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![Page 1: Section 3: Risk Management for Investment Banks Financial Risk Management UC Berkeley Prof. Jeff (YuQing) Shen.](https://reader031.fdocuments.us/reader031/viewer/2022012922/56649efd5503460f94c1110e/html5/thumbnails/1.jpg)
Section 3: Risk Management for Investment Banks
Financial Risk Management
UC Berkeley
Prof. Jeff (YuQing) Shen
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Financial Risk Management
Prof. Jeff (YuQing) Shen, UC Berkeley
2
The BIG Picture
• Theory
•Efficient Frontier
•Asset/Liability Framework
•Alpha/Beta risk budgeting
• Practice
•Pension Crisis
•GM Pension Plan
•Yale Endowment
• Theory
•VaR
•Stochastic behavior of asset
returns
•Worst case scenario
• Practice
•Investment bank VaR
•Bear Stearns
• The Theory
•Active Portfolio Management
•Quantitative equity models
•Hedge fund investment
• The Practice
•Hedge fund industry
•Aug 2007 crisis for quant
Investment BankInstitutional Investor Hedge Fund
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Financial Risk Management
Prof. Jeff (YuQing) Shen, UC Berkeley
![Page 4: Section 3: Risk Management for Investment Banks Financial Risk Management UC Berkeley Prof. Jeff (YuQing) Shen.](https://reader031.fdocuments.us/reader031/viewer/2022012922/56649efd5503460f94c1110e/html5/thumbnails/4.jpg)
Financial Risk Management
Prof. Jeff (YuQing) Shen, UC Berkeley
4
Event risk is on the rise
-0.08
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
0.08
1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001
10/27/97 8/31/984/14/00
3/12/01
These four extreme movements should happen only once per 120 years!!!
S&P 500 Daily Return
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Financial Risk Management
Prof. Jeff (YuQing) Shen, UC Berkeley
OUTLINE
Introduction to VaR
– Statistical framework. Risk and diversification: some examples. Possible applications. Visual interpretation.
The Stochastic Behavior of Asset Returns
– Time variations in volatility. VaR: approaches and comparison. The Hybrid Approach to VaR.
Beyond Volatility Forecasting
– The VaR of derivatives and interest rate VaR. Structured Monte Carlo. Extreme events and correlation breakdown. Stress testing and scenario analysis. Worst case scenario
![Page 6: Section 3: Risk Management for Investment Banks Financial Risk Management UC Berkeley Prof. Jeff (YuQing) Shen.](https://reader031.fdocuments.us/reader031/viewer/2022012922/56649efd5503460f94c1110e/html5/thumbnails/6.jpg)
Financial Risk Management
Prof. Jeff (YuQing) Shen, UC Berkeley
OUTLINE
Introduction to VaR– Statistical framework. Risk and diversification: some examples. Possible
applications. Visual interpretation.
The Stochastic Behavior of Asset Returns
– Time variations in volatility. VaR: approaches and comparison. The Hybrid Approach to VaR.
Beyond Volatility Forecasting
– The VaR of derivatives and interest rate VaR. Structured Monte Carlo. Extreme events and correlation breakdown. Stress testing and scenario analysis. Worst case scenario
![Page 7: Section 3: Risk Management for Investment Banks Financial Risk Management UC Berkeley Prof. Jeff (YuQing) Shen.](https://reader031.fdocuments.us/reader031/viewer/2022012922/56649efd5503460f94c1110e/html5/thumbnails/7.jpg)
Financial Risk Management
Prof. Jeff (YuQing) Shen, UC Berkeley
Types of risk
Market risk– interest rate, currency, equity, commodity, spread, volatility,…
– example: P(bond) decline as interest rates rise
Credit risk– default, downgrade
– example: P(bond)=recovery upon default
Other– liquidity, regulatory, political, model, execution,…
We focus primarily on market risk, and to a lesser extent on credit risk (especially the link btw market and credit risk for derivatives)
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Financial Risk Management
Prof. Jeff (YuQing) Shen, UC Berkeley
Risk Measurement
Address the question:
“ HOW MUCH CAN WE LOSE ON HOW MUCH CAN WE LOSE ON OUR TRADING PORTFOLIO BY OUR TRADING PORTFOLIO BY TOMORROW’S CLOSE? ”TOMORROW’S CLOSE? ”
Risk MEASUREMENT <==?==><==?==> Risk MANAGEMENT
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Financial Risk Management
Prof. Jeff (YuQing) Shen, UC Berkeley
VaR is a measure of the maximum potential change in the value of a portfolio with a given probability (confidence interval) over a pre-set horizon.
