Directory UMM :Data Elmu:jurnal:J-a:Journal Of Business Research:Vol49.Issue3.2000:

Impact of Correlation of Asset Value and Interest
Rates upon Duration and Convexity of Risky Debt
Vance P. Lesseig
UNIVERSITY OF TENNESSEE AT CHATTANOOGA

Duane Stock
UNIVERSITY OF OKLAHOMA

Early analysis of duration ignored default risk. Of course, many debt
instruments have at least some potential for default. Recent analyses have
partially filled this void. In our more complete model, we consider that
the default potential of debt issued by many firms is at least partially
dependent upon interest rates. The lower the credit quality of the debt,
the more important this relation becomes. We explore the duration and
convexity of both senior and junior debt. We find that the relation of
asset values to interest rates affects both duration and convexity of debt.
Additionally, the effect of this relationship on convexity is significantly
affected by the shape of the term structure. J BUSN RES 2000. 49.289–
301.  2000 Elsevier Science Inc. All rights reserved.

B

ecause of its importance in bond price volatility measurement, bond portfolio immunization, and financial
institution management, duration has been analyzed a
great deal. Most, but not all analysis has been of defaultfree debt. Fisher and Weil (1971), among others, analyzed
immunization under a non-flat term structure. Bierwag (1977)
was one of the first to provide a rigorous analysis of duration
for various types of changes in term structures. More recently,
Chambers, Carleton, and McEnally (1988), advocated the use
of duration vectors to improve immunization performance
and Stock and Simonson (1988) examined the duration of
amortizing instruments. Prisman and Shores (1988) analyzed
duration measures for specific term structure estimations. In
their modelling of risky debt, Longstaff and Schwartz (1995)
maintain that the duration of risky debt may decline as it
reaches maturity and depends on the correlation of assets with
interest rates.
Although convexity has attracted less attention, it is also
an important tool of bond analysis. Some early literature on

Address correspondence to Dr. D. Stock, Finance Division, 205-A Adams Hall,

University of Oklahoma, Norman, OK 73019. Tel.: (405) 325-5591; fax: (405)
325-1957; E-mail: dstock@ou.edu
Journal of Business Research 49, 289–301 (2000)
 2000 Elsevier Science Inc. All rights reserved.
655 Avenue of the Americas, New York, NY 10010

convexity states simply that the greater the convexity, the more
valuable the bond (Grantier, 1988). This is logical because
convexity essentially compares the positive price movement
from a decline in interest rates to the negative price movement
from an increase in rates. The greater the convexity, the greater
the upside potential compared to the downside potential from
interest rate movements. Thus many portfolio strategies encouraged maximizing convexity, ceteris paribus, although the
effectiveness of this has been questioned [see Schnabel (1990)
and Kahn and Lochoff (1990)]. Notwithstanding some of the
criticisms, convexity is still considered an important characteristic of a bond. Therefore it is important to consider the impact
of the relation between interest rates and asset value upon
both the duration and convexity of a firm’s debt.
The purpose of this research is to analyze the asset value
effect upon duration and convexity. That is, we go beyond

the work of other models by including and focusing upon
the relation between interest rates and asset value. This is
done by utilizing a binomial model for pricing debt and then
computing duration and convexity for bonds with default
risk. We include analysis of both senior and subordinated
debt. The greater riskiness of subordinated (junior) debt enhances the effects we observe for senior debt. Our model
shows that junior duration and convexity are more sensitive
to various parameters than senior debt.
Our results indicate that an increasingly negative relation
between asset value and interest rates causes both duration and
convexity to increase, while an increasingly positive relation
causes both to decrease. Increasing the default risk of the
issuer causes this effect to be enhanced. When junior debt is
considered, the risk and the sensitivity become even more
important. For instance, we show that convexity values can
vary by as much as 50% by simply altering the issuer’s sensitivity to interest rates. Additionally, in our model, this effect is
directly affected by the shape of the term structure. We find
that the steeper the slope of the term structure the greater
the impact of asset interest-rate sensitivity. A flat structure
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PII S0148-2963(99)00016-8

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V. P. Lesseig and D. Stock

substantially reduces the impact of this asset sensitivity and
provides duration and convexity measures quite close to riskless measures.
One of the more interesting findings is that default risk
alone does not have a strong effect on duration and convexity
in the absence of asset sensitivity. Even with high leverage
and volatility and using junior debt, if the issuer’s assets are
insensitive to interest rates the debt displays duration and
convexity similar to that of riskless debt. This result reinforces
the importance of the interest-rate sensitivity of an issuer’s
assets on the performance of debt. However, in cases where
assets are interest-rate sensitive, greater leverage and volatility

have a quite strong impact on the difference between senior
and junior duration and convexity.
The rest of the article is organized as follows. The next
section provides a discussion of the important attributes of
duration and convexity as well as a discussion of interest-rate
sensitivity. The third section describes the model we have
developed, and the fourth section presents hypotheses regarding duration and convexity with respect to our model. The
fifth section discusses the results and compares them to those
hypothesized. The last section concludes the article.

