Back Matter

Appendix I. Risk Assessment Matrix

article image
article image
article image

Appendix II. Satellite Models

Large Banks

article image

Small and Medium-Sized Banks

article image

Difference between customer loans and deposits in percent of total assets, lagged

Appendix III. Treatment of Fixed-Income Securities

69. The calculation of haircuts on fixed income holdings under different macro scenarios is based on an IMF-developed model for the valuation of sovereign debt using information from CDS markets. Sovereign bond prices for each year under each scenario are calculated using market expectations of default risk as reflected in forward rates on five-year sovereign CDS contracts. Five-year bonds are assumed to be representative of the maturities of banks’ bond holdings. Bonds for which market quotes from Bloomberg were available were selected consistent with the approach taken by CEBS for the 2010 Europe-wide stress test by choosing maturities between 4.5 and 6.5 years.

70. The standard pricing formula for coupon-bearing bonds is reconciled with the zero-coupon bond pricing formula exp((− rT)(1 − LGD x PD(T))) with the cumulative probability of default (PD) and loss given default (LGD), in order to project bond prices contingent on changes in idiosyncratic risk (irrespective of changes in the term structure of yields). Since the sample bonds carry regular coupon payments, the cash flow pricing formula

Pb,Tt=Πk=1Ttc(1+rt)(Tt)/n+f(1+rt)Tt

of the bond b in year t and time to maturity T-t is stripped of coupon payments c (with payout frequency n) and set equal to the quasi-zero coupon price at the last observable sample date, after controlling for changes in market valuation over the course of 2010 in excess of baseline expectations of each country-specific yield-to-maturity according to CEBS. Thus, one can write

Pb,Tt=f(1+rt)Tt=exp((rft+(YTMend2010,baselineYTMend2010,actual)+SCDS10,000)(Tt)),

where rt is the yield in each year, f is the face value and the five-year cash CDS spread sCDS, j = − ln (1 − LGD x PD (t))/T of country j, which replaces the term for default risk in exp((− rT)(1 − LGD x PD(T))).42 Note that the actual end-2010 YTM and the five-year cash CDS spread refer to values observed at end-2010. The equation above is then solved for the risk-free rate rf (before the first forecast year) by maximum likelihood.

71. For all bonds of each sample country, the future bond prices Pb,t,j (up to five years) are calculated by applying the forward five-year sovereign CDS spread F (sCDS,j)t to the modified zero-coupon pricing formula

Pb,t,j=exp((rtT+F(SCDS,j)t/10,000)t)

in order to inform estimates of default risk (and its impact on future bond values relative to end-2010) for each year of the forecast horizon based on the previously estimated risk-free rate rt. This is done for several bonds of each sample country (with a residual maturity T of about five years).

72. More specifically, the past dynamics of expected default risk are used to determine parametric estimates of future haircuts. The monthly variations of forward rates on CDS spreads F (sCDS,j)t between January 2009 and December 2010 are parametrically calibrated as a generalized extreme value distribution with point estimates x^t,a=μ^j+σ^j/ξ^j((1n(a))ξ^j1),where1+ξj(xμj)/σj>0, scale parameter σj > 0, location parameter μj, and shape parameter ξj = 0.5.43 The higher the absolute value of shape parameter, the larger the weight of the tail and the slower the speed at which the tail approaches its limit.44 For the baseline scenario, the median (50th percentile) for x^t,a=0.5 is chosen. Since haircuts under the adverse scenario should reflect the volatility of market expectations, country-specific shocks to F (sCDS, j)t are assumed at the 75th percentile (for the mild “double dip” scenario and the slow growth scenario (“adverse 1”), and 90th percentile (for the severe “double dip” scenario (“adverse 2”) of the probability distribution. Thus, for each year over the forecast horizon, there are three “stressed” bond prices {Pb,t,jbaseline;Pb,t,jadverse1;Pb,t,jadverse2;} based on three different forward CDS rates {F(SCDS,j)tbaseline;F(SCDS,j)tadverse1;F(SCDS,j)tadverse2}.

73. Corresponding haircuts were calculated for each bond from changes in bond prices relative to the base year 2010, using the following specification

ΔPb,t,j=(Pb,t,j/Pb,0,j1)×100,

where Pb,0 is the bond price in the base year.45

74. The haircut h for each sovereign j is calculated as an issuance size-weighted average of individual projected haircuts applied to a K-number of bonds outstanding,46 so that

ht,j=max(Σb=1kΔPb,t,j×Amtb,j(Σb=1kAmtb,j)1,0),

where ΔPb,t,j is the haircut on bond b, and Amtb is the outstanding amount of bond b issued by country j. These haircuts should then be applied to banks’ sovereign bond exposures to countries47 jJ held in both the banking and trading books as of end-2010. The sovereign bond losses or changes in valuation in each year t over the forecast horizon are calculated as ΣjJht,j×exposure0,j, based on a bank’s total exposure to country j at end-2010. Sovereign exposure gains, should they materialize, are ignored for stress test purposes.

