Systematic risk, leverage, and default risk

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BEBR

FACULTY WORKING
PAPER NO. 914.

Systematic Risk, Leverage, and Default Risk
K. C. Chen

College of Commerce and Business Administration
Bureau of Economic and Business Research
University of Illinois. Uroana-Chamoaign

BEBR

FACULTY WORKING PAPER NO. 914
College of Commerce and Business Administration
University of Illinois at Urbana-Champaign
November 1982

Systematic Risk, Leverage, and Default Risk

K. C. Chen, Assistant Professor
Department of Finance

Abstract
The purpose of this note is to investigate the theoretical rela-
tionship between the systematic risk of equity, the systematic risk of
debt, the systematic risk of the unlevered firm, and leverage in the
presence of default risk. The cash-flow approach is adopted in con-
trast to the literature. The analysis demonstrates that a truncation
factor (or survival probability) exists in addition to Hamada and
Rubinstein's traditional formulation. Hence, the result derived here
is more general.

SYSTEMATIC RISK, LEVERAGE, AND DEFAULT RISK

In their classical paper, Modigliani and Miller (M&M) [15, 16],
based upon the risk-class assumption and the arbitrage argument, have
shown the famous propositions I and II. By integrating M&M's proposi-
tion I with the mean-variance, Hamada [9] and Rubinstein [18] have
shown that the systematic risk of a firm's equity should be positively
correlated with the firm's leverage. Numerous subsequent studies have
empirically and theoretically investigated the effect of financial
leverage on the systematic risk of equity [3, 4, 7, 8, 10, 14], How-
ever, only few of them have incorporated default risk in the analysis
[7, 8].

The purpose of this note is to investigate the theoretical rela-
tionship between the systematic risk of equity and leverage in the
presence of default risk within a framework of one-period Capital Asset
Pricing Model (CAPM) under uncertainty. We adopt cash-flow approach
which distinguishes from [2] and [8] with option-pricing approach and
[7] with expected-rate-of-return approach. In Section I, we discuss
the pricing of market values of different claims, which is borrowed
from Chen [6].* In Section II, we derive the relationship between
systematic risks and leverage. Section III presents the conclusion.

I. Market Values of Different Claims
Sharpe [19], Lintner [12], and Mossin [17] have derived the fol-
lowing two-parameter equilibrium valuation raoidel, referred to as the
Capital Asset Pricing Model, in a hypothetical world with three key
assumptions.

-2-

where

V ■ the equilibrium value of asset j;

E(Y.) = the expected value of the end-of-period cash
flows to the owners of asset j;

R • 1 + R~» where R^ is the risk-free interest rate;

Cov(Y.:,R ) = the covariance between the total cash flows of
asset j and the return on the market portfolio;

A = the market price of risk.

Equation (1) states that in equilibrium the value of asset j is

the present value of the certainty-equivalent (CEQ) of the asset's

random cash flow.

A. The Market Value of All- Equity Firm

Denote X as the firm's operating income which is assumed to be
jointly normally distributed with the return on the market portfolio
so that

X = N(X, o£)

— 2

for any given assessment of R and a . The after-tax cash flows to

mm

the owners of the unlevered firm are

X(1-t) if X > 0

(2)

if X < 0

'"I o

-3-

where x is the proportional corporate income tax. Therefore, the mar-

2

ket value of the unlevered firm is given by

Vu = (1-t)[E0(X) - XCov0(X,Rm)](R)"1. (3)

where

EQ(X) =

Xf(X)dX; Covn(X,R ) = E{ [Xn-En(X) ] [R -E(R) ]},

U m u u m m

"3
the partial covariance between X truncated from 0 upward and R

m

B. The Market Value of Debt

For simplicity, we assume that the total promised payment to bond-
holders is tax deductible. Bondholders receive their contractual

claims of D at the end of the period if the firm is solvent, and the

4
entire value of the firm if the firm is declared bankrupt. Hence,

the total cash flows to bondholders at the end of the period are

" D if X >_ D

\-\ . (4)

X if 0 < X < D

0 if X < 0.
The market value of debt can be expressed as

VQ = {D[1-F(D)] + [EJj(X) - ACov°(X,Rm)]} (R)"1 (5)

|0 - -

where F(D) = I f(X)dX, the probability that the firm is declared

bankrupt.

-4-

C. The Market Value of Equity

At the end of the period shareholders receive the after-tax
residual value of the firm if it remains solvent, and they receive

■

nothing if the firm goes bankrupt. Therefore, the end-of-period cash
flows to shareholders are

(1-t)(X-D) if X > D

h'\ - (6>

0 if X 1 D.