assets depends on default risk. More specifically, duration is
a weighted average of the firm’s asset duration and duration
of a risk free pure discount bond. (The weights are the elasticity
of a default risky bond with respect to firm assets and the
elasticity of a default risky bond with respect to a risk free
bond.) Nawalkha (1996) shows that the duration of a risky
bond must then be something between that of the duration
of assets and that of the pure discount bond. He also finds
that if asset duration is positive, Chance’s (1990) measure of
duration is biased downward. Furthermore, if asset duration

is greater than bond maturity, then duration of the risky bond
is greater than maturity. However, Nawalkha (1996) does not
consider duration of junior versus senior debt nor does he
consider the convexity of risky debt or the impact of term
structure shape upon duration and convexity. Furthermore,
he includes no empirical analysis of his theory (as we do in
Appendix B).
To compute duration our approach follows that of Garman
(1985) where duration will be computed from the following
[Eq. (1)]:

Duration and Convexity

Here DP is the price change due to the shift in rates, Dr is
the size of the parallel term structure shift, and Po is the initial
price of the bond. In our simulation Dr will be a five basis
point parallel shift over the entire term structure. Convexity
is computed with Equation (2).

Duration is often defined as the weighted average timing of

cash flows; it is frequently developed from the first derivative
of the price function (P) with respect to yield (y) and thus
measures the sensitivity of a security’s value to changes in
yield. Convexity compares the relative impact of both a positive and negative shift in yields on bond value. The functional
forms of both duration and convexity are derived from a
Taylor series expansion of the price equation. Duration, (]P/
]y)/P, is represented by the first term and convexity, (]2P/]y2)/
P, by the second. Because duration measures price sensitivity,
convexity measures the change in that sensitivity. The concepts arise from the tradeoff between prices and yields of
bonds as shown in Figure 1. The general shape is convex and
the Taylor expansion provides a close approximation to the
shape of the curve.
The pricing of zero coupon bonds is a simple equation,
P 5 F[e2yT], where F is the face value, y is the yield, and T
is the time to maturity. From this equation duration is equal
to maturity, T, and convexity is T2. [This is because the formula
for duration is 2(]P/]y)(1/P) and that for convexity is (]2P/
]y2)(1/P)]. These values apply to riskless zeros, but when
default-risky zero coupon bonds are analyzed the durations
and convexities can vary significantly from T and T2. We

demonstrate how sensitivity of a firm’s assets to interest rates
affects the value of risky zeros and, in turn, duration and
convexity of the bonds. The convexity of risky debt has not,
to our knowledge, been previously analyzed.
Nawalkha (1996) develops a model where the duration of
default risky bonds issued by firms with interest-rate sensitive

D5

C5

DP 1
.
Dr Po

P1 1 P2 2 2Po 1
(Dr)2
Po

(1)

(2)

where P1 is the price after a positive shift in the term structure,
P2 is the price after a negative shift in the term structure, and
Po is the original price of the bond.
Because we model asset value as a function of interest rates,
both duration and convexity measures will be influenced by
the asset sensitivity to interest rates. The importance of this
relation is perhaps most apparent when considering debt issued by financial institutions whose assets are dominated by
rate-sensitive instruments. For industrial firms the issue of
correlation is not as often recognized but can be significant
for the value of a firm’s debt. Consider industries such as
automobiles and housing which are very interest-rate sensitive. We would expect a strong negative relation between
interest rates and firm assets because demand for their products declines as interest rates increase.
On the other hand, consider firms that have more financial
liabilities than financial assets i.e. net debtor firms. [See DeAlessi (1964) among others.] Increases in interest rates, occurring due to high inflation, result in advantageously low
capital costs, assuming low fixed-rate interest costs are locked
in. Thus higher interest rates encourage enhanced earnings
and can result in a positive relation between interest rates and

asset value. Furthermore, some firms may enjoy dramatically

Duration and Convexity of Risky Debt

higher profit margins if they have large supplies of raw materials purchased prior to large increases in inflation and interest
rates.

The Model
The development of option theory has generated new techniques for pricing contingent securities. However, Merton
(1974) and some other early works applying option theory
to the pricing of risky debt are limited due to their simplifying
assumptions of a flat term structure and constant risk-free
interest rate. While these assumptions often allow a closedform solution, the widely held view is that both the shape of
the term structure and interest-rate volatility are important in
the practical pricing of debt. Black and Cox (1976) and Cakici
and Chatterjee (1993) are examples of important work on
debt options subsequent to Merton (1974). Also, Chance
(1990) and others show that a default risky discount bond
can be represented as a risk-free bond less a put option on
the firm’s assets. We use this technique for valuing risky

bonds.
Specifically, we use the Kishimoto (1989) model with some
small adjustments. Kishimoto (1989) builds upon the Ho and
Lee (1986) model to include the impact of interest rates upon
asset values. The basic model we use assumes the Ho and Lee
(1986) process for interest rates and then applies processes
for (1) changes in asset value due to interest rate changes,
and (2) changes in asset value unrelated to interest rates.
Following Kishimoto (1989), we use the following assumptions:
A1: The time to expiration is N periods of equal length,
each of which is divided into two subperiods. The first
subperiod consists of changes in asset value caused
by interest rate movements which follow the process
suggested by Ho and Lee (1986). The second subperiod represents rate-independent movement resulting
from firm-specific factors.
A2: A frictionless market is assumed with no taxes, transactions costs, or restrictions on short sales. All securities
are perfectly divisible.
A3: The bond market is complete in that a pure discount
bond exists with times to maturity (n) of 0, 0.5, 1,
1.5, 2, . . ., N.
A4: During the first subperiod, interest rate uncertainty is
resolved where the discount function either moves to
an “upstate” or a “downstate” of the term structure.
The discount function is unchanged for the second
subperiod.
A5: Time and the number of upstates completely determine the shape of the discount function. Furthermore,
the price of a risky asset at any time n is completely
determined by the number of upstates of the term
structure and the number of upstates of the asset price.