Appendix IV. The Contingent Claims Analysis Approach—Standard Definition

75. The CCA is used to construct risk-adjusted balance sheets, based on three principles. The principles are: (i) the values of liabilities (equity and debt) are derived from assets; (ii) liabilities have different priority (i.e., senior and junior claims); and (iii) assets follow a stochastic process. Assets (present value of income flows, proceeds from assets sales, etc.) are stochastic and over a horizon period may be above or below promised payments on debt which constitute a default barrier. Uncertain changes in future asset value, relative to the default barrier, are the driver of default risk which occurs when assets decline below the barrier. When there is a chance of default, the repayment of debt is considered “risky,” to the extent that it is not guaranteed in the event of default (risky debt = risk-free debt minus guarantee against default). The guarantee can be held by the debt holder, in which case it can be thought of as the expected loss from possible default or by a third party guarantor, such as the government.

76. In the first structural specification, commonly referred to as the Black-Scholes-Merton framework (or in short, the “Merton model”) of capital structure-based option pricing theory (Black and Scholes, 1973; Merton, 1973 and 1974), total value of firm assets follows a stochastic process may fall below the value of outstanding liabilities. Thus, the asset value A(t) at time t describes a continuous asset process so that the physical probability distribution of the end-of-period value is

A(Tt)A(t)exp{(rA+σ2A/2)(Tt)+σATtz},

for time to maturity T-t. More specifically, A(t) is equal to the sum of its equity market value, E(t), and its risky debt, D(t), so that A(t) = E(t) + D(t). Default occurs if A(t) is insufficient to meet the amount of debt owed to creditors at maturity, which constitute the bankruptcy level (“default threshold” or “distress barrier”). The equity value E(t) is the value of an implicit call option on the assets, with an exercise price equal to default barrier. It can be computed as the value of a call option E(t) = A(t)Φ(d1) – Be–r(T–t)Φ(d2), with d1=(ln(A(t)/B)+(r+σ2A/2)(Tt))(σATt)1,d2=d1σATt, asset return volatility σA, and the cumulative probability Φ(.) of the standard normal density function. Both the asset, A(t), and asset volatility, σA, are valued after the dividend payouts. The value of risky debt is equal to default-free debt minus the present value of expected loss due to default,

D(t)=Ber(Tt)PE(t).

77. Thus, the present value of market-implied expected losses associated with outstanding liabilities can be valued as an implicit put option, which is calculated with the default threshold B as strike price on the asset value A(t) of each institution. Thus, the present value of market implied expected loss can be computed as

PE(t)=Ber(Tt)Φ(d2)A(t)Φ(d1),

over time horizon T –t at risk-free discount rate r, subject to the duration of debt claims, the leverage of the firm, and asset volatility. Note that the above option pricing method for PE (t) does not incorporate skewness, kurtosis, and stochastic volatility, which can account for implied volatility smiles of equity prices. Since the implicit put option PE (t) can be decomposed into the PD and the LGD,

PE=Φ(d2)(1Φ(d1)A(t)Φ(d2)BerT)Ber(Tt)=PD×LGD,

there is no need to introduce potential inaccuracy of assuming a certain LGD. As a consequence of the assumptions on the underlying asset price process, this would imply the risk-neutral probability distribution (or state price density (SPD)) of A(t) is a log-normal density

ft*(A(T))=ert,TtTt2E(t)B2|B=A(T)=1A(T)2πσ2(Tt)exp[[ln(A(T)/A(t))(rt,Ttσ2/2)(Tt)]22σ2(Tt)]

with mean (rσA2/2)(Tt) and variance σA2(Tt) for ln (A(T)/A(t)), where rt,T–t and f* (.) denote the risk-free interest rate and the risk-neutral probability density function (or state price density (SPD)) at time t, with risk measures

Δ=defE(t)A(t)=Φ(d1)andΓ=def2E(t)A(t)2=Φ(d1)A(t)σTt.