The market value of equity can be expressed as

V£ = (1-T){ED(X) - XCovD(X,Rm) - D[1-F(D)]} (R)"1. (7)

D. The Market Value of the Levered Firm

The market value of the levered firm is simply the sum of market
values of its debt and equity. Adding V_ in (5) and V_ in (7), we get

D J-

V = {(1-t)[E0(X) - XCovQ(X,Rm)] + t[E°(X) - ACov^X.R^ ]

+ tD[1-F(D)]} (R)"1 (8)

Then, substituting Vu in (3) and VD in (5) into (8), the market value
of the levered firm can be expressed as

V - VE + VD = Vu + TV (9)

Equation (9) shows that the market value of the levered firm is the sum
of the market value of the unlevered firm plus the tax subsidy on debt.
This result is consistent with M&M [16] within a framework of risky
debt.

-5-

II. Systematic Risks and Leverage
In this section we are trying to develop the theoretical linkage
between systematic risks of equity, debt, and the unlevered firm and
leverage.

A. The Systematic Risk of Equity, the Systematic Risk of the Unlevered
Firm, and Leverage

From Sharpe-Lintner-Mossin's CAPM, the systematic risk of equity

is defined as

s Cov(Y Rm)
g , - (10)

2 .
where a is the variance of market portfolio's returns. Substituting

m ™

(6) into this definition yields

(l-T)Cov(X,Rm)
8 ~ • [l-F(D)] (11)

E m

By the same token, substituting (2) into the definition of the
systematic risk of the unlevered firm yields

Cov(Y ,R )

„u u m

V a2
u m

(1-t)Cov(X,R )
m

u m

[l-F(O)] (12)

" " 2
By solving (11) and (12) for (l-T)Cov(X,Rm) /a , we can derive the

relationship between the systematic risk of equity and the systematic

risk of the unlevered firm as follows

-6-

E
This result shows that the systematic risk of the levered firm is
equal to the systematic risk of the unlevered firm adjusted for the
difference in equity value of the two firms and the survival proba-
bility (the bracket in (13)). When no bankruptcy risk (or no truncation
of the firm's operating income distribution) is assumed, (13) is identical
to Hamada's [9] result. Furthermore, substituting the accounting identity
in (9) for V , we derive the following expression:

This result states that the systematic risk of equity is equal to
the systematic risk of the same firm without leverage times one plus
the leverage ratio (debt to equity) multiplied by one minus tax rate
and times the survival probability. If no bankruptcy risk is assumed,
the second bracket in (13) disappears and (14) is identical to what
Hamada [9] and Rubinstein [18] have shown. Hence, the model we derive
here is claimed to be more general.

To further study the comparative statics of (14), we use numerical
analysis instead of mathematic analysis for the sake of simplicity.
The data for the numerical example is given in table I.

Insert Table I

Figure 1 illustrates the effect of leverage (debt ratio) on the
systematic risk of equity. As is expected from this figure, the systematic
risk of equity increases monotonically with leverage.

-7-

Insert Figure 1

Figure 2 shows the effect of the face value of debt on the systematic
risk of equity. Not surprisingly, the systematic risk, of equity is a
positive function of the face value of debt.

Insert Figure 2

In figure 3, the impact of business risk on the systematic risk of
equity is depicted, where business risk is represented by standard deviation
of the firm's operating income. To isolate the leverage effect, we
designate the face value of debt equal to 150,000. As is evident from
this figure, the more risky the firm (the higher the standard deviation) ,
the smaller the systematic risk of equity because stockholders profit
from the probability that the value of the firm will exceed the face
value of debt.

Insert Figure 3

In the option pricing literature, Black and Scholes [2] and Galai
and Masulis [8] have shown that

3 = nsB

VD 3VE V

= (1 + r> t? s (15)

E

where

-8-

3VE V
nc = ~^T7 * TT~ » the elasticity of equity value with respect

E to firm value;

V
6 = the systematic risk of the firm,

Comparing (14) with (15) without corporate tax, both equations are
quite similar in the sense that the truncation factor in (14) and the
partial-derivative factor in (15) both reflect the default risk, and
the relationship between the systematic risk of equity and leverage
is curvilinear. However, (15) with elasticity concept is not as empir-
ically appealing as (14) with truncated distribution. The latter can
be estimated in a way similar to Aharony, Jones, and Swary [1J, who
estimate the probability of bankruptcy from a truncated normal distribu-
tion. Omitting the truncation factor is (14) which is always less than
one with positive leverage will cause the systematic risk of equity
overestimated. Hence, the implication of this model stands along the
same line as Hamada [9, p. 445] in the sense that it should be pos-
sible to improve the forecast of a stock's systematic risk by fore-
casting the total firm's systematic risk first, and then make adjust- .
ments on leverage and survival probability.

