Valuation of Interest Payment Guarantees on Developing Country Debt

Eduardo Borensztein; George Pennacchi

doi:10.5089/9781451930788.024.A004

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Valuation of Interest Payment Guarantees on Developing Country Debt

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Mr. Eduardo Borensztein

Mr. Eduardo Borensztein
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George Pennacchi

George Pennacchi
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Publication Date:: 01 Jan 1990

eISBN:: 9781451930788

Language:: English

Keywords:: SP; interest rate equation; interest rate; deficit; interest payment guarantee; market price; guarantee contract; interest guarantee estimate; secondary market; Interest payments; Securities markets; Treasury bills and bonds; Asset prices

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The cost of interest payment guarantees is difficult to evaluate because guarantees are a contingent obligation that become effective only if a certain condition is met (that is, the debtor fails to make a certain payment). In this paper a technique to value interest payment guarantees on developing country debt is developed and preliminary estimates are presented. Using results from option pricing theory, the market price that an interest payment guarantee would have if it were traded in financial markets is estimated from the characteristics of secondary market prices of developing country debt.

Abstract

Many proposals for solving the foreign debt problem of developing countries contain some kind of contract in which part of the repayment to creditor banks is insured by a third party, such as a donor country or a multilateral organization. Evaluating the cost of issuing these guarantees is more complicated than for other schemes (such as cash buy-backs) in which the actual cost is directly observable because it involves a cash payment. A guarantee on future payments, in contrast, is a contingent obligation that only becomes effective if a certain condition is met—that is, if the debtor fails to make a certain payment.¹ This paper develops a technique to value these guarantees and provides some preliminary estimates of their cost.

The strategy that we adopt is to estimate the market price that an interest payment guarantee would have if the contract were traded in financial markets. Many other contingent liabilities with similar characteristics are traded in financial markets, most notably options, which means that we can use a number of results in finance theory. We show that an interest payment guarantee can be modeled as a portfolio of two put options. This model is then used to estimate the value of interest payment guarantees for a number of highy indebted countries.

The main problem in trying to price a guarantee is how to specify the random structure of the debtor country’s payments. For a “problem” debtor, in particular, net payments to the private banking system are difficult to predict, and are probably the result of a complicated bargaining process, which is itself affected by factors such as terms of trade changes, the economic and political evolution of the debtor country, and lending policies of official creditors. Without loss of generality, we assume the existence of an unobservable state variable that determines debtor country repayments to private banks. This state variable expresses the result of the bargaining process just described. We also make the relatively nonrestrictive assumption that this variable can be modeled as a random variable that follows a certain stable stochastic time-series process. We further assume that this process would not be altered by the issuance of a guarantee on payments. This means that we are abstracting from “moral hazard” problems—both from the point of view of countries and banks—that might reduce actual payments once a guarantee has been issued, or the conceivable opposite case in which payments on guaranteed debt are higher because of the possibly stronger bargaining power of the donor country or the multilateral organization involved.

Under the above assumptions, we are able to identify the stochastic process followed by the state variable (and therefore to determine the theoretical market price of the guarantee) by using available observations on the price of developing country debt in secondary markets. This procedure minimizes the number of specific assumptions that need to be made regarding the nature of the random structure of payments to creditor banks, or the kinds of variables that determine those payments. Previous work on this and related problems has been based on alternative approaches. Dooley and Symansky (1989) and Lamdany (1989) implicitly assume a given probability structure for debt repayments, and Claessens and van Wijnbergen (1989) assume that the price of oil is perfectly correlated with Mexico’s external debt payments.² The advantage of the technique followed in this paper is that no such special assumptions are necessary in order to arrive at the price of the guarantee.

Another major source of risk to the issuance of an interest guarantee on floating-rate securities is interest rate volatility. To evaluate this risk along with that associated with the debtor country’s uncertain repayment, we use an option model with stochastic interest rates inspired by Merton (1973) gy developed by Marcus and Snaked>As a first step, we estimate a model of the term structure of interest rates based on Vasicek (1977) using data on U.S. Treasury bills. We then use the estimated parameters that describe the stochastic processes followed by interest rates at different maturities, together with the parameters that describe the stochastic process of the market value of debt, to obtain all the statistical moments necessary for the option price equation.

The results of the estimation indicate that the cost of hypothetical interest payment guarantees for four years would fluctuate between close to the full value of interest payments for most countries with a market price of debt at or below 30 cents on the dollar, to nearly half that amount for countries whose debt sells at about 60 cents on the dollar. Loosely speaking, the estimates indicate that the cost of guarantees is high because debt prices are low and do not have a very large variance. This implies that there is little hope that payments would be high enough to avoid a substantial use of guarantee money.

The remainder of the paper proceeds as follows: Section I warns about the risks of using a simple back-of-the-envelope calculation for pricing guarantees. Section II derives a mathematical formula for pricing guarantees, inspired by option pricing theory. Section III discusses the technique for the estimation of guarantee values on the basis of the prices of debt in secondary markets alone. Section IV presents the estimation results, and Section V contains concluding remarks.

I. Is the Market Price a Sufficient Statistic?

It is tempting to adopt a simple, back-of-the-envelope calculation to estimate the cost of guarantees on debt payments. Such a calculation would use the market price of debt as the estimated probability that the country makes each and every payment. This calculation implies setting the value of the guarantee equal to 1 minus the market price of debt times the value of the guaranteed payments. However, the result could be far off the true theoretical price of the guarantee. Although the guarantee does not represent an independent risk in the sense that all the risk derives only from the randomness of the payments of the original debt, the random distributions of all the future cash flows of the debt contract are in principle necessary to determine the value of the guarantee. In particular, knowledge of the price at which debt is transacted is by no means sufficient to determine (or even rank) the cost of guarantees, although it is an essential piece of information; in fact, our pricing of the guarantee will be based as much as possible on the market price of debt, as opposed to using other variables.

Let us consider an example to illustrate why the market price of debt does not convey sufficient information for pricing the guarantee. Consider a two-period obligation, with equal payments due each period. To abstract from issues of asset pricing, let us assume that investors are risk-neutral and the interest rate is fixed and equal to zero. The face value of the debt is 100, and therefore the two payments are equal to 50. Suppose that the market price of debt, in this case equal to the expected value of payments, is equal to 40. There are many different random structures for the debt payments that may support the same market price for debt, but different prices for a guarantee on the first-period payment. Let us consider two of them. Country A makes, with certainty, a fraction p of every payment that is due. It is clear that the market price per dollar of face value of claims on country A is going to be equal to p, (and p = 0.4 in our example). The expected cost of a guarantee on the first-period debt payment for country A is going to be equal to

E (G^{A}) = (1 - p) 50 = 30,

with certainty, because a fraction 1-p will have to be paid by the guarantor.

Country B’s debt has a different payoff structure. In the first period, it is known with certainty that payments are going to be equal to zero. In the second period, payment will be determined with the following probability distribution:

\begin{array}{l} 50 with probability Π \\ 0 with probability 1 - Π, \end{array}

where ∏ = 2p; that is, ∏ = 0.8 in our example. It is clear that the certain cost of a guarantee on the first-period debt payment of country B is

E (G^{B}) = 50.

