Bond Valuation Research Articles

An Empirical Comparison of Convertible Bond Valuation Models

This paper empirically compares three convertible bond valuation models. We use an innovative approach where all model parameters are estimated by the Marquardt (1963) algorithm using a subsample of convertible bond prices. The model parameters are then used for out-of-sample forecasts of convertible bond prices. The mean absolute deviation is 1.86% for the Ayache-Forsyth-Vetzal (2003) model, 1.94% for the Tsiveriotis-Fernandes (1998) model, and 3.73% for the Brennan-Schwartz (1980) model. For this and other measures of fit, the Ayache-Forsyth-Vetzal (2003) and Tsiveriotis-Fernandes (1998) models outperform the Brennan-Schwartz (1980) model.

A Dynamic Programming Approach for the Valuation of Callable Corporate Bonds within the CIR Framework

We show how to price corporate callable (res. putable) bonds in a model where interest rates are described by the Cox, Ingersoll and Ross model and default is described by a stochastic intensity model. Pricing is done by mean of a dynamic programming procedure, recently proposed for non-defaultable bonds, that is shown to be fast and accurate. The procedure is easy to implement and constitute an alternative to the use of similar precision but more elaborate difference elements methods. The interplay between default and callability is also discussed in the case of European options where closed form solutions are available.

A Methodological Approach for the Valuation of Callable Bonds in Emerging Markets: The TGI Example

The purpose of this paper is to clarify some of the difficulties that a practitioner may find in implementing the binomial model for valuing a corporate bond with multiple embedded options in emerging markets. Especially, when faced with the dilemma of determining which should be the proxy variables for the risk-free rate, sovereign risk and country specific risk. In order to clarify some of the challenges that the practitioners face, the paper will present the reader a practical example that can serve as a guide through the required steps needed to value a callable bond in an emerging market. The callable bond used in this example is issued by the Transportadora de Gas del Interior International Ltd., which is a company located in Colombia and its economic activity is the transportation of natural gas and has four embedded call options by the issuer until its final maturity in October 3, 2017. Our conclusion is that by using the binomial model to find the option adjusted spread of the bond is also possible to find a more reliable measure of specific or unique risk attributable to the company economic activities.

Optimal mortgage refinancing: application of bond valuation tools to household risk management

Despite the enormous volume of refinancing activity in conventional residential mortgages, reaching record levels during recent years of historically low interest rates, the solution to the problem of how to time refinancing decisions optimally has remained elusive. It is recognized that the decision should depend, among other factors, on the ‘call’ options of the outstanding and the new mortgage. Determining the value of these options is a challenge in the absence of an observable optionless mortgage yield curve. We solve this by calibrating a benchmark interest rate process to the value of the new mortgage and then apply the notion of refinancing efficiency to the timing decision. In particular, risk-averse decision makers can use refinancing efficiency to measure how close to optimal a refinancing is. We analyse the sensitivity of the decision to interest rate volatility and also show how to incorporate homeowner-specific considerations, namely borrowing horizon and income taxes. While calibration and ...

Flight-to-Quality or Flight-to-Liquidity? Evidence from the Euro-Area Bond Market

Do bond investors demand credit quality or liquidity? The answer is both, but at different times and for different reasons. Using data on the Euro-area government bond market, which features a unique negative correlation between credit quality and liquidity across countries, we show that the bulk of sovereign yield spreads is explained by differences in credit quality, though liquidity plays a nontrivial role, especially for low credit risk countries and during times of heightened market uncertainty. In contrast, the destination of large flows into the bond market is determined almost exclusively by liquidity. We conclude that credit quality matters for bond valuation but that, in times of market stress, investors chase liquidity, not credit quality.

