An option - pricing approach to secondary market debt : applied to Mexico

Policy, Planning, and Research WORKING PAPERS LDebt and International Finance ____ ___ __ j International Economics Department and Latin America and the Caribbean Country Department 11 The World Bank January 1990 WPS 333 An Option-Pricing Approach to Secondary Market Debt (Applied to Mexico) Stijn Claessens and Sweder van Wijnbergen This pricing model for secondary market debt is designed to assess the impact of debt reduction on the value of remaining claims and the market value of different types of guarantees. The Policy, Planning, and Research Con'olex distnbutes PPR Workung Papers to disseminate the findings of work in progress and to encourage the exehange of ideas among ;lank staff and all others interested in dcvclopment issues. These papers carry the names of the authors, reflect only their views, and should be used and cited accordingly. Tle findings. inlerpretaions, and conclusions arc the authors' own. They should not be atributcd so the World [lank, its lIoard of Directors, its management, or any of its member countries.  Plc,Planning, and Research | Debt and International Finance| Claessens and van Wijnbergen present a pricing Claessens and van Wijnbergen apply the model for secondary market debt designed to model to Mexico. They first price the value of a assess the market value of various forms of general obligation claim. They then price claims guarantees and the impact of debt reduction on with fixed and rolling interest guarantees. Thc) the value of remaining claims. derive specific market values for general obliga- tion debt and for collateralized exit bonds and Their model is more flexible and realistic show the impact of different debt reduction than other models. The technique used -- shemes on the secondary market price. They option pricing - accounts explicitly for the conclude that the terms of the new bonds are in sources and nature of risks on sovereign debt. accord with recent secondary market prices of By so doing it is possible to assess the market the existing debt. value of various forms of guarantees as weU as the impact of debt reduction on secondary The authors show that the three debt restruc- market pricing. turing options offered to individual ba.iks are not equivalent if the newly created exit bonds are The model is extremely flexible in handling senior to new-money claims. The new-money different maturity schedules, differences in option is wonh considerably less. seniority, and expectations about the availability of foreign exchange and willingness to pay. This paper is a product of the Debt and International Finance Division, International Economics Department and the Country Operations Division, Latin America and the Caribbean Country Department H. Copies are available free from the World Bank, 1818 H Street NW, Washington DC 20433. Please contact Sheilah King-Watson, room S7-033, extension 33730 (21 pages with figures and tables). The PPR Working Paper Scries disseminates the findings of work under way in the Bank's Policy, PManning, and Rcscarch Complex. An objective of the series is to get thcsc findings out quickly, even if presentations are less than fully polished. The findings, interprctations, and conclusions in these pipers do not necessarily represent official policy of the Bank. Produced at the Pl'R Dissemination Centcr  An Option-Pricing Approach to Secondary Market Debt (Applied to Mexico) by Stijn Claessens World Bank and Sweder van Wijnbergen* World Bank and CEPR Table of Contents I. Introduction 2 II. Pricing Secondary Market Debt Using Option Pricing 3 II.1 A Secondary Market Model 3 II.2 Pricing Exit Bonds with Fixed Guarantees 6 II.3 Pricing Exit Bonds with Rolling Interest 7 Guarantees II.4 Pricing Bonds with Recapture Clauses 9 III. Secondary Market Pricing and the Value of 10 Guarantees: Mexico 1989 III.1 Behavior of Foreign Exchange Available 10 III.2 Secondary Market Pricing 11 III.3 Valuation of Fixed Interest Guarantees 15 III.4 The Value of Rolling Interest Guarantees 16 IV. Mexico's 1989 External Debt Agreement 17 IV.1 Outline of the July 1989 Debt Agreement 17 IV.2 Debt Relief 18 IV.3 Attractiveness of the Various Options for 19 the Creditors V. Conclusions 20 References 21 * We are indebted to Leonardo Auernheimer, Michael Dooley, Reuben Lamdany and other participants in two World Bank seminars for comments. The second author also thanks Daniel Cohen and Ricardo Martin for helpful discussions. Paul van der Eyck provided inv.aluable assistance in debugging the computer programs used in this paper. Copies of the computer programs used are available on request from the second author.  