Bargains in the corporate convertible bond market
The embedded option implicit in Treasury issues has recently attracted a lot of attention. Studies in the callable US Treasury bond market suggest that prices of callable bonds often imply negative values for the call option embedded in them. Although options embedded in Treasury issues have been the focus of a lot of research, options embedded in corporate bonds have not received as great attention. This is the first study which conducts a complete examination of the pricing behaviour of convertible bonds with respect to pricing conditions implied by the presence of the embedded option. Results indicate that convertible bonds are often underpriced to the extent that negative conversion option values are implied.
L’option enchassee implicate dans les emissions du Tresor a recemment suscite beaucoup d’attention. Des etudes sur le marche des obligations remboursables a vue du Tresor americain suggerent que les coats des obligations remboursables a vue sous-entendent souvent des valeurs negatives pour l’option d’achat qui y est enchassee. Bien que les options enchassees dans les emissions du Tresor ont ete le sujet central de plusieurs recherches, les options enchassees dans les obligations de societes n’ont pas refu une aussi grande attention. Ceci est la premiere etude qui conduit un examen complet du comportement de la valorisation des obligations convertibles relativement aux conditions de valorisation sous-entendues par la presence des options enchassees. Les resultats indiquent que les obligations convertibles sont souvent en vente a un prix inferieur a leurs vraies valeurs a tel point que des valeurs negatives d’option de conversion sont sous-entendues.
In today’s complex markets, borrowers and investors demand products with option-like features embedded in them. Given the proliferation of such products, it is important that players understand their behaviour and risk characteristics, and make sure that these products are priced property. The focus of this paper is the examination of the pricing behaviour of one of these products: the corporate convertible bond. It is the first study of this nature in the sense that it provides a complete examination of the convertible bond pricing behaviour with respect to pricing bounds implied by the presence of the embedded conversion option.
The pricing behaviour of Treasury bonds with embedded options has recently attracted a lot of attention, with conflicting results. Longstaff (1992), Edleson, Fehr, and Mason (1993), and Carayannopoulos (1995) examine the pricing behaviour of callable Treasury bonds and document what seems to constitute a puzzle: market prices for these bonds often imply negative values for the implicit call, put options, or both, a result at odds with one of the most fundamental propositions of finance. Moreover, the negative values frequently exceed the spread-related costs of an arbitrage strategy designed to exploit the mispriced callable bond. In contrast, Jordan, Jordan, and Jorgensen (1995) and Jordan, Jordan, and Kuipers (1998) find little or no evidence of profitable arbitrage opportunities involving misvalued implicit options. They suggest that what seems to be a puzzle is only the result of tax effects associated with the cash flow replication techniques used in the previous studies. However, as Bliss and Ronn (1998) suggest, the tax argument relies implicitly on the unlikely absence of a tax-exempt entity which can ignore the tax effects and arbitrage the pricing errors. Using a larger sample period than previously noted and a completely different methodology, which allows them to determine not only overpriced but also underpriced callable bonds, Bliss and Ronn (1998) show that the frequency of mispriced callable bonds is time-varying. Furthermore, Carayannopoulos (1998) suggests that tax effects cited as an explanation for the puzzle cannot fully explain the magnitude and the frequency of the violations.
Negative option values are not unique to callable U.S. Treasury bonds. Athanassakos, Carayannopoulos, and Tian (1997) also find negative values for options embedded in Canadian Treasury extendible bonds. The authors demonstrate that these negative values cannot be explained by tax effects, bid-ask spreads, or nonsynchronous trading.
While the pricing behaviour of Treasury bonds with embedded options has been extensively studied in the past, the pricing behaviour of their corporate counterparts has received little attention. Corporate convertible bonds are such an example. Convertible bonds can be viewed as regular corporate bonds with an option attached that gives the holder the right to convert the bond into a predetermined number of common (or preferred) shares. While a number of theoretical studies have modeled the valuation of convertible bonds, relatively little empirical research has focused on their pricing behaviour (see Ingersoll, 1977a, 1977b, Brennan & Schwartz, 1977, 1980, and Ho & Pfeffer, 1996). Most research effort has been directed at analyzing and rationalizing the seemingly abnormal call behaviour of firms that call the convertible bonds they have issued (see Ingersoll, 1977a, Harris & Raviv, 1985, and Asquith & Mullins, 1991). King (1986) provides an empirical test of a contingent claims valuation model and argues that, when market prices are compared to model valuations, 90% of model predictions fall within 10% of market valuations. His results, however, are specific model dependent and constitute a test of the particular model employed rather than a test of the pricing behaviour of convertible bonds.
This paper provides an examination of the pricing behaviour of convertible bonds. Unlike previous empirical research, the results of this study do not depend on a particular pricing model but rather are developed within a simple no-arbitrage framework. An empirical investigation is conducted in order to determine whether convertible bond prices violate the pricing bounds implied by the presence of the embedded option. The results suggest that corporate convertible bonds are often underpriced to the extent that the implied conversion option has, at times, a negative value.