B. The Systematic Risk of Equity, the Systematic Risk of the Unlevered
Firm, and the Systematic Risk of Debt

Like the systematic risk of equity, the systematic risk of debt

Q

can be defined by the CAPM as

jj Cov(VV

2
D ra

Cov(X,R )

m

D m

f- [F(D) - F(0)] (16)

-9-

Given the result shown in (16), we can further demonstrate the

linkage between the systematic risk of equity, the systematic risk of

the unlevered firm, and the systematic risk of debt (the proof is

shown in the Appendix) .

V V

6S = BU[1+(1-t) =£] - 6D[(1-t) =2] (17)

E E

This result is consistent with Conine [7] in the presence of

risky corporate debt. The same result without corporate tax can be

9

derived from the option pricing model. The model says that the system-
atic risk of equity is a weighted average of the systematic risk of
the unlevered firm and the systematic risk of debt (with negative
weight), which is intuitively appealing in a portfolio sense. By using
the same numerical example, figure 2 illustrates that the systematic
risk of debt not only is a positive function of the face value of debt
but cannot in equilibrium exceed the systematic risk of the unlevered
firm. Under this kind of formulation, the truncation of the distribution
due to default risk is not shown in (17), instead is embedded in 6
and leverage. When corporate debt is riskfree, 0 is equal to zero
and (17) is identical to the traditional formulation shown by Hamada
and Rubinstein.

III. Conclusion
The purpose of this note is to investigate the theoretical rela-
tionship between systematic risks and leverage in the presence of default
risk. The cash-flow approach is used in contrast to the literature.
The analysis shows that a truncation factor (or survival probability)
exists in addition to Hamada [9] and Rubinstein's [18] traditional for-
mulation. Hence, the result derived here is claimed to be more general.

-10-

Footnotes

*University\of Illinois at Urbana/Champaign.

(1) There exists a fixed risk-free interest rate in perfectly
competitive capital markets; (2) all investors have homogeneous ex-
pectations with respect to the probability distributions of future
yields on risky assets; and (3) all investors are risk-averse and the
expected utility of terminal wealth maximizers.

2
Because of the existence of default risk, the assumption of

quadratic utility is implicitly required to apply the CAPM.

3

For discussion of truncation, refer to Lintner [13] and Chen [6].

4
This is an agency-cost issue. A numerical example can illus-
trate why bondholders will not receive the entire after-tax value of
the firm if the firm is declared bankrupt. Let D = $100, t = 50%,
and X = $99. In this case, the firm is declared bankrupt because
X < D. If bondholders had to receive the after-tax value of the
firm, $49.5, they would be better off by making side payments of the
one dollar short to stockholders to persuade them not to go bankrupt.
Hence, bondholders would net $99, which is exactly equal to X.

We assume that there are no costs of voluntary liquidation or
bankruptcy, e.g., court or reorganization costs.

We should expect to get identical results as shown by Galai
and Masuli [8] in an option pricing context.

No corporate and personal taxes are assumed.

Q

Sfnce the debt by nature is a single-period discount bond, the
problem of duration on the systematic risk of debt does not arise.

9

Black and Scholes [2] and Galai and Masulis [8] have shown that

6S = N(d )rp 6V (18)

E

6D = [1-N(d )] ^- 6V (19)

D

where N(») is the standardized normal cumulative probability density

V
function. Then, multiplying (19) by—, adding (18), and rearranging

E
yields

-11-

E VE

6va+^)-sDA.

E VE

Q.E.D,

-12-

Ref erences

1. J. Aharony, C. P. Jones, and I. Swary. "An Analysis of Risk and

Return Characteristics of Corporate Bankruptcy Using Capital
Market Data." Journal of Finance (September 1980).

2. F. Black and M. Scholes. "The Pricing of Options and Corporate

Liabilities." Journal of Political Economy (May /June 1973).

3. A. J. Boness, A. H. Chen, and S„ Jatusipitak. "Investigations of

Non-Stationarity in Prices." Journal of Business (October
1974).

4. R. Bowman. "The Theoretical Relationship Between Systematic Risk

and Financial (Accounting) Variables." Journal of Finance
(June 1979).

