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Pricing the implicit contracts in the Paris Club debt buybacks at par
Arnaud Manas1 – Laurent Daniel2
In 2005, Russia3, Poland4 and Peru5 bought back more than 30 billion dollars in debt from Paris Club
members. The buybacks consisted of prepayment of debts at par and with no penalties. These
transactions were carried out at a discount of more than 20% compared to their net present value. The
total loss incurred by creditors in the three buybacks can be estimated at more than 5 billion dollars.
It is worthwhile to ask why the Paris Club creditors agreed to the buybacks voluntarily. It appears that
these buybacks are the result of the exercise of specific contracts previously agreed with the debtors in
the 1990s, without getting any compensation for that and without assessing the consequences. These
implicit contracts make it possible to formalise the respective interests for creditors and debtors. Their
pricing requires the use of tools taken from financial mathematics (derivatives) and stochastic models
for interest rates (Vasicek), but applied in the Paris Club framework.
On the one hand, as we shall see below, the value of these contracts is highly dependent upon market
interest rates. Hull and White have developed a methodology to calculate the value of assets that
depend on an interest rate following a stochastic model and Vasicek has proposed a model that tracks
interest rate developments. On the other hand, the use of derivative pricing models for buybacks has
been studied in particular in the case of “callable bonds” by Büttler, Waldvogel, Brennan and
Schwartz. Leaning on these works, we have developed a pricing model for debt buybacks at par that
fits in with the Paris Club functioning.
We shall present in the first part of the paper the players, the functioning of the market, the notation
and assumptions taken to assess contracts. In the second part, we shall price these contracts in four
cases (mandatory versus optional and collective versus selective buybacks). In the last part, we shall
present our numerical simulations.
**
*
Part 1: the debt buyback scheme in the Paris Club
The Paris Club was created by the main sovereign creditors in 1956 to maximise their bargaining
power vis-à-vis debtor countries. The Club acts together to reschedule the debt of countries facing
liquidity problems.
The key feature of debt in the Paris Club is illiquidity. Unlike bond issues, the claims are not
transferable. Therefore, debtors hold a buyback monopoly with regard to their creditors. However,
creditors can securitise their claims in the market with a discount. The price of a buyback is
theoretically the result of a negotiation between the debtor and the cartel within an Edgeworth box.
1
Banque de France – Research and Foreign Relations Department – Debt Unit
2
Banque de France – Research and Foreign Relations Department – International Monetary Relations Unit
3
For a face value of USD 15 billion (http://www.clubdeparis.org/fr/news/page_detail_news.php?FICHIER=com11159910510)
4
For a face value of EUR 12.6 billion.
5
Peru's finance minister, Pedro Pablo Kuczynski, expects the Paris Club to accept the country’s offer to buy back some US$1.5 billion of
debt, according to The Financial Times. The country hopes to save US$300 million in debt servicing charges annually in retiring the debt
early. […] The government will finance the buy-back by selling longer-term sovereign debt both on the local and international capital
markets. Peru's total debt with the US$8.5 billion, according to the paper.
http://bankrupt.com/periodicals/may05.pdf and
http://www.clubdeparis.org/fr/news/page_detail_news.php?FICHIER=com11189306920
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To avoid these negotiations, the Paris Club has put in place rules based on fair treatment between
debtors and creditors and on reconciliation of diverging interests. On the one hand, debtors were eager
to have the right to buy back their debt with a discount as had been made possible for some countries
in the case of commercial credits (Brady’s bonds). On the other hand, some creditors, in case of
anticipated reimbursement, wanted to get some penalty fees in compensation for the breakage costs
similar to the standards recommended by the OECD:
“In case of an anticipated and voluntary reimbursement of the totality or a part of a credit, the debtor
compensates the governmental institution, which gives its financial support, for all the costs and losses
coming from this anticipated reimbursement and, in particular, for the cost due to the replacement of
of flows at fixed rate stopped by the anticipated reimbursement.”
These two positions were difficult to reconcile. Indeed, for some creditors, the principle of a discount
was unacceptable because they considered discounts as concessional treatment. Their position was not
to add granting to granting as some initial contracts provided for preferential treatments of debtors
(decrease of debt stock or interest rate charges) and considered discounted buybacks as a “second
discount”. This accounting position did not take into account financial market developments. The
opponents to the discount principle refused any modification of debt value from its historical value.
Reciprocally, the payment of breakable costs through penalty fees was unanimously rejected by
debtors.
For these reasons, Paris Club members chose a median way and a compromise “neither discount nor
penalty fees”, allowing, by late 1990s, anticipated debt buybacks at par (in addition to Debt
Investment Swaps with discounts of nearly 50%). In fine, the combination of improved refinancing
conditions, compared to those prevailing during the 1980s and the neither-nor rule of the 1990s led to
buybacks in 2005.
During the negotiations, a pool of united creditors and one single debtor gathered under the auspices
of the Paris club (there is no such link between debtors in spite of unsuccessful trials in the past to
federate debtors in a “counter-cartel”). Processed claims are both loans that have commercial or aid
origins and debts that have already been rescheduled. Only the debts due by a sovereign country or the
ones that benefit from its guarantee are likely to be processed by the Paris Club. The rescheduling
negotiation between a debtor and the Paris Club results in a new homogeneous scheduling, with a
view to lengthening and homogenising the maturity of initial loans (interest rates are rarely modified).
The structure for exchanging debt is specific to the Paris Club, since no market exists. The creditors’
claims are not liquid. The debtors hold a buyback monopoly with regard to their creditors, since the
debts cannot be sold. The debtors alone can take the initiative of proposing a buyback. However,
creditors may securitize these debts and sell them on the market, if they are willing to accept a
discount on their claims. In its simplest form, a buyback means that a debtor redeems all of its debt at
face value. Once the buyback offer has been announced, the creditors cannot reject it and the buyback
must be carried out, regardless of the outstanding amount of the securities they hold. The group of
creditors acts jointly and severally inside the cartel. In this form (we shall see below that other forms
exist), the contract is simply a conventional call option. We will consider the situation of a single
debtor (no debtors counter cartel) and his creditors.
We assume that the debtor’s debts for a total face value K are represented by n bullet bonds indexed
from 1 to n. To simplify, we assimilate claim to creditor, considering that each creditor owns a single
claim on the debtor. This claims previously rescheduled by Paris Club have the following
characteristics:
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These fixed-rate bonds all have the same residual maturity D because of past restructurings. They have
a reimbursement type similar to “bullet bond” with capital reimbursement at the end and annual
payments of interests. The debts have the same face value K/n. These assumptions are not restrictive.
In the case of a mixed portfolio containing fixed-rate and variable-rate debt, the portfolio can be
treated as one that contains only fixed-rate debt (see equivalence in the appendix). The assumption of
identical residual maturities stems from the rationale behind the rescheduling agreements, which align
repayments. In addition, the largest debts can be broken up to make debts of similar amounts.
Furthermore, each debt carries an initial rate of interest. This rate is denoted qi.
Characteristics for the ith debt ( 1  i  n ):
Face value K n
Residual maturity D
Initial interest rate (rate) qi
The value of the ith debt is the present value of future flows (i.e. the d interest payments and the final
principal repayment) from the debt at the actualisation rate ract .
D
qi K n
K n

