Risk Management in the Real World

Jonathan Schachter

Delta Vega, Inc.

Apr. 19, 2018

1Fordham University

Outline

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Part 1: Theoretical Underpinnings

1. Market Risk Management Overview

2. VaR and Expected Shortfalla) Flavors of VaRb) Coherent Risk Measures

3. Position-level Risk4. Precision of Estimators5. Tweaks to Historical VaR/ES6. Performance of VaR/ES Models

Part 2: Wall Street Examples

1. Real World Riska) Bank 1b) Bank 2c) Bank 3d) Bank 4e) Bank 5

2. Other Risksa) Operational Riskb) Counterparty Credit Risk/CVA

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Abstract

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Risk management in the post-crisis world provides regulators with job security, and its red meat provides financial employment opportunities for mathematical finance students. This talk gives examples of actual risk systems at 4 large banks. The material indicates the variations in risk methodology on The Street.

The majority of the talk discusses statistical measures of loss. It also touches on the perspective of international regulators.

Finally, it provides background on the speaker's career prior to finance. Having an experiment fly on the Space Shuttle is the ultimate test of real-world risk management.

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Part 1Theoretical Underpinnings

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Risk Management Essentials

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Greeks • Primary use by traders: Δ, Γ, Vega• Feeds sensitivity-based VaR calc

Stress Testing

• Historical and hypothetical scenarios• Non-statistical, so somewhat subjective• Regulators require annual reporting (DFAST)

Statistical Measures

• Value-at-Risk (VaR)• Expected Shortfall• General spectral measures

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Dodd Frank Act Stress Tests

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•10-year Treasury yield•BBB corporate yield•Mortgage rate•Prime rate

DFAST Asset Prices/Market Conditions•Housing price index•Dow Jones total stock market index•Commercial real estate (CRE) price index•U.S. market volatility index (VIX)

DFAST Economic Activity Metrics•Real gross domestic product (GDP) growth•Nominal GDP growth•Real disposable income growth•Nominal disposable income growth•Unemployment rate•Consumer price index (CPI) inflation rate

DFAST Interest Rate Metrics•3-month Treasury rate•5-year Treasury yield

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Definition of VaR: Statistical Measure of Potential Loss

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“<Horizon> VaR at <Confidence> level with <lookback> years lookback”

• Horizon: usually 1 day (or 𝑡 scaling thereof) • Confidence: commonly 95%, 99%, or 99.9%.• Lookback: varies among banks from 1 to 4 yrs (tradeoffs)• Always stated as a POSITIVE number.

Note: loss is EXACTLY AT the confidence level, not above or below it. Insensitive to loss tail.

Reflects amount of time to hold a position before sale

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Historical Background of VaR

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• Late 1980s: JP Morgan first uses term “value risks” in context of long-maturity bond portfolio

• c. 1990: “4:15 report” at JP Morgan requested by Chairman.At close of trading, provide one number describing Bank’s risk over next day

• 1993: The term “value at risk” appears for the first time in a report of the Group of 30 (G30), with contributions from a JP Morgan member.

• Mid 1990s: VaR widely adopted as the most common measure of financial risk

• 1996: First use for calculating regulatory capital (Basel I)

• Present Day: paradigm shift to expected shortfall – driven by regulators

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Flavors of VaR

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Parametric

Historical

Monte Carlo

E.g., normal distribution: 𝜎 × 𝛼, α=2.33 for 99% VaR

• Implicit covariance matrix• Limited to number of days in lookback

• Explicit covariance matrix• Arbitrarily many realizations

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CAPM Refresher

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Idealization of a simple portfolio of linear assets (like stocks) with:• (unitless) weight vector w, returns R (portfolio return = 𝑤 ∙ 𝑅 = 𝑅𝑃)

• Currency weight vector x and value (wealth) W

• variance-covariance matrix Σ so that:

Variance of R: 𝜎𝑃2= 𝑤𝑇Σw

Currency variance: 𝜎𝑃2𝑊2= 𝑥𝑇Σx

• The return of position i is modelled linearly:

𝑅𝑖 = α𝑖 + 𝛽𝑖 𝑅𝑃

It is easy to show that 𝛽 = Σ𝑤/(𝑤𝑇Σ𝑤).

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Parametric VaR

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• Assuming normal distribution,

• VaR = α 𝑥𝑇Σx

• For example, 𝛼=2.33 for 99% confidence.