Convertible Bond Prices and the No-Arbitrage Bounds
A long position in a convertible bond can be considered the same as a portfolio consisting of a long position in a nonconvertible bond with the same coupon, maturity, and risk and a long position in an Americantype call option. The call option gives the holder the right to acquire a predetermined number of the firm’s common (or preferred) shares in exchange for the bond that he or she currently holds. Thus, where t is the current date, T is the date of the maturity of the bond and the option, B^sub conv^(t, T) denotes the value of the convertible bond, B^sub nconv^(t, Tj denotes the value of an equivalent nonconvertible bond (equivalent in the sense that it has the same coupon, maturity, and risk as the convertible bond) and C(t, T) represents the value of the corresponding embedded call option. Given the nonnegativity of option values, Equation 1 implies the following condition for the value of the convertible bond:
Condition 2 suggests that the value of a convertible bond should be greater than or equal to the value of a nonconvertible bond with the same coupon, maturity, and risk. If this is not the case, a negative value is implied for the embedded option.
Since at any time, and at the holder’s option, the convertible bond can be converted into a predetermined number of the issuing firm’s common shares, the convertible bond should be worth at least as much as its conversion value, i.e., the value of the equity it can be converted into. Thus, where q represents the number of common shares each bond can be converted into, and S(t) represents the issuing firm’s current stock price. If Condition 3 does not hold, an investor can exploit the arbitrage opportunity by purchasing the convertible bond and immediately converting it into common equity.1
Methodology for Examining Violations of the No Arbitrage Pricing Bounds
Convertible bond prices can be easily examined for violations of Condition 3. The conversion ratio, q, is known, while the price per share of common stock, S(t), and the price of the convertible bond are observable in the market place. Thus, any violations of Condition 3 can be documented easily. However, violations of Condition 2 cannot be as easily verified. Study of such violations requires knowledge of the value of the equivalent nonconvertible corporate bond, Le, the bond without the embedded conversion option attached to it.2 Since the nonconvertible bond portion of the convertible bond and the embedded conversion option do not trade separately, the equivalent nonconvertible bond value is not directly observable and has to be estimated. The estimation procedure followed in this paper involves estimating the yield to maturity of the equivalent nonconvertible bond and then discounting the future cash flows of the bond at a rate equal to the estimated yield to maturity in order to obtain its price. This equivalent nonconvertible bond price is then compared to the convertible bond price. The yield to maturity estimation procedure is discussed in more detail in the next section.
An Approach for Estimating the Yield to Maturity for Risky Corporate Bonds
An efficient way for estimating the yield to maturity of a nonconvertible corporate bond is to decompose the yield to maturity into a riskless yield, which is the yield to maturity of an equivalent government bond, that is, a government bond with the same coupon and maturity, and a yield spread. Thus, where YTM^sub nconv^(t, T) represents the yield to maturity of a nonconvertible corporate bond and YTM^sub gov^(t, T) the yield to maturity of the government issue that has the same coupon and maturity as the nonconvertible corporate bond. The yield spread represents the additional compensation investors require due to differences in default risk and other factors unique to the corporate bond. Each component in Equation 4 is estimated in this paper. By adding the two components, a final estimate of the yield to maturity of the nonconvertible corporate bond is obtained.
Estimating the Yield to Maturity of the Equivalent Government Bond
Since it is unlikely that it exists, the equivalent government bond has to be constructed and its value estimated using other existing Treasury issues. For this matter, the term structure of interest rates implied by prices of U.S. Treasury STRIPS is used to construct the equivalent Treasury issue.
U.S. Treasury STRIPS are zero-coupon instruments created from either the principal (principal STRIPS) or individual coupons (coupon STRIPS) of U.S. Treasury notes or bonds. STRIPS maturities span a period of about 30 years, with STRIPS maturing on the 15th day of February, May, August, and November of each year. Thus, spot interest rates for maturities in each of the above months every year for approximately 30 years can easily be calculated from the observed prices of the STRIPS. Since cash flows from corporate bonds may materialize during any month and day of the year, the spot rate for any date other than the date when STRIPS mature is found by fitting a curve to the spot rates implied by the prices of STRIPS.3,4 The price of an equivalent government bond is found by discounting the cash flows of the corresponding corporate bond at the spot rates implied by the fitted Treasury STRIPS spot rate curve.5 From this price, the yield to maturity of the equivalent government bond is calculated using standard industry practices.