J Busn Res
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291

Interest Rate Process
Discount factors, Dij(k), are obtained from existing spot rates
where k is the duration of the rate. Dij(k) represents the jth
discount factor that can occur at time i covering k periods.
Ho and Lee (1986) restrict rate movements to an up or down
movement at each node (i). The movements have the form
Di11,j,11(k) 5

Dij (k 1 1)
h (k)
Dij (1)

for an up movement, and
Di11,j(k) 5

Dij (k 1 1)
h* (k)
Dij (1)

for a down movement, with
h (k) 5

1
p 1 (1 2 p) dk

and
h* (k) 5

dk
p 1 (1 2 p) dk

Here p is the predetermined probability of an up movement
and d is a predetermined parameter related to the volatility
of interest rates. d 5 1 implies no volatility of rates, while
lower values imply greater volatility. Hull (1989) attempts to
relate the value of d to the standard deviation of interest rates
because d completely explains this volatility in the Ho and
Lee model. We set d 5 0.995 and p 5 0.5 for our simulations.

Asset Value Process Due to Interest Rates
Asset value movement in the first subperiod is caused by a
change in interest rates. To model the relation between interest
rate changes and asset value, we provide the following asset
value move, gnj [Eq. (3)]:
gnj 5 exp [φ (Rnj (1))]

(3)

where Rnj(1) is the terminal period (period n) one-year rate
computed from the model for each interest rate level j where
j is the number of term structure upstates. φ is a sensitivity
parameter (not a correlation) relating the interest rate to the
change in asset value and can be negative or positive. This
term is multiplied by the initial asset value to represent the
asset value change caused by interest rates. Equation 3 uses
only rates in the terminal period because zero coupon bonds
pay no cash flows until maturity. Therefore the value of the
assets, which determines the put value for bond valuation, is
important only at the maturity of the debt. Because Equation
3 is an exponential function it is consistent with the type of
moves resulting from the asset-specific process. The magnitude of the rate change as well as φ dictates the effect of
interest rates on asset value. The higher the absolute value of
φ the stronger the impact of rates in either a positive or negative
direction. Under this formulation the percentage change in

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asset value, V, will be equal to φ[Rnj(1)]. For instance, with
φ 5 11.5, a terminal one-year rate of 3%, will cause gnj to
be approximately 1.045 producing a positive 4.5% change in
the value of the assets. The arbitrage-free nature of the model
is preserved with use of Equation 3 as the interest rate move
can be considered simultaneous with the asset-specific move
(see Appendix A). Because this form of interest-rate sensitivity
does not allow us to describe the relation between asset value
and interest rates as a correlation, the sensitivity term, φ, is
not limited to the range of 11 and 21.

V. P. Lesseig and D. Stock

subtracting this put value from the initial value of a risk-free
bond.
With the construction of our interest-rate sensitivity, we
have created a recombining sequential binomial tree. This
allows us to compute the time-zero put value directly from
the following formula [Eq. (4)]:
Puto 5

oj oi (jn)(in)[FV 2 (Vognjun2idi)]pn2j
3 (1 2 p)jqn2i(1 2 q)iPDij(1)

(4)

where

Asset Value Changes Unrelated to
Interest Rates
For the second subperiod changes in asset value are unrelated
to interest rates. This process for asset values is computed
based on the work of Rendleman and Bartter (1979, 1980)
(RB). The RB model uses a binomial process that approaches
a log-normal distribution defining up and down movements
in asset value of the following form:
u 5 es√Dt
d 5 e2s√Dt
q5

a2d
u2d

a 5 euDt
where
u
d
q
Dt
s

5
5
5
5
5

size of the up movement
size of down movement
probability of up movement
length of the compounding period
standard deviation of changes in asset value.

The expected change in asset value per unit of time is the
drift term (m). This second subperiod movement is combined
with the move from the first subperiod (gnj) for each period
until maturity to provide the ending asset value. This ending
value is then subtracted from the face value of the debt. The
maximum of this difference and zero represents the ending
put value for each possible interest rate and asset value at the
maturity date of the debt.
It should be noted that having part of the asset value
unrelated to interest rates does not allow us to describe the
relationship between total asset value and interest rates as a
correlation. The sensitivity term (φ) merely describes the impact interest rate changes will have on the value of the assets.

Put Valuation
The value of the time-zero put option is found by discounting
each ending put value through the tree at the risk-free rate,
as this is a risk-neutral process. After the initial put value
is obtained, the initial value of the risky bond is found by

