Financial Risks, Stability, and Globalization
Chapter

4 Calculating Counterparty Credit Exposure When Credit Quality Is Correlated with Market Prices

Author(s):
Omotunde Johnson
Published Date:
April 2002
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Wrong-Way Risk

Counterparty risk management involves tracking both the credit quality of the counterparty and the actual amount of exposure on the derivative contract. Unlike exposure on traditional credit products, such as loans and commitments, derivatives credit exposure is constantly changing with the mark-to-market value of the contract.

Most models of derivatives credit exposure assume that the credit quality of the counterparty is unrelated to the value of the contract. This is not always the case, however. For example, the default of a large, government-sponsored counterparty in an emerging economy is likely to coincide with a depreciation of its home currency. Assume such a counterparty has entered into a currency forward with a U.S. bank in which it will pay dollars and receive its local currency. The forward is most in-the-money to the U.S. bank when the foreign currency depreciates substantially. The same scenario that produces the greatest upside for the U.S. bank also involves the greatest risk of counterparty default.

Most derivatives exposure models do not capture the additional credit risk created by the correlation between the value of a contract and the credit quality of the counterparty. Contracts that introduce additional credit risk due to such correlations are said to be “wrong-way” deals. Wrong-way risk can come from any number of factors that drive the value of derivative products, including interest rates, foreign exchange rates, equity prices, and credit spreads. This paper will discuss basic types of wrong-way transactions, describe techniques for measuring the true credit exposure on such contracts, and comment briefly on new frontiers in wrong-way risk research.

Quantification of Wrong-Way Risk

Credit Exposure on Derivative Contracts

The credit exposure on any contract represents what the bank stands to lose—in its own currency—in the event of counterparty default.1 In the case of derivative contracts, exposure is equal to either zero or the replacement value of the contract at the time of default,2 whichever is greater (i.e., MAX[0, replacement value]). Like the value of the contract, the exposure will change as the underlying market factors move.

Because the value of the contract depends on variables that are stochastic in nature, it is useful to consider the expected exposure on the contract at any given time in the future. The expected exposure at that time is simply the expected value of MAX[0, replacement value] over all possible market outcomes, and will depend on where the underlying market factors are expected to be at that time. By calculating expected exposure over a range of future times, an exposure profile can be created.

The counterparty’s exposure profile can be used to assess a credit charge for the derivative contract. The exposure profile is analogous to the notional exposure profile of an accreting or amortizing loan, and can be priced accordingly.

Wrong-Way Credit Exposure

As stated above, the credit exposure on a contract represents what the bank stands to lose in the case of default. For deals that exhibit correlation between the value of the contract and the default of the counterparty, it is necessary to measure explicitly the credit exposure conditional on counterparty default. In general, conditional exposure can be estimated by replacing the expected value of the contract with its conditional expected value given default.3 Wrong-way exposure is thus the expected value of MAX[0, replacement value given default].

Consider the following specific case of the previous example: a U.S. bank has contracted to buy dollars one year forward from a large, quasi-government counterparty in an emerging market country. We wish to calculate the exposure in three months’ time. The unconditional, expected exposure will be relatively small, since on average we do not expect the forward rates to have moved significantly one way or the other. Given that our counterparty has defaulted, however, we expect the local currency to be worth less than it would be otherwise. A weaker currency means the forward is more valuable to the U.S. bank at the time of counterparty default, making the conditional, expected exposure far higher than the unconditional, expected exposure.

The calculation of credit exposure is relatively straightforward and will be described in the next section. To find wrong-way exposure, the only additional step required is to determine the expected value of the derivative contract given default. Once the conditional expected value is known, it can be used with the appropriate credit exposure model to determine the conditional exposure.

Finding conditional future values for different types of contracts requires different techniques. Procedures for “conditioning” exchange rate contracts exist and will be described below. Methodologies for “conditioning” interest rates and equity prices are still being developed. Other sections of this paper will present a description of that work, as well as a few comments on the calculation of conditional credit spreads and default probabilities.

Calculation of Credit Exposure

The expected credit exposure on any given contract at a certain point in time is equal to the expected value of MAX[0, replacement value given default], as stated above. If the counterparty’s credit quality is uncorrelated with the value of the deal, the default condition is superfluous and can be eliminated. In such cases, the exposure can be averaged over all possible market outcomes. The following formulation describes the general case:

E[exposure(τ)] = E[MAX[0,Φ(x(τ))]]

Φ(x(τ)) is the value of the contract at time τ.

x(t) is (are) the underlying variable(s) for the contract Φ

exposure(τ) is the credit exposure on Φ at time τ

The expected value of the MAX function, and hence expected credit exposure, can be calculated directly using no-arbitrage option pricing theory or estimated using Monte Carlo simulation techniques. Both methods are described below.

Using Option Valuation

An intuitive way to estimate expected exposure is to use an option model. Consider the previous example: a forward contract to buy dollars in exchange for local currency in one year. The expected exposure on the forward at a given time τ is equal to the price of an option, expiring at time τ, to “put” the remaining contract to the bank at time τ.

To see intuitively why this is so, assume that the counterparty is bankrupt at τ = 3 months. The counterparty now has the option to “put” the forward back to the U.S. bank and pay nothing, effectively canceling the deal. By defaulting, the counterparty has essentially received the put option for free. In this sense, the bank stands to lose the value of this option if the counterparty defaults at time τ = 3 months. This value is therefore the bank’s credit exposure on the forward contract at τ. The same calculation can be performed for any point in time during the life of the contract.

To be specific, expected exposure on Φ at time τ is equal to the future value of a European option, expiring at time τ, to put the remaining portion of the contract.

E[exposure(τ)] = Put{Φ(x(τ));0,τ}·FV.