Estimation of the Yield Spread
As noted earlier, the yield spread represents the additional compensation investors require from the corporate bond over and above the return they demand from an equivalent government issue. Following research and findings on the determinants of yield spreads by Garbade and Silber (1976), Cook and Hendershott (1978), Litterman and Then (1991), Ederington, Yawitz, and Roberts (1987), Reiter and Ziebart (1991), and Duffee (1996), the yield spread at a specific point in time is assumed to be a function of default risk, call risk (if bond callable), and any differential tax treatment that the corporate bond is subject to. Thus,
Yield spread=f (default risk, call risk, taxation), (5)
where Yield spread= YTM^sub nconv^(t, T) – YTM^sub gov^(t, T).6
Default risk refers to the risk that the issuer of a bond may be unable to make the periodic interest payments on the issue and the timely principal payment. Call risk, if the bond is callable, stems from the fact that the issuing firm may call (buy) the bonds back at a predetermined call price at a point in time when interest rates are low and refinancing is attractive. In addition to the above risks that may cause differences between corporate and government bond yields, spreads may also be affected by taxes. Income from corporate bonds is taxed at the federal, state, and local levels, while income from Treasury issues is taxed only at the federal level. As long as the marginal investor faces a positive state and local marginal tax rate, this tax wedge will affect the yield spread between corporate and Treasury issues (see Duffee, 1996).
Proxies of variables in the yield spread model. The Moody’s rating and time to maturity are used as proxies for default risk. The role of bond ratings in the determination of yields of corporate bonds has been examined extensively in the past. The majority of recent research (see, for example, Reiter & Ziebart, 1991, and Hite & Warga, 1997) suggests that ratings do contain private as well as public information and play an important role in the determination of corporate bond yields. Only Moody’s ratings are used in this study, but as Ederington, Yawitz, and Roberts (1987) suggest, the market views Moody’s and Standard and Poor’s ratings as equally reliable measures of risk. Default risk, and thus the yield spread, of a corporate bond is also assumed to be a positive function of the bond’s maturity. On an intuitive level, the risk premium on shorter term bonds should be lower than the risk premium of longer maturity bonds. Litterman and Then (1991) provide evidence that yield spreads increase with maturity. Fons (1994), on the other hand, argues that as the maturity of a corporate bond increases, its credit spread versus a comparable maturity Treasury bond may widen or narrow, depending on the bond’s credit risk. Lower rated (smaller, younger, more highly leveraged) issuers tend to have wider credit spreads that narrow with maturity. Higher rated (more mature, stable) firms tend to have narrower credit spreads that widen with maturity.
Due to the fact that all convertible bonds examined in this study are also callable, the sample of nonconvertible corporate bonds is also limited to callable ones. However, the value of the call option, and thus its impact on the yield spread, varies with respect to the degree to which the call option is in- or out-of-the-money and the maturity of the bond itself. To capture residual effects of callability on the yield spread, in addition to the time to maturity which is already being used as a proxy variable, coupon size is also used. Both time to maturity and coupon size are related to the degree to which the option is in- or out-of-the-money. Since the higher the coupon size the higher the taxes paid on the corporate bond, as compared to the equivalent government issue, pre-tax yield spread is also affected by the bond’s coupon. Thus, coupon size also serves as a proxy for tax effects.
An approach to estimating the yield spread. A rather simple approach is used to estimate the yield spread of the equivalent nonconvertible bond corresponding to each convertible issue under consideration. First, for each examination date, each convertible bond is matched with all nonconvertible corporate bonds in the sample that have the same (or similar) values for the proxy variables (or attributes) in Equation 5, that is, rating, coupon, and maturity. Second, the average yield spread of the nonconvertible corporate bonds matched to the convertible issue is calculated and used as an estimate of the yield spread of the equivalent nonconvertible bond. The advantage of such an approach is its simplicity. It makes no assumption about the functional form of Equation 5 and, instead, directly compares issues with similar attributes. The approach, referred to from this point as the matrix approach, is discussed in more detail next.
A three-dimensional matrix, with Moody’s rating, coupon size, and time to maturity in each dimension, is developed for each cross section of bonds in the sample. The Moody’s rating dimension is divided into 23 intervals, one for each rating. The coupon size dimension is divided into 10 intervals (less than 4, greater than 12, and intervals in increments of one in between). Eight intervals in the time to maturity dimension are also created (time to maturity greater than 14 years, and 7 intervals in increments of 2 years between 0 and 14 years). This approach results in 23 x 10 x 8 (or 1,840) cells in the three-dimensional matrix. The (observable) yield to maturity of each bond in the cross section of nonconvertible corporate bonds is decomposed into a riskless yield (with the method described earlier) and a yield spread (as the difference between the yield to maturity and the riskless yield). Each calculated yield spread is placed in the cell corresponding to the bond’s rating, coupon, and maturity. After all the nonconvertible corporate bonds have been considered and placed in cells, an average yield spread for each cell is calculated. Subsequently, for each convertible bond in the cross section of convertible bonds in the sample, the yield spread of the equivalent nonconvertible bond is assumed to be equal to the average yield spread in the cell with the same attributes as the convertible bond in question.