VaR provides answer to questions such as: – what is the maximum a client can lose over the next week, 95% of the
time?
VaR provides a summary of the risk profile of an instrument or portfolio which allows for measuring risk uniformly across different instruments and portfolios.
VaR definition
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Financial Risk Management
Prof. Jeff (YuQing) Shen, UC Berkeley
VaR: Example
Consider a spot equity position worth $1,000,000
Suppose the daily standard deviation of the S&P500 is 100 basis points per day
How do we make an informative statement about risk?
We can only make a probabilistic statement:
Assume St,t+1 is distributed normally ( 0 , 100bp2 )
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Financial Risk Management
Prof. Jeff (YuQing) Shen, UC Berkeley
“Value at Risk” (VaR): (First Look)
From the normal dist’n tables:– -1STD to +1STD 68.3%– -2STD to +2STD 95.4%– What is the “value” of one standard deviation?– What are the amounts on the X-axis?
-1sd-1sd +1sd+1sd==-$10,000-$10,000 ==$10,000$10,000
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Financial Risk Management
Prof. Jeff (YuQing) Shen, UC Berkeley
“Value at Risk” (VaR):(First Look)
From the normal dist’n tables:Prob(Z< -1.65)=5%, Prob(Z< -2.33)=1%
““With probability 95% we will not see a loss greater than ___? on our position”With probability 95% we will not see a loss greater than ___? on our position”
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Financial Risk Management
Prof. Jeff (YuQing) Shen, UC Berkeley
OUTLINE
Introduction to VaR
–Statistical framework. Risk and diversification: some examples. Possible applications. Visual interpretation.
The Stochastic Behavior of Asset Returns
– Time variations in volatility. VaR: approaches and comparison. The Hybrid Approach to VaR.
Beyond Volatility Forecasting
– The VaR of derivatives and interest rate VaR. Structured Monte Carlo. Extreme events and correlation breakdown. Stress testing and scenario analysis. Worst case scenario
![Page 14: Section 3: Risk Management for Investment Banks Financial Risk Management UC Berkeley Prof. Jeff (YuQing) Shen.](https://reader031.fdocuments.us/reader031/viewer/2022012922/56649efd5503460f94c1110e/html5/thumbnails/14.jpg)
Financial Risk Management
Prof. Jeff (YuQing) Shen, UC Berkeley
Quantifying the Exposure:Calculating the standard deviation
Three ways to define the change in the spot rate
– Absolute change St,t+1 = St+1 - St
– Simple rate of returnSt,t+1 = St+1/ St
– Cont’ comp’ change St,t+1 = ln( St+1/St )
Which definition of “St,t+1” is most appropriate?
To answer this we must recognize that we are going to make a strong assumption:
the past is representative / predictive of the futurethe past is representative / predictive of the future
The question is : which aspect of the past?
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Financial Risk Management
Prof. Jeff (YuQing) Shen, UC Berkeley
Stationarity
We are going to assume “stationarity”
Consider the following statements– a 80pt change in the Dow is as likely at 3000 as it is at 9000– a 2% change in the Dow is as likely at 3000 as it is at 9000– ...which one is more likely to hold ?
