Martin Goldberg, Citigroup June, 2006 Page 1 of 45 ECONOMIC CAPITAL FOR TRADING MARKET RISK PRESENTED TO:Risk Magazine's Training Course: Economic Capital

Martin Goldberg, Citigroup June, 2006 Page 1 of 45

ECONOMIC CAPITAL FOR TRADING MARKET RISK

PRESENTED TO: Risk Magazine's Training Course: Economic Capital

BY: Martin Goldberg, [email protected] of Model ValidationRisk ArchitectureCitigroupNew York, New York

DATE: June 29 - 30, 2006

PLACE: New York City


Caveats

The views expressed in this talk are my own, and may not be at all similar to the Citigroup methodology.

Which calculations or methodologies are allowed by the regulators for Basel II is not the subject of this talk. This is my personal opinion of the best approach.


Outline

● What is Economic Capital?

● Risk factors versus historical simulation

● Estimating economic capital using scaled VaR

● Estimating economic capital based on stress testing or Extreme Value Theory

● Correlation and contagion

● Integration of market and credit risks – assessing the various parametric and copula approaches and the difficulties associated with them


Outline








Regulatory Capital and Economic CapitalCapital is a “Rainy Day Fund” to ensure your institution does

not go bankrupt in a bad periodPotential unexpected loss, over a specified time horizon, at a

specified confidence level.Regulatory Capital = VaR Economic Capital - 99% ten day

• Worst 2 weeks of 4 yearsBasel 2 Economic Capital - 99.9% one year

• Worst year of the millenium - somewhat hard to backtest

(Citigroup AA Standard) - 99.97% one year• Based on the .03% transition probability from AA-> D• Third worst year in 10,000• Very hard to backtest with actual historical data

What was the effect on your bank of the invention of agriculture? Of writing?

What happened to LIBOR when Rome was sacked?


-37.00-38.00-39.00-40.00-41.00-42.00-43.00-44.00-45.00-46.00-47.00-48.00-49.00-50.00-51.00-52.00-53.00-54.00-55.00-56.00-57.00-58.00-59.00-60.00-61.00-62.00-63.00-64.00-65.00-66.00

Probability Distribution of Potential Credit Loss for a Portfolio of Many Obligors

0.0%

0.5%

1.0%

1.5%

2.0%

2.5%

3.0%

0-20-40-60-80-100-120-140-160

Potential Credit Loss ($mm)

Pro

bab

ilit

y o

f C

red

it L

oss

A Reminder: EC Definition And Capital HorizonInsolvency Risk rather than Undiversified Volatility of Returns

EC measures risk from an insolvency or debt holders perspective (potential loss of value) rather than from an equity investment perspective (undiversified volatility of returns).

Insolvency risk is calculated by comparing EC to the Available Financial Resources of bank(i.e. AFR = book capital – goodwill + fair market adjustment).

Here is an example of EC for a loan portfolio(not market risk, but a good example anyay):

Expected Loss

Loss atvery high CL

Economic Capital

Economic capital

= Unexpected Loss

= Loss at very high CL – Expected loss.

Expected loss should be covered by reserves and/or pricing.


Timing Issues In Measuring EC for Market Risk

Capital Horizon (TCap): The time horizon over which Economic Capital is assessed.

One year forBasel II capital. Liquidity (Defeasance) Holding Period (Tliq):

Time over which portfolio is held before it can be defeased.

If TCap > Tliq , then need to analyze how risk is taken over multiple holding periods. Rolled over, held fixed, or other.

If portfolio is not held constant, is there serial dependency?

Incremental Annualized Risk Is the market assumed to be stationary? If so, you can scale up

by The difference, assuming no structural changes, between the

annualized risk scaling up a very short holding period and the annualized risk scaling a longer holding period. For example, what is difference in annualized EC between:

1)250 independent draws of 1-day risk2)250 dependent draws of 1-day risk with one of the days

disastrous3)5 independent draws of 50-day risk

T


Potential Assumptions Underlying Different

Methods of Annualization Constant Position means that EC is calculated over the capital horizon under the assumption that the position is kept fixed (or that in one year the portfolio will have the identical composition, including time to maturity, of today’s portfolio). This is the typical assumption for EC for credit risk.

Constant Risk means that the positions in each holding period can materially change but that the level of risk of each holding period is assumed constant over the entire capital horizon.

Constant Level of Risk with Knock-Out Option means there are established policies and procedures that mandate that market risk limits will be decreased for some long period of time if cumulative losses exceed a threshold. (Serial dependency)

Changed Market means the distribution of market-related P/L in period T+1 conditional on disastrous losses in period T is different from the unconditional distribution.