78. In this analysis, the Merton model is refined without altering the analytical form by means of the closed-form Gram-Charlier (GC) model of Backus and others (2004), which allows for kurtosis and skewness in returns and does not require market option prices to implement, but is constructed using the same diffusion process for asset prices.48 The above option pricing method, however, does not incorporate skewness and kurtosis, which can account for implied volatility smiles of equity prices. Thus, the Merton model above is enhanced – without altering the analytical form by means of the closed-form GC model of Backus and others (2004), which allows for kurtosis and skewness in returns based on the same diffusion process for asset prices, and does not require option prices in its calibration. The model is constructed around the Gram-Charlier expansion of the density function of asset changes defined by a standard normal random variable in the Merton model. For default threshold B as strike price on the asset value A(t) of each institution, the price of a European put option can be written as

PE(t)=Ber(Tt)Φ(d2)A(t)Φ(d1)A(t)Φ(d1)σA[γ(t)13!(2σAd1)γ(t)24!(1d12+3dσA3σ2A)]

79. over time horizon T−t at risk-free discount rate r, subject to the duration of debt claims, the leverage of the firm, the sensitivity d1=(ln(A(t)/B)+(rA+σ2A/2)(Tt))/σTt ofthe option price to changes in A(t), d2=d1σATt, and asset volatility σA, and the Gram-Charlier correction for t-period skewness γ1 and kurtosis γ2 in returns based on the same diffusion process for asset prices.49

80. Since the Merton model also contains empirical irregularities that can influence the estimation of implied assets (which also affects the calibration of implied asset volatility), we estimate the SPD of implied asset values using equity option prices without any assumptions on the underlying diffusion process (Box 1 below). Using equity option prices, we can derive the risk-neutral probability distribution of the underlying asset price at the maturity date of the options. We determine the implied asset value as the expectation over the empirical SPD by adapting the Breeden and Litzenberger (1978) method, together with a semi-parametric specification of the Black-Scholes option pricing formula (Aït-Sahalia and Lo, 1998). More specifically, this approach uses the second derivative of the call pricing function (on European options) with respect to the strike price (rather than option prices as identifying conditions). Estimates are based on option contracts with identical time to maturity, assuming a continuum of strike prices. Since available strike prices are always discretely spaced on a finite range around the actual price of the underlying asset, interpolation of the call pricing function inside this range and extrapolation outside this range are performed by means nonparametric (local polynomial) regression of the implied volatility surface (Rookley, 1997).

Estimation of the Empirical SPD

Breeden and Litzenberger (1978) show Arrow-Debreu prices can be replicated via the concept of the butterfly spread on European call options. This spread entails selling two call options at strike price K and buying two call options with adjacent strike prices K = K − ΔK and K+ = K + ΔK respectively, with the stepsize ΔK between the two call strikes. If the terminal underlying asset value A(T) = K then the payoff Z (.) of 1/ΔK of such butterfly spreads at time T − τ (and time to maturity τ) is defined as

Z(A(T),K;ΔK)=price(A(Tτ),τ,K;ΔK)|r=0=u1u2ΔK|A(T)=K,T=0=1,

with

u1=C(A(Tτ),τ,K+ΔK)C(A(Tτ),τ,K)

and

u2=C(A(Tτ),τ,K)C(A(Tτ),τ,KΔK).

C(A, τ, K) denotes the price of a European call option with an underlying asset price A, a time to maturity τ and a strike price K. As ΔK → 0, Price(A(Tτ), τ, K; ΔK) of the position value of the butterfly spread becomes an Arrow-Debreu security paying 1 if A(T) = K and zero in other states. If A(T)+ is continuous, however, we obtain a security price

limΔK0(Price(A(t),τ,K;ΔK)ΔK)|K=A(T)=f*(A(T))ert,τ,

where rt, τ and f* (.) denote the risk-free interest rate and the risk-neutral probability density function (or state price density (SPD)) at time t. On a continuum of states K at infinitely small ΔK a complete state pricing function can be defined. Moreover, as ΔK → 0, this price

limΔK0(Price(A(t),τ,K;ΔK)ΔK)=limΔK0u1u2(ΔK)2=2Ct(.)K2

will tend to the second derivative of the call pricing function with respect to the strike price evaluated at K, provided that C (.) is twice differentiable. Thus, we can write

2Ct(.)K2|K=A(T)=ft*(A(T))ert,ττ

across all states, which yields the SPD

ft*(A(T))=ert,ττ2Ct(.)K2|K=A(T)

under no-arbitrage conditions and without assumptions on the underlying asset dynamics. Preferences are not restricted since no-arbitrage conditions only assume risk-neutrality with respect to the underlying asset. The only requirements for this method are that markets are perfect, i.e., there are no transactions costs or restrictions on sales, and agents are able to borrow at the risk-free interest rate.