j
(1  ract ) D
j 1 (1  ract )
viact  
Taking into account the continuous time hypothesis
D
viact   qi K n e ract j  K n e ract D
j 1
Therefore, in the first order equation, if ract is low,


viact  K / n 1  D ract  qi
 .
The market value in t of the debt for the creditor is the present value of the flows with the debtor’s
refinancing rate estimated in t at the present time (t=0). This interest rate is rt  st where rt is the
spot rate that should prevail in t and s t the spread that should be applied to the debtor in t.
Furthermore, the market estimates the debtor’s probability of default at the level defined by the spread
s compared to the risk-free interest rate. The expectation for the future spread is equal to the present
spread. This spread is exogenous. As the best estimator for the spread in t is the spread in t=0, the
actualisation rate should therefore be rt  s0 . The remaining maturity at the time t is D-t. In addition to
r t
get the value for t=0, one must apply the discount factor e 0 where r0 is the spot interest rate in t=0.
This gives.


vimarket (t )  K n 1  D  t  rt  s0  qi
 e
 r0t
The spot risk-free interest rate rt is determined exogenously by the market, since neither the debtors
nor the creditors have enough influence to determine such rates. Similarly, putting the debt on the
market through securitisation or by eliminating the debtor’s monopoly would only have a minor
impact on market rates.
The personal value of the debt for the creditor is the present value of the flows with his personal rate
that reflects his own risk perception and his preference for liquidity. This rate is equal to the sum of
spot rate in t rt and of the creditor’s personal spread on the debtor z i ,t . This spread reflects the
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subjective willingness to get rid of the debt (because of the probability of default and because of the
need for liquidity). Furthermore, a risky interest rate r+zi is defined for each debt. This rate depends on
the risk-free interest rate r and a specific spread zi. The spread is made up of two components: the
creditor’s subjective perception of the probability of default on the debt in question and the creditor’s
preference for liquidity over the risk-free interest rate7. The best estimator for z i ,t in t= 0 is z i , 0 .
Taking into account the discount factor, this gives



