3 Contingent Claims Approach to Risk Assessment and Pricing of Sovereign Debt

Assessing sovereign country risk and pricing it have been at the forefront of the literature on international credit markets. Several authors have modelled sovereign default risk and proposed methods of pricing it. See Cohen (1991, 1993), Duffee (1999), KMV Corporation (2002), Duffee et al. (2003), Arellano (2008), Borensztein and Panizza (2008) and Hilscher and Nosbuch (2010). One specific approach, the contingent claims analysis, seems appropriate for assessment of the riskiness of sovereign debt of developing countries. It is based on the pricing of options proposed by Black and Scholes (1973) and Merton (1973, 1974) and has been developed by Grossman and Van Huyck (1985), Gray et al. (2007, 2008), Gapen et al. (2008), Francois et al. (2011) and Jobst and Gray (2013).

Gray et al. (2007) present a simple model of the balance sheet approach to the contingent claims risk assessment and pricing of sovereign debt. They portray the economy of the borrower country as a combined balance sheet of Government and monetary authorities. Assets of the balance sheet include (i) Foreign reserves, (ii) Net fiscal asset and (iii) Other public assets. The Foreign reserves consist of the public sector's net international reserves. Net fiscal assets are the difference between the present value of taxes and revenues on the one hand and the present value of non-discretionary expenditures on the other hand. Other public assets include equity in public enterprises, value of the public sector's monopoly on the issue of money and other financial and non-financial assets.

The liabilities included in the country's balance sheet comprise (i) Base money, (ii) Local currency debt, (iii) Foreign currency debt and (iv) Guarantees. Base money consists of currency in circulation and bank reserves. Local currency debt is owed to domestic creditors outside Government and monetary authorities. Foreign currency debt is sovereign and denominated in foreign currency and owed to foreigners. Guarantees are extended by Government to domestic financial and non-financial entities.

Gray et al. define a distress barrier as the present value of the promised payment related to sovereign debt denominated in foreign currency and propose to measure it as the country's short term debt plus one-half of long term debt plus interest payment up to time t. Distress or default occurs when the country's sovereign assets fall below the distress barrier, which may happen considering that the country's foreign assets are stochastic. Therefore the country's debt is risky.

The borrower country's balance sheet can be written as follows: Assets =Equity+ Risky Debt, or

(1)

A(t) is the value of assets at time t

J(t) is the value of the country's equity at time t and

D(t) is the country's risky debt at time t.

Based on the contingent claims approach the equity can be considered as an implicit call option on the assets with an exercise price that is equal to the promised payments, B, that will mature in T-t periods. The risky debt can be considered as a risk-free debt minus a guarantee against default which is equal to a put option on the assets with an exercise price equal to B. Therefore,

Risky debt = Default-free Debt Debt guarantee and

(2)

Where P(t) is the value of the debt guarantee.

Assuming t = 0, Black and Scholes's formula for the value of a call option (the equity) gives

(3)

(4)

(5) σ2

r is the risk-free rate

σ is the asset return volatility

N(d ) is the cumulative probability of the standard normal density function below d

The “risk-neutral” or “risk-adjusted” default probability is N(–d2).

The formula for the “delta” of the put option is N(d1) – 1.

The yield to maturity on the risky debt, y, is defined by:

(6)

(7)

And the credit spread is:

(8)

The Value of Assets at Time (t)

Gray et al. depict the process of asset return as follows:

(9)

where μA is the drift rate or asset return on A,

σA is the volatility of the return on asset A.

e is a normally distributed random variable with zero mean and unit variance.

As indicated earlier, default occurs when assets, A, fall to or below the promised payments, Bt. Therefore, the probability of default is the probability that At ≤ Bt which is:

(10)

Considering that ε : N(0, 1), the “actual” probability of default is N(-d2, μ), where

(11)

Two variables in (11) are of key interest with respect to the determination of a borrower country's sovereign risk. Asset A is expected to increase by μA. If μA

increases, it increases d2,μ, and lowers the probability of default N(-d2, μ), A higher level of volatility, σA, lowers the numerator in (11), increases the denominator and results in a lower probability of default. In summary, the riskiness of a borrower country's sovereign debt can result from a higher level of the assets with which it services its debt or from lower volatility of the return on the assets.

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