n 5 the number of periods to maturity
i 5 the number of independent down moves of
asset value
j 5 the number of down moves in the Ho-Lee
structure
FV 5 the face value of the debt
Vo 5 the initial asset value
gnj 5 the interest-rate sensitivity factor
p 5 probability of an up move in discount factors
from Ho-Lee
q 5 probability of an up move from the RB model.
The term in brackets, [FV-(Vognjun-idi)], represents the put
value at maturity for a particular j. Vognjun-idi represents the
asset value at maturity. Both combinatorial terms, (ij) and
(ni), are required since the model combines two binomial
processes. The probability terms p and q represent the probability of each put value’s occurrence and are necessary for
discounting. The discount term Pij(1) represents the discount
path for a particular node ij. Each possible put value is then
summed across term structure levels, j, and asset-specific
movements, i. To compute the value of junior debt, the senior
face value is subtracted from the ending asset value at each
node to reflect absolute priority. These put values are then
discounted just as those for senior debt.
The formula allows us to compute the put values and their
probabilities without enumerating each node individually,
thus greatly reducing the necessary calculations. However,
because the discount paths of each node are still distinct,
discounting each individual node would require the computation of 22n discount paths. Since this is beyond current computer capabilities, we use Ho’s Linear Path Space technique
(Ho, 1992) for discounting the nodes. This technique selects
groups of nodes to be discounted along an optimal path,
greatly reducing the number of discount paths being considered while imposing no bias in the chosen path.
The model provides an explicit and easily understood relationship between asset value changes and interest rates while
also including the whole term structure of interest rates.
Chance (1990) and Nawalkha (1996) do not discuss the impact of term structure. That is, we show the change in asset
value to be a simple exponential function of interest rates. If
φ 5 1 and interest rates change x%, asset value change will

Duration and Convexity of Risky Debt

be x%. No such relationship is given in other papers in this
area. In Cakici and Chatterjee (1993), for example, their valuation results are produced by a finite difference procedure to
numerically solve the differential equation. Such a solution
makes intuitive explanation and interpretation of results more
difficult and, potentially, less clear and understandable.

Hypotheses
Duration Hypotheses
From Equation 1 it is easily seen that the greater the magnitude
of DP, for a given change in rates, the greater the duration of
the bond. For a positive shock to rates, we typically expect
the price of a bond to fall regardless of the relation between
asset value and interest rates due to the increase in the discount
rates used to value the bond. If asset value is positively related
to rates, however, this price decline will be diminished because

J Busn Res
2000:49:289–301

293

the increase in rates increases the value of the assets, thus
reducing default risk. With a negative relation and a positive
shift, the decrease in the bond price should be accentuated
since asset value will decline due to the shift in rates.
For a negative shock to interest rates, we should see a
reduction in the price increase if asset value is positively
related to rates. The decline in rates will generally make DP
positive, but asset value should decline to at least partially
neutralize the first effect. A negative relation should enhance
the effect of the negative shock and further increase the size
of DP. We anticipate the impact of asset valuation on duration
will be stronger as credit risk increases. Here we emphasize
comparisons between junior and senior debt and specific analyses of junior debt. As debt becomes riskier more terminal
put values will be non-zero. A non-zero put value will reflect
any change in asset value while a zero put value can only be
affected in one direction, and even that effect is limited if the

Figure 1. Price of zero coupon bond of 30 years maturity, initial yield of 8% with continuous compounding.

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V. P. Lesseig and D. Stock

Figure 2. Price/yield relationship for different sensitivities.

put is deeply “out-of-the-money.” Therefore as credit risk
increases (junior bonds) and more nodes become sensitive to
interest rate changes, there will be a greater impact on duration
and convexity. Thus the increased riskiness of junior debt
should cause duration to be more heavily impacted by interestrate sensitivity. Extensive empirical testing for a sensitivity
effect on bond returns has been performed in Appendix B.
Regressions of bond returns on duration and a proxy for
sensitivity (φ) show a strong significant impact.
Altering the initial term structure directly affects the future
rates determined by the Kishimoto (Ho and Lee) model. In
particular, increasing the upward slope causes future rates to
be higher. In our model we examine how the slope interacts
with the interest-rate sensitivity of assets to affect duration
and convexity. The higher rates implied by the steeper slope
should make the sensitivity factor more important, as the
interest rate level and the sensitivity parameter determines the
size of the asset value movement due to interest rates (see
Equation 3). Because we expect a negative relation between

asset values and interest rates to increase duration and a positive relation to decrease duration, we expect a greater difference between the durations of the two issuer types the steeper
the term structure.

Convexity Hypotheses
The impact of interest-rate sensitivity on convexity is more
complex than for duration. As discussed previously a negative
relation should amplify any price change due to rate changes
while a positive relation will lessen the change. This makes
the impact upon duration easy to predict, but with convexity
we are concerned with the magnitude of the respective positive
and negative shift effects.
Because convexity is the sensitivity of duration to changes
in rates, we can expect the negative relation between asset
value and interest rates that increases duration to increase
convexity as well. The reason is that the relation between
asset value and interest rates should change the price/yield
relationship shown in Figure 1. Figure 2 demonstrates the

Duration and Convexity of Risky Debt

J Busn Res
2000:49:289–301

295

Table 1. Duration Calculations
Base Parameters: m 5 0.02, s 5 0.15, V0 (senior) 5 1500, V0 (junior) 5 2500, d 5 0.997, p 5 0.5
Maturity

2

5

10

15

20

30

Riskless
1) Senior Base
φ 5 11.5
φ 5 21.5
2) Senior V0 5 1000
φ 5 11.5
φ 5 21.5
3) Junior Base
φ 5 11.5
φ 5 21.5
4) Junior V0 5 2000
φ 5 11.5
φ 5 21.5
5) Junior V0 5 2000, s 5 0.20
φ 5 11.5
φ 5 21.5
6) Junior φ 5 11.5
V0 5 2500
V0 5 1800
7) Junior V0 5 1800, s 5 0.20
φ 5 11.5
φ50
φ 5 21.5
8) Junior Base, b 5 1
φ 5 11.5
φ 5 21.5
9) Junior, V0 5 1800, b 5 0
φ 5 11.5
φ 5 21.5