Many Large Losses vs. Huge Loss in One Tliq


Are Trading Desks an ongoing business? Mutually exclusive assumptions

1. The desks are not going to reduce their level of risk taking because of one shock in market rates. However, the positions will change often, as is usual in trading.

2. Market risk limits would be materially reduced for the remainder of the year if the cumulative trading losses exceeded some level.

3. After the 99.9% worst loss, the desk will be shut down.

Assume Gaussian P/L Distribution, or Fat Tails, or Skew? If Gaussian, then 99.9%ile is 1.33 times 99%ile. If not Gaussian, then

what? Checkable for 1 day holding periods - 99.9%ile is worst trading day in

4 years. What was shape of P/L tail historically from 99% to 99.9%?

Assume Efficient Markets and Ignore Customer Flow Revenue We assume an efficient market, i.e. the expected change in value of

any end-of-day position is the time decay . We also ignore customer/counterparty revenues in our calculation of Economic Risk Capital. Consequently the expected trading revenue is each day.

Assume All Positions Equally Liquid This makes scaling uniform for the whole book. Alternatively one

could have subportfolios split by Tliq

Assumptions Underlying Annualized EC For Market Risk


Outline








Analogy to Value at RiskThere are only 3 ways in the literature to calculate

VaR for a real-sized portfolio:Variance-covariance - No bank uses this since it cannot

handle options

Historical simulation • Captures all non-linear correlations, fat tails, and

autoregression• Non-parametric• No clue as to how to extrapolate beyond sample size• Assumes stationarity

Monte Carlo• User must decide which non-linear correlations, fat tails,

and autoregression to capture, and how to model them• Parametric• Easy to extrapolate beyond sample size• Assumes stationarity


Amount of Historical Data NeededHistorical Simulation

VaR regulatory minimum 1 year ~ 250% coverage for a 99th percentile

Infer then that a 99.9th percentile needs 10 years daily data to get same 250% coverage.

This raises major issues of stationarity. Examples:• Was the equity market dynamics the same in the mid-1990’s

at the high point of the tech boom?• Should one correct corporate spreads for changes in the

economic cycle? Harder to find suitable proxies from that long ago.

• Company XYZ before downgrade is an unsuitable proxy for same company XYZ today

• Pre-€ dataMonte Carlo

Stationarity less serious an issue since parametric form allows extrapolation beyond observed quantiles using only recent data

Is the parametric form appropriate for the fat tail effects (EVT)? Is the parametric form appropriate for non-linear correlation

(contagion)?


Number of time series needed for VaRIn a variance-covariance Value-at-Risk calculation, a key

component is a correlation matrix. In a historical simulation, a key component is proxying one time series with some time series assumed to be highly correlated with it.

A daily returns covariance matrix calculated from a year of daily returns has rank at most 252, no matter how many timeseries you put in.

Random Matrix Theory [1], [2] describes the principal components you would get if the timeseries were pure noise. How does it compare to the eigenvalues of, say, the S&P 500 correlation matrix? About 94% of the spectrum is random, leaving the

meaningful signal a rank of about 30.The rest of the data is probably noise with little or no

out-of-sample predictive power.At the 99.9% level it is likely that even fewer distinct

signals are present in the noise.


Number of time series needed at the 99.9% level

MC: Simulations are done by assuming some multivariate process for the underlying market factors, calibrating that process to historical data, and running scenarios (Citi uses 5000) of that process.

Historical simulations are done by collecting time series of the underlying market factors, and treating each observed daily multivariate change as a scenario - with N day’s data there are N-1 daily scenarios, but only (N/10)-1 independent fortnightly scenarios, and only (N/252)-1 independent annual scenarios.

In both, each instrument is valued in each scenario as a (possibly very model-dependent) functional of the underlying market factors.

A large bank should expect to have on the order of 50,000 - 200,000 historically observed market factors, but many of these add no new information at the 99.9% level.

MC: An eigen-analysis of the factor loadings, principal components, or some other estimate [3] should be compared with a random matrix to see how many of them are independent at this extreme level - probably very few.

Historical: I have no good suggestions here - the point of historical simulation is to avoid the sort of modeling described above for MC, and many markets are not comparable today to where they were in the 1980’s or 90’s, or may not even have existed then.


Number of time series needed at the 99.9% level Received wisdom is that correlations between stocks goes to one in a crash.

One example of the studies of correlation in a crash is [4]. Note that they always use ordinary Pearson correlations.

