P2 T5 Meissner Correlation
P2 T5 Meissner Correlation
P2 T5 Meissner Correlation
We differentiate (a) static financial correlations, which measure how two or more
financial assets are associated within a certain time period, and (b) dynamic financial
correlations, which measure how two or more financial assets move together in time.
Wrong-way risk exists when there is a tendency for both the credit exposure and the
credit risk to increase. For example, wrong-way risk exists if Deutsche Bank sells a put on
itself. In this case the put buyer has wrong-way risk: If the Deutsche Bank stock
decreases in price, the credit exposure increases (since the put is more valuable). But if
the Deutsche Bank stock decreases in price, this typically also means that the default
probability of Deutsche Bank increases; hence the credit risk for the put buyer with
respect to Deutsche Bank also increases.
It means that the value of a variable such as a credit default swap (CDS) or a
collateralized debt obligation (CDO) can increase or decrease for a continued increase or
decrease of correlation values (see Figure 1.2).
Financial correlations are critical in (1) investments, (2) trading, (3) risk management, (4)
financial crises, (5) regulation, and more.
5) High diversification is related to low correlation. Why is this considered one of the few
“free lunches” in finance?
Following the capital asset pricing model (CAPM) it can be shown that a higher
diversification leads to a higher return/risk ratio of the assets of a portfolio. This effect is
enhanced with lower correlation values between the assets. This can be seen from
2 2 2 2
equation (1.5), σP wXσX wYσY 2wXwYCOVXY . The lower the correlation,
expressed as the covariance CovXY, the lower the portfolio risk σP.
6) Create a numerical example and show why a lower correlation results in a higher
return/risk ratio.
µP = wX µX + wY µY (1.1) and
σ P w 2X σ 2X w 2Y σ 2Y 2wX w Y COVXY
(1.5)
From equation (1.5) we can already observe that a lower covariance (Cov) (i.e., a lower
correlation) leads to a lower portfolio volatility σP and hence a higher return/risk ratio
µP/σP. Let’s confirm this in a numerical example.
Asset A Asset B
Table 1
For an equal weighting (i.e., wx = wy = 0.5), from Table 1 it follows that μP = 25.30%. The
1 n
(sample) covariance COVXY (x t μ X )(yt μ Y ) = –0.1525. From equation
n 1 t 1
(1.5), σP = 15.79%. Hence the return/risk ratio = 25.30%/15.79% = 160.27.
Let’s now increase the value of asset A in 2013 to 350.00 as displayed in Table 2.
Asset A Asset B
Table 2
This results in a lower correlation expressed as the (sample) covariance of –0.1783. The
portfolio return in Table 2 is slightly higher than in Table 1 and results in 27.87%. The
portfolio volatility (i.e., the portfolio risk) σP = 16.74%. Hence the return/risk ratio is
27.87%/16.74% = 166.47. The reader may look at the spreadsheet at
www.dersoft.com/CAPM.xls for these results.
Correlation trading is trying to exploit the changes in correlation between two or more
financial assets.
In pairs trading a type of statistical arbitrage is performed. Let’s assume assets X and Y
have moved in a highly correlated manner in time. If now asset X performs poorly with
respect to Y, then asset X is bought and asset Y is sold with the expectation that the gap
will narrow.
9) Name three correlation options in which a lower correlation results in a higher option
price.
n
- Portfolio of basket options. Payoff n iSi K, 0 , where ni is the weight of
i 1
assets i
10) Name one correlation option where a lower correlation results in a higher option
price.
11) Create a numerical example of a two-asset portfolio and show that a lower
correlation coefficient leads to a lower VaR number.
In example 1.2, we derived the 10-day VaR for a two-asset portfolio with a correlation
coefficient of 0.7, daily standard deviation of returns of asset 1 of 2%, asset 2 of 1%, and
$10 million invested in asset 1 and $5 million invested in asset 2, on a 99% confidence
level. The VaR came out to $1.7513 million.
Let’s now decrease the correlation coefficient between asset 1 and asset 2 to 0.6. The
VaR then reduces to 1.7174. This can be verified with the spreadsheet at
www.dersoft.com/2-asset VAR.xlsx.
In a systemic crash, typically the “herding mentality” of traders and investors sets in,
resulting in panic selling. As a result many assets decrease in price simultaneously,
resulting in a high price correlation of the assets.
13) In 2005, a correlation crisis with respect to CDOs occurred that led to huge losses of
several hedge funds. What happened?
In 2005 General Motors was downgraded to BB and Ford was downgraded to BB+, so
both companies were now in “junk” status. A downgrade to junk typically leads to a
sharp bond price decline, since many mutual funds and pension funds are not allowed
to hold junk bonds.
14) In the global financial crisis of 2007 to 2009, many investors in the presumably safe
super-senior tranches got hurt. What exactly happened?
Many investors had leveraged the super-senior tranches, termed leveraged super-senior
(LSS) tranches, to receive a higher spread. This leverage was typically 10 to 20 times,
meaning an investor paid $10,000,000 but had risk exposure of $100,000,000 to
$200,000,000. What made things technically even worse was that these LSSs came with
an option for the investors to unwind the super-senior tranche if the spread had
widened (increased). Therefore many investors started to purchase the LSS spread at
very high levels, realizing a loss and increasing the LSS tranche spread even further.
The objective of the Basel accords is to “provide incentives for banks to enhance their
risk measurement and management systems” and “to contribute to a higher level of
safety and soundness in the banking system.” In particular, Basel III is being developed
to address the deficiencies of the banking system during the financial crisis 2007 to
2009.
16) The Basel accords have no legal authority. Why do most developed countries implement
them?
About 100 countries have implemented Basel II as a regulatory requirement for their
banks. The reason is mainly twofold: (1) it increases the soundness and safety of the
country’s banking system and (2) it results in a higher credit rating (from Standard &
Poor’s, Moody's, Fitch, and Dagong) of the country and its banks, which allows
international trade at a lower cost of capital.
17) How is correlation risk related to market risk and credit risk?
Correlation risk can be categorized as its own type of risk. However, correlation risk is
also an integral part of market risk and credit risk. In particular, market risk is typically
quantified with the value at risk (VaR) concept. VaR is highly sensitive to the correlation
between the assets in the portfolio (see section 1.3.3 and Chapter 9, section 9.4).
Arguably, correlation risk is even more critical in credit risk. Default correlation and
correlated migration to a lower credit state are not only important in structured
products such as collateralized debt obligations (CDOs), but also critical in the loan
portfolios of mortgage lenders, for credit card debt, and in all loan portfolios in general.
A combined, correlated default or migration to a lower credit state of the debtors can
lead to huge unexpected losses for the creditors.
18) How is correlation risk related to systemic risk and to concentration risk?
Systemic risk can be defined as the risk that a financial market or an entire financial
system collapses. Naturally, if an entire financial market collapses, such as the mortgage
market and stock market from 2007 to 2009, the correlation between the prices in
decline increases. See also Figure 1.8.
Concentration risk can be defined as the risk of financial loss due to a concentrated
exposure to a particular group of counterparties. Correlation risk and concentration risk
are related. The lower the concentration risk, the lower is the worst-case scenario for
the creditor. Similarly, the lower the correlation between the debtors, the lower is also
the worst-case scenario for the debtor. This can be seen from equation (1.11),
the lower the probability of joint default, P(X Y) . See also the numerical example 1.3.