viperso (t )  K / n 1  D  t  rt  zi ,0  qi er0t
The differential between the market value and the creditor’s personal value represents the Pareto gain
that the creditor could realise by selling his security at the market price. The debtor’s monopoly means
that the creditor cannot realise the total Pareto gain represented by the differential between the market
price of the debt and his personal value. The gain is shared between the creditor and the debtor as a
function of the selling price (in this case, the face value K / n ):
 iPareto  vimarket  viperso   icreditor   idebtor  kd ( zi  s)
We also assume that the qi rates are normally distributed and Qi is the associated VAR i.i.d.):
Qi ~ N ( µQ ,  Q )
with µQ for expectation and  Q for standard deviation.
We also assume that the individual zi spreads are normally distributed and Zi is the associated VAR.
We assume that spreads and interest rates are not correlated (zero covariance).
Z i ~ N ( µZ ,  Z )
with µZ for expectation and  Z for standard deviation.
At this point, we assume that the spot interest rate follows a classical Vasicek stochastic process 8. This
model has a number of advantages, since it has mean-reverting and symmetric distribution properties.
The spot rate is normally distributed, and not lognormally distributed, as it is in the Black-76 model9.
These properties are fully justified for pricing contracts over the long term (maturities over 15 years).
If Rt is the associated VAR:
Rt ~ N ( µRt ,  Rt )
with µRt  r0e t  r (1  e t ) and  Rt  
1  e 2 t

If rt denotes the spot interest rate at t expected at the current instant (t=0), we get rt  E ( Rt )  µRt .
We also write  Rt   t . In the first approximation rt  r0  (r0  r )t and  t   2t
Part 2 : Pricing the contracts
7
The Growth and Stability Pact may mean that EU countries with excessive deficits and debt may have a stronger liquidity
preference, because the Pact creates asymmetry on debt since it only counts gross debt. This means that it encourages
countries to “liquidify” their external claims in order to reduce their debt.
8
Vasicek O. (1977) “An equilibrium characterization of the term structure”; Journal of Financial Economics 5, 177-188
9
Black (1976) “The pricing of commodity contracts”, Journal of Financial Economics, 3, 167-179
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In its simplest form, the debtor’s contract is like a call option. However, there are other, more
complex, forms that constitute sui generis contracts. In practical terms, the debtor informs the Paris
Club of its intention to buy back its debts, but the Club determines the buyback procedures.
The Club informally specifies the type of option, depending on the clout and the superior interests of
the creditors and the clauses in the multilateral and bilateral agreements. Depending on their
interpretation of the “de minimis” clause, the creditors may make the buyback optional or mandatory.
Depending on the wording and interpretation of bilateral agreements, creditors may grant the debtor
the right to choose which debts it wants to buy back. Therefore:

The buyback may be mandatory or optional for creditors. In other words, the creditors may or
may not have the right to reject the debtor’s buyback offer.

The buyback offer may be collective or selective. The debtor may be bound to make a
buyback offer to all of its creditors (i.e. the Paris Club members) or, it may select only some of
its creditors.

The buyback offer may be revocable or irrevocable. Either the debtor may retract its offer, if it
deems it helpful to do so, or else it may be required to go ahead with the offer once it is
announced, even if it turns out to be unfavourable for the debtor.
This means that there are four possible types of contracts:
Mandatory (M) offers that Optional (O) offers
creditors cannot reject
creditors can reject
Collective buybacks (C) that the
Type I
Type II
debtor must offer to all creditors CM contract / “standard form”
CO contract
Selective buybacks (S) where
the debtor selectively offers to
Type III
Type IV
buy back the debts held by
SM contract
SO contract
individual creditors
that
The collective-mandatory or type I contract (CM) is the standard form for prepayment buybacks. It is
akin to a conventional call option (see above).
Furthermore, the attractiveness and the value of each option contract depend on the specific rights
included in each option. It is already clear that the options that let the debtor select which debts it
wants to buy back (SM and SO) are more advantageous for the debtor. On the other hand, the CM and
CO options are more advantageous for creditors because they can use the cartel power of the Paris
Club. In the same vein, mandatory buybacks (CM and SM) are advantageous for the debtor, while the
CO and SO options are the least disadvantageous for creditors. All in all, the SM contract is the most
advantageous for debtors and the SO contract is the least disadvantageous for creditors.
1) Pricing of type I contracts : collective and mandatory buybacks (CM)
The market value of each debt is normally distributed:


Vi market (t )  K n e  r0t  K n D  t  Qi  Rt  s0 e  r0t
r T
The buyback of all debts is equivalent to the buyback of a single debt at par Ke 0 at time T with an
interest rate of µQ (average of the initial rates on the individual debts). The value of this composite
asset ST is therefore equal to the conditional expectation on RT of the cumulative market value of the
individual debts at the time T . The price of this asset depends on the spot interest rate rT at the time
T.
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

