Impact of Correlation of Asset Value and Interest Rates upon Duration and Convexity of Risky Debt

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: [email protected] 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 ISSN 0148-2963/00/$–see front matter PII S0148-2963(99)00016-8

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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.

Duration and Convexity 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 ⫽ F[e⫺yT], 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 ⫺(⳵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

V. P. Lesseig and D. Stock

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)]: D⫽

⌬P 1 . ⌬r Po

(1)

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

P⫹ ⫹ P⫺ ⫺ 2Po 1 (⌬r)2 Po

(2)

where P⫹ is the price after a positive shift in the term structure, P⫺ 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.

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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 Di⫹1,j,⫹1(k) ⫽

Dij (k ⫹ 1) h (k) Dij (1)

for an up movement, and Di⫹1,j(k) ⫽

Dij (k ⫹ 1) h* (k) Dij (1)

for a down movement, with h (k) ⫽

1 p ⫹ (1 ⫺ p) ␦k

and h* (k) ⫽

␦k p ⫹ (1 ⫺ p) ␦k

Here p is the predetermined probability of an up movement and ␦ is a predetermined parameter related to the volatility of interest rates. ␦ ⫽ 1 implies no volatility of rates, while lower values imply greater volatility. Hull (1989) attempts to relate the value of ␦ to the standard deviation of interest rates because ␦ completely explains this volatility in the Ho and Lee model. We set ␦ ⫽ 0.995 and p ⫽ 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, ␥nj [Eq. (3)]: ␥nj ⫽ 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 ⫽ ⫹1.5, a terminal one-year rate of 3%, will cause ␥nj 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 ⫹1 and ⫺1.

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 ⫽

兺j 兺i (jn)(in)[FV ⫺ (Vo␥njun⫺idi)]pn⫺j ⫻ (1 ⫺ p)jqn⫺i(1 ⫺ q)i⌸Dij(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 ⫽ e␴√⌬t d ⫽ e⫺␴√⌬t q⫽

a⫺d u⫺d

a ⫽ eu⌬t where u d q ⌬t ␴

⫽ ⫽ ⫽ ⫽ ⫽

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 (␮). This second subperiod movement is combined with the move from the first subperiod (␥nj) 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 ⫽ the number of periods to maturity i ⫽ the number of independent down moves of asset value j ⫽ the number of down moves in the Ho-Lee structure FV ⫽ the face value of the debt Vo ⫽ the initial asset value ␥nj ⫽ the interest-rate sensitivity factor p ⫽ probability of an up move in discount factors from Ho-Lee q ⫽ probability of an up move from the RB model. The term in brackets, [FV-(Vo␥njun-idi)], represents the put value at maturity for a particular j. Vo␥njun-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 ⌸ij(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 ⫽ 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 ⌬P, 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

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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 ⌬P 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 ⌬P. 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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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

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Table 1. Duration Calculations Base Parameters: ␮ ⫽ 0.02, ␴ ⫽ 0.15, V0 (senior) ⫽ 1500, V0 (junior) ⫽ 2500, ␦ ⫽ 0.997, p ⫽ 0.5 Maturity

2

5

10

15

20

30

Riskless 1) Senior Base φ ⫽ ⫹1.5 φ ⫽ ⫺1.5 2) Senior V0 ⫽ 1000 φ ⫽ ⫹1.5 φ ⫽ ⫺1.5 3) Junior Base φ ⫽ ⫹1.5 φ ⫽ ⫺1.5 4) Junior V0 ⫽ 2000 φ ⫽ ⫹1.5 φ ⫽ ⫺1.5 5) Junior V0 ⫽ 2000, ␴ ⫽ 0.20 φ ⫽ ⫹1.5 φ ⫽ ⫺1.5 6) Junior φ ⫽ ⫹1.5 V0 ⫽ 2500 V0 ⫽ 1800 7) Junior V0 ⫽ 1800, ␴ ⫽ 0.20 φ ⫽ ⫹1.5 φ⫽0 φ ⫽ ⫺1.5 8) Junior Base, ␤ ⫽ 1 φ ⫽ ⫹1.5 φ ⫽ ⫺1.5 9) Junior, V0 ⫽ 1800, ␤ ⫽ 0 φ ⫽ ⫹1.5 φ ⫽ ⫺1.5