2 1 INTRODUCTION The Brady plan, with its emphasis on negotiated debt reduction with possible support from official sources, has increased the importance of a good understanding of the pricing of external debt in secondary markets. It is clearly difficult to assess the feasibility of different debt restructuring schemes without a better understanding of the pricing of existing debt in the secondary market, and the likely effects of different debt reduction strategies on secondary market prices. After all, these prices represent the opportunity cost to holders of the claims being restructured and define the limits within which the bargaining process can produce an outcome. Three major issues are at stake. First the impact of debt reduction per se on the valuation of the claims that remain. Second the extent to which credit enhancements, through collateralization and forms of official guarantees for newly created claims, increase the market value of the instruments to which such enh.ancements are attached. Third, the impact of changes in seniority structure due to the newly created claims. These problems are interrelated in that debt reduction itself may affect the valuation of credit enhancements on remaining or newly created instruments. Existing modals fall short of providing the necessary consistent approach to secondary market pricing of existing debt and newly created, partly enhanced claims. We will argue below that most existing models are insufficiently equipped to discuss the dynamics of secondary market prices under alternative debt reduction strategies and few are able to provide insights on the value of credit enhancement. Most models used for pricing secondary market debt can be classified in one of two classes: 1) the all-or-nothing approach: in each year, the debtor pays its debt service obligation in full with a certain probability and pays nothing with one minus that probability. Special cases are a constant probability p over time, in which case the secondary market price equals p (for instance, Martin and van Wijnbergen (1989)); or a geometrically decl:ning probability, which would warrant use of a constant risk adjusted discount rate to calculate the present value of contractual repayments (for instance, Lamdany (1988)). 2) the certainty approach: here the secondary market price is the present value of future trade balances (with some corrections), taken as exogenous and deterministic, and divided by the face value of debt (for instance, Dooley and Symansky (1989), and Rodriguez (1988)). Both approaches have their weaknesses. The all-or-nothing approach does not allow one to discuss the effects of a debt reduction on the secondary market price: this price remains the exogenous probability of (willingness of) repayment and is not affected by any amount of debt reduction. Marginal and average price of debt are equal by assumption under this approach. Neither does it allow for a full discussion of guarantees that are tied to debt reduction schemes, as the value of a guarantee is independent of any debt reduction taking place. The certainty approach ignores the impact of debt reduction on the present value of expected repayments completely and can by construction not be used to evaluate guarantees or the impact of changes in the seniority  3 structure, since these issues are inherently related to existing uncertainty about likelihood and magnitude of repayment. The approach can however account for different price paths in response to debt reduction. The weaknesses of these models point to the importance of modelling explicitly the sources of uncertainty driving secondary market prices. The improved understanding of secondary market prices that would result is, in turn, important for an assessment of debt reduction schemes; an evaluation of the market value of new instruments is essential for an assessment of the feasibility of any given proposal. Finally, explicit modeling of the sources of uncertainty allows for an assessment of the impact of different repayment schedules and seniority structure on secondary market prices. This explicit modeling is the more important as some debt reduction schemes involve enhancements through guarantees and collaterals whose values are state-contingent since they depend on the stochastic pattern of amounts available for repayments. This paper presents a model for pricing and evaluating existing and new (possibly credit enhanced) claims using option pricing techniques which explicitly models sources and natures of risks on sovereign debt. The paper is structured as follows. Section 2 sets out the basic approach; we present a pricing model of secondary market debt using option pricing tools. In Section 3 we apply the approach to pricing of claims with fixed and/or rolling interest and principal guarantees. We also show how to price recapture clauses that can be associated with newly created debt claims. Section 4 presents an application to Mexico and discusses the valuation and likely impact on secondary markets of the recent agreement between Mexico and its commercial creditors. Seetion 5 concludes. II. PRICING SECONLARY MARKET DEBT USING OPTION PRICING II.1 A Secondary Market Model