• Results unrealistic for large bank portfolio.

• But closed form solutions provide intuition when we discuss VaR tools later.

• A t distribution with small number of degrees of freedom could account for tails.

• Variance and correlation time series modelling, discussed later on, is parametric but not normal.

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Historical VaR

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• Idea: the distribution of past P&Ls is (at least approximately) IID.

• Obtain historical time series at least as far back as one lookback period before now.• Clean and remove artifacts• Proxy missing data• Make as stationary as possible so homoscedastic (demean, take logs and other

transformations)

• Compute scenario returns over the chosen horizon – usually daily

• Subject today’s portfolio to scenarios over the entire lookback period

• Compute VaR from the empirical distribution

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Monte Carlo VaR

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• Choose a large number of paths (> 10,000) to simulate the risk in N random variables.

• The N variables are correlated via an assumed or historical variance-covariance matrix

• We need to have an algorithm to create the correlation structure from uncorrelated random variables.

• Let φ be the desired vector of correlated variables, A an N×N matrix, and ε a vector of uncorrelated variables such that 𝜑 = 𝐴𝜖.

• It is straightforward to show that the variance-covariance matrix C = 𝐴𝐴𝑇 so that A is the square-root matrix of C, known as its Cholesky decomposition.

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Monte Carlo VaR, cont.

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• The simplest case is 2 variables, with correlation 𝜌, where

A = 1 0

𝜌 1 − 𝜌2

• To check, set X = AZ, with Z = [Z1, Z2] ~ N(0, 1). Then E(X) = [0, 0], var(X) = [1, 1]. Thus, X ~N(0, 1)• Furthermore, cov(X1,X2) = 𝜌. And when 𝜌 = 0, X = Z (uncorrelated).

• Computing costs (memory and CPU) are high, requiring the use of variancereduction techniques – essentially doing part of a numerical integration in closed form.

• Example: Geometric Brownian Motion represented as normal variables (Z) can use a trick that the mean of Z(x) and Z(-x) has lower variance than either variable individually.

• Variance reduction methods targeted to finance are discussed in Glasserman (2003).

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Interlude: Expected Shortfall

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• Risk measure that has some advantages over VaR

• Measures average loss above a certain percentile:

1

1 − ∝σ𝑝=0∝ (𝑝th loss)

• VaR tells you how much you will not lose.

• ES tells you if you have a loss, how large the loss is.

• Synonyms: Conditional VaR (CVaR), Expected Tail Loss (ETL), and—rarely--Average VaR (AVaR)

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Coherent Risk Measures

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Most important. Intuition: diversification

Positions

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Is VaR a Coherent Risk Measure? Is ES?

Trader ACombined Portfolio

Trader B

VaR(A)

ES(A)

+=

VaR(B)

ES(B)

VaR(A + B)

ES (A + B)

Idealistic Situationtwo traders: Trader A, Trader Beach same type of 1 period loaneach loan principal $150 MM$200 K profit if no defaultPD each loan 1.25%if one loan defaults, other doesn’tLGD each loan 0%-100% unif

+

+

? >=

? >=

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Details of Loan Portfolio Risk Calcs

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Parameter Trader A Trader B Portfolio

asset $10 MM Loan $10 MM Loan, indep $20 MM loan total

prob of default (PD) 1.25% 1.25% 2.5% (one loan)

loss given default (LGD) uniform 0%-100% uniform 0%-100% uniform 0%-100%

PD x (1-LGD) for 1% loss 1.25% x 80% 1.25% x 80% 2.5% x 40%

LGD in dollars $2 MM $2 MM $6 MM

profit on undefaulted loan - - ($0.2 MM)

99% VaR $2 MM $2 MM $5.8 MM INCOHERENT

loss above VaR uniform $ 2MM-$ 10 MM uniform $ 2MM-$ 10 MM uniform $ 5.8 MM - $9.8 MM

Exp Shortfall $6 MM $6 MM $7.8 MM COHERENT

Note: Although not subadditive in general, VaR is for normally distributed returns, since for positions A and B,

𝜎2 𝐴 + 𝐵 ≤ 𝜎2 𝐴 + 𝜎2(𝐵)

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Generalization: Spectral Risk Measures

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𝑀ϕ = න0

1

ϕ(𝑝) 𝑞𝑝 ⅆ𝑝

Where ϕ(𝑝) ≥ 0, is normalized, and is a well behaved probability weight

For VaR:, ϕ(𝑝) = 𝛿(𝑝 = 𝛼) a Delta function. Unity weight at confidence level α, zero elsewhere.