81. The implied asset value is estimated directly from option prices (in tandem with an option pricing approach that takes into account higher moments of the underlying asset diffusion process). This avoids the calibration error of using two-equations-two unknowns in the traditional Merton model in solving both implied asset value and asset volatility simultaneously. Thus, asset volatility can be derived from:

σA[1+γ(t)13!(2σAd1)γ(t)24!(1d12+3dσA3σA2)]=E(t)A(t)Φ(d1)σE.
50

Using Systemic CCA to Calculate a Possible Systemic Risk Surcharge

There is an ongoing debate about the advantages and disadvantages of systemic risk capital add-ons or premium-based fees, whether such levies should be charged ex-post or ex-ante, and whether proceeds would go to special funds or to general government revenue.

In the context of presented stability analysis, the Systemic CCA model-derived expected losses can be used to calculate a “fair value” price of a systemic risk surcharge for all sample banks. To illustrate this, the fair value (in basis points) of a risk-based surcharge that would compensate for expected losses arising from systemic solvency risk on an actuarial basis can be written as

1Tln(11TGμ,σ,ξ1(a;Pp,T)ΣjpBjerT)×10,000,

where B represents the aggregate default barrier of all p-institutions in the sample, r is the risk-free rate, T is time horizon of the surcharge, and Gμ,σ,ξ1() is the multivariate density function (with location, scale and shape parameters μ, σ and ξ of individual expected losses Pp, T (equity put option).

As an illustration, using the results obtained from the Systemic CCA analysis above, the estimated average annual “through-the-cycle” systemic surcharge for systemically important financial institutions would at least be 3 basis points (see Table 9), which seems to be in line with the German bank levy, with took into effect on January 1, 2011, as part of the newly adopted Bank Restructuring Act. This charge would be on debt liabilities excluding insured deposits and regulatory capital instruments.51 A reasonable systemic surcharge for systemically important financial institutions during stress periods would be about 30 basis points per year if estimations are based on observations during the recent financial crisis following the collapse of Lehman Brothers in September 2008.

Appendix V. Heuristic Approximation of Contingent Liabilities from the Financial Sector

82. The market-implied expected losses calculated for each financial institution from equity market and balance sheet information using the CCA can be combined with information from CDS markets to estimate the government’s contingent liabilities. The put option value PCDS (t) using CDS prices reflects the expected losses associated with default net of any financial guarantees, i.e., residual default risk on unsecured senior debt and can be written as

PCDS(t)=(1exp((SCDS(t)/10,000)(B/D(t)1)(Tt)))Ber(Tt).

The linear adjustment (B/D (t)–1) is needed if outstanding debt B trades either above (below) par value D, which decreases (increases) the CDS spread sCDS (t) (in bps) due to an implicit recovery rate of the CDS contract at notional value and below (above) the recovery rate implied by the market price D(t). This negative (positive) difference (“basis”) between the CDS spread and the corresponding bond spread represents the ratio between recovery at face value, which underpins the CDS spread calculation, and recovery at market value, which applies to the commensurate bond spread.52

PCDS (t) above is derived by rearranging the specification of the CDS spread

SCDS(t)=(Tt)1ln(1PCDS(t)/Ber(Tt))×(B/D(t)1)×10,000

under the risk-neutral measure, assuming a survival probability

1p¯=exp(0th(u)du)=exp(ht)

at time t with cumulative default rate p, and a constant hazard rate s(t)CDSh. Then PCDS (t) can be used to determine the fraction

α(t)=1PCDS(t)/PE(t)

of total potential loss due to default, PE (t), covered by implicit guarantees that depress the CDS spread below the level that would otherwise be warranted for the option-implied default risk.53 In other words, α(t)PE (t) is the fraction of default risk covered by the government (i.e., its contingent liability) and (1 − α (t))PE (t) is the risk retained by an institution and reflected in the CDS spreads. Thus, the time pattern of the government’s contingent liability and the retained risk in the financial sector can be measured.