2.00

5.00

10.00

15.00

20.00

30.00

2.00
2.00

4.97
5.18

9.82
10.11

14.83
15.21

19.68
20.09

29.52
30.45

1.72
3.01

4.41
5.51

9.64
10.65

14.43
15.40

19.54
20.51

29.40
30.40

2.00
2.59

4.60
6.25

9.70
10.74

14.37
16.11

19.50
20.74

29.28
30.72

1.42
4.31

3.74
6.20

9.22
11.59

13.79
16.05

19.07
21.45

28.85
31.43

2.00
4.49

3.57
6.41

9.06
11.97

13.44
16.46

18.74
22.18

28.19
32.92

2.00
20.23

4.60
3.76

9.70
8.37

14.37
13.82

19.50
18.36

29.28
28.14

20.38
2.00
4.68

3.57
4.99
7.16

8.02
9.98
12.05

13.43
14.94
16.46

17.70
19.90
22.33

26.86
29.80
33.37

2.00
2.58

4.62
6.25

9.74
10.81

14.43
16.34

19.62
21.11

29.54
31.66

20.43
4.70

3.55
6.45

8.08
12.07

13.68
16.41

18.31
21.94

28.54
31.94

impact on this relationship when the issuing firm’s asset value
is impacted by interest rates.
Consider three distinct issuers, each with a different relation to interest rates: one with φ equal to zero, one with
sensitivity less than zero, and the last one greater than zero.
Assuming all three firms have equal, nonzero default risk, an
interest rate of zero will cause the debt of each to be of equal
value (something less than face value due to the positive
probability of default). However, as the interest rate increases
each issue will be affected differently. The bond with φ 5 0
(no asset value sensitivity) represents the standard price/yield
tradeoff and its curve will lie between the other two. If φ .
0, asset value rises with interest rates. This should increase
the value of the debt relative to that of the insensitive firm’s
debt. The tradeoff will still be convex but flatter than with
φ 5 0. Therefore the convexity of this bond should be lower
than that for an issuer unrelated to interest rate changes.
For firm assets negatively related to rates, the increasing
rate has a greater effect on bond value. The bond is discounted
at higher rates and the asset value is reduced by the higher
rates. Therefore its price/yield curve falls faster than the other
two as rates increase (higher duration) and is more convex
or bent than the other two (higher convexity). This should
cause the positive relation between interest rates and asset

value to reduce both duration and convexity and a negative
relation to increase both duration and convexity.
The assumption of equal default risk is necessary to have
the bond values of all three sensitivities begin at the same
point on the graph in Figure 2. To prove that the curvature
changes we must verify that all three converge at the “end”
of the graph, where yields become extremely large. Zero value
is only achieved asymptotically when the interest rate becomes
infinite. Therefore all three lines will converge to zero and
will have imperceptibly different values at extremely high
rates. If all three begin at the same point, have different slopes
(durations), but the same values at the “end,” they must have
different curvatures (convexities).
The slope of the initial term structure should impact the
degree to which convexity is affected by the relation between
asset value and interest rates. The Ho and Lee (1986) process
used in the Kishimoto (1989) model bases the size of future
interest rate movements on the spread of rates in the initial
term structure. A more steeply sloped term structure will
lead to larger movements in rates. Thus a steeper initial term
structure will result in a larger spread of terminal period rates
computed by the model. This larger spread should cause the
sensitivity parameter (φ) to have a greater effect on the assets
of the firm simply because there will be a wider range of rates

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V. P. Lesseig and D. Stock

Table 2. Difference between Junior and Senior Duration (Using
Base Values)
Maturity
s 5 0.10, φ 5 21.5
A
Vary s only
from Table 1
Senior
Junior
s 5 0.20, φ 5 21.5
Senior
Junior
s 5 0.22, φ 5 21.5
Senior
Junior
s 5 0.25, φ 5 21.5
Senior
Junior
B
Vary s and Change
Leverage from
Table 1
V0 (senior) 5 1000 and V0 (junior) 5 2000
Senior
Junior
s 5 0.20, φ 5 21.5
Senior
Junior
s 5 0.22, φ 5 21.5
Senior
Junior
s 5 0.25, φ 5 21.5
Senior
Junior

S(T) 5 a 1 b

1ln(T)
100 2

20

30

20.22
20.61

30.35
30.71

20.61
21.30

30.81
31.85

where S(T) is the spot rate for T periods, a and b are parameters set at 0.05 and 1.5, respectively, in the base case and
adjusted as described. b represents the slope of the initial
term structure and ranges from 20.5 to 11, while a ranges
from 0.03 to 0.08. A b of 1.0 results in a spread of 3.4%
between 1-year and 30-year spot rates. (This spread was 4.7%
in September 1992 for zero coupon U.S. Treasury strips.)
The use of the equation allows easy adjustments of the term
structure and keeps the shape relatively simple.

20.63
21.47

30.85
32.31

Duration Results

20.65
21.80

30.90
33.76

20.53
21.05

30.61
31.06

20.84
22.43

31.05
33.46

20.88
22.86

31.11
34.87

20.93
23.83

31.20
43.11

used in the model. Therefore a steeper slope should magnify
the difference between convexities of negatively and positively
sensitive assets. As with duration we should see a greater effect
for riskier junior debt in all cases.