19) How can we measure the joint probability of occurrence of a binomial event as default
or no-default?
We can apply the binomial correlation measure of Lucas 1995, equation (1.11):
20) Can it be that two binomial events are negatively correlated but they have a positive
probability of joint default?
Yes. This can be seen from equation (1.11) above. A slightly negative correlation
coefficient ρXY can result in a positive joint default probability P(X Y) , if the product of
the standard deviations of the binomial events PX (1 - PX ) PY (1 - PY ) is fairly low and the
product of the default probabilities PX PY is high. For example, if the standard deviation
of both entities is 30% and a default probability of both entities is 10%, together with a
negative correlation coefficient of –0.1, then, following equation (1.11), this leads to a
joint default probability of +0.1%.
21) What are value at risk (VaR) and credit value at risk (CVaR)? How are they related?
VaR measures the maximum loss of a portfolio with respect to a certain probability for a
certain time frame with respect to (correlated) market risk. Market risk is the risk that a
price or a rate in the (a) equity market, (b) interest rate market, (c) currency market, or
(d) commodity market changes unfavorably.
CVaR measures the maximum loss of a portfolio of (correlated) debt with a certain
probability for a certain time frame.
Hence VaR is the risk related to market risk and CVaR is the risk related to credit risk. For
both VaR and CVaR, correlation is critical. For the exposure of VaR to correlation, see
section 1.3.3, example 1.2, and Chapter 9, section 9.4. In Chapter 12, section 12.1, we
show that a higher (copula) correlation between assets results in a higher CVaR.
22) Correlation risk is quite broadly defined in trading practice, referring to any co-
movement of assets in time. How is the term correlation defined in statistics?
23) What do the terms measure of association and measure of dependence refer to in
statistics?
In statistics, the terms measure of association and measure of dependence typically refer
to associations of variables measured by concepts other than the Pearson correlation
measure.
Answers to Questions and Problems of Chapter 2:
Generally in bad economic times. However, in our study, the correlation volatility was
only slightly lower in normal economic periods (83.40%) than in a recession (80.48%).
Correlation volatility was lowest in expansionary periods (71.17%).
Traders and risk managers should take the higher correlation level and higher
correlation volatility in bad economic times into consideration.
Mean reversion is the tendency of a variable to be pulled back to its long-term mean.
To find the mean reversion rate a, we can run a standard regression analysis of the form
Y=α+βX
St - St -1 a μ S a St -1
(2.5)
Y = α + βX
From equation (2.5), we observe that the regression coefficient β is equal to the
negative mean reversion parameter a.
7) For equity correlations, we see the typical decrease of autocorrelation with respect to
time lags. What does that mean?
It means that correlation levels can be better explained by more recent correlations
than by correlations that existed farther back in time.
Mean reversion and autocorrelation are inversely related: The higher mean reversion is
(i.e., the more strongly a variable is pulled back to its long-term mean), the lower is
autocorrelation (i.e., the degree to which a variable is correlated to its past values).
Our empirical studies showed that equity correlations are not normally or lognormally
distributed. Rather they can be best approximated with the Johnson SB distribution. See
the spreadsheet “Correlation Fitting” at www.wiley.com/go/correlationriskmodeling,
under “Chapter 2.”
10) When modeling stocks, bonds, commodities, exchange rates, volatilities, and other
financial variables, we typically assume a normal or lognormal distribution. Can we do
this for equity correlations?
No, as mentioned under question 9, equity correlation cannot be approximated well
with a normal or lognormal distribution. One reason is that most correlations in finance
are positive and increase the more positive they are, with the exception of very high
correlation values. So a rather complex distribution such as the Johnson SB with four
parameters can approximate equity correlations well. See Figure 2.6.
Answers to Questions and Problems of Chapter 3:
No financial model can or will ever be able to replicate the immensely complex financial
system perfectly. However, financial models are useful tools to help us understand the
financial system and to give approximations. The value at risk (VaR) model can give us a
good estimate about our market risk. The multivariate copula model can give us a good
estimate about the credit value risk of a portfolio, CVaR. The Black-Scholes-Merton
(BSM) model can give us a good idea about the value of an option.
The general limitation of any financial model is its inability to model the immensely
complex financial environment perfectly. In particular, financial models may have these
limitations:
a) Financial models may have known conceptual deficiencies such as the slim tails of
the VaR underlying normal distribution, constant volatility in the Black-Scholes-
Merton model, or a simplistic single correlation parameter in the one-factor
Gaussian copula model.
b) Sometimes financial models have difficulty in calibration. This means that it is
difficult to find parameter values for the model so that the model output can match
market prices.
Generally, no financial model should be trusted uncritically. Traders and risk managers
should be constantly be aware of the limitations of every financial model.
Physics is a “hard” science. This means that physical relationships such as E = mc2 are
correct in any normal physical environment and will be correct in the future. Financial
models may give incorrect values (such as the Black-Scholes-Merton model for an
option) or incorrect expected values, such as the value at risk (VaR) model for expected
future market risk.
4) Name three critical aspects that have to be considered when applying financial models
in reality!
5) What problems with financial modeling occurred in particular in the Great Recession of
2007 to 2009?
The main problem regarding financial modeling in the Great Recession of 2007 to 2009
was inadequate calibration. Benign default probabilities and default correlations were
input into the credit value at risk (CVaR) models. It cannot be expected that wrong input
data can generate correct output data. Garbage in, garbage out, in programming
terminology.
1) Linear dependencies, which are evaluated in the Pearson model, do not appear
often in finance. Most financial relationships are nonlinear; see for example Figures
1.2 to 1.4, and 1.6 to 1.8.
2) As a consequence of point 1, zero correlation derived in the Pearson model does not
necessarily mean independence. For example, the parabola Y = X2 will lead to ρ = 0,
which is arguably misleading. See Appendix 1A of Chapter 1 for details.
1
An elliptical distribution is a generalization of multivariate normal distributions.
= 0 – 0 E(X2)
9) In the Pearson correlation model, what values do covariances take, and what values
does the correlation coefficient take?
1 n
The covariance of equation (1.3) COVXY (x t μ X )(yt μ Y ) takes values
n 1 t 1
between –∞ and +∞. Conveniently, the correlation coefficient in equation (1.4)
COVXY
ρ XY
σ X σ Y takes values between –1 and +1.
The Pearson correlation model has severe limitations; most critically, it measures only
the linear association between variables. There it can at best serve as an approximation
for the association between financial variables, which are typically nonlinear.
11) What is the main difference between cardinal correlation measures such as the Pearson
model and ordinal correlation measures such as Spearman’s rank correlation and
Kendall’s τ?
The Pearson correlation model, as a cardinal correlation measure, takes the numerical
values of the variables into consideration.
12) What is a severe limitation when applying Spearman’s rank correlation and Kendall’s τ
to finance?
The problem with applying ordinal rank correlations to cardinal observations is that
ordinal correlations are less sensitive to the outliers. For example, doubling the outliers
will have no effect on the correlation measure (see section 3.2.2). So the rank
correlation measure is less sensitive to very high losses that may have occurred. This
may result in a distorted, seemingly better performance than the actual performance.
13) When should we apply Spearman’s rank correlation and Kendall’s τ in financial
modeling?
We should apply the ordinal Spearman’s rank correlation and Kendall’s τ measure only if
the input data is ordinal, for example for rating categories.