83. While this definition of market-implied contingent liabilities provides a useful indication of possible sovereign risk transfer, the estimation of the alpha-value depends on a variety of assumptions that influence the assessment of the likelihood of government support, especially at times of extreme stress during the credit crisis. Some caveats regarding the estimation of expected losses (and contingent liabilities) are in order:

  • Equity prices might not only reflect fundamental values due to both shareholder dilution and trading behavior that obfuscate proper economic interpretation. During the credit crisis, rapid declines in market capitalization of financial firms were not only a signal about future solvency risk, but also reflected a “flight to quality” motive that was largely unrelated to expectations about future firm earnings or profitability. Assuming that CDS pricing was efficient, the definition of the alpha-value would erroneously flag implicit government support due to extremely low equity valuations but not as a result of depressed CDS spreads (in expectation of possible guarantee to short-term creditors). However, empirical evidence does not concur with such a “denominator effect” of equity prices on the alpha-value. For the given sample, a high cointegration and weaker negative correlation between equity prices and CDS spreads during stress periods suggest consistent co-movement but lower sensitivity of CDS spreads to changes in default risk over time amid rapidly declining levels of market capitalization.

  • The equality condition of default probabilities derived from equity prices and from CDS spreads eliminate the possibility of positive alpha-values. Carr and Wu (2007) and Zou (2003) show that for many corporations the put option values from equity options and CDS are closely related.54 Arbitrage trading between both price shows in the synthetic replication of credit protection on guaranteed bonds using equity (i.e., a long position in an equity option “straddle” combined with a short CDS position). However, in stress situations, the implicit put options from equity markets and CDS spreads differ in their price sensitivity to the impact of changes in the underlying capital structure on the implicit default probability and, thus, should be priced differently.

Appendix VI. The Systemic CCA Methodology—Calculating the Systemic Worst-Case Scenario Using Multivariate Generalized Extreme Value

84. The Systemic CCA framework (Gray and Jobst, 2010; Gray and Jobst, forthcoming) is predicated on the quantification of the systemic financial sector risk. It is applied in this context to generate a multivariate extreme value distribution (MGEV) that formally captures the potential of tail realizations of market-implied joint expected losses. The analysis of dependence is completed independently from the analysis of marginal distributions, and, thus, differs from the classical approach, where multivariate analysis is performed jointly for marginal distributions and their dependence structure by considering the complete variance-covariance matrix, such as the multivariate MGARCH approach. We first define a nonparametric dependence function of individual expected losses. We then combine this dependence measure with the marginal distributions of these individual expected losses, which are assumed to be generalized extreme value (GEV). These marginal distributions estimated via the LRS method, which identifies possible limiting laws of asymptotic tail behavior of normalized extremes (Coles and others, 1999; Poon and others, 2003; Stephenson, 2003; Jobst, 2007). The dependence function is estimated iteratively on a unit simplex that optimizes the coincidence of multiple series of cross-classified random variables – similar to a Chi-statistic that measures the statistical likelihood of observed values to differ from their expected distribution.

85. More specifically, we first specify the asymptotic tail behavior of the vector-valued series Xi, jPi, j = (P1n,...,Pmn) of expected losses (i.e., put option values) of an m number of financial sector entities j as the limiting law of an n-sequence of normalized maxima (over rolling window estimation period of τ=120 days and daily updating), so that the jth univariate marginal

yj=yj(x)=(1+ξj(xμj)/σj)+1/ξj(forj=1,...,m)

lies in the domain of attraction of the generalized extreme value (GEV) distribution, where 1+ξj (xμj)/σj > 0, scale parameter σj > 0, location parameter μj, and shape parameter ξj. The higher the absolute value of shape parameter, the larger the weight of the tail and the slower the speed at which the tail approaches its limit.

86. Second, the multivariate dependence structure of joint tail risk of expected losses is derived nonparametrically as the convex function

A(ω)=min(1,max{n{Σi=1nj=1myi,j/y^jωj}1,ω,1ω}),

over the same estimation window, where y^j=Σi=1nyi,j/n and

0 ≤ max(ω1,...,ωm–1)≤ A(ωj)≤1 for all 0≤ ωj ≤1, subject to the optimization of the (m-1)-dimensional unit simplex

Sm={(ω1,...,ωm1)+n:ωj0,1jm1;Σj=1m1ωj1andωm=1Σj=1m1ωj}.

87. Finally, after estimation of the marginal distributions and the dependence structure over the a rolling window of τ number of days, we obtain the multivariate distribution

Gt,ξ^,μ^,σ^(x)=exp{(Σj=1myj)A(ω)}

at time t = τ + 1, using the maximum likelihood estimation θ^MLE=argθmaxΠi=1ng(x;θ).

88. We then obtain the Expected Shortfall (ES) (or conditional VaR) as the probability-weighted residual density beyond a prespecified statistical confidence level (say, a=0.95) of maximum losses, where point estimate of joint expected losses is defined as55

x^t,a=Gt,ξ^,μ^,σ^1(a)=μ^j+σ^j/ξ^j((ln(a)A(ω))ξ^j1).

