Based on proposition 4.2 and 4.3, we furthermore explore the factors to influence the slope of earnings on dividend
D X
, and hence to change optimal dividend policy.
The results of proposition 4.2 state that optimal dividend policy is different from that of D
is represented by
When existing heterogeneous beliefs among investors, D X
is always smaller than 1.
That is, the slope is flatter than that of homogeneous case D X
=1. The degree of
deviation depends on two terms:
1
These two terms respectively represent the ratio of optimistic investors to pessimistic ones, and earnings of a firm viewed by theses two divergent investors. Unlike to previous research such as Miller and Rock (1985), John and Williams (1987), optimal dividend policy is changed not because of information asymmetry but heterogeneous beliefs. In addition, optimal dividend policy is changed when considering heterogeneity among investors not only by the ratio of pessimistic to the optimistic investors
1
2
, but also divergent beliefs, ao and ap. This finding has pushed one
stage deeper in this topic regarding optimal dividend policy with diversity of opinions.
According to proposition 4.4, an increase in ratio of pessimistic to optimistic
investors will result in a higher dividend. This is easy to explain. For a firm’s manager, he will shave investment and pay more dividends when a depression occurs.
On the other hand, in investors’viewpoints they become more conservative and lead in asking more cash in the form of dividend. Accordingly, a high dividend is paid out. This finding is the same as that of previous research work. However, the interpretations we propose are very different from the Miller and Rock theory. They interpreted a higher dividend policy as a signaling. A good firm must pay a level of dividends that is sufficiently high to make it unattractive for bad firms to reduce their investment enough to achieve the same level. Firms shave investment to make dividends higher and signal high earnings. However, they have not considered the investors’divergent beliefs.
In order to better understand the impact of the ratio of pessimistic to optimistic investors on
, we use one example to illustrate. Three
conditions including
Condition 1. When the ratio approaches to zero
1
2
0, the investors are
almost optimistic. According to Equation (4.23), we get
D
Condition 2. When the ratio approaches to 1
1
2
0, the optimistic investors
are the same as that of pessimistic ones. Substituting
1 2
=1 into Equation (4.23),
we have
D
Condition 3. When the ratio approaches to infinite,
1
2
, the investors are
almost pessimistic. Substituting
1
Comparing Equation (4.29), (4.30) and (4.31), we get 1- ( )
This means that as the ratio
1
2
approaches to 0, the slope D X
is the largest
(closest to 1) regardless of what values F(Io) and F(Ip) are. Conversely, when
1
2
approaches to infinite, D X
is the smallest. To summarize, as pessimistic
investors increases, i.e. a depression occurs, the slop of earnings on dividends
D X
becomes flatter than that of homogeneous condition. A high dividend policy is
appropriate. On the contrary, when optimistic investors increase, the slop of earnings
on dividend becomes steeper. A low dividend policy is appropriate. We represent the preceding results graphically in Figure 4.6. As shown in the figure, the solution of heterogeneous beliefs is below the homogeneous beliefs case because
D X
must
be smaller than 1.
Figure 4.6 The solution to heterogeneous beliefs
For a given earnings X, dividend paid out are D0, D1, corresponding to homogeneous and heterogeneous beliefs, respectively. Apparently, we could find once a deviation from homogeneous beliefs the firm’s dividend policy is changed towards higher dividend policy given the same earnings X. The degree depends on the ratio of pessimistic to optimistic investors. As the ratio increases, the curve shifts towards the right-down direction. In managers’viewpoints, more cash is paid out because there are no investment projects good enough when a depression occurs.
This will result in an increase in dividend. In investors’viewpoints, high dividend is
D
X
Homogeneous beliefs
D0 D1
Heterogeneous beliefs
asked as a depression occurs because they become conservative. More pessimistic investors make dividend paid out higher.
Next, we check the impact of divergent beliefs of optimistic and pessimistic investors, aoand ap, on optimal dividend policy.