ST  E  Vi market (T ) RT 
 1in

Hence
ST  Ker0T  K ( D  T )( µQ  RT  s0 )e r0T
r T
The buyback value being Ke 0 . The debtor’s payoff from the buyback is
GainT (rT )  ST  K   K ( D  T )( µQ  RT  s0 ) 

Value
Strike rate
µQ  s0
Debtor’s payoff
( ST  Ke r0T ) 
Spot rate
Debtor’s profit expectation
ST  Ke  r0T
The debtor will exercise his option contract if the market value of this composite asset is greater than
its face value ST  K . Meaning, if the debtor’s expected payoff is positive. This condition is fulfilled
when the interest rate that the debtor would have to pay for refinancing, which is the risk-free market
rate, plus a spread10, is lower than the interest rate on the debt.
In this case, the prepayment buyback option is the same as an embedded call option in the Paris Club
debts. Such options are found in certain types of bonds, such as saving bonds11 and parity bonds. They
enable the issuer to call the bond.
The call price is a function of time and the price of the underlying asset Ct  C (t , St ) . The asset price
per se is determined by time and the spot interest rate St  S (t , rt ) .
All we have to do to determine the value of the call option is to calculate the expected payoff from a
buyback at the expiry of the option in a risk-neutral universe. The value of the call option at the
maturity of the underlying asset (T=D) is zero by definition, since the asset value is equal to the
principal (K=SD), hence CD  C( D, S D )  0 . Furthermore, the value of the call option at the outset is
equal to the intrinsic value C0  C (0, S0 )  KD(µQ  r0  s)  .
Therefore, the expected payoff at the expiry of the option is:
10
It can be assumed that the debtor’s actuarial calculations do not incorporate the possibility of default. If this were the case,
the debtor would not offer prepayment of its debt early since it would be more advantageous to default.
11
“Saving bonds, retractable bonds and callable bonds”, M. Brennan & E. Schwartz, Journal of Financial Economics 5 (1977) 67-88.
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
E (GainT ) 
f


RT
(r )GainT (r )dr   f RT (r ) K ( D  T )e r0T µQ  r  s0  dr


So, we get the value of the call option:

 µQ  s0   rT
 r0T 
I
CT  K ( D  T )e  µQ  s0   rT N 

 RT





  RT 

e

2

µQ s0 rT 2 

2 RT 2



µQ rT  s0 





µ

r

s

2 RT 2
R
Q
T
0
r T
T

As  rT  rT (see above) : CTI  K ( D  T )e 0  µQ  rT  s0 N 
e


 RT
2




2





When the interest rate is much lower than the strike rate, the value of a call option is equal to the
intrinsic value. On the contrary, when the interest rate is much higher than the strike rate, the value of
the call option is zero. The time value is greatest at the strike rate.
Value
of call
option
µQ  s
Interest rate rT
In the first order equation, the value of the call option can be broken down very simply into two
components: the intrinsic value of the call option when it is exercised and the time value, which is
linked to the volatility of interest rates.



1
2
 r0T
I


µQ rT 
CT  K ( D  T )e
 s0    RT
 
2

 

 
 intrinsicvalue
time
value


CT
 0 at T  Topt shortly after the
T
initial date. This value is an absolute maximum: if 0  T  D then CT  CT Opt
As a general rule, the call option reaches its maximum value
Numerical simulations show this maximum value after 1 to 2 years for typical parameters (duration 10
to 20 years). The time value is zero at maturity (T=0) and increases as a function of the square root of
time. Furthermore, the value of the call option shows a quasi-linear decrease over time to reach zero at
maturity (T=D).
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CTOpt
C0  KD(µQ  s  r0 ) 
CD  0
In the most probable case14, where the intrinsic value of the call option is small or even zero at the
outset, we can approximate the value of a European-style call option:
CTI 
 