For Expected Shortfall:

ϕ 𝑝 = ቐ

0, 𝑝 < α1

1 − α, 𝑝 ≥ α

Constant weight above confidence level, zero elsewhere.

General case: user can specify a function – subjective, but similar to utility function.

A risk-averse user would have a weighting function giving higher losses higher weights.

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Analysis of VaR* at the position level

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* These results also hold for Expected Shortfall.

Marginal VaRIncremental

VaRComponent

VaR

• VaR change on small change in position

• Intuitive result for historical VaR

• VaR change on addition of position

• “Best hedge” concept

• Contribution to VaR of a position

• Truly additive

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Marginal VaR

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Definition: the change in VaR for a small difference in one position. For the idealized portfolio,

𝑚𝑉𝑎𝑅 = ∆𝑉𝐴𝑅𝑖 =𝜕𝑉𝐴𝑅

𝑊𝜕𝑤𝑖

For normally distributed returns, this reduces to𝛼 cov(𝑅𝑖 , 𝑅𝑃)/𝜎𝑃

which is similar to the beta factor in CAPM:

𝛽𝑖 = 𝛼 cov(𝑅𝑖 , 𝑅𝑃)/𝜎𝑃2 ≅ 𝛽 = Σ𝑤/(𝑤𝑇Σ𝑤)

The positional VaRs on the path (historical or Monte Carlo) with the largest overall loss are the mVaRs.

Nice Fact

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Incremental VaR

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Definition: the change in VaR when adding one position.• iVaR > 0 means the position has added to the portfolio risk• iVaR < 0 means the position is hedging the portfolio risk => there is an optimum position size (“best

hedge”). For simple portfolio, size of variance minimizing position is

−𝑊𝛽𝑖𝜎𝑝2

𝜎𝑖2

• How to estimate without pricing portfolio multiple times?

• Methodology using mVaR framework – essentially a Taylor expansion: ∇𝑉𝑎𝑅𝑃dw

where ∇ is the vector 𝜕𝑉𝑎𝑅/𝜕𝑤𝑖 and dw is the transpose of the vector of infinitesimal weight changes when the position is added.

• Normally distributed PnL, mean=0: ∇𝑉𝑎𝑅𝑃 = αΣ𝑤

𝑤𝑇Σ𝑤

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Component VaR

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Definition: VaR additive decomposition

For the simple portfolio, we dollarize the mVaR:

𝑐𝑉𝑎𝑅𝑖 = 𝑚𝑉𝑎𝑅𝑖𝑤𝑖𝑊 = 𝑤𝑖 𝛽𝑖𝑉𝑎𝑅𝑃

so that

σ𝑖=1𝑁 𝑐𝑉𝑎𝑅𝑖 = 𝑉𝑎𝑅𝑃

Putting together the building blocks of VaR

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Precision of VaR Estimates

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Normal Distribution

Large sample error for sample SD, ො𝜎,

is ො𝜎/ 2𝑇. The denominator is ~22 for 1 yr.

Known Dist.

Error in quantile c and known distribution f with quantile value q is

𝑐(1−𝑐)

𝑇𝑓(𝑞)2

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Precision of VaR Estimates, cont.

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Monte Carlo • Limited by number of paths.

• Variance reduction techniques helpful.

Historical Simulation

• Bootstrapping

• Fit to a known distribution and use percentile formula, above.

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Tweaks to Historical VaR Methodology

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Motivation: Most banks use historical VaR, but it has some undesirable features

Feature #1: lookback window is a box, so VaR can spike on days markets have large shiftsSolution: change box so has softer tail (e.g., exponential)Problems: Don’t know decay parameter. Shouldn’t all P&L’s contribute equally to VaR?

Feature #2: Non-stationarity – variances change over time, often blow up in market crashesSolution: Fit a parametric time series model such as EWMA or GARCHProblems: Don’t know decay parameter.

Feature #3: correlations change over time, and all go to 1 (unity) in market crashesSolution: Multivariate GARCH (many parameters) or pairwise EWMA (few)Problems: Calculated correlations may fall outside [-1,1].

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Backtesting

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• Idea: compare number of actual exceptions (or “breaks”) to true P&L to number predicted by VaR. Essential test of any VaR model.