One disadvantage of the approach is that its accuracy depends on the choice of the interval size in each of the maturity and coupon dimensions.7,8 As the interval size decreases, accuracy increases since the convertible bond is matched to other nonconvertible issues with maturity and coupon values closer to the convertible bond’s maturity and coupon. However, as the interval size decreases, the inability of the nonconvertible corporate bonds to span the entire matrix also increases, that is, there may not be enough nonconvertible corporate bonds in the sample to obtain yield spread information for all the cells in the matrix. Thus, the number of convertible bond prices that can be examined is limited to the availability of nonconvertible corporate bonds in the same cell the convertible bond belongs in.
The methodology for estimating the price of the equivalent nonconvertible bond described in the previous sections requires data for both corporate convertible and nonconvertible bonds as well as STRIPS.
Quarterly corporate convertible bond data are collected from the Wall Street Journal for the period from September 1990 to March 1996. Information with respect to conversion ratios, callability, and stock price is collected from Moody’s Bond Record and the Wall Street Journal. Our sample is limited to convertible bonds that traded during the last trading day of the particular quarter considered. This is done in order to minimize nonsynchronous trading problems among the convertible bond, the common share of the underlying firm, and the STRIPS data used in the yield to maturity estimation methodology. All of the convertible bonds in the sample are also callable. In all, 1,239 convertible bond prices are collected for the period under consideration.
Information about nonconvertible corporate bonds is extracted from the University of Houston fixed income data base.9 The data base contains detailed pricing information as well as ratings and industry codes on the individual bonds that make up the Lehman Brothers Bond Index. Since all the convertible bonds in our sample are also callable, noncallable corporate bonds were excluded from the sample. Furthermore, since all the convertible bonds to be examined come from the industrial, utilities, and transportation sectors, our sample of nonconvertible corporate bonds excludes issues from other industries. As a result, our data base consists of 59,583 nonconvertible corporate callable bond prices.
STRIPS prices for each quarter between September 1990 and March 1996 are collected from the Wall Street Journal, which has been publishing STRIPS prices monthly since mid-1989. Prior to July 1990, STRIPS quotations were obtained from the Bloomberg system. From July 1990 onwards, quotations are provided by Bear Stearns. Thus, for consistency reasons, the first quarter considered is September 1990.10
An Examination of the Pricing Behaviour of Corporate Convertible Bonds
As noted earlier, the examination of the convertible bond pricing behaviour with respect to Condition 3 involves the comparison of the convertible bond’s conversion value to the convertible bond price. A violation occurs when the conversion value exceeds the convertible bond price.
In order to examine the convertible bond pricing behaviour with respect to Condition 2, an estimate of the price of the equivalent nonconvertible bond is needed. The estimation process described is used to derive the yield to maturity and, thus, the price of the equivalent nonconvertible, which is then compared to the price of the corresponding convertible issue.
Violations of the Pricing Bounds
Violations of Conditions 2 and 3 are reported in Table 1. They are based on a convertible bond face value of $100. With respect to the pricing bound implied by the conversion value, only 31 violations (or 2.50% of all observations) are reported. The mean violation is $2.57, the median $2.18, and 75% of the violations are less than $3.81. However, violations of the no-arbitrage condition implied by the value of the equivalent nonconvertible bond are much more frequent and greater in magnitude. There are 83 (21.23% of the entire sample) violations out of 391 cases of convertible bonds for which the construction of the equivalent nonconvertible was possible with the matrix technique.11 The mean violation is $9.97 and the median $6.56; 25% of the violations exceed $11.59.
The difficulty of implementing an arbitrage transaction with respect to the documented violations may explain the frequency and magnitude of the violations of the two conditions. Arbitrage, when the pricing bound implied by the convertible bond’s conversion value is violated, is relatively easy. All an investor has to do is buy the convertible bond, immediately convert it, and sell the shares received. Bid-ask spreads are the main costs involved in such a strategy. These costs would probably eliminate a considerable portion of the $2.57 average arbitrage opportunity that is documented in this case. Furthermore, some of the violations documented in this case can also be the result of nonsynchronous trading between the convertible bond and the underlying stock. It is a well known fact that the bond market has much less liquidity than the stock market. As noted earlier, data were collected for convertible bonds that traded during the last trading day of the particular quarter. However, this trade may have taken place any time during the day and likely at an earlier time than the last trade for the underlying stock. Thus, a violation might have not actually existed when the convertible bond was traded. Assume, for example, that the price of the underlying stock went up by $0.50 between the time the corresponding convertible bond was traded and the time of the last trade of the stock. If the conversion factor is four, then the conversion price calculated in this study would be $2.00 more than the actual conversion value of the bond at the time it traded. Therefore, transaction costs and nonsynchronous trading problems can eliminate at least some of the already low number of arbitrage opportunities observed with respect to the conversion value of the bond.