KD 
2
1  
 µQ  r0  s0    R
T 1   r0  T  

2 

D  
 
Immediately
Topt 
1
3(r0  1 / D)
Pricing an American-style call option is a much more delicate matter. The upper bound of the
European-style call option is the lower bound of the American-style call option for which, as a general
rule, there is no accurate analytical solution. This means that a numerical solution or an approximation
are the only conceivable means of pricing callable bonds.
Furthermore, the maximum value of European-style call options is not the same as for American-style
call options.
Furthermore, the interest rate that corresponds to the strike rate seems to depend on the rates in the
initial contracts and the spread that the market requires from the debtor. The only endogenous factor is
the spread s0 , since the interest rate on the composite asset is exogenous and determined by the initial
contracts. The spread is determined by the markets, according to their perception of the risk of default.
Since it is in the debtor’s interest is to have the lowest strike price possible, this mechanism constitutes
a virtuous cycle, because it encourages the debtor to reduce his spread by improving his behaviour.
2) Pricing other contracts
a) Pricing of type II contracts : Collective optional buybacks (CO)
As in the previous case, all debt is treated as a composite asset. However, under this contract, the Club
of creditors may reject the buyback if its expected profit is negative.
The debtor’s payoff depends on the Club’s profit. At the expiry of the option, the payoff is positive
and equal to the debtor’s profit expectation when the debtor’s expectation and the Club’s expectation
14
µr  r in the long term, the average rate on the debt of a given debtor is not very
different from the equilibrium rate plus the market spread µQ  r  s . Furthermore, the initial rate r0 is also normally distributed
Since rates fluctuate around their equilibrium level
around the equilibrium rate
r .
On average, there is little difference between these two rates ( r0
approximation can legitimately be used.
 r ).
Consequently, an
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9
are both positive. Otherwise, the payoff is zero, when the debtor’s profit expectation or that of the
Club is negative.
GainT ( RT )  K ( D  T )( µQ  RT  s0 )er0T
if RT  µQ  µZ and RT  µQ  s0
GainT ( RT )  0
if RT  µQ  µZ or RT  µQ  s0
The buyback can only be carried out if it is attractive to both the Club and the debtor. This means
when. µQ  µZ  RT  µQ  s0 This condition can only be fulfilled if µZ  s , meaning there is a
possible Pareto gain (positive sum game). In other words, this condition can only be met if the
creditors feel that the risk on the debtor is greater than market risk.
Value
Strike rate
µQ  s
Debtor’s payoff
Market rate rT
Club’s profit expectation
Debtor’s profit
expectation
This contract combines the plain-vanilla call option under the collective mandatory contract at the
interest rate and an “asset-or-nothing put option. This “asset-or-nothing put option is broken down into
a plain-vanilla put option and a cash-or-nothing put option for an amount equal to the Pareto gain.
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Plain vanilla call
Plain vanilla put
r
Strike price: Ke 0 T
Strike rate: µQ  s0
r
Strike price: Ke 0 T
Strike rate: µQ  µZ
Cash or nothing put
Strike price: K ( D  T )e  r0T ( µZ  s)
Strike rate: µQ  µZ
E (GainT )  K ( D  T )e r0T
µQ  s0

f RT (r )µQ  r  s0 dr 
µQ  µZ
µQ  s0
K ( D  T ) r0T
e
 µQ  r  s0 e
2  RT
µQ  µZ
The net present value of the contract is thus

  µQ  rT  s0 
 µ  r  µZ

  N Q T
CTII  K ( D  T )e r T  ( µQ  rT  s0 ) N 


 
R
R



 


( r   RT ) 2
2 RT 2
dr
µQ rT s0 
µQ rT µZ 

   RT   2 R 2
2 RT 2
0
T

 
e
e


2 
T
T


Outside of the central zone between µQ  µZ and µQ  s the contract value is almost zero.
2
2
Value
of the
call
option
µQ  µZ
µQ  s
Interest rate rT
In the first order approximation, the value of the contract is CT  K ( D  T ) exp( r0t )
1
r t
Hence, C  K ( D  T )e 0
2
II
T
1 µZ  s0 
 RT
2
2
The differential with the standard contract (CM) is negative if there is a Pareto gain:
µZ  s 2
2 T 2
.





1/07/2005
11
µQ µZ rT 


 µQ  µZ  rT   T  2 T 2
 µQ  µZ  rT  

 
 
T  K ( D  T )e  ( µQ  µZ  rT ) N 
e
 µZ  s N 
T
T
2







This differential breaks down into the value of the plain-vanilla put option and the cash-or-nothing put
option on the Pareto gain.
2
r0T
It should be noted that the approximated value of the contract shows no first order dependence on the
level of future interest rates.
Furthermore, this contract encourages virtuous behaviour in appearance only. The debtor’s gainmaximising strategy does indeed consist of reducing the market spread s0 , which constitutes virtuous
behaviour. But it is also in the debtor’s interest to increase his creditors’ spread µZ , meaning their
perception of the debtor’s specific risk. This type of behaviour may give rise to contradictory specific
signals, or even putting on a virtuous front for the markets, while trying to make creditors wary at the
same time. Therefore, this form of contract is likely to give rise to a specific default behaviour.
b) Pricing of type III contracts : Selective mandatory buybacks (SM)
Under a selective mandatory buyback contract, the debtor may choose to prepay only those debts
r T
where its profit will be positive  idebtor  Vi market (T )  Ke 0  0 , meaning debts where the initial rate
is higher than the current rate, plus the spread. Thus, the debtor has a set of call options.

