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Working Paper 78-4

MEASURING THE DEFAULT RISK OF BONDS
USING YIELDS TO MATURITY

Thomas A. Lawler

Federal Reserve Bank of Richmond
October 1978

The views expressed here are solely those of
the author and do not necessarily reflect the
views of the Federal Reserve Bank of Richmond.

-l-

I.

INTRODUCTION

In both the theoretical and empirical literature of finance the relative
riskiness of two debt instruments identical in all respects save the likelihood
of default on payments of principal and/or interest has generally been measured
by the difference between the yields to maturity of the two debt instruments.l
In a recent paper Benson and Rogowski 111 argue that the relative yield spread,
defined as the yield spread divided by the less risky (or riskless) yield, is a
better measure of default risk, because the value of the expected loss due to
default risk should be "greater the higher are interest rates."2 Unfortunately,
those using the yield spread or the relative yield spread as default risk measures
have not discussed the relationship between their default risk measure and the
way in which investors adjust future promised payments for default risk.3 The
purpose of this paper is to examine this relationship. In Section II the relationship is examined in the context of a simple model where investors are risk-neutral
and where debt instruments differ enly in the probability of default on future
promised payments, and an alternative measure of default risk is proposed.
Section III uses the results of Section II to explain previous empirical findings
concerning the behavior of yield spreads over time, especially the relationship
found between default risk and the level of interest rates.

II.

ADJUSTING FUTURE PROMISED PAYMENTS FOR RISK
To simplify the analysis of investors' adjustment of future cash payments

for default risk, the following assumptions are made:4

lE.g., see Fisher [4], Jaffee [5], Cook and Hendershott [3], and Van Horne [7]
2[11, p. 352.
3An exception is Cook and Hendershott [3, p. 11801.
4These assumptions are not critical, but they do make the exposition easier.
For example, if investors were risk-averse, then pt would incorporate both the
expected value of the future payment and some premium demand for risk-bearing. In
either case, pt represents the dollar amount of a certain future payment in t years
that investors see as equivalent to a risky future payment in t years of $1.

-2a)

investors are risk-neutral;

b)

default is an all-or-nothing event--i.e., if default occurs, there
is no payment;

cl

the probability of default on a future risky payment in t years is
dt, and the probability of payment is pt = l-d,, where 0 <dtcl.

A.

The Single Payment Case
The present value of a future risky payment of $1 in t years is given by
(1) v = (1+i)t ,

where r is the yield to maturity. Risk-neutral investors will be indifferent
between a risky payment in t years of $1 and a certain future payment of $pt.
Consequently, the present value of the future risky $1 payment in t years can also
1

be written
Pt
(2) " = (l+i)t )
where i is the default-free yield, with r>i, because pt<l.

(3) pt =

Equating (1) and (2),

1+i t
(l+r)
.
w

It seems clear that any measure of "default risk" should be dependent only
on the probability of payment pt (or the probability of default dt).

It is easily

demonstrated that neither the yield spread (r-i) nor the relative yield spread
are such measures.

Solving (3) for the yield spread and the relative yield

spread, one finds that

(4) r-i = (l+i)[l-(p+)l'tl
(Pt)l't

and
(5) y

= (1+i) [l-(pt)l'tl
i
(Pt)l't

5
l

Treating pt as a constant, it is obvious from (4) and (5) that the yield spread
(r-i) is positively related to the level of interest rates, and the relative yield
spread 9

is negatively related to the level of interest rates. That is,

5These relationships are also derived in Bierman and Hass [2] and Yawitz [8].

-3(6) a(r-i)
ai

rl-(pt)l't] > o since o<p <le
t '
(PtW

and

(7)

a[+1

ai

=

-'w;4;~1

< o

since O<pt<l.

Thus neither the yield spread nor the relative yield spread is an adequate measure
of default risk alone because each is dependent on the level of interest rates.
To illustnate, suppose that initially the yield on a riskless one-year
no coupon discount bond is 5 percent, and that the probability of default on a
risky one-year no-coupon discount bond is 0.03 (or pt=0.97). From (4) and (5) it
can be seen that r-i = (1*05)(1-*g7) = 3.25 percentage points, and
.97
= 0.65. Now suppose the riskless yield i increases to 10
percent, while the probability of default remains 0.03. The yield spread will rise
slightly to

is now

r-i
i=

(l.lO)(l-.97)
- 3.40 percentage points, while the relative yield spread
.97
(l.lO)(l-.97)
= 0.34, a drop of almost half.
.l
.97

Figure 1 further illustrates the relationship between these two risk
measures and the level of interest rates, by showing how the yield spread and
relative yield spread vary with the level of rates for three different values of pt,
0.96, 0.98, and 0.99.