Therefore, the cost of the guarantee is 66 percent higher for country B than for country A, even though they both have the same market price of debt. The divergence could be even greater if the maturity of the debt were longer, because there would be more potential for different probability distributions of guaranteed payments.

There is one case in which the guarantee could be easily priced on the basis of the market price of debt—when the guarantee covers the full term of the debt. The pricing of a full guarantee would be straightforward: the guarantee would convert the risky debt into a default-free one, and its value would therefore be the difference between the price of a default-free bond and the market price of the risky debt. However, although this insight is useful, in practice it is not sufficient to price contracts that imply limited-term insurance.

Even greater variation in the value of guarantees may arise if we drop the assumption of investor risk neutrality. The proper valuation of a guarantee contract cannot be made only in terms of expected payments but must also include the risk premium associated with random guarantee payments. In other words, expected guarantee payments must be discounted by the appropriately risk-adjusted interest rates, which increases the potential for divergence in the value of guarantees for different countries with the same market price of debt but a different random structure of payments.

II. A Framework for Pricing Guarantees

We will assume that all debt takes the form of floating-rate infinite maturity contracts; that is, floating-rate perpetuities.³ We will assume that there exists a random variable S(t) that represents the state of nature and that determines the amount paid by the debtor country to the holders of its foreign debt. Each contractual payment that becomes due at time j is given by i_jD, and i_j is the interest rate applicable to the time j payment, and D is the contractual value (principal) of the debt. Without loss of generality, we assume that the actual payment, V_j made by the debtor country is determined according to the following schedule:

\begin{matrix} V_{j} = i_{j} D & if S (j) \geq D (1 + i_{j}) \end{matrix}

\begin{array}{l} V_{j} = S (j) - D & if D \leq S (j) \leq D (1 + i_{j}) \\ V_{j} = 0 & if S (j) \leq D . \end{array}

Consider now a contract that guarantees a single interest payment, i_j - D. At the maturity date,_j, the payoff of this contract, G_j will be

\begin{array}{l} G_{j} = 0 & if S (j) \geq D (1 + i_{j}) \\ G_{j} = D (1 + i_{j}) - S (j) & if D (1 + i_{j}) - S (j) \\ G_{j} = i_{j} D & if S (j) \leq D . \end{array}

This means that the value of the contract at maturity can be written as

\begin{matrix} G_{j} = \max [0, D (1 + i_{j}) - S (j)] - \max [0, D - S (j)], & (1) \end{matrix}

which is equivalent to a portfolio of two put options written on the underlying variable S; a long position on a put with exercise price D (1 +i_j), and a short position on a put with exercise price D.

If the guarantee covered only a fraction a of the interest due, the exercise price of the first put would be D(1 + αi_j). In the case of a partial interest guarantee, however, a better contract would be one that covered a fraction α of the shortfall in interest payments; that is, α(i_jD - V_j). Although incentive considerations are beyond the scope of this paper, guaranteeing a fraction of the shortfall would likely reduce the effects of moral hazard. In this case, when the debtor country makes one dollar of payment, it “loses” a fraction α from the guarantee contract, whereas in the previous case it “loses” one full dollar of potential guarantee money. In the case in which the guarantee covers a fraction α of the shortfall, it is clearly seen that its value will be equal to α times the value of the full-interest guarantee; all estimations were, therefore, done for the case of full-interest guarantee, since the value of partial guarantees can be computed easily from that basis.

Equation (1) above suggests the possibility of using option pricing theory to obtain the value of the guarantee G. However, the problem is more complicated than the standard Black-Scholes case, because the floating-rate feature makes the applicable interest rate a stochastic variable (until the time at which it is set for the next payment). This means that the exercise price itself will be a stochastic variable. However, using results by Merton (1973) on option pricing with stochastic interest rates and by Vasicek (1977) on the term structure of interest rates, it is possible to derive a formula for the first and second put options in equation (1). The details are provided in the Appendix. The derivation requires the assumptions that the rate of change of state variable S follow a continuous-time process with a constant variance per unit of time, and that the instantaneous interest rate follow an Ornstein-Uhlenbeck process.⁴

III. Measuring the Characteristics of the State Variable

The state variable S, which represents the repayment prospects by the debtor country, will, in general, depend on a number of variables. First, it will depend on variables affecting the debtor country’s economic situation such as random shocks affecting gross domestic product, terms of trade changes, and government policies. Second, S will be affected by policies adopted by creditor countries or by international initiatives, such as the tax and regulatory environment for banks or proposals to deal with the debt situation. Finally, S will depend on variables affecting the outcome of the bargaining process between the country and its creditor banks, such as, for example, the state of negotiations between creditor banks and other debtor countries.

The measurement of S thus poses a significant problem. Our strategy, following the methodology developed by Marcus and Shaked (1984) and Pennacchi (1987), is to obtain a measurement of S using data on the secondary market prices for debt. We start by noting that the value of a single interest payment on developing country debt in secondary markets equals the value of a default-free payment minus the value of a contract that guarantees full payment of interest; that is

\begin{matrix} V_{j} (t) = F_{j} (t) - G_{j} (t), & (2) \end{matrix}

where V_j(t) indicates the time t value of the debtor country’s interest payment that is contracted to be paid at date j F(t) is the time t value of a default-free, floating-rate payment of D i_j received at time j; and G_j(t) is the time f value of a guarantee on this interest payment at date j, which we described previously.⁵ Assuming that developing country debt can be modeled as a perpetuity, the total value of this debt, V(t), can be written as

\begin{matrix} V (t) = \sum_{j = 1}^{\infty} V_{j} (t) \\ \begin{matrix} = \sum_{j = 1}^{\infty} F_{j} (t) - \sum_{j = 1}^{\infty} G_{j} (t) . & (3) \end{matrix} \end{matrix}

Given that we have a solution for the value of a guarantee for each payment, G_j, and the value of a default-free payment, F_j, then equation (3) represents a formula for the market value of developing country debt. Since each guarantee contract G_j is equal to the value of the portfolio of two put options given in equation (1), V(t) will be a (nonlinear) function of S, the standard deviation of the rate of change in S, and the correlation between S and the instantaneous interest rate (see equations (14) and (15) in the Appendix).⁶ In the context of deposit insurance valuation, Pennacchi (1987) has shown that it is possible to estimate those three values by solving a three-equation system: the option price formula, the expression for the variance of the price of debt, and the expression for the correlation of the price of debt and default-free bond prices. This procedure is outlined in the Appendix.

IV. Estimation Results

The general indication of the estimation results is that, based on the information derived from secondary markets for developing country debt, the cost of guarantees appears to be pretty high. The reason is that debt of heavily indebted countries carries low prices with relatively low volatility (including volatility arising from the floating interest rate feature of the contracts). The low price of debt means that the situation looks bleak, and the low volatility of debt prices suggests that a sufficiently large improvement is unlikely. Table 1 presents descriptive statistics for secondary market prices of foreign debt for ten major highly indebted countries.

