of the bargaining region

In his diagrammatic approach, Kulpa (see, for example, [15], [16], and [17])
develops a useful methodology for representing bounded intervals
$\left[a,b\right]\subset\mathbb{R},a\leq b$ as points on a plane. This is
done, as Kulpa writes, citing [18], in [15], with the idea that “solving a
problem simply means representing it so as to make the solution transparent”.

Based on this approach, each non-empty interval of the form $\left[a,b\right]$
is set in a one-to-one correspondence with point

$(x,y)=\left(\frac{a+b}{2},\;\frac{b-a}{2}\right)\ \in\mathbb{R}^{2}$

where the two coordinates represent, respectively, the mid-point and the
radius, or half-length, of $[a,b]$. An interval which collapses to the
singleton $\\{a\\}$ corresponds, according to Kulpa’s representation, to the
point $(a,0)$, and lies on the horizontal axis.

Consider now intervals

$\left[\underline{r}_{\mu}(\mu_{M}),\,\overline{r}_{\mu}(\mu_{M})\right]\;\;\;\;\;\;\text{and}\;\;\;\;\;\;\left[\underline{r}_{\rho}(\rho_{M}),\,\overline{r}_{\rho}(\rho_{M})\right],$
(14)

which, as described in Section 2, are the sets of exchange ratios accepted by
both kinds of shareholders that guarantee, respectively, an increase of the
value and a decrease of the overall risk of the transaction.

According to the diagrammatic approach, they are represented, respectively, by
the points

$\begin{split}P_{\mu}\left(\mu_{M}\right)&\equiv\left(\frac{\underline{r}_{\mu}(\mu_{M})+\overline{r}_{\mu}(\mu_{M})}{2},\,\frac{\overline{r}_{\mu}(\mu_{M})-\underline{r}_{\mu}(\mu_{M})}{2}\right)\\\
&=\left(\frac{r_{\ast}}{2}\left(\frac{\mu_{A}}{\mu_{M}-\mu_{B}}+\frac{\mu_{M}-\mu_{A}}{\mu_{B}}\right),\;\frac{r_{\ast}}{2}\left(\frac{\mu_{M}-\mu_{A}}{\mu_{B}}-\frac{\mu_{A}}{\mu_{M}-\mu_{B}}\right)\right)\end{split}$
(15)

and

$\begin{split}P_{\rho}\left(\rho_{M}\right)&\equiv\left(\frac{\underline{r}_{\rho}(\rho_{M})+\overline{r}_{\rho}(\rho_{M})}{2},\,\frac{\overline{r}_{\rho}(\rho_{M})-\underline{r}_{\rho}(\rho_{M})}{2}\right)\\\
&=\left(\frac{r_{\ast\ast}}{2}\left(\frac{\rho_{M}-\rho_{A}}{\rho_{B}}+\frac{\rho_{A}}{\rho_{M}-\rho_{B}}\right),\;\frac{r_{\ast\ast}}{2}\left(\frac{\rho_{A}}{\rho_{M}-\rho_{B}}-\frac{\rho_{M}-\rho_{A}}{\rho_{B}}\right)\right)\end{split}$
(16)

The positions of $P_{\mu}(\mu_{M})$ and $P_{\rho}(\rho_{M})$ in the plane vary
with the admissible values of $\mu_{M}$ and $\rho_{M}$222Recall that $\mu_{M}$
and $\rho_{M}$ are admissible when $\mu_{M}\geq\mu_{A}+\mu_{B}$ and
$\max\\{\rho_{A},\rho_{B}\\}<\rho_{M}\leq\rho_{A}+\rho_{B}$., thus defining
two parametric curves, that we analyze in the next subsection.

We observe that, consistently with
$\mathcal{BR}_{\mu}\left(\mu_{A}+\mu_{B}\right)=\left\\{r_{\ast}\right\\}$ and
$\mathcal{BR}_{\rho}\left(\rho_{A}+\rho_{B}\right)=\left\\{r_{\ast\ast}\right\\}$,
the corresponding points become

$P_{\mu}\left(\mu_{A}+\mu_{B}\right)=\left(r_{\ast},0\right)\ \ \ \ \text{ and
}\ \ \ \
P_{\rho}\left(\rho_{A}+\rho_{B}\right)=\left(r_{\ast\ast},0\right)\text{,}$

that is, the two curves intersect the horizontal axis at the points
$\left(r_{\ast},\,0\right)$ and $\left(r_{\ast\ast},\,0\right)$, since they
represent the singletons $\\{r_{\ast}\\}$ and $\\{r_{\ast\ast}\\}$.

The points $P_{\mu}\left(\mu_{M}\right)$ and $P_{\rho}\left(\rho_{M}\right)$,
depending respectively on $\mu_{M}$ and $\rho_{M}$ describe the curves
$\gamma(\mu_{M})$ and $\delta(\rho_{M})$.

In the next subsection (see also Appendix A.2) we show that $\gamma$ and
$\delta$ represent two hyperbolas.

The explicit equations of these two curves allow us to represent the
intersection of intervals in (14). This achievement allows to make readable
and intelligible the set of exchange ratios on which an agreement is possible.

The points of each hyperbola, restricted to those satisfying the conditions
imposed on $\mu_{M}$ and $\rho_{M}$, represent the set of exchange ratios
accepted by all the shareholders with respect to one of the two aspects (i.e.,
expected wealth or overall risk). Since we eliminate one dimension, it becomes
easier and more intuitive to determine the accepted exchange ratios by varying
$\mu_{M}$ and $\rho_{M}$, as well as those belonging to the bargaining region.

###  3.1 Analysis of the curves $\gamma$ and $\delta$

Consider point $P_{\mu}(\mu_{M})$ in (15). As proved in Appendix A.2,
observing that

$x(\mu_{M})-y(\mu_{M})=\frac{r_{\ast}\mu_{A}}{\mu_{M}-\mu_{B}},$ (17)

using trivial algebra, and removing for the sake of simplicity the dependence
from $\mu_{M}$, the parametric curve $\gamma$ is described by hyperbola

$x^{2}-y^{2}+\alpha_{1}(x-y)-\alpha_{2}=0,$

where

$\alpha_{1}=r_{*}\left(\frac{\mu_{A}}{\mu_{B}}-1\right)\;\;\;\text{and}\;\;\;\alpha_{2}=\frac{r_{*}^{2}\mu_{A}}{\mu_{B}}.$

We restrict this curve to those points that satisfy
$\mu_{M}\geq\mu_{A}+\mu_{B}$. This condition, together with formula (17),
implies that

$0<x(\mu_{M})-y(\mu_{M})\leq r_{\ast}.$

Therefore, the set of points in the plane that represent the exchange ratios
through which stockholders of both parties share an improvement in their
expected wealth is

$\\{(x,y)\in\mathbb{R}^{2}\mid\,x^{2}-y^{2}+\alpha_{1}(x-y)-\alpha_{2}=0\;\text{and}\;x-r_{\ast}\leq
y<x\\}.$

The condition $y<x$ is strict because $y=x$ is an asymptote of the hyperbola

A graphical depiction of these curves is provided in Figure 2.

Figure 2: Kulpa’s diagrammatic representation of an hyperbola depicting a
$\mathcal{BR}$ \- $p_{A}=4$, $p_{B}=2$, $N_{A}=20$, $N_{B}=10$,
$r_{\ast}=0.5$, $V_{1}\equiv\left(-1.75;-0.75\right)$,
$V_{2}\equiv\left(0.25;-0.75\right)$, case $\mu_{A}>\mu_{B}$. - The relevant
portion of the hyperbola is the one that starts from $r_{\ast}$ and belongs to
the first quadrant. The equation of the downward sloping asymptote is
$y=-x-1.5$.

Along the same lines, using the fact that

$x(\rho_{M})-y(\rho_{M})=\frac{r_{\ast\ast}(\rho_{M}-\rho_{A})}{\rho_{B}},$

the parametric curve $\delta$ is described by

$x^{2}-y^{2}+\beta_{1}(x+y)-\beta_{2}=0,$

where

$\beta_{1}=r_{**}\left(\frac{\rho_{A}}{\rho_{B}}-1\right)\;\;\;\text{and}\;\;\;\beta_{2}=\frac{r_{**}^{2}\rho_{A}}{\rho_{B}}.$

Again, we restrict the curve to the points satisfying
$\max\left\\{\rho_{A},\,\rho_{B}\right\\}<\rho_{M}\leq\rho_{A}+\rho_{B}$. This
condition yields

$0<x(\rho_{M})-y(\rho_{M})\leq r_{\ast\ast}.$

Therefore, the set of points that represent the exchange ratios on which
shareholders agree because of the associated reduction of risk is

$\\{(x,y)\in\mathbb{R}^{2}\mid\,x^{2}-y^{2}+\beta_{1}(x+y)-\beta_{2}=0\;\text{and}\;x-r_{\ast\ast}\leq
y<x\\}.$

###  3.2 The bargaining region under Kulpa’s perspective

We choose now an admissible level of post-merger risk,
$\max\\{\rho_{A},\rho_{B}\\}<\hat{\rho}\leq\rho_{A}+\rho_{B}$, or, in other
words, we fix a point $P_{\rho}(\hat{\rho})$ along the curve $\delta$, which
represents all the exchange ratios to which correspond the amount of risk
$\hat{\rho}$:

$P\left(\hat{\rho}\right)=\left(\frac{\underline{r}_{\rho}(\hat{\rho})+\overline{r}_{\rho}(\hat{\rho})}{2},\,\frac{\overline{r}_{\rho}(\hat{\rho})-\underline{r}_{\rho}(\hat{\rho})}{2}\right).$

We now try to address the following questions. Under which conditions does
there exist an interval of exchange ratios that makes an agreement among all
the shareholders acceptable? In such a case, what is the shape of this set? Is
it possible to describe its shape? To answer these questions, we consider the
points of intersection between the interval represented by
$P_{\rho}\left(\hat{\rho}\right)$, and the admissible intervals on which there
is an agreement with respect to the expected wealth, that is, those intervals
represented by the points $P_{\mu}\left(\mu_{M}\right)$ along the curve
$\gamma$, with $\mu_{M}\geq\mu_{A}+\mu_{B}$. Following Kulpa, the intersection
between intervals $[a,b]$ and $[c,d]$ is non-empty if

$\max\\{a,\,c\\}\leq\min\\{b,\,d\\}.$

In such a case, Kulpa’s representation of the intersection set is given by the
point $(x,y)$, where

$x=\frac{\max\\{a,\,c\\}+\min\\{b,\,d\\}}{2}\;\;\;\;\;\;\text{and}\;\;\;\;\;\;y=\frac{\min\\{b,\,d\\}-\max\\{a,\,c\\}}{2}.$

Therefore, according to Kulpa’s approach, the intersection between the
intervals represented by $P_{\rho}\left(\hat{\rho}\right)$ and
$P_{\mu}\left(\mu_{M}\right)$ is non-empty if the condition

$\max\left\\{\underline{r}_{\mu}(\mu_{M}),\,\underline{r}_{\rho}(\hat{\rho})\right\\}\leq\min\left\\{\overline{r}_{\mu}(\mu_{M}),\,\overline{r}_{\rho}(\hat{\rho})\right\\},$
(18)

is satisfied, and each point belonging to the bargaining region has
$x$-coordinate

$x(\mu_{M},\hat{\rho})=\frac{\max\left\\{\underline{r}_{\mu}(\mu_{M}),\,\underline{r}_{\rho}(\hat{\rho})\right\\}+\min\left\\{\overline{r}_{\mu}(\mu_{M}),\,\overline{r}_{\rho}(\hat{\rho})\right\\}}{2}$

and $y$-coordinate

$y(\mu_{M},\hat{\rho})=\frac{\min\left\\{\overline{r}_{\mu}(\mu_{M}),\,\overline{r}_{\rho}(\hat{\rho})\right\\}-\max\left\\{\underline{r}_{\mu}(\mu_{M}),\,\underline{r}_{\rho}(\hat{\rho})\right\\}}{2}.$

We now consider all the cases that may arise from condition (18). For each
possible scenario, we explicit what condition $\mu_{M}$ (respectively
$\rho_{M}$) has to satisfy, and describe the type of intersection accordingly
to Table 1. We refer to Appendix A.4 for the details.