Results
For our simulations we use the following base case variables
for senior debt: firm-specific asset growth (m) of 2%, standard
deviation of growth (s) of 15%, sensitivity (φ) of 0, initial
asset value of $1,500, and face value of debt of $1,000. For
junior debt the only changes are that beginning asset value
is raised to $2,500 and the face value of both junior and
senior debt is $1,000. The leverage is similar to that used by
Cakici and Chatterjee (1993) in their analysis of bank debt.
Of course, many banks are even more leveraged, and the
leverage used is similar to that of many nonfinancial firms
with large issues of high yield (junk) bonds. See Cakici and
Chatterjee (1993, footnote 5) for examples of high leverages.
The initial term structure used in our simulations is created
by the following equation:

The size of the sensitivity effect is dependent upon the riskiness
of the debt, the steepness of the term structure, and the
magnitude of the sensitivity parameter. Table 1 summarizes
the results for duration calculations where five of the seven
cases address junior debt. Case 1 demonstrates the difference
in durations for senior debt with base case parameters and
sensitivities of 11.5 and 21.5. It shows that the debt with
sensitivity of 21.5 has greater duration for maturities greater
than two but the difference is less than one year at a maturity
of 30. Case 2 shows the effect of increasing the riskiness of
the debt by lowering the asset value for the senior debt issuer
to $1,000. Note that the firm still has some equity as the
present value of debt is less than face value. Here the difference
between sensitivities illustrates the impact of increased riskiness. The difference in durations is one full year or more for
almost all maturities which is a large proportionate difference
for shorter maturities. For example, at a maturity of two years
the duration with φ 5 21.5 is 75% greater than for φ 5 11.5
(3.01 versus 1.72). Importantly, duration exceeds maturity for
negative φ cases but not when φ is positive in all Case 2
examples.
In general junior debt displays a greater response to changes
in parameters. Case 3 shows a greater difference than that
occurring for senior debt. With the base case parameters,
junior debt has a duration difference of about 1.5 years between sensitivities of 11.5 and 21.5 for maturities of 15
years or more. The proportional difference for short maturities
(under 10 years) is quite large. Case 4 shows that as initial
asset value is decreased to $2,000 the difference in duration
is very close to 2.5 years even at very short maturities. At a
maturity of 2 years the duration of the negative sensitivity is
more than double that of positive sensitivity. When the standard deviation is increased to 0.20 (Case 5) the duration
spread increases slightly with maturity, reaching more than
4 full years at a maturity of 30 and is 2 or more across all
maturities. Note that duration substantially exceeds maturity
(for example, 4.49 versus a maturity of 2) when a negative
sensitivity is used in Vases 3, 4, and 5.
We find that asset value (leverage) and volatility (s), parameters had comparatively little impact on duration in some

Duration and Convexity of Risky Debt

J Busn Res
2000:49:289–301

297

Table 3. Convexity Calculations
Base Parameters: m 5 0.02, φ 5 0.15, V0 (senior) 5 1500, V0 (junior) 5 2500, d 5 0.997, p 5 0.5
Maturity
Riskless
1) Senior Base
φ 5 11.5
φ 5 21.5
2) Senior V0 5 1000
φ 5 11.5
φ 5 21.5
3) Junior Base
φ 5 11.5
φ 5 11
φ 5 21
φ 5 21.5
4) Junior V0 5 1800, s 5 0.20
φ 5 11.5
φ 5 11
φ50
φ 5 21
φ 5 21.5
5) Junior b 5 11
φ 5 11.5
φ 5 21.5
6) Junior Debt φ 5 21.5
b 5 1.5
b 5 2.5
7) Junior V0 5 1800, s 5 0.20, b 5 0
φ 5 11.5
φ 5 21.5