ES defines the average estimated value z of the aggregate expected losses over estimation days τ in excess of the statistical confidence limit. Thus, we can write ES at time t as

ESt,τ,a=E[zτ|zτGt1(a)=VaRt,a]

at a threshold quantile value

VaRt,τ,a=sup{Gt1()|Pr[zτ>Gt1()]a=0.95}.

ES can also be written as a linear combination of individual ES values, where the relative weights (in the weighted sum) are given by the second order cross-partial derivatives of the inverse of the joint probability density function Gt1(a) to changes in both the dependence function and the individual marginal severity of expected loses. Thus, by re-writing ESt, τ, a above, we obtain the sample ES

ESt,τ,a=ΣjmMCj,τ,aE[zj,τ|zm,τGt1(a)=VaRt,qa],

where the relative weight of institution j is defined as the marginal contribution

MCj,τ,a=2Gt1(a)yj,aA1(ω)s.t.ΣimMCj,τ,a=1andMCj,τ,azi,τzm,τ

to expected shortfall

ESt,l,a=Σjmψj,l,aE[zj,l|zm,lGt1(a)=VaRt,qa],

attributable to the joint effect of both the marginal distribution yj, a and the change of the dependence function A(.) absent institution j.

1

Recent capital measures by large German banks focus on those exceptions.

2

The key role of interest rate income has been documented in Deutsche Bundesbank’s 2010 Financial Stability Review.

3

According to Bundesbank’s latest Financial Stability Review, German banks would need to raise about €50 billion to adjust to the impending economic and regulatory conditions.

4

The analysis used bank-by-bank data published by CEBS in July 2010 and aggregate information for the German banking system as of end-2010.

5

The note was prepared by Andreas Jobst and Christian Schmieder.

6

Changes in the eligibility of capital under Basel Ill during the forecast horizon were taken into account based on publicly available data.

7

As a caveat, it has to be taken into account that confidence levels become wider with longer time horizons.

8

Oil prices have already gone up significantly, and additional geopolitical conflicts in the Middle East could trigger an additional increase in oil prices towards levels of $200, which was the underlying assumption.

9

In historical terms (i.e., referring to the last 30 years and accounting for the German unification), the simulated shock corresponds approximately to a shock of 2 standard deviations (cumulative deviation of 5 PPs during the first two years compared to a historical average of 2.5 percent (over two years)).

10

Specifications with bank-specific portions of exposure to broad sectors (corporate, financial institutions, public sector) showed limited differences across sectors.

11

For the large banks, a model using net interest income was employed.

12

This statistical mapping technique via a high-dimensional parametric fit function to align GDP with trading helped reduce discontinuity of the matched series.

13

The dividend payout ratio is defined as the percentage of “dividend payable in a year” to “net income during the year.”

14

The pay-out ratio for the Sparkassen is lower, but if one includes social spending then 40 percent becomes a valid benchmark.

15

In case of a bank-level loss ratio of 1 percent, for example, the expected loss for bank exposure would be set at 0.25 percent and for corporate at 1.5 percent. For a bank with a loss level of 2 percent, the asset class specific expected losses would be twice as high.

16

The leverage ratio was not explicitly taken into account, thereby accounting for the fact that it will only come into effect by 2018. See http://www.bis.org/press/p100912.htm.

17

Behavioral adjustments to reduce RWAs do not necessarily denote deleveraging, but will also happen with a reduction of activities that consume a substantial amount of capital (i.e., risky activities), in line with the purpose of Basel III.

18

For the universe of banks participating in the QIS 6, the increase of RWAs was computed to be 23 percent for large (Group 1 banks) and 4 percent for smaller banks (Group 2 banks).

19

Hence, total eligible capital was reduced by about 3 percent each for the large banks in 2013, 2014, and 2015.

20

Basel III envisages the phase out of certain items (such as deferred tax assets and minority interests) from Core Tier I capital (BCBS, 2010a).

21

It was assumed that 25 percent of the cross-border bank exposure to the GIIPSB countries (as reported by Bundesbank) are medium and long-term debt securities. Official data on this ratio are not available, but indications are available from banks’ annual reports and other public data sources (aggregate data). The assumption is considered conservative on the system level, but for specific banks the portion could be higher.

22

That is, the 14 large banks and the other two Landesbanken.

23

The group of Sparkassen and cooperative banks does not include the banks that were classified as large banks.