We develop a more complete model for pricing a country's secondary market commercial debt using option pricing techniques 1/. The setup is the following. Due to uncertainty in the country's export earnings, import requirements and net scheduled capital in-or outflows, the net amount of financing available each period to service foreign commercial debt is uncertain. The uncertainty in the amount of resources available to service foreign obligations can be due to ability to pay as well as willingness to pay factors. For convenience, we lump these factors together and assume that the creditors have appropriability of any resources falling short of contractual debt service, or, alternatively and equivalently, that the country is a perfectly willing, but sometimes unable payer. Thus, each period the country will pay as much as its financial resources allow to the commercial banks, but 1 Option pricing has been used before in the pricing of LDC debt by Kharas et alii (1987); Cohen (1989) gives an analytical solution to the pricing problem they solve numerically. These papers focus on the option a creditor has to call a default, whereas we focus on the option the country has not to service its debt in periods of low foreign exchange availability.  4 never more than its contractual obligations in the period. Consequently, repayments may fall short of commercial debt service obligations due. We can represent this repayment behavior by the following: (la) R*(t)- min (Rt,FX.) with Re(t) equal to the repayment in period t; R, equals the contractual debt service in period t and FX. the resources available to service commercially held debt, also in period t. Rt is assumed known, although it is straightforward to extend the methodology to stochastic contractual debt service, such as in the case of floating interest rate debt (see for instance, Fischer (1978) and Margrabe (1978)). (la) can be rearranged to yield: (lb) R*(t) -Rt - max[O,R, - MXt] But max[O,R, - FX.J equals the value of a put, with a strike price of Rt, which is written on the value of the foreign exchange available, FXt. 2/ Thus equation lb shows that the uncertain repayment can be represented by a certain repayment Rt minus a put, with a strike price of R,, which is written on the value of the foreign exchange available, FXt. FIG. la R* .,.".. .. .... EX 2 The state variable FX is a non-traded asset and not as such priced in the market. But if the state variable is spanned by other traded instruments, one can price the non-traded asset and all results go through identically as in the case of traded assets. See also section III.  5 This is shown graphically in Figures la and lb. In Figure la, the shaded area represents the value of a put written on FXt with exercise price R,. The put pays max[O,Rt - FX>]: whenever FXt falls below Rt, the put is in the money and its value is equal to Rt - FXt; and whenever FX, is above Rt, the put is out of the money and thus worthless. Figure lb below shows first of all the payment obligation, Rt, which is independent of FXt and thus represented by a horizontal line (FX is on the horizontal axis). Subtracting the put (shaded area) from the fixed payment Rt, yields the desired payoff function, R*t - Rt - max[O,Rt - FXJ]. This is represented by the heavy line in Fig.lb, the line that goes from the origin out at a 45-degree angle until it cuts R, and then moves horizontally to the right. For any outcome of FXt above Rt, full repayment results and thus R*t - Rt. For a value of FXt below Rt, only FX, is paid and hence R*t - FXt. Thus R*t clearly also equals min(Rt,FXt). FIG.lb Rt / _t Now that we have replicated the payoff stream at maturity, it is easy to calculate the current value of the uncertain payoff stream as the current value of the certain future obligation Rt minus the current value of the put. This equals the discounted value of Rt, exp(-rt)*Rt (where r is the (continously compounding) interest rate), minus the current value P of a put with an exercise price of Rt, written on FXt. 3/ If V(Rt, is the present value of the claim, we can represent this as: 3 The formula assumes a constant interest rate r for notational convenience only. The empirical application presented below allows for different maturity structures of interest rates.  6 (2) V(Rt) - exp(-rt)*Rt - P(FXt,Rt,r,t,a). where P(FXt,Rt,r,t,a) is the current value of a put written on FXt with exercise price Rt, intezest rate r, maturity t and standard deviation a. If one furthermore assumes that FX behaves lognormally, the pricing of the put can be done using the Black and Scholes option pricing formula (see Black and Scholes (1973)). 