Recall: three ways to define the change in the spot rate
– Absolute change St,t+1 = St+1 - St
– Percentage change (rate of return) St,t+1 = St+1 / St
– C.C. change St,t+1 = ln( St+1/St )
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Financial Risk Management
Prof. Jeff (YuQing) Shen, UC Berkeley
Time - consistency
Consider the cont’ comp’ two day returnSt,t+2 = ln( St+2/St )
= ln{ (St+2/St+1 ) * (St+1/St ) }
= ln(St+2/St+1 ) + ln(St+1/St ) = St,t+1 + St+1,t+2
Suppose St+i,t+i+1 is distributed N(0, 2)
The sum, St,t+J, is also normal: N(0, J*2)
(under certain assumption, to be discussed later)
Easy to extrapolate VaR:
J J day VaR = SQRT(day VaR = SQRT(JJ) * (1 day VaR)) * (1 day VaR)
With any other definition of returns normality is not preserved (i.e., the product of normals is non-normal)
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Financial Risk Management
Prof. Jeff (YuQing) Shen, UC Berkeley
Non-negativity
Consider the cont’ comp’ i-day return
St,t+i= ln( St+i/St )
Since St,t+i is distributed N(0, i *), the value of any possible St+i is guaranteed to be
non-negative:
St+i = St *exp{St,t+i }
This is the standard log-normal diffusion process (such as in
Black/Scholes): log(returns) are normal
With other definitions of returns positivity of asset prices is not guaranteed
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Financial Risk Management
Prof. Jeff (YuQing) Shen, UC Berkeley
Interest rates and spreads
The exceptions to the rule are interest rates and spreads
(e.g., zero rates, swap spreads, Brady strip spreads,…)
For these assets the “change” is it,t+i= it+i-it usually measured in basis points
This is an added complication in terms of calculating risk
– for stocks, commodities, currencies etc, there is a 1-for-11-for-1 relation between the risk index and the portfolio value
– with interest rates there is a 1-for-D1-for-D relation, where D is the duration:
a 1b.p. move in rates ==> D b.p. move in bond value
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Financial Risk Management
Prof. Jeff (YuQing) Shen, UC Berkeley
Quantifying the Exposure:Calculating Standard deviation (cont’d)
The STD of change can be calculated easily
– “Volatility” (vol)
STD(St,t+1 ) =SQRT[ VAR(St,t+1) ]
– ...where VAR(St,t+1) is the “Mean squared deviation”
VAR(St,t+1) = AVG[ (S-avg(S))2 ]
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Financial Risk Management
Prof. Jeff (YuQing) Shen, UC Berkeley
OUTLINE
Introduction to VaR
– Statistical framework. Risk and diversification: some examples. Possible applications. Visual interpretation.
The Stochastic Behavior of Asset Returns
– Time variations in volatility. VaR: approaches and comparison. The Hybrid Approach to VaR.
Beyond Volatility Forecasting
– The VaR of derivatives and interest rate VaR. Structured Monte Carlo. Extreme events and correlation breakdown. Stress testing and scenario analysis. Worst case scenario
![Page 21: Section 3: Risk Management for Investment Banks Financial Risk Management UC Berkeley Prof. Jeff (YuQing) Shen.](https://reader031.fdocuments.us/reader031/viewer/2022012922/56649efd5503460f94c1110e/html5/thumbnails/21.jpg)
Financial Risk Management
Prof. Jeff (YuQing) Shen, UC Berkeley
A two asset example Consider the following FX position
(where vol($/Euro)=75bp ==> VaR=75*1.65=123,
vol($/GBP)=71bp ==> VaR=71*1.65=117 )
Position* 95% move = VaR
(FX in $MM) (in percent) (in $MM)
Euro 100 1.23% $1.23
GBP -100 1.17% -$1.17
Undiversified risk $2.40
(absolute sum of exposures, ignoring the effect of diversification)
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Financial Risk Management
Prof. Jeff (YuQing) Shen, UC Berkeley
Portfolio variance: a quick review
X, Y are random variables c, d are parameters
VAR(X+Y) =VAR(X)+VAR(Y)+2COV(X,Y)VAR(c*X) =c2 VAR(X)==>VAR(cX+dY) =c2 VAR(X)+d2 VAR(Y)+2 c d COV(X,Y)
...applied to portfolio theory:
A portfolio of two assets: Rp= w Ra + (1-w) Rb
and the vol of a portfolio, VAR( Rp ) , can now be calculated
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Financial Risk Management
Prof. Jeff (YuQing) Shen, UC Berkeley
Correlation and covariance
Correlation:Correlation: the tendency of two variables to co-move
COV(Ra, Rb) = Ra,RbRa,Rb * Ra * Rb
The volatility of a portfolio in percent:
%STD=sqrt[Wa2Ra
2 +Wb2 Rb
2 +2WaWb Ra,RbRa,Rb Ra Rb ]
The volatility of a portfolio in $ terms:
$STD=sqrt[$Ra2 +$Rb
2 + 2 Ra,RbRa,Rb $Ra $Rb ]
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Financial Risk Management
Prof. Jeff (YuQing) Shen, UC Berkeley
From vol to VaR
How do we move from vol to VaR?