• Example: 1 day, 99% VaR with 1 year lookback (say, 252 days)• Should produced approximately 2 to 3 exceptions• If actual number of exceptions < 2, VaR model is too conservative.• If > 3, VaR model may be underestimating loss. Needs to be checked.

• The probability of exactly 4 exceptions is

2524

40.012480.99 = 13.6%

while the probability of 4 or more exceptions is ~ 25%.

• Consider market events • Analyze models at position level• Upside breaks (profit; right tail) are also in scope

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Backtesting, cont.

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What is “true P&L” for comparison of VaR? Some definitions:

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“Dirty” P&L: actual profit/loss on books, including

1. Day trading2. Commissions and other fees

“Clean” P&L: excludes trading and fees. Positions constant during the trading day.

Regulators require backtesting with respect to both dirty and clean P&L.

From a mathematical standpoint, the clean P&L comparison makes more sense.

Backtesting, cont. 2

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Internal Backtesting

Regulatory Backtesting

Typically computed daily at 95% conf.

For review of CRO and senior management.

Used to reject/accept proposed trades and set position size limits.

Computed quarterly at 99% conf and reported.

Too many exceptions will result in punitive multiplier.

Too few exceptions will require further explanation.

vs.

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Backtesting, cont. 3

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• Multiplier shown is for newest regulation framework (“FRTB”), due 2022

• Color coding is commonly used: RAG = “red, amber, green”

• Regulatory capital is based on the scaled value of VaR

• More than 10 exceptions likely will result in shutting down trading

1.50 1.50 1.50 1.50 1.50

1.70

1.761.81

1.881.92

2.00

1.40

1.50

1.60

1.70

1.80

1.90

2.00

2.10

0 1 2 3 4 5 6 7 8 9 10

Multiplier vs Number of Exceptions

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Ongoing Performance Assessment

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• OPA goes beyond backtesting by examining more than simple count of breaks• Size of breaks• Time between breaks (“duration”)• Clustering of breaks• Distribution of breaks• Probability of a break tomorrow given a break today

Example of last: Conditional coverage test [Likelihood ratio (LR) approach (Christoffersen 1998)]:LR (break tomorrow, given break today) = LR(break tomorrow) + LR(breaks tomorrow

and today are independent)LR(conditional) = LR(unconditional) + LR(independent)LR(unconditional) ~ LR(independent) ~ χ2 1

Therefore, LR(conditional) ~ χ2 2

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Practical Considerations: Approaches on the Street

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• Returns• Actual (“dirty”)• Hypothetical (“clean”)

• Valuation• Full reval -- computationally intensive• Factors -- Front Office and Risk Mgt models will differ

• Historical VaR requires typically 100,000+ time series• Missing data/Flatlines• Low quality data

• Monte Carlo likely will use a historical variance-covariance matrix• May not be positive definite. Need to “cure.”

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Part 2Wall Street Examples

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Public VaR information from annual reports

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JP Morgan Chase – Annual Report, April 2018

Bank Profiles -- Anonymized

Name Bank Total Assets Headquarters in US?

Bank 1 > $2 T YES

Bank 2 > $800 B YES

Bank 3 > $1.5 T YES

Bank 4 > $800 B NO

Bank 5 > $200 B YES

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Full Reval

Own TS

Sensitivities

Own TS

Full Reval

TS from Backbone

Sensitivities

Own TS

Asset Class 1 Asset Class 2 Asset Class 3 Asset Class 4

Bank 1 Risk System

Own TSComputes Sensitivities VaR

Also accepts P&LsConvert P&Ls to VaR

TS

Calc.Sens.

Asset Class 5

Sens.

P&Ls

Backbone

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Bank 1 Risk Calculation

P&L Vector – 263 daily P&Ls

Profit/Loss Amount Loss Rank

-1,000,000 1

-900,000 2

-500,000 3

-300,000 4

-100,000 5

-50,000 6

-10,000 7

-8,000 8

-2,000 9

… …

1,000,000 263

Risk metric:

• Average tail losses: Expected Shortfall

• N=7 equivalent to 99th

Pctile. for Normal Distribution

• Change sign (loss quoted as positive)

−1

𝑁

𝑖=1

𝑁

𝑃&𝐿𝑖

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Bank 1 Risk Calculation, cont.