4/ P iS then equal _o ne following expression: (3) P(FXt,Rt,r,t,a) - - FXO*exp((p-r)t)*N(dl) + exp(-rt)R*N(d2' where dl - [-log(FX0*exp(pt)/Rt) - (a2/2)*t]/(ajt) d2 - dl + ajt p - the drift in FXt over tha peLiod 0.. t5/ The current value of a loan with the series Rt falling due over tine is imply the sum of the current values of a series of these claims over the maturity of the contract. The present value VL of a series of contractual obligations R., for a maturity T, is thus equal to: (4) VL - EtRtexp(-rt) - EtP(FXt,Rt,r,t,a) where Rt can be different for each period depending on the terms on the loan and the summation is over t-l,..,T. Note that this implies that we can study the implications of different maturity structures on the price of debt, something which in most other pricing models by assumDtion does not affect the price of debt. II.2 Pricing Exit Bonds with Fixed Guarantees Thp methodology explained above can also be used to price guarantees that are provided by a third party for a specific payment falling due at a specific maturity date. Assume that the third party provides a guarantee for full payment of K at maturity date r. Following a similar line of reasoning, one can represent the guarantee as a put option with an exercise price of K, a maturity date r and written on an underlying asset FX. Such a put can again be priced using the Black and Scholes formula: (5) VFG - P(FX,,K,,r,r,a) 4 Other dens , functions can easily be incoporated using numerical integration techniques. 5 The formula assumes a constant drift p for notational convenience only. The empirical application presented below allows for time varying drift parameter p. The valuation formula differs from the Black-Scholes equation in that we do not assume p-r.  7 Define the set of years r over which guarantees are provided as ('T); furthermore, assume for simplicity that K, - R, for all re(r'). Then the value of such a set of guarantees equals: (6) VFG - E:,,)?(FX,,RT,r,r,O) and the value of the loan with this set of guarantees attached becomes: (7) VL.FG - LtRtexp(-rt) - EtP(FX,tRt,r,t,a) + E{.)P(FXX,,R,,r,r,a) Any type of fixed guarantee, whether of principal or interest and whether single or multiple years, can be priced using this methodology. II.3 Pricing Exit Bonds with Rolling Interest Guarantees A bond with rolling guarantees can also be priced using the same option pricing methodology. Assume the following rules. The guarantee is at time zero extended for coverage of one year of interest. If the country remains current on the guaranteed obligation, the guarantee will be extended for another year, and so on. 6/ In terms of our model, the guarantee will cover next year's debt service obligation provided the foreign exchange available in each of the previous periods was above the debt service obligation of the corresponding year. As before, it is assumed that in case of partial repayment the claimholders acquire all the foreign exchange available in this period if it falls below the debt service obligation and can at most retain their debt service obligation if the -tate of nature is better this period. In period one the repayment of R1 is assured through the guarantee, implying that the current value of the debt service obligation is exp(-rtl)*Rl. In period two the repayment is assured provided the country did not default in period one on its obligation, in which case the guarantee would have been called. If however the guarantee was called, the repayment in period two will be min[FX2,R2] as under the regular claim without any guarantee. This implies that the current value of the second period obligation will be equal to exp(-rt2)*R2 minus the current value of a put on FX2 with exercise price R2, plus a put which is conditional on the guarantee not being called the first period PC: (8) V(R2)RG - exp(-rt2)*R2 - P(FX2,R2,r,t,a) + PC(FX2,R2,r,t,a) The first two terms are equal to the standard expression for a claim on a country, the contractual obligation discounted minus the value of a put. The third term represents the value of the guarantee, which is the value of a put conditional on no prior calls so that the guarantee is indeed effective. If FXt is serially independent over time, an assumption we make, the pricing of this last conditional put is particularly simple and yields: (9) PC(FX2,R2,r,t,a) - 0(2,R1,FXj,a)*P(FX2,R2,r,t,a) 6 The pricing is done for a guarantee. Identical results obtain for an escrow account as long as the interest earnings on the escrow account are not retained in the account.  