Arbitrage with respect to violations of the pricing bound implied by the value of the equivalent nonconvertible bond is very difficult. The equivalent nonconvertible bond does not exist and its construction may not be possible. Nevertheless, while the difficulty of arbitrage may explain differences in the frequency and the magnitude of the violations of the two conditions examined, it cannot explain why these violations appear in the first place.
A more systematic analysis of the convertible bond pricing behaviour is conducted next. Table 2 reports conversion option values on the basis of the extent to which the embedded option is in- or out-of-the-money. The value of the conversion option is calculated by subtracting the value of the equivalent nonconvertible bond from the observed convertible bond price. The measure used to partition the sample is the ratio of the convertible bond’s conversion value to the value of the equivalent nonconvertible bond. A ratio greater (less) than one indicates that the option embedded in the convertible bond is in- (out-of)-the-money.12 As a result, the higher the value of such a ratio, the higher the value of the embedded conversion option. In other words, when the conversion option is deep-in-the-money, any changes in the value of the equity of the issuing firm should directly affect the value of the convertible bond almost dollar for dollar. Therefore, in such cases, the behaviour of the convertible bond should be more equity-like and less bond-like and violations of Condition 2 should be very rare. For low ratio values, since the option is out-of-the money, the value of the conversion option should be small and the convertible bond should behave more like a regular nonconvertible bond rather than equity. In cases of extremely low conversion values (relative to the value of the equivalent nonconvertible bond), the value of the embedded option should be close to zero and the price of the convertible bond should be close to the price of the equivalent nonconvertible issue.
As expected, when the conversion option is in-themoney (ratios greater than one) no violations exist. Similarly, for conversion value to nonconvertible bond value ratios of between one and 0.8 the embedded conversion option has a high value and the frequency of violations (or equivalently negative option values) is very small. These results, in conjunction with the absence of any significant violations of Condition 3 as demonstrated in Table 1, demonstrate clearly the equity-like behaviour of the convertible bond in this ratio range. However, for ratios less than 0.6, a significant percentage of the embedded options have negative values. As a matter of fact, for conversion ratios in the intervals (0.3, 0.4) and (0, 0.3) the average option value is -$5.02 and -$11.43 respectively, while 60.87 and 68.75% of option prices in the two conversion ratio intervals respectively have negative values. Ten percent of the time the option value is lower than (exceeds in absolute terms) -$21.79 and -$34.42 respectively. The results in Tables 1 and 2 suggest a severe underpricing of convertible bonds, at times, to the extent that negative conversion option values are often implied.
The number and magnitude of lower boundary violations of the embedded conversion option found above are quite surprising. In this section a number of potential explanations are examined.
Yield Spread Estimation Process and the Number of Bonds Examined
As noted earlier, a major advantage of the matrix approach to the yield spread estimation is that it makes no assumption about the functional form of the yield spread. Instead, it directly matches the equivalent nonconvertible bond to existing nonconvertible corporate bonds with similar attributes. However, the unavailability of nonconvertible corporate bond data for all the cells in the matrix prevents the examination of the pricing behaviour of all the convertible bonds in the sample. Thus, only the examination of 391 convertible bond prices was possible out of a total sample of 1,239. The question remains whether results would persist if the examination of a greater number of convertible bonds was possible. One way of examining a greater number of convertible bonds is to reduce the number of cells in the matrix by increasing the size of the interval in either the coupon dimension, the maturity dimension, or both. However, as discussed earlier, such an increase could have an adverse effect on the accuracy of the estimated yield spread for each of the cells in the matrix. To mitigate such effects, a regression model is used in combination with the matrix approach. The approach, which makes an assumption about the functional form of the yield premium, drastically increases the number of convertible bonds examined.
A Yield Spread Regression Model. The regression model discussed in this section is a functional form of Equation 5. It is postulated that the functional form of Equation 5 is not linear. A number of functional expressions were tested and the exponential function was found to be the one that provided the best fit. The final testable form of Equation 5 employed in this paper is as follows:
In addition to the proxy variables discussed previously, the dollar amount of bonds outstanding (Amnt_out^sub i^) is used as an explanatory variable. The particular variable serves as a proxy for liquidity risk not included in the original Equation 5. Liquidity risk refers to the ease with which an issue can be sold at a reasonable price. For an investor who plans to hold the bond until maturity, liquidity risk is less important. However, for someone who is uncertain about his or her investment horizon, liquidity risk plays an important role in his or her investment decision and, thus, may affect yield spreads. Elton and Green (1997) suggest that the best proxy for liquidity is trading volume. However, in the absence of such information in our data base, the dollar amount of bonds outstanding is used in this study. This proxy for liquidity has been suggested by Fisher (1959) and Garbade and Silber (1976); its advantages and disadvantages have been extensively discussed by Sarig and Warga (1989). It is used as a proxy on the basis of the potential high correlation between the dollar amount of bonds outstanding and the trading volume of the bond. Thus, a negative relationship is expected between the Amnt_out^sub i^ and the yield spread. Consistent with Reiter and Ziebart (1991), the effects from the ratings are captured by converting the Moody’s rating into a numerical value. Thus, Rating^sub i^ is assigned values from 1 (Aaa bonds) to 23 (bonds rated D). The (Coup^sub i^)2 and (Coup^sub i^)(TTM^sub i^) terms are included in Regression Equation 6 in order to capture any nonlinear effects arising from the capability and differential taxation treatment of corporate bonds.