Case 1: Consistency region $\mathcal{CR}_{B}\left(\mu_{M},\hat{\rho}\right)$

Assume that $\underline{r}_{\mu}(\mu_{M})\geq\underline{r}_{\rho}(\hat{\rho})$
and $\overline{r}_{\mu}(\mu_{M})\geq\overline{r}_{\rho}(\hat{\rho})$. This
situation is consistent, together with $\mu_{M}\geq\mu_{A}+\mu_{B}$, with the
condition

$\max\left\\{\dfrac{r_{\ast\ast}}{r_{\ast}}\cdot\dfrac{\mu_{B}\rho_{A}}{\hat{\rho}-\rho_{B}}+\mu_{A},\,\dfrac{r_{\ast}}{r_{\ast\ast}}\cdot\dfrac{\mu_{A}(\hat{\rho}-\rho_{B})}{\rho_{A}}+\mu_{B}\right\\}\leq\mu_{M}\leq\dfrac{r_{\ast}}{r_{\ast\ast}}\cdot\dfrac{\mu_{A}\rho_{B}}{\hat{\rho}-\rho_{A}}+\mu_{B}.$

The same condition, written in terms of expected synergy, becomes

$\displaystyle\max$
$\displaystyle\left\\{\mu_{B}\left(\dfrac{r_{\ast\ast}}{r_{\ast}}\cdot\dfrac{\rho_{A}}{\hat{\rho}-\rho_{B}}-1\right),\,\mu_{A}\left(\dfrac{r_{\ast}}{r_{\ast\ast}}\cdot\dfrac{\hat{\rho}-\rho_{B}}{\rho_{A}}-1\right)\right\\}\leq\mu_{M}-\mu_{A}-\mu_{B}$
$\displaystyle\leq\mu_{A}\left(\dfrac{r_{\ast}}{r_{\ast\ast}}\cdot\dfrac{\mu_{A}\rho_{B}}{\hat{\rho}-\rho_{A}}-1\right).$

The set of exchange ratios on which all shareholders agree, represented,
according to Kulpa, by the points

$\left(\frac{\underline{r}_{\mu}(\mu_{M})+\overline{r}_{\rho}(\hat{\rho})}{2},\,\frac{\overline{r}_{\rho}(\hat{\rho})-\underline{r}_{\mu}(\mu_{M})}{2}\right),$

lies on the straight line of equation

$y=-x+\frac{r_{\ast\ast}\rho_{A}}{\hat{\rho}-\rho_{B}},$

which corresponds to the region
$\mathcal{CR}_{B}\left(\mu_{M},\hat{\rho}\right)$ in Table 1.

The diagrammatic representation for this case is in Figure 3.

Figure 3: Kulpa’s diagrammatic representation of set $\mathcal{BR}_{\mu,\rho}$
\- $p_{A}=4$, $p_{B}=2$, $\varphi_{A}=4$, $\varphi_{b}=3$, $N_{A}=20$,
$N_{B}=10$, $r_{\ast}=0.5$, $r_{\ast\ast}=0.6667$, $\mu_{M}=120$,
$\rho_{M}=94$. The resulting interval $\mathcal{CR}_{B}$ is denoted with the
blue dot and segment.

Case 2: Bargaining region $\mathcal{BR}_{\mu}\left(\mu_{M}\right)$

Assume that $\underline{r}_{\mu}(\mu_{M})\geq\underline{r}_{\rho}(\hat{\rho})$
and $\overline{r}_{\mu}(\mu_{M})\leq\overline{r}_{\rho}(\hat{\rho})$. This
situation is consistent, together with $\mu_{M}\geq\mu_{A}+\mu_{B}$, with the
condition

$\mu_{A}+\mu_{B}\leq\mu_{M}\leq\min\left\\{\frac{r_{\ast}}{r_{\ast\ast}}\cdot\frac{\mu_{A}\rho_{B}}{\hat{\rho}-\rho_{A}}+\mu_{B},\,\frac{r_{\ast\ast}}{r_{\ast}}\cdot\frac{\mu_{B}\rho_{A}}{\hat{\rho}-\rho_{B}}+\mu_{A}\right\\}$

The same condition, written in terms of expected synergy, becomes

$0\leq\mu_{M}-\mu_{A}-\mu_{B}\leq\min\left\\{\mu_{A}\left(\frac{r_{\ast}}{r_{\ast\ast}}\cdot\frac{\rho_{B}}{\hat{\rho}-\rho_{A}}-1\right),\,\mu_{B}\left(\frac{r_{\ast\ast}}{r_{\ast}}\cdot\frac{\rho_{A}}{\hat{\rho}-\rho_{B}}-1\right)\right\\}.$

The set of exchange ratios on which all shareholders agree on lies on the
curve $\gamma$, and corresponds to the region
$\mathcal{BR}_{\mu}\left(\mu_{M}\right)$ in Table 1.

The diagrammatic representation for this case is in Figure 4.

Figure 4: Kulpa’s diagrammatic representation of set $\mathcal{BR}_{\mu,\rho}$
\- $p_{A}=4$, $p_{B}=2$, $\varphi_{A}=4$, $\varphi_{b}=3$, $N_{A}=20$,
$N_{B}=10$, $r_{\ast}=0.5$, $r_{\ast\ast}=0.6667$, $\mu_{M}=120$,
$\rho_{M}=86$. The resulting interval $\mathcal{BR}_{\mu}$ is denoted with the
blue dot and segment.

Case 3: Bargaining region $\mathcal{BR}_{\rho}\left(\hat{\rho}\right)$

Assume that $\underline{r}_{\mu}(\mu_{M})\leq\underline{r}_{\rho}(\hat{\rho})$
and $\overline{r}_{\mu}(\mu_{M})\geq\overline{r}_{\rho}(\hat{\rho})$. This
situation is consistent, together with $\mu_{M}\geq\mu_{A}+\mu_{B}$, with the
condition

$\mu_{M}\geq\max\left\\{\frac{r_{\ast}}{r_{\ast\ast}}\cdot\frac{\mu_{A}\rho_{B}}{\hat{\rho}-\rho_{A}}+\mu_{B},\,\frac{r_{\ast\ast}}{r_{\ast}}\cdot\frac{\mu_{B}\rho_{A}}{\hat{\rho}-\rho_{B}}+\mu_{A}\right\\}$

The same condition, written in terms of expected synergy, becomes

$\mu_{M}-\mu_{A}-\mu_{B}\geq\max\left\\{\mu_{A}\left(\frac{r_{\ast}}{r_{\ast\ast}}\cdot\frac{\rho_{B}}{\hat{\rho}-\rho_{A}}-1\right),\,\mu_{B}\left(\frac{r_{\ast\ast}}{r_{\ast}}\cdot\frac{\rho_{A}}{\hat{\rho}-\rho_{B}}-1\right)\right\\}.$

The (unique) interval of exchange ratios on which all shareholders agree on
corresponds to a point on the curve $\delta$, represented by

$\left(\frac{\underline{r}_{\rho}(\hat{\rho})+\overline{r}_{\rho}(\hat{\rho})}{2},\;\frac{\overline{r}_{\rho}(\hat{\rho})-\underline{r}_{\rho}(\hat{\rho})}{2}\right),$

and corresponds to the region $\mathcal{BR}_{\rho}\left(\hat{\rho}\right)$,
which shrinks to a singleton, in Table 1.

The diagrammatic representation for this case is in Figure 5.

Figure 5: Kulpa’s diagrammatic representation of set $\mathcal{BR}_{\mu,\rho}$
\- $p_{A}=4$, $p_{B}=2$, $\varphi_{A}=4$, $\varphi_{b}=3$, $N_{A}=20$,
$N_{B}=10$, $r_{\ast}=0.5$, $r_{\ast\ast}=0.6667$, $\mu_{M}=120$,
$\rho_{M}=98$. The resulting interval $\mathcal{BR}_{\rho}$ is denoted with
the blue dot and segment.

Case 4: Consistency region $\mathcal{CR}_{A}\left(\mu_{M},\hat{\rho}\right)$

Assume that $\underline{r}_{\mu}(\mu_{M})\leq\underline{r}_{\rho}(\hat{\rho})$
and $\overline{r}_{\mu}(\mu_{M})\leq\overline{r}_{\rho}(\hat{\rho})$. This
situation is consistent, together with $\mu_{M}\geq\mu_{A}+\mu_{B}$, with the
condition

$\max\left\\{\dfrac{r_{\ast}}{r_{\ast\ast}}\cdot\dfrac{\mu_{A}\rho_{B}}{\hat{\rho}-\rho_{A}}+\mu_{B},\,\dfrac{r_{\ast\ast}}{r_{\ast}}\cdot\dfrac{\mu_{B}(\hat{\rho}-\rho_{A})}{\rho_{B}}+\mu_{A}\right\\}\leq\mu_{M}\leq\dfrac{r_{\ast\ast}}{r_{\ast}}\cdot\dfrac{\mu_{B}\rho_{A}}{\hat{\rho}-\rho_{B}}+\mu_{A}.$

The same condition, written in terms of expected synergy, becomes

$\displaystyle\max$
$\displaystyle\left\\{\mu_{A}\left(\dfrac{r_{\ast}}{r_{\ast\ast}}\cdot\dfrac{\rho_{B}}{\hat{\rho}-\rho_{A}}-1\right),\,\mu_{B}\left(\dfrac{r_{\ast\ast}}{r_{\ast}}\cdot\dfrac{\hat{\rho}-\rho_{A}}{\rho_{B}}-1\right)\right\\}\leq\mu_{M}-\mu_{A}-\mu_{B}$
$\displaystyle\leq\mu_{B}\left(\dfrac{r_{\ast\ast}}{r_{\ast}}\cdot\dfrac{\rho_{A}}{\hat{\rho}-\rho_{B}}-1\right).$

The set of exchange ratios on which all shareholders agree, represented,
according to Kulpa, by the points

$\left(\frac{\underline{r}_{\rho}(\hat{\rho})+\overline{r}_{\mu}(\mu_{M})}{2},\,\frac{\overline{r}_{\mu}(\mu_{M})-\underline{r}_{\rho}(\hat{\rho})}{2}\right),$

lies on the straight line of equation

$y=x-\frac{r_{\ast\ast}(\hat{\rho}-\rho_{A})}{\rho_{B}},$

which corresponds to the region
$\mathcal{CR}_{A}\left(\mu_{M},\hat{\rho}\right)$ in Table 1.