Put{Φ; strike, maturity}is the current value of a European put option of a given maturity, with a given strike, on the contract Φ4
FVτis the future-value discount factor to time τ (e.g., e).

In the case of wrong-way transactions, credit exposure must be explicitly calculated given counterparty default. To estimate wrong-way exposure, the conditional value for the contract Φ at time τ given default should be used in place of its expected value:

E[exposure(τ) | default] = Put{Φ(x(τ)) | default;0,τ}

Consider our example. The expected exposure on the 12-month forward at τ = 3 months is equal to the price of a 3-month option on a 9-month forward contract:

E[exposure(3m)] = Put{9m fwd(FX(3m));0,3m}.

To find the exposure conditional on default of the counterparty, we must find the price of a three-month option on the same nine-month forward contract given counterparty default:

E[exposure(3m) | default] = Put{9m fwd(FX(3m)) | default;0,3m}.

Standard option-pricing calculations are based on the forward value of the underlying. For estimating wrong-way exposure, it is sufficient to replace this forward value with the conditional forward value of the option’s underlying given counterparty default, and assume that the volatility in the contract’s value is unchanged. The option’s underlying in this case is the remaining contract, i.e. the 9-month forex forward.

Using Monte Carlo Simulation

A more advanced way to estimate derivatives credit exposure is through Monte Carlo simulation. Using diffusion models to generate different future paths for the market variables, the simulation can evolve many different portfolio “paths” based on these variables. Each complete history will result in an exposure profile for each instrument, which can be averaged over all paths to calculate the expected exposure profile.

Consider again the contract Φ. The Monte Carlo process will generate N equally probable paths j for the evolution of the underlying variable(s) x(t). At any given time τ, the value of the contract (and hence its credit exposure), can be found for each path j given the value of the underlying(s) xj(τ). The expected exposure is simply the arithmetic mean of the calculated exposure at time τ for each path j:

To estimate the exposure conditional on default of the counterparty, the underlying variable(s) xj(τ) on each path j must be “conditioned” for default at time τ.5 For example, if the contract is a currency forward, x(t) will be forward forex rates. For each path j, the forward xj(τ) can be adjusted by the appropriate “residual value” factor to estimate its value conditional on default:

RV is the average ratio of the conditional to unconditional values of x(τ)

By using the average residual values for the underlying parameters, there is an implicit assumption made that this ratio is not path-dependent. A more detailed simulation might weight the unconditional outcomes xj(τ) by the probability of their realization given default of the counterparty. To do so would require either obtaining the full probability distribution for each xj(τ) given default, or simulating the evolution of market variables and credit quality simultaneously (e.g., Iscoe, Krenin, and Rosen, 1999).

Calculation of Credit Charge

To calculate credit charge for a derivative contract with varying exposure, it is convenient to decompose the exposure profile into loan equivalents. Credit charges can then be assessed using any standard loan pricing techniques.

loan equivalent (t) = E[exposure(t) | default]− E[exposure(t + Δt)] | default]

loan equivalent (t)is the exposure on a loan maturing at time t, such that the total exposure profile for all loan equivalent exposures matches the expected exposure profile.

Practical Issues in Wrong-Way Risk

The following section contains examples of some key implementation issues to be considered when developing a business model that incorporates wrong-way risk.

Parameterization

The main objective in measuring wrong-way exposure is to calculate the appropriate credit charge for wrong-way transactions. Any credit charge model is fundamentally a pricing model and should be parameterized as such. Implied and risk-neutral parameters, including implied volatilities and risk-neutral default probabilities, should be used wherever applicable.

Risk-neutral default probabilities can be determined from market spreads, as shown below. Implied volatilities can sometimes be obtained from the options market, but are not available for all parameters. Fortunately, at the time when credit exposure is highest, the contract is so deeply in the money that the accuracy of the volatility parameter has little impact. Historical volatilities are adequate in most cases.

Option Valuation Versus Simulation

Using option valuation to calculate credit exposure offers a clean, closed-form solution that is easy to understand intuitively. However, it is not well suited for portfolio-level calculations. Most banks that trade in derivatives use Monte Carlo simulation models to calculate credit exposure.

The Monte Carlo method works well for large portfolios of derivatives, as it is able to capture correlations between the market variables that drive valuations. For example, the value of a fixed/floating currency swap depends on both forex rates and local interest rates. The simulation evolves these variables simultaneously, capturing the correlation between short-term interest rates and exchange rates.

Modeling these correlations properly is critical when determining the total, netted credit exposure to a single counterparty. Standard option models cannot accommodate the large number of correlated market variables needed to simultaneously value multiple transactions with a single counterparty. If the bank has entered more than one type of contract with a given counterparty, it is necessary to use a Monte Carlo simulation to calculate the netted credit exposure correctly.

The simulation can also capture changes in portfolio composition due to factors such as option exercise and collateral calls that depend on underlying market moves (see Levy and Clarke, 2000, p. 5). For example, if the derivatives portfolio includes a swaption, certain paths might lead to its exercise while others will not. The expected credit exposure on the resulting swap depends on whether the option is exercised and is best modeled using a simulation.

Collateral Exposure as a Wrong-Way Risk

A counterparty that posts collateral against a credit exposure may incur wrong-way risk for its creditor on the value of the collateral. For example, consider a company that has fully collateralized a bank loan with shares of its own stock. If the company were to default, it is unlikely that the equity collateral would have substantial value.