Answers to Questions and Problems of Chapter 4:
1) The original Heston (1993) model correlates the Brownian motion of which two financial
variables? What is the most significant result of the original Heston model?
The two variables that are correlated in the Heston model are the stock price S and the
volatility σ. One of the most significant results is that the Heston model can replicate
volatility skews found in reality well.
2) To create negative correlation between asset 1 and asset 2 in the Heston (1993) model,
what value does the correlation coefficient α take in equation
3) The Heston model is one of the most widely applied correlation models in finance. Why?
4) What is the difference between the Pearson correlation model and the binomial
correlation model of Lucas (1995)?
The binomial correlation model of Lucas is a special case of the Pearson model. The
Pearson model’s correlation coefficient is
E(XY) E(X)E(Y)
ρ1 (X, Y) (3.3)
E(X 2 ) (E(X)) 2 E(Y 2 ) (E(Y)) 2
P(XY) - P(X)P(Y)
ρ(1{τ X T}
,1{τ Y T}
) (4.8)
(P(X) (P(X))2 (P(Y) (P(Y)) 2
By construction, equation (4.8) can only model binomial events, for example default and
no default. With respect to equation (3.3), we observe that in equation (3.3) X and Y are
sets of i = 1, ..., n variates, with i . P(X) and P(Y) in equation (4.8), however, are
scalars, for example, the default probabilities of entities X and Y for a certain time
period T, respectively, 0 ≤ P(X) ≤ 1, and 0 ≤ P(Y) ≤ 1. Hence the binomial correlation
approach of equation (4.8) is a limiting case of the Pearson correlation approach of
equation (3.3).
5) What are the limitations of the binomial correlation model of Lucas (1995)?
As in the Pearson model, the most severe limitation is that the binomial model analyzes
only linear relationships between variables. However, most financial relationships are
nonlinear.
7) Why is the Gaussian copula model the most popular copula model in finance?
8) What does “In the copula mapping process, the marginal distributions are preserved”
mean?
The statement means that the percentiles are preserved. For example, the 5th
percentile of the marginal distribution (i.e., the original distribution) is mapped to the
5th percentile of the new standard normal distribution, the 10th percentile of the
marginal distribution is mapped to the 10th percentile of the standard normal
distribution, and so on.
9) Given are the marginal 1-year default probabilities of 5% for asset 1 and 7% for asset 2.
If the Gaussian correlation coefficient is 0.3, what is the joint probability of default in
year 1, assuming asset 1 and asset 2 are jointly bivariately distributed?
From equation (4.14), the joint default probability is
10) Given are the 5-year default probability of entity i of 10% and the 6-year default
probability of entity i of 13%. What is the forward default probability in year 6 and what
is the default intensity in year 6?
11) What are the limitations of the Gaussian copula for financial applications?
a) The Gaussian copula has low tail dependence. Following a definition by Joe (1999), a
bivariate copula has lower tail dependence if
lim y 1 0, y 2 0
P[(τ 1 N 11 (y1 ) | (τ 2 N 21 (y 2 )] 0
(4.22)
where τi is the default time of asset i, yi is the marginal distribution of asset i, and N–1
is the inverse of the standard normal distribution. However, it can be easily shown
that the Gaussian copula has no tail dependence for any correlation parameter ρ:
lim y 0, y
1 2 0
P[(τ1 N11 (y1 ) | (τ 2 N 21 (y 2 )] 0, ρ {-1,1}.
b) Some researchers report that the one-factor version of the Gaussian copula has
problems in calibrating CDO tranches. However, other studies of Hull and White
(2004), Andersen and Sidenius (2004), and Burtschell et al. (2008) find no problems
in calibration.
c) The Gaussian copula is principally static and consequently allows only limited risk
management. In particular, there is no stochastic process for the critical underlying
variables’ default intensity and default correlation. However, back-testing and
stress-testing the variables for different time horizons can give valuable sensitivities;
see Whetten and Adelson (2004) and Meissner et al. (2008b).
12) Since the Gaussian copula has low tail dependence, which other copulas seem more
suitable to model financial correlations?
Other copulas such as the Student’s t or the Gumbel copula have higher tail dependence
and may be more suited to model financial correlations.
13) Can the copula model be blamed for the Great Recession of 2007 to 2009?
The copula model is a rigorous statistical correlation model. For finance, the limitations
discussed in question 11 apply. The critical problem with respect to the 2007–2009
global financial crisis was not the financial models themselves, but inadequate
calibration. Benign default probabilities and default correlations from 2003 to 2006
were input into the models. It cannot be expected that models produce correct outputs
if the inputs are flawed. Garbage in, garbage out in programming terminology.
Contagion correlation modeling, pioneered by Davis and Lo (1999 and 2001) and Jarrow
and Yu (2001), is based on the idea that the default of one entity impacts the default
intensity of another entity. Hence contagion default modeling incorporates
counterparty risk (i.e., the direct impact of a defaulting entity on the default intensity of
another entity).
Introducing symmetric contagion among all entities creates the problem of circularity. In
this case, the construction of a joint distribution is rather complex. One solution is to
model “asymmetric dependence” (i.e., the default of primary entities impacts the
default intensity of secondary entities, but not vice versa).
As with all financial models, the user has to be aware of the general limitations of
financial models. Models can serve as an approximation of the complex financial reality,
but will never exactly replicate it.
16) See the model www.dersoft.com/choleskyexample.xlsm for the answer.
Answers to Questions and Problems of Chapter 5:
The three main types of CDOs are cash CDOs, synthetic CDOs, and nonfunded CDOs.
3) Which are the three main players in a CDO? Why is the SPV typically AAA rated?
1) The originator (or protection buyer), who transfers the credit risk,
2) The investor, who assumes the credit risk, and
3) The SPV (special purpose vehicle), which manages the CDO.
The SPV is AAA rated in order to reduce the credit risk for the investor.
The motivation for the originator is naturally to transfer the credit risk, which improves
the protection buyer’s credit rating, frees credit lines, reduces regulatory capital, and
lowers funding cost. The motivation for the investor is to receive high yields. The
motivation for the SPV is fee income.
5) Name the three financial principles that are incorporated in a CDO and explain them
briefly.
6) What is the default probability of an entity based on the Merton 1974 model, if the
current asset value V0 = $4,000,000, the debt value D = $3,000,000, the maturity T of the
debt is in 1 year, the risk-free interest rate r is 2%, and the volatility of the assets σ is
20%?
It is N(–d2) in the Merton 1974 model, which comes out to 7.53%. A simple model that
derives the answer can be found at www.dersoft.com/Mertonmodel.xlsx.
7) In the Merton 1974 model, there is a closed form solution for the default probability.
What is it?
8) The elegant Merton 1974 model principally serves as a basis for more realistic
extensions. What are the limitations of the Merton 1974 model?
The ingenious Merton 1974 model outlines the principles of a company’s default using
structural properties such as assets and debt. The main limitations of the model are that
only one form of debt D is modeled and that default can occur only at debt maturity T.
Naturally, numerous extensions of the model have been created to bring the model in
line with the complexities of reality.
9) The Merton 1974 model is the basis for all structural models. Why is the Merton model
called structural?
Because it uses the structural data of a company such as assets and liabilities as inputs.
10) When valuing the default probability in a CDO, why do we map the default probability of
asset i, λi, to standard normal via N–1(λi)?