If one assumes a CAPM-type beta, On an average fortnight,

• beta times the market moves each stock by, say, 1% • Alpha moves each stock by a few percent uncorrelated to the market

In a 99.9% worst-fortnight crash,• Same beta times the crashing market moves each stock by, say, 26%• Alpha moves each stock by a few percent uncorrelated to the market• All the stocks have gone down 26%± 2%• It looks like contagion but the dynamics have not changed

This implies that idiosyncratic movement relative to the broader markets may be unimportant at the 99.9% or 99.97% level in a diversified portfolio.

MC: Use fewer time series, only broad markets, but model them more carefully.

Historical: Use broad markets as proxies, but get longer time series. Beware of non-stationarity.

Broad market aggregates are more likely to have longer time series. The above should also apply to most other asset classes, but don’t just take

my word for it.


Outline








EC_Market_Risk = K1 * K2 * VARTLiq

TABLE OF COMBINED SCALING FACTORS: K1*K2For different combination of holding period and CL

Confidence Level

CL = 99.90%(Basel)

K1 = 1.33

CL = 99.97%(Economic Capital)

K2 = 1.48

TLiq

1 dayK2 = 15.8

21.0 23.3

10 daysK2 = 5

6.6 7.4

NH daysK2 =

sqrt(250/NH)

21/sqrt(NH) 23.3/sqrt(NH)

If Basel were calculating an annualized Risk Capital for market risk at the 99.9%CL the formula to transform a 99%, 10-day VAR into a 99.9% annualized Risk Capital would be:

Annualized RC = 6.6 * VAR_10day_99%CL

Implicit in the current Basel scaling factor of 3 is a short Risk Capital Time Horizon:

TCap Basel = (3/6.6)2 yr TCap Basel = 21% of year or 2.5 months.

Transforming VAR99%CL From TLiq To An

Annualized EC at Different CL Assuming Stationarity and Gaussianity

Transform from:

99%CL Targeted Conf Level

TLiq One Year


Actual Example - S&P 500 daily since 1984

Is the data Gaussian?

Daily P/ L Histogram for S&P

0

500

1000

1500

% Loss per day

99.97% Worst loss = 20.5%/ day

99.9% worst loss = 6.7%/ day

99% worst loss =2.6%/ day

from 99 from 99.9 from 99.97 from data2.6 5.0 13.9 2.63.5 6.7 18.4 6.73.8 7.5 20.5 20.5


Actual Example Continued - Sqrt(T) 10 business days

Ten-day loss Histogram

% Loss

99.9% ten-day loss= 25.9%

99% ten-day loss= 7.4%


Actual Example Continued - Sqrt(T) 63 business days

S&P Quarterly P/ L Histogram

99th percentilequarterly loss= 26%

Scaling from Scaling from Actual99% 1-day 99.9% 1-day Data

99% 10-day 8.2 15.9 7.499.9% 10-day 10.9 21.2 25.999% 63-day 20.6 40.0 26


Indication

Based on the one time series above, it seems

sqrt(T) scaling is not that bad in this toy problem,

but the tails are much fatter than Gaussian


Outline








Stress test estimatesAnnual market data for the entire past millenium does not existThe nature of the market has clearly changed since Leif Ericson

and his Viking crew discovered VinlandOnce-per-thousand-year events are unlikely to be similar to

each otherOne possibility is to be purely subjective, and construct

plausible scenarios of these hypothetical disaster years, tuned to your firm’s current holdings

Involve the firm’s economists and analysts in this exercisePros:

As many disaster scenarios as you wantCan cover many kinds of market meltdownLimited only by your firm’s imagination

Cons:The regulators may not be happy with this method -

too disconnected with actual experiencesEasy to game this system to get the results you wanted


Extreme Value TheoryEVT says that, for quantiles outside the observation data set,

the marginal (one asset) pdf can take only 3 different shapes:Gaussian decayAbrupt cutoff (Gumbel)Power-law decay (Frechet)

Most financial time series have Frechet tails. Estimate the exponent, and fit it with the bulk of the distribution, and you are done. You can use this to predict the 99%ile (VaR), the 99.9%ile (Basel 2), or the 99.97%ile (economic capital for a AA firm.) The result will differ from scaling up using the assumption of a multivariate normal density

At a low enough quantile you can backtest the tail parameter to show it fits reality for the data you have