Regression Equation 6 is run on each cross-section of nonconvertible corporate callable bond yield spreads for the end of each quarter from September 1990 to March 1996. Each cross-section is examined separately since the focus of the model is the estimation of the yield spread at a particular point in time rather than its time series behaviour. Furthermore, and in order to capture additional effects caused by the differential taxation between corporate and government bonds and the way capability risk influences yield spreads in different price ranges, each quarterly cross section of data is partitioned into four different categories according to the degree of discount (or premium) of each bond.13 The first category contains bonds selling at par or at a premium; the second includes bonds selling between 85% and 100% of par; the third includes bonds selling between 70% and 85% of par; and, finally, the fourth category includes bonds selling at below 70% of par. Regression Equation 6 is run separately for each of the above sub-groups of data. Pricing differences between actual and estimated prices for all nonconvertible corporate bonds in the sample used to estimate the parameters in Equation 6 are reported in Table 3.14.15 A face value of $100 is assumed for all bonds. The overall average pricing difference is -$0.14 with a standard deviation of $3.33 and a median of -$0.07. The 90% range of the pricing differences is ($4.80, -$5.03). Results are very similar within each group of prices considered and no significant biases exist among them. Results (not-reported) are also similar on a quarter by quarter basis.
While results reported in Table 3 suggest that pricing differences are relatively small for the nonconvertible corporate bonds in the sample used to estimate the regression parameters, it is still to be seen whether the technique works well for an out-of-sample set of nonconvertible corporate observations. To assess the performance of the model with an out-of-sample set of corporate bond observations, approximately 28% (16,589 out of 59,583) of observations in the original sample are randomly eliminated. The remaining observations are used in the estimation of the regression parameters and the results are employed to estimate the prices of the bonds that were initially eliminated. Pricing differences between the actual and estimated out-of-sample bond prices are reported in Table 4. The overall results are very similar to the in-sample ones (reported in Table 3). Overall average pricing difference is 40.11 with a standard deviation of $3.45 and a median of -$0.02. The 90% range of the pricing differences is ($4.79, -$5.02). Results are relatively similar within each group of prices considered and no significant biases exist among them.
After the parameters of Regression Equation 6 are estimated for each cross section of nonconvertible corporate bonds and each subcategory of prices, the results are used to estimate the yield spread of the equivalent nonconvertible bond for each convertible bond in the sample. However, the estimation of this yield spread with the use of the price-category-specific versions of Regression Equation 6 requires the knowledge of what subcategory of prices the equivalent nonconvertible bond belongs in, that is, whether the price is below 70% of par, between 70% and 85% of par, between 85% and 100% of par, or 100% of par or above. This complication is circumvented by using the matrix approach described earlier to obtain a first crude estimate of the price of the equivalent nonconvertible bond. The coupon size grid in the matrix is divided into six intervals (coupon sizes less than 4, more than 12, and 4 intervals in increments of 2 in between) and the time to maturity grid is divided into three intervals (maturity less than 5 years, between 5 and 10 years, and more than 10 years). The larger interval size used reduces the number of cells in the matrix to 23 x 6 x 3 or 414 cells. This allows us to get a first yield spread estimate and, thus, a first price estimate of the equivalent nonconvertible bond for a greater number of convertible bonds. Given the increased interval size in the matrix, one expects this estimate to be less accurate than the estimate obtained with the same process with a finer grid. However, Regression Equation 6 is used to refine this yield estimate in the following manner: The first crude price estimate from the matrix approach serves as the means to determine the category-specific version of Regression Equation 6 to be used in the estimation of a new yield spread and, following that, a new price. Recursively, using the most recent estimated price in the determination of the category-specific version of Regression Equation 6, the process is repeated until there is no difference between the newly obtained price and the previous one. This price represents the final estimate of the price of the equivalent nonconvertible bond which is then compared to the price of the corresponding convertible issue in order to determine any violations of Condition 2.
