1 (4.13)
The solution is depicted graphically in Figure 4.3.
Figure 4.3 The relationship between earnings of a firm and cash dividend (homogeneous beliefs)
4.3.2 Heterogeneous beliefs on earning
Relaxing the preceding assumption, investors have common information but agree to disagree about the future earnings of a firm. Divergent investors receive common information, but differ in the way in which they interpret this information.
We assume that there are two types of investors including the optimistic and pessimistic ones. Each type of investor’s information is independent of the other type of investors, and of the common information.
D
X
–1+
i a
1
X=D–1+
i a
1
Investors have heterogeneous beliefs in the future earning of the firm because the investors have different priors, they will interpret information differently. They have common information, but agree to disagree about X. The reasons are as follows: a.
People who respect each other’s opinions and nevertheless disagree about subject probabilities as Aumann (1976) proposed. b. As Gollier (2003) argued, heterogeneity of beliefs does not come from asymmetric information but rather from intrinsic differences in how to view the world. People agree to disagree, which implies that prices and observed behaviors of other market participants do not generate any Bayesian updating of individual beliefs. c. Investors share common prior beliefs and receive common information but differ in the way in which they interpret this information.
According to previous assumption, investors receive common information, but differ in the way in which they interpret this information. The different information about earnings lead in turn to differences in the perceived value of the firm. To model the differences of opinion, we assume that there are two classes (types) of investors agreeing to disagree about the future earnings, with the population of class-i investors being πi ＞0, i=1,2, π1+π2=1. More specifically, class i=o,p. Recalling Equation (4.2)
V1m=D1+
i 1
1 [ E1mF(I1)+γ｜φ~1 m]
To better understand the effect of heterogeneous beliefs of investors, we sort the market outsider into two types of investors based on V1m.
Viewed by optimistic investors, the value of a firm is given by Vo=D+
i 1
1 {ao[ln(X-D+1)]+γε} (4.14)
For pessimistic investors, the value of a firm is given by Vp=D+
i 1
1 {ap[ln(X-D+1)]+γε} (4.15)
Equation (4.14) and (4.15) state that the value of a firm is estimated differently by divergent types of investors. The parameter ao estimated by type-o investors is assumed to be nonnegative. As shown in Figure 4.4, the solution is away from that
of MM dividend invariance theorem. The real line depicts the relationship between earnings X and cash dividend D without considering heterogeneous beliefs of investors; the dotted line is with considering heterogeneous beliefs.
Figure 4.4 The relationship between earnings of a firm and cash dividend (heterogeneous beliefs)
In this case, the firm’s investment expenditure is increased from (-1+
i
1 ). For a given X, the dividend is hence decreased from (X-i
1 +1). The cash dividend paid in difference between the MM theorem and this case is given by
ΔD= (
The MM theorem Heterogeneous beliefs
Where ΔD is the reduction in dividend caused by the heterogeneous beliefs of the optimists, and proportional to the magnitude of ao. The result is different from that of MM dividend invariance theorem. Although the slope remains the same, the intercept is changed. That is, the cash dividend is not a constant but depends on the value of ao.
Figure. 4.5 The difference in cash dividend paid between MM and heterogeneous beliefs of investors
To furthermore explore the effect of dividend policy on a firm’s value, a stronger assumption is made. We assume that the function of V(D) is concave. The following proposition is to derive the relationship between dividend and valuation by different type of investors.
Proposition 4.1 The sign of D Vo
is opposite to that of D Vp
.
D
X
The MM theorem Heterogeneous beliefs ΔD
X-i ao
1
+1
X
X-i a
1
+1
Proof. One objective function for the firm compatible with that requirement is a function in which the firm’s managers attach weights to the interest of each group proportional to the values of their holdings. The manager’s problem can then be written as
D
MaxV=π1Vo+π2Vp
The first-order condition for a maximum with respect to D requires π1
Re-arranging the preceding equation, we have
D
Because both π1 and π2 are larger than 0 according to the previous assumption, the sign of
D Vo
must be opposite to that of D Vp
. Equation (4.16) means as dividend
paid increases, the value of a firm viewed by the optimistic investors increases but decreases by the pessimistic ones.
Proposition 4.2 Optimal dividend policy with heterogeneous beliefs of investors is different from that of homogeneous case.
Proof. For the optimistic investors, the value of a firm is given by Vo=D+
i 1
1 {aoln[(X-D+1)]+γε}
For the pessimistic, the value is given by Vp=D+
i 1
1 {apln[(X-D+1)]+γε}
The manager’s problem is described as
D
Taking the first-order partial derivate with respect to D. Assuming that ao, ap are independent of D, we have
π1{1+[
Re-arranging the preceding equation, we have π1+π1
In order to compare dividend policy with heterogeneous beliefs to that of homogeneous one, we collect terms of
D X
to the left-hand side of Equation (4.18)
and have
Furthermore, we get
D
As known previously, π1+π2=1, the preceding Equation is then reduced to
D
Because π1, π1,
are all positive, D X
must
be smaller than 1. In other words, the slope of the earnings X against dividend D is not linear the same as that of homogeneous case. The differences between heterogeneous and homogeneous case depend on the magnitude of the following term,
)
It is a deviation from that of homogeneous beliefs. Therefore, optimal dividend policy with heterogeneous beliefs of divergent investors is different from that of homogeneous case. Based on previous proposition, we furthermore study the reasons why heterogeneous beliefs lead in diversity of opinions among investors on a firm’s future earnings.
Proposition 4.3 Optimal dividend policy is changed not only by the ratio of pessimistic to optimistic investors, but also divergent beliefs.
Proof. Based on Equation (4.19), we let F(Io)= Divided by (4.20), Equation (4.21) is then reduced to
F(Ip)= o
p
a
a F(Io) (4.22)
Substituting Equation (4.20) ,(4.21) and (4.22) into the preceding Equation, we have 1+ F(Io) [
Collecting terms of D X
to the left hand-side in Equation, we have
D
To replace F(Ip) with F(Io), we substitute Equation (4.22) into the preceding one and get
Substituting (4.20) into the above Equation, we get
D
To simply the equation furthermore, we let A = o
Substituting (4.24) and (4.25) into Equation (4.23), we have
D
The result shows that D X
is affected not only by the ratio of pessimistic investors
to the optimistic C (=
1
a ). This completes the proof.
Then, we check the degree how the ratio of pessimistic to optimistic investors impacts on dividend policy.
Proposition 4.4 An increase in the ratio of pessimistic to optimistic investors
1
2
,
that is, pessimistic investors increases will result in a higher dividend.
Proof. In order to check the impact of
1
, we take first-order partial
derivate with respect to C in Equation (4.26), other things equal, we get
C
Holding A constant, we take first-order partial derivate with respect to C and have
C
There are three conditions of A to influence
C
This case does not exist because o
p
a
a >1 is not reasonable according to the previous assumption. Second, when A<1, Substituting into Equation (4.27)
C
must be smaller than zero. This means
that an increase in C(=
1
. That is, when pessimistic
investors increase, i.e. a depression in economy occurs, the slope of earnings on dividend
D X
becomes flatter than that of normal condition. We infer that high
dividend policy is appropriate to a depression.
Third, when A(= o
p
a a )=1
C D
X
2
=- ( ) 1
Io
F [ 2
) 1 (
) 1 (
CA A
]=0
The result is the same as that of homogeneous case.
C D
X
2
=0 means that D X
is
not affected by the ratio of the pessimistic investors to the optimistic ones. To summarize, an increase in the ratio of pessimistic to optimistic investors
1 2
, that is,
pessimistic investors increases will result in a higher dividend.