This is done mainly for mathematical and computational convenience. The default
barrier Mn is (multivariate) standard normal, so to compare it with the default
probability of asset i, λi, we map λi to standard normal via N–1(λi).
11) The multivariate copula function Mn serves as the default threshold. How is the default
of asset i derived in the copula model?
To derive the threshold, typically the popular Gaussian copula model is applied. We
derive the default threshold as
12) The credit triangle is s ≈ λ (1 – R), where s is the credit spread, λ is the default intensity,
and R is the recovery rate. When R = 0, we have s ≈ λ. Explain the intuition of s ≈ λ.
The relationship s ≈ λ is intuitive since the default probability λ is the risk that the
investor takes, and the investor should be compensated for this risk by receiving a
similar amount, the spread s. The relationship s ≈ λ (1 – R) was formally derived by
Lando (1998) with R = 0 and by Duffie and Singleton (1990) with R ≠ 0.
13) The recovery rate is often modeled as being higher, the lower the credit rating of an
asset. This seems counterintuitive. But why is it rational?
Recovery rates are often approximated using historical recovery rates of defaulted
companies. Interestingly, typically rating agencies assign a lower recovery rate to higher-
rated entities. The logic is that higher-rated entities will default only in a recession, in
which recovery rates are lower. Lower-rated entities are assigned a higher recovery
rate, since they can default even in an economic expansion.
14) Can the Gaussian copula be blamed for the global financial crisis of 2007 to 2009?
No. The Gaussian copula is a sound statistical correlation model. Applied to finance, the
limitations of slim tails, occasional calibration problems, and the principally static nature
are of concern. However, the main problem in 2007 and 2008 when valuing CDOs with
the Gaussian copula was inadequate calibration. Benign default probability functions
were applied, and low default correlations between the assets in the CDO were inputted
in correlation matrices. When data from non-crisis periods are inputted into a model, it
cannot be expected that the model will produce correct outputs in a crisis!
15) What were the main reasons for the misevaluation of CDOs before and during the crisis?
As mentioned in question 14, the main reason for the misevaluation of CDOs in 2007
and 2008 was inadequate calibration. No model can produce correct results when fed
incorrect inputs.
Answers to Questions and Problems of Chapter 6:
1) Name the three strongly simplistic assumptions of the one-factor Gaussian copula
(OFGC) model.
Those assumptions are justifiable only for very homogeneous portfolios, for example for
portfolios that have the same or similar credit ratings and/or that belong to the same
sector.
3) The correlation concept of the OFGC is incorporated in the simple equation (6.1)
The key property of equation (6.1) is that we do not model the default correlation
between the assets i in the portfolio directly, but instead we condition defaults on M.
We assume that ρ is identical for all asset pairs. Therefore, we have the same
relationship between every asset i and M: If ρ is one, every asset i has a perfect
correlation with M; hence all assets are perfectly correlated. If ρ is zero, all assets i
depend only on their idiosyncratic factor Zi; hence the assets are independent. For a ρ of
0.7071 (and therefore ρ 0.5 ), all xi are determined equally by M and Zi. Importantly,
4) Equation (6.1) applies the conditionally independent default (CID) correlation approach.
Explain the term conditionally independent!
As mentioned under question 3, once we have drawn M and Zi, the correlation between
the assets is determined not directly, but indirectly, by conditioning xi on M. Hence the
assets are not directly dependent, but “conditionally independent.”
5) Why are the variables M, Zi, and the resulting xi in equation (6.1) called latent and frailty
variables?
They are called latent (meaning hidden) because they can’t be observed in reality.
They are called frailty variables because the lower M, Zi, and the resulting xi are, the
earlier the default of company i.
6) In equation (6.1) the xi are standard normally distributed. How are the xi transformed
into probabilities?
The xi are transformed into probabilities simply by using N(xi) = Pi, 0 ≤ Pi ≤ 1, where N is
the cumulative standard normal distribution function.
t t
7) In equation (6.2), s i = 1 – Pi, s i is the survival probability of asset i at time t, and 1 – Pi is
the default threshold, which includes the correlation. Solve equation (6.2) for the
default time t of asset i.
t
What is the default time of asset i if s i = 80% and Pi = 50%?
8) Calculate the fair equity tranche spread of a CDO for the following CDO with a three-
year maturity: The starting notional is $2,000,000,000, with 125 equally weighted
companies. Hence each asset has a notional value of $16,000,000.
Let’s assume spread payments and payouts are annually in arrears. The recovery
rate for every asset is 30%. Interest rates are constant at 5%. We consider an equity
tranche with a detachment point of 3%. Hence the equity tranche has a starting notional
value of $60,000,000.
Let’s assume that we have derived that one asset defaults after 1.5 years and
one asset defaults at 2.5 years. Hence the starting notional of $60,000,000 reduces to
$44,000,000 for t2 (end year 2) and to $28,000,000 for t3 (end of year 3).
Hence the fair equity tranche spread, paid annually in arrears, is:
$19,774,108/$120,986,435 = 16.34%
9) The tranche spread of the equity tranche and the senior tranche behave very differently
with respect to changes in the correlation of the assets in the CDO. Draw a graph
showing the tranche spread–correlation dependence for the equity tranche and a senior
tranche.
Tranche Spread
Equity tranche
Mezzanine tranche
Senior tranche
Default
0% 50% 100% Correlation
Evenly distributed Clustering of
defaults defaults
The negative relationship between the equity tranche spread and default correlation is
intuitive: The higher the default correlation of the companies in the CDO, the higher the
probability of extreme events; that is, the probability of either many or no defaults is
high. The high probability of no defaults reduces the riskiness and hence reduces the
equity tranche spread. The high probability of many defaults at the same time increases
the riskiness of the equity tranche, which increases the equity tranche spread. However,
this effect does not impact the equity tranche significantly, since the losses are capped
at the detachment level.
The opposite logic applies to the senior tranche: If default correlation is high, many
defaults may occur at the same time. Therefore, the senior tranche may be impacted;
hence the riskiness and the spread are both high.
11) Name the main differences between the standard Gaussian copula and the OFGC.
x i ρ M 1 ρ Zi (6.1)
where Qi(t) is the cumulative default distribution of asset i with respect to t, and ρM is
the correlation matrix of the assets in the portfolio.
Let’s look at the differences in the OFGC of equations (6.1) and (6.1a), and standard
copula of equation (4.12).
a) The correlations between the assets i in equation (6.1) are modeled indirectly by
conditioning the auxiliary variable of asset i, xi, on a common factor M. In contrast,
equation (4.12) applies the typical correlation matrix ρM (for an example see Chapter
1, Table 1.3).
b) As a consequence, in the OFGC all asset pairs in the portfolio have the same
correlation. The standard Gaussian copula is richer as it can model asset pair
correlation individually in the correlation matrix.
c) The cumulative normal distribution in equation (6.1a), which includes the correlation
between the assets i via xi, is conveniently one-dimensional. The cumulative normal
distribution in equation (4.12) is n-dimensional.
d) The bivariate case of the standard Gaussian copula is equivalent with the OFGC:
Sampling from equation (4.12) is achieved by Cholesky decomposition (as explained
in Appendix 4A of Chapter 4). In the bivariate case, Cholesky sampling of two
correlated variables x1 and x2 from equation (4.12) reduces to
x1 = ε1
x 2 ρ ε1 1 ρ ε 2
12) The OFGC is a first, simplistic approach to derive the tranche spread in a CDO and the
credit risk in portfolios. Name three extensions of the OFGC.