Collateral that is delivered up front, as in a repurchase agreement, acts like a deliverable forward contract with negative credit exposure. Many collateral agreements, however, call for the periodic posting of collateral based on the mark-to-market value of the exposure. The exposure at any given time will depend on how far the market has moved since the most recent collateral posting. This generates new challenges for wrong-way risk calculations. Given the default of the counterparty, the market move since the most recent collateral call will depend strongly on short-term volatility of the underlying. In certain notable cases, such as financial counterparties and counterparties in countries with pegged currencies, the default of the counterparty often coincides with periods of extremely high short-term volatility. This correlation increases the likelihood of a large market move between default and the most recent collateral call, decreasing the effectiveness of the collateral agreement.6

There are many legal and practical aspects of collateral agreements that can influence how wrong-way counterparty credit exposure is calculated and charged for.7 Such details, including enforceability issues, operational issues and client relationship issues, are beyond the scope of this paper.

“Right-Way” Exposure

Transactions are said to exhibit “right-way” relatedness when the contract’s expected exposure increases as the creditworthiness of the counterparty improves. In some cases, the credit exposure on right-way transactions can be close to zero.

Consider again our example with the emerging market counterparty, but assume instead that the U.S. bank had agreed to pay dollars and receive local currency. Given the default of the counterparty, this new swap is expected to be out-of-the-money to the U.S. bank. Therefore, the bank’s credit exposure is minimal.

A common misconception about right-way exposure is that its effect will “cancel out” the effect of wrong-way exposure in a well-diversified portfolio. Wrong-way risk can produce extremely large credit exposures. An important feature of right-way exposure, on the other hand, is that its upside is limited. The most favorable right-way transaction can only limit credit exposure to zero.

Consider the sample calculation presented by Levin and Levy (1999a), which appears in Appendix II to this paper. The credit exposure calculated using wrong-way relatedness is about $32 million. Ignoring wrong-way relatedness gives an expected exposure of $5.5 million. If the transaction were right-way, the exposure would be no lower than zero. A bank that enters this wrong-way contract and the corresponding right-way contract will have total credit exposure of at least $32 million. By contrast, a bank that enters two equivalent contracts without right- or wrong-way relatedness would have total credit exposure of only $11 million.

Wrong-Way Currency Risk

Wrong-way forex exposure is significant when an international bank enters a contract, such as a forex swap or forward, to pay the currency of an emerging market country to a counterparty in that region in exchange for hard currency. Rapid declines in the value of emerging market currencies can make it prohibitively expensive for banks in those countries to buy other currencies, as well as being symptomatic of general economic distress.

Wrong-way forex exposure has been the most closely studied case of wrong-way risk to date. Technology for estimating wrong-way forex exposure has been developed and a number of quantitative models exist.8 This section will outline the model used at J.R Morgan, described in the July 1999 issue of Risk magazine.

A significant amount of credit risk in emerging markets is due to the possibility of sovereign default. If the central bank is unsuccessful in defending its own currency in the open market, it may place a moratorium on foreign currency outflows. In addition to sovereign obligations, any local bank obligations in a foreign currency will be forced into technical default regardless of the credit quality of the obligor.

When our derivatives counterparty is the sovereign itself, calculating wrong-way exposure is straightforward. If the conditional forward forex rate is known at any given time, the conditional exposure at that time can be calculated using either an option valuation model or a Monte Carlo simulation. Conditional spot rates given sovereign default can be measured empirically through historical default studies.9 These conditional rates are expressed as residual value factors, defined as a ratio of the conditional forex rate given default to the unconditional forex rate.

Corporate counterparties introduce additional complications. It is assumed that the default of the sovereign will force all corporate counterparties into default with 100 percent probability. Aside from sovereign-induced “co-default,” there is also the possibility of independent default of the corporate obligor. When calculating credit exposure, it is useful to consider the two cases separately and weight the two results according to their relative probability.

To do so, the probability of each case and the residual value factor in each case must be calculated. The implied risk-neutral default probability of sovereigns and corporations with public debt can be determined from the yield spread over the risk-free rate on traded credit instruments, given an assumed recovery rate. From these unconditional probabilities, we can determine the probability of both co-default and independent default, as will be discussed in the next section. The residual value for sovereign default (and thus for co-default) has been measured empirically.10 The residual value for independent default is challenging to measure empirically, but can be parameterized using a structural model of default.11

Quantification of Wrong-Way Forex Risk

The objective in quantifying wrong-way forex risk is to determine conditional spot and/or forward forex rates at the time of default. If the contract is a currency basis swap (floating for floating), its value will depend only on the spot rate. If the contract is a forward, its value will depend on the forward forex rate to the remaining maturity of the contract.

The forward forex rate will be a function of the spot rate and the differential between forward interest rates in the two currencies. If the possible wrong-way relatedness of interest rates is ignored for now,12 the conditional forward rate is a linear function of the conditional spot rate. In this case, only the conditional spot rate need be modeled.

Consider again the previous example of the forward with the emerging market counterparty. In order to compute conditional exposure, it was necessary to find the nine-month forward rate three months forward given default of the counterparty. This particular rate will be the product of the conditional spot rate three months forward and the differential between nine-month local deposit rates three months forward in the two currencies.

As in the previous section, conditional parameters are expressed in terms of residual value, or the ratio of the expected value given default to the unconditional expected value. The residual value of an emerging market currency in the case of sovereign default has been measured historically for over 100 credit events across a number of countries, and is given as a function of the sovereign’s credit rating.

RatingAAAAAABBBBBB
Residual value1317%17%22%27%41%62%

In the case of corporate default, it is necessary to consider the cases of co-default and independent default separately. Methodologies for determining the probability of each case and the residual factor in each case are shown below.