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Bank 1’s Stressed VaR Conundrum

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• Stressed VaR required by regulators (Appendix A) after financial crisis:• Use existing VaR model on one-year lookback period of largest losses• Add to ordinary VaR to compute capital requirements

• Time series issues• Stressed period was hardwired --- Jan 2008 to Jan 2009• Jan 2007 to Jan 2008 data had to be analyzed, organized, cleaned, and

proxied as necessary• Statistical testing: ACF, CCF with benchmarks• Return calculations: absolute or relative?

𝑅𝑇+1 − 𝑅𝑇 or (𝑃𝑇+1/𝑃𝑇) − 1 ?

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Bank 2 VaR system -- Jan. 1, 2012Four-year Lookback Historical VaR

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Volatility spiked as prices sank

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Clearly, the P&L’s are not IID.

How do we capture the time dependence of volatility?

0.0%

10.0%

20.0%

30.0%

40.0%

50.0%

60.0%

70.0%

80.0%

10/10/2006 2/22/2008 7/6/2009 11/18/2010 4/1/2012 8/14/2013 12/27/2014

annualized 60 day moving sd

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The case for an EWMA Model

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We really want a time series model for conditional variance.• Heteroscedastic (time-varying variance) models have a storied history (Engle 2001),

particularly GARCH (Generalized Autoregressive Conditional Heteroscedastic) – at least 2 parameters.

• Exponentially weighted moving average (EWMA) models have a single parameter:

σ𝑇2 = λσ𝑇−1

2 + (1 − λ)𝑟𝑇−12

where 𝑟𝑇 =𝑃𝑇

𝑃𝑇−1− 1 .

The parameter λ ≈ 0.90 − 0.99.

Sometimes this range is referred to as “slow” to “fast.”

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The case for an EWMA Model, cont.

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A reformulation showing the explicit effect of the decay is:

σ𝑇2 = (1 − λ)

𝑖=1

𝑛

λ𝑖−1 𝑟𝑇−𝑖2

The effective number of days of variance used in the model is approximately

ln(1 − 𝑐𝑜𝑛𝑓)

ln λ

As the speed increases from 0.90 to 0.99, the number of days increases from 44 to 458 at 99% confidence.

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The case for an EWMA Model, cont. 2

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Morgan Stanley approach was to scale P&L’s like this:

(𝑃&𝐿)𝑇,𝑛𝑒𝑤 = (𝑃&𝐿)𝑇×𝜎𝑡𝑜𝑑𝑎𝑦

𝜎𝑇

How do we figure out value of λ?

If heteroscedasticity is removed, new P&L distribution should look more IID than before.

Risk management department selected key indices across all major asset classes. Performed hypothetical backtesting and OPA on old and new P&L distributions.

Result: most OPA tests supported λ=0.97. Backtesting supported 0.99. Risk managers said they wanted 0.99 “because Goldman uses it.”

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Bank 3 VaR

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“… uses a single, independently approved Monte Carlo simulation VaR model for both Regulatory VaR and Risk Management VaR. Such model incorporates the volatilities and correlations of 300,000 market factors, making use of 180,000 time series, with risk sensitivities updated daily and model parameters updated daily in some instances, and weekly for all others.“

• Three year lookback for correlations

• Volatilities: max 𝜎3 𝑌𝑒𝑎𝑟 , 𝜎30 𝐷𝑎𝑦

• Sensitivities approach to repricing

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Bank 4 VaR

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1. Assumption that the representation of market risk is a decreasing function of lookback time.

2. Exponentially weight actual P&L’s, rather than volatilities (Bank 2).

3. Two year lookback.

How do we exponentially weight P&L’s?

VaR Computation with Weighted P&Ls

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Source: Boudoukh, Richardson and Whitelaw.

Example parameters (not used by Bank 3): • λ = 0.90• Lookback period K = 100

VaR Computation with Weighted P&Ls, cont.

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Day Number in Lookback

Var P&L Weight

1 81.1178112 0.0231

2 138.347506 0.0226

3 -192.091658 0.0221

4 -36.2155789 0.0217

5 -74.0996651 0.0213

6 -214.858759 0.0208

… … …

95 38.1013432 0.0035

96 21.8068199 0.0034

97 4.37945675 0.0033

98 -49.5863638 0.0032

99 -49.3772317 0.0032

100 -66.0083849 0.0031

Step 1: Compute Weights

Day Number in Lookback

Var P&L Exp. WeightCumul Exp.