The technique allows us to increase the number of convertible bonds examined for violations of Condition 2. Overall the pricing behaviour of 1,090 convertible bonds is examined and results reported in Tables 5 and 6. The results are very similar to the ones reported earlier for the smaller sample. Overall, Condition 2 is violated 27.43% of the time (299 violations in a sample of 1,090 observations) while the mean option value is negative for conversion ratios less than 60%. For conversion ratios in the intervals (0.3, 4) and (0, 0.3), in particular, the average option value is -$3.95 and -$4.56 respectively, while the percentage of negative values is 70.79% and 61.54% respectively. Ten percent of the time the option value is lower than (exceeds in absolute terms) -$16.12 and -$21.62 respectively. The results are very similar to the ones in Table 2 for the smaller sample examined with the matrix-only technique. Furthermore, a comparison of these values to the values denoting the accuracy of the technique, that is, mean pricing differences, percent negative and so forth for both the in-sample and out-of-sample nonconvertible corporate bonds reported in Tables 3 and 4 again suggests a severe underpricing of convertible bonds with relatively low conversion values.
The findings indicate that violations of Condition 2 are persistent regardless of the estimation technique used in the analysis and the number of convertible bonds examined.
All of the convertible bonds in our sample are also callable at the discretion of the issuer. The value of the embedded call option is a complex function of the bond’s coupon rate, maturity, level and volatility of interest rates, stock volatility, and other attributes such as call price and call protection period. As noted earlier, the yield spread estimation procedure involves only nonconvertible corporate bonds that are callable. The procedure accounts for factors influencing the call option such as coupon rate, maturity, and level of interest rates. However, other factors such as volatility of stock price, call price, and call protection period are not accounted for in the estimation procedure. It is possible that if the process underestimates the impact of the embedded call option, what appears to be a negative option value is nothing other than an underestimation of the call option embedded in the bond.
However, as results in Table 6 indicate, most of the serious violations occur when the convertible bond sells at deep discount, that is, at a price less than $85. When the price of the convertible bond is at such discount, the call option is also deep out-of-the-money and, consequently the call option value is close to zero. Therefore, any underestimation of the impact of the call option in such cases could not be severe and, certainly, could not produce the serious pricing violations observed.
Nonsynchronous quotes due to the differential source of data for nonconvertible and convertible bonds cannot possibly explain the pattern of the violations observed. First, interest rate, and thus bond price, movements are much less volatile during the course of a trading day vis-A-vis stock price movements and thus no serious pricing errors due to the nonsynchronicity of data should have resulted in this study. Furthermore, prices for the convertible bonds in our sample were obtained only for bonds that traded during the last trading day of each quarter under consideration, thus eliminating the possibility of stale convertible bond prices. Second, any errors due to nonsynchronicity of data should have had a random pattern and should not have resulted in the significant negative means observed for the two lower conversion value to equivalent nonconvertible bond price ratio ranges documented in Tables 3 and 5.
This paper examines the pricing behaviour of corporate convertible bonds for the period September 1990 to March 1996. The evidence provided in this paper suggests an underpricing of convertible bonds to the extent that negative conversion option prices are often implied. The findings cannot be attributed to capability and nonsynchronous trading issues. The reason, but not the explanation, for such a behaviour could be the fact that, unlike regular bonds which are traded by bond traders, convertible bonds are traded by equity traders who may attach more emphasis to the security’s equity than its straight debt value in determining overall value. A similar argument has been brought forward by Merrill Lynch sources in a recent Business Week article (“Bargain Time”, 1998). According to the article, a company’s convertible bond price tends to rise (or fall) along with its common stock price. The same sources argue that, at the time the article was written, convertible bonds traded at prices that were, on average, 4% below their fair value, with a few issues trading for as much as 10% below their fair value.
The results also seem to support previous findings that suggest that convertible bonds usually behave more like equity and less like bonds. Kihn (1996), in a comparison of returns on open-end convertible bond funds versus returns on nonconvertible low-grade corporate bonds, suggests that, in general, convertible bonds are significantly more equity-like and significantly less bond-like than low-grade bonds and they display a strong January effect.
The findings of this study certainly have implications for participants in the corporate convertible bond market. Managers of issuing firms should be aware that the proceeds from a convertible bond issue with a low initial conversion value, could be less than an otherwise equivalent straight debt issue. Buyers of convertible bonds may also be able to take advantage of arbitrage opportunities which likely exist in certain instances due to the underpricing of the instrument by taking offsetting positions in the bond and the underlying equity. Testing the above mentioned implications are beyond the scope of this paper and could be the subject of future research, the objective of which would be to study the efficiency of the bond market.
1. Conditions 2 and 3 can be combined into a no arbitrage lower bound: B^sub com^.(t, 7) >= max [B^sub nconv^.(t, T), qS(t)]. The exercise price of the embedded call option is B^sub nconv^.(t, T) and the conversion value qS(t) can be considered as the underlying asset.