8 where (1(2,R1,FX1,o) is the probabil'--y tha' the guarantee is not called prior to time 2. The value of the put which is conditional on no prior call simplifies to the value of an unconditional put multiplied by the probability of no prior call in any previous periods. Similar expressions follow for later periods. Multigeriod Guarantees More general expressions for N-period rolling guarantees can easily be derived using similar methodology. For an N-year rolling guarantee, the first N repayments are fully guaranteed and thus valued without any credit risk discount. The claim value for period N+l is the discounted contractual value minus the value of an unconditional put, plus the value of tne guarantee. The value of the guarantee in that period equals the value of a put which is conditional on less than N calls in the preceding N periods. This last put option can similarly be priced as the conditional put derived for the one-year rolling guarantee. The only difference is that, for a N-period rolling guarantee, n now refers to the cumulative probability of at most N-1 prior calls. It is convenient to index 0 by the number of years covered by the rolling guarantee: ON. Define, furthermore, WN(t) as the amount left in the guarantee fund at the start of year t, expressed in number of years of interest covered, for a fund that originally covered N years. Thus the following holds by definition: (10) N(t) > 0 for t ? N ANat time t depends on all R and FX of the periods preceding t. Call (t') the set of t' preceding t. From the definition of ON and SN it is clear that: (11) ON(t,R{t.l,FX{t.),a) - Prob(fN(t) > 0) (10) and (11) together imply: (12) ON(t,R{t.},FX{t,},a) - 1 for t s N < 1 for t > N and a > 0 Martin and van Wijnbergen (1989) show that the value of QN(t,R(t.,)FXpt..,a) can be derived using a simple recursion formula in conjunction with the initial conditions in (12). This recursion formula greatly simplifies the numerical analysis and is incorporated in the computer programs used for the empirical analysis presented below. With all this machinery developed, one can express the increment of the value of rolling guarantees with N years coverage over the value of a N-year fixed guarantee: (13) VL,RG.N - VL,FG-N - Et.N nN(t,R(t0)FX(t,},)*P(FXt,Rt,r,t,a) 2- 0  9 with obvious definitions of VLRG.N and VL .G-N. Also, Rt in equ. (13) should be undarstood to only include interest payments. The inequality in (13) shows that, for the creditors, rolling guarantees are at worst of equal value to a corresponding fixed interest guarantee; and if there is any positive QN(t,R(t.),FX{te),a) for t>N, even if only one, rolling guarantees are strictly preferable from the creditors' point of view over fixed length guarantees with similar coverage. Equ.(13) suggests that the incremental value of switching from fixed to rolling guarantees is influenced by the initial level of debt Do, through the impact of Do on Rt.: (14) S(VL,RG-N - VL,FG N)/61)O Et,N 6(N(t,R(t,),FX(t,),a)/6Do*P(FXt,Rttr,t,a) < 0 + ZtN fN(t,R{t.),FX{t,),o)*6P(FXt,Rt,r,t,O)/6DO > 0 The first set of terms is negative since higher debt and thus high-r RL implies greater credit ri'-t and thus smaller fl; the fund is more likely to be exhausted at any given ti.- beyond period N. However the second term is positive, since the value of the put increases with an increase in the striking price Rt. The net effect is a priori ambiguous and thus needs to be addressed empirically (cf. Section III). II.4 Pricing Bonds with RecaRture Clauses The methodology used above is also easily extended to account for the possibility of recapture clauses, where future payments obligations depend in some fashion on the amount of foreign exchange available in each individual period. Assume, for instance, that in exchange for a certain amount of debt reduction at time zero, the creditors receive o recapture clause which entitles them, whenever foreign exchange exceed! a certain level L, to a share a of the excess foreign exchange over L in every period after time T. Assume further that the maximum amount tha%,. creditors can receive per period under this sharing rule is limited by an amount M. 7/ The Mexican debt package negotiated in the summer of 1989 contains a similar sharing rule. Such a sharirg rule can easily be represented in terms of option terminology: the creditors hold, in addition to their regular claim, a fraction a of a series of calls that are written on FX with exercise prices L, maturity dates r+l, r+2,..,T, and are short a fraction a of a series of calls that are written on FX with excercise price U-L+M/a and maturity date r+1, r+2,..,T. To see the equivalence between the sha-e g rule and the portfolio of options just described, consider the payoff scructure for the recapture clause, which we call I. 7 L, M and a can be made time dependent. In addition, L, M and a can be made dependent on other stochastic variables, such as world inflation rates in case of indexed clauses; in that case, one needs to use the stochastic option pricing formula of Fischer (1978) and Margrabe (1978).  10 (15) I -,., max[amax[FXq,.)-L,0],M] - .>,a*(max[FX(,.)-L,O] - max[FX%1.)