1. The OFGC is principally static (i.e., it has a one-period time horizon). However, the
static property of the OFGC can be relaxed, as in Hull et al. (2005), who apply a
dynamic OFGC model. Hence they modify equation (6.1) and model
2. Furthermore, more common factors M can be modeled. In this case equation (6.1)
generalizes to m m
x i p i,k M k Zi p i,k
k 1 k 1
m
and the correlation between xi and xj is k 1
p i,k p j,k
_ (6.7)
x i ρ M 1 ρ Zi
_
where M and Zi are independent and n ~ (0,1). x i x i W where W follows an
inverse gamma distribution. It follows that the latent variable xi is Student’s t
distributed.
13) Should we apply the OFGC to value CDOs? Should we apply the OFGC to value credit risk
in portfolios?
To answer these questions, we have to address the assumptions of the OFGC (i.e., same
default probability of all assets in a CDO or any other portfolio, and same correlation for
all asset pairs in a CDO or another portfolio). Are these assumptions too simplistic to
derive the credit risk in a CDO or other portfolios? The answer is: Only in rare cases, if
the assets in the portfolio are very homogeneous (i.e., they have similar default
probabilities and similar default correlations), is the OFGC an adequate model.
Most portfolios of investment banks, however, are highly diversified with assets from
different sectors and different geographical regions and hence have different default
probabilities and default correlations. In this case the OFGC is an inappropriately
simplistic model!
Traders deliberately alter the tranche correlations to derive desired tranche spreads,
which violates the basic principle of the OFGC, which assumes a constant CDO-wide,
tranche-nonspecific default correlation.
15) Explain the correlation smile that traders apply to derive tranche spreads. How is the
correlation smile related to the volatility smile when pricing options?
Traders use a higher correlation for the equity tranche and the super-senior tranches
than the mezzanine tranche, constituting the correlation smile (see Figure 6.9).
However, there is a crucial difference between the volatility smile of options and the
correlation smile of CDOs. While an increase in the implied correlation increases the
senior tranche spread, an increase in the implied correlation decreases the equity
tranche spread. This is because the equity tranche spread has a negative dependence on
implied correlation (see Figure 6.8). Hence CDO traders arbitrarily decrease the equity
tranche spread and arbitrarily increase the senior tranche spread.
Answers to Questions and Problems of Chapter 7:
3) Why can the one-factor Gaussian copula (OFGC) be considered a top-down model?
The one-factor Gaussian copula model can be considered a top-down correlation model
since it abstracts from the individual default intensities of each asset i. Rather, one
default intensity is assumed for all assets in the portfolio.
4) Markov processes are “memoryless.” What does this mean? Give an example!
This means that only the present information, not the past, is relevant. For example,
rolling a die or playing roulette is a memoryless process. Whatever the result was in a
trial, that result is irrelevant for future results of trials.
In credit risk modeling, transition rates are probabilities to move from one credit state
to another.
6) Why does higher transition rate volatility mean higher default correlation in the
Schӧnbucher 2006 model?
A higher transition rate volatility means higher default correlation since a higher
transition rate volatility means a higher transition rate of all entities n to a lower state,
hence a higher default correlation; likewise, a lower transition rate volatility means a
lower transition rate of all entities n to a lower state, hence lower default correlation.
7) Why does an increase in stochastic time change mean a higher default correlation in the
Hurd-Kuznetsov 2006 model?
In the Hurd-Kuznetsov 2006 model, correlation is induced by the speed of the stochastic
clock τt. An increase in the speed of the clock increases the speed of migration of all
entities and hence increases the probability of simultaneous defaults. If the stochastic
clock jumps, the probability of simultaneous defaults is even higher.
8) What is the random thinning process in top-down models, and what does it accomplish?
Random thinning allows the allocation of the portfolio intensity to the sum of the
individual entities’ intensities. In addition, the name of the entity that defaults is
revealed at the default time, highlighting the fact that random thinning allocates the
portfolio intensity to the individual entities.
Answers to Questions and Problems of Chapter 8:
3) Why does it seem like a good idea to model financial correlations as a stochastic
process? Name two reasons.
The DCC model is a concept in the ARCH framework. Here, one of the core equations
is:
where:
ri,t : Return of asset i at time t
σi,t : Standard deviation of the return of asset i at time t (also called volatility)
εi,t : Random drawing of a standard normal distribution for asset i and time t, ε = n ~
(0,1)
Hence stochasticity enters the model via εi,t.
In addition, the variance σ2 or the standard deviation σ in equation (8.7) is modeled
with an ARCH process (or one of many extensions such as GARCH, NGARCH,
EGARCH, TGARCH,2 and more) of the form
5) The geometric Brownian motion (GBM) is applied to model many financial variables,
such as stock prices, commodities, and exchange rates. What are two of the
limitations of the GBM to model financial correlations?
dρ
μ dt σ ε dt (8.2)
ρ
where:
ρ : Correlation between two or more variables
µ : Expected growth rate of ρ
σ : Expected volatility of ρ
ε : Random drawing from of standard normal distribution. Formally, ε = n ~ (0,1).
We can compute ε as =normsinv(rand()) in Excel/VBA and norminv(rand) in
MATLAB.
b) Equation (8.2) is not bounded, meaning correlation ρ can take values bigger than 1
and smaller than –1. From equation (8.2) we see that a value of ρ > 1 is more likely
to happen when the growth rate µ is high, if the volatility σ is high, and if we have a
2
The N in NGARCH stands for nonlinear, the E in EGARCH stands for exponential, and the T in TGARCH stands for
truncated. See Bollerslev (2008) for a nice overview of all ARCH extensions.
high value of ε in a simulation. Conversely, values of ρ < –1 are more likely to occur
for low values of µ and high values of σ and ε.
6) The Vasicek model is an improvement over the GBM to model financial correlations.
Why?
7) The bounded Jacobi process seems like a good choice to model financial
correlations. What advantage does it have over the GBM and the Vasicek model?
dρ a (m ρ ρ t ) dt σ ρ (h ρ t )(ρ t f ) ε t dt (8.15)
where h = upper boundary level, and f = lower boundary level (i.e., h ≥ ρ ≥ f).
The bounded Jacobi process is restricted to correlation values between –1 and +1,
when the boundary conditions
σ 2 (h f)/2 (8.17)
α
(mρ f)
σ 2 (h f)/2
α (8.18)
(h - mρ )
are met.
8) In the Buraschi, Porchia, and Trojani 2010 stochastic correlation model, which two
stochastic processes are correlated?
As in the Heston model, the Buraschi, Porchia, and Trojani 2010 stochastic
correlation model correlates the relative change of the variable of interest S with the
change in volatility σ:
dS
μ dt σ t dz1 (8.22)
S
dσ2t a (mσ2 σ2t ) dt ξ σt dz2 (8.23)
The correlation between the stochastic processes (8.22) and (8.23) is achieved by the
identity
where dz2(t) and dz3(t) are independent, and dz(t) and dz(t’) are independent, t ≠ t’.
9) In the Buraschi, Porchia, and Trojani model, which financial properties can be
replicated? Name two.
1) When we talk about “market risk,” which four markets are typically included?
1) Equity market
2) Fixed income market
3) Commodity market
4) Exchange rate market
2) Name several other markets not included in the four markets mentioned in question 1.
Other markets are the energy market, real estate market, weather market, economic
variables, and the derivatives market.