An identifying assumption is needed in order to proceed to the estimation. The reason is that 5 is not really a traded asset that would be held in an investor’s portfolio and whose rate of return is determined in accordance with asset market equilibrium, but rather a state variable whose expected rate of change may differ from that of an asset with the same systematic risk. Therefore, the estimation requires an assumption regarding the expected rate of change of S, or more precisely, the difference between the rate of return on an asset with the same risk as S and the true expected rate of change in S. This parameter is denoted as c.⁷ It is important to note that the effect of the assumed value of c over the estimated cost of guarantees is somewhat weaker than it might appear. The reason is that there is some trade-off between the assumed vaiue for c and the estimated value for 5. If one assumes a higher expected rate of growth for S—lower c—the estimate of S will be lower, partially offsetting the effect on the valuation of the guarantee.

Table 1.

Secondary Market Prices of Debt

Sources: Salomon Brothers, Inc., Indicative Prices for Less Developed Country Bank Loans (various issues), and International Monetary Fund, International Financial Statistics (various issues).

^{^a}Annualized standard deviation of the rate of return on the market price of debt.

^{^b}Covariance between the rate of return on the market price of debt and the change in the log price of six-month Treasury bills.

Table 1.

Secondary Market Prices of Debt

	Market Price		Standard	Covariance with
Country	Mar. 2, 1989	Jan. 18. 1990	Deviation^a	Six-Month Bond^b
Argentina	17.25	11.75	0.3294	-1.402E-05
Brazil	26.75	27.50	0.3030	2.081E-05
Chile	55.25	64.25	0.1628	1.270E-05
Colombia	50.00	59.75	0.1425	2.034E-07
Costa Rica	13.50	18.75	0.3816	-1.597E-05
Ecuador	12.00	14.50	0.2847	6.335E-06
Mexico	33.00	38.00	0.1871	-6.407E-06
Philippines	36.00	48.00	0.1534	2.121E-05
Uruguay	57.00	50.00	0.0758	2.974E-06
Venezuela	27.25	35.25	0.2456	8.690E-06

Sources: Salomon Brothers, Inc., Indicative Prices for Less Developed Country Bank Loans (various issues), and International Monetary Fund, International Financial Statistics (various issues).

^{^a}Annualized standard deviation of the rate of return on the market price of debt.

^{^b}Covariance between the rate of return on the market price of debt and the change in the log price of six-month Treasury bills.

Table 1.

Secondary Market Prices of Debt

	Market Price		Standard	Covariance with
Country	Mar. 2, 1989	Jan. 18. 1990	Deviation^a	Six-Month Bond^b
Argentina	17.25	11.75	0.3294	-1.402E-05
Brazil	26.75	27.50	0.3030	2.081E-05
Chile	55.25	64.25	0.1628	1.270E-05
Colombia	50.00	59.75	0.1425	2.034E-07
Costa Rica	13.50	18.75	0.3816	-1.597E-05
Ecuador	12.00	14.50	0.2847	6.335E-06
Mexico	33.00	38.00	0.1871	-6.407E-06
Philippines	36.00	48.00	0.1534	2.121E-05
Uruguay	57.00	50.00	0.0758	2.974E-06
Venezuela	27.25	35.25	0.2456	8.690E-06

Sources: Salomon Brothers, Inc., Indicative Prices for Less Developed Country Bank Loans (various issues), and International Monetary Fund, International Financial Statistics (various issues).

^{^a}Annualized standard deviation of the rate of return on the market price of debt.

^{^b}Covariance between the rate of return on the market price of debt and the change in the log price of six-month Treasury bills.

The estimation, reported in Tables 2 to 5, was carried out for two values of c: 0 and 0.09. These are two natural assumptions of values for c. A value of 0 for c, as in Tables 2 and 4, implies that the expected rate of change of S(t) equals the (unknown) expected rate of return on a marketable asset with the same risk as S(t). Thus, in this case, the value of the interest guarantee can be intepreted as the difference between two put options written on an asset. A value of 0.09 for c, as in Tables 3 and 5, is an approximation for the case in which the expected rate of growth of S is zero. This is because a value of 0.09 for c only means that the expected rate of change of S(t) equals that of a marketable asset with the same risk as S(t) less a 9 percent annual rate of change, 9 percent being approximately equal to the risk-free interest rate. In general, the expected rate of return on an asset with the same risk as S could be higher or lower than the risk-free rate, but in the case of S representing a risk uncorrected with other assets in the market, or in the case of a risk-neutral economy, a value of 0.09 for c would represent an expected rate of change in S(t) of approximately zero.

Table 2.

Value of Interest Payment Guarantees: Pre-Brady Plan Announcement, April 7, 1987 to March 2, 1989

(with c = 0)

Note: Guarantee refers to a four-year guarantee on a floating-rate perpetuity with semi-annual payments tied to the yield on the six-month U.S. Treasury bill rate. The initial six-month Treasury bill yield was 9.0625 percent; S is the current level of the state variable; c is the difference between the expected rate of growth of an asset with the same risk as S and that of S ; σ_s is the standard deviation of the rate of growth of S ; and ρsr, is the correlation of the rate of growth of S and the short-term (instantaneous) U.S. rate of interest.

Table 2.

Value of Interest Payment Guarantees: Pre-Brady Plan Announcement, April 7, 1987 to March 2, 1989

(with c = 0)

	Market Price				Value of
Country	of Debt	S	σ_s	ρst	Guarantee
Argentina	17.25	23.90	0.3369	-0.0300	36.39
Brazil	26.75	35.92	0.3125	-0.0480	35.27
Chile	55.25	61.41	0.1680	-0.0577	30.76
Colombia	50.00	54.07	0.1455	-0.0356	33,77
Costa Rica	13.50	20.45	0.3908	-0.0325	36.43
Ecuador	12.00	15.28	0.2881	-0.0166	36.79
Mexico	33.00	37.24	0.1903	-0.0233	36.19
Philippines	36.00	39.12	0.1555	-0.0270	36.32
Uruguay	57.00	58.36	0.0764	-0.0229	33.85
Venezuela	27.25	33.24	0.2509	-0.0324	36.10

Table 2.

Value of Interest Payment Guarantees: Pre-Brady Plan Announcement, April 7, 1987 to March 2, 1989

(with c = 0)

	Market Price				Value of
Country	of Debt	S	σ_s	ρst	Guarantee
Argentina	17.25	23.90	0.3369	-0.0300	36.39
Brazil	26.75	35.92	0.3125	-0.0480	35.27
Chile	55.25	61.41	0.1680	-0.0577	30.76
Colombia	50.00	54.07	0.1455	-0.0356	33,77
Costa Rica	13.50	20.45	0.3908	-0.0325	36.43
Ecuador	12.00	15.28	0.2881	-0.0166	36.79
Mexico	33.00	37.24	0.1903	-0.0233	36.19
Philippines	36.00	39.12	0.1555	-0.0270	36.32
Uruguay	57.00	58.36	0.0764	-0.0229	33.85
Venezuela	27.25	33.24	0.2509	-0.0324	36.10

Table 3.