The debtor’s profit expectation is therefore. E  idebtor  idebtor  0



GainT (rT )  E  idebtor  idebtor  0   E ( debtor  debtor  0)
1 i  n
 1i  n

r T
As the debtor’s profit is normally distributed:  idébiteur  K ( D  T )(Qi  RT  s0 )e 0 , the lemma
demonstrated in the appendix can be applied :

 µ r s
 Q T
0
 r0T 
debtor
debtor
E i
i
 0  K n D  T  e  µQ  rT  s0 N 
 Q2  R 2

T


Hence the value of the contract


CTIII 
 E 
1i  n
debtor
i
 idebtor  0
µQ rT  s0 2

 Q 2   RT 2  2  Q 2  R 2  

T
e


2






µQ rT  s0 2 
2
2

 µ r s 





Q
RT
2  Q 2  RT 2  
 Q T
0 
CTIII  K D  T  e r0T  µQ  rT  s0 N 

e

2
  Q 2   R 2 


T




debtor
debtor
The payoff function stems from the conditional expectancy: Gain( RT )   E  i
i
 0 RT
1i  n

µQ  RT s0 




µ

R

s


2 Q 2
Q
T
0
Q

Gain( RT )  K D  T  e r0T  µQ  RT  s0 N 
e


Q
2




2






1/07/2005
12
Gain
Debtor’s payoff (selective)
E ( debtor  debtor  0)
Debtor’s payoff
(mandatory)
(ST  Ke r0T ) 
µQ  s
Market rate rT
Debtor’s profit expectation
ST  K
In the first approximation
CTIII 


1
2
K ( D  T )e r0T  µQ  rT  s0  
 Q 2   RT 2 
2



A comparison with the collective contract shows that the “intrinsic” terms are identical. The
differential between the two contracts is positive by construction, since the selective contract is more
advantageous:
T  CTIII  CTI  CTIII 
1
2 
2
2
K ( D  T )e r0T
    Q   RT 

2
  RT
Such contracts encourage virtuous behaviour since the spread is the only incentive factor. It should
also be noted that the value of the contract increases with the number of creditors.
c) Pricing of type IV contracts : Selective optional buybacks (SO)
Under a selective optional buyback, the debtor offers to buy back debts where both the debtor’s profit
and the creditor’s profit are positive.
 icreditor  K / n e  r0T  Vi perso (T )
The debtor’s profit expectation is therefore the cumulative positive part of the debtor’s profit
expectation on debts where the creditor’s profit is positive too.
CTIV 
 E 
1i  n
debtor
i
 icreditor  0 and  idébtor  0

Since the debtor’s profit and the creditor’s profit are normally distributed, we can apply the lemma by
writing:
E ( idebtor  icreditor  0 et  idébtor  0)  K / n ( D  T )e  r0T E Qi  RT  s0 0  Qi  RT  s0  Z i  s0 
1/07/2005
13
As first approximation18, we get
EQi  RT  s0 0  Qi  RT  s0  Z i  s0   EQi  RT  s0   EQi  RT  s0 Qi  RT  s0  0  EQi  RT  s0 Qi  RT  s0  Z i  s0 
Using twice the lemma with the conditional expectancy, we get the contract’s value.
CTIV


 
  µQ  rT  s 0
 r0T 
 K ( D  T )e  µQ  rT  s 0  N 
   Q 2   RT 2
 


µQ  rT  µZ 
  µQ  rT  s0 


2  Q 2  RT 2 
2 ( Q 2  RT 2  Z 2 )
e
e



2
2
 Q 2   RT 2   Z 2
  Q   RT

2


µQ  rT  µZ


  N

  Q2   Z 2   R 2
T


Thus, the buyback payoff at expiry T is GainT ( RT ) 
 E 
1i  n
  2   2

Q
RT
 
2



debtor
i
2
 icreditor  0 and  idebtor  0 RT




 


Value
Strike rate
Debtor’s payoff
Spot rate
Hence


 
µ R µ
µ  RT  s0 

T
Z
  N Q
GainT ( RT )  K ( D  T )e r0T  µQ  RT  s0  N  Q


2
2

Q
 




Q
Z




µQ  RT µZ 
  µQ  RT 2 s0 

  2 
2 Q
2 ( Q 2  Z 2 )
e
e


Q 

   2 
Q

 Q2   Z 2



2
2






Clearly, the payoff is zero for extreme rates. As in the case of the collective optional contract, the
buyback can only be carried out if the creditors’ profit and the debtor’s profit are both positive. This
means that there is a Pareto gain.
In approximation (third order for µZ  s  )
CTIV


K ( D  T )e r0T 
 µQ  rT  s0 2   Q 2   RT 2

 Q 2   RT 2 2 





1  2µ  r  s    µ


2
Q
T
0
2
Q
2
RT


Q

 rT  s0 µQ  rT  µZ    Q   RT
2
1
2

µ
If  Z is equal to 0, the value of this contract is equal to contract of the second type (II). When  Z is
small in comparison with à  Q   RT , we get the following approximation:
2
18
2