Consider the $ volatility
{ $STD=sqrt[$Ra2 +$Rb
2 + 2 Ra,Rb $Ra $Rb ] }*1.65*1.65
and we get
$VaR=sqrt{$VaR$VaR=sqrt{$VaRRaRa2 2 + $VaR+ $VaRRbRb
2 2 + 2+ 2Ra,Rb Ra,Rb $VaR$VaRRa Ra $VaR$VaRRbRb}}
...or we could calculate the %vol and
VaR= %STD * value * 1.65VaR= %STD * value * 1.65
The two approaches are EQUIVALENT
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Financial Risk Management
Prof. Jeff (YuQing) Shen, UC Berkeley
Portfolio VaR
Suppose $/Euro,$/GBP = 0.800.80
For simplicity forget cont’ comp’ returns for the moment
$VaR=sqrt{$VaRRa2 + $VaRRb
2 +2Ra,Rb $VaRRa$VaRRb}
=sqrt{1.232 + (-1.17)2 + 2*0.80*(-1.17)(1.23) }
=sqrt{1.5129 + 1.3689 - 2.3025 }
= $0.76Mil
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Financial Risk Management
Prof. Jeff (YuQing) Shen, UC Berkeley
The portfolio effect Compare: Undiversified VaR $2.40MM
Diversified VaR $0.76MM
==> Portfolio effect $1.64MM
Risk reduction due to diversification depends on the correlation of assets in the portfolio
As the number of assets increases, portfolio variance becomes more dependent on the covariances and less dependent on variances
The “marginal” risk of an asset when held as a small portion of a large portfolio, depends on its return covariancecovariance with other securities in the portfolio
Exercise: take an equally weighted portfolio with N uncorrelated asset. Assume all assets have equal volatility. What is the portfolio’s volatility? Take N to infinity. what happens to volatility?
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Financial Risk Management
Prof. Jeff (YuQing) Shen, UC Berkeley
Diversification example: prop trading desk
Consider a risk manager examining 9 positions taken by 9 different prop traders
For simplicity assume that:
– the positions are valued at $100MM$100MM each, with a an annual VaR 16.5% (i.e., vol = 10% per annum)
– the strategies are uncorrelated (e.g., high yield FX, special situation, fixed income arb, Japanese warrants arb, spread trading,...)
The undiversified VaR is 9*$16.5MM = $148.5MM, on a $900MM investment
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Financial Risk Management
Prof. Jeff (YuQing) Shen, UC Berkeley
Prop trading desk’s VaR The VaR of N uncorrelated assets:
VaR port = sqrt{ 9* VaR strat 2 + zerozero}
= sqrt{9} VaR strat
= 3 * $16.5MM =$49.5MM$49.5MM
==> the risk reduction due to diversification is 66%
Now suppose each strategy had an annualized Sharpe ratio of E[R]/STD[R]=2 2 ==> E[R]=20%, or $180MM. The
portfolio’s Sharpe ratio would be 180/30 = 6 6
What would the Sharpe Ratio and VaR be if we have the entire $900MM in only one strategy by one prop trader?
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Financial Risk Management
Prof. Jeff (YuQing) Shen, UC Berkeley
Bond portfolio VaR
The VaR of a portfolio of $100 of face of the 1yr and the 10yr can now be calculated as usual (how?)
What is the VaR of a 10% s.a. coupon bond w/ 10yr to maturity?
Note that the coupon bond VaR involves– 20 volatilities – 190 correlations
==> MATURITY BUCKETS
(cash flow mapping)
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Financial Risk Management
Prof. Jeff (YuQing) Shen, UC Berkeley
Spread VaR
YY
tt
SpreadSpread
10yr AA 10yr AA
10yr treasury 10yr treasury
%%VaR=2.5bp/dayVaR=2.5bp/day
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Financial Risk Management
Prof. Jeff (YuQing) Shen, UC Berkeley
The (approx) VaR of a 10yr AA bond
VaR(treasury)=11.5bp
VaR(spread)=2.5bp
CORR(spread, treasury)=0
==>VaR(AA Bond)=sqrt{11.52 + 2.52 } = 11.77bp/day
Note:
VaR(AA Bond) / VaR(treasury) = 11.77/11.5 = 1.023,
only 2.3% higher VaR!
