Bierman and Hass (1975)

8/17/2019 Bierman and Hass (1975)

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default on its bonds and second on the marketability of the bonds. The three

variables chosen by Fisher as determinants of the risk of default were earnings

variability (as measured by the coefficient of variation of the firm's net in-

come over the past nine

years),

financial leveraqe (as measured by the firm's

market value of equity to book value of debt ratio), and reliability in meeting

obligations (as measured by the length of time the firm has been operating with-

out forcing creditors to take a loss).

Fisher defines the risk of default as the probability that the firm's

earnings will not be sufficient for the firm to meet the payments on its debts.

The variables noted above are those measurable variables he deems investors con-

sider in estimating the aforementioned probability. In the next section of

this paper we will present a graphical model of this interest rate setting pro-

cess. In the third section we will relate this process to the cost of debt

capital schedule and the debt capacity of a firm under some assumed investor

preferences. We will conclude with some mathematical models that extend the

graphical process described earlier.

II. The Risk of Default: A Graphical Analysis

The first objective of this section is to determine the size of the risk

differential on a bond necessary to give an investor an expected present value

equal to the present value he would earn if his funds were invested in a default-

free security. For simplicity let us assume that (1) the bond under discussion

is a perpetual bond and (2) the probability of the firm's survival (and conse-

quently meeting its interest obligations) from one period to the next is p, in-

dependent of the length of time of survival. Then the expected present value

of a bond that promises to pay annual interest of I dollars is the discounted

value of the expected interest payments, using the default-free interest rate i:

(1)

(2)

B -

(ffj,

[ i

+

(

JB_,

2

Fisher, Lawrence. "Determinants of Risk Premiums on Corporate Bonds,"

The Journal of Political Economy (June 1959), pp. 217-237. Since the capital

asset pricing literature uses the term "risk premium" to describe the differ-

ence between the expected rate of return and the risk-free rate, we use the

term "risk differential" to describe the difference between the contractual

rate and the risk-free rate.

Both of these assumptions will be relaxed later in the paper.

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Probability

of Survival

M

M

Required Contractual

Rate of Return

Figure 1

Investor Required Contractual Rate of Return

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for all possible levels of interest payments. This is shown in quadrant III

of Figure 2, assuming the firm has

no

cash resources other than earned income.

Let us now turn to the options available to the firm issuing the debt.

Most important is the fact that, for a given level of outstanding debt, the

probability

of

survival

is

inversely related

to

the contractual interest rate

paid

on

the debt: the larger the interest rate, the higher the promised annual

interest payments, and the lower the probability of survival. The relationship

between the interest rate (r) and probability of survival (p) is depicted in

quadrant

I of

Figure 2.

To

determine this relationship, let us choose

an

in-

terest rate, say r , on the horizontal axis of either Quadrant I or IV, and

o

trace clockwise around the four quadrants. The implied interest payment for

B of debt is I and the corresponding probability of survival is p . Thus if

the firm has

B

par value dollars

of

debt outstanding and pays interest rate

r , the probability of survival is p . Quadrant II has a 45° line that enables

us to move from Quadrant III to I. In Quadrant I the point r , p ) is the

o o

first point

to be

determined

of an

interest rate-survival probability feasibil-

ity curve QQ. By successive tracings for varying interest rates, one can de-

velop the interest rate-survival probability set shown in. the first quadrant

of Figure

2

for

B of

debt.

Finally, Figure 3 shows how to combine the investor's required rate of

return set from Figure 1 (denoted MM) with the interest rate-survival probabil-

ity feasible set from the first quadrant of Figure 2 (denoted QQ) to ascertain

r*, the minimum interest rate required by the market for B of debt.

The default-risk differential is (r* - i) and from equation 4)

r

* . i = i±L _ . i

P*

where p is taken from the equilibriu m solu tion in Figure 3.

Example

Suppose the probability distribution

of

earnings before interest and taxes

is uniform between $0 and $1000 and the firm wishes to sell $300 worth of bonds

at par. If the default-free interest rate is 4 percent and investors behave

according

to

equation (4), what

is

the minimum acceptable equilibrium interest

rate and what is the default risk differential the company must pay?

The solution will be obtained algebraically. From the above suppositions

Should

MM

and

QQ

cross more than once, the lower

or

lowest

r

is,

of

course,

the correct interest rate since the firm would have to pay higher in-

terest costs at all other r's.