Market correlation risk is the risk that the correlations between the prices in one market
or between these markets change unfavorably.
4) We can measure market correlation risk with Cora. What information does Cora give us?
Cora measures how much a dependent variable changes if the correlation between two
or more independent variables changes by an infinitesimally small amount. Formally,
V
Cora (9.1)
ρ(x i 1,..., x n )
Cora is the first mathematical derivative of a function (such as value at risk [VaR]) with
respect to correlation between the underlying variables (such as the assets in the VaR
portfolio).
7) Measuring correlation risk is not totally new. In option theory, a Vanna exists. What
information does Vanna give us?
Vanna tells us how much the option price V* changes if the correlation between the
future option price F and the volatility of the future price σ changes. Formally:
V*
Vanna (9.6)
ρ(F, σ)
Vanna is a special case of Cora, with the dependent variable being the option price V*
and the correlated variables being the forward price or rate F and the volatility of F, σ.
Gora measures how much Cora changes if the correlation between two or more
independent variables changes by an infinitesimally small amount. Formally,
Cora 2V
Gora (9.2)
ρ(x i 1 ,..., x n ) ρ 2 (x
i 1 ,..., x n )
Gora is the first partial derivative of the Cora function or the second partial derivative of
the original function V with respect to correlation. That is, Gora measures the curvature
of a function with respect to correlation.
11) Okay, here is a tough one: Differentiate the price function of an exchange option
S2 e q 2 T 1 2 2 S2 e q 2 T 1 2 2
ln q T (σ1 σ 2 2ρ σ1σ 2 )T ln q T (σ1 σ 2 2ρ σ1σ 2 )T
Se 1 2 Se 1 2
E S2 e q 2 T N 1 S1e q1T N 1
σ1 σ 2 2ρ σ1σ 2 T
2 2
σ1 σ 2 2ρ σ1σ 2 T
2 2
with respect to the correlation coefficient ρ. Try doing this yourself first. After
rearranging, you can just use the chain rule. If you give up, look at
www.wiley.com/go/correlationriskmodeling, under “Chapter 9,” for the answer.
12) Arguably, the most important application of correlation risk management is in risk
management. In practice, the risk of a portfolio is often measured with the value at risk
(VaR) concept. Is VaR sensitive to changes in the correlation of the assets in the
portfolio? That is, what is the Cora of VaR?
CoraVaR measures how much VaR changes for an infinitesimally small change in all
pairwise correlations of all asset returns in the portfolio. Formally,
VaR
Cora VaR (9.15)
ρ(x i 1,..., x n )
Gora for VaR measures how much Cora of VaR changes if the correlation of all assets in
the portfolio changes, or what the curvature of the original VaR function is. Formally,
b) Another type of Gap-Cora measures the correlation exposure of VaR with respect to
a change of the correlation between the correlation of just two assets i and j;
formally:
VAR
Gap-Corai, j
ρ(xi, j ) (9.17)
We can also derive the Gap-Gora for equations (9.16) and (9.17). This would tell us how
much the Gap-Coras change if the correlation changes.
Answers to Questions and Problems of Chapter 10:
Credit risk is the risk of financial loss due to an unfavorable change in the credit
quality of a debtor.
2. Which two types of credit risk exist? What is the relationship between these two
types of credit risk?
There are principally two types of credit risk: (1) migration risk and (2) default risk.
See Figure 10.1 for a graph.
Default risk is a special case of migration risk for a migration of the debtor into the
default state. Default risk exists only for a long credit position, for example being
long a bond or long a tranche in a CDO.
However, migration risk and default risk have different dynamics. For example, if a
bond migrates to a lower state, for example CCC, the bond investor just suffers a
paper loss and will receive the principal investment back at maturity if the bond
does not default. However, if a bond defaults and stays in default, the bond investor
will not receive the principal investment back, just the recovery rate of the bond.
Credit correlation risk is the risk that credit quality correlations between two or
more counterparties change unfavorably.
4. Name three financial products that are exposed to credit correlation risk.
All loan portfolios of financial institutions, as well as all structured products such as
collateralized debt obligations (CDOs) and mortgage-backed securities (MBSs), are
exposed to credit correlation risk. In addition, all portfolios that apply derivatives as
a hedge also include credit correlation risk.
5. A CDS that is used as a hedge has three parties: (1) the investor (CDS buyer), (2) the
counterparty (CDS seller), and (3) the underlying asset. The default correlation
between which two entities is most significant for the valuation of a CDS?
Most significant is the default correlation between the counterparty c and the
reference asset r.
6. For the counterparty, the default correlation between the investor and the
underlying asset is also of importance. What is the worst-case scenario for the
counterparty from a risk perspective?
The worst-case scenario for the counterparty is when the default correlation
between the investor i and the reference entity r is negative, with the credit quality
of the investor decreasing and the credit quality of the reference asset increasing:
An increase in the credit quality of the reference entity increases the present value
of the CDS for the counterparty since the counterparty is now receiving an above-
market spread (this is beneficial from a profit perspective but negative from a risk
perspective since a higher present value means more credit exposure). In addition,
the decrease in the credit quality of the investor means higher credit risk for the
counterparty with respect to the investor.
7. When valuing a CDS, we can also include the default intensity correlation of all three
entities. Draw a Venn diagram that displays the default intensity correlation’s
properties.
Reference
entity r
Counterparty Investor
c i
Figure 10.9
CDS
CDSCora 1 (10.11)
λ(r c)
9. Since there are three entities in a CDS, there are principally three Coras. Name them
and interpreted them. Which one is the most critical?
CDS
b) CDSCora 2 (10.12)
λ(r i)
CDSCora2 measures how much the value of the CDS changes if the default
correlation between the reference asset r and the investor i, λ(r∩i), changes by a
very small amount.
CDS
c) CDSCora 3 (10.13)
λ(i c)
CDSCora3 measures how much the value of the CDS changes if the default
correlation between the investor i and the counterparty c, λ(i∩c), changes by a
very small amount. However, CDSCora3 is close to 0, since the effects of a change
in the correlation between the investor i and the counterparty c net.
CDS
d) CDSCora 4 (10.14)
λ(r c i)
We can also derive the sensitivity of the CDS value with a change in the joint
default correlation of all entities in a CDS. The default intensity correlation
λ(r∩c∩i) can be simulated by a trivariate copula as in Brigo and Pallavicini (2008)
or can be simulated by Monte Carlo simulation as discussed in the book in
section 10.3. The numerical values for CDSCora4 are complex and depend on the
default intensity input parameter values λ(r), λ(c), λ(i), the volatilities of λ(r), λ(c),
λ(i), and the correlation λ(r∩c∩i). Different sensitivities of the CDS spread result
for different combinations of the input parameters.
The Gora of a CDS tells us how much the Cora of a CDS changes if the correlation
between the entities in question changes by an infinitesimally small amount. Since
we have four different Coras for a CDS, we have four different Goras.
Credit risk and correlation risk are the two main risks in a CDO.
12. The value of a CDO and its tranches depends critically on the correlation of the
assets in the CDO. Draw a graph showing the equity tranche value, mezzanine
tranche value, and a senior tranche value with respect to correlation.