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 ⫺0.23

4.60 3.76

9.70 8.37

14.37 13.82

19.50 18.36

29.28 28.14

⫺0.38 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

⫺0.43 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 ⫽ 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 ⫽ 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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Table 2. Difference between Junior and Senior Duration (Using Base Values) Maturity ␴ ⫽ 0.10, φ ⫽ ⫺1.5 A Vary ␴ only from Table 1 Senior Junior ␴ ⫽ 0.20, φ ⫽ ⫺1.5 Senior Junior ␴ ⫽ 0.22, φ ⫽ ⫺1.5 Senior Junior ␴ ⫽ 0.25, φ ⫽ ⫺1.5 Senior Junior B Vary ␴ and Change Leverage from Table 1 V0 (senior) ⫽ 1000 and V0 (junior) ⫽ 2000 Senior Junior ␴ ⫽ 0.20, φ ⫽ ⫺1.5 Senior Junior ␴ ⫽ 0.22, φ ⫽ ⫺1.5 Senior Junior ␴ ⫽ 0.25, φ ⫽ ⫺1.5 Senior Junior

S(T) ⫽ ␣ ⫹ ␤

冢ln(T) 100 冣

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, ␣ and ␤ are parameters set at 0.05 and ⫹.5, respectively, in the base case and adjusted as described. ␤ represents the slope of the initial term structure and ranges from ⫺0.5 to ⫹1, while ␣ ranges from 0.03 to 0.08. A ␤ 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 (␮) of 2%, standard deviation of growth (␴) 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 ⫹1.5 and ⫺1.5. It shows that the debt with sensitivity of ⫺1.5 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 ⫽ ⫺1.5 is 75% greater than for φ ⫽ ⫹1.5 (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 ⫹1.5 and ⫺1.5 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 (␴), parameters had comparatively little impact on duration in some

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Table 3. Convexity Calculations Base Parameters: ␮ ⫽ 0.02, φ ⫽ 0.15, V0 (senior) ⫽ 1500, V0 (junior) ⫽ 2500, ␦ ⫽ 0.997, p ⫽ 0.5 Maturity Riskless 1) Senior Base φ ⫽ ⫹1.5 φ ⫽ ⫺1.5 2) Senior V0 ⫽ 1000 φ ⫽ ⫹1.5 φ ⫽ ⫺1.5 3) Junior Base φ ⫽ ⫹1.5 φ ⫽ ⫹1 φ ⫽ ⫺1 φ ⫽ ⫺1.5 4) Junior V0 ⫽ 1800, ␴ ⫽ 0.20 φ ⫽ ⫹1.5 φ ⫽ ⫹1 φ⫽0 φ ⫽ ⫺1 φ ⫽ ⫺1.5 5) Junior ␤ ⫽ ⫹1 φ ⫽ ⫹1.5 φ ⫽ ⫺1.5 6) Junior Debt φ ⫽ ⫺1.5 ␤ ⫽ ⫹.5 ␤ ⫽ ⫺.5 7) Junior V0 ⫽ 1800, ␴ ⫽ 0.20, ␤ ⫽ 0 φ ⫽ ⫹1.5 φ ⫽ ⫺1.5

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

⫺0.67 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 ⫹1.5 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 ⫹1.5 and ⫺1.5 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 (␴ ⫽ 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 ␴ and leverage (initial asset value). Table 2, panel A, contains duration as ␴ varies for long maturities if φ is ⫺1.5 where greater volatility increases the difference and junior duration always exceeds senior. All other parameters are unchanged from Table 1. For example, if ␴ 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 ␴ of 0.25. Table 2, panel B, contains variation in ␴ and also lowers asset values (greater leverage) so that the difference in duration grows even more dramatically as ␴ increases. That is, the difference in duration is almost 12 when maturity is 30 and ␴ is 0.25. Examination of the table reveals that the increase in duration difference is due to junior duration being quite sensitive to ␴ 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