Extrapolate out to the data you do not haveUse small enough time series to avoid large non-stationaritiesAssociation between different time series discussed later in

this talk - not same as their correlation in the bulk


Extreme Value Theory IIFinance is now using EVT - see [5]Still assumes stationarity, but you can restrict your estimator

to data since the last structural market changeCitigroup now uses EVT to estimate some forms of Economic

Capital


Outline








What is contagion?Suppose we have a “toy” world with only 2 assets The changes in each asset follow a generalized jump-diffusion

nContagatioddEdE

nCorrelatiodwdwEdwE

ddwtssdttds

ddwtssdttds

)()(

)()(

(..),...),,(,...)(

(..),...),,(,...)(

211

211

22212222

11121111


Measures of AssociationIt is useful to separate out (parametric) estimation of the

univariate (marginal) probabilities from the connection between time series.

The common way to do this is by using Copula theory.


Quick Introduction to Copula TheoryCopula theory[6],[7] is a generalization of the concept of

correlation.A copula is expressed as a quantile of the distribution in N

dimensions.A copula in one dimension is a tautology - x% of the data are at

or below the x% quantile.The one-dimensional copula density, also called a marginal,

is a uniform distribution from 0 to 1, with the copula density of each point being its quantile in the data series.

The copula is related to the copula density by

),(

),Pr(),(),(

)0,0( VUc

vVuUvuCvu


Properties of CopulasAny multivariate density can be expressed as a copula

connecting marginal densities. The copula and the marginals are completely separate - any

marginal pdf’s can be connected by any valid copula.For distributions with continuous marginals the copula is unique.The average of two copulas may not be a copula, but the average

of two copula densities is a copula density.A Gaussian copula connecting two Gaussian marginals is a

Pearson (ordinary) correlation.Pearson Correlation is not a good measure if the copula is not

Gaussian or any marginal is not symmetric.The rank correlation is a non-parametric copula equivalent, for

any distributions, of Pearson correlations for multivariate normals. Spearman rho is easier to compute than Kendall Tau.Spearman rank correlation in 2-D is the Pearson

correlation between ranks of the entries of the 2 data series.

Easy to compute in Excel using the RANK() function.


An Example

In two dimensions, the copula density of changes in USD Libor and JPY Libor looks like

Note that it is not actually continuous, since some days are unchanged.

3m Libor Weekly Changes Copula Density Rank Correlation 10.6%

0.0%

20.0%

40.0%

60.0%

80.0%

100.0%

0.0% 20.0% 40.0% 60.0% 80.0% 100.0%

JPY Libor

US

D L

ibor


Actual changes - Corelation = -7%

Scatterplot of 5300 Daily changes

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

-0.3 -0.2 -0.1 0 0.1 0.2

SPX

LIB

OR


Rank CorrelationsIn the absence of jumps, any amount of skewness or kurtosis

leaves the rank correlation between time series unchanged.This is one of the best reasons to use copula theory.Rank correlations are very little extra effort compared to

Pearson correlations, and do not assume Gaussian marginals.Enhancing your model of the univariate marginals does not

require redoing the copula / rank correlation matrix.Non-Gaussian copulas can capture tail dependence and not

confuse it with the tail shapes.


Rank Plot of SPX and LIBOR DataRank correlation = -2%

Rank Scatterplot

SPX

US

D L

ibor

Possible banana-shaped copula density?


Tail DependenceThe Upper Tail Dependence is defined for any

copula density c(U1,U2) as

By flipping the < to > and replacing the u with 1-u, we can get a Lower Tail Dependence

More generally, in N dimensions, the hypercube has 2N

corners, so we can define 2N Tail DependencesTail Dependence is my preferred definition of contagion.A Gaussian Copula has zero tail dependence - a very extreme

move in one dimension is never simultaneous with a similarly big move in any other dimension.

)(u

)1/(),Pr(1

),()( 21 uuUuU

uuuC

u


Distinguishing Correlation from Contagion

Non-Linear CorrelationContagion can be defined as a significant difference in the association between large moves (tail events) relative to the association between smaller moves (ordinary days). This is Tail Dependence. As an example study, here is a test of contagious increases in Pearson correlation between Brent oil and kerosene, using 215 pairs of weekly historical spot data. Since you always should use a control, I have also used 215 pairs of random numbers with the same correlation of 63%.

Random fake normals correlated 62.82%

-4

-3

-2

-1

0

1

2

3

4

-4 -3 -2 -1 0 1 2 3 4

Brent vs Kerosene 62.82% CorrelationAll weekly data

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

-0.35 -0.3 -0.25 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15

Brent % Moves

Ker

o %

Move

s


Tail Dependence for Brent and Kerosene

Upper Tail Dependence - Brent and Kero

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1

U

Lam

bda

Lower Tail dependence - Brent and Kero

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

00.10.20.30.40.50.6

u

Lam

bda

�Upper tail dependence, but not lower, is found in the US Treasury curve[8].