2

5

10

15

20

30

4.00

25.00

100.00

225.00

400.00

900.00

4.35
4.35

24.93
27.43

96.97
101.89

221.26
233.34

392.37
408.74

884.41
913.26

3.97
9.49

20.20
31.01

92.92
113.90

209.58
239.02

387.30
426.44

876.86
936.92

4.35
3.01
6.33
6.99

21.56
22.43
27.40
39.60

94.24
88.71
110.02
116.26

207.65
213.62
235.07
261.42

385.78
376.45
424.28
436.61

869.05
858.34
937.08
958.25

20.67
0.00
4.25
12.36
17.97

12.87
15.63
27.28
34.97
48.24

61.41
72.77
96.43
126.69
145.89

179.40
195.40
226.58
253.58
273.86

314.35
339.57
398.06
468.36
494.43

717.78
775.91
860.85
1041.03
1118.60

4.10
7.37

23.02
38.41

93.84
115.68

207.73
265.14

380.79
440.45

856.51
983.79

6.99
6.82

39.60
28.57

116.26
115.57

261.42
238.32

436.61
423.72

958.25
930.06

0.00
17.06

10.86
39.22

61.03
143.22

180.32
263.06

320.12
462.12

789.23
985.50

cases. Case 6 shows the duration for two issues with sensitivity
parameters of 11.5 each but where one has an asset value of
$1,800 and the other $2,500. Note that the difference between
the two is typically just over 1 year past a maturity of 5 years.
Also note that a negative value for duration is present at the
2-year maturity. This occurs because the reduction in default
risk from the increased asset value dominates the effect of
higher discount rates on the bond price.
The most striking results are achieved when the junior
debt is used with asset value of $1,800 and standard deviation
of growth of 0.20 [parameters which are consistent with those
used by Cakici and Chatterjee (1993)]. Case 7 illustrates this
result where the difference in duration between sensitivities
of 11.5 and 21.5 is never less than three periods and grows
to over six at a maturity of 30. At shorter maturities the
negatively related duration can be more than double the positively related.
Case 7 also displays the results of zero sensitivity. Note
that even with the low asset value and the high standard
deviation (s 5 0.20) the durations are still very close to
that of the riskless shown at the top of the table. This again
demonstrates the importance of the sensitivity, φ, relative to
other risk factors.
Cases 8 and 9 illustrate the hypothesized impact of term
structure. Compare Case 8 to Case 3 and Case 9 to Case 7

where the only difference is Case 8 assumes a greater term
structure slope (than Case 3) and Case 9 assumes a lesser
slope (than Case 7). The spread between 30-year durations
is greater in Case 8 (than Case 3) but less in Case 9 (than
Case 7).
The difference between senior and junior duration is quite
sensitive to s and leverage (initial asset value). Table 2, panel
A, contains duration as s varies for long maturities if φ is 21.5
where greater volatility increases the difference and junior
duration always exceeds senior. All other parameters are unchanged from Table 1. For example, if s is 0.10, the difference
in duration for a 30-year maturity is only 0.36 (30.71 versus
30.35) but 2.86 (33.76 versus 30.90) for s of 0.25. Table 2,
panel B, contains variation in s and also lowers asset values
(greater leverage) so that the difference in duration grows
even more dramatically as s increases. That is, the difference
in duration is almost 12 when maturity is 30 and s is 0.25.
Examination of the table reveals that the increase in duration
difference is due to junior duration being quite sensitive to
s while senior is not very sensitive.

Convexity Results
The initial results for convexity are found in Table 3. Seven
cases are given where five are for junior debt. In every case
the convexity with a negative sensitivity exceeds that of the

298

J Busn Res
2000:49:289–301

V. P. Lesseig and D. Stock

Table 4. Difference between Junior and Senior Convexity (Using
base values)
Maturity
s 5 0.10, φ 5 21.5
A
Vary s only
from Table 3
Senior
Junior
s 5 0.20, φ 5 21.5
Senior
Junior
s 5 0.22, φ 5 21.5
Senior
Junior
s 5 0.25, φ 5 21.5
Senior
Junior
B
Vary s and Change
Leverage from
Table 1
V0 (senior), 1000, V0 (junior) 5 2000
Senior
Junior
s 5 0.20, φ 5 21.5
Senior
Junior
s 5 0.22, φ 5 21.5
Senior
Junior
s 5 0.25, φ 5 21.5
Senior
Junior

20

30

405.77
422.29

907.34
928.37

421.76
448.76

933.49
988.75

422.50
454.91

936.70
1021.89

422.96
465.53

938.55
1105.65

418.97
440.37

923.12
949.91

429.84
491.37

948.77
1092.57

432.66
512.41

949.93
1161.62

435.22
554.77

954.77
1623.85

positive. Case 1 shows the senior base case convexities with
sensitivities of 11.5 and 21.5. Notice that each convexity is
quite close to that of a riskless bond shown at the top of the
table. Case 2 reveals that as initial value is decreased to $1,000
the difference between sensitivities grows significantly and
increases with maturity. At a maturity of two, the convexity
of the negative φ is more than twice the positive.
Junior debt shows much greater difference in convexities
and is again much more sensitive to parameter changes than
senior. Case 3 shows the base case positions with sensitivities
of 11.5, 11, 21, and 21.5 where, again, convexity differences increase with maturity. Changing the initial asset value
provides significant changes to the convexity of junior debt.
Case 4 shows that when initial asset value is reduced to $1,800
and the standard deviation is raised to 0.20 the difference
between convexities becomes very large—over 400 between
the high and low sensitivities at 30-year maturities. Notice
that when the sensitivity is equal to zero, convexity is much
closer to that of the riskless bond. This again verifies the
importance of the relation between asset value and interest
rates.
We can also analyze the term structure effects on junior

debt. Case 5 uses b 5 11 instead of 10.5 and shows the
convexity differences resulting from the increased slope of the
initial term structure. Note that with the steeper slope the difference due to the sensitivities is greater. The difference in
convexities grows from about 90 for Case 3 to almost 130 in
Case 5, at a maturity of 30 years. However, the direction of
this term structure slope is not as important as the magnitude.
Case 6 compares convexities of negative sensitivity issues under both an upward (b 5 0.5) and a downward sloping term
structure (b 5 20.5). The two convexities are similar because
each slope creates similar differences between spot rates for
1 and 30 years which will make the spread of terminal rates
projected by the Kishimoto (1989) model approximately the
same under each slope. Thus, while the different slopes may
affect the relative value of the debt, it has less impact on the
debt’s sensitivity to rate changes.
Case 7 in Table 3 shows the impact of a flat term structure
(b 5 0). The flat term structure creates a set of convexities
that have considerably less variation in value for long maturities than in Case 4 because a flat term structure implies no
change in expected interest rates. This term structure shape
does not result in convexities exactly equal to the riskless case
due to nonzero asset value volatility.
Similar to Table 2, the difference between junior and senior
convexity is quite sensitive to s and leverage. Table 4, panel
A, shows convexity as s varies for long maturities if φ is
21.5 where greater volatility increases the difference quite
dramatically and junior convexity always exceeds senior. All
other parameters are the same as in Table 3. For example, if
s is 0.10, the difference in convexity for a 30-year maturity
is only 21.03 (928.37 versus 907.34) but 167.10 (1105.65
versus 938.55) if s is 0.25. Table 4, panel B, again varies s
but at a lower asset value so that the difference in convexity
grows even more dramatically with s. That is, the difference
in convexity is 669 when s is 0.25 and maturity is 30. Examination of the table for a 30-year maturity reveals that the
increase in convexity difference is due to junior convexity
being quite sensitive while senior is not very sensitive.

