Appendix I: Proof of the MM cost of capital proposition (pre-tax)
Appendix II: Proof of the MM cost of capital proposition (after tax)
The following symbols will be used:
VU=SU=market value of the ungeared (all equity) business
SG=market value of the equity of the geared business
BG=market value of the loan notes of the geared business
VG=SG+BG=market value of the geared business
i =interest rate
X =net operating income of both businesses
α =the proportion of either business’s securities owned by an individual.
Suppose that an individual owns proportion αof the shares of the ungeared busi-
ness. This is an investment valued at αVU (or αSU), which will produce income of αX.
The same income could be obtained (with, according to our assumption, the same
risk) by buying proportion αof both the equity and loan notes of the geared business:
Investment required Income producedBuy proportion aof geared business’s equity aSG=a(VG−BG) a(X−iBG)
Buy proportion aof geared business’s loan notes aBG aiBG
Total aVG aX
If VU> VG, the investor would sell the equity shares in the ungeared business and
buy equity shares and loan notes. Thus the investor would end up with the same
income and make a profit on the difference between the price at which the equity
shares of the ungeared business could be sold and what would have to be paid to buy
the equity shares and loan notes of the geared business.
Such action by our investor (and by others similarly placed) will have the effect of
reducing the value of the ungeared business’s equity and increasing that of the geared
one’s securities. This would occur until the value of the two businesses was equal.
Efficient capital market research suggests that such a process would not take long.
The commodity that is being traded here (as far as our individual is concerned) is
annual income of amount αX. MM argued that it would be irrational for this to be
priced differently merely because it is not packaged in the same way. They contended
that the packaging of the income is irrelevant since the individual investor can repack-
age it personally without (according to MM’s assumption on frictionless capital mar-
kets) any personal cost.
MM also tackled the proof from the other starting point.
Suppose that an individual owns proportion αof the equity shares of a geared busi-
ness. This would have a market value of αSG[or α(VG−BG), since VG=SG+BG] and
produce annual income α(X−iBG).
In order to get the same income the individual could buy proportion αof the
ungeared business’s equity and finance it partly by personal borrowing:
Investment required Income producedBuy proportion aof ungeared business’s shares aSU=aVU aX
Borrow (personally) an amount equal to aBG −aBG −aiBG
Total a(VU−BG) a(X−iBG)