24

However, all action to increase capital was taken into account, including, for the supplementary tests based on core tier 1 capitalization, that two large banks have increased their capitalization during the EBA tests.

25

The scenario doubles the macroeconomic severity of the CEBS stress test, i.e., 2.6 standard deviations in historical terms.

26

Sovereign CDS rates for these countries can be used to derive estimates of market expectations of sovereign “haircuts.” The “haircuts” used here are set at the 75th percentile of the distribution of these market expectations, starting from the 2010Q4 average CDS rates.

27

To illustrate the sensitivity of results, a hypothetical severe write down of 60 percent of sovereign claims on Greece, Ireland, and Portugal is estimated to cost these banks €42 billion in 2011, and €36 billion by 2015; the hypothetical additional Tier 1 capital shortfall would be €14 billion.

28

See Gray and Jobst (2010 and forthcoming) as well as Gray and others (2010).

29

Shareholders also have the option to default if their firm’s asset value (“reference asset”) falls below the present value of the notional amount of outstanding debt (“strike price”) owed to bondholders at maturity. So, corporate bond holders effectively write a European put option to equity owners, who hold a residual claim on the firm’s asset value in non-default states of the world. Bond holders receive a put option premium in the form of a credit spread above the risk-free rate in return for holding risky corporate debt (and bearing the potential loss) due to the limited liability of equity owners.

30

The value of the put option is subject three principles: (i) the values of liabilities (equity and debt) are derived from assets; (ii) liabilities have different priority (i.e., senior and junior claims); and (iii) assets follow a stochastic process.

31

The Merton model has shown to consistently underpredict spreads (Jones and others, 1984; Ogden, 1987; Lyden and Saranti, 2000), with more recent studies pointing to considerable pricing errors due to its simplistic nature (Eom and others, 2004).

32

This approach here is an alternative to other proposed extensions aimed at imposing more realistic assumptions, such as the introduction of stationary leverage ratios (Collin-Dufresne and Goldstein, 2001) and stochastic interest rates (Longstaff and Schwartz, 1995). Incorporating early default (Black and Cox, 1976) does not represent a useful extension in this context given the short estimation and forecasting time window used for the CCA analysis.

33

Note that a bank’s CDS spread captures only the expected loss retained by the bank after accounting for the implicit government guarantee.

34

The contribution to systemic risk is derived as the partial derivative of the multivariate density relative to changes in the relative weight of the univariate marginal distribution of individual expected losses at the specified percentile. More specifically, the total expected shortfall can be written as a linear combination of expected shortfalls of individual banks, whose relative weights (in the weighted sum) are given by the second order cross-partial derivatives of the inverse of the joint probability density function to changes in both the dependence function and individual expected losses. Since point estimates of systemic risk are derived from a time-varying multivariate distribution, it is more comprehensive than the current exposition of both CoVaR (Adrian and Brunnermeier, 2008) and Marginal Expected Shortfall (Acharya and others, 2009) (as well as extensions thereof, such as Huang and others, 2009).

35

This charge is intended to support the reorganization of systemically important activities of distressed institutions, but not to absorb losses from activities subject to ordinary insolvency proceedings. After considering the time-variation of expected losses (and their distributional behavior), it would be possible to devise a counter-cyclical surcharge by combining estimates at different percentile levels of statistical confidence.

36

The levy is capped at 15 percent of unconsolidated income over the assessment period (i.e., the previous year), which materially reduces the payment amount for the largest banks and mitigates the procyclical impact in times of low profitability.

37

The analysis was based on daily data from January 1, 2005, to end-January 2011. Key inputs used were the daily market capitalization of the two largest firms (from Bloomberg), the default barrier (Appendix IV) estimated for each firm based on quarterly financial accounts (for all sample firms), and the one-year CDS spreads from Markit.

38

The level of outflows of liabilities and the liquidity of assets under stress was set in accordance with empirical evidence, assumptions used in other FSAPs and upcoming regulatory changes.

39

For further information see Schmieder and others (forthcoming).

40

Other assumptions were (i) share of high quality liquid assets needed to satisfy margin calls: 10 percent; (ii) Market value change of derivatives (20); (iii) share of asset-backed securities maturing within the next 30 days (10); (iv) share of undrawn but committed liabilities that are drawn (50); and (v) share of assets reinvested (80).

41

Recent (unpublished) work by Bundesbank shows that the margins of deposits rates relative to the riskless level has dropped by 75 bps since the onset of the financial crisis (i.e., 2009).