Calculation of Co-Default and Independent Default Probabilities

The term structure of default probability for both corporate and sovereign borrowers can be determined by observing credit spreads on traded obligations and assuming a recovery rate. Given the term structure of bond spreads, it is possible to bootstrap a default probability “forward curve” indicating the (noncumulative) probability that the obligor defaults for the first time during each period:14

P(τ) ≡ P(default{τ − dt,τ})

P(default{τ − dt,τ}), or p(τ)is the probability that the counterparty defaults for the first time during the period {τ − dt,τ}.
dtis the length of each period.
P(default{0,τ})is the probability that the counterparty defaults by time τ

It is assumed that if the sovereign defaults, all corporates will default as well. If the counterparty’s default probability is similar to that of the sovereign, or if both are very small, the following are good approximations for the probabilities of co-default and independent default within the time period {τ - dt,τ} (see Appendix I).

Psov&corp(τ) = Psov(τ)

Pcorp&no sov(τ) = Pcorp(τ) − Psov(τ)

Psov&corp(τ) = Psov(τ)is the probability that both the corporate counterparty and the sovereign default for the first time during the interval {τ - dt,τ}.
Pcrop&no sov(τ)is the probability that the corporate counterparty defaults for the first time during the interval {τ - dt,τ} but the sovereign does not default.

Calculation of Residual Value Factors for Co-Default and Independent Default

Given that the sovereign has defaulted, the knowledge of any additional corporate defaults are unlikely to have any effect on the expected currency value. Therefore, the residual value in co-default is effectively equal to the residual value given sovereign default only.

On the other hand, given that the sovereign has not defaulted, the default of the counterparty may have some correlation with a decline in the currency value. At the same time, the nondefault of the sovereign may have a positive implication for the currency value. We can approximate the total residual value as the product of two residual value factors representing the two conditions:

RVcorp&no sov = RVcorp·RVno sov;

RVcorpis the residual value of the currency given corporate default.
RVno sovis the residual value of the currency given sovereign non-default.

The residual value given corporate default can be estimated using a structural model to capture the correlation between the value of the counterparty’s assets and the forex rate. Levin and Levy (1999) propose the following formulation for corporate residual value:

ρFX, assetsis the correlation between normalized changes in the local forex rate and normalized changes in the counterparty’s asset value.
σFXis the annualized volatility of the forex rate.
N-1[q]is the inverse of the standard normal distribution given the probability q.
τis a fixed time parameter.15

The residual value given nondefault of the sovereign can be determined as follows. The unconditional residual value (i.e., 100 percent of the forward value) must be equal to the probability-weighted average of the residual value given sovereign default and the residual value given sovereign nondefault:

Therefore, the residual value given sovereign nondefault can be found as follows:

Example of Credit Charge Calculation

Levin and Levy (1999a) give a sample calculation to illustrate the impact of wrong-way risk, included in Appendix II to this paper. The results are summarized below.

Average ExposureCredit Charge
No relatedness$5.5 million$150,000
Wrong-way$32 million$850,000
Right-way (best case)$0 (see p. 84)$0

Practical Issues in Wrong-Way Forex Risk

J.P. Morgan has incorporated wrong-way forex risk into its credit exposure systems. The following section contains examples of some key implementation issues to be considered when developing a business model that incorporates wrong-way forex risk.

Parameterizing Rvcrop Using Correlation

The structural model employed by Levin and Levy (1999a) invokes two new parameters:

  • σ, the volatility of the currency, can be measured from readily available market data.

  • ρ, the correlation between the counterparty’s assets and the local currency, is difficult to observe. The correlation of the share price to the forex rate can be used as a starting point and refined according to the practitioner’s judgment. Levin and Levy observe that bank share prices tend to be highly correlated with the local currency, while shares of exporters can have negative correlation with the currency.16

Parameterizing P(default)

The spread on a traded debt obligation is reflective of the market’s assessment of its probability of default plus a premium for the uncertainty of default losses. By using market spreads, which includes this premium for investor risk aversion, one measures risk-neutral default probabilities that are generally higher than historical default frequencies. The following formula applies for one-year spreads:17

The default probabilities measured as shown above will depend on the assumption made about the recovery rate of the obligation in default. Recovery rates can be quite variable and difficult to predict from financial or economic models. Because credit charge models are fundamentally credit pricing models, however, they are relatively insensitive to the choice of recovery rate.

Furthermore, the spread parameter in the above equation is meant to reflect only the credit spread. Liquidity and tax factors will cause many types of debt obligation, including risk-free bonds such as those issued by U.S. government agencies, to trade at a discount to their fair credit value. To account for this effect, it is necessary to subtract a liquidity spread from the observed market spread to find the pure credit spread. The spread on agency bonds is sometimes used as a proxy for the liquidity spread. For example, if a BBB- bond trades at a 125 basis point spread to the Treasury curve and agency bond spreads are trading at a 20 basis point spread, the counterparty credit spread is estimated to be approximately 105 basis points.

Relatedness Due to Regional Correlations

A forex transaction does not have to be referenced on the counterparty’s home currency to have wrong-way relatedness. For example, consider the situation of a U.S. bank receiving dollars from a Malaysian counterparty in a forward on the Thai currency (baht). Wrong-way relatedness between such forwards and the Malaysian currency (ringgit), combined with correlation between the ringgit and the baht, leads to wrong-way exposure on the deal.

J.P. Morgan’s solution to the problem of cross-border relatedness is to tabulate residual value factors for currencies other than the counterparty’s home currency, including all currencies with an anticipated correlation under a stressed scenario. These factors are estimated qualitatively, leveraging the input of our traders to extend the existing residual factor table (shown on p. 88).

Wrong-Way Exposure to Counterparties in Developed Nations

There are several obstacles to measuring conditional credit exposure to counterparties in developed nations. Sovereign Relatedness

  • From a practical standpoint, it is difficult to separate credit risk from liquidity risk. For example, it is unclear what risk-free rate should be subtracted from the yield of a U.S. treasury note to find its credit spread.