If we fix instead $\hat{\mu}\geq\mu_{A}+\mu_{B}$, we can consider the point
$P_{\mu}(\hat{\mu})$, corresponding to the bargaining region
$\mathcal{BR}_{\mu}(\hat{\mu})$. Recall that the intersection of the sets is
not empty if (18) holds, then, again, this represents the starting point for
the analysis of the possible scenarios. The details can be found in Appendix
A.4.

The diagrammatic representation for this case is in Figure 6.

Figure 6: Kulpa’s diagrammatic representation of set $\mathcal{BR}_{\mu,\rho}$
\- $p_{A}=4$, $p_{B}=2$, $\varphi_{A}=4$, $\varphi_{b}=3$, $N_{A}=20$,
$N_{B}=10$, $r_{\ast}=0.5$, $r_{\ast\ast}=0.6667$, $\mu_{M}=112$,
$\rho_{M}=99$. The resulting interval $\mathcal{CR}_{A}$ is denoted with the
blue dot and segment.

Case 1: Consistency region $\mathcal{CR}_{B}\left(\hat{\mu},\rho_{M}\right)$

Assume that $\underline{r}_{\mu}(\hat{\mu})\geq\underline{r}_{\rho}(\rho_{M})$
and $\overline{r}_{\mu}(\hat{\mu})\geq\overline{r}_{\rho}(\rho_{M})$. This
situation is consistent, together with
$\max\\{\rho_{A},\rho_{B}\\}<\rho_{M}\leq\rho_{A}+\rho_{B}$, with the
condition

$\frac{r_{\ast\ast}}{r_{\ast}}\cdot\frac{\mu_{B}\rho_{A}}{\hat{\mu}-\mu_{A}}+\rho_{B}\leq\rho_{M}\leq\min\left\\{\frac{r_{\ast}}{r_{\ast\ast}}\cdot\frac{\mu_{A}\rho_{B}}{\hat{\mu}-\mu_{B}}+\rho_{A},\frac{r_{\ast\ast}}{r_{\ast}}\cdot\frac{(\hat{\mu}-\mu_{B})\rho_{A}}{\mu_{A}}+\rho_{B}\right\\}.$

The same condition, written in terms of reduction of the overall risk, becomes

$\displaystyle\max$
$\displaystyle\left\\{\rho_{B}\left(1-\frac{r_{\ast}}{r_{\ast\ast}}\cdot\frac{\mu_{A}}{\hat{\mu}-\mu_{B}}\right),\,\rho_{A}\left(1-\frac{r_{\ast\ast}}{r_{\ast}}\cdot\frac{\hat{\mu}-\mu_{B}}{\mu_{A}}\right)\right\\}\leq\rho_{A}+\rho_{B}-\rho_{M}$
$\displaystyle\leq\rho_{A}\left(1-\frac{r_{\ast\ast}}{r_{\ast}}\cdot\frac{\mu_{B}}{\hat{\mu}-\mu_{A}}\right).$

The set of exchange ratios on which all shareholders agree, represented,
according to Kulpa, by the points

$\left(\frac{\underline{r}_{\mu}(\hat{\mu})+\overline{r}_{\rho}(\rho_{M})}{2},\,\frac{\overline{r}_{\rho}(\rho_{M})-\underline{r}_{\mu}(\hat{\mu})}{2}\right),$

lies on the straight line of equation

$y=x-\frac{r_{\ast}\mu_{A}}{\hat{\mu}-\mu_{B}},$

which corresponds to the region
$\mathcal{CR}_{B}\left(\hat{\mu},\rho_{M}\right)$ in Table 1.

Case 2: Bargaining region $\mathcal{BR}_{\mu}\left(\hat{\mu}\right)$

Assume that $\underline{r}_{\mu}(\hat{\mu})\geq\underline{r}_{\rho}(\rho_{M})$
and $\overline{r}_{\mu}(\hat{\mu})\leq\overline{r}_{\rho}(\rho_{M})$. This
situation is consistent, together with
$\max\\{\rho_{A},\rho_{B}\\}<\rho_{M}\leq\rho_{A}+\rho_{B}$, with the
condition

$\max\\{\rho_{A},\rho_{B}\\}\leq\rho_{M}\leq\min\left\\{\dfrac{r_{\ast}}{r_{\ast\ast}}\cdot\dfrac{\rho_{B}\mu_{A}}{\hat{\mu}-\mu_{B}}+\rho_{A},\,\dfrac{r_{\ast\ast}}{r_{\ast}}\cdot\dfrac{\rho_{A}\mu_{B}}{\hat{\mu}-\mu_{A}}+\rho_{B}\right\\}.$

The same condition, written in terms of overall risk, becomes

$\max\left\\{\rho_{B}\left(1-\dfrac{r_{\ast}}{r_{\ast\ast}}\cdot\dfrac{\rho_{B}\mu_{A}}{\hat{\mu}-\mu_{B}}\right),\,\rho_{A}\left(1-\dfrac{r_{\ast\ast}}{r_{\ast}}\cdot\dfrac{\rho_{A}\mu_{B}}{\hat{\mu}-\mu_{A}}\right)\right\\}\leq\rho_{A}+\rho_{B}-\rho_{M}<\min\\{\rho_{A},\rho_{B}\\}.$

The (unique) interval of exchange ratios on which all shareholders agree on
corresponds to the point on the curve $\gamma$

$\left(\frac{\underline{r}_{\mu}(\hat{\mu})+\overline{r}_{\mu}(\hat{\mu})}{2},\;\frac{\overline{r}_{\mu}(\hat{\mu})-\underline{r}_{\mu}(\hat{\mu})}{2}\right),$

representing the region $\mathcal{BR}_{\mu}\left(\hat{\mu}\right)$ in Table 1.

Case 3: Bargaining region $\mathcal{BR}_{\rho}\left(\rho_{M}\right)$

Assume that $\underline{r}_{\mu}(\hat{\mu})\leq\underline{r}_{\rho}(\rho_{M})$
and $\overline{r}_{\mu}(\hat{\mu})\geq\overline{r}_{\rho}(\rho_{M})$. This
situation is consistent, together with
$\max\\{\rho_{A},\rho_{B}\\}<\rho_{M}\leq\rho_{A}+\rho_{B}$, with the
condition

$\rho_{M}\geq\max\left\\{\dfrac{r_{\ast}}{r_{\ast\ast}}\cdot\dfrac{\rho_{B}\mu_{A}}{\hat{\mu}-\mu_{B}}+\rho_{A},\,\dfrac{r_{\ast\ast}}{r_{\ast}}\cdot\dfrac{\rho_{A}\mu_{B}}{\hat{\mu}-\mu_{A}}+\rho_{B}\right\\}.$

The same condition, written in terms of overall risk, becomes

$\rho_{A}+\rho_{B}-\rho_{M}\leq\min\left\\{\rho_{B}\left(1-\dfrac{r_{\ast}}{r_{\ast\ast}}\cdot\dfrac{\rho_{B}\mu_{A}}{\hat{\mu}-\mu_{B}}\right),\,\rho_{A}\left(1-\dfrac{r_{\ast\ast}}{r_{\ast}}\cdot\dfrac{\rho_{A}\mu_{B}}{\hat{\mu}-\mu_{A}}\right)\right\\}.$

The set of exchange ratios on which all shareholders agree on lies on the
curve $\delta$, and corresponds to the region
$\mathcal{BR}_{\rho}\left(\rho_{M}\right)$ in Table 1.

Case 4: Consistency region $\mathcal{CR}_{A}\left(\hat{\mu},\rho_{M}\right)$

Assume that $\underline{r}_{\mu}(\hat{\mu})\leq\underline{r}_{\rho}(\rho_{M})$
and $\overline{r}_{\mu}(\hat{\mu})\leq\overline{r}_{\rho}(\rho_{M})$. This
situation is consistent, together with
$\max\\{\rho_{A},\rho_{B}\\}<\rho_{M}\leq\rho_{A}+\rho_{B}$, with the
condition

$\frac{r_{\ast}}{r_{\ast\ast}}\cdot\frac{\rho_{B}\mu_{A}}{\hat{\mu}-\mu_{B}}+\rho_{A}\leq\rho_{M}\leq\min\left\\{\frac{r_{\ast\ast}}{r_{\ast}}\cdot\frac{\rho_{A}\mu_{B}}{\hat{\mu}-\mu_{A}}+\rho_{B},\frac{r_{\ast}}{r_{\ast\ast}}\cdot\frac{\rho_{B}(\hat{\mu}-\mu_{A})}{\mu_{B}}+\rho_{A}\right\\}.$

The same condition, written in terms of reduction of the overall risk, becomes

$\displaystyle\max$
$\displaystyle\left\\{\rho_{A}\left(1-\frac{r_{\ast\ast}}{r_{\ast}}\cdot\frac{\mu_{B}}{\hat{\mu}-\mu_{A}}\right),\,\rho_{B}\left(1-\frac{r_{\ast}}{r_{\ast\ast}}\cdot\frac{\hat{\mu}-\mu_{A}}{\mu_{B}}\right)\right\\}\leq\rho_{A}+\rho_{B}-\rho_{M}$
$\displaystyle\leq\rho_{B}\left(1-\frac{r_{\ast}}{r_{\ast\ast}}\cdot\frac{\mu_{A}}{\hat{\mu}-\mu_{B}}\right).$

The set of exchange ratios on which all shareholders agree, represented,
according to Kulpa, by the points

$\left(\frac{\underline{r}_{\rho}(\rho_{M})+\overline{r}_{\mu}(\hat{\mu})}{2},\,\frac{\overline{r}_{\mu}(\hat{\mu})-\underline{r}_{\rho}(\rho_{M})}{2}\right),$

lies on the straight line of equation

$y=-x+\frac{r_{\ast}(\hat{\mu}-\mu_{A})}{\mu_{B}},$

which corresponds to the region
$\mathcal{CR}_{A}\left(\hat{\mu},\rho_{M}\right)$ in Table 1.

Two more cases are displayed in this Table: in both, sets
$\mathcal{BR}_{\mu,\rho}\left(\mu,\rho\right)$ is empty. Figure 7 shows one of
them. As it is clear from the above presentation, this set is empty due to the
limited increase in expected equity value and/or limited reduction in overall
risk.

The last case resembles the one just presented, with the only difference that
$r_{\ast\ast}<r_{\ast}$. For sake of paucity, its diagrammatic representation
is omitted.

Figure 7: Kulpa’s diagrammatic representation of set $\mathcal{BR}_{\mu,\rho}$
\- $p_{A}=4$, $p_{B}=2$, $\varphi_{A}=4$, $\varphi_{b}=3$, $N_{A}=20$,
$N_{B}=10$, $r_{\ast}=0.5$, $r_{\ast\ast}=0.6667$, $\mu_{M}=104$,
$\rho_{M}=106$. The resulting set is empty.