Value of Interest Payment Guarantees: Pre-Brady Plan Announcement, April 7, 1987 to March 2, 1989

(with c = 0.09)

Note: Guarantee refers to a four-year guarantee on a floating-rate perpetuity with semi-annual payments tied to the yield on the six-month U.S. Treasury bill rate. The initial six-month Treasury bill yield was 9.0625 percent; S is the current level of the state variable; c is the difference between the expected rate of growth of an asset with the same risk as S and that of S ; σs is the standard deviation of the rate of growth of S ; and ρsr is the correlation of the rate of growth of S and the short-term (instantaneous) U.S. rate of interest.

Table 3.

Value of Interest Payment Guarantees: Pre-Brady Plan Announcement, April 7, 1987 to March 2, 1989

(with c = 0.09)

	Market Price				Value of
Country	of Debt	S	σ_s	ρsr	Guarantee
Argentina	17.25	54.41	0.0543	-0.5059	36.76
Brazil	26.75	67.15	0.0624	-0.5303	35.91
Chile	55.25	92.59	0.0536	-0.7476	24.01
Colombia	50.00	88.92	0.0503	-0.7928	27.35
Costa Rica	13.50	48.24	0.0542	-0.4544	36.81
Ecuador	12.00	45.63	0.0411	-0.5720	36.82
Mexico	33.00	74.12	0.0501	-0.7115	34.75
Philippines	36.00	77.11	0.0475	-0.7787	33.89
Uruguay	57.00	93.58	0.0434	-0.9263	23.20
Venezuela	27.25	67.78	0.0543	-0.6134	35.93

Note: Guarantee refers to a four-year guarantee on a floating-rate perpetuity with semi-annual payments tied to the yield on the six-month U.S. Treasury bill rate. The initial six-month Treasury bill yield was 9.0625 percent; S is the current level of the state variable; c is the difference between the expected rate of growth of an asset with the same risk as S and that of S ; σs is the standard deviation of the rate of growth of S ; and ρsr is the correlation of the rate of growth of S and the short-term (instantaneous) U.S. rate of interest.

Table 3.

Value of Interest Payment Guarantees: Pre-Brady Plan Announcement, April 7, 1987 to March 2, 1989

(with c = 0.09)

	Market Price				Value of
Country	of Debt	S	σ_s	ρsr	Guarantee
Argentina	17.25	54.41	0.0543	-0.5059	36.76
Brazil	26.75	67.15	0.0624	-0.5303	35.91
Chile	55.25	92.59	0.0536	-0.7476	24.01
Colombia	50.00	88.92	0.0503	-0.7928	27.35
Costa Rica	13.50	48.24	0.0542	-0.4544	36.81
Ecuador	12.00	45.63	0.0411	-0.5720	36.82
Mexico	33.00	74.12	0.0501	-0.7115	34.75
Philippines	36.00	77.11	0.0475	-0.7787	33.89
Uruguay	57.00	93.58	0.0434	-0.9263	23.20
Venezuela	27.25	67.78	0.0543	-0.6134	35.93

Note: Guarantee refers to a four-year guarantee on a floating-rate perpetuity with semi-annual payments tied to the yield on the six-month U.S. Treasury bill rate. The initial six-month Treasury bill yield was 9.0625 percent; S is the current level of the state variable; c is the difference between the expected rate of growth of an asset with the same risk as S and that of S ; σs is the standard deviation of the rate of growth of S ; and ρsr is the correlation of the rate of growth of S and the short-term (instantaneous) U.S. rate of interest.

Table 4.

Value of interest Payment Guarantees: Post-Brady Plan Announcement, April 7, 1987 to January 18, 1990

(with c = 0)

Note: Guarantee refers to a four-year guarantee on a floating-rate perpetuity with semi-annual payments tied to the yield on the six-month U.S. Treasury bill rate. The initial six-month Treasury bill yield was 9.0625 percent; S is the current level of the state variable; c is the difference between the expected rate of growth of an asset with the same risk as S and that of S ;σs is the standard deviation of the rate of growth of S ; and ρsr is the correlation of the rate of growth of S and the short-term (instantaneous) U.S. rate of interest.

Table 4.

Value of interest Payment Guarantees: Post-Brady Plan Announcement, April 7, 1987 to January 18, 1990

(with c = 0)

	Market Price				Value of
Country	of Debt	S	σs	σsr	Guarantee
Argentina	11.75	18.51	0.4142	-0.0328	36.48
Brazil	27.50	41.04	0.3722	-0.0643	33.89
Chile	64.25	71.48	0.1644	-0.0780	26.11
Colombia	59.75	65.57	0.1549	-0.0604	29.16
Costa Rica	18.75	27.60	0.3709	-0.0423	35.91
Ecuador	14.50	20.21	0.3427	-0.0312	36.58
Mexico	38.00	45.26	0.2290	-0.0413	34.55
Philippines	48.00	54.96	0.1953	-0.0548	32.58
Uruguay	50.00	51.29	0.0813	-0.0178	35.68
Venezuela	35.25	45.44	0.2825	-0.0555	33.85

Note: Guarantee refers to a four-year guarantee on a floating-rate perpetuity with semi-annual payments tied to the yield on the six-month U.S. Treasury bill rate. The initial six-month Treasury bill yield was 9.0625 percent; S is the current level of the state variable; c is the difference between the expected rate of growth of an asset with the same risk as S and that of S ;σs is the standard deviation of the rate of growth of S ; and ρsr is the correlation of the rate of growth of S and the short-term (instantaneous) U.S. rate of interest.

Table 4.

Value of interest Payment Guarantees: Post-Brady Plan Announcement, April 7, 1987 to January 18, 1990

(with c = 0)

	Market Price				Value of
Country	of Debt	S	σs	σsr	Guarantee
Argentina	11.75	18.51	0.4142	-0.0328	36.48
Brazil	27.50	41.04	0.3722	-0.0643	33.89
Chile	64.25	71.48	0.1644	-0.0780	26.11
Colombia	59.75	65.57	0.1549	-0.0604	29.16
Costa Rica	18.75	27.60	0.3709	-0.0423	35.91
Ecuador	14.50	20.21	0.3427	-0.0312	36.58
Mexico	38.00	45.26	0.2290	-0.0413	34.55
Philippines	48.00	54.96	0.1953	-0.0548	32.58
Uruguay	50.00	51.29	0.0813	-0.0178	35.68
Venezuela	35.25	45.44	0.2825	-0.0555	33.85

Note: Guarantee refers to a four-year guarantee on a floating-rate perpetuity with semi-annual payments tied to the yield on the six-month U.S. Treasury bill rate. The initial six-month Treasury bill yield was 9.0625 percent; S is the current level of the state variable; c is the difference between the expected rate of growth of an asset with the same risk as S and that of S ;σs is the standard deviation of the rate of growth of S ; and ρsr is the correlation of the rate of growth of S and the short-term (instantaneous) U.S. rate of interest.