42

Note that this is a simplified CDS pricing formula, which does not take into account the valuation effects of credit events between (quarterly/semiannual) CDS premium payment dates.

43

The upper tails of most (conventional) limit distributions (weakly) converge to this parametric specification of asymptotic behavior, irrespective of the original distribution of observed maxima (unlike parametric value-at-risk (VaR) models).

44

All raw moments are estimated by means of the Linear Combinations of Ratios of Spacings (LRS) estimator.

45

Note that the haircut estimation is not fully accurate, because in each year over the projected time horizon, the projected YTM is imposed on an unchanged set of bonds. This implies no new government issuance (and time-invariant coupon), which overstates the actual haircut (unlike in cases when the sample of bonds changes and the remaining maturity is kept constant over the projected time period).

46

Haircuts cannot take negative values when price appreciation occurs between years (e.g., in response to “safe haven flows”).

47

Austria, Belgium, France, Germany, Greece, Ireland, Italy, Luxembourg, the Netherlands, Portugal, Spain, Sweden, Switzerland, the UK, and the US.

48

Further refinements of this model would include various simulation approaches at the expense of losing analytical tractability. The ad hoc model of Dumas, Fleming, and Whaley (1998) is designed to accommodate the implied volatility smile and is easy to implement, but requires a large number of market option prices. The pricing models by Heston (1993) and Heston and Nandi (2000) allow for stochastic volatility, but the parameters driving these models can be difficult to estimate. Many other models have been proposed, to incorporate stochastic volatility, jumps, and stochastic interest rates. Bakshi and others (1997), however, suggest that most of the improvement in pricing comes from introducing stochastic volatility. Introducing jumps in asset prices leads to small improvements in the accuracy of option prices. Other option pricing models include those based on copulas, Levy processes, neural networks, Generalized AutoRegressive Conditional Heteroskedasticity (GARCH) models, and nonparametric methods. Finally, the binomial tree proposed by Cox, Ross and Rubinstein (1979) spurned the development of lattices, which are discrete-time models that can be used to price any type of option—European or American, plain-vanilla or exotic.

49

The advantage of the GC model is that it is only slightly more complicated to implement than the Merton model because only two additional parameters—skewness and kurtosis—need to be estimated. The disadvantage is that it is assumes that these parameters are constant.

50

The two-equations-two-unknowns approach is based on Jones and others (1984), which was subsequently extended by Ronn and Verma (1986) to a single equation to solve two simultaneous equations for asset value and volatility as two unknowns. Duan (1994), however, shows that the volatility relationship between implied assets and equity is redundant if equity volatility is stochastic. An alternative estimation technique for asset volatility introduces a maximum likelihood approach (Ericsson and Reneby, 2004 and 2005), which generates good prediction results.

51

After considering the time-variation of expected losses (and their distributional behavior), it would be possible to devise a counter-cyclical surcharge by combining estimates at different percentile levels of statistical confidence.

52

We approximate the change in recovery value based on the stochastic difference between the standardized values of the fair value CDS (FVCDS) spread and the fair value option adjusted spread (FVOAS) reported by Moody’s KMV (MKMV). Both FVOAS (FVCDS) are credit spreads (in bps) over the London Interbank Offered Rate for the bond (CDS) of a particular company, calculated by MKMV’s valuation model based on duration (term) of t years (where t=1 to 10 in one-year increments). Both spreads imply an LGD determined by the industry category. In practice, this adjustment factor is very close to unity for most of the cases, with a few cases where the factor is within a 10 percent range (0.9 to 1.1).

53

Note that the estimation assumes a European put option, which does not recognize the possibility of premature execution. This might overstate the actual expected losses inferred from put option values in comparison with the put option derived from CDS spreads.

54

Carr and Wu (2007) find that equity options used in a modified CCA seem to produce risk-neutral default probabilities (RNDP) matching fairly closely RNDPs derived from CDS (sometimes higher, sometimes lower, and differences seem to predict future movements in both markets). Zou (2003) finds that divergences of default probabilities derived from equity options used in CCA model and CDS disappear or revert driven by capital arbitrage relationships and trading impacts. The paper by Yu (2006) uses a less sophisticated model based on CreditGrades, which contains some simplifying assumption that are currently being revised by RiskMetrics.

55

ES is an improvement over VaR, which, in addition to being a pure frequency measure, is “incoherent”; i.e., it violates several axioms of convexity, homogeneity, and subadditivity found in coherent risk measures. For example, subadditivity, which is a mathematical way to say that diversification leads to less risk, is not satisfied by VaR.