##  4 Conclusions

In this article, we present an attempt to extend, in a stochastic setting,
deterministic models for exchange ratios determination for merger agreements.

Under a financial point of view, an important result we achieve is that the
introduction of a risk measure changes the attitude shareholders of the
merging companies have toward the agreed exchange ratio. For shareholders of
the acquiring company, smaller exchange ratios result in both larger expected
equity value and equity risk. For shareholders of the acquired company, larger
exchange ratio yield both a larger expected equity value and equity risk. This
introduces a trade-off that makes the negotiation for settling on the
definitive exchange ratio more fitting to a risky environment.

As far as the bargaining region is concerned, being here dependent on two
quantities, its effective representation is obtained exploiting the Kulpa’s
diagrammatic technique as it transforms bounded intervals into points in a bi-
dimensional plane and allows to plainly represent all possible cases.

Even if it can be argued that, in real financial markets, imposing that a
merger creates both a positive expected synergy and a reduction in the overall
equity value risk rarely occurs, our contribution can be intended as a
theoretical framework under which mergers can be analyzed, a result that lacks
in financial literature.

To conclude, results presented here must be intended as a first step further
research development in this field. For instance, ranges of acceptable
exchange ratios might be determined following an expected utility approach,
where a trade-off between expected value and risk of an investor’s wealth can
be established.

## Acknowledgments

The idea of exploiting the diagrammatic approach was suggested by Nicolae
Popovici, wonderful friend, excellent researcher, and highly esteemed
professor at the Department of Mathematics of the Faculty of Mathematics and
Computer Science, Babeș-Bolyai University in Cluj-Napoca, Romania. Sadly,
Nicolae suddenly passed away in June 2022. Without Nicolae’s help, this
article would lack its best part.

La revedere, Nicolae; și mulțumesc pentru tot!

##  Appendix A Appendices

###  A.1 Appendix 1

To justify the claim about Company $B$’s stockholders in subsection 2.2,
consider at first what happens in the left-hand side of (8) if $r=0$: company
$B$’s shareholders receive no stock, carry no risk, and experience a reduction
in their riskiness.

As

$\frac{\partial}{\partial
r}\left[\frac{r\cdot\rho_{M}}{N_{A}+rN_{B}}\right]=\frac{\rho_{M}N_{A}}{(N_{A}+rN_{B})^{2}}>0\text{,}$

the left-hand side of expression (8) is strictly increasing with respect to
$r\geq 0$ so that the larger $r$, the larger the risk for $B$’s stockholders.

Finally, being, in accordance with (10), $\rho_{M}>\rho_{B}$,

$\lim_{r\to+\infty}\frac{r\cdot\rho_{M}}{N_{A}+rN_{B}}=\frac{\rho_{M}}{N_{B}}>\frac{\rho_{B}}{N_{B}}\text{.}$

Then, it results that sufficiently large values for $r$ bring about larger
risk to company $B$’s stockholders when compared to the risk level they bore
before companies merged.

###  A.2 Appendix 2

Consider point $P_{\mu}(\mu_{M})$ in (15). The difference between its
coordinates is

$x(\mu_{M})-y(\mu_{M})=\frac{r_{\ast}\mu_{A}}{\mu_{M}-\mu_{B}},$ (19)

from which (removing the dependence from $\mu_{M}$ for the sake of simplicity)
we get

$\mu_{M}=\frac{r_{\ast}\mu_{A}}{x-y}+\mu_{B}.$

Recalling that

$\displaystyle x$
$\displaystyle=\frac{r_{\ast}}{2}\left(\frac{\mu_{A}}{\mu_{M}-\mu_{B}}+\frac{\mu_{M}-\mu_{A}}{\mu_{B}}\right)$
$\displaystyle y$
$\displaystyle=\frac{r_{\ast}}{2}\left(\frac{\mu_{M}-\mu_{A}}{\mu_{B}}-\frac{\mu_{A}}{\mu_{M}-\mu_{B}}\right)$

and replacing the expression of $\mu_{M}$ found above, we obtain

$x+y=r_{\ast}\left(\frac{r_{\ast}\mu_{A}}{\mu_{B}}\cdot\frac{1}{x-y}+1-\frac{\mu_{A}}{\mu_{B}}\right)$

and multiplying by $x-y$ the following equation is obtained

$x^{2}-y^{2}=r_{\ast}\left(1-\frac{\mu_{A}}{\mu_{B}}\right)(x-y)+r_{\ast}^{2}\frac{\mu_{A}}{\mu_{B}},$

so that $\gamma$ is given by

$x^{2}-y^{2}-r_{\ast}\left(1-\frac{\mu_{A}}{\mu_{B}}\right)(x-y)-r_{\ast}^{2}\frac{\mu_{A}}{\mu_{B}}=0.$
(20)

This is the equation of the quadratic curve

$c_{1}x^{2}+c_{2}xy+c_{3}y^{2}+c_{4}x+c_{5}y+c_{6}=0$

with $c_{1}=1$, $c_{2}=0$, $c_{3}=-1$,
$c_{4}=-r_{\ast}\left(\mu_{B}-\mu_{A}\right)/\mu_{B}$,
$c_{5}=r_{\ast}\left(\mu_{B}-\mu_{A}\right)/\mu_{B}$, and
$c_{6}=-r_{\ast}^{2}\mu_{A}/\mu_{B}$.

As $c_{2}^{2}-4c_{1}c_{3}=4>0$, equation (20) identifies a hyperbola whose
eccentricity is $\sqrt{2}$ and that can be rewritten as

$\frac{\left(x-r_{\ast}\frac{\mu_{B}-\mu_{A}}{2\mu_{B}}\right)^{2}}{r^{2}_{\ast}\frac{\mu_{A}}{\mu_{B}}}-\frac{\left(y-r_{\ast}\frac{\mu_{B}-\mu_{A}}{2\mu_{B}}\right)^{2}}{r^{2}_{\ast}\frac{\mu_{A}}{\mu_{B}}}=1$
(21)

Its two vertices are

$V_{1}\equiv\left(\frac{r_{\ast}\left(\mu_{B}-\mu_{A}\right)}{2\mu_{B}}-r_{\ast}\sqrt{\frac{\mu_{A}}{\mu_{B}}},\frac{r_{\ast}\left(\mu_{B}-\mu_{A}\right)}{2\mu_{B}}\right)\text{,}$

and

$V_{2}\equiv\left(\frac{r_{\ast}\left(\mu_{B}-\mu_{A}\right)}{2\mu_{B}}+r_{\ast}\sqrt{\frac{\mu_{A}}{\mu_{B}}},\frac{r_{\ast}\left(\mu_{B}-\mu_{A}\right)}{2\mu_{B}}\right)$

The abscissa of $V_{1}$ is smaller then the abscissa of $V_{2}$. In terms of
diagrammatic representation (see Figure 2), the relevant branch of hyperbola
(21) is the one whose vertex is $V_{2}$. This is the case as there exist
points on the hyperbola’s branch that contains $V_{1}$ with negative
abscissas, a condition that is not compatible with the fact that points
belonging to $P_{\mu}\left(\mu_{M}\right)$ must have positive $x-$values.

Focusing on $V_{2}$, its abscissa is surely positive if $\mu_{B}>\mu_{A}$. If,
instead, $\mu_{B}<\mu_{A}$ then the abscissa of $V_{2}$ is positive as long as
$\mu_{B}\geq\left(3-2\sqrt{2}\right)\mu_{A}$.

As far as the ordinate of $V_{2}$ is concerned, it is positive (respectively
negative) when $\mu_{B}>\mu_{A}$ $\left(\text{respectively
}\mu_{B}<\mu_{A}\right)$.

The asymptotes of (21) are

$y=x,\qquad y=-x+\frac{r_{\ast}\left(\mu_{B}-\mu_{A}\right)}{\mu_{B}}.$

Similarly, consider point $P_{\rho}(\rho_{M})$ in (16). The sum of its
coordinates is

$x(\rho_{M})+y(\rho_{M})=r_{\ast\ast}\frac{\rho_{A}}{\rho_{M}-\rho_{B}},$

from which (removing the dependence from $\rho_{M}$ for the sake of
simplicity) we get

$\rho_{M}=\frac{r_{\ast\ast}\rho_{A}}{x+y}+\rho_{B}.$

Recalling that

$\displaystyle x$
$\displaystyle=\frac{r_{\ast\ast}}{2}\left(\frac{\rho_{M}-\rho_{A}}{\rho_{B}}+\frac{\rho_{A}}{\rho_{M}-\rho_{B}}\right)$
$\displaystyle y$
$\displaystyle=\frac{r_{\ast\ast}}{2}\left(\frac{\rho_{A}}{\rho_{M}-\rho_{B}}-\frac{\rho_{M}-\rho_{A}}{\rho_{B}}\right)$

and replacing the expression of $\rho_{M}$ found above, we obtain

$x-y=r_{\ast\ast}\left(\frac{r_{\ast\ast}\rho_{A}}{\rho_{B}}\cdot\frac{1}{x+y}+1-\frac{\rho_{A}}{\rho_{B}}\right)$

and multiplying by $x+y$ the following equation is obtained

$x^{2}-y^{2}=r_{\ast\ast}\left(1-\frac{\rho_{A}}{\rho_{B}}\right)(x+y)+r_{\ast\ast}^{2}\frac{\rho_{A}}{\rho_{B}},$

so that $\delta$ is given by

$x^{2}-y^{2}-r_{\ast\ast}\left(1-\frac{\rho_{A}}{\rho_{B}}\right)(x+y)-r_{\ast\ast}^{2}\frac{\rho_{A}}{\rho_{B}}=0,$
(22)

which is, again, a hyperbola with equation

$\frac{\left(x-r_{\ast\ast}\frac{\rho_{B}-\rho_{A}}{2\rho_{B}}\right)^{2}}{r^{2}_{\ast\ast}\frac{\rho_{A}}{\rho_{B}}}-\frac{\left(y+r_{\ast\ast}\frac{\rho_{B}-\rho_{A}}{2\rho_{B}}\right)^{2}}{r^{2}_{\ast\ast}\frac{\rho_{A}}{\rho_{B}}}=1.$

Its vertices are

$V_{3}\equiv\left(\frac{r_{\ast\ast}\left(\rho_{B}-\rho_{A}\right)}{2\rho_{B}}-r_{\ast\ast}\sqrt{\frac{\rho_{A}}{\rho_{B}}},\frac{r_{\ast\ast}\left(\rho_{B}-\rho_{A}\right)}{2\rho_{B}}\right)$

and

$V_{4}\equiv\left(\frac{r_{\ast\ast}\left(\rho_{B}-\rho_{A}\right)}{2\rho_{B}}+r_{\ast\ast}\sqrt{\frac{\rho_{A}}{\rho_{B}}},\frac{r_{\ast\ast}\left(\rho_{B}-\rho_{A}\right)}{2\rho_{B}}\right)$

while the asymptotes are

$y=-x,\quad y=x-\frac{r_{\ast\ast}\left(\rho_{B}-\rho_{A}\right)}{\rho_{B}}.$

The abscissa of $V_{3}$ is smaller then the abscissa of $V_{4}$. In terms of
diagrammatic representation (see Figure 2), the relevant branch of hyperbola
(22) is the one whose vertex is $V_{4}$. This is the case as there exist
points on the hyperbola’s branch that contains $V_{3}$ with negative
abscissas, a condition that is not compatible with the fact that points
belonging to $P_{\rho}\left(\rho_{M}\right)$ must have positive $x-$values.