Table 5.

Value of Interest Payment Guarantees: Post-Brady Plan Announcement, April 7, 1987 to January 18, 1990

(with c =0.09)

Note: Guarantee refers to a four-year guarantee on a floating-rate perpetuity with semi-annual payments tied to the yield on the six-month U.S. Treasury bili rate. The initial six-month Treasury bill yield was 9.0625 percent; S is the current level of the state variable; c is the difference between the expected rate of growth of an asset with the same risk as S and that of S ; σs is the standard deviation of the rate of growth of S and ρsr is the correlation of the rate of growth of S ; and the short-term (instantaneous) U.S. rate of interest.

Table 5.

Value of Interest Payment Guarantees: Post-Brady Plan Announcement, April 7, 1987 to January 18, 1990

(with c =0.09)

	Market Price				Value of
Country	of Debt	S	σs	ρsr	Guarantee
Argentina	11.75	45.04	0.0532	-0.4356	36.82
Brazil	27.50	67.99	0.0718	-0.4632	35.65
Chile	64.25	97.89	0.0527	-0.7497	17.75
Colombia	59.75	95.34	0.0515	-0.7721	21.05
Costa Rica	18.75	56.60	0.0607	-0.4702	36.70
Ecuador	14.50	49.99	0.0509	-0.5051	36.81
Mexico	38.00	79.03	0.0580	-0.6413	32.94
Philippines	48.00	87.48	0.0568	-0.6983	28.26
Uruguay	50.00	88.88	0.0436	-0.9186	27.56
Venezuela	35.25	76.38	0.0646	-0.5634	33.75

Table 5.

Value of Interest Payment Guarantees: Post-Brady Plan Announcement, April 7, 1987 to January 18, 1990

(with c =0.09)

	Market Price				Value of
Country	of Debt	S	σs	ρsr	Guarantee
Argentina	11.75	45.04	0.0532	-0.4356	36.82
Brazil	27.50	67.99	0.0718	-0.4632	35.65
Chile	64.25	97.89	0.0527	-0.7497	17.75
Colombia	59.75	95.34	0.0515	-0.7721	21.05
Costa Rica	18.75	56.60	0.0607	-0.4702	36.70
Ecuador	14.50	49.99	0.0509	-0.5051	36.81
Mexico	38.00	79.03	0.0580	-0.6413	32.94
Philippines	48.00	87.48	0.0568	-0.6983	28.26
Uruguay	50.00	88.88	0.0436	-0.9186	27.56
Venezuela	35.25	76.38	0.0646	-0.5634	33.75

The estimation was carried out with data corresponding to a date just before the announcement of U.S. Treasury Secretary Nicholas Brady’s initiative (in March 1989), and after it. The reason is that the Brady plan itself, by affecting the expected return on debt, may have had a major impact for certain countries, thus distorting our estimate of the actual payment capacity of debtor countries. Therefore, Tables 2 and 3 give estimates of the value of interest payment guarantees based on market prices of debt observed just before the announcement of the Brady plan, and Tables 4 and 5 give estimates that make use of all available data. In each table, we have assumed a debt principal level of D = 100 and an initial U.S. six-month, default-free interest rate of 9.0625 percent.⁸ Ten debtor countries with relatively more active secondary markets were selected for estimation. The first column of each table gives secondary market prices of debt quoted by Salomon Brothers, Inc. (see footnote to Table 1). The following three columns give the parameter estimates that were inferred from the market prices of this debt, the variance of the rate of return on the debt, and the correlation of the return with U.S. interest rates. Finally, the fourth column uses these parameter estimates to calculate the value of a four-year guarantee on semi-annual floating-rate interest payments, where the floating rate is equal to the yield on a six-month, default-free U.S. discount bond issued six months prior to the interest payment date.

When the expected rate of change in S(t) is assumed to equal that of an asset with the same risk as S(t)—that is, c — 0—the estimated value of the guarantee is in fact close to its upper bound of 36.82 (the full amount of the promised interest payments), with the exception of the three countries with higher-priced debt (Chile, Colombia, and Uruguay).⁹ When c is assumed to be equal to 9 percent, there is not a significant difference for countries with low secondary market prices of debt, but for the three with higher-priced debt the cost of guarantees falls significantly, because the larger estimated value for S makes them more likely to service their debt.

In general, the interest guarantee estimates based on secondary debt prices observed after the announcement of the Brady plan are somewhat lower than those based on prices observed prior to the announcement. For most debtor countries, debt prices increased after the plan was announced, and the estimated standard deviation of prices also increased. (The exceptions are Argentina and Uruguay.) In many cases, however, despite a sharp increase in market prices of debt, the value of the guarantee has not shown a significant decrease (for example, Costa Rica, Mexico, the Philippines, and Venezuela). The reason is that when the level of the state variable S is sufficiently low, increases in its estimated value do not generate significant increases in the probability of debt service by the debtor country and thus do not translate into much lower guarantee values.

It is interesting to compare these estimates with those that would be obtained by applying the rule of thumb referred to in Section I of this paper. The estimate of the value of a guarantee by the rule of thumb is simply the product of 1 minus the market price of debt times 36.82. It can be seen that although the ranking of the cost of the guarantees is roughly the same under both methods, the rule-of-tbumb estimates considerably undervalue the cost of guarantees. The ranking is similar under both methods, because the statistical moments of the market prices for the different countries are roughly similar (see Table 1)—a reflection of the casual observation of substantial co-movement of prices in this market. The underestimation of the true value of the guarantee responds to a more fundamental shortcoming of the rule-of-thumb method. When payments follow a stochastic process of the kind identified through the market price of debt, there is always some probability that the debtor country will make substantial payments at some point in the future, especially for long-term debt or with rescheduling of missed payments. Therefore, low prices of debt (such as those observed for most debtor countries) can be sustained even if essentially nothing is expected to come out as payments in the short run. The rule of thumb, by ignoring uncertainty altogether, is unable to capture this effect.

To check the robustness of the results, a different specification was estimated by using a fixed value for the standard deviation of the state variable S, and estimating both the level and expected rate of change of S. This allowed us to estimate the value of the parameter c. The standard deviation of the state variable was assumed to be equal to the standard deviation of gross national product (GNP) of each debtor country, which was generally higher than the estimated variance of S in the previous exercise. The results, which are reported in Tables 6 and 7, do not differ significantly from those obtained using the original specification. The estimated values of c range between minus 3 percent and plus 6 percent, and the differences with respect to previous estimates of the value of guarantees are therefore small.

Table 6.

