San José State University
Department of Economics

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Capital Structure and the Value of the Firm

The Effect of Leverage on the Value of the Firm


The Market Line and the Effect of Leverage

The expected rate of return on equity, requity, and the beta risk on equity, βequity, both satisfy from their definitions the same sort of equation; i.e.,

requity = rassets + L(rassets - rdebt)
βequity =βassets + L(βassets -βdebt),

where L is the debt/equity ratio.

The Capital Asset Pricing Model (CAPM) says that the expected rate of return and risk for the common stock should fall on the Market Line:

requity = rf +βequity(rm - rf),


where rf is the riskfree interest rate and rm is the expected rate of return on the market portfolio.

Suppose the expected rate of return and risk fall on the Market Line for an unleveraged firm; i.e., L=0. It is shown below that under some standard assumptions, the expected rate of return and risk will fall on the market line for any leverage ratio.

If the unleveraged firm return and risk fall on the market line this means that
(1)

rassets = rf + βassets(rm - rf).


Now let us consider whether the return and risk for a leverage ratio L will satisfy the equation of the market line. (2)

rassets + L(rassets - rdebt) =
rf + (βassets + L(βassets -βdebt))(rm - rf).


If we subtract equation (1) from equation (2) we are left with

L(rassets - rdebt) =
L(βassets -βdebt)(rm - rf).


We can divide through by L to get

rassets - rdebt = (βassets -βdebt)(rm - rf).


This can be rewritten as

rassets =
rdebt + βassets(rm - rf) -βdebt(rm - rf).


Now, if rdebt = rf and βdebt=0 then this equation is just equation (1),

rassets = rf +βassets(rm - rf).


This is just the equation of the Market Line. thus the expected rate of return and risk will fall on the Market Line for any leverage ratio L.

The Effect of Leveraging on the Value of a Company
When Taxation is Taken Into Account


Let t be the tax rate on corporate profits. Let Y be the earnings before interest and taxes (EBIT). Without leverage,

βequity =βassets.


The required rate of return on equity is then

requity = rf +βassets(rm - rf)


Without debt the income to equity holders is Y(1-t) so the value of the company is

VU = Y(1-t)/requity
= Y(1-t)/[rf +βassets(rm - rf)]

With a leverage ratio of L and riskfree debt (βdebt=0)

βequity = (L+1)βassets


so the required rate of return on equity is

requity = rf + (L+1)β assets(rm - rf).

The income after interest (taxable profits) is

Y - rdebtD.

The debt share of total capital D/(D+E) is equal to L/(L+1). Therefore debt is equal to (L/(L+1)) times total capital, but total capital is the same as VU. Thus,

Y - rdebtD = Y - rf(L/(L+1))VU

After taxes the income to equity holder is

(1-t)(Y - rdebtD)

and the valuation of the company is

EL =
(1-t)(Y - rdebtD)/
(rf + (L+1)βassets(rm - rf)).

When the assumption of rdebt = rf and the relations for D and VU are substituted into this expression one gets:

EL =
(1-t)(Y - rf(L/(L+1)))VU)/
[rf + (L+1)βassets(rm - rf))]

It is convenient to express Y(1-t) as VU(rf + βassets(rm - rf)) so

EL =
[VU(rf +βassets(rm - rf)
- (1-t)rf (L/(L+1))VU]/
[rf + (L+1)βassets(rm - rf))]

For convenience now βassets is expressed as βa. Thus,

EL/VU =
[rf + βa(rm-rf)-(1-t)rf(L/(L+1))]/
[rf+(L+1)βa(rm-rf)]
=
[(L+1)rf + (L+1)βa(rm-rf))-(1-t)rfL)]/
[(L+1)(rf + (L+1)βa(rm-rf)]
=
[rf + (L+1)βa(rm-rf) +trfL)]/
[(L+1)(rf + (L+1)βa(rm-rf))]
= (1/(L+1))[1 + trfL)/
(rf + (L+1)βa(rm-rf))]

The value of the company is then

VL = EL + D
= EL + (L/(L+1))VU
= [1 + trfL)/
[rf + (L+1)βa(rm-rf)) + L]VU)/(L+1))
and
VL/VU =
[1 + L[1 + trf/[rf + (L+1)βa(rm-rf)]] (1/(L+1))

Example: Let L=3, t=0.4, rf=0.05, rm-rf=0.08, andβa=0.8.

Then

VL/VU =
[1+3(1+0.02)/(0.05+4(0.8)(0.08))]/4
= [1+3(1+0.02)/0.306)) )]/4
= 1.04902

Effect of Leverage on the Valuation of Shares

The effect of leverage on the valuation of shares can also be determined from the above analysis. Let N be the number of share in the unlevered corporation. Then the price per share, pU is given by:

pU = VU/N.


In achieving a leverage ratio of L, the equity share of capital is reduced from 1 to 1/(L+1) so the number of shares is reduced from N to N/(L+1)). The price per share for the leveraged firm is then

pL = EL/(N/(L+1))) and
pL/pU = (L+1)EL)/VU))

This reduces to

pL/pU = [1+trfL/ [rf+(L+1)βa(rm-rf)]


For example let L=10, t=0.4, rf=0.05, rm=0.09 and βa=0.5.

Then

pL/pU =
1+(0.4)(0.05)(10)/
0.05+(10+1)(0.5)(0.04)
= 1+0.2/0.27) = 1.74.

Thus if PU = $50 then PL = $87.

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