xid-2722972_2 - All As promised here is the algebraic proof...

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Unformatted text preview: All, As promised here is the algebraic proof that shows that equity beta is a function of financial leverage, and thus an equity beta for a firm with leverage is called a "levered beta". I'm putting this up not to stress the importance of this proof but because some of you mentioned that it would be better to slow down and go step by step through some of the detailed calculations. The most important part here is understanding conceptually that an equity beta is driven by financial leverage because 1. leverage magnifies both the upside and the downside and this increases variability or risk and 2. since equity holders are residual claimants when debtholders enter the picture they only are paid after debtholders are serviced. Please email with any questions. Josh Given the following facts: The first line says that the expected return on a portfolio of assets is the sum of the weighted return for each component of the portfolio. The following three lines say that the expected return of any asset is the risk free rate plus the beta of the asset times the market risk premium. I will now substitute each of the bottom three into the top and I get the first hand written line. E E[RAl = yE[rEl+ = E[RAJ E [RD ;{+ ~A = 'f + fJ D (R m - QV\ Bj) +~~ kvM5 \.fo'-\. Me f¥-OM. bD~ ~lcle5 Bit t: -5' BE ~ rJ M\A.I tiply (0.(' ~[1+~o(0~ (~~+ C (ZM -1~") cA S ld.eS Ql'\ct (ctV\cd. ~ If) - -l~() LR//rr)~ ~~ Th e r+ Su~¥'ad- + fJ.,q (Rm If + fJE(Rm • E[RE]= ( rj D yE[rDl Ore. fottV\d ieH uJllt-t ()ote-~::-o ') b~~ by Jf ~ ~ ~(\"~ 011\. bc+t-, ~t: ct'AcA "fcu.. ho.ve. BA-~ Y>E t% Bb -~ £. S tV\. ce, i"VI e(.Q I VI-4'N'S f /) pcu.rtY\e(J\~ M a plAt CL C I -T<.) be hY-e L d ~~C I- TJ (1+1r)::~C ~qu '-\-y ~oldet3 v-e9vl~ (1')1,'1 asse.13 ore (P;>A) plu.5 0- blJl<\p "p tv{ Delat. OS~ 1V\\~()([qV\rlt t1t,fs sCUfs {ehul\. 1 o: tu..~ slA \ el J (Yv""", D/~ ~A - fcf ho.u \:t\ltt- ~~ ...
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xid-2722972_2 - All As promised here is the algebraic proof...

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