Chapter 1 • Introduction
Thus by investing α(VU−BG) the investor can get the same income as had been
obtained from an investment valued at α(VG−BG). If the market value of the geared
business (VG) is greater than that of the ungeared one (VU), it will benefit our inves-
tor to sell the equity stake in the geared business and then go in for some ‘personal’
gearing.
Once again the action of investors in selling the geared business’s equity and buy-
ing that of the ungeared business would, by the laws of supply and demand, quickly
drive the two market values (VUand VG) back into equality.
The assumptions on which the above analysis is based are stated and discussed in
the chapter.APPENDIX II Proof of the MM cost of capital proposition (after tax)
It is proved below that:VG=VU+TBG*Suppose that an individual owns proportion αof the shares of an ungeared busi-
ness. This would be an investment valued at αVU(or αSU), which will produce income
αX(1 −T), that is, proportion αof the after-tax income of the business.
The same income could be obtained by buying proportion αof the equity and pro-
portion α(1 −T) of the loan notes of the geared business:Investment required Income produced
Buy proportion aof geared business’s equity aSG=a(VG−BG) a(X−iBG)(1 −T)
Buy proportion a(1 −T) of the geared
business’s loan notes aBG(1 −T) aiBG(1 −T)
Total a(VG−TBG) aX(1 −T)If α(VG−TBG) < αVU(that is, if VG< VU+TBG), it means that the investor could
retain the same income [αX(1 −T)] by selling the ungeared equities and replacing
them with the cheaper geared business’s shares and loan notes. The action of investors
making this switch will tend to cause VG=VU+TBG.
Tackling the proof from the other starting point, suppose that an investor holds
proportion αin the equity of a geared business. This would give an income of
α(X−iBG)(1 −T). (This is because the interest on borrowings (iBG) is tax deductible.)
The investor could obtain the same income from selling these shares and buying pro-
portion αof the shares of the ungeared business and by borrowing amount α(1 −T)BG
to help finance the purchase of the shares.* Note that =TBG
that is, the present value of the saving on loan interest (iTBG) is TBG.iTBG
(1 +i)nα
∑
n= 0