  • Correlations between developed nations are likely to be driven by global systematic risks, with one result being the apparent reduction of wrong-way relatedness. For example, if our counterparty is a bank in the euro area, the conditional forex rate in default might be close to the forward, reflecting weakness of both the single European currency (euro) and the U.S. dollar.

Corporate Relatedness

  • The effect of global variables on corporate counterparties will be determined to a large extent by the idiosyncratic nature of the counterparty. For example, importers and exporters will be impacted differently by rising exchange rates. In developed markets, sovereign defaults are rare and corporate relatedness becomes much more significant, making it more important to capture such idiosyncrasies.

As a result, the decision was made at J.P. Morgan to implement wrong-way forex methodology in two steps, with emerging markets given priority. Wrong-way charges in developed nations will be implemented in the future as justified by business needs.

Wrong-Way Interest Rate Risk

Occasionally, interest rate contracts can produce wrong-way exposure due to relatedness between the movement of interest rates and the credit quality of the counterparty. Wrong-way interest rate risk is significant when international banks enter local market transactions, such as interest rate swaps, to receive floating-rate payments in emerging countries. In times of economic distress, inflation or liquidity problems can trigger extremely high nominal interest rates in these markets.

For example, assume that the foreign subsidiary of a U.S.-based bank has entered an interest rate swap to receive floating-rate payments from a local counterparty. The bank subsidiary’s credit quality is unlikely to be affected by the local economy because it is backed by the parent. The counterparty’s credit quality, on the other hand, is likely to be dependent on local economic conditions. In an emerging market country, economic distress is often accompanied by high inflation and interest rates. Given default of the counterparty, chances are good that the local economy is in decline and real interest rates are high.

Wrong-way interest rate risk often comes hand in hand with right-way forex risk. Consider the example above. If the counterparty is in default, interest rates are expected to be high and the value of the currency, low. The high interest rates increase the bank’s exposure on the swap, while the low currency value decreases the value of the exposure in dollar terms.

Some contracts exhibit wrong-way relatedness in both forex rates and interest rates. Consider the previous example of a cross-border forex forward with an emerging market counterparty. As noted earlier, the conditional value of the forex forward contract depends on both the spot forex rate and the short-term interest rates in both currencies. Given the default of the counterparty, the value of the local currency is expected to be low, decreasing the conditional forward forex rate. Furthermore, local interest rates in the emerging country are expected to be high, meaning that forward rates relative to spot rates will be even lower.

Quantification of Wrong-Way Interest Rate Risk

The key to determining wrong-way exposure on interest rate products is to find their conditional forward value given default. For most products, it is sufficient to value the contract using conditional values for the underlying variables given counterparty default.18 For products such as swaps, however, the underlying variables comprise a series of forward interest rates. It is then necessary to find the entire forward curve conditional on counterparty default, rather than a single rate (as in the case of wrong-way forex risk).

Since interest rate swap valuation requires the entire forward curve, wrong-way analysis must consider the shape of the forward curve conditional on default. Counterparty default is likely to be symptomatic of economic distress, which is often accompanied by very high interest rates. Because such high rates cannot persist forever, it is likely that the forward curve will be downward sloping.

Once the conditional forward curve has been estimated, it can be used in either an option model (using swaption valuation) or a Monte Carlo simulation to find the wrong-way credit exposure on a swap. Also, knowing the forward curve conditional on default makes it possible to calculate conditional forex forwards that capture wrong-way relatedness in both interest rates and exchange rates.

As mentioned above, the expected conditional forward curve will be quite different in the case of sovereign default (as opposed to corporate default). This effect makes it necessary to treat sovereign and corporate counterparties separately, as for wrong-way forex exposure. The details of that methodology will not be repeated here.

There are, however, a few notable complications in the treatment of sovereign versus corporate counterparties. For one, the default of the sovereign on its local debt does not necessarily force local corporate counterparties to default, as was the case for foreign exchange contracts. Also, the sort of correlation that may exist between corporate defaults (without default of the sovereign) and the local interest rate is unclear. These are topics of ongoing research in wrong-way interest rate risk.

Example: Interest Rate Behavior in the Case of Sovereign Default

Consider the case of Russia, which defaulted on its local debt at the end of August 1998. From the beginning of 1998 through mid-May, local interest rates fluctuated around the 20 percent level. (See Figure 4.1) Between May and August, overnight rates went through a period of extremely high volatility. The default itself was preceded by a steep rise in rates, which reached 160 percent on August 18. When Russia defaulted on its GKO bonds at the end of August, overnight rates were around 120 percent.

Figure 4.1.Moscow Interbank Offered Rate

(Annualized percent, November 1997 to April 1999)

Source: Bloomberg screen.

After the default, the overnight rate dropped quickly. By mid-October 1998, rates were back down around 40 percent, and by March 1999 rates were hovering around 20 percent. (Overnight rates in Moscow were 22.1 percent as of April 27, 2000.) If rational investors could have foreseen the rapid decrease in interest rates that followed the default, the forward interest rate curve would have been steeply downward sloping at the time of default. Unfortunately, the default itself makes direct observation of complete interest rate curves very difficult.

Practical Issues in Wrong-Way Interest Rate Risk

J.P. Morgan has not yet incorporated wrong-way interest rate risk into its credit exposure systems. The model described above is still in an exploratory stage and has not been implemented. The following section contains examples of some key implementation issues that would need to be considered when using such a model.

Estimating Conditional Forward Interest Rates

The forward overnight rate, which anchors the forward curve, may be quite high given default of the counterparty. For sovereign counterparties, the conditional forward rate might be estimated by observing the behavior of local interest rates during sovereign defaults. An economic model may also be useful in parameterizing the conditional forward rate. (However, as noted above, the expected behavior of interest rates given corporate default is unclear.)