Proposition 4.5 An increase in ao, that is, beliefs of optimistic investors increases will result in an increases in
D X
and hence a relative lower dividend policy.
Proof. According to Equation (4.23) and (4.25)
D
, I take the first-order partial derivate with
respect to aoassuming that aois independent of apand F(Io), other things equal, I get
ao
is always positive because C, I, i, F(Io), 1and apare all larger than zero. The result shown in the preceding Equation says that an increase in aowill result in an increase in
D X
and hence a relative lower dividend policy. This completes the
proof.
Proposition 4.6 An increase in ap, that is, the beliefs of pessimistic investors increases
will result in an increases in D X
and hence a relative lower dividend policy.
Proof. According to Equation (4.23) and (4.25)
D
, I take first-order partial derivate with
respect to apassuming that apis independent of aoand F(Io),other things equal, I have
ap
An increase in ap, that is, the beliefs of pessimistic investors increases will result in an increases in
D X
and hence a relative lower dividend policy. This completes the
proof. According to proposition 4.5 and 4.6. An increase in ao and ap, that is, the beliefs of both optimistic and pessimistic investors become larger simultaneously, other things equal, there will be an increases in
D X
and hence a relative lower
dividend policy. For example, a new medicine is innovated. This new biotechnology will simultaneous stimulate both optimistic and pessimistic investors, and hence change production functions in Figure 4.7.
Figure 4.7 Shifted curve of a production function Fbio(I)=abi
[ln(I+1)]
as creating a new biotechnology
Graphically, they are shown in the form of increasing parameter of production function ao and ap. F(Io)=aoln[(I+1)] and F(Ip)=apln[(I+1)] are replaced by Fbio(Io)=abo
ln[ (I+1)] and Fbio(Ip)=abp
ln[ (I+1)] respectively after a new medicine is innovated, where Fbio(Io) and Fbio(Ip) are new production functions estimated by optimistic and pessimistic investors after a new biotechnology is innovated.
On the other hand, according to Equation (4.21)
D
When an increase in ao and ap simultaneously because of innovation of a new biotechnology, other things equal,
D X
will become larger given the sameπ1andπ2
I
F(I)
F(Ip)=ap[ln(I+1)]
F(Io)=ao[ln(I+1)]
Shifting towards this direction
Fbio(Io)=abo
[ln(I+1)]
Fbio(Ip)=abp[ln(I+1)]
according to the preceding Equation. As a new biotechnology is created, a firm will increase investment in it at the expense of shaving dividend and expects more earnings in the next period. Hence, dividend paid out is decreased. In investors’
viewpoints, they become more aggressive and result in asking less cash in the form of dividend. Therefore, a low dividend policy is appropriate.
Another interpretation is depicted in Figure 4.8. As shown in the figure, we could find as a new medicine is innovated, the slope of earnings X on dividend D will be shifted. Dividend paid out by a firm is hence decreased from D1 to D2. Consequently, a lower dividend policy will be appropriate as the beliefs of both optimistic and pessimistic investors are simultaneously stimulated when a new biotechnology is created.
This chapter has analyzed heterogeneous beliefs among investors-the two period model in leading to changing a firm’s optimal dividend policy. A firm’soptimal dividend policy is different from that of the homogeneous beliefs. It is changed not only by the ratio of the pessimistic to optimistic investors, but also heterogeneous beliefs. An increase in the pessimistic investors will result in a higher dividend because a depression in economy occurs. On the other hand, as the beliefs of both optimistic and pessimistic investors increase, i.e. a new biotechnology is innovated, a relative low dividend policy is appropriate because all of them are stimulated simultaneously. Heterogeneous beliefs among investors changes dividend policy given the same information even under full information.
Figure 4.8 The solution to heterogeneous beliefs as innovating a new medicine
D
X
–1+
i a
1
homogeneous beliefs creating a new medicine
before creating a new medicine
D2 D1