2. The term “equivalent nonconvertible bond” will be used here on to denote a bond which is equivalent (i.e., same risk, coupon, and maturity) to the convertible under con
sideration but without the conversion option attached to it.
3. The nonlinear equation that is used to fit the spot rates implied by the prices of STRIPS has the following form: r^sub it^ = a^sub 0^ + a^sub l^x^sub it^ + a^sub 2^x^sub it^ ^sup 1/2^ + a^sub 3x^sub it^ 2 + a^sub 4^x^sub it^ ^sup 3^ + a^sub 5lnx^sub it^, where r^sub it^, is the spot rate implied by the STRIPS maturing at time i, and x^sub it^, represents the time to maturity of the STRIPS measured in days. Other, simpler specifications of the model were also used, but the results were somewhat inferior to the above specification. R^sup 2^ values consistently above 97% were obtained. A similar nonlinear equation was employed by Ananthanarayanan and Schwartz (1980).
4. Only strips derived from coupons are used in this exercise. It has been shown in the past that STRIPS derived from principal contain liquidity effects and as a result their prices are, in general, different than the prices of their equivalent coupon counterparts. For a discussion see Daves and Ehrhardt (1993).
5. In general, when strips are used to replicate the cash flows of other Treasury issues, resulting pricing differences are relatively small. Using monthly observations from July 1989 to March 1994, Carayannopoulos (1996) reports average pricing differences between seasoned bonds and equivalent STRIPS portfolios of $0.08 per $100 face value Treasury bond. Using daily observations from 1990 to 1994, Jordan, Jordan, and Kuipers (1998) find an average difference of $0.11.
6. Yield spreads can be measured as the difference in yields between two bonds or on a relative basis by taking the ratio of the yield spread to the yield level. The first approach is used in this paper, as the yield spread is better understood and applied as a spread rather than a ratio. In addition, yield spreads are, in practice, typically measured this way (see Fabozzi, 1996, p. 82). A similar approach is used by Duffee (1996).
7. The number of intervals for the rating dimension is fixed at 23, one for each of the Moody’s ratings.
8. Although the grid in our analysis seems to be wide with respect to coupon size and maturity, it should be noted that it is the yield spread that is estimated and not the yield to maturity. Yield spread should be less sensitive (as compared to yield to maturity) to variations within a relative coupon size and time to maturity range.
9. The fact that the convertible and nonconvertible bond price data are from different sources should not introduce a bias in this study (see Sarig & Warga, 1989).
10. Prices provided by the Bloomberg system were characterized by larger bid-ask spreads.
11. It should be noted that although the overall number of convertible bond observations available is 1,239, due to lack of nonconvertible corporate data in a number of cells in the matrix, the value of the equivalent nonconvertible bond was estimated in only 391 cases.
12. The underlying asset of the embedded call option is the
conversion value qS(t) and the exercise price is the price of the equivalent nonconvertible bond, B^sub nonc^(t, T) (see note 3).
13. A first run of the yield spread model and the subsequent comparison of actual to estimated prices for all nonconvertible corporate bonds in the sample suggests the existence of significant biases in the model for deep discount bonds. This is likely due to the complex relationship between the depended variable in the model and such explanatory variables as the differential taxation between corporate and government bonds and callability. It seems this relationship is not constant within the entire range of prices considered and the behaviour depends on the degree of discount (or premium) of the bond under consideration. These biases result in estimated prices exceeding actual prices by a considerable amount when the bond is selling at a deep discount. This would have distorted any inferences with respect to violations of Condition 2
when estimating the price of a nonconvertible bond equivalent to a convertible one.
14. Only pricing differences are reported here as they are key to our discussion and findings about the pricing behaviour of convertible bonds later on in our analysis. However, regression results for each cross section of data are also available from the authors upon request.
15. Estimated prices are obtained by using the regression results from category-specific versions of Regression 6 to first obtain an estimate of the yield spread for each corporate convertible bond in the sample. Then, the estimated yield spread is added to the yield to maturity of the equivalent government bond to obtain an estimate of the yield to maturity of the nonconvertible corporate corporate bond in the sample. The final step involves calculating the estimated price by discounting the corporate bond’s actual future cash flows at a rate equal to the estimated yield to maturity.
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George Athanassakos* Peter Carayannopoulos*
Wilfrid Laurier University
We thank research assistants Romeo Simone and Sebastien Maussion. Without their help, this project would have been almost impossible to complete. Funding for this project was provided by a Social Sciences and Humanities Research Council of Canada (SSHRC) grant.
*School of Business and Economics, Wilfrid Laurier University, Waterloo, ON, Canada, N2L 3C5. E-mail: email@example.com or firstname.lastname@example.org.
Copyright Administrative Sciences Association of Canada Jun 2000
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