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Financial Risk Management
Prof. Jeff (YuQing) Shen, UC Berkeley
OUTLINE
Introduction to VaR
– Statistical framework. Risk and diversification: some examples.
Possible applications. Visual interpretation.
The Stochastic Behavior of Asset Returns
– Time variations in volatility. VaR: approaches and comparison. The Hybrid Approach to VaR.
Beyond Volatility Forecasting
– The VaR of derivatives and interest rate VaR. Structured Monte Carlo. Extreme events and correlation breakdown. Stress testing and scenario analysis. Worst case scenario
![Page 33: Section 3: Risk Management for Investment Banks Financial Risk Management UC Berkeley Prof. Jeff (YuQing) Shen.](https://reader031.fdocuments.us/reader031/viewer/2022012922/56649efd5503460f94c1110e/html5/thumbnails/33.jpg)
Financial Risk Management
Prof. Jeff (YuQing) Shen, UC Berkeley
Uses and applications
Corporates
Financial Institutions– Internal uses External uses
- Trading limits - Reporting- Capital Allocation - Capital requirements
- Self regulation
Self regulation and market disclosure– An alternative to the BIS's and the Fed's proposals, which may result in capital
inefficiency and mixed incentives
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Financial Risk Management
Prof. Jeff (YuQing) Shen, UC Berkeley
Trading Limits
Full system may include VaR limits, notional limits, types of securities, types of exposures,…
Management information and resource allocation– A unified measure of exposure at the trader, desk, group,... level– An input for capital allocation and reserve decisions
Implementation isn’t simple
Consider the case of using VaR for compensation/performance evaluation of trading divisions/desks
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Financial Risk Management
Prof. Jeff (YuQing) Shen, UC Berkeley
Example of UBS VaR
35
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Financial Risk Management
Prof. Jeff (YuQing) Shen, UC Berkeley
VaR for Reporting: UBS’s VaR
The 1996 time series of daily estimated VaR (defined as 2*STD)
Source: 1996 annual report
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Financial Risk Management
Prof. Jeff (YuQing) Shen, UC Berkeley
UBS’s realized P&L
AVG(P&L)=CHF9.2MM,
STD(P&L)=CHF6.8MM
Source: 1996 annual report
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Financial Risk Management
Prof. Jeff (YuQing) Shen, UC Berkeley
UBS Investment Bank VaR
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Financial Risk Management
Prof. Jeff (YuQing) Shen, UC Berkeley
UBS VaR
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Financial Risk Management
Prof. Jeff (YuQing) Shen, UC Berkeley
Did UBS manage its risk well?
40
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Financial Risk Management
Prof. Jeff (YuQing) Shen, UC Berkeley
Regulatory Environment
Objective: “To provide an explicit capital cushion for the price risks to which banks are exposed, particularly those arising from their trading activities… important further step in strengthening the soundness and stability of the international banking system and of financial markets generally” (Jan96 amendment to the BIS capital accord)
The use of internal models will be conditional upon explicit approval of the bank’s supervisory authority.
Criteria– General criteria re risk management systems– Qualitative criteria– Quantitative criteria– Criteria for external validation of models– Stress testing
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Financial Risk Management
Prof. Jeff (YuQing) Shen, UC Berkeley
General criteria
Risk mgt system conceptually sound
Sufficient skilled staff
(use of pricing, hedging, risk and trading models)
Models have proved track record in measuring risk
The bank regularly conducts stress tests
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Financial Risk Management
Prof. Jeff (YuQing) Shen, UC Berkeley
Qualitative Criteria Independent risk control unit
Regular back testing
Senior mgt actively involved in risk control
Consistency between risk measurement system and internal trading and exposure limits
Existence of a documented set of internal policies, controls and procedures concerning the operation of a risk measurement system
regularly independent review of risk measurement system by internal auditing
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Financial Risk Management
Prof. Jeff (YuQing) Shen, UC Berkeley
Quant Standards
No particular type of model required
(e.g., VarCov, HistSim, SMC…)
VaR on a daily basis– 99th %ile – 10 day horizon– Lookback at least 1yr
Discretion to recognize empirical corr within broad risk categories. VaR across these categories is to be aggregated (simple sum…)
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Financial Risk Management
Prof. Jeff (YuQing) Shen, UC Berkeley
Capital Requirements
CapRequ= MAX[VaR t-1, (Mult+AddOn)*AVG(VaRt,t-60)]
Mult=3.