5. M. J. Brennan and E. Schwartz, "Corporate Income Taxes, Valuation,

and the Problem of Optimal Capital Structure." Journal of
Business (January 1978).

6. A. H. Chen. "Recent Developments in the Cost of Debt Capital."

Journal of Finance (June 1978).

7. T. E. Conine, Jr. "Corporate Debt and Corporate Taxes: An

Extension." Journal of Finance (September 1980).

8. D. Galai and R-.-W. Masulis. "The Option Pricing Model and the Risk

Factor of Stock." Journal of Financial Economics (January/
March 1976).

9. R. Hamada. "The Effect of the Firm's Capital Structure on the

Systematic Risk of Common Stocks." Journal of Finance
(May 1972).

10. N. C. Hill and B. K. Stone. "Accounting Betas, Systematic

Operating Risk, and Financial Leverage: A Risk-Composition
Approach to the Determinants of Systematic Risk." Journal
of Financial and Quantitative Analysis (September 1980).

11. E. H. Kim. "A Mean-Variance Theory of Optimal Corporate Structure

and Corporate Debt Capacity." Journal of Finance (March 1978).

12. J. Lintner. "The Valuation of Risk Assets and the Selection of

Risky Investments in Stock Portfolios and Capital Budgets."
Review of Economics and Statistics (February 1965).

13. . "Bankruptcy Risk, Market Segmentation, and

Optimal Capital Structure," in Risk & Return in Finance,

I. Friend and J. Bicksler (eds.), Ballinger Publishing Co.,
1977.

-13-

14. G. Mandelker and S. G. Rhee. "The Impact of Financial and

Operating Leverages on the Systematic Risk of Common Stocks."
(Forthcoming, Journal of Finance).

15. F. Modigliani and M. Miller. "The Cost of Capital, Corporate

Finance, and the Theory of Investment." American Economic
Review (June 1958).

16. and . "Corporate Income Taxes and

the Cost of Capital: A Correction." American Economic
Review (June 1963).

17. J. Mossin. "Equilibrium in a Capital Asset Market." Econometrica

(October 1966).

18. M. Rubinstein. "A Mean-Variance Synthesis of Corporate Finance

Theory." Journal of Finance (March 1973).

19. W. Sharpe. "Capital Asset Price: A Theory of Market Equilibrium

Under Conditions of Risk." Journal of Finance (September
1964).

M/E/286

Table I
Parameters for Numerical Example

Corporate tax rate (t)° =0.5

Expected market return (R ) = 0.15

m

One plus risk free rate (R) =1.05

Standard deviation of market return (a ) = 0.2

m

Standard deviation of operating income (a ) = 80,000

Mean of operating income (X) = 120,000

Correlation coefficient between the firm and the market = 0.5

B
E
T
R

BU

1.4

-

BS

12

-

•

•

1C

:'

8

'm

;'

c

'

b

4

,.-'"

2

c

i i !

1 I.I

i 1 1 1

0 0. 1 0. 2 0. 3 G. 4 0. 5 0. 6 0. 7 0. 8 0.9 I

DEBT RRTIQ

Figure 1. The relationship between systematic risks and debt ratio

B
E
T
R

BD

14

BU

BS

/
/

12

/
/
/

10

/
/

s

/

8

•

6

4

._--"'

^" "

2

_ — -""

— — — —

0

I i l

\

1 1 1 1 1

25 50 75 100 125 150

FACE VALUE OF DEBT

175 200 225

THOUSANDS

Figure 2. The relationship between systematic risks and face value of debt

B

E
T

10

BS

a

i i '" i ~ i ~ I i 1^ i i i I I i

30 35 40 45 50 55 60 65 70 75 80 85 90

STANDARD DEVIATION THOUSANDS

Figure 3. The relationship between systematic risk and standard deviation

Appendix
Let's restate equation (14) as follows

RS _ fiu ,V rl-F(D),
S " 6 X} [1-F(0)J

a ,Vux „u ,V rF(D)-F(0).

- 8 M - .8 (v~) [ 1-F(0) 1 (18)

E E

We also can derive the relationship between the systematic risk

of the unlevered firm and the systematic risk of debt by solving (12)

and (16) for Cov(X,Rm)/a2.

m

V

ftu = «D t\-r\ -2. r_l=£i2i_i (19)

& - 3 (1-t) v LF(D)_F(o)J K^>

u
Then, substituting (19) into the second term of (18) yields

E E

V V

= eu[i+(i-x) -2.] - sd[(i-t) =2.]

E E

Q.E.D.

;CKMAN

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