Estimating Conditional Shape of the Forward Curve

Approximating the entire forward curve using empirical measurements would be extremely challenging. Instead, the forward curve can be characterized by a function, the value of which decreases with time. For simplicity, it is convenient to use an exponential function decaying to a long-term mean reversion level.19 In the Russian example shown above, the realized forward rates indeed followed an approximately exponential trend, leveling off at their precrisis level of around 20 percent.

Wrong-Way Equity Risk

Contracts that reference equity prices often generate wrong-way exposure. Wrong-way equity risk is significant whenever banks purchase downside protection using equity derivatives, such as put options and total-return swaps. Since all stock prices are vulnerable to systematic risk, there will generally be some correlation between the price of two stocks. Holding equity collateral also produces wrong-way risk.

For example, a counterparty that offers its own stock as collateral generates direct wrong-way risk for its creditor. In fact, a counterparty that offers any equity as collateral generates indirect wrong-way risk. The degree of wrong-way relatedness is driven largely by the magnitude of the correlation between the two stocks. Consider a bank that makes a loan to an Internet startup and receives stock in General Motors as collateral. Although not directly related, the dot-com industry and the automobile industry are nonetheless correlated due to systematic risk. To the extent that large moves coincide with counterparty default, wrong-way exposure is generated.

Quantification of Wrong-Way Equity Risk

To model the correlation between equity and credit quality, it is useful to use the Merton default model. Briefly, the Merton model proposes that bankruptcy occurs when the value of the firm’s assets (stochastic) falls below the level of its liabilities (stable).

The firm’s asset returns over any given time period are assumed to have a normal distribution. The left tail of this distribution includes negative returns which cross the “default barrier,” causing assets to drop below liabilities. Integrating the probability distribution in this region yields the probability of default of the firm over that time.

The firm’s risk-neutral default probability can be inferred from market spreads, as shown on pp. 89-90. The inverse-normal of P(default) will indicate the distance from the mean asset return to the default barrier in units of standard deviation. Given the correlation between the firm’s assets and a particular reference stock, it is possible to determine the expected return on the reference stock, and thus its expected value.

The residual equity value given counterparty default, as a fraction of the forward value, can be written in terms of the reference stock’s implied volatility σref the counterparty’s cumulative default probability, and the asset correlation ρref between the two firms:20

The residual equity value as calculated here can be used with an option valuation model or a Monte Carlo simulation to determine wrong-way exposure on an equity contract.

Practical Issues in Wrong-Way Equity Risk

J.P. Morgan has incorporated same-name, wrong-way equity risk into its credit exposure systems. In emerging markets, related-name, wrong-way risk is also measured, using subjective estimates of residual value. The following section contains examples of some key implementation issues that need to be considered when using such a model.

Parameterization

The structural model of default employed above invokes three parameters: σ, the volatility of the reference stock; ρ, the correlation between the counterparty’s assets and the reference stock; and P(default). The first can be readily measured from available market data. The last can be determined from market spreads, as shown on pp. 89-90.

The needed pairwise asset/equity correlations can be approximated by the correlation between the stock prices of the two firms. These correlations can be tedious to measure for each counterparty/equity pair; instead, it is helpful to use a single-factor model. Consider the example above of the Internet start-up company. The dot-com might have a correlation of 0.8 with the S&P 500. The GM stock it is offering as collateral might have a correlation of 0.5 with the S&P. The cross-correlation is simply the product of the two correlations, or 0.4. In this case, the S&P was used as the single factor. The choice of an appropriate factor will increase the accuracy of the correlation estimate. KMV Corp. has developed a more sophisticated multifactor approach.21

Challenges of the Merton Model

The Merton model was chosen in this situation because it can readily accommodate correlations. However, the model itself is not ideal for the task of conditional measurements. It is not especially well suited to problems involving multiple time horizons or nonzero-drift situations. It has also shown itself to be a poor predictor of the term structure of market credit spreads. It is unclear whether these problems will lead to unintuitive results when applied to the problem of wrong-way credit exposure.

Potential Enhancements

For the sake of computational expediency, this model has made a couple of notable simplifications. Enhancements to the model could improve upon these points (potentially at the expense of the simplicity of the result). The value of these enhancements in terms of the accuracy of the model has not yet been assessed.

First, volatility of the reference stock has been held constant. Intuitively, the volatility of the stock can be expected to be higher than average given the default of the counterparty. Significant downward moves in equity markets are associated with both high volatility and increased corporate defaults. Even if the equity markets have no strong bearish trend, periods of high volatility can be coincident with increased defaults.

Second, the Merton model implicitly assumes that the firm’s asset value follows a continuous “random walk” diffusion process. In reality, equity prices often gap significantly in times of stress. Models that incorporate jump-diffusion processes into the Merton framework may reflect the behavior of stock prices more accurately.

Third, asset returns and stock returns are used interchangeably. In the Merton model, equity is represented as a call option on the firm’s assets struck at the level of its liabilities. For investment-grade firms this option is deep in-the-money. In this case, the equity behaves like an outright position in the firm’s assets and their returns become virtually indistinguishable. For subinvestment-grade names, downward moves in assets aren’t fully reflected in equity moves due to the equity’s optionality.22

Wrong-Way Credit Risk

Contracts whose values depend on the credit quality of a third party often generate wrong-way exposure. Wrong-way credit risk is significant whenever banks purchase credit derivatives, such as credit default swaps, that are used as protection against credit deterioration and default. Since the assets of all firms are vulnerable to systematic risk, there will always be some correlation between the credit quality of two names. Holding credit instruments (e.g., bonds) as collateral also produces wrong-way risk.