Weight

29 -285.27 0.0131 0.0131

80 -281.89 0.0047 0.0178

76 -258.63 0.0051 0.0228

47 -257.51 0.0091 0.0319

6 -214.86 0.0208 0.0528

3 -192.09 0.0221 0.0749

… … … …

82 -170.76 0.0045 0.0990

56 -145.66 0.0076 0.1066

93 -133.57 0.0036 0.1102

78 -127.40 0.0049 0.1151

44 -125.25 0.0097 0.1248

36 -112.73 0.0114 0.1361

Step 2: Sort by P&L

VaR Computation with Weighted P&Ls, cont.

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Day Number in Lookback

Var P&L Exp. WeightCumul Exp.

Weight

29 -285.27 0.0131 0.0131

80 -281.89 0.0047 0.0178

76 -258.63 0.0051 0.0228

47 -257.51 0.0091 0.0319

6 -214.86 0.0208 0.0528

3 -192.09 0.0221 0.0749

… … … …

82 -170.76 0.0045 0.0990

56 -145.66 0.0076 0.1066

93 -133.57 0.0036 0.1102

78 -127.40 0.0049 0.1151

44 -125.25 0.0097 0.1248

36 -112.73 0.0114 0.1361

Extrapolated 99% VaR: -287.51Equal weighted: -285.27

Interpolated 95% VaR: -220.56Equal weighted: -214.86

Step 3: Calculate VaR at Given Percentiles

Issues with Weighted P&Ls

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1. Two parameters to estimate, vs. just one for EWMA

2. Lookback time (K) may need to be different in different economic regimes.

3. Choose of K may be subjective.

Bank 5 VaR

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• Crucial difference: computation of overnight VaR of client portfolios for client risk mgrs.

• Full revaluation historical simulation: capture FX and fixed income exotics in hedge fund clients.

• Lookback period selected by client.

• Can be compared with vendor approaches: Algorithmics, Bloomberg, RiskMetrics

Bloomberg methodology?

Operational Risk: Categories

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Operational Risk Implementation

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High Medium Low

High

Medium

Low

Frequency

Seve

rity

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• Model risk is considered a part of operational risk.

• The Fed and OCC have strict model risk management standards: SR 11-7.

Fed/OCC SR11-7 Standards on Model Risk

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Components to every model

Two types of model risk

Fundamental Errors (Bad Model) Model Misuse (Good Model, Applied to Wrong Situation)

Model validation (independence, effective challenge)

Conceptual Soundness

Information input component

Ongoing Monitoring, including Benchmarking

Outcomes analysis, including backtesting

Processing component Reporting component

Fed/OCC SR11-7 Standards on Model Risk, cont.

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A guiding principle for managing model risk is "effective challenge" of models, that is, critical analysis by objective, informed parties who can identify model limitations and assumptions and produce appropriate changes.

Effective challenge depends on a combination of incentives, competence, and influence. Incentives to provide effective challenge to models are stronger when there is greater separation of that challenge from the model development process and when challenge is supported by well-designed compensation practices and corporate culture.

Competence is a key to effectiveness since technical knowledge and modeling skills are necessary to conduct appropriate analysis and critique.

Finally, challenge may fail to be effective without the influence to ensure that actions are taken to address model issues. Such influence comes from a combination of explicit authority, stature within the organization, and commitment and support from higher levels of management.

Counterparty Credit Risk

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• History: During crisis, traders added an additional haircut to products reflecting credit worthiness of counterparty

• The haircut came to be known formally as a “credit valuation adjustment” (CVA).

• Basically follows the usual breakdown of credit risk:

𝐿𝑜𝑠𝑠 𝑔𝑖𝑣𝑒𝑛 ⅆ𝑒𝑓𝑎𝑢𝑙𝑡 × 𝐸𝑥𝑝𝑜𝑠𝑢𝑟𝑒 𝑎𝑡 ⅆ𝑒𝑓𝑎𝑢𝑙𝑡 × (𝑃𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦 𝑜𝑓 ⅆ𝑒𝑓𝑎𝑢𝑙𝑡)

Or

where R is the recovery given default, EE* is the expected exposure discounted at the risk neutral rate, and PD is the risk neutral probability (e.g., from CDS spreads).

