Appendix II: Proof of the MM cost of capital proposition (after tax)Investment required Income producedBuy proportion aof the ungeared aSU=aVU aX(1 −T)
business’s shares
Borrow (personally) a(1 −T)BG −a(1 −T)BG −a(1 −T)iBG
Total a[VU−(1 −T)BG] a(X−iBG)(1 −T)
If α[VU−(1 −T)BG] < αSG, then VU< SG+BG(1 −T), which would encourage holders
of the geared business’s equity to switch to the ungeared business’s equity (with per-
sonal borrowing) since they could continue with the same income but make capital
gains on the switch. Such switching would cause
VU=SG+BG(1 −T)Therefore this equality must hold.