As with wrong-way equity risk, the degree of wrong-way credit relatedness is driven largely by the correlation between the two names. For example, consider a bank that buys credit protection in the form of a credit default swap. Given that the counterparty has defaulted, there is a greater chance that the reference name has also defaulted and the contract is payable. The bank’s loss in such cases is the settlement value. If the reference name has not defaulted, the bank will have to repurchase the credit protection from another counterparty. The cost of the protection will depend on the default-swap spread for the reference name in the market, conditional on counterparty default.

Quantification of Wrong-Way Credit Risk

Counterparty credit exposure on single-name credit derivatives (e.g., credit default swaps, total return swaps, and asset swaps) can be significant when the bank has bought credit protection. On the other hand, when the bank has sold protection, it only stands to lose the premium; credit exposure in this case is negligible.

There are three scenarios that need to be considered when calculating credit exposure on credit derivatives used to purchase protection.

  • Both the counterparty and the reference name default during the period;

Credit exposure: settlement value of contract.

  • The counterparty defaults during the period and the reference name survives;

Credit exposure: replacement cost of contract.

  • The counterparty defaults during the period, but the reference name has defaulted prior to the beginning of the period;

Credit exposure: zero (contract settled prior to the beginning of the period).

In order to estimate the expected credit exposure, it is necessary to find the probability of all scenarios and the credit exposure in each case.

The probabilities in each scenario must sum to the total probability of counterparty default. The counterparty’s risk-neutral default probability in each time period can be derived from credit spreads, as illustrated earlier in this paper. The methodologies for finding the probability of joint default and the relative probabilities of the other two scenarios are straightforward and will not be presented here.

The credit exposure in the case of co-default is equal to the notional value of the contract times one minus the assumed recovery rate on the reference name. If the counterparty defaults and the reference name does not, the replacement value of the contract will depend on the spread of the reference name at the time of counterparty default. Methodologies for estimating conditional forward spreads do exist, but remain a topic of active research.

Practical Issues in Wrong-Way Credit Risk

J.P. Morgan has incorporated wrong-way credit risk into its credit exposure systems as a necessary feature of credit derivative pricing. The following section contains examples of some key implementation issues to be considered when developing a business model that incorporates wrong-way credit risk.

Early Settlement of Contracts

Because most single-name credit derivative contracts terminate upon the default of the reference name, the maturity of the contract is itself stochastic. In such cases, the option-valuation method for calculating credit exposure is invalid; a Monte Carlo simulation must be used to capture the effects of early settlement.

Conditional Forward Spread Modeling

As mentioned above, there is no widely accepted model for estimating conditional forward spreads. Structural models (such as the Merton model and discretized CreditMetrics-style models) and reduced-form models (which use spread-based default estimates) both have significant drawbacks. Structural models are poor predictors of the term structure of spreads; reduced-form models have difficulty capturing correlations.

J.P. Morgan has developed both a discretized structural model and a reduced-form model for wrong-way exposure on investment-grade, single-name credit derivatives. One of the main problems with the discretized model is that it requires a risk-neutral transition matrix, which is not directly observable in the market. Recent developments in the parameterization of risk-neutral transition matrices23 may make this model more viable in the future.

Appendix I. Calculation of Co-Default and Independent Default Probabilities for Wrong-Way Forex

The term structure of default probability for both corporate and sovereign borrowers can be determined by observing credit spreads on traded obligations and assuming a recovery rate. Given the term structure of bond spreads, it is possible to bootstrap a default probability “forward curve” indicating the (noncumulative) probability that the obligor defaults for the first time during each period:24

P(τ) ≡ P(default{τ − dt, τ}

P(default{τ−dt,τ}), or p(τ)is the probability that the counterparty defaults for the first time during the period {τ−dt,τ}.
dtis the length of each period.
P(default{0,τ})is the probability that the counterparty defaults by time τ

It is assumed that if the sovereign defaults, all corporates will default as well. Thus the probability of co-default in the interval {τ - dt,τ} will be equivalent to the joint probability of sovereign default before τ and no corporate default before τ - dt:

Psov&corp(τ) = P(def{0,τ}∩no def{0,τ − dt}};

Psov&corp(τ)is the probability that both the corporate counterparty and the sovereign default for the first time during the interval {τ - dt,τ}.
Pcorp&no sov(τ)is the probability that the corporate counterparty defaults for the first time during the interval {τ - dt,τ}, but the sovereign does not default.

Using Bayes’ Rule, this is shown to be equivalent to

Psov&corp(τ) = P(defsov{0,τ}| no defcorp{0,τ − dt})·P(no defcorp{0,τ − dt}).

The first factor—that is, the conditional probability of sovereign default during {τ-dt,τ} given no corporate default before τ-dt—is difficult to measure. However, sovereign default after τ—dt is largely independent of idiosyncratic corporate defaults before τ-dt. The only corporate defaults before τ-dt that will significantly influence sovereign default after τ-dt are systematic (sovereign-driven) defaults. Thus the condition of no corporate default before τ-dt has virtually the same effect as the condition of no sovereign default before τ-dt.25

The following substitution is therefore possible:

P(defsov{0,τ} | nodefcrop{0,τ− dt} = P(defsov{0,τ} | no defsov{0,τ − dt})

Psov&crop(τ) = P(defsov{0,τ} | nodefsov{0,τ − dt}·P(no defcorp{0,τ − dt}).