Value of Interest Payment Guarantees; Pre-Brady Plan Announcement, April 7, 1987 to March 2, 1989

(with σs = volatility of GNP)

Note: Guarantee refers to a four-year guarantee on a floating-rate perpetuity with semi-annual payments tied to the yield on the six-month U.S. Treasury bill rate. The initial six-month Treasury bill yield was 9.0625 percent; S is the current level of the state variable; c is the difference between the expected rate of growth of an asset with the same risk as S and that of S; σs is the standard deviation of the rate of growth of S ; and σsr is the correlation of the rate of growth of S and the short-term (instantaneous) U.S. rate of interest.

Table 6.

Value of Interest Payment Guarantees; Pre-Brady Plan Announcement, April 7, 1987 to March 2, 1989

(with σs = volatility of GNP)

					Value of
Country	σs	S	c	σsr	Guarantee
Argentina	0.393	23.80	-0.0104	-0.0196	36.12
Brazil	0.125	48.12	0.0529	-0.2676	36.35
Chile	0.226	57.66	-0.0254	0.0311	30.13
Colombia	0.086	67.11	0.0408	-0.3890	32.71
Costa Rica	0.177	25.33	0.0390	-0.1444	36.81
Ecuador	0.168	19.12	0.0259	-0.1278	36.82
Mexico	0.200	36.61	0.0033	-0.0072	36.15
Philippines	0.129	42.45	0.0124	-0.1205	36.35
Uruguay	0.213	43.62	-0.0832	0.4800	33.46
Venezuela	0.136	41.88	0.0364	-0.2133	36.52

Note: Guarantee refers to a four-year guarantee on a floating-rate perpetuity with semi-annual payments tied to the yield on the six-month U.S. Treasury bill rate. The initial six-month Treasury bill yield was 9.0625 percent; S is the current level of the state variable; c is the difference between the expected rate of growth of an asset with the same risk as S and that of S; σs is the standard deviation of the rate of growth of S ; and σsr is the correlation of the rate of growth of S and the short-term (instantaneous) U.S. rate of interest.

Table 6.

Value of Interest Payment Guarantees; Pre-Brady Plan Announcement, April 7, 1987 to March 2, 1989

(with σs = volatility of GNP)

					Value of
Country	σs	S	c	σsr	Guarantee
Argentina	0.393	23.80	-0.0104	-0.0196	36.12
Brazil	0.125	48.12	0.0529	-0.2676	36.35
Chile	0.226	57.66	-0.0254	0.0311	30.13
Colombia	0.086	67.11	0.0408	-0.3890	32.71
Costa Rica	0.177	25.33	0.0390	-0.1444	36.81
Ecuador	0.168	19.12	0.0259	-0.1278	36.82
Mexico	0.200	36.61	0.0033	-0.0072	36.15
Philippines	0.129	42.45	0.0124	-0.1205	36.35
Uruguay	0.213	43.62	-0.0832	0.4800	33.46
Venezuela	0.136	41.88	0.0364	-0.2133	36.52

Note: Guarantee refers to a four-year guarantee on a floating-rate perpetuity with semi-annual payments tied to the yield on the six-month U.S. Treasury bill rate. The initial six-month Treasury bill yield was 9.0625 percent; S is the current level of the state variable; c is the difference between the expected rate of growth of an asset with the same risk as S and that of S; σs is the standard deviation of the rate of growth of S ; and σsr is the correlation of the rate of growth of S and the short-term (instantaneous) U.S. rate of interest.

Table 7.

Value of interest Payment Guarantees: Post-Brady Plan Announcement, April 7, 1987 to January 18, 1990

(with σs = volatility of GNP)

Note: Guarantee refers to a four-year guarantee on a floating-rate perpetuity with semi-annual payments tied to the yield on the six-month U.S. Treasury bill rate. The initial six-month Treasury bill yield was 9.0625 percent; S is the current level of the state variable; c is the difference between the expected rate of growth of an asset with the same risk as S and that of S ; σs is the standard deviation of the rate of growth of S ; and σsr is the correlation of the rate of growth of S and the short-term (instantaneous) U.S. rate of interest.

Table 7.

Value of interest Payment Guarantees: Post-Brady Plan Announcement, April 7, 1987 to January 18, 1990

(with σs = volatility of GNP)

Country	σs	S	c	σsr	Value of Guarantee
Argentina	0.393	18.48	0.0032	-0.0350	36.55
Brazil	0.125	53.44	0.0621	-0.2753	36.01
Chile	0.226	67.80	-0.0301	0.0125	25.58
Colombia	0.086	79.01	0.0474	-0.4236	27.25
Costa Rica	0.177	33.12	0.0398	-0.1568	36.70
Ecuador	0.168	25.57	0.0358	-0.1550	36.81
Mexico	0.200	46.88	0.0094	-0.0765	34.83
Philippines	0.129	62,40	0.0301	-0.2225	32.79
Uruguay	0.213	37.97	-0.0712	0.4648	35.26
Venezuela	0.136	55.66	0.0464	-0.2471	35.12

Note: Guarantee refers to a four-year guarantee on a floating-rate perpetuity with semi-annual payments tied to the yield on the six-month U.S. Treasury bill rate. The initial six-month Treasury bill yield was 9.0625 percent; S is the current level of the state variable; c is the difference between the expected rate of growth of an asset with the same risk as S and that of S ; σs is the standard deviation of the rate of growth of S ; and σsr is the correlation of the rate of growth of S and the short-term (instantaneous) U.S. rate of interest.

Table 7.

Value of interest Payment Guarantees: Post-Brady Plan Announcement, April 7, 1987 to January 18, 1990

(with σs = volatility of GNP)

Country	σs	S	c	σsr	Value of Guarantee
Argentina	0.393	18.48	0.0032	-0.0350	36.55
Brazil	0.125	53.44	0.0621	-0.2753	36.01
Chile	0.226	67.80	-0.0301	0.0125	25.58
Colombia	0.086	79.01	0.0474	-0.4236	27.25
Costa Rica	0.177	33.12	0.0398	-0.1568	36.70
Ecuador	0.168	25.57	0.0358	-0.1550	36.81
Mexico	0.200	46.88	0.0094	-0.0765	34.83
Philippines	0.129	62,40	0.0301	-0.2225	32.79
Uruguay	0.213	37.97	-0.0712	0.4648	35.26
Venezuela	0.136	55.66	0.0464	-0.2471	35.12

Note: Guarantee refers to a four-year guarantee on a floating-rate perpetuity with semi-annual payments tied to the yield on the six-month U.S. Treasury bill rate. The initial six-month Treasury bill yield was 9.0625 percent; S is the current level of the state variable; c is the difference between the expected rate of growth of an asset with the same risk as S and that of S ; σs is the standard deviation of the rate of growth of S ; and σsr is the correlation of the rate of growth of S and the short-term (instantaneous) U.S. rate of interest.

V. Conclusions

This paper has outlined a framework for valuing guarantees on floating-rate debt payments of developing countries. The main advantage of this method is that it derives the current level and the parameters of the stochastic process that determine repayments by using data on the market value of debt and interest rates only. It therefore requires no assumptions regarding the economic or political determinants of repayments, which, as mentioned above, could be diverse and hard to measure in any comprehensive way.

