4 Credit Value Adjustment (CVA) Approach without Wrong-Way Risk (WWR) in The Basel Accord
Tải bản đầy đủ
WEBC12
11/25/2013
15:50:53
Page 262
CORRELATION RISK MODELING AND MANAGEMENT
262
Credit-Risky
Derivative
=
Default-Free
Derivative
–
CVA
(12.10)
CVA is by deﬁnition ³0; see also equations (12.11) and (12.12) below.
Hence from equation (12.10) it follows that a credit-risky derivative has a
lower price than a derivative without credit risk. This is because the buyer of
the credit-risky derivative (often referred to as the dealer) lowers the price of
the derivative since he assumes the credit risk of the counterparty (the
derivatives seller). In particular, if the counterparty defaults, the buyer of
the derivative will not receive the payout of the derivative. CVA is an
adjustment since the derivatives buyer adjusts (lowers) the price of the
derivative due to credit risk.
CVA is an integral part of the Basel III accord. Figure 12.4 shows CVA and
the associated wrong-way risk (WWR), which will be discussed in section 12.5.
FIGURE 12.4 Credit Value Adjustment (CVA) and Wrong-Way Risk (WWR) in the
Basel III Framework
Source: Moody’s Analytics, 2011.
WEBC12
11/25/2013
15:50:54
Page 263
Correlation and Basel II and III
263
From Figure 12.4 we observe that CVA has a market risk component and
a credit risk component. We formalize this in equation (12.11).
+
; PDc )
CVAa;c = f (Da;c
|{z} |{z}
Market
risk
(12.11)
Credit
risk
where
CVAa,c: credit value adjustment of entity a with respect to the counterparty c
D+a,c: netted, positive derivatives portfolio value of entity a with counterparty c
PDc: default probability of counterparty c
In equation (12.11),
we take
only the positive netted derivative portfolio
+
= max
value, Da;c
D ; 0 , into consideration. This is because entity a has
∑ a;c
credit exposure only if the netted derivatives portfolio between a and c is
positive for a. (In simple terms, a has credit exposure with respect to c only if
c is a’s debtor.) Figure 12.5 shows this property.
Figure 12.5 shows that for positive credit exposure, the credit exposure is
identical with the netted portfolio value, also called portfolio marked-tomarket (MtM) value. Credit risk is the risk that the credit exposure changes.
For example, the credit risk for entity a with respect to c would increase if the
credit exposure increases due to an increase in the market value of the
derivatives D+
a,c or an increase in the default probability of c, PDc.
Equation (12.11) also shows that CVA can be viewed as a derivative
itself. It is a complex derivative since it has two underlyings, D+ and PDc,
Credit exposure of entity a
with respect to c
Netted portfolio value
from the viewpoint of
a with respect to c
FIGURE 12.5 Credit Exposure of Entity a with Respect to Counterparty c
WEBC12
11/25/2013
264
15:50:55
Page 264
CORRELATION RISK MODELING AND MANAGEMENT
which may be correlated! If they are not correlated, we can multiply the
market risk component and the credit risk component. Adding a recovery rate
of the counterparty c, Rc, we can write:
+
´ PDc )(1 − Rc )
CVAa;c = (Da;c
(12.12)
Let’s look at an example of equation (12.12).
EXAMPLE 12.2: CALCULATING CVA, ASSUMING
NO CORRELATION BETWEEN MARKET RISK AND
CREDIT RISK
Entity a has a derivatives portfolio with counterparty c, which has a
present value of +$100,000,000 for a. The default probability of c for a
1-year time horizon is 5%. The recovery rate of counterparty c in case of
default of c is expected to be 30%. What is the CVA from the viewpoint
of a with respect to counterparty c for a 1-year time horizon?
Following equation (12.12), it is CVAa,c = $100,000,000 ´ 0.05 ´
(1 − 0.3) = $3,500,000.
Equation (12.12) is the basis for calculating CVA in the Basel III accord,
when the correlation between market risk and credit risk is assumed to be zero.
However, this is a simplistic assumption, which we will alter in the next section.
12.5 CREDIT VALUE ADJUSTMENT (CVA) WITH
WRONG-WAY RISK IN THE BASEL ACCORD
As mentioned earlier, equation (12.12) assumes that market risk of the
derivative D and credit risk PD are not correlated. However, this is not a
realistic assumption. Market risk and credit risk are clearly related. For
example, if the equity market declines (maybe due to a recession), the default
probabilities of companies typically increase (since debt-to-equity ratios
increase). Conversely, if the default probability of a company increases
(maybe due to bad management or increased competition), the stock price
of the company will decline.
The Basel accord recognizes the correlation between market risk and
credit risk. The Basel accord deﬁnes two types of wrong-way risk (WWR),
general wrong-way risk and speciﬁc wrong-way risk. Let’s look at general
wrong-way risk ﬁrst.
WEBC12
11/25/2013
15:50:55
Page 265
Correlation and Basel II and III
265
GENERAL WRONG-WAY RISK (WWR)
Exists when the probability of default of counterparties is positively
correlated with general market risk factors.5
Following the Basel II accord, general market risk factors are interest
rates, equity prices, foreign exchange rates, commodity prices, real-estate
prices, and more.
Let’s discuss an example of general wrong-way risk regarding the market
risk factor interest rates, which can be positively correlated with default
probability. We will explain general wrong-way risk with the practical
example of a long bond position, which is displayed in Figure 12.6.
In Figure 12.6 only the bond investor has credit risk with respect to bond
issuer. This is because in case of default of the issuer, the bond investor will not
receive the coupon payments, and, most important, will just receive the recovery
rate of the principal investment of $1,000,000. The bond issuer does not have
credit exposure to the bond investor, since the bond issuer has received all
contractual payments (i.e., the initial investment of $1,000,000 at t0).
A bond price B is mainly a function of the market interest rate level i and
the default probability of the issuer PDc; hence B = f (i, PDc,:::). There is a
negative relationship between the bond price B and market rates i: The higher
the market interest rates i, the lower is the bond price B, since the coupon of
the bond price is now lower compared to the market interest rate i; formally:
∂B £ 0. There is also a negative relationship between the bond price B and the
∂i
B
£ 0.
default probability of the issuer PDc: ∂∂PD
c
The relationship of B, i, and PDc constitutes general wrong-way risk: In a
weakening economy, typically interest rates i decrease and default probabilities
$1,000,000 in t0
Bond
Investor
Coupon from t1 to T and
$1,000,000 at T
Bond
Issuer
FIGURE 12.6 Cash Flows of a Standard Bond Purchase with Maturity T
5. BCBS, “Annex 4 (to Basel II),” 2003, 211, www.bis.org/bcbs/cp3annex.pdf.