30.00%
25.00%
20.00%
S pread
15.00%
10.00%
5.00%
0.00%
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.99
-5.00%
C orre la tion
0% -3% 3% -7% 7% -10% 10% -15% 15% -30%
Figure 10.15: Tranche spread with respect to correlation in the one-factor Gaussian
copula (OGFC) model; 125 credits, 1% default intensity rate, 5-year maturity, 30,000
MC simulations
It tells us how much the spread of the tranche changes if the correlation
between all assets in the CDO changes by an infinitesimally small amount.
14. Which tranche in a CDO has the highest correlation risk (i.e., the highest
Cora)?
Definitely the equity tranche (i.e. the 0-3% tranche) as seen in Figure 10.15
above. We observe that the equity tranche spread changes strongly for a
change in the correlation, since the slope of the equity function (i.e., the Cora)
is highly negative.
15. CDOs and their correlation properties are sometime termed “toxic.” Do you
agree with this view?
Not at all. The correlation properties of the CDO as displayed in Figure 10.15 are very
intuitive. In addition, the correlation risk can be quantified with Cora and Gora and
hedged accordingly. CDOs are not toxic, but traders are, who overinvested in them,
did not hedge the risks, and did not want to take the blame for their irresponsible,
high-risk trading.
Answers to Questions and Problems of Chapter 11:
1. What is hedging?
1) Eliminate the original position. So when a bond was bought, just sell the bond.
2) Hedge the position with a derivative such as a future, swap, or option.
3) Enter into a position that is negatively correlated with the original position.
For example, if the investor bought a Greek bond, the investor can sell a
Spanish bond. This is a good hedge as long as the Greek bond price and the
Spanish bond price are positively correlated.
3. Name two reasons why it is more difficult to hedge correlation risk compared to
equity risk or currency risk.
a) Hedging correlation risk involves two or more assets, since the correlation is
measured between at least two assets.
b) Hedging financial correlation is challenging because there is principally no
underlying instrument that trades in the market and that can be bought or
sold as a hedge.
Yes. If the correlation does not change or does not move in the unfavorable
direction, it would, in retrospect, have been a good idea not to hedge.
5. In a delta hedge, the delta amount of the exposure is sold or bought. Give an
example of delta hedging.
Example 11.1a
An option trader at Goldman Sachs buys a call on IBM. The call option premium
is $10,000 (e.g., the trader bought 1,000 calls with a call premium of C0 = $10).
IBM trades at S0 = 100. The trader decides to delta hedge the IBM price risk of
the option. The delta, derived from an option pricing model such as the Black-
Scholes-Merton model, comes out to 60%. Formally:
C 0.6
ΔC (11.1a)
S 1
where:
∆C = Delta of the call
C = Call price
S = Price of the underlying stock IBM
= Partial derivatives operator
Equation (11.1a) reads: How much does the call price C change if S changes by an
infinitesimally small amount, assuming all other variables influencing the call
price are constant? For practical purposes, the change in S can be approximated
by a change of 1, as done in equation (11.1a).
How much IBM stock does the option trader have to sell to stay delta-neutral,
meaning the option trade has no price risk with respect to IBM?
The option trader has to sell IBM stock in the delta amount, hence 60% of the
option premium of $10,000. Therefore the option trader sells 60 shares at $100
each and receives $6,000. The option trader now has no IBM price risk.3 Let’s
show this.
If IBM increases by 1%, following equation (11.1a) the call price increases by
0.60%. Therefore the profit on the call is:
3
This is true for small changes in the IBM price. If the IBM changes by a large amount, the delta changes and has to
be adjusted.
S1 60 – S0 60 = $101 60 – $100 60 = $60
Hence the option trade is hedged against the price risk of IBM. What the option
trader gains on the call is lost on the hedge, and vice versa.
6. Delta hedges are typically not constant. Give an example of dynamic delta
hedging.
Let’s assume the option trader has hedged his call option exposure as in Example
11.1a in question 5. Now let’s assume that the delta has increased from 0.6 to
0.7.4 In this case the option trader has to increase his hedge to 70%; that is, he
has to sell 70 shares of IBM. Now the option trader is hedged against the price
risk of IBM:
C 0.7
ΔC (11.1b)
S 1
If IBM increases by 1%, following equation (11.1b) the call price increases by
0.70%. Therefore the profit on the call is:
Hence the option trade is hedged against the price risk of IBM. What the option
trader gains on the call is lost on the hedge, and vice versa.
4
An increase in the delta can happen for several reasons: (a) The price of the underlying increases, (b) the implied
volatility decreases, or (c) the maturity of the option decreases.
Principally every instrument that can replicate the correlation risk of the original
trade can serve as a hedge for correlation risk. In particular, we can use all
correlation-dependent options, correlation futures, and correlation swaps.
We should add the correlation exposure of the original trade and the correlation
exposure of the hedge. If the sum of the correlation exposures is close to zero, it
is a good correlation hedge.
9. When an investor has a perfect hedge, doesn’t this mean that the profit
potential is zero?
Yes. The more the correlation exposure is hedged, the lower are both the
potential profit and the potential loss from the hedged position.
10. Generally, when should we hedge with forwards, futures, and swaps, and when
should we hedge with options?
The more confident an investor is that the undesirable event will occur (for
example, a price decline of a bond if the bond was bought), the more
appropriate it is to hedge with a future, forward, or swap. In this case, no option
premium is wasted.
The less confident an investor is that the undesirable event will occur (a price
decline if the bond was bought), the more appropriate it is to hedge with a put
option. In this case, a put option premium has to be paid. However, if the
underlying goes in the right direction (increases in price if the bond was bought),
the investor participates in the bond price increase. The (put) option will expire
worthless.
Answers to Questions and Problems of Chapter 12:
Credit value at risk (CVaR) measures the maximum loss of a portfolio due to credit
risk with a certain probability for a certain time frame.
2) Why don’t we just apply the market value at risk (VaR) concept to value credit risk?
3) Which correlation concept underlies the CVaR concept of the Basel II and III
approach?
Basel II and III apply the simplistic one-factor Gaussian copula (OFGC) model with the
4) In the Basel committee CVaR approach, what follows for the relationship between
the CVaR value and the average probability of default, if we assume the correlation
between all assets in the portfolio is zero?
N -1[PD(T)] ρ N -1(X)
From equation (12.7) CVaR(X, T) N
we can observe that for ρ =
1 ρ
1 ρ
model at www.dersoft.com/CVaR.xlsm.)
6) In the Basel committee CVaR model, the default correlation is an inverse function of
the average probability of the default of the assets in the portfolio. Explain the
rationale for this relationship.
25%
20%
ρ 15%
10%
5%
0%
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24
PD(T)
In Figure 12.3 it is assumed that highly rated companies with a low default
probability have a higher correlation of default since they are mostly prone to
systematic factors such as a recession, in which they default together. However,
companies with a high default probability are more affected by their own
idiosyncratic factors and less by systematic risk; hence they are assumed to be less
correlated.
7) In the Basel committee approach, the required capital to be set aside for credit risk
is the CVaR minus the average probability of default. Explain why.
The required capital RC is derived in the Basel accord with equation (12.8):
As seen in equation (12.8), the capital charge RC is reduced by the expected loss,
which is measured by PD(T). The rationale is that banks cover the expected loss with
their own provisions, such as the interest rate that they charge. Naturally, low-rated
debtors have to pay a higher interest rate on their loans than highly rated debtors
pay.
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
From equation (12.11) we can observe that CVA is derived from the underlying
instruments D+a,c and PDc. Importantly, the two underlyings, D+ and PDc, may be
correlated, which makes CVA a rather complex derivative to evaluate.