�Barry Schachter’s gloriamundi.org website has many other such tail dependence papers.


Causes of Non-constant CorrelationUse of a standard Pearson correlation assumes multivariate

normal distributions.The changes in correlation could be due to contagion, or just to

skewed or fat-tailed underlyings. Kurtosis alone still has elliptical distribution and does not have

much effect on correlation.

Copula theory - not well developed for n>2. For example, a Student-t copula has the same degrees of freedom for every pair of variables.

If two Gaussian marginals are associated by a Gaussian copula, the rank-correlation is constant, and equal to the Pearson correlation.

Copulas can be made flexible enough to represent arbitrarily screwy associations.

Mathematical theories use the copula, which is the cumulative function; easier to visualize copula density.


Problems with Pearson Correlation

High leverage data points


Outline








Credit Risk PortfoliosCredit Risk Capital usually based on extrapolated Unexpected

Loss estimate based on a transition matrix Often using MC for simulating macroeconomic cyclesNot usually a historical time series approachPortfolio usually mainly long positionsLoans have a maximum payoff of par, and a minimum of zero

Large unexpected loss, much smaller unexpected gainMarket risk portfolios typically hedged and have shorts

and longsCredit risk portfolios more hedged than they used to be -

some loan books may be short enough credit protection to have no net exposure to macroeconomic cycles

Credit risk portfolios usually have very few nonlinear instruments (options)

Deep out-of-the-money protective credit default options not liquid enough to hedge most loan portfolios

Double default issues


Combining Credit and Market Risk Capital

Naïve approach - add the two capital numbers togetherAssumes 100% tail dependenceAssumes both portfolios are net long linear instrumentsPoor assumption - realistic market risk portfolio may not have

worst loss in worst market• Could be net short and make huge windfall profit in market

crash• Could be hedged against extreme movements based on

stress testsFlight-to-quality is bad for junk loan book but good for govvy

desk

Infeasible - MC sweep through all possible scenariosOnly a few per 1000 of the simulations relevant to the

99.9%ileToo much compute time Correlations estimated for bulk may be irrelevant to contagion

estimate


Combining Credit and Market Risk Capital 2

Still not useful - scaled historicalMisses contagions if data set too smallMisses structural market changes if data set too largeNot enough data (1000 years!) if not scaled up from

existing history

Better - explore many stress scenarios to locate vulnerabilitiesWeighted average of stress scenario losses

(somewhat subjective weightings) gives a stress test eonomic capital combining market and credit risks

MC average of sweeps through extremes of parameter space (contagion estimates still a bit subjective) gives an EVT-based combined EC

Do not use historical covariance as a proxy for contagion estimates


ConclusionNobody said this would be easy

No magic bullets

Not enough data to do purely objectively and get a realistic answer

Easy to get confused

Contagion and EVT


References5 Embrechts, editor,”Extremes and integrated Risk Management,” RISK

Books, 20001. http://arxiv.org/abs/cond-mat?papernum=0111503 Noisy Covariance

Matrices2. http://arxiv.org/abs/cond-mat?papernum=0205119 Noisy Covariance

Matrices II3. Hettmansperger and McKean,“Robust Non-Parametric Statistical

Methods,” Arnold, 19984. http://arxiv.org/abs/cond-mat?papernum=0302546 Dynamics of market

correlations: Taxonomy and portfolio analysis Authors: J.-P. Onnela, A. Chakraborti, K. Kaski, J. Kertesz, A. Kanto

5. Embrechts, editor,”Extremes and integrated Risk Management,” RISK Books, 2000

6. http://gro.creditlyonnais.fr/content/wp/copula-survey.pdf Copulas for Finance - A Reading Guide and Some Applications by Bouyé, Durrleman, Nikeghbali, Riboulet, Roncalli

7. H. Joe, “Multivariate Models and Dependence Concepts” Chapman&Hall, 1997

8. http://gloriamundi.org/picsresources/mjasnw.pdf Nonlinear Term Structure Dependence by M Junker, A Szimayer, N Wagner

Documents

Martin Goldberg, Citigroup June, 2006 Page 1 of 45 ECONOMIC CAPITAL FOR TRADING MARKET RISK PRESENTED TO:Risk Magazine's Training Course: Economic Capital