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1 -

Figure 2

The Interest Rate - Survival Probability Set

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ir*

M

Figure 3

Equilibrium Interest Rate Determination

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the following relationships follow:

I = 300 r . (quadrant IV)

I = 1000 - 1000 p (quadrant III)

(a) The investor requirement curv e, MM, can be obtained directly from

equation (4) :

(b) The firm's tradeoff set, QQ , is obtained by solving the equations

I = 300 r and I = 1,000 - 1,000 p

for p in terms of r:

300 r = 1000 - 1000 p

1000 - 300 r , „.

or p = — = 1 - .3 r (00)

(c) Taking the two equations from (a) and (b) above and solving simul-

taneously for r we find that the two roots are

r = (.058,

2.275)

so that, ignoring the higher root,

r* = 5.8%, and the risk differential = 5.8% - 4% = 1.8%.

The probability of survival is:

p*

= 1 - .3 r* = 1 - .3(.O58) = 0.983.

IV. The Cost of Debt and Debt Capacity

The model presented in the preceding section can be used to determine both

the cost of debt capital schedule and the debt capacity for a given firm. One

approach would be to solve repeatedly for r* as we systematically vary the debt

outstanding, B, to find the cost of debt schedule, r*(B). As B increases, the

firm's survival probability feasibility curve 0.Q of Figure 3 shifts inward.

Hence for a given investor preference curve, MM, r*(B) will rise as B increases

until MM and QQ are tangent, and the corresponding B, denoted B' , is the

max

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firm's debt capacity. While the market is willing to hold B debt at inter-

mix

est rate r*(B

) ,

it will not purchase any more debt than B because the firm

max max

cannot meet the investors' return-probability requirements.

An equivalent method would be to place the investor preference schedule,

MM, in Quadrant I (see Figure

4 ) .

Choosing a point on MM, say (r ,p ) , let us

move counterclockwise through Quadrant II and Quadrant III to determine the mar-

ket interest payment (I ) that can be paid with probability p . The point

o o

(r ,1 ) in Quadrant IV is one point defining a feasible amount of interest for

o o

given investor preference and a given interest rate r . By starting in Quad-

rant I with different values of r and repeating the process, we obtain the

curve labeled iR which is the set of points (r, I) that the firm could offer

to the market via bond issues and the market would accept. For any given level

of debt the equilibrium promised interest rate is determined by the intersec-

tion of the debt curve and iR curve in Quadrant IV. Thus if B debt is issued

9

it would have to carry a promised rate of r*(B ) . The firm's debt capacity is

o

equal to B since a larger amount of debt will result in an interest require-

max

ment that is not feasible.

The cost of debt schedule is, of course, a necessary input into any capi-

tal structure (mix of debt and equity financing) decision. Aside from provid-

ing such a schedule, the above analysis also sheds some light on the extent to

which the firm can substitute debt for. equity financing. B is not the maxi-

max

mum market value of the firm. Implicit in the above model is the assumption

that any earnings over and above the required interest payment I are paid as

equity dividends.

Thus there is an expected dividend stream for any debt level B (including

B ) that will vary in value inversely with the debt level. While further

max

J

Referring to Figure 3, as B increases and QQ shifts inward, the relevant

intersection of QQ and MM slides down MM, so that r*(B) increases as B increases.

For a general discussion of corporate debt capacity see G. Donaldson,

Corporate Debt Capacity

(Harvard University, 1961).

g

Richard Holman suggested this method of analysis.

9

Since r*(B ) < r , the firm will issue bonds at r*(B ) rather than r .

o o o o

See footnote 5.

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M

max

r*(B ) r*(B )

o max

Figure 4

The Cost of Debt and Debt Capacity

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exploration of this facet of the problem is inappropriate here, a few comments

are in order:

(1) If excess earnings are retained in order to assist in meeting future

obligations in peri ods when income is not sufficient to meet debt

payments, investors would revise upward the probability of survival

over time reflecting this "cushion" effect. B would correspond-

nicLX

ingly be increased. We approach this complexity indirectly in the

next section by examining investor preference behavior when the prob-

ability of survival increases over time with successful operations

rather than, as previously assumed, remaining constant.

(2) In discussing optimal capital structure considerations, many theorists

have discussed the virtually "all debt-financed firm." Income

limi-

tations and investor preferences will limit the amount of debt a firm

can issue but an equity residual may still remain and carry some mar-

ket value as long as some probability of a dividend on stock remains.