Focusing on $V_{4}$, its abscissa is surely positive if $\rho_{B}>\rho_{A}$.
If, instead, $\rho_{B}<\rho_{A}$ then the abscissa of $V_{4}$ is positive as
long as $\rho_{B}\geq\left(3-2\sqrt{2}\right)\rho_{A}$.

###  A.3 Appendix 3

In terms of risk-adjusted performance, the resulting company performs better
then the pre-existing ones when

$\lambda_{M}=\frac{\mu_{A}+\mu_{B}+s}{\rho_{A}+\rho_{B}-v}\geq\frac{\mu_{A}}{\rho_{A}}=\lambda_{A}\text{,}$

and

$\lambda_{M}=\frac{\mu_{A}+\mu_{B}+s}{\rho_{A}+\rho_{B}-v}\geq\frac{\mu_{B}}{\rho_{B}}=\lambda_{B}\text{.}$

Recalling the notation for expected synergy and risk reduction introduced in
Subsection (2.2), these inequalities are respectively equivalent to

$\frac{\mu_{B}+s}{\rho_{B}-v}\geq\lambda_{A}\text{,}$

and

$\frac{\mu_{A}+s}{\rho_{A}-v}\geq\lambda_{B}\text{.}$

If, now, $s=0$ and $v=0$, the first inequality becomes

$\frac{\mu_{B}}{\rho_{B}}=\lambda_{B}\geq\lambda_{A}$

while the second reads

$\frac{\mu_{A}}{\rho_{A}}=\lambda_{A}\geq\lambda_{B}\text{.}$

These expressions explain why, in the absence of any improvement in terms of
either positive expected synergy or risk reduction, stockholders of company
$A$ $\left(\text{respectively }B\right)$ experience an improvement in the
risk-corrected performance of the shares they own if company $B$
$\left(\text{respectively }A\right)$ better behaves, in terms of risk-
corrected performance.

###  A.4 Appendix 4

Consider $\max\\{\rho_{A},\rho_{B}\\}<\hat{\rho}\leq\rho_{A}+\rho_{B}$. In
Case 1, the intersection is the set
$\mathcal{CR}_{B}\left(\mu_{M},\hat{\rho}\right)=\left[\underline{r}_{\mu}(\mu_{M}),\overline{r}_{\rho}(\hat{\rho})\right]$.
This interval is represented, under Kulpa’s approach, by the point

$(x,y)=\left(\frac{\underline{r}_{\mu}(\mu_{M})+\overline{r}_{\rho}(\hat{\rho})}{2},\,\frac{\overline{r}_{\rho}(\hat{\rho})-\underline{r}_{\mu}(\mu_{M})}{2}\right),$

that is,

$x=\frac{1}{2}\left(\frac{r_{\ast}\mu_{A}}{\mu_{M}-\mu_{B}}+\frac{r_{\ast\ast}\rho_{A}}{\hat{\rho}-\rho_{B}}\right),$
$y=\frac{1}{2}\left(\frac{r_{\ast\ast}\rho_{A}}{\hat{\rho}-\rho_{B}}-\frac{r_{\ast}\mu_{A}}{\mu_{M}-\mu_{B}}\right).$

The sum of its coordinates is

$x+y=\frac{r_{\ast\ast}\rho_{A}}{\hat{\rho}-\rho_{B}};$

that is, on varying $\mu_{M}$ we move along the straight line of equation

$y=-x+\frac{r_{\ast\ast}\rho_{A}}{\hat{\rho}-\rho_{B}}.$

The conditions under which this situation is possible are

$\left\\{\begin{array}[]{@{}l@{}}\vskip 3.0pt plus 1.0pt minus
1.0pt\underline{r}_{\mu}(\mu_{M})\geq\underline{r}_{\rho}(\hat{\rho})\\\
\vskip 3.0pt plus 1.0pt minus
1.0pt\overline{r}_{\mu}(\mu_{M})\geq\overline{r}_{\rho}(\hat{\rho})\\\
\underline{r}_{\mu}(\mu_{M})\leq\overline{r}_{\rho}(\hat{\rho}),\end{array}\right.$

that is,

$\left\\{\begin{array}[]{@{}l@{}}\vskip 6.0pt plus 2.0pt minus
2.0pt\dfrac{r_{\ast}\mu_{A}}{\mu_{M}-\mu_{B}}\geq\dfrac{r_{\ast\ast}(\hat{\rho}-\rho_{A})}{\rho_{B}}\\\
\vskip 6.0pt plus 2.0pt minus
2.0pt\dfrac{r_{\ast}(\mu_{M}-\mu_{A})}{\mu_{B}}\geq\dfrac{r_{\ast\ast}\rho_{A}}{\hat{\rho}-\rho_{B}}\\\
\dfrac{r_{\ast}\mu_{A}}{\mu_{M}-\mu_{B}}\leq\dfrac{r_{\ast\ast}\rho_{A}}{\hat{\rho}-\rho_{B}},\end{array}\right.\;\;\;\;\;\text{that
is,}\;\;\;\;\;\left\\{\begin{array}[]{@{}l@{}}\vskip 6.0pt plus 2.0pt minus
2.0pt\mu_{M}\leq\dfrac{r_{\ast}}{r_{\ast\ast}}\cdot\dfrac{\mu_{A}\rho_{B}}{\hat{\rho}-\rho_{A}}+\mu_{B}\\\
\vskip 6.0pt plus 2.0pt minus
2.0pt\mu_{M}\geq\dfrac{r_{\ast\ast}}{r_{\ast}}\cdot\dfrac{\mu_{B}\rho_{A}}{\hat{\rho}-\rho_{B}}+\mu_{A}\\\
\mu_{M}\geq\dfrac{r_{\ast}}{r_{\ast\ast}}\cdot\dfrac{\mu_{A}(\hat{\rho}-\rho_{B})}{\rho_{A}}+\mu_{B},\end{array}\right.$

which can be summarized as follows:

$\max\left\\{\dfrac{r_{\ast\ast}}{r_{\ast}}\cdot\dfrac{\mu_{B}\rho_{A}}{\hat{\rho}-\rho_{B}}+\mu_{A},\,\dfrac{r_{\ast}}{r_{\ast\ast}}\cdot\dfrac{\mu_{A}(\hat{\rho}-\rho_{B})}{\rho_{A}}+\mu_{B}\right\\}\leq\mu_{M}\leq\dfrac{r_{\ast}}{r_{\ast\ast}}\cdot\dfrac{\mu_{A}\rho_{B}}{\hat{\rho}-\rho_{A}}+\mu_{B}.$

To obtain the condition about the expected synergy, we subtract
$\mu_{A}+\mu_{B}$:

$\max\left\\{\dfrac{r_{\ast\ast}}{r_{\ast}}\cdot\dfrac{\mu_{B}\rho_{A}}{\hat{\rho}-\rho_{B}}-\mu_{B},\,\dfrac{r_{\ast}}{r_{\ast\ast}}\cdot\dfrac{\mu_{A}(\hat{\rho}-\rho_{B})}{\rho_{A}}-\mu_{A}\right\\}\leq\mu_{M}-\mu_{A}-\mu_{B}\leq\dfrac{r_{\ast}}{r_{\ast\ast}}\cdot\dfrac{\mu_{A}\rho_{B}}{\hat{\rho}-\rho_{A}}-\mu_{A},$

that is,

$\displaystyle\max$
$\displaystyle\left\\{\mu_{B}\left(\dfrac{r_{\ast\ast}}{r_{\ast}}\cdot\dfrac{\rho_{A}}{\hat{\rho}-\rho_{B}}-1\right),\,\mu_{A}\left(\dfrac{r_{\ast}}{r_{\ast\ast}}\cdot\dfrac{\hat{\rho}-\rho_{B}}{\rho_{A}}-1\right)\right\\}\leq\mu_{M}-\mu_{A}-\mu_{B}$
$\displaystyle\leq\mu_{A}\left(\dfrac{r_{\ast}}{r_{\ast\ast}}\cdot\dfrac{\mu_{A}\rho_{B}}{\hat{\rho}-\rho_{A}}-1\right).$

In Case 2, the intersection is the interval
$\mathcal{BR}_{\mu}\left(\mu_{M}\right)=\left[\underline{r}_{\mu}(\mu_{M}),\overline{r}_{\mu}(\mu_{M})\right]$.
This is represented, under Kulpa’s approach, by the point

$(x,y)=\left(\frac{\underline{r}_{\mu}(\mu_{M})+\overline{r}_{\mu}(\mu_{M})}{2},\,\frac{\overline{r}_{\mu}(\mu_{M})-\underline{r}_{\mu}(\mu_{M})}{2}\right),$

which, varying $\mu_{M}$, leads to the curve $\gamma$ (see A.2). The
conditions under which this situation is possible are

$\left\\{\begin{array}[]{@{}l@{}}\vskip 3.0pt plus 1.0pt minus
1.0pt\underline{r}_{\mu}(\mu_{M})\geq\underline{r}_{\rho}(\hat{\rho})\\\
\overline{r}_{\mu}(\mu_{M})\leq\overline{r}_{\rho}(\hat{\rho}),\end{array}\right.$

that is,

$\left\\{\begin{array}[]{@{}l@{}}\vskip 6.0pt plus 2.0pt minus
2.0pt\dfrac{r_{\ast}\mu_{A}}{\mu_{M}-\mu_{B}}\geq\dfrac{r_{\ast\ast}(\hat{\rho}-\rho_{A})}{\rho_{B}}\\\
\dfrac{r_{\ast}(\mu_{M}-\mu_{A})}{\mu_{B}}\leq\dfrac{r_{\ast\ast}\rho_{A}}{\hat{\rho}-\rho_{B}},\end{array}\right.\;\;\;\;\;\text{that
is,}\;\;\;\;\;\left\\{\begin{array}[]{@{}l@{}}\vskip 6.0pt plus 2.0pt minus
2.0pt\mu_{M}\leq\dfrac{r_{\ast}}{r_{\ast\ast}}\cdot\dfrac{\mu_{A}\rho_{B}}{\hat{\rho}-\rho_{A}}+\mu_{B}\\\
\mu_{M}\leq\dfrac{r_{\ast\ast}}{r_{\ast}}\cdot\dfrac{\mu_{B}\rho_{A}}{\hat{\rho}-\rho_{B}}+\mu_{A},\end{array}\right.$

which can be summarized as follows:

$\mu_{A}+\mu_{B}\leq\mu_{M}\leq\min\left\\{\dfrac{r_{\ast}}{r_{\ast\ast}}\cdot\dfrac{\mu_{A}\rho_{B}}{\hat{\rho}-\rho_{A}}+\mu_{B},\,\dfrac{r_{\ast\ast}}{r_{\ast}}\cdot\dfrac{\mu_{B}\rho_{A}}{\hat{\rho}-\rho_{B}}+\mu_{A}\right\\}.$