The main caveat of the technique applied here is the necessity of making an identifying assumption about the expected rate of growth of the unobservable variable that determines payments to banks. The problem arises because it is not possible to measure all the random factors affecting payments made by the debtor country, and there is no traded financial asset that is perfectly correlated with the unobservable state variable that determines a debtor country’s payments. However, it is possible to compute the value of guarantees for reasonable boundaries for the value of the unknown parameter. In any event, the assumptions required by the methodology applied in this paper are certainly much weaker than those That are implicit in the application of a rule of thumb of the type discussed in Section I.

The estimated values for the guarantee contracts may perhaps appear to be on the high side, especially for the six or seven debtor countries for which the prices of debt are lower, but this is merely a reflection of the low market valuation of the debt of these countries. It opens the question however, of whether the estimated specification of the process followed by debt prices is consistent with the record of payments by debtor countries, or whether it might be too pessimistic relative to that record, possibly reflecting fears of dramatic bad news for payments to banks, arising from political or institutional considerations.

APPENDIX Derivation of Interest Payment Guarantee Formula

The stochastic process for the state variable, S(t). is assumed to take the form

\begin{matrix} \frac{d s (t)}{S (t)} = α_{s} (t) d t + σ_{s} d z, & (4) \end{matrix}

where αs(f) is the (possibly time-varying) instantaneous expected rate of change of S(t); dz is a standard Wiener process; and σs, the instantaneous standard deviation of S(t), is assumed to be a constant. We will also assume that σs may(t) differ from the expected rate of return on a marketable asset with the same risk as S(t. The difference between the rate of return on this marketable asset and σs(t) is assumed to equal a constant, c.

Let the price at time t of a default-free discount bond that pays $1 at time t + τ be given by P(t,τ). As in Merton (1973), we assume this bond price has the following dynamics:

\begin{matrix} \frac{d P (t, τ)}{P (t, τ)} = α_{p} (t) d t + σ_{p} (τ) d q, & d z d q = ρ d t, & (5) \end{matrix}

where σ_p (t) is a function only of the bond’s time to maturity.

We further assume that interest rates are described by Vasicek’s (1977) model, which is consistent with the assumed dynamics for bond prices given in equation (5). This implies that the instantaneous nominal interest rate, r(t), follows the process

\begin{matrix} d r (t) = α [γ - r (t)] d t - σ d q . & (6) \end{matrix}

The parameter γ is the steady-state mean of r(t), and σ² represents its instantaneous variance. The parameter α measures the magnitude of mean reversion in the short-term interest rate.

Given equation (6), and assuming the market price of interest rate risk is a constant,φ Vasicek (1977) showed that the equilibrium price of a default-free discount bond is of the form

\begin{matrix} P (r (t), τ) = A (τ) \exp [- \frac{1}{α} (1 - e^{α τ}) r (t)], & (7) \end{matrix}

where

A (τ) = \exp [(\frac{1 - e^{- α τ}}{α} - τ) (γ + \frac{σ ϕ}{α} - \frac{σ^{2}}{2 α^{2}}) - \frac{σ^{2}}{4 α^{3}} {(1 - e^{- α τ})}^{2}] .

Using Ito’s lemma, the standard deviation of this bond’s rate of return is given by

\begin{matrix} σ_{p} (τ) = \frac{σ}{α} (1 - e^{- α τ}) . & (8) \end{matrix}

Now consider the value of a guarantee on a single floating-rate interest payment, where the interest payment is tied to the yield on a default-free discount bond with a maturity τr. It is assumed that the interest reset date of the floating-rate debt is also exactly τr periods prior to the interest payment date. More specifically, the debt interest payment equals a spread, S, plus the yield on a default-free bond with a maturity τr, that was issued τr periods prior to the interest payment date. Under these assumptions, the maturity value of the interest guarantee is given by equation (1) in the text where the promised interest payment is

\begin{matrix} 1 + i_{t + τ} = \frac{e^{s τ_{r}}}{P (t + τ - τ_{r}, τ_{r})} . & (9) \end{matrix}

A straightforward extension of Merton’s (1973) work on valuing options when interest rates are stochastic yields the value of the guarantee, G_r+τ(t):

\begin{matrix} G_{t + τ} (t) = e^{s τ_{t}} D P (t, τ - τ_{r}) N (- d_{12}) - e^{- c τ} S (t) N (- d_{11}) \\ \begin{matrix} - D P (t, τ) N (- d_{22}) + e^{- c τ} S (t) N (- d_{21}), & (10) \end{matrix} \end{matrix}

where

\begin{matrix} d_{11} = \ln (\frac{e^{- c τ} S (t)}{e^{s τ_{r}} D P (t, τ - τ_{r})}) / T_{1}^{1 / 2} + \frac{1}{2} T_{1}^{0 / 2} \\ d_{12} = d_{11} - T_{1}^{1 / 2} \\ d_{21} = \ln (\frac{e^{- c τ} S (t)}{D P (t, τ)}) / T_{2}^{1 / 2} + \frac{1}{2} T_{2}^{1 / 2} \\ d_{22} = d_{21} - T_{2}^{1 / 2}, \end{matrix}

and where

\begin{matrix} T_{1} = \int_{0}^{τ_{r}} σ_{s}^{2} + σ_{p}^{2} (ω) - 2 ρ σ_{s} σ_{p} (ω) d ω \\ + \int_{0}^{τ - τ_{r}} σ_{s}^{2} + σ_{p}^{2} (ω) - 2 ρ σ_{s} σ_{p} (ω) d ω \\ T_{2} = \int_{0}^{τ} σ_{s}^{2} + σ_{p}^{2} (ω) - 2 ρ σ_{s} σ_{p} (ω) d ω . \end{matrix}

Using the expression for σp(ω) from equation (8), the integration in the formulas for T₁, and T₂ can be easily carried out.

Estimation of Parameters of the Term Structure

Let B(t) = ln P(r(t),τ); that is, the log of a given maturity bond price. Then, using equation (7) and Ito’s lemma, one can show that B(t), given a constant maturity t, will follow the process

\begin{matrix} \begin{matrix} d B (t) = (α \ln A (τ) - γ (1 - e^{- α τ}) - α B (t)) d t + \frac{σ}{α} (1 - e^{- α τ}) d q & (11) \end{matrix} \\ \equiv (K (τ) - α B (t)) d t + \frac{σ}{α} (1 - e^{- α τ}) d q . \end{matrix}

This continuous time process has a discrete time AR(1) representation of the form

\begin{matrix} B (t + δ) = K^{'} (δ) + e^{- α δ} B (t) + v_{t} (δ), & (12) \end{matrix}

where v_t, is normally distributed with mean zero and variance equal to

\begin{matrix} var (v_{t} (δ)) = \frac{σ^{2}}{2 α^{3}} {(1 - e^{- α τ})}^{2} (1 - e^{- 2 α δ}) . & (13) \end{matrix}