To compute the exact value, one must take into account the E Q  R  s Q  R  s  Z  s  0 factor, but it is
i
T
0
i
T
0
i
0
negligible in comparison with the other factors as a Pareto profit exists ( µZ  s ). This approximation gives a lower bound
for the value of the contract.
 rT  µZ 


2(   RT   Z ) 


Z2
1 2

2

 Q   RT

Q
2
Q
2
2
2
1/07/2005
CTIV 
14
1
K ( D  T )e r0T
2

2
2
1  µZ  s0 
3 Z

2   Q 2   R 2
 Q 2   RT 2
T





The incentive is similar to that produced by the selective mandatory contract (SM). This type of option
may give rise to “hypocritically” virtuous behaviour.
The value for the contracts can be summarized as follows : :
µQ  s0  rT 2 

 µQ  rT  s 0   RT  2 R 2 

T

C  K ( D  T )e  µQ  rT  s 0 N 
e



 RT
2







  µQ  rT  s 0 
 µ  µ Z  rT    RT

  N Q
 
CTII  K ( D  T )e  r0T  µQ  rT  s 0  N 





 RT
 RT
2




 

µQ rT  s0 2

 µ r s 
 Q 2   RT 2  2  Q 2  R 2  

 Q T
0 
T
CTIII  K D  T  e r0T  µQ  rT  s0 N 

e

2
  Q 2   R 2 


T




 r0T
I
T
IV
T
C
 K ( D  T )e
 r0T


 
  µQ  rT  s0

 µQ  rT  s0  N 

 2   RT 2

  Q



The volatility depends on  RT
“Virtuous” contract
The intrinsic value depends on µQ  rT  s0
The volatility depends on  RT and  Q
“Virtuous” contract
2
2





µQ  rT  µZ 
  µQ  rT  s0 

 2  Q 2  RT 2 
2 ( 2  2  2 )
e Q RT Z
e


2
2
 Q 2   RT 2   Z 2
  Q   RT

2


µQ  rT  µZ


  N 
2
2
2


 Q   Z   RT
The intrinsic value depends on µQ  rT  s0

µQ  µZ  rT 
  µQ  s0  rT 


2 RT 2
2 RT 2
e

e




  2   2

Q
RT
  
2


2




 

The intrinsic value depends on µz  s0 
Low volatility
Hypocritical contract
The intrinsic value depends on
µz  s0 
The volatility depends on  Z
“Hypocritical” contract
As regards irrevocable vs. revocable buyback offers, the following point must be stressed. Logically,
the debtor will only exercise the option when his payoff is positive. Consequently, the debtor is not
concerned about sustaining losses (except if exogenous variables were to change between the time the
buyback is announced and the time it is completed). Therefore, the debtor enjoys an implicit right to
retract the buyback offer. This reasoning holds for mandatory contracts where all of the information is
known.
Part 3 : Numerical simulations
The numerical simulations with the following parameters confirm that the maximal value for the
European type I and III contracts occur after approximately two years. The computed value for type IV
contract is effectively a lower bound value as it should be higher than the type II’s value.
11,7% (T=2,6 years)
12,9% (T=2,2 years)
3,5% (T=0,2 years)
>3,4% (T=0 year)
The average initial rates of the debt contracted in the 90swas at approximately 8% ( µQ ) with a
standard deviation of 2% ( Q ). The market spread could amount to 300 basis points ( s0 ). For such
parameters, the strike rate for a type I contract was 5% (8%-3%). When taking a long term (spot)
1/07/2005
15
interest rate of 5% ( r  r0 ) equal to the initial spot rate –implying a zero intrisic value-with a
standard deviation of 2% (  R ) and a residual maturity of 10 years, the value of the type I contract
amounts to at least 11.7% of the face value.
With an average personal spread of 500 basis point ( µZ ) and a standard deviation of 100 basis points
(  Z ),the type II contract’s value represents 3.5% of the principal. The values for the type III and IV
are above 12.9% and 3.4%.
0.12
0.1
0.08
0.06
0.04
0.02
2
4
6
8
10
From top to bottom: type III, I, II and IV. As abscissa : time to exercise (in years). As ordinate ; the
value of the contract as a fraction of the principal.
1/07/2005
16
The approximate values are for the four types of contracts :
CTI 

1
2
K ( D  T )e  r0T  µQ  rT  s 0  
 RT
2


CTIII 





1
2
K ( D  T )e  r0T  µQ  rT  s 0  
 RT 2   Q 2
2






CTII 
2
1
2  1 µZ  s0  
K ( D  T )e r0t

2
  2  RT

CTIV 

2
Z2
1
2  1 µZ  s0 
3
K ( D  T )e r0T


2
  2  2  2 2  2  2
Q
RT
Q
RT

Type II
Type I
Value
Value
Strike rate
Espérance de
profit du Club
Strike rate
Debtor’s Payoff
Debtor’s Payoff
Spot rate
Spot rate
Debtor’s profit
expectation
Debtor’s profit
expectation
Value
Type III
Value
Type IV
Debtor’s Payoff
Debtor’s Payoff
Spot rate
Debtor’s profit
expectation
Spot rate