Pricing Parisians and barriers by hitting time simulation

Parisian options are not exchange traded, but there are various applications of Parisian optionality in the fields of real option theory, convertible bond valuation and credit risk. Especially the valuation of consecutive Parisian options is complicated and there exist no explicit formulas for these contracts. So far valuation can be done by numerically inverting Laplace transforms or by PDE methods. This paper develops a Monte Carlo method by exploiting the Markovian nature of the underlying value process. As a result, the Parisian option value can be written as an expression that can be solved by Monte Carlo integration, where the Parisian times are the random variables that need to be simulated. The Parisian times cannot be simulated directly as there exists no explicit distribution function. Therefore, these times are approximated by the simulation of hitting times in a special way. The quality of this approximation can be controlled and is a trade-off between accuracy and computation time.

Default Risk Explains Main Part of Corporate Credit Spreads: Evidence from Multi-Period Non-Merton Model

The paper describes model of a new type for valuation of risky bonds and loans that we call Bayesian Multi-Period (BMP) model. BMP is neither structural model nor reduced form and not a Merton-type model at all. BMP proceeds from concept of a risky bond (loan) value as Net Present Value (NPV) of a cash flow, generated by a bond. For a defaultable bond NPV is random value, and BMP identifies “fair” price of a risky bond as its mean NPV. Statistical properties of a (random) difference between NPV of a risky bond and NPV of risk-free bond with the same terms of issuance characterize riskness of a bond. BMP supposes that a borrower (e.g. a firm) generally has several debt issues (bonds, loans) simultaneously - with different terms of issuance (interest rates, maturity horizons, payment schedules etc.) and calculates risk characteristics for each debt issue separately. It considers exact contractual cash flow schedule of each specific debt issue and combines it with probabilities of a borrower’s default at all stages of cash flow process. Default prognosis in turn accounts for joint influence of all outstanding debt of a firm. BMP uses multi-period default prognosis of Bayesian type based on indices of borrower’s current financial position with accounting for predictive abilities of repayment schedule of a firm’s long-term debt. This type prognosis can additionally incorporate other predictive variables like familiar market factor - “distance to default”. BMP calculates “fair” interest rates for newly issued risky corporate bonds, “fair” prices and “fair” yield to maturity for risky bonds at intermediate moments of bond’s life. We compare them with observed market prices, rates and spreads. The model explains on average about 70% of observed interest rates, credit spreads and market prices of a bond. That is much more, than usually explain Merton-type models. The paper discusses relation between multi-period default probabilities and credit ratings.

Reduced-Form Valuation of Callable Corporate Bonds: Theory and Evidence

Reduced-Form Valuation of Callable Corporate Bonds: Theory and Evidence

Default-risky bond prices with jumps, liquidity risk and incomplete information

This paper provides a default-risky bond valuation model, which assumes that the issuer’s credit quality, modelled by the intensity of default, is driven by a continuous-time Markov chain. The model accounts for default and liquidity risk as well as incomplete information. A full-information semimartingale representation of a liquid defaultable bond price, which separates three different types of risks—default, interest-rate and credit-quality, is obtained. The illiquidity is modelled as exogenously specified stochastic reduction in the price of the bond, which adds more risks for the investors. A model of a market with partially informed investors, belonging to specific investor classes and having access to discrete information sets about credit quality, was specified. Valuations of defaultable bonds in this market were provided as well as price impacts of the new information releases.

Using Utility Functions to Model Risky Bonds

This paper prices defaultable bonds by incorporating inherent risks with the use of utility functions. By allowing risk preferences into the valuation of bonds, nonlinearity is introduced in their pricing. The utility‐function approach affords the advantage of yielding exact solutions to the risky bond pricing equation when familiar stochastic models are used for interest rates. This can be achieved even when the default probability parameter is itself a stochastic variable. Valuations are found for the power‐law and log utility functions under the interest‐rate dynamics of the extended Vasicek and CIR models.