AddOn related to past performance– Green zone 4/250 exceptions 1%VaR OK– Yellow zone up to 9/250 AddOn =0.3 To 1– Red zone 10plus/250 Investigation starts
Model must capture risk associated with options
Must specify risk factors and a price-factor mapping process a priori
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Financial Risk Management
Prof. Jeff (YuQing) Shen, UC Berkeley
Implications
Tradeoff between capital charge and accuracy– too low -- less capital needed, but risky– …and v.v.
Starts to affect portfolio choice, and thus traders/money managers, – move away from leverage– PRICES AFFECTED (we will hear more about that…)
Makes bank think about– business risk– stress risk (extreme moves)– catastrophic risk (stability)
… for more, see www.bis.org
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Financial Risk Management
Prof. Jeff (YuQing) Shen, UC Berkeley
OUTLINE
Introduction to VaR
– Statistical framework. Risk and diversification: some examples. Possible
applications. Visual interpretation.
The Stochastic Behavior of Asset Returns
– Time variations in volatility. VaR: approaches and comparison. The Hybrid Approach to VaR.
Beyond Volatility Forecasting
– The VaR of derivatives and interest rate VaR. Structured Monte Carlo. Extreme events and correlation breakdown. Stress testing and scenario analysis. Worst case scenario
![Page 48: Section 3: Risk Management for Investment Banks Financial Risk Management UC Berkeley Prof. Jeff (YuQing) Shen.](https://reader031.fdocuments.us/reader031/viewer/2022012922/56649efd5503460f94c1110e/html5/thumbnails/48.jpg)
Financial Risk Management
Prof. Jeff (YuQing) Shen, UC Berkeley
Visual interpretation
Negative CorrelationNegative Correlation
Positive CorrelationPositive Correlation
Ra
RbRbRpRp
RpRp
RbRb
Ra
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Financial Risk Management
Prof. Jeff (YuQing) Shen, UC Berkeley
Visual interpretation - corp’ bond example
treasury
spreadspreadAAbond
treasury
AAbond
Think, similarly, on the total risk of an FX equity investment
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Financial Risk Management
Prof. Jeff (YuQing) Shen, UC Berkeley
Visual interpretation - FX example
Long DEM Long GBPLong GBP
Short GBPShort GBPPositionPosition
VaRVaR
![Page 51: Section 3: Risk Management for Investment Banks Financial Risk Management UC Berkeley Prof. Jeff (YuQing) Shen.](https://reader031.fdocuments.us/reader031/viewer/2022012922/56649efd5503460f94c1110e/html5/thumbnails/51.jpg)
Financial Risk Management
Prof. Jeff (YuQing) Shen, UC Berkeley
OUTLINE
Introduction to VaR
– Statistical framework. Risk and diversification: some examples. Possible applications. Visual interpretation.
The Stochastic Behavior of Asset Returns– Time variations in volatility. VaR: approaches and comparison. The
Hybrid Approach to VaR.
Beyond Volatility Forecasting
– The VaR of derivatives and interest rate VaR. Structured Monte Carlo. Extreme events and correlation breakdown. Stress testing and scenario analysis. Worst case scenario
![Page 52: Section 3: Risk Management for Investment Banks Financial Risk Management UC Berkeley Prof. Jeff (YuQing) Shen.](https://reader031.fdocuments.us/reader031/viewer/2022012922/56649efd5503460f94c1110e/html5/thumbnails/52.jpg)
Financial Risk Management
Prof. Jeff (YuQing) Shen, UC Berkeley
The Stochastic Behavior of Asset Returns
1. The problem of f-a-t t-a-i-l-s
Time variations in volatility
3. VaR: approaches and comparison
4. The Hybrid Approach to VaR
5. Long horizon VaR
6. Benchmarking and backtesting VaR engines
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Financial Risk Management
Prof. Jeff (YuQing) Shen, UC Berkeley
How can we obtain the 5% tail move?