(3) While the above analysis assumes investor preferences are solely the

result of expected monetary value considerations, the methodology and

qualitative concl usion s do not rely on that assumption. Risk prefer-

ences can be incorporated into the investor's decision process and the

resultant MM curve would shift to the right. The extent of the shift

would depend on the extent to which the investor is risk adverse and

on diversification possibilities (portfolio considerations) implicit

in the firm's income stream.

V. Relaxing the Survival Assumptions

In this section we shall change the assumptions of the model in two ways:

first we shall relax the assumption that the bond has infinite maturity by al-

lowing a principal repayment with a probability measure attached; second, after

ascertaining the equilibrium interest rate assuming finite maturity, we shall

allow for the revision of probabilities of survival, conditional upon past sur-

vival and examine how this modification affects the equilibrium interest rate.

These two extensions of the previous model affect the market trade-off curve

MM above, making it a more realistic investor indifference locus.

Let p denote the proba bility of surviving each period (and consequently

If the bondholders were risk averters, the MM curve in quadrant I of

Figure 4 would be flatter; its corresponding RR curve would

fall-

below the

existing one in quadrant IV and the consequent maximum level of debt and the

promised interest payme nt (I) would be less than it would in the absence of

risk aversion and the expected equity dividend greater.

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meeting debt interest payments) while the debt is outstanding, p denote the

probability of collecting the final interest payment and principal at maturity,

and N denote the length of time to maturity. The market price of the debt sell-

ing at par is B. Assuming investors make decisions on the basis of expected

monetary value, they will be indifferent between investing in a riskless se-

curity yielding the default-free rate i and this debt if the promised interest

rate on the debt, r, is such that

fc

P?

1

P,d+r)B

(5) B = I -*• " +

Dividing both sides by B and rewriting equation (5) using known geometric sum

formulas,

^1 H_1 M_1

P,(l+r)

N-l

- 1

or

P. r p" "

1

-

(l+i)

11

"

1

+

P

Solving (6) for r we find

(7)

[ 1+i)

- p£ p

2

l (p

x

- 1 - i)

p?

1

- d+i)

1

] + P ~ V

Note that if we divide the numerator and denominator of equation (7) by (1+i)

we find

and

liro

iLA

N

which is our earlier equation (4). Equation (4), in fact, holds for a bond of

any length of maturity (N) if the probability of survival is constant over time.

If p..=p_=p» equation (7) reduces to equation (4) as follows:

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p)

[(14-J)

N

- p

N

] (p-l-i)

(p-l-i)

- p

N

] (p-l-i)

- P

N

] (p-l-i)

.. .,N

- (l+l) p + p

N+l

- p

N

]

or

(4)

p - l - i 1 + i

- ~ - - p -

Thus, the risk differential is invariant with the life of the bond when inves-

tors maximize expected net monetary value and the probability of survival is a

constant over time. The result that, when p = p = p, the risk differential

is invariant to change in the length of time to maturity is not intuitively ob-

vious. One explanation is that the risk differential does not change as maturity

changes since the change in the expected present value of the principal repay-

ment is exactly offset by the change in the expected present value of the inter-

est payments.

Suppose the maturity date is put off from N to N + M. Then the change

(loss) in the net expected present value of the principal repayment is

P

N + M

B

- 1]

while the change (gain) in the net expected present value of the interest pay

ment is

1+i

P - (1+i)

Equating these two differences we find that

rp

p - d+ i)

- 1

or

(4) as

we

anticipated.

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Table 1 shows the risk differentials for different values of N, p and

p_

for i = .05; these values were computed using equation (7) and defining the

difference between the contractual risky rate (r) and the riskless rate (i) as

the risk differential. The table shows that when p > p , as length of time

to maturity increases, the risk differential falls. Also, as p decreases for

a given p and maturity, the risk differential increases.

We now turn to an examination of the effect of revising the probability

of survival over time. As Fisher's empirical analysis suggests, a history of

past survival enhances the probability of future survival and consequent inter-

est and/or principal collection.

While it is undoubtedly dangerous to claim that probabilities of survival

will always increase with length of survival, if earnings are retained in pro-

fitable years and invested in risk-decreasing investments, this will change the

prbbability of survival curve for a given amount of debt. The proportion of

excess earnings retained, as well as the rate of return earned on their rein-

vestment and the nature of the investment, will determine the rate at which the

probabilities of survival are revised upward.

While numerous formulations of the probability revision process could be

developed, we will use a simple one that reflects the spirit of the procedure.

w + t

Let us assume the probability of surviving in period t is if the firm

O T t

has survived

in all

previous

t-1

periods.