Conclusion
This article has demonstrated some of the important effects
that relating asset value to interest rates can have on a firm’s
debt. We have incorporated a non-flat term structure and
volatile interest rates, and then related these to the assets of
the firm. The results indicate that if asset value is negatively
related to interest rates, debt will be more sensitive to changes
in rates than an issue with a positive or no relation. The junior
debt durations and convexities for risky debt can be much
different than those of riskless debt and senior debt where
this difference becomes more pronounced when interest-rate
sensitivity is included. No previous research has analyzed the
convexity of default risky bonds. With a negative relation we
show that both duration and convexity increase from the

Duration and Convexity of Risky Debt

riskless levels, and duration can exceed maturity. When a
positive relation exists duration and convexity are reduced
from those of riskless debt. Additionally, the results displayed
by junior debt are more dramatic than those of senior debt.
This inherent riskiness of junior debt seems to magnify the
impact of the asset’s sensitivity to interest rates. Greater leverage and volatility magnify the difference between junior and
senior duration and convexity. The shape of the term structure
can play a critical role in determining duration and convexity.
Thus duration defined as the sensitivity of bond price is
clearly inconsistent with duration defined as the weighted
average timing of cash flows. As the risk of the issue increases,
these effects are dramatically enhanced. Although this is obviously important for financial institutions, many other firms
are affected by interest rates in some way making many of
the issues discussed in this article important for any bond
investor. A bond portfolio manager using simple Macaulay
duration computed from a weighted average of cash flows
may well have an inaccurate measure of price volatility.
The authors thank Ajay Madwesh for his exceptional work in programming
the model used in this article. Also, we thank Louis Ederington for helpful
comments.

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Appendix A
The excess returns to a bond are functions of the perturbation
terms h(1) and h*(1). Given that we model varying degrees
of sensitivity (φ) of asset value to interest rates, the excess
returns become fairly complex functions of h(1) and h*(1) as
shown below.
In our model, asset value and interest rate movements are
related in the following manner
gnj 5 eφ(Rnj(1)) but Rnj(1) 5 2lnDnj(1).
Thus gnj 5 eφ(2lnDnj(1)) 5 Dnj (1)2φ.
Looking at the first subperiod of the first period
g11 5 D11(1)2φ 5 gup, and g10 5 D10(1)2φ 5 gdown.
Here,
D11(1)2φ 5

2φ
D00(2)
h(1)
and
D00(1)

1

2

Ho, Thomas S. Y., and Lee, Sang-Bin: Term Structure Movements
and Pricing Interest Rate Contingent Claims. Journal of Finance
41 (Dec. 1986): 1011–1029.

D10(1)2φ 5

Hull, John: Options, Futures and Other Derivative Securities, PrenticeHall, Englewood Cliffs, NJ. 1989.

Thus gup and gdown are the only factors determining the

1

2φ
D00(2)
h*(1) .
D00(1)

2

300

Table A1.

Aaa and
Aa
A-Ba
Aaa-Ba

J Busn Res
2000:49:289–301

1DP

i,t

V. P. Lesseig and D. Stock

2

1 ci/12
2 rt 5 b0 1 b1Di,t(2Drt) 1 b2(φi) 1 b3(SPDi,t22) 1 ei,t
Pi,t
b0

b1

b2

b3

0.00077
(2.288)*
0.0024
(10.527)
0.0020
(7.899)

0.8588
(69.11)
0.7201
(60.149)
0.6345
(44.273)

20.000309
(24.112)
20.00014
(22.456)
20.000201
(23.228)

0.0900
(7.271)
0.1624
(17.958)
0.2077
(21.004)

Adjusted
R2

Number of
Observations

0.7954

1,371

0.4143

8,299

0.3934

9,670

* t-statistics in parentheses

change in asset value due to interest rate changes from one
period to the next. Within gup and gdown, both Doo(2) and
Doo(1) are given by the initial term structure used to price the
risk-free bond. Thus the only arbitrage possibility results from
the perturbation functions h(1)2φ and h*(1)2φ giving the no
arbitrage requirement that
ph(1)2φ 1 (12p)h*(1)2φ 5 1
Exhaustive simulations testing that this requirement holds
have been performed. The simulations assume realistic values
for d, φ values within the range used in this research, and
varying p values. Realistic d values are those that yield realistic
interest volatilities using Hull’s (1989) formula for per annum
interest rate volatility. ds less than 0.995 are thus very unusual
because smaller ds give volatilities over 50% which are unrealistically high. φ values of close to zero and d values close to
one give a result ph(1)2φ plus (1 2 p)h*