So far the answer was: VaR(5%)= 1.65 * VaR(5%)= 1.65 *
Asset returns are assumed to be
Stable ...but vol varies through time...but vol varies through time
and
Normal ...but they are not...but they are not
How do we make this determination?
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Financial Risk Management
Prof. Jeff (YuQing) Shen, UC Berkeley
3 months T-bill rate
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Financial Risk Management
Prof. Jeff (YuQing) Shen, UC Berkeley
Interest rate changes: are they normal?
N( 0, 7.3bp N( 0, 7.3bp 2 2 ))
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Financial Risk Management
Prof. Jeff (YuQing) Shen, UC Berkeley
The Tails of the Distribution
There ate, say 2500 ’s
Order them in ascending order
The 1%-ile is, under normality
7.3*2.33=17bp
Where should you find this “17”?
What do you actually find there?
11
22
33
24992499
25002500
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Financial Risk Management
Prof. Jeff (YuQing) Shen, UC Berkeley
Fat tails
If IR changes were normal: Prob( IR change >17bp ) = 1%
... but in reality Prob( IR change >21bp ) = 1%
===> “F -- A -- TF -- A -- T Tails”
This is especially true for
return series such as oil,
EM debt, some currencies...
The Effect subsides
gradually by aggregation– through time – cross sectionally
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Financial Risk Management
Prof. Jeff (YuQing) Shen, UC Berkeley
Why are tails so fat?
In trying to explain the fat tails, it could be the case that returns are simply fat tailed relative to the normal dist’n benchmark, or that returns are CONDITIONALLY NORMAL, but
1. expectations vary through time(...maybe, but not enough to explain the tails)
2. volatility varies through time(…it does, but is it enough to explain the tails?)
We follow 2 . . . but go back to “square one”
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Financial Risk Management
Prof. Jeff (YuQing) Shen, UC Berkeley
The Effect of Cyclical Vol.
We measure vol as 7.3bp/day
Suppose now that in fact
If in a given day 22bp, thendo we interpret is as
22/7.3=3sd, or 22/15=1.5sd ?
This is the key goal of dynamic VaR engines
1515
7.37.3
55tt
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Financial Risk Management
Prof. Jeff (YuQing) Shen, UC Berkeley
The Difficulty in Estimating Cyclical Vol.
Need few days of data to realize change in vol.measure vol
Question: how adaptable do you want to be
Tradeoff exists, and we shall elaborate on it now
“This is the key goal of dynamic VaR engines”
... And also the key difficulty...
1515
7.37.3
55tt
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Financial Risk Management
Prof. Jeff (YuQing) Shen, UC Berkeley
OUTLINE
Introduction to VaR
– Statistical framework. Risk and diversification: some examples. Possible applications. Visual interpretation.
The Stochastic Behavior of Asset Returns
–Time variations in volatility. VaR: approaches and comparison. The Hybrid Approach to VaR.
Beyond Volatility Forecasting
– The VaR of derivatives and interest rate VaR. Structured Monte Carlo. Extreme events and correlation breakdown. Stress testing and scenario analysis. Worst case scenario
![Page 62: Section 3: Risk Management for Investment Banks Financial Risk Management UC Berkeley Prof. Jeff (YuQing) Shen.](https://reader031.fdocuments.us/reader031/viewer/2022012922/56649efd5503460f94c1110e/html5/thumbnails/62.jpg)
Financial Risk Management
Prof. Jeff (YuQing) Shen, UC Berkeley
Modeling time-variations in VaR
PARAMETRIC APPROACH:
estimate the parameters of a given distribution– STD - simple historical vol + Conditional normality– Declining weights + Conditional normality (RiskMetrics)– Mixture of normals, t-distribution, GARCH...