1/07/2005
17
**
*
Conclusion
A comparison of different types of European-style call options shows a large difference in the value of
the premium. Intuitively, the maximum premium applies to SM buybacks and the minimum premium
applies to CO buybacks.
Mandatory buyback contracts carry a relatively high implicit cost. On the other hand, optional
buybacks have a fairly low value, but they may have a perverse effect in that the debtor may be
tempted to put on a reassuring front for the markets and another, more troubling, front for its main
creditors. This is why the arrangement for buybacks at par does not seem appropriate for optimal
management of the liquidity of the Club’s claims. The discounted buyback arrangement conceived by
the Club Secretariat seems clearly preferable, since it may be carried out at any time and it limits risks
and moral hazard.
Several further developments can be considered. More specifically, it might be a good idea to price
American-style call options for these four types of contracts. Furthermore, we could investigate the
dynamics between the debtor and the creditors from the point of view of the debtor’s strategy for
influencing the market and the creditors’ perception of its risk. We could also look at pricing the
supplementary option resulting from the revocability of buyback offers.
1/07/2005
18
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1/07/2005
19
Appendix
Variable-rate debt
The presence of variable-rate debt does not affect the analysis or the typology of call options. It merely
entails a change in the value of the individual spreads and the market spreads to be used.
In the case of floating-rate bonds, the contract rate is by construction equal to the market rate, plus the
contractual spread mi ( qi  r  mi ).
Let g denote the proportion of principal carrying a fixed rate k f and the proportion of principal
carrying a variable rate kv .
 idebtor  kd r  s  qi   gkd ( s  m)  kd (r  s  g ( s  m)   qi )  kd (r  s'qi )
 icreditor   ki di r  zi  qi   gkd ( zi  m)  ki di r  zi  qi 
with s '  s (1  g )  m and zi  zi (1  g )  m
Demonstration of the main findings
If X and Y are normal distributions, X ~ N (µX ,  X ) and Y ~ N (µY ,  Y )
 µ µ
X
Y
E ( X X  Y )  µX N 
  2  2
X
Y

 µ µ
X
Y
E ( X X  Y )  µX N 
  2  2
X
Y

By definition E ( X X  Y ) 



2




2

 xf
X
X2
 X 2  Y 2
X2
 X 2  Y 2
 µX  µY 2 

exp  
2
2 
 2( X   Y ) 
 µX  µY 2 

exp  
2
2 
2
(



)
X
Y


( x) fY ( y)dxdy
 y  x
 x
Now
 (u  v) f
X
( x) fY ( y)dxdy  2
 y x
 x
 (u  v) f
X
(u  v) fY (u  v)dudv
0v
u 
 1u vµ
X
 
(
u

v
)
f
(
u

v
)
f
(
u

v
)
dudv

exp
X
Y




2


2

X Y
X

0v 

u  
uv
2
 1  u  v  µY
  
Y
 2



2

dudv


After a few calculations, we get
E ( X X  Y )  µX


1
µX  µY
1  Erf  
2
2

2
2  X  Y



 µX  µY 2 
X2
 

exp  
2
2 
2
2

2
(



)
2




X
Y


X
Y

The second finding is trivially derived from the equality E( X X  Y )  E( X X  Y )  E( X )
Furthermore, we can approximate the expectation for µX  µY
2
 µX  µY 2 
µX µX µX  µY    X

E( X X  Y ) 

exp  
2
2 
2
2
2
2
(



)
2  X   Y
X
Y




1/07/2005
20



 µX  µY 2 
µX  µY
X2
1  Erf 
 


exp
 2( 2   2 ) 

2
2
 2( 2   2 )  
2




X
Y


X
Y
X
Y



2
2
If µX  µY , we can use the approximation Erf ( x) 
x ex
E( X X  Y ) 
µX
2

 1

 1
 µX  µY 2
µX  µY
µX  µY
  1  Erf 
  

Exp
 2  2  2
2
2
 2 
 2  2   2   2
2




X
Y

X
Y
X
Y





 µ µ 
  1 E( X X  Y )  µ
X
Y
Furthermore, if µX  µY N 
X
  2  2 
X
Y 

 µ µ 
  0 E( X X  Y )  0
X
Y
Conversely, µY  µX N 
  2  2 
X
Y 

As E( X X  Y )  E( X X  Y )  E( X )
 µ µ
X
Y
N
  2  2
X
Y




2
 µX  µY 2 
µX µX µX  µY    X

E( X X  Y ) 

exp  
2
2 
2
2
2
2
(



)
2  X   Y
X
Y