Using Bayes’ Rule again, we can re-express the equation in terms of forward default probabilities (as defined above):

Logically, the sum of the probabilities of co-default and independent default must add up to the total probability of counterparty default obtained from bond spreads. The probability of independent corporate default is therefore the difference between the total corporate default probability and the probability of co-default:

If the counterparty’s default probability is similar to that of the sovereign, or if both are very small, the last factor in the above equation is close to unity. The following are good approximations in such cases:

Psov&corp(τ) = Psov(τ)

Pcorp&no sov(τ) = Pcorp(τ) − Psov(τ).

Appendix II. Example: Wrong-Way Forex Charge Calculation Using Option Valuation26

Transaction:

  • par currency swap

  • $100 million notional

  • floating to floating

  • three-year maturity

  • BBB-rated foreign counterparty

  • AA-rated sovereign

Parameters:

  • 25 basis point sovereign spread

  • 100 basis point counterparty spread

  • 50 percent recovery rate

  • 40 percent correlation between forex rate and counterparty asset value

  • 10 percent implied forex volatility (constant)

  • 5 percent risk-free rate in both currencies

Table 4.A1Wrong-Way Forex Charge Calculation Option Using Valuation
ResultComputation
Psov(year 1) = 0.5%

Pcorp(year 1) = 2%
p(year 1)=P(default1-yr)͌ spread/(1 - recovery rate) spreads: 25 basis points; 100 basis points recovery rate: 50%
Psov&crop(year 1) = 0.5%Psov&corp(τ) = Psov(τ)

Psov(year 1): 05.%
Pcorp&no sov(year 1) = 1.5%Pcorp&no sov(τ) = Pcorp(τ) − Psov(τ)

Psov(year 1): 0.5% Pcorp(year 1) = 2%
RVsov&corp = 17%RVsov&corp(τ) = RVsov

RVsov from table: 17%
RVno sov = 100%RVnosov(τ)=1RVsovP(defaultsovτ)1P(defaultsovτ)forsmallP(d)RVsovfromtable:17%;P(dsov):0.5%(above)
RVcorp = 85%RVcorp=1+ρFX,ASSESTSσFXN1[P(default)corpτ)/2].τρ:40%σ:10%τ:4P(defcrop1yr):1.5%(above)P(default4yr)4*P(default1yr):6%
RVcorp&no sov = 85%RVcorp&no sov = RVcorp · RVno sov

RVno sov: 100% RVcorp: 85%
E[exposure | defaultsov&corp] = 83%E[exposure(τ) | defaultsov] « E[fwd| defsov] = 1 - RVsov The low residual value indicates the option is deep in-the-money and can be approximated linearly. The ratio of interest rates is 1, and therefore does not affect the value of the forward relative to spot.
E[exposure | defaultcorp&no sov]͌ 15%E[exposure(τ) | default] = Put{Φ(x(τ)) | default;0,τ} Option valuation gives expected exposure ̴15%.
E[exposure | defaultcorp]͌32%E[exp|defcorp]=Psov&corpPcorpE[exp|defsov&corp]+Pcorp&nosovPcorpE[exp|defcorpnosov]psov&corp:0.5%pcorp&nosov:1.5%pcorp:2%E[exp|defsov&corp]:83%E[exp|defcorp&nosov]:15% The ratios of co-default and independent default probability to total default probability are assumed to be the same as year 1 in all subsequent years.
Charge ͌ 0.85%charge=Σeytt=13E[exposure|defcorp]spreadE[exposure|defcorp]:32%Riskfreerate:5%spread:1%yield:6%

When wrong-way risk is considered, the average expected exposure is about 32 percent of notional, or $32 million. The up-front credit charge is 0.85 percent of notional, or $850,000. If wrong-way risk were not accounted for, the average expected exposure would be approximately $5.5 million and the credit charge only about $150,000.

Reference

Levy and Clarke (2000), p. 2. For the purposes of exposure calculation, losses include all defaulted amounts, regardless of the possibility of recovery.

See Levy and Clarke (2000), p. 4, for a more detailed discussion of default timing and collateral settlement.

A more precise calculation would also consider the conditional distribution of the contract’s value given default. However, that approach introduces a great deal of complexity and offers little marginal benefit.

The type of contract under consideration will determine which option-pricing model is appropriate. For example, Black-Scholes models are used for options on forwards and swaption valuation models are used for swaps.

The knowledge of default actually implies a distribution of possible values around xjτ. The mean of the distribution is used here as a first-order approximation.

For example, Manos (1998).

It is also possible to estimate the residual currency value using a structural model; however, such models are difficult to parameterize.

A discussion of wrong-way interest rate risk on forex forwards can be found on pp. 92-93

For a discussion of the bootstrapping procedure, see Levin and Levy (1999b).

It is generally assumed that the residual forex value given corporate default is independent of time. If τ is not fixed, RV as calculated here may change with time due to the construction of the model. To correct for this variation, τ is fixed (somewhat arbitrarily) at four years, independent of the exposure’s maturity. See Levin and Levy (1999a), p. 7.

The actual risk-neutral default probability will depend on the tenure of the obligation. Levin and Levy (1999b) give a thorough treatment of the term structure of default probabilities.

For nonlinear products such as options, this is only an approximation.

The decay rate would need to be parameterized as well, perhaps judgmentally.

This is analogous to the residual forex value described on p. 88, but assumes a lognormal distribution of stock prices. As noted in that section, τ is fixed at four years.

For a comprehensive treatment of asset correlations, see Crosbie and Bohn (1999) [KMV] or Gupton, Finger, and Bhatia (1997).

For a comprehensive treatment of the relationship between equity and assets, see Crosbie and Bohn (1999).

For a discussion of the bootstrapping procedure, see Levin and Levy (1999b).

Although this assumption may not be strictly true, it may lead to a higher probability of co-default and thus a more conservative estimate of wrong-way exposure. See Levin and Levy (1999a).

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