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Table 4. Difference between Junior and Senior Convexity (Using base values) Maturity ␴ ⫽ 0.10, φ ⫽ ⫺1.5 A Vary ␴ only from Table 3 Senior Junior ␴ ⫽ 0.20, φ ⫽ ⫺1.5 Senior Junior ␴ ⫽ 0.22, φ ⫽ ⫺1.5 Senior Junior ␴ ⫽ 0.25, φ ⫽ ⫺1.5 Senior Junior B Vary ␴ and Change Leverage from Table 1 V0 (senior), 1000, V0 (junior) ⫽ 2000 Senior Junior ␴ ⫽ 0.20, φ ⫽ ⫺1.5 Senior Junior ␴ ⫽ 0.22, φ ⫽ ⫺1.5 Senior Junior ␴ ⫽ 0.25, φ ⫽ ⫺1.5 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 ⫹1.5 and ⫺1.5. 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 ⫹1.5, ⫹1, ⫺1, and ⫺1.5 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 ␤ ⫽ ⫹1 instead of ⫹0.5 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 (␤ ⫽ 0.5) and a downward sloping term structure (␤ ⫽ ⫺0.5). 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 (␤ ⫽ 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 ␴ and leverage. Table 4, panel A, shows convexity as ␴ varies for long maturities if φ is ⫺1.5 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 ␴ 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 ␴ is 0.25. Table 4, panel B, again varies ␴ but at a lower asset value so that the difference in convexity grows even more dramatically with ␴. That is, the difference in convexity is 669 when ␴ 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.

References Bierwag, G. O.: Immunization, Duration, and the Term Structure of Interest Rates. Journal of Financial and Quantitative Analysis 12 (December 1977): 725–742. Black, Fischer, and Cox, John: Valuing Corporate Securities: Some Effects of Bond Indenture Provisions. Journal of Finance 31 (1976): 361–376. Cakici, Nusret, and Chatterjee, Sris: Market Discipline, Bank Subordinated Debt, and Interest Rate Uncertainty. Journal of Banking and Finance 32 (Aug. 1993): 747–762. Chambers, Don, Carleton, W. T., and McEnally, R. W.: Immunizing Default Free Bond Portfolios with a Duration Vector. Journal of Financial and Quantitative Analysis 23 (March 1988): 89–104. Chance, Don M.: Default Risk and the Duration of Zero Coupon Bonds. Journal of Finance 45 (March 1990): 265–274. DeAlessi, Louis: Do Business Firms Gain from Inflation? Journal of Business 48 (April 1964): 162–166. Fisher, Lawrence, and Weil, Roman L.: Coping with the Risk of Interest-Rate Fluctuations: Returns to Bondholders from Naive and Optimal Strategies. Journal of Business 20 (Oct. 1971): 408– 431. Garman, Mark B.: The Duration of Option Portfolios. Journal of Financial Economics 14 (June 1985): 309–316. Grantier, Bruce J.: Convexity and Bond Performance: The Benter, the Better. Financial Analysts Journal 44 (Nov./Dec. 1988): 79–82. Ho, Thomas S. Y.: Managing Illiquid Bonds and the Linear Path Space. Journal of Fixed Income 2 (March 1992): 80–93.