To obtain the condition about the expected synergy, we subtract
$\mu_{A}+\mu_{B}$:

$0\leq\mu_{M}-\mu_{A}-\mu_{B}\leq\min\left\\{\dfrac{r_{\ast}}{r_{\ast\ast}}\cdot\dfrac{\mu_{A}\rho_{B}}{\hat{\rho}-\rho_{A}}-\mu_{A},\,\dfrac{r_{\ast\ast}}{r_{\ast}}\cdot\dfrac{\mu_{B}\rho_{A}}{\hat{\rho}-\rho_{B}}-\mu_{B}\right\\},$

that is,

$0\leq\mu_{M}-\mu_{A}-\mu_{B}\leq\min\left\\{\mu_{A}\left(\dfrac{r_{\ast}}{r_{\ast\ast}}\cdot\dfrac{\rho_{B}}{\hat{\rho}-\rho_{A}}-1\right),\,\mu_{B}\left(\dfrac{r_{\ast\ast}}{r_{\ast}}\cdot\dfrac{\rho_{A}}{\hat{\rho}-\rho_{B}}-1\right)\right\\}.$

In Case 3, the intersection is the interval
$\mathcal{BR}_{\rho}\left(\hat{\rho}\right)=\left[\underline{r}_{\rho}(\hat{\rho}),\overline{r}_{\rho}(\hat{\rho})\right]$.
This is represented, under Kulpa’s approach, by the point

$(x,y)=\left(\frac{\underline{r}_{\rho}(\hat{\rho})+\overline{r}_{\rho}(\hat{\rho})}{2},\,\frac{\overline{r}_{\rho}(\hat{\rho})-\underline{r}_{\rho}(\hat{\rho})}{2}\right).$

This point is unique, that is, the bargaining region does not depend on
$\mu_{M}$. The conditions under which this situation is possible are

$\left\\{\begin{array}[]{@{}l@{}}\vskip 3.0pt plus 1.0pt minus
1.0pt\underline{r}_{\mu}(\mu_{M})\leq\underline{r}_{\rho}(\hat{\rho})\\\
\overline{r}_{\mu}(\mu_{M})\geq\overline{r}_{\rho}(\hat{\rho}),\end{array}\right.$

that is,

$\left\\{\begin{array}[]{@{}l@{}}\vskip 6.0pt plus 2.0pt minus
2.0pt\dfrac{r_{\ast}\mu_{A}}{\mu_{M}-\mu_{B}}\leq\dfrac{r_{\ast\ast}(\hat{\rho}-\rho_{A})}{\rho_{B}}\\\
\dfrac{r_{\ast}(\mu_{M}-\mu_{A})}{\mu_{B}}\geq\dfrac{r_{\ast\ast}\rho_{A}}{\hat{\rho}-\rho_{B}},\end{array}\right.\;\;\;\;\;\text{that
is,}\;\;\;\;\;\left\\{\begin{array}[]{@{}l@{}}\vskip 6.0pt plus 2.0pt minus
2.0pt\mu_{M}\geq\dfrac{r_{\ast}}{r_{\ast\ast}}\cdot\dfrac{\mu_{A}\rho_{B}}{\hat{\rho}-\rho_{A}}+\mu_{B}\\\
\mu_{M}\geq\dfrac{r_{\ast\ast}}{r_{\ast}}\cdot\dfrac{\mu_{B}\rho_{A}}{\hat{\rho}-\rho_{B}}+\mu_{A},\end{array}\right.$

which can be summarized as follows:

$\mu_{M}\geq\max\left\\{\dfrac{r_{\ast}}{r_{\ast\ast}}\cdot\dfrac{\mu_{A}\rho_{B}}{\hat{\rho}-\rho_{A}}+\mu_{B},\,\dfrac{r_{\ast\ast}}{r_{\ast}}\cdot\dfrac{\mu_{B}\rho_{A}}{\hat{\rho}-\rho_{B}}+\mu_{A}\right\\}.$

To obtain the condition about the expected synergy, we subtract
$\mu_{A}+\mu_{B}$:

$\mu_{M}-\mu_{A}-\mu_{B}\geq\max\left\\{\dfrac{r_{\ast}}{r_{\ast\ast}}\cdot\dfrac{\mu_{A}\rho_{B}}{\hat{\rho}-\rho_{A}}-\mu_{A},\,\dfrac{r_{\ast\ast}}{r_{\ast}}\cdot\dfrac{\mu_{B}\rho_{A}}{\hat{\rho}-\rho_{B}}-\mu_{B}\right\\},$

that is,

$\mu_{M}-\mu_{A}-\mu_{B}\geq\max\left\\{\mu_{A}\left(\dfrac{r_{\ast}}{r_{\ast\ast}}\cdot\dfrac{\rho_{B}}{\hat{\rho}-\rho_{A}}-1\right),\,\mu_{B}\left(\dfrac{r_{\ast\ast}}{r_{\ast}}\cdot\dfrac{\rho_{A}}{\hat{\rho}-\rho_{B}}-1\right)\right\\}.$

In Case 4, the intersection is the interval
$\mathcal{CR}_{A}\left(\mu_{M},\hat{\rho}\right)=\left[\underline{r}_{\rho}(\hat{\rho}),\overline{r}_{\mu}(\mu_{M})\right]$.
This is represented, under Kulpa’s approach, by the point

$(x,y)=\left(\frac{\underline{r}_{\rho}(\hat{\rho})+\overline{r}_{\mu}(\mu_{M})}{2},\,\frac{\overline{r}_{\mu}(\mu_{M})-\underline{r}_{\rho}(\hat{\rho})}{2}\right),$

that is,

$x=\frac{1}{2}\left(\frac{r_{\ast\ast}(\hat{\rho}-\rho_{A})}{\rho_{B}}+\frac{r_{\ast}(\mu_{M}-\mu_{A})}{\mu_{B}}\right),$
$y=\frac{1}{2}\left(\frac{r_{\ast}(\mu_{M}-\mu_{A})}{\mu_{B}}-\frac{r_{\ast\ast}(\hat{\rho}-\rho_{A})}{\rho_{B}}\right).$

The difference between its coordinates is

$x-y=\frac{r_{\ast\ast}(\hat{\rho}-\rho_{A})}{\rho_{B}};$

that is, on varying $\mu_{M}$ we move along the straight line of equation

$y=x-\frac{r_{\ast\ast}(\hat{\rho}-\rho_{A})}{\rho_{B}}.$

The conditions under which this situation is possible are

$\left\\{\begin{array}[]{@{}l@{}}\vskip 3.0pt plus 1.0pt minus
1.0pt\underline{r}_{\mu}(\mu_{M})\leq\underline{r}_{\rho}(\hat{\rho})\\\
\vskip 3.0pt plus 1.0pt minus
1.0pt\overline{r}_{\mu}(\mu_{M})\leq\overline{r}_{\rho}(\hat{\rho})\\\
\underline{r}_{\rho}(\hat{\rho})\leq\overline{r}_{\mu}(\mu_{M}),\end{array}\right.$

that is,

$\left\\{\begin{array}[]{@{}l@{}}\vskip 6.0pt plus 2.0pt minus
2.0pt\dfrac{r_{\ast}\mu_{A}}{\mu_{M}-\mu_{B}}\leq\dfrac{r_{\ast\ast}(\hat{\rho}-\rho_{A})}{\rho_{B}}\\\
\vskip 6.0pt plus 2.0pt minus
2.0pt\dfrac{r_{\ast}(\mu_{M}-\mu_{A})}{\mu_{B}}\leq\dfrac{r_{\ast\ast}\rho_{A}}{\hat{\rho}-\rho_{B}}\\\
\dfrac{r_{\ast\ast}(\hat{\rho}-\rho_{A})}{\rho_{B}}\leq\dfrac{r_{\ast}(\mu_{M}-\mu_{A})}{\mu_{B}},\end{array}\right.\;\;\;\;\;\text{that
is,}\;\;\;\;\;\left\\{\begin{array}[]{@{}l@{}}\vskip 6.0pt plus 2.0pt minus
2.0pt\mu_{M}\geq\dfrac{r_{\ast}}{r_{\ast\ast}}\cdot\dfrac{\mu_{A}\rho_{B}}{\hat{\rho}-\rho_{A}}+\mu_{B}\\\
\vskip 6.0pt plus 2.0pt minus
2.0pt\mu_{M}\leq\dfrac{r_{\ast\ast}}{r_{\ast}}\cdot\dfrac{\mu_{B}\rho_{A}}{\hat{\rho}-\rho_{B}}+\mu_{A}\\\
\mu_{M}\geq\dfrac{r_{\ast\ast}}{r_{\ast}}\cdot\dfrac{\mu_{B}(\hat{\rho}-\rho_{A})}{\rho_{B}}+\mu_{A},\end{array}\right.$

which can be summarized as follows:

$\max\left\\{\dfrac{r_{\ast}}{r_{\ast\ast}}\cdot\dfrac{\mu_{A}\rho_{B}}{\hat{\rho}-\rho_{A}}+\mu_{B},\,\dfrac{r_{\ast\ast}}{r_{\ast}}\cdot\dfrac{\mu_{B}(\hat{\rho}-\rho_{A})}{\rho_{B}}+\mu_{A}\right\\}\leq\mu_{M}\leq\dfrac{r_{\ast\ast}}{r_{\ast}}\cdot\dfrac{\mu_{B}\rho_{A}}{\hat{\rho}-\rho_{B}}+\mu_{A}.$

To obtain the condition about the expected synergy, we subtract
$\mu_{A}+\mu_{B}$:

$\max\left\\{\dfrac{r_{\ast}}{r_{\ast\ast}}\cdot\dfrac{\mu_{A}\rho_{B}}{\hat{\rho}-\rho_{A}}-\mu_{A},\,\dfrac{r_{\ast\ast}}{r_{\ast}}\cdot\dfrac{\mu_{B}(\hat{\rho}-\rho_{A})}{\rho_{B}}-\mu_{B}\right\\}\leq\mu_{M}-\mu_{A}-\mu_{B}\leq\dfrac{r_{\ast\ast}}{r_{\ast}}\cdot\dfrac{\mu_{B}\rho_{A}}{\hat{\rho}-\rho_{B}}-\mu_{B},$

that is,

$\displaystyle\max$
$\displaystyle\left\\{\mu_{A}\left(\dfrac{r_{\ast}}{r_{\ast\ast}}\cdot\dfrac{\rho_{B}}{\hat{\rho}-\rho_{A}}-1\right),\,\mu_{B}\left(\dfrac{r_{\ast\ast}}{r_{\ast}}\cdot\dfrac{\hat{\rho}-\rho_{A}}{\rho_{B}}-1\right)\right\\}\leq\mu_{M}-\mu_{A}-\mu_{B}$
$\displaystyle\leq\mu_{B}\left(\dfrac{r_{\ast\ast}}{r_{\ast}}\cdot\dfrac{\rho_{A}}{\hat{\rho}-\rho_{B}}-1\right).$