10) How can CVA without correlation between market risk and credit risk be calculated?
If we do not assume that market risk and credit risk are correlated, CVA in equation
(12.11) can simply be derived by multiplying the components D+a,c and PDc:
11) Including the correlation between market risk and credit risk, the concept of wrong-
way risk (WWR) arises. What is general wrong-way risk, and what is specific wrong-
way risk?
5
Basel Committee on Banking Supervision (BCBS), “Annex (to Basel II),” www.bis.org/bcbs/cp3annex.pdf, p. 211.
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 the bond price B, since the coupon of the bond price is now lower
Figure 12.7: General wrong-way risk: Decreasing interest rates i lead to higher credit
exposure via a higher bond price B. Decreasing interest rates i in a recession also
mean increasing default probability PDc of the bond issuer. Hence the higher the
exposure, the higher the credit risk (i.e., the higher the risk that the issuer can meet
its obligation to pay coupons and principal).
A bank is exposed to specific wrong-way risk (WWR) if future exposure to a specific
counterparty is positively correlated with the counterparty’s probability of default.6
$M million coupon k
Reference asset
of obligor o
Figure 12.8: Cash flows of an investor i, who has credit exposure to an obligor o,
which is hedged with a credit default swap (CDS) with the guarantor g.
R = Recovery rate
In Figure 12.8, the investor has specific wrong-way risk if there is a positive
correlation between the default probability of the obligor o and the guarantor g (i.e.,
the CDS seller). This means that the higher the default probability of the obligor PDo,
the higher is also the default probability of the guarantor PDg.
In particular, if the default probability of the obligor increases, the market spread of
the CDS increases. Therefore the present value for the CDS buyer increases, since
6
See BCBS, “Basel III: A Global Regulatory Framework for More Resilient Banks and Banking Systems,”
www.bis.org/publ/bcbs189.pdf, p. 38.
the fixed spread s is now lower than the market spread. If the CDS is market to
market, this is nice from a profit perspective, but from a risk perspective it means
that the credit exposure for the CDS buyer i increases.
Also, with increasing default probability of the guarantor, the credit risk increases,
since it is less likely that the guarantor can pay the payoff in default. Hence we have
increased credit exposure together with increased credit risk, constituting specific
wrong-way risk.
b) A further example of specific wrong-way risk (which is mentioned in the Basel III
accord7) is if a company sells put options on its own stock. This is displayed in Figure
12.9:
Figure 12.9: Example of specific wrong-way risk: Deutsche Bank selling a put on its
own stock
Deutsche Bank selling a put on its own stock constitutes specific wrong-way risk,
since the lower the stock price, the more the put is in the money (i.e., the higher is
the credit exposure for the put option buyer with respect to the put option seller,
Deutsche Bank). But the lower the Deutsche Bank stock price, the higher is typically
also the default probability of Deutsche Bank. This means that the higher the credit
exposure (when the put is deeper in the money), the higher the credit risk (the
probability that Deutsche Bank defaults), constituting specific wrong-way risk.
7
Ibid.
13) How does the Basel committee address wrong-way risk?
Basel II and III have a simple approach to address general wrong-way risk and
specific wrong-way risk. A multiplier α is applied to increase the derivatives exposure
D+a,c. The multiplier α is set to 1.4, which means the credit exposure D+a,c is increased
by 40% compared to assuming credit exposure D+a,c and credit risk PDc are
independent, as was expressed in equation (12.12). Banks that use their own
internal models are allowed to use an α of 1.2, meaning the credit exposure is
increased by 20% to capture wrong-way risk. Banks report an actual alpha of 1.07 to
1.1; hence the α of 1.2 to 1.4 that Basel III requires is conservative.
Debt value adjustment (DVA) allows an entity to adjust the value of its portfolio by
taking its own default probability into consideration.
a) An entity like entity a would benefit from its own increasing default probability
PDa, since a higher default probability would increase DVA via equation (12.18),
which in turn increases the value of the derivatives portfolio via equation
(12.19)!
b) Entity a could realize the DVA benefit only if it actually defaults!
Both (a) and (b) defy financial logic. Therefore the Basel accord has principally
refrained from allowing DVA to be recognized. In 2008 several financial firms had
actually reported huge increases in their derivatives portfolios due to DVA. This is no
longer possible.
1. GPU technology originated in the gaming industry and has been modified to solve
complex financial problems. What is the general approach of graphical processing
units (GPUs) to solve financial problems?
The advantages of GPU technology are (a) speed, (b) increasing user-friendliness,
and (c) structural efficiency (i.e., no compiling is necessary). It is debatable whether
GPU technology is also more accurate than applying mathematical algorithms.
a) Although there have been efforts to combine central processing unit (CPU) and
GPU technology, GPUs still have a different architecture and require their own
distinct structure of programming and specialized programming languages such
as CUDA, OpenGL, and OpenCg.
b) GPUs provide efficient and fast solutions for problems that are complex but can
be represented in matrices, since matrix multiplication and manipulation are
easy to execute. However, for nonlinear, path-dependent structures, GPUs may
not be well suited.
Altogether, the GPU technology is a promising new approach to derive fast, real-
time results for complex financial problems, for which analytical solutions are
questionable or not at all available. However, one can argue that if mathematical
techniques are available, although slower, they should have preference over brute-
force nonanalytical iterative search procedures.
5. Artificial neural networks mimic the human brain and therefore have the ability to
learn. How do they learn?
Naturally, neural networks have their drawbacks. First, they are so-called black
boxes; that is, the mathematical algorithm, which optimizes the output, is hidden.
Second, neural networks often have quite a slow convergence rate. Third, and most
important, neural networks can get stuck at local optima, not deriving the general
optimum. These reasons have limited the usage of neural networks in reality.
7. Fuzzy logic is cool since it alters the traditional logic of a statement being either true
or false. What logic does fuzzy logic apply?
For example, a professor asks the question: “What are the prime numbers between
10 and 20?” The correct answer is the set {11, 13, 17, 19}. If a student gives the
answer {11, 13, 17}, traditional logic would argue that the answer is false. However,
fuzzy logic would argue that the answer is partially true, actually 75% true.
8. Which three main concepts of evolution do genetic algorithms apply? Explain them
briefly.
10. Which properties of chaos theory translate well to finance, and which do not?
From our analysis, we find that chaos criteria 3 and 4, the self-similarity principle
and the regime changes, are also found in finance. However, criteria 1, 2, and 5, the
dependence on initial conditions, the short-term but not long-term predictability,
and the deterministic nature of chaos theory are typically not properties in finance.
The critical question is whether criteria 3 and 4 are sufficient to support financial
trading decisions.
11. Which concept does Bayesian logic share with fuzzy logic?
The Bayesian approach reinterprets and extends classical probability reasoning. The
Bayesian theorem is algebraically identical with the classical frequentist probability
theory (called “frequentist” since it draws its conclusions from the frequency of
data). However, the probabilities are reinterpreted. Bayesian probabilities are
related to fuzzy logic, since both apply the concept of partial truth.
12. What are prior probabilities and posterior probabilities in the Bayesian theory?
A prior initial probability, P(A), is the hypothesis before accounting for evidence.
P(A) can be a personal subjective belief, rather than an objectively derived
probability. The posterior probability P(A|B), is the probability of A given that the
evidence B is observed.