Bond durations: Corporates vs. Treasuries

We compare the durations (the percentage price sensitivity with respect to the default-free short rate) of corporate and Treasury bonds in the reduced-form, intensity-based credit risk modeling framework. In a frequently used intensity-based model for corporate bond valuation we provide an example showing that, given the parameter estimates found in empirical studies, the duration of a corporate coupon bond may very well be larger than the duration of a similar Treasury bond. This finding contrasts with conclusions of previous studies. In a general, intensity-based recovery of market value framework we provide a simple sufficient condition for when the duration of a corporate bond will be smaller than that of a similar Treasury bond. We also provide an upper bound on the duration of the corporate coupon bond.

Reduced-Form Valuation of Callable Corporate Bonds: Theory and Evidence

We develop a reduced-form approach for valuing callable corporate bonds by characterizing the call probability via an intensity process. Asymmetric information and market frictions justify the existence of a call-arrival intensity from the market's perspective. Our approach extends the reduced-form model of Duffie and Singleton (1999) for defaultable bonds to callable bonds and can capture some important differences between call and default decisions.We also provide one of the first comprehensive empirical analyses of callable bonds using both our approach and the traditional approach of valuing callable bonds as American options. Empirical results show that the reduced-form model fits callable bond price data well and outperforms the traditional approach in both in-sample and out-of-sample applications.

The Valuation of Callable Bonds with Floored CMS-spread Coupons

A new type of structured bond has recently been introduced with enormous success - primarily among private investors - in many countries in Europe. The bonds are medium term and with fixed and very high initial coupons. The remaining coupons are determined as a constant multiplier times the spread between a long and a short swap interest rate. These coupons are floored at or near zero, and the bond investment can thus be seen as a bet on the steepening of future term structure curves. However, if the term structure becomes too steep, the bonds may be called by the issuer. The paper studies the pricing and the optimal call strategy of these highly exotic bonds in a stochastic interest rate framework. We implement two versions of the LIBOR Market Model as well as a Gaussian two-factor short rate model. We show how to adapt the Least-Squares Monte Carlo procedure to handle the callability of the product in a numerically efficient manner. We also calculate lower bounds for the product as well as delta and vega ratios.

On the Effects of Dimension Reduction Techniques on Some High-Dimensional Problems in Finance

Many problems in finance can be formulated as high-dimensional integrals, which are often attacked by quasi-Monte Carlo (QMC) algorithms. To enhance QMC algorithms, dimension reduction techniques, such as the Brownian bridge (BB) and principal component analysis (PCA), are used to reduce the effective dimension. This paper explores in depth the effects of these techniques on the dimension structure of some typical high-dimensional problems from finance: the pricing of path-dependent options and bond valuation according to term structure models. By deriving explicit expressions for the underlying integrands and the associated weights that control the relative importance of different variables, and by investigating the variance ratios, the effective dimensions, the mean dimension, and their limiting behavior as the nominal dimension tends to infinity, we show theoretically and empirically how and to what extent the BB and PCA algorithms change the dimension structure (including the degree of additivity) of the underlying functions. They change the functions to be strongly weighted and substantially reduce the effective dimensions and the mean dimension; and they enhance the degree of additivity, which is particularly important for QMC. Moreover, the resulting functions are of low effective dimension, not only in the superposition sense, but also in the truncation sense. The variance ratios, the effective dimensions, and the mean dimension associated with these techniques are very insensitive to the nominal dimension (they are essentially constant), which highlights the possibility of removing the curse of dimensionality when dimension reduction techniques are used in combination with QMC. A counterexample is also shown for which the BB and PCA may increase the effective dimension. The investigation provides further insight into the effects of dimension reduction techniques.