The figure shows that the yield spread is linearly related

to the level of rates when the probability of default is constant,6 while the
relative yield spread is linearly related to the inverse of the riskless rate.
Thus the relative yield spread is an especially poor measure of default risk,
fluctuating greatly when the level of zates changes and the probability of default
remains constant.

6Thus the inclusion of the riskless interest rate as an independent variable
in an equation where the dependent variable is the spread between risky and riskless
rates is theoretically correct([3, p. llgO]).However, because the level of rates
moves cyclically, its inclusion as a regressor may lead to spurious estimates when
examining the cyclical variability of default risk.

-4It can be seen from (4) that an adequate measure of default risk, at least
for no-coupon discount bonds, is &,

i.e., the yield spread divided by one plus

the riskless yield (not simply the riskless yield), because it is a function of the
probability of default only.7
B.

The Multiperiod Payments Case
Deriving a single measure of default risk for bonds with multiperiod pay-

ments

is more difficult because there may be a different probability of default

associated with each future payment. For example, the probability that next year's
coupon payment will be defaulted may differ from, and may have no relation to, the
probability that the coupon payment due in 10 years will be defaulted.
If there exist bonds in identical risk classes but of maturities of 1, 2,
3,...

N years, as well as riskless bonds corresponding to each risky bond

maturity, then it would be possible to solve for all of the pt's (since there
would be N equations and N unknowns), and thus obtain a default risk measure for
each future promised payment. Unfortunately, such bonds seldom if ever exist, so
the use of this procedure is limited.8
One way to analyze the problem is to make a plausible assumption
regarding the functional relationship between pt and t.

In the following

discussion the relationship between a risky and a riskless bond, both with
coupon payments of C, face value at maturity of $100, and maturity of N years
is examined under the assumption that the conditional probability of payment
on any future promised payment is a constant p, and that when default occurs no
future promised payments will be paid.

In other words, the probability that

'Another adequate default risk measure for no-coupon discount bonds would
be the spread between continuously compounded rates of return. For example, if a
one-year risky no-coupon bond with a par value of $100 sells for $X, then its continuously compounded rate of return is r = ln(lOO)-in(X), the continuously compounded riskless rate of return will be i = ln(lOOpl)-in(X) = ln(lOO)+ln(pl)-in(X), and
r-i = -ln(pl)>O since O<pl<l.

8Silvers [6] uses the prices of bonds in similar risk classes but of maturitic
of 5, 15, and 30 years to estimate the "certainty equivalent coefficients" (the pt
discussed in footnote 4 above) for these years.

-5-

a promised payment in t years will be paid is conditional on the payment of all
previous promised payments. This implies that the actual probability that a payment due in t years will be paid (p,) will equal pt--i.e., the probability of payment falls exponentially over time.'

An assumption of a declining p, over time is

intuitively appealing, at least for most risky bonds, and empirical evidence
suggests that pt does indeed fall in an approximately exponential fashion over
time for rated bonds.10
The present value of the risky bond described above is given by
N
100
\'
1
(8) V = ct& (l+rlt + (l+r>m
.
Because risk-neutral investors are indifferent between a risky payment in t years
of $1 and a certadn payment in t years of $pt, (8) can also be written
t

N

100
-.
>:
(9) v = ctzl (l:*)t + o+oN

.

Rearranging (9) yields
.N
(10)

v

=

c

>:

t=l

1
[1+V]t

100
+ [l+(l+i-PQN
P

l

Setting (10) equal to (8), one finds that
(lla) r-5 = (l+ii(lBP)
and

Thus one finds that if the probability of payment falls exponentially over time,
then the yield spread

is

positively (and linearly) related to the level of interest

rates, and the relative yield spread is negatively related to the level of rates.
The relative yield spread continues to be an especially poor measure of default
9Bierman and Hass [2] and Yawitz [8] also use this assumption to examine the
relationship between risky and riskless yields.
loSee Silvers [6], pp. 947-949.

-6r-i
risk, and l+i is again an acceptable measure.11

III. PREVIOUS EMPIRICAL FINDINGS

The results in the previous section suggest that if default risk remains
constant, then an increase in the level of interest rates will a) raise the spread
between a risky bond yield and a riskless bond yield, and b) lower the reiative
yield spread of these two bonds.