Consider now $\hat{\mu}\geq\mu_{A}+\mu_{B}$. In Case 1, the intersection is
the interval
$\mathcal{CR}_{B}\left(\hat{\mu},\rho_{M}\right)=\left[\underline{r}_{\mu}(\hat{\mu}),\overline{r}_{\rho}(\rho_{M})\right]$.
This is represented, under Kulpa’s approach, by the point

$(x,y)=\left(\frac{\underline{r}_{\mu}(\hat{\mu})+\overline{r}_{\rho}(\rho_{M})}{2},\,\frac{\overline{r}_{\rho}(\rho_{M})-\underline{r}_{\mu}(\hat{\mu})}{2}\right),$

that is,

$x=\frac{1}{2}\left(\frac{r_{\ast}\mu_{A}}{\hat{\mu}-\mu_{B}}+\frac{r_{\ast\ast}\rho_{A}}{\rho_{M}-\rho_{B}}\right),$
$y=\frac{1}{2}\left(\frac{r_{\ast\ast}\rho_{A}}{\rho_{M}-\rho_{B}}-\frac{r_{\ast}\mu_{A}}{\hat{\mu}-\mu_{B}}\right).$

The difference between its coordinates is

$x-y=\frac{r_{\ast}\mu_{A}}{\hat{\mu}-\mu_{B}};$

that is, on varying $\rho_{M}$ we move along the straight line of equation

$y=x-\frac{r_{\ast}\mu_{A}}{\hat{\mu}-\mu_{B}}.$

The conditions under which this situation is possible are

$\left\\{\begin{array}[]{@{}l@{}}\vskip 3.0pt plus 1.0pt minus
1.0pt\underline{r}_{\mu}(\hat{\mu})\geq\underline{r}_{\rho}(\rho_{M})\\\
\vskip 3.0pt plus 1.0pt minus
1.0pt\overline{r}_{\mu}(\hat{\mu})\geq\overline{r}_{\rho}(\rho_{M})\\\
\underline{r}_{\mu}(\hat{\mu})\leq\overline{r}_{\rho}(\rho_{M}),\end{array}\right.$

that is,

$\left\\{\begin{array}[]{@{}l@{}}\vskip 6.0pt plus 2.0pt minus
2.0pt\dfrac{r_{\ast}\mu_{A}}{\hat{\mu}-\mu_{B}}\geq\dfrac{r_{\ast\ast}(\rho_{M}-\rho_{A})}{\rho_{B}}\\\
\vskip 6.0pt plus 2.0pt minus
2.0pt\dfrac{r_{\ast}(\hat{\mu}-\mu_{A})}{\mu_{B}}\geq\dfrac{r_{\ast\ast}\rho_{A}}{\rho_{M}-\rho_{B}}\\\
\dfrac{r_{\ast}\mu_{A}}{\hat{\mu}-\mu_{B}}\leq\dfrac{r_{\ast\ast}\rho_{A}}{\rho_{M}-\rho_{B}},\end{array}\right.\;\;\;\;\;\text{that
is,}\;\;\;\;\;\left\\{\begin{array}[]{@{}l@{}}\vskip 6.0pt plus 2.0pt minus
2.0pt\rho_{M}\leq\dfrac{r_{\ast}}{r_{\ast\ast}}\cdot\dfrac{\rho_{B}\mu_{A}}{\hat{\mu}-\mu_{B}}+\rho_{A}\\\
\vskip 6.0pt plus 2.0pt minus
2.0pt\rho_{M}\geq\dfrac{r_{\ast\ast}}{r_{\ast}}\cdot\dfrac{\rho_{A}\mu_{B}}{\hat{\mu}-\mu_{A}}+\rho_{B}\\\
\rho_{M}\leq\dfrac{r_{\ast\ast}}{r_{\ast}}\cdot\dfrac{\rho_{A}(\hat{\mu}-\mu_{B})}{\mu_{A}}+\rho_{B},\end{array}\right.$

which can be summarized as follows:

$\dfrac{r_{\ast\ast}}{r_{\ast}}\cdot\dfrac{\rho_{A}\mu_{B}}{\hat{\mu}-\mu_{A}}+\rho_{B}\leq\rho_{M}\leq\min\left\\{\dfrac{r_{\ast}}{r_{\ast\ast}}\cdot\dfrac{\rho_{B}\mu_{A}}{\hat{\mu}-\mu_{B}}+\rho_{A},\,\dfrac{r_{\ast\ast}}{r_{\ast}}\cdot\dfrac{\rho_{A}(\hat{\mu}-\mu_{B})}{\mu_{A}}+\rho_{B}\right\\},$

or, equivalently,

$\max\left\\{-\dfrac{r_{\ast}}{r_{\ast\ast}}\cdot\dfrac{\rho_{B}\mu_{A}}{\hat{\mu}-\mu_{B}}-\rho_{A},\,-\dfrac{r_{\ast\ast}}{r_{\ast}}\cdot\dfrac{\rho_{A}(\hat{\mu}-\mu_{B})}{\mu_{A}}-\rho_{B}\right\\}\leq-\rho_{M}\leq-\dfrac{r_{\ast\ast}}{r_{\ast}}\cdot\dfrac{\rho_{A}\mu_{B}}{\hat{\mu}-\mu_{A}}-\rho_{B}.$

To obtain the condition about the overall risk, we sum $\rho_{A}+\rho_{B}$:

$\max\left\\{\rho_{B}-\dfrac{r_{\ast}}{r_{\ast\ast}}\cdot\dfrac{\rho_{B}\mu_{A}}{\hat{\mu}-\mu_{B}},\,\rho_{A}-\dfrac{r_{\ast\ast}}{r_{\ast}}\cdot\dfrac{\rho_{A}(\hat{\mu}-\mu_{B})}{\mu_{A}}\right\\}\leq\rho_{A}+\rho_{B}-\rho_{M}\leq\rho_{A}-\dfrac{r_{\ast\ast}}{r_{\ast}}\cdot\dfrac{\rho_{A}\mu_{B}}{\hat{\mu}-\mu_{A}},$

that is,

$\displaystyle\max$
$\displaystyle\left\\{\rho_{B}\left(1-\dfrac{r_{\ast}}{r_{\ast\ast}}\cdot\dfrac{\mu_{A}}{\hat{\mu}-\mu_{B}}\right),\,\rho_{A}\left(1-\dfrac{r_{\ast\ast}}{r_{\ast}}\cdot\dfrac{\hat{\mu}-\mu_{B}}{\mu_{A}}\right)\right\\}\leq\rho_{A}+\rho_{B}-\rho_{M}$
$\displaystyle\leq\rho_{A}\left(1-\dfrac{r_{\ast\ast}}{r_{\ast}}\cdot\dfrac{\mu_{B}}{\hat{\mu}-\mu_{A}}\right).$

In Case 2, the intersection is the interval
$\mathcal{BR}_{\mu}\left(\hat{\mu}\right)=\left[\underline{r}_{\mu}(\hat{\mu}),\overline{r}_{\mu}(\hat{\mu})\right]$.
This is represented, under Kulpa’s approach, by the point

$(x,y)=\left(\frac{\underline{r}_{\mu}(\hat{\mu})+\overline{r}_{\mu}(\hat{\mu})}{2},\,\frac{\overline{r}_{\mu}(\hat{\mu})-\underline{r}_{\mu}(\hat{\mu})}{2}\right).$

This point is unique, that is, the bargaining region does not depend on
$\rho_{M}$. The conditions under which this situation is possible are

$\left\\{\begin{array}[]{@{}l@{}}\vskip 3.0pt plus 1.0pt minus
1.0pt\underline{r}_{\mu}(\hat{\mu})\geq\underline{r}_{\rho}(\rho_{M})\\\
\overline{r}_{\mu}(\hat{\mu})\leq\overline{r}_{\rho}(\rho_{M}),\end{array}\right.$

that is,

$\left\\{\begin{array}[]{@{}l@{}}\vskip 6.0pt plus 2.0pt minus
2.0pt\dfrac{r_{\ast}\mu_{A}}{\hat{\mu}-\mu_{B}}\geq\dfrac{r_{\ast\ast}(\rho_{M}-\rho_{A})}{\rho_{B}}\\\
\dfrac{r_{\ast}(\hat{\mu}-\mu_{A})}{\mu_{B}}\leq\dfrac{r_{\ast\ast}\rho_{A}}{\rho_{M}-\rho_{B}},\end{array}\right.\;\;\;\;\;\text{that
is,}\;\;\;\;\;\left\\{\begin{array}[]{@{}l@{}}\vskip 6.0pt plus 2.0pt minus
2.0pt\rho_{M}\leq\dfrac{r_{\ast}}{r_{\ast\ast}}\cdot\dfrac{\rho_{B}\mu_{A}}{\hat{\mu}-\mu_{B}}+\rho_{A}\\\
\rho_{M}\leq\dfrac{r_{\ast\ast}}{r_{\ast}}\cdot\dfrac{\rho_{A}\mu_{B}}{\hat{\mu}-\mu_{A}}+\rho_{B},\end{array}\right.$

which can be summarized as follows:

$\max\\{\rho_{A},\rho_{B}\\}<\rho_{M}\leq\min\left\\{\dfrac{r_{\ast}}{r_{\ast\ast}}\cdot\dfrac{\rho_{B}\mu_{A}}{\hat{\mu}-\mu_{B}}+\rho_{A},\,\dfrac{r_{\ast\ast}}{r_{\ast}}\cdot\dfrac{\rho_{A}\mu_{B}}{\hat{\mu}-\mu_{A}}+\rho_{B}\right\\},$

or, equivalently,

$\max\left\\{-\dfrac{r_{\ast}}{r_{\ast\ast}}\cdot\dfrac{\rho_{B}\mu_{A}}{\hat{\mu}-\mu_{B}}-\rho_{A},\,-\dfrac{r_{\ast\ast}}{r_{\ast}}\cdot\dfrac{\rho_{A}\mu_{B}}{\hat{\mu}-\mu_{A}}-\rho_{B}\right\\}\leq-\rho_{M}<\min\\{-\rho_{A},-\rho_{B}\\}.$

To obtain the condition about the overall risk, we sum $\rho_{A}+\rho_{B}$:

$\max\left\\{\rho_{B}-\dfrac{r_{\ast}}{r_{\ast\ast}}\cdot\dfrac{\rho_{B}\mu_{A}}{\hat{\mu}-\mu_{B}},\,\rho_{A}-\dfrac{r_{\ast\ast}}{r_{\ast}}\cdot\dfrac{\rho_{A}\mu_{B}}{\hat{\mu}-\mu_{A}}\right\\}\leq\rho_{A}+\rho_{B}-\rho_{M}<\min\\{\rho_{A},\rho_{B}\\},$

that is,

$\max\left\\{\rho_{B}\left(1-\dfrac{r_{\ast}}{r_{\ast\ast}}\cdot\dfrac{\rho_{B}\mu_{A}}{\hat{\mu}-\mu_{B}}\right),\,\rho_{A}\left(1-\dfrac{r_{\ast\ast}}{r_{\ast}}\cdot\dfrac{\rho_{A}\mu_{B}}{\hat{\mu}-\mu_{A}}\right)\right\\}\leq\rho_{A}+\rho_{B}-\rho_{M}<\min\\{\rho_{A},\rho_{B}\\}.$