-U,O]); U-L+M/a The expressions in tae two brackets in the last equation are the two calls mentioned above, with exercise prices L and U-L+M/a. The value of the calls can once again be evaluated using the Black-Scholes formula. Alternative recapture clauses, whir', may dep id in a more complicated manner on FX, can be handled similarly. III SECONDARY MARKET PRICING t D THE VALUE OF GUARPANTEES: MEXICO 1989 In this Section, we first assess the characteristics of the stochastic process governing foreign exchange availability in Mexico. The results are used in an analysis of the detetm.inants of s6condary market prices. We then assess the valuation of different forms of interest guarantees (fixed versus "rolling" guarantees). This is done within the context of Mexico's situation mid-1989. Section IV analyses the Mexican debt package negotiated over the summer of 1989. III.1 Behavior of Foreign Exchange Available The availability of foreign exchange to service Mexico's commercial bank debt depends predominantly on the behavior of Mexico's non-interest current account, which in turn depends to a large extent on the behavior of oil export earnings. Thus the variability of the financial resources available to service external debt is in the case of Mexico predominantly a result of the uncertainty of the price of oil. Even though the foreign exchange earnings of Mexico are non-traded assets, and as such not priced directly in the market, they are li.Aely spanned by assets which are traded and whose current values are known. For example, Mexico's oil earnings can easily be spanned through forward or futures contracts traded on over-the-counter and exchange markets Consequently, the pricing methodology underlying the option valuation, which assumed traded assets, can be used. The behavior of Mexico's future oil earnings will depend on price behavior and expected quantity. It is projected that the quantity of oil produced will remain at its current level over the near future (1.2 million barrels per day) and will decrease in the late 1990s (to 0.8 million barrels a day). The standard deviation of the average price of Mexican oil over the last 8 years has been 23%. Similar standard deviations are observed for prices that are close substitutes of Mexican oil, such as Borneo light (25% over 87-89), and for the average OPEC oil price (40% over 87-89, 21 percent over 85-89). The standard deviation of the annual changes in most (nominal) oil prices over the period 1975-1988 has been at least 20% annually. Correcting for any trend in oil prices does not change these estimates significantly. Another way to get an estimate of expected standard deviation is to use market information, such as actual prices of oil options. Given a pricing model, observed option prices can be used to back out volatilities that are consistent with those prices. Doing that one finds that the historical estimates of the standard deviation of oil prices are in fact consistent with those implied by the prices of options on oil traded on exchanges. Using thc  11 Black and Scholes formula on recent option prices implies volatilities of around 20%. Thus historical values for the volatility of oil prices closely ajpproximate the market's assessment of future volatility. We therefcre use the historical volatility in our pricing exercise. 8/ Commercial banks claims are de facto junior to many other claims on Mexico, e.g. official sector claims and bonds. Thus, the resources available to service the commercial debt have to be determined after these other creditors are serviced. This implies that amount of foreign exchange available for commercial bank debt servicing contains a component which is dependent on oil revenues and another, more deterministic part. The following procedure is therefore used. First, the non-oil, non-interest current account is projected in a deterministic fashion. The projections are based on the model reported in van Wijnbergen (1989) and van Wijnbergen and Pena (1989). Second, the non-oil, non-interest current account is adjusted for debt service to more senior claim holders, for foreign direct investment flows and for capital account transactions such as resetve accumulation (see van vlijnbergen and Pena (1989) for details). Third, oil earnings are added to the flow, thus introducing the stochastic element in FXt. III.2 Secondary Market Pricing Using the option pricing model outlined above, we calculated the value of existing commercial bank claims on Mexico. At the "base case" values for the distribution of FXt and a2, the model predicts that the secondary ma-:ket price of the part of Mexico's commercial debt under negotiation in 1989 ($52.7 bUS), is 37 cents before any debt reduction. The secondary market price in February, 1989, just before the Brady plan was proposed, was in fact around that value. The matrix below ptesents prices for alternative combinations of foreign exchange expected to be available to service commercial bank held debt and different degrees of uncertainty regarding these expected values. The bold numbers in the matrix represent combinations of the expected present value of foreign exchange available (PV(FX)) and oAl price variance a2 that yield a valuation close to the pre-Brady price, between 35 and 40 cents on the dollar. 