Back Matter

### APPENDIX

For lower-rated issuers, default risk is a major factor linking an issuer’s bond and equity prices. As default risk increases, equity prices fall as the residual claimants of the firm, the shareholders, face the possibility that the firm may go bankrupt. Similarly, bond prices fall because debtholders face the possibility of not being paid in full. For higher-rated issuers, default risk is not a major driving factor of bond and equity prices because the current value of the firm’s assets largely exceed its debt, This appendix formalizes these arguments in Merton’s corporate debt model (Merton, 1974) and explains why it holds true for both corporate and sovereign issuers.

#### A. Equity and Debt Prices Linkages in Merton’s Corporate Debt Model

Merton (1974) assumes that a firm’s debt consist on a zero-coupon bond with a face value F maturing T periods in advance. The corporation defaults at maturity if its asset value, V, is less than the amount it owes to bondholders. Assuming risk neutrality, Merton shows that the equity price is equivalent to a call option on the assets of the firm. The strike price of the option is the face value of the debt because shareholders get paid only if bondholders are paid in full.

It follows that the firm’s bond and equity prices are linked by the equation:

$\frac{B}{E}=\frac{1}{N\left({d}_{1}\right)-\mathit{dN}\left({d}_{2}\right)}-1,$

where B is the debt price, E is the equity price, d is the leverage of the corporation measured as the ratio between the face value of the bond discounted at the risk-free rate, r, and the asset value of the firm:

$d=\frac{F{e}^{-rT}}{V},$

N is the cumulative normal distribution and

$\begin{array}{l}{d}_{1}=\left(-{log}\left(d\right)+1/2{\sigma }^{2}T\right)/\sigma \sqrt{T},\\ {d}_{2}=\left(-{log}\left(d\right)-1/2{\sigma }^{2}T\right)/\sigma \sqrt{T}.\end{array}$

In the model, bond and equity prices are always positively correlated. The degree of correlation, however, depends directly on the firm’s leverage. In the extreme case that B/E0, the correlation is also negligible. This discussion is summarized graphically in the figure below.

#### B. Why Merton’s Model Applies For Sovereign Issuers

One major objection to applying Merton’s model directly to sovereign issuers is that a country may choose to default on its debt even if it is technically solvent—the country’s assets are enough to pay bondholders back but the country chooses not to pay. This situation is known as a country’s “willingness-to-pay.” In this section, we use a heuristic approach using a conceptual model to rationalize the willingness-to-pay. We also explain why the willingness-to-pay does not affect the linkages between debt and equity prices implied by Merton’s model.

The model assumes that debt consists of a single bond that matures T periods ahead. The debt is held by foreigners, and the equity held by the country. For simplicity, the model also assumes that the country pays nothing if the country’s assets are worth less than the face value of the country’s debt. Default is defined as the event of the country paying bondholders less than the face value of the bond. Default may occur even if the country is technically solvent.

Feasible functions for the value of debt and equity in the model should satisfy three conditions. The first condition is the balance sheet identity requiring that the country’s asset value is equal to the sum of its debt and equity values. The second condition is that countries are worse off when they default. Therefore, the country’s equity value is worth less when the country chooses to default than when it continues to honor its debt. This incentive compatibility condition guarantees that countries avoid default whenever they can. The third condition is a limit condition. Namely, the country always honor its obligations if its assets largely exceed its debt.

The main characteristics of the feasible functions for debt and equity can be inferred using a trial-and-error approach. We first guess that the country pays nothing if the country’s assets are worth less than the country’s nominal debt, as assumed above. Otherwise, the country pays its debt fully. The value of debt, hence, is equivalent to a cash-or-nothing call option (Figure A2, panel A). The value of equity, however, is discontinuous and violates the incentive compatibility condition: just prior to default, equity is worth nothing and upon default, its value is equal to the country’s nominal debt.

Our first guess can be modified to satisfy the incentive compatibility condition. One such modification of our first guesses for the debt and equity value functions that satisfy the balance-sheet, compatibility, and limit conditions is shown in Figure A2, panel B. These value functions illustrate three main characteristics shared by all feasible functions for the value of debt and equity. First, the country defaults when it is technically solvent. Thus, the model captures the willingness to pay factor. Second, the value of debt and equity are increasing functions of the country’s asset value. This implies positive correlation between bond and equity prices, as in Merton’s model. Third, the value of debt is less sensitive to changes in country value when the value of the country’s assets is large relative to the face value of debt. The opposite is true for the value of equity. The second and third features imply that correlation is stronger when the country’s asset value is barely above the default point, as in Merton’s original model.

Our second guess for the value of debt and equity is interesting on its own right. The incentive compatibility condition requires that feasible solutions to the value of equity must be non-decreasing in the range bounded by the face value of debt and the default point. Therefore, the solution depicted in panel B is the lower bound for all feasible solutions to the value of equity. It follows that the corresponding solution to the value of debt is an upper bound for all feasible solutions to the value of debt.

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We would like to thank Todd Groome, Donald Mathieson, Michele Nicoletta, and David Xu for their comments and suggestions. We remain responsible for any errors or omissions.

Underlying assets may be bank loans or commercial bonds. Or it could be other financial obligations on which the cash flows and/or credit event for credit derivatives are based.

Credit risk is a risk that the counterparties to transactions will fail to make obligated payments. Credit risk is sometimes called default risk. Market risk refers to movements in interest rates, exchange rates, stock prices, or commodity prices.

It should be noted here that in the United States restructuring is accepted as a credit event while it is not accepted in Europe. Therefore, sometimes, the U.S. contract may not be substitutable for European contract.

In September 2003, banks sold US$1.3 trillion of gr/oss credit protection though they were net buyers of protection in the amount of US$229 billion. Insurance companies sold protection in a net amount of US\$303 billion (FitchRatings, 2003).

For more information on these instruments, see Tavakoli (1998).

Although a large number of contracts are written on 5-year maturity, investment banks also write CDS contracts on the notional amounts and maturities specified by their clients. For corporate and financial institutions, however, trading activity is greatest for five-year contracts.

The main assumptions are: first, the risky bond and the risk-free bond are par-floating rate securities. Second, there are no transaction costs, and tax effects are negligible. Third, the payment of the CDS spread stops if a credit event occurs. Finally, protection buyers are paid on the next coupon date following the occurrence of the credit event.

The interested reader should refer to the original citations, Baillie and others (2002), and Lehman (2002). See Blanco, Brennan, and Marsh (2003) for an application to CDS and bond spreads at the corporate level,

The negative coefficients further suggest that the equity market in these countries are prone to price bubbles.

Equity Prices, Credit Default Swaps, and Bond Spreads in Emerging Markets
Author: Mr. Jorge A Chan-Lau and Ms. Yoon Sook Kim