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Ilmanen, Antti, McGuire, Donald, and Warga, Arthur: The Value of Duration as a Risk Measure for Corporate Debt. Journal of Fixed Income 4 (March 1994): 70–76. Kahn, Ronald N., and Lochoff, Roland: Convexity and Exceptional Returns. The Journal of Portfolio Management 16 (Winter 1990): 43–47. Kishimoto, Naoki: Pricing Contingent Claims under Interest Rate and Asset Price Risk. Journal of Finance 44 (July 1989): 571–589. Longstaff, Francis A., and Schwartz, Eduardo S.: A Simple Approach to Valuing Risky Fixed and Floating Rate Debt. Journal of Finance 50 (July 1995): 789–819. Merton, Robert C.: On the Pricing of Corporate Debt: The Risk Structure of Interest Rates. Journal of Finance 29 (May 1974): 449–470. Nawalkha, Sanjay: A Contingent Claims Analyses of Interest Rate Characteristics of Corporate Liabilities. Journal of Banking and Finance 20 (2 1996): 227–245. Prisman, Eliezer, and Shores, Marilyn: Duration Measures for Specific Term Structure Estimation and Applications to Bond Portfolio Immunization. Journal of Banking and Finance 12 (Sept. 1988): 493–503. Rendelman, Richard J., Jr., and Bartter, Brit J.: Two State Option Pricing. Journal of Finance 34 (Dec. 1979): 1093–1110. Rendleman, Richard J., Jr., and Bartter, Brit J.: The Pricing of Options on Debt Securities. Journal of Financial and Quantitative Analysis 15 (March 1980): 11–24. Schnabel, Jacques A.: Is Benter Better? A Cautionary Note on Maximizing Convexity. Financial Analysts Journal 46 (Jan./Feb. 1990): 78–79. Stock, Duane, and Simonson, Don: Tax Adjusted Duration for Amortizing Debt Instruments. Journal of Financial and Quantitative Analysis 23 (Sept. 1988): 313–327.

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 ␥nj ⫽ eφ(Rnj(1)) but Rnj(1) ⫽ ⫺lnDnj(1). Thus ␥nj ⫽ eφ(⫺lnDnj(1)) ⫽ Dnj (1)⫺φ. Looking at the first subperiod of the first period ␥11 ⫽ D11(1)⫺φ ⫽ ␥up, and ␥10 ⫽ D10(1)⫺φ ⫽ ␥down. Here, D11(1)⫺φ ⫽

⫺φ D00(2) h(1) and D00(1)

冢

冣

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)⫺φ ⫽

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

Thus ␥up and ␥down are the only factors determining the

冢

⫺φ D00(2) h*(1) . D00(1)

冣

300

Table A1.

Aaa and Aa A-Ba Aaa-Ba

J Busn Res 2000:49:289–301

冢⌬P

i,t

V. P. Lesseig and D. Stock

冣

⫹ ci/12 ⫺ rt ⫽ ␤0 ⫹ ␤1Di,t(⫺⌬rt) ⫹ ␤2(φi) ⫹ ␤3(SPDi,t⫺2) ⫹ ⑀i,t Pi,t ␤0

␤1

␤2

␤3

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

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

⫺0.000309 (⫺4.112) ⫺0.00014 (⫺2.456) ⫺0.000201 (⫺3.228)

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 ␥up and ␥down, 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)⫺φ and h*(1)⫺φ giving the no arbitrage requirement that ␲h(1)⫺φ ⫹ (1⫺␲)h*(1)⫺φ ⫽ 1 Exhaustive simulations testing that this requirement holds have been performed. The simulations assume realistic values for ␦, φ values within the range used in this research, and varying ␲ values. Realistic ␦ values are those that yield realistic interest volatilities using Hull’s (1989) formula for per annum interest rate volatility. ␦s less than 0.995 are thus very unusual because smaller ␦s give volatilities over 50% which are unrealistically high. φ values of close to zero and ␦ values close to one give a result ␲h(1)⫺φ plus (1 ⫺ ␲)h*(1)⫺φ equals 1 with accuracy beyond the 8th decimal place (e.g., 1.00000006). If φ is, say, 1.5, ␦ ⫽ 0.995, and ␲ ⫽ 0.5, the result is accuracy to the sixth decimal place (e.g., 1.000002).