Consequently, in empirical work relating yield

spreads and relative yield spreads to the level of interest rates, our a priori
expectation is for a positive correlation between yield spreads and the level of rates
and a negative correlation between relative yield spreads and the level of rates.
These expectations are realized in the empirical work of others. In one of
the most widely cited articles on the cyclical variability of bond yield spreads,
Jaffee [5] found that the spread between lower-rated and higher-rated bond yields
(e.g., BAA VS. AAA) was positively related to the level of interest rates. Benson
and Rogowski [l] found that the spread between the yields on risky and riskless (AAA)
municipal bonds was positively related to the level of municipal bond rates. They
also found that this municipal bond yield spread divided by the riskless yield--i.e.,
the relative yield spread--was negatively (and very strongly) related to the level
of rates.
If, as was argued in the previous section, neither the yield spread nor the
relative yield spread is an adequate default risk measure, then using these measures
as proxies for default risk may lead to unwarranted conclusions. For example, Benson
and Rogowski argue that the negative relationship between the relative municipal

llOf course, it should be realized that (lla) and (llb) will hold only for
the case where pt=pt, and may not hold for all risky bonds. However, if one wishes
to use the yields to maturity to measure default risk some functional relationship
between pt and t must be assumed, and the above relationship seems quite plausible.

-7bond yield spread and the level of bond rates suggests that "Commercial bank
withdrawal from the market during tight credit conditions causes yields on highgrade bonds to rise faster than low-grade bonds , resulting in narrower yield
lb2 The results of this paper, however, suggest that this negative respreads.
lationship is to be expected simply because of the way in which investors adjust
future promised payments for risk.

Consequently, Benson and Rogowski's results

say nothing concerning the effect of bank portfolio behavior on yield spreads.
One final comment should be made concerning the use of default risk
proxies.

If the default risk measure employed is sensitive not only to default

risk but also to the level of interest rates, then it may be difficult to separate
the cyclical variation in default risk from the cyclical variation in interest
rates. Thus any finding that yield spreads or relative yield spreads have a
cyclical component may be the result of a misspecified dependent variable, and not
of increased riskiness.

IV.

CONCLUSION

This paper has examined the relationship .betweenpreviously used measures
of default risk and the way in which investors adjust future promised payments for
default risk.

The paper argues that if investors are risk-neutral then any default

risk measure should depend solely on the probability of default. The paper then
demonstrated that neither the yield spread nor the relative yield spread satisfy
this criterion, with the spread between risky and riskless yields being positively
related to the level of interest rates and the ratio of this spread to the riskless
yield being negatively related to the level of rates when the probability of default
remains constant. In contrast, the ratio of the yield spread to one plus the riskless yield was found to be dependent solely on the probability of default, and was

relative

%I,
p. 355. Actually, this statement should read ". . . in narrower
yield spreads."

-8-

thus judged to be an adequate measure of default risk.

Finally, the paper indicates

that recent empirical work supports the findings of this paper.

-9REFERENCES

1.

E. Benson and R. Rogowski. "The Cyclical Behavior of Risk Spreads on New
Municipal Issues," Journal of Money, Credit and Banking, X (August 1978),
pp. 349-362.

2.

H. Bierman, Jr. and J. Hass. "An Analytical Model of Bond Risk Differentials,"
Journal of Financial and Quantitative Analysis, X (December 1975), pp. 757-773.

3.

T. Cook and P. Hendershott. "The Impact of Taxes, Risk and Relative Security
Supplies on Interest Rate Differentials," Journal of Finance, XXIII (September
1978).

4.

L. Fisher. "Determinants of Risk Premiums on Corporate Bonds," Journal of
Political Economy, LXVIII (June 1959), pp. 217-237.

5.

D. Jaffee. "Cyclical Variations in the Risk Structure of Interest Rates,"
Journal of Monetary Economics, 1 (July 1975), pp. 309-325.

6.

J. Silvers. "An Alternative to the Yield Spread as a Measure of Risk,"
Journal of Finance, XXVIII (September 1973), pp. 933-955.

7.

J. Van Horne. The Function and Analysis of Capital Market Rates. Englewood Cliffs
New Jersey, Prentice-Hall, 1970.

8.

J. Yawitz. "An Analytical Model of Interest Rate Differentials and Different
Default Recoveries," Journal of Financial and Quantitative Analysis, XII
(September 1977), pp. 481-490.

c

.

Figure 1. Yield Spread and Relative Yield Spread vs. Riskless Interest Rate
For Different Probabilities of Payment
Yield Spread (r-i)
(r-i)
(Percentage Points)
Relative Yield Spread
i

(I)

4
Riskless Rate (1)
(Percent)

6
Riskless Rate (I)
(Percent)


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