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Conclusion

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• Statistical predictors of market risk losses are widespread in finance, though the specific methodologies vary

• Banks customize risk solutions to their own needs, while at the same time within the constraints of legacy systems

• Neither VaR not Expected Shortfall is a perfect risk metric

• Risk models should be continually tested in order to ensure that the probabilistic assumptions that create them continue to have validity

• Such performance testing is crucial for review by senior management and regulators

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References

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1. https://www.crowehorwath.com/insights/banking-performance/2017-stress-testing-scenarios.aspx

2. http://www.care-web.co.uk/blog/seven-operational-risk-event-types-projected-basel-ii/3. Hull, J.C., Risk Management and Financial Institutions, 4th Edition, Wiley.4. Jorion, P., Value at Risk: The New Benchmark for Managing Financial Risk, 3rd Edition, McGraw-

Hill.5. Dowd, K., Measuring Market Risk, 2nd Edition, Wiley.6. https://www.sciencedirect.com/science/article/pii/S2212567115006073 (“The History and Ideas

Behind VaR”)7. http://www.citigroup.com/citi/investor/data/b25d140331.pdf?ieNocache=1578. http://vassarstats.net/binomialX.html9. http://www.ims.nus.edu.sg/Programs/econometrics/files/kw_ref_4.pdf (Christophersen 1998)10. Engle, Robert F. (2001). GARCH 101: The Use of ARCH/GARCH Models in Applied Econometrics.

Journal of Economic Perspectives. 15 (4): 157–168. doi:10.1257/jep.15.4.157. JSTOR 2696523.11. Wikipedia – various topics.12. http://www.springer.com/us/book/9780387004518 (Glasserman 2003)13. Longerstaey, J., RiskMetrics Technical Document, 1996, 4th Edition, JP Morgan.14. The Best of Both Worlds, Boudoukh, J., Richardson, M., and Whitelaw, R.F.

(www.faculty.idc.ac.il/kobi/thebestrisk.pdf)

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Appendix A: Risk Regulatory TimeLine

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• 1988: Bureau of International Settlements (BIS) sets first international capital adequacy requirements – concept of risk-weighted assets (RWA)• Came to be referred to as Basel I, from the city in Switzerland where BIS met• Recognition of differences between trading book (dynamic) banking book (static)

• 1996: Amendment to Basel I: introduced capital charge for trading book. • Multiple approaches to calculation – standardized (SA) and internal models approaches

(IMA; intended for large, complex banks)• First use of VaR• Capital charge for specific risk• Backtesting requirement

• 1999: Basel II -- Extend to credit and operational risk, with SA and IMA• Post Crisis: Basel II.5 -- Stressed VaR, credit default and migration, correlation• 2009: Basel III – Extend to liquidity risk, counterparty credit risk (CVA)• Present: Fundamental Review of the Trading Book (FRTB; “Basel IV”) – position-specific liquidity

horizons (holding periods), replacing VaR with Expected Shortfall

Caution!

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Appendix B: My Astrophysics Career

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• Ph.D., Berkeley, 1990. Dissertation: “A Study of Bowen Fluorescence in Accretion-Powered Sources• Taking data on a laser process present in the ionized gaseous material surrounding black

holes, neutron stars, and white dwarves (dead and dying stars)• Data analysis: estimates of size and distribution of unseen X-ray emitting material

• 1990-2000: Research Associate, Harvard Astronomy Department; and Postdoctoral Researcher, Smithsonian Astrophysical Observatory (Cambridge)

• 1990-1995: Einstein Slew Survey – identification of 50+ new “radio loud” galaxies, analysis of the relation between intensity and velocity of magnetically active stars

• 1996-2000: Software testing, Chandra X-ray Observatory satellite telescope. Chandra is the X-ray analog to the Hubble Space Telescope. It was launched on Space Shuttle Columbia in 1999.

Pretty pictures: https://www.nasa.gov/mission_pages/chandra/images/index.html

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About the Speaker

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Jon Schachter is founder of Delta Vega, Inc., an independent consultancy in mathematical finance. Currently, he is partnering with Renaissance Risk Management Labs to provide high-performance derivatives pricing tools (using GPUs and adjoint algorithmic differentiation) to banks, insurance companies, and family offices. Jon’s past experience includes valuation and risk roles at JP Morgan, Goldman Sachs, State Street, Lehman Brothers (post bankruptcy), and Morgan Stanley. He is a 2002 graduate of the Columbia mathematics of finance program. Jon began his career as a postdoctoral researcher in astrophysics at Harvard, and was part of the team that launched the Chandra X-ray Observatory satellite telescope. He is a native New Yorker, but never imagined working here.

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