In Case 3, the intersection is the interval
$\mathcal{BR}_{\rho}\left(\rho_{M}\right)=\left[\underline{r}_{\rho}(\rho_{M}),\overline{r}_{\rho}(\rho_{M})\right]$.
This is represented, under Kulpa’s approach, by the point

$(x,y)=\left(\frac{\underline{r}_{\rho}(\rho_{M})+\overline{r}_{\rho}(\rho_{M})}{2},\,\frac{\overline{r}_{\rho}(\rho_{M})-\underline{r}_{\rho}(\rho_{M})}{2}\right),$

which, varying $\rho_{M}$, leads to the curve $\delta$ (see A.2). The
conditions under which this situation is possible are

$\left\\{\begin{array}[]{@{}l@{}}\vskip 3.0pt plus 1.0pt minus
1.0pt\underline{r}_{\mu}(\hat{\mu})\leq\underline{r}_{\rho}(\rho_{M})\\\
\overline{r}_{\mu}(\hat{\mu})\geq\overline{r}_{\rho}(\rho_{M}),\end{array}\right.$

that is,

$\left\\{\begin{array}[]{@{}l@{}}\vskip 6.0pt plus 2.0pt minus
2.0pt\dfrac{r_{\ast}\mu_{A}}{\hat{\mu}-\mu_{B}}\leq\dfrac{r_{\ast\ast}(\rho_{M}-\rho_{A})}{\rho_{B}}\\\
\dfrac{r_{\ast}(\hat{\mu}-\mu_{A})}{\mu_{B}}\geq\dfrac{r_{\ast\ast}\rho_{A}}{\rho_{M}-\rho_{B}},\end{array}\right.\;\;\;\;\;\text{that
is,}\;\;\;\;\;\left\\{\begin{array}[]{@{}l@{}}\vskip 6.0pt plus 2.0pt minus
2.0pt\rho_{M}\geq\dfrac{r_{\ast}}{r_{\ast\ast}}\cdot\dfrac{\rho_{B}\mu_{A}}{\hat{\mu}-\mu_{B}}+\rho_{A}\\\
\rho_{M}\geq\dfrac{r_{\ast\ast}}{r_{\ast}}\cdot\dfrac{\rho_{A}\mu_{B}}{\hat{\mu}-\mu_{A}}+\rho_{B},\end{array}\right.$

which can be summarized as follows:

$\rho_{M}\geq\max\left\\{\dfrac{r_{\ast}}{r_{\ast\ast}}\cdot\dfrac{\rho_{B}\mu_{A}}{\hat{\mu}-\mu_{B}}+\rho_{A},\,\dfrac{r_{\ast\ast}}{r_{\ast}}\cdot\dfrac{\rho_{A}\mu_{B}}{\hat{\mu}-\mu_{A}}+\rho_{B}\right\\},$

or, equivalently,

$-\rho_{M}\leq\min\left\\{-\dfrac{r_{\ast}}{r_{\ast\ast}}\cdot\dfrac{\rho_{B}\mu_{A}}{\hat{\mu}-\mu_{B}}-\rho_{A},\,-\dfrac{r_{\ast\ast}}{r_{\ast}}\cdot\dfrac{\rho_{A}\mu_{B}}{\hat{\mu}-\mu_{A}}-\rho_{B}\right\\}.$

To obtain the condition about the overall risk, we sum $\rho_{A}+\rho_{B}$:

$\rho_{A}+\rho_{B}-\rho_{M}\leq\min\left\\{\rho_{B}-\dfrac{r_{\ast}}{r_{\ast\ast}}\cdot\dfrac{\rho_{B}\mu_{A}}{\hat{\mu}-\mu_{B}},\,\rho_{A}-\dfrac{r_{\ast\ast}}{r_{\ast}}\cdot\dfrac{\rho_{A}\mu_{B}}{\hat{\mu}-\mu_{A}}\right\\},$

that is,

$\rho_{A}+\rho_{B}-\rho_{M}\leq\min\left\\{\rho_{B}\left(1-\dfrac{r_{\ast}}{r_{\ast\ast}}\cdot\dfrac{\mu_{A}}{\hat{\mu}-\mu_{B}}\right),\,\rho_{A}\left(1-\dfrac{r_{\ast\ast}}{r_{\ast}}\cdot\dfrac{\mu_{B}}{\hat{\mu}-\mu_{A}}\right)\right\\}.$

In Case 4, the intersection is the interval
$\mathcal{CR}_{A}\left(\hat{\mu},\rho_{M}\right)=\left[\underline{r}_{\rho}(\rho_{M}),\overline{r}_{\mu}(\hat{\mu})\right]$.
This is represented, under Kulpa’s approach, by the point

$(x,y)=\left(\frac{\underline{r}_{\rho}(\rho_{M})+\overline{r}_{\mu}(\hat{\mu})}{2},\,\frac{\overline{r}_{\mu}(\hat{\mu})-\underline{r}_{\rho}(\rho_{M})}{2}\right),$

that is,

$x=\frac{1}{2}\left(\frac{r_{\ast\ast}(\rho_{M}-\rho_{A})}{\rho_{B}}+\frac{r_{\ast}(\hat{\mu}-\mu_{A})}{\mu_{B}}\right),$
$y=\frac{1}{2}\left(\frac{r_{\ast}(\hat{\mu}-\mu_{A})}{\mu_{B}}-\frac{r_{\ast\ast}(\rho_{M}-\rho_{A})}{\rho_{B}}\right).$

The sum of its coordinates is

$x+y=\frac{r_{\ast}(\hat{\mu}-\mu_{A})}{\mu_{B}};$

that is, on varying $\rho_{M}$ we move along the straight line of equation

$y=-x+\frac{r_{\ast}(\hat{\mu}-\mu_{A})}{\mu_{B}}.$

The conditions under which this situation is possible are

$\left\\{\begin{array}[]{@{}l@{}}\vskip 3.0pt plus 1.0pt minus
1.0pt\underline{r}_{\mu}(\hat{\mu})\leq\underline{r}_{\rho}(\rho_{M})\\\
\vskip 3.0pt plus 1.0pt minus
1.0pt\overline{r}_{\mu}(\hat{\mu})\leq\overline{r}_{\rho}(\rho_{M})\\\
\underline{r}_{\rho}(\rho_{M})\leq\overline{r}_{\mu}(\hat{\mu}),\end{array}\right.$

that is,

$\left\\{\begin{array}[]{@{}l@{}}\vskip 6.0pt plus 2.0pt minus
2.0pt\dfrac{r_{\ast}\mu_{A}}{\hat{\mu}-\mu_{B}}\leq\dfrac{r_{\ast\ast}(\rho_{M}-\rho_{A})}{\rho_{B}}\\\
\vskip 6.0pt plus 2.0pt minus
2.0pt\dfrac{r_{\ast}(\hat{\mu}-\mu_{A})}{\mu_{B}}\leq\dfrac{r_{\ast\ast}\rho_{A}}{\rho_{M}-\rho_{B}}\\\
\dfrac{r_{\ast\ast}(\rho_{M}-\rho_{A})}{\rho_{B}}\leq\dfrac{r_{\ast}(\hat{\mu}-\mu_{A})}{\mu_{B}},\end{array}\right.\;\;\;\;\;\text{that
is,}\;\;\;\;\;\left\\{\begin{array}[]{@{}l@{}}\vskip 6.0pt plus 2.0pt minus
2.0pt\rho_{M}\geq\dfrac{r_{\ast}}{r_{\ast\ast}}\cdot\dfrac{\rho_{B}\mu_{A}}{\hat{\mu}-\mu_{B}}+\rho_{A}\\\
\vskip 6.0pt plus 2.0pt minus
2.0pt\rho_{M}\leq\dfrac{r_{\ast\ast}}{r_{\ast}}\cdot\dfrac{\mu_{B}\rho_{A}}{\hat{\mu}-\mu_{A}}+\rho_{B}\\\
\rho_{M}\leq\dfrac{r_{\ast}}{r_{\ast\ast}}\cdot\dfrac{\rho_{B}(\hat{\mu}-\mu_{A})}{\mu_{B}}+\rho_{A},\end{array}\right.$

which can be summarized as follows:

$\dfrac{r_{\ast}}{r_{\ast\ast}}\cdot\dfrac{\rho_{B}\mu_{A}}{\hat{\mu}-\mu_{B}}+\rho_{A}\leq\rho_{M}\leq\min\left\\{\dfrac{r_{\ast\ast}}{r_{\ast}}\cdot\dfrac{\mu_{B}\rho_{A}}{\hat{\mu}-\mu_{A}}+\rho_{B},\,\dfrac{r_{\ast}}{r_{\ast\ast}}\cdot\dfrac{\rho_{B}(\hat{\mu}-\mu_{A})}{\mu_{B}}+\rho_{A}\right\\},$

or, equivalently,

$\max\left\\{-\dfrac{r_{\ast\ast}}{r_{\ast}}\cdot\dfrac{\mu_{B}\rho_{A}}{\hat{\mu}-\mu_{A}}-\rho_{B},\,-\dfrac{r_{\ast}}{r_{\ast\ast}}\cdot\dfrac{\rho_{B}(\hat{\mu}-\mu_{A})}{\mu_{B}}-\rho_{A}\right\\}\leq-\rho_{M}\leq-\dfrac{r_{\ast}}{r_{\ast\ast}}\cdot\dfrac{\rho_{B}\mu_{A}}{\hat{\mu}-\mu_{B}}-\rho_{A}.$

To obtain the condition about the overall risk, we sum $\rho_{A}+\rho_{B}$:

$\max\left\\{\rho_{A}-\dfrac{r_{\ast\ast}}{r_{\ast}}\cdot\dfrac{\mu_{B}\rho_{A}}{\hat{\mu}-\mu_{A}},\,\rho_{B}-\dfrac{r_{\ast}}{r_{\ast\ast}}\cdot\dfrac{\rho_{B}(\hat{\mu}-\mu_{A})}{\mu_{B}}\right\\}\leq\rho_{A}+\rho_{B}-\rho_{M}\leq\rho_{B}-\dfrac{r_{\ast}}{r_{\ast\ast}}\cdot\dfrac{\rho_{B}\mu_{A}}{\hat{\mu}-\mu_{B}},$

that is,

$\displaystyle\max$
$\displaystyle\left\\{\rho_{A}\left(1-\dfrac{r_{\ast\ast}}{r_{\ast}}\cdot\dfrac{\mu_{B}}{\hat{\mu}-\mu_{A}}\right),\,\rho_{B}\left(1-\dfrac{r_{\ast}}{r_{\ast\ast}}\cdot\dfrac{\hat{\mu}-\mu_{A}}{\mu_{B}}\right)\right\\}\leq\rho_{A}+\rho_{B}-\rho_{M}$
$\displaystyle\leq\rho_{B}\left(1-\dfrac{r_{\ast}}{r_{\ast\ast}}\cdot\dfrac{\mu_{A}}{\hat{\mu}-\mu_{B}}\right).$

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