If the

bond sells

at par and

pays

interest r on par value, and if investors are using the expected present value

of

the

future promised payments discounted

at the

default-free rate

i as the

basis of setting the market pri ce, then:

_ _

N

"

1

(W-t-t) I SI rB (W+N)

SI B(l+r)

t N

t=il (S+t) 1 W (1+i) (Wl) (S+N)

(1+i)

or

(8) B = rBA + B(l+r)G

The calculations for all the tables in this section were performed b y

David Downes.

We could also interpret

the

situation

as one

where

we do not

know

the

probability of survival, make an estimate of the prior probability, and revise

the probability as new information is obtained.

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T

1

V

O

T

H

R

S

K

D

F

A

F

R

M

E

Q

O

7

W

=

.

0

T

i

m

e

t

o

M

a

t

u

r

i

t

y

(

5

1

0

2

0

3

0

0

0

P

2

=

9

9

9

.

0

0

1

.

0

0

1

.

0

0

1

.

0

0

1

.

0

0

1

P

l

P

2

=

9

9

O

.

0

0

3

.

0

0

2

.

0

0

1

.

0

0

1

.

0

0

1

.

9

9

9

P

2

=

9

5

O

.

0

1

0

.

0

0

5

.

0

0

3

.

0

0

2

.

0

0

1

P

2

=

9

O

O

.

0

2

0

.

0

0

9

.

0

0

4

.

0

0

3

.

0

0

1

P

2

=

9

5

O

.

0

5

5

.

0

5

5

.

0

5

5

.

0

5

5

.

0

5

5

P

l

=

P

2

=

9

.

0

6

8

.

0

5

9

.

0

5

6

.

0

5

6

.

0

5

5

.

9

5

0

P

2

=

8

O

O

.

0

8

4

.

0

6

6

.

0

5

8

.

0

5

6

.

0

5

5

P

2

=

7

.

1

0

5

.

0

7

3

.

0

6

0

.

0

5

7

.

0

5

5

p

=

.

9

0

0

p

.

. . .

.

P

l

=

8

0

0

.

1

3

6

.

1

2

3

.

1

1

8

.

1

1

7

.

1

1

7

.

9

0

0

p

=

.

7

0

0

.

1

5

5

.

1

2

9

.

1

1

9

.

1

1

7

.

1

1

7

P

2

=

6

O

O

.

1

7

5

.

1

3

5

.

1

2

0

.

1

1

7

.

1

1

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16/17

. .

N

I

(W+t) 1

SI ., ..-t . _

(W+N)

S „ ..-N

where A = J ^

( s + t )

,

w

, (1+D and G =

( S + N )

,

w

, d+i ) .

Solving equation (8) for r we have

Table 2 displays the risk differential required by investors implied by

equation (9) for i = .05 and selected values of N, W, and S. There are three

aspects of the table we wish to highlight:

(i) For any given initial probability of survival, denoted p in Table

w+i

2, there are infinite numbers of W and S pairs such that rrr = p..

S+JL J.

For any given p , the lower the value of W (or S, since S = W/p ) ,

the faster the rate of revision. Hence, for any given p in Table

2, the risk differentials for the lower W are lower than for higher

W's.

(ii) The initial probability of survival, p , does not alone determine

the risk differential. In comparing two situations, it is possible

that the one with the higher initial probability of collection has

higher risk differentials for some N if the probability of survival

revises slowly. For example, compare the values in the columns under

p = .85 and p = 4/5 = .8 in Table 2 for N = 10. The p = .85 has

a higher risk differential than the p = .8 column since probability

of survival revises very slowly with W/S =

85/100.

(iii) For any given W and S, the risk differential decreases with length

of time to maturity. Earlier we found that when probabilities of

survival are held constant over time, the risk differential was in-

variant to length of time to maturity. Here, the probability of the

principal being repaid is higher than with the earlier model when

maturity is lengthened since the probability of survival for each

period revises upward over time; also the expected interest payment

is higher for the same reason, so the risk differential falls in this

case.

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TABLE 2

VALUES OF RISK DIFFERENTIAL FROM EQUATION (9)

WHEN

i = .05

Time

to

Maturity (N)

2

5

10

20

30

P

l

W+l= 9

S+l=10

.111

.099

.088

.077

.073

= .90

W+l= 90

S+l=100

.116

.115

.113

.111

.110

Pj_

= -85

'

W+l

Documents

Bierman and Hass (1975)