Using equations (12) and (13), maximum likelihood estimation of the parameters ρ,σγ, and φ can be carried out. This was done using end-of-month prices of 30-, 90-, 180-, and 345-day Treasury bills over the period 1970 through 1986. The estimates and standard errors are

\begin{matrix} α & σ & γ & ϕ \\ 0.1961 & 0.0452 & 0.0889 & 0.3146 \\ (0.1210) & (0.0032) & (0.0525) & (1.1329) \end{matrix}

Estimation of the Parameters of the State Variable S(t

In this section we describe a technique that allows us to estimate the level of S(t, its rate of return variance, σ²_s and the correlation parameter ρ, using data on secondary market prices of developing country debt. The developing country debt is assumed to be equal to a floating-rate perpetuity. Let V_t+τ(r) equal the value of time t of a single floating-rate payment to be received in τ periods that is subject to default risk; that is, it is not guaranteed. Then its value must equal the value of a default-free, floating-rate payment, less the value of the guarantee on this floating-rate payment:

\begin{matrix} V_{t + τ} = e^{s τ_{r}} D P (t, τ - τ_{r}) - D P (t, τ) - G_{t + τ} (t) \\ = D (e^{s τ_{r}} P (t, τ - τ_{r}) N (d_{12}) - P (t, τ) N (d_{22})) \\ \begin{matrix} + e^{- c τ} S (t) (N (- d_{11}) - N (- d_{21})) . & (14) \end{matrix} \end{matrix}

Therefore, the market value of this floating-rate perpetuity, V, is given by

\begin{matrix} V (t) = \sum_{τ_{i}}^{\infty} V_{t + τ} (t) . & (15) \end{matrix}

Using Ito’s lemma, we can solve for the instantaneous variance of V(t), as well as its covariance with the rate of return on a t-period discount bond.

\begin{matrix} σ_{V}^{2} = {(\frac{\partial V}{\partial S} σ_{s} \frac{S}{V})}^{2} + {(\frac{\partial V}{\partial r} \frac{σ}{V})}^{2} + 2 ρ \frac{\partial V}{\partial S} \frac{\partial V}{\partial r} σ_{s} σ \frac{S}{V^{2}} & (16) \end{matrix}

\begin{matrix} σ_{V r} = - \frac{\partial V}{\partial S} {ρσ}_{s} \frac{σ}{α} (1 - e^{- α τ}) \frac{S}{V} - \frac{\partial V}{\partial r} \frac{σ^{2}}{α} (1 - e^{α τ}) \frac{1}{V}, & (17) \end{matrix}

where ωV/∂S and ∂V/∂r are evaluated in a straightforward (but lengthy) manner using equations (15) and (14). By using secondary market prices of developing country debt as well as prices of U.S. Treasury bills, we can observe V(t) as well as estimate σ²_v and σ_Vr. Given these estimates, equations (15), (16), and (17) are a system of three nonlinear equations in the three unknowns, S(t), σ_s and ρ. Numerical methods can then be used to solve this system.

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Eduardo Borensztein, an Economist in the Research Department, received his doctorate from the Massachusetts Institute of Technology. George Pennacchi, Associate Professor at the University of Illinois, is a graduate of Brown University and received his doctorate from the Massachusetts Institute of Technology. Part of this paper was written while he was a visiting scholar in the Research Department. The authors are grateful to David Folkerts-Landau, Mohsin Khan, and seminar participants at the Research Department and the University of Pennsylvania.

The cost of a guarantee should not be confused with the maximum potential liability of the guarantor; even if that amount is required to be set aside as collateral, the true value of the guarantee is a smaller amount given by the economic cost of providing the contingent payments that might be made out of the collateral.

For other useful insights into the pricing of guarantees, see Nocera (1989) and Clark (1990).

Although developing country loans have fixed contractual maturities, in practice principal repayments have tended to be rescheduled. Besides, prices of long-term bonds (30 years or so) do not differ much from perpetuities.

This means that the rate of return on S follows the continuous-time analogy of a random walk with a possibly stochastic drift and the short-term interest rate follows the continuous-time analogy of a first-order autoregressive process.

Because of interest rate uncertainty, i_j will, in general, be random. The formula for F_j(t) is given implicitly in equation (14) in the Appendix.

The total value of the debt, V(_t), will also depend on the difference between the rate of return on a marketable asset with the same risk as S(t) and the expected rate of change of S(t), This variable is denoted as c in the Appendix. Estimates of interest guarantees are carried out under alternative assumptions regarding the value of c.

Alternative identifying assumptions are possible. We also report, below, the results from fixing the value of the volatility of S.

As described in the Appendix, our estimates used the Vasicek (1977) model of the term structure, which assumes that the instantaneous (short-term) rate of interest follows a mean-reverting process. This interest rate was estimated to have a long-run mean of 8.89 percent.

The figure of 36.82 is the sum of the market values of eight default-free bonds with maturities at six-month intervals. The prices of these bonds are derived from the Vasicek term-structure model whose estimation is reported in the Appendix.

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IMF Staff papers: Volume 37 No. 4

Author:

International Monetary Fund. Research Dept.

Volume/Issue:: Volume 1990: Issue 004

Publisher:: International Monetary Fund

ISBN:: 9781451930788

ISSN:: 1020-7635

Pages:: 180

DOI:: https://doi.org/10.5089/9781451930788.024

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Table 7.

Value of interest Payment Guarantees: Post-Brady Plan Announcement, April 7, 1987 to January 18, 1990

(with σs = volatility of GNP)

Country	σs	S	c	σsr	Value of Guarantee
Argentina	0.393	18.48	0.0032	-0.0350	36.55
Brazil	0.125	53.44	0.0621	-0.2753	36.01
Chile	0.226	67.80	-0.0301	0.0125	25.58
Colombia	0.086	79.01	0.0474	-0.4236	27.25
Costa Rica	0.177	33.12	0.0398	-0.1568	36.70
Ecuador	0.168	25.57	0.0358	-0.1550	36.81
Mexico	0.200	46.88	0.0094	-0.0765	34.83
Philippines	0.129	62,40	0.0301	-0.2225	32.79
Uruguay	0.213	37.97	-0.0712	0.4648	35.26
Venezuela	0.136	55.66	0.0464	-0.2471	35.12

Note: Guarantee refers to a four-year guarantee on a floating-rate perpetuity with semi-annual payments tied to the yield on the six-month U.S. Treasury bill rate. The initial six-month Treasury bill yield was 9.0625 percent; S is the current level of the state variable; c is the difference between the expected rate of growth of an asset with the same risk as S and that of S ; σs is the standard deviation of the rate of growth of S ; and σsr is the correlation of the rate of growth of S and the short-term (instantaneous) U.S. rate of interest.

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Abstract

I. Is the Market Price a Sufficient Statistic?

II. A Framework for Pricing Guarantees

III. Measuring the Characteristics of the State Variable

IV. Estimation Results

V. Conclusions

APPENDIX Derivation of Interest Payment Guarantee Formula

Estimation of Parameters of the Term Structure

Estimation of the Parameters of the State Variable S(t

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