Appendix B Sensitivity of Asset Value to Interest Rate Changes and the Impact on Bond Returns Our model suggests that a bond’s sensitivity to interest rate changes can be significantly affected by the relation to interest rates. If assets are negatively (positively) related to interest rates, its debt will be more (less) sensitive to interest rate changes. To provide empirical testing of such an hypothesis, we follow Ilmanen, McGuire, and Warga (1994) (IMW), who examined the ability of duration to measure interest-rate risk in corporate debt. Their basic procedure was to regress excess returns upon duration, convexity, and default risk. They conclude that duration explains a large proportion of returns although the proportion declines as credit rating declines. To test whether asset sensitivity to interest rate changes affects bond returns, we add an explanatory variable which is the result of regressing monthly equity (stock) returns (EQRET) for firm i upon changes in the one-year U.S. Treasury rate (rt)

EQRETi,t ⫽ ␣0 ⫹ ␣1 ⌬rt. The regression coefficient ␣1 is then used to represent sensitivity (φ) to interest rate changes. Stock returns are from the Center for Research in Security Prices (CRSP) where stock price behavior is used as a proxy for asset value. The above equation is estimated over a minimum of four years depending on availability of continuous stock and bond prices from January 1985 to December 1993. Firms without continuous prices were eliminated. The specification as used in IMW is

冢⌬P

i,t

冣

⫹ ci/12 ⫺ rt ⫽ ␤0 ⫹ ␤1Di,t (⫺⌬rt) ⫹ ␤2(φi) Pi,t ⫹ ␤3(SPDi,t⫺2) ⫹ ⑀i,t

⌬Pi,t is bond price change for bond “i” from month “t-1” to “t,” Pi,t is price in month “t,” and ci is the annual coupon. Di,t is the duration of the bond, ⌬rt is the change in the one year Treasury bill rate, and φi is the sensitivity of stock price to changes in interest rates, ␣1 from above. SPD is a default risk variable used by IMW and is calculated as return on the bond less the risk free rate for time t-2. IMW use a lag to eliminate potential bias because there will be no contemporary prices on the left and right sides of the regression. (IMW find convexity does not add much explanatory power and we found similar results.) The database of bond prices and returns used was the Fixed Income Data Base (University of Wisconsin at Milwaukee). Due to an expectation of greater interest-rate sensitivity, only bonds from financial firms were used. The time period used was the same as that used to estimate the sensitivity in the EQRET regression. Thus 71 firms were used where some firms had more than one issue included. Callable and convertible bonds were excluded. As in IMW, bonds of less than one year maturity were not included because duration for such short-term bonds is greatly affected by the passing of one month. Our model predicts a negative relation between excess returns and φ such that ␤2 is expected to be negative. Table A1 contains the results where regressions for both high grade (Aaa and Aa) and lower grades (A, Baa, Ba) are given. Like the IMW results, the R2 value is lower for lesser quality bonds. As in IMW, separate regressions for different

Duration and Convexity of Risky Debt

grades of bonds were estimated. Small sample size discourages regressions for each rating class. As hypothesized, the sensitivity measure is significantly and negatively related to excess returns for both sets of bonds. Thus, bond returns are in fact sensitive to asset values. Also, the return attributed to this sensitivity can be a large proportion of total excess return. For example, the average monthly excess return is 0.0033 for Aaa and Aa bonds and the minimum φ value is ⫺10.747. Thus, ⫺10.747 times the coefficient ⫺0.000309 is 0.0033 which is 100% of the average return. If the average φ of ⫺3.664 is used, the result is ⫺3.664 times ⫺0.000309 or 0.0012 which is more than one-third the average monthly return.

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In this formulation, the parameter φ is measured with error. This error could potentially bias the estimate of the standard errors in the second regression. To address this potential, simulations were run based on the standard error of estimate of ␣1 in the first regression equation. In all, 300 different values of φ for each firm were used with the values randomly selected from a normal distribution with mean equal to the least squares estimate of ␣1 in the first regression and standard deviation equal to the standard error of ␣1. The residual errors for the simulations were averaged across all 300 regressions and compared to those from the second regression. The process was repeated three times and in no case were the values of the parameters, or their significance, affected.