Avaliação de títulos conversíveis com opções de compra e venda: uma aplicação dos métodos de simulação GVW e MQMC

The present paper values callable and redeemable convertible bonds by means of two of the most modern and efficient simulation methodologies for derivative pricing: Least Squares Monte Carlo (LSM) and Grant, Vora & Weeks (GVW). An attempt was made at the evaluation of the efficiency of the Monte Carlo Simulation Methodology considering its more recent features applicable to derivative pricing. A major challenge in this paper was the use of these two models in the valuation of complex convertible bonds, given that both were originally developed and used by their authors for regular American options. Therefore, the models were applied, and their results were compared to those

Binomial pricing of fixed-income securities for increasing and decreasing interest rate cases

The option features embedded in many bonds and fixed-income securities have made the binomial interest rate tree approach to bond valuation a valuable model for pricing debt securities. Fundamental to the application of the binomial model to bond valuation is the assumption about the underlying stochastic process. There are two general approaches to modelling stochastic interest rate movements using a binomial model – the equilibrium model and the calibration model. Both models assume that the interest rate's logarithmic return is normally distributed. However, a number of empirical studies have provided evidence that the return distributions of a number of securities exhibit persistent skewness. In modelling interest rate patterns, the existence of skewness in a binomial process impacts not only the values of the up and down parameters, but also the probabilities of the underlying rate increasing or decreasing each period. The purpose of this study is to show how a binomial model of interest rates can be...

Liquidity and Credit Risk

ABSTRACTWe develop a structural bond valuation model to simultaneously capture liquidity and credit risk. Our model implies that renegotiation in financial distress is influenced by the illiquidity of the market for distressed debt. As default becomes more likely, the components of bond yield spreads attributable to illiquidity increase. When we consider finite maturity debt, we find decreasing and convex term structures of liquidity spreads. Using bond price data spanning 15 years, we find evidence of a positive correlation between the illiquidity and default components of yield spreads as well as support for downward‐sloping term structures of liquidity spreads.

A Tactical Implication of Predictability

This article confirms that high earnings yields portend high equity returns. Absolute valuation levels of equity have predictive power over future long-run equity returns. The predictability is far less powerful in the short term. On a tactical investment horizon, investors tend to rely on the relative valuation of equity versus bonds to gauge whether equity markets are attractive. The FED model, which compares earnings yield and bond yield, is the preferred yardstick in the finance profession. First, this article examines the FED model and shows that it is not only theoretically flawed, but also unable to predict equity returns over long sample periods. Second, Salomons improves the model by adding corrections for perceived risk, enabling a better fit of the data. The article concludes with its main innovation, testing a tactical asset allocation model constructed on the basis of the improved FED model. A model portfolio taking advantage of the short-term deviation in relative value, corrected for risk, leads to superior performance. <b>TOPICS:</b>Security analysis and valuation, fundamental equity analysis

Valuation of stochastic interest rate securities with time-dependent variance

We consider the problem of how to prize general securities whose payoff at maturity only depends on the interest rate r T at the time of exercise, where r t is supposed to be a stochastic Feller process. We show how to generalize the results of Cox et al. [Econometrica 53 (2) (1985) 385] regarding bond valuation to a situation where the stochastic evolution of r t under the martingale probability involves time-dependent coefficients and the payoff is arbitrary. The solution to this problem is given in terms of the propagator for the heat operator with a potential. This propagator is constructed in terms of a classical harmonic oscillator with time-dependent frequency.

A closed-form analytical solution for the valuation of convertible bonds with constant dividend yield

AbstractIn this paper, a closed-form analytical solution for pricing convertible bonds on a single underlying asset with constant dividend yield is presented. A closed-form analytical formula has apparently never been found for American-style convertible bonds (CBs) of finite maturity time although there have been quite a few approximate solutions and numerical approaches proposed. The solution presented here is written in the form of a Taylor's series expansion, which contains infinitely many terms, and thus is completely analytical and in a closed form. Although it is only for the simplest CBs without call or put features, it is nevertheless the first closed-form solution that can be utilised to discuss convertibility analytically. The solution is based on the homotopy analysis method, with which the optimal converting price has been elegantly and temporarily removed in the solution process of each order, and consequently, the solution of a linear problem can be analytically worked out at each order, resulting in a completely analytical solution for the optimal converting price and the CBs' price.