8 Consistent with our assumption of no serially dependence of FXt, we modelled in the application not the uncertainty in the change in the price of oil but instead the uncertainty in the level of the price of oil. The standard deviation of annual changes in the price of oil is therefore converted into the standard deviation of (the logarithm of) the price of oil.  12 Table 1: Secondary Market Price as a function of PV(FX) and 02 IPX \ a2 0.25 0.36 0.49 0.75 0.37 0.31 0.26 1.00 0.43 0.37 0.31 1.25 0.49 0.41 0.35 Note: PV(FX) is presented as a share of the base case value and variances are expressed relative to the logarithm of the oil price. The Table demonstrates the sensitivity of the secondary market price to the expected value and the variance of resources to service commercial debt. Consider the impact of the variance first. The value of any put increases with the degree of uncertainty: (16) 6p/5a2 _ FX t*/t*N' > 0, Thus, since the secondary market value equals the discounted face value minus the value of a put, the value of the claim decreases as uncertainty increases, something Table 1 confirms. The Table also demonstrates that the value of the put falls with the initial value of the underlying asset; therefore, and not surprizingly, the secondary market evaluation in fact rises with a higher expected ability to pay. Figure 2 illustrates the impact of the structtre of payments on secondary market evaluation, something that cannot be assessed with most existing models of debt pricing. The figure shows the within period price of a claim falling due in that period; this within period price can be derived from formula (2) by dividing through the discounted face value of the claim falling due in year t: (17) P..,(t) - (exp(-rt)*Rt - P(FXt,R,,r,t,a))/(exp(-rt)*Rt) 1 - P(FXt,Rt,r,t,a)*exp(rt)/Rt The figure shows the P...(t) associated with the original amount of debt under negotiation in 1989, $52.7 bUS, but with the amortization assumed due as a bullet payment at the end of thirty years. P.0c(t) declines gradually over time because uncertainty increases as time progresses; this increases the value of the put constituting the discount and thus depresses the price. This gradual decline would seem to lend some respectability to the practice of using a risk-adjusted discount rate. That respectability is lost, however, once we look at the last period, the period in which the bullet payment comes due. Using a risk-adjusted discount rate implies a declining probability to repay, but one that is independent of the amount due in any given period. Thus there is no difference in valuation, using that method, between a period in which scheduled debt service just consists of $5 bUS in interest, and a period in which, in addition, over $50 bUS principal comes due. But Figure 2 shows what common sense would also suggest, that such a large difference in scheduled debt service has a major impact on relative valuation. P..c(t),  13 which was falling gradually at less than one point per period, suddenly drops by 11 points in the period in which the bullet payment is due. FIG.2 Within-Perlod Market Va lue PsecCt) 0.0Ooc i W OXIIgstImu) 0.6 0., 0.4 0.2 0.2 a , ,,,.,,,,,,.,,., .,,, , 1 2 3 4 5S 678 a1O1121314151I1716I92021222324252827282930 Year o DIS01. + DISC-OS.5 The impact of the size of an obligation on its relative value explains something that one misses using risk adjusted rates, the importance of full collateralization of principal in any exchange offer. With a repayment probability independent of the size of the obligation, such collateralization seems inefficient; it ties up funds for an event that takes place so far in the future so as to be of little importance. Fig. 2 shows that, while collateralization guarantees a payment far in the future, it could still be valuable and an efficient use of enhancement resources because the credit risk in the bullet period is so much larger than in other periods. The impact of the size of an obligation on its relative value is also clear from the line in Figure 2 which depicts the P...(t) in case the contractual interest payments are halved. The relative values P.,,C(t) are considerably higher. One can derive the secondary market price of a bond from the series ot within-period valuations P,..(t) as follows: (18) Pec,L- Et P,,,(t)*Rte-rt/(Et Rte-rt) Figure 3 shows the sensitivity of this secondary market price with respect to the amount of debt. This figure is especially illustrative since it can show the effects of debt reduction on the secondary market price, something other models did not account for. A debt service reduction of 50 percent for instance, increases the price from 37 cts to 50 cts. As a consequence, the 50% debt reduction does reduce the market value, from $19.3 billion to $13.2 billion, but by 32% only, much less than 50%. The difference between the 50%  14 face value reduction and the 32% reduction in market value is caused by the increase in unit value from 37 cts to 50 cts. FIG.3 Secondary Market Value versus Discount 52 - CPerc. of face value) 51 - 50 - 49 49 1 49- 4,