BUSF_A01.qxd

Appendix: Proof of the MM dividend irrelevancy proposition

APPENDIX Proof of the MM dividend irrelevancy proposition

In the proof of their assertion, MM considered business jduring a period that starts at

time tand ends at time t+1. They used the following symbols:

Djt=total dividend paid during the period starting at time t

djt =dividend per share paid during that period

Pjt=price per share at time t

rt =return per share during the period starting at time t

nt =number of shares of the business at time t

mt=number of new shares issued during the period starting at time t

Vjt=value of the business at time t(that is, Vjt=nt×Pjt).

Now:

rt= (A12.1)

• that is, the return per share during the period is the dividend paid during it plus
  the increase in price per share (Pj(t+1)−Pjt) expressed as a fraction of the price at the start
  of the period. Rearranging (A12.1) gives:

Pjt= (A12.2)

• that is, the price per share is the discounted value of the dividend for the period
  plus the price per share at the end of the period.
  Market forces will tend to ensure that equations (A12.1) and (A12.2) hold to the
  extent that the current price Pjtwill be such as to cause rtto be similar to that of other
  shares of similar risk to those of business j.
  Further:

n(t+1)−nt=m(t+1) (A12.3)

• that is, the difference between the number of shares at the start and end of the
  period is accounted for by those issued (or withdrawn) during the period:

Vjt= (A12.4)

From equation (A12.4), the value of the whole business is the discounted value of

the total dividend payable during the period plus the value of the business at the end

of the period.

Expressed another way:

Vjt= (A12.5)

• that is, the value of the business at the start of the period is the discounted value
  of the dividend for the period plus its value at the end of the period less the value of
  any share capital raised during the period. This is because

V(t+1)=ntP(t+1)+mtP(t+1) (A12.6)

Djt+V(t+1)−mtP(t+1)

1 +rt

Djt+ntPj(t+1)

1 +rt

djt+Pj(t+1)

1 +rt

djt+Pj(t+1)−Pjt

Pjt