NONPARAMETRIC APPROCH:
let the data talk --> estimate the entire distribution – Historical simulation
The Hybrid Approach
------------------------------------------------------------------------------------------------------
Finance-based forecasts (e.g., implied vol)
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Financial Risk Management
Prof. Jeff (YuQing) Shen, UC Berkeley
Volatility is cyclical
Long Run MeanLong Run Mean
volatilityvolatility
tt
Mandelbrot (1963)
“...large changes tend to be followed by large chages -- of either sign -- and small chages by small changes’’
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Financial Risk Management
Prof. Jeff (YuQing) Shen, UC Berkeley
Historical STD
Simple historical STDDEV is estimated by calculating the average of squared changes
t2=(1/K)(t
2+t-12 + t-2
2 +...+t-k+12 )
note that the weights sum up to one (why?)– Note1: it is common to use 1/(K-1)– Note2: the “suqared change” t-i
2 can be demeaned
squared returns (S-avg(S))2 , or non-demeaned (S)2
The choice of K involves a tradeoff between– acuuracy– adaptability
Weight
on
St,t+1 2
todaytoday
11//kk
kk
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Financial Risk Management
Prof. Jeff (YuQing) Shen, UC Berkeley
Time-varying vol
Three window sizes K=30, 60, 150 days,
are used to estimate STD
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Financial Risk Management
Prof. Jeff (YuQing) Shen, UC Berkeley
OUTLINE
Introduction to VaR
– Statistical framework. Risk and diversification: some examples. Possible applications. Visual interpretation.
The Stochastic Behavior of Asset Returns
– Time variations in volatility. VaR: approaches and comparison. The Hybrid Approach to VaR.
Beyond Volatility Forecasting
– The VaR of derivatives and interest rate VaR. Structured Monte Carlo. Extreme events and correlation breakdown. Stress testing and scenario analysis. Worst case scenario
![Page 67: Section 3: Risk Management for Investment Banks Financial Risk Management UC Berkeley Prof. Jeff (YuQing) Shen.](https://reader031.fdocuments.us/reader031/viewer/2022012922/56649efd5503460f94c1110e/html5/thumbnails/67.jpg)
Financial Risk Management
Prof. Jeff (YuQing) Shen, UC Berkeley
Exponential smoothing: the idea
Examples: RiskMetrics, GARCH
Idea: recent observations convey more
current information
==> use declining weights
Pros:– More weight on recent observations– Uses all the data
Cons:– Strong assumptions– Many parameters
to estimate
Weight
on St,t+1 2
todaytodaykk
11//kk
low low
HighHigh
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Financial Risk Management
Prof. Jeff (YuQing) Shen, UC Berkeley
Exponential smoothing: RiskMetricsTM
Simple historical STDDEV is estimated by calculating the average of squared changes
In RM volatility is a weighted average (with exp declining weights) of past changes squared:
t2=(1- )(t
2+t-12 + t-2
2 + t-32 +... )
note that the weights sum up to one (why?)
t2 can be also presented as
t2 = (1- ) t
2 + t-12
this is an “updating scheme” given last period’s estimate of vol and the news from last period till now
“Optimal” is “estimated”
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Financial Risk Management
Prof. Jeff (YuQing) Shen, UC Berkeley
Picking the “Best” Smoothing Parameter
For each day we have t and St-1,t .
The “error” is t2 - St-1,t
2
Mean Squared Error = Average[ (t2-St-1,t
2 ) 2]
– note RiskMetrics “alternative”
We look for lambda such that is minimizes the MSE
= MIN{ MSE( ) }
Would for oil be the same as for interest rates?
How do we reconcile the s for correlation
Solution: pick to fit all assets (???)
What is the problem?
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Financial Risk Management
Prof. Jeff (YuQing) Shen, UC Berkeley
RiskMetrics example
t2=(1- )(t
2+t-12 + t-2
2 + t-32 +... )
Let – weight1 (1-) =(1-.94) = 6.00%
– weight2 (1-) =(1-.94)*.94 = 5.64%
– weight3 (1-) =(1-.94)*.942 = 5.30%
– weight4 (1-) =(1-.94)*.943 = 4.98%
– ...– weight100 (1-) =(1-.94)*.9499 = 0.012%
– ...
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Financial Risk Management
Prof. Jeff (YuQing) Shen, UC Berkeley
RiskMetrics vol
Two smoothing params: 0.96 and 0.90
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Financial Risk Management
Prof. Jeff (YuQing) Shen, UC Berkeley
ARCH/GARCH(Engle 82, Engle Bollerslev 88)
Generalized AutoRegressive Conditional Heteroskedasticity
GARCH(1,1)
t2 = a+b t
2 +c t-12
Note the close relation to riskMetrics(let a=0, b=1- , c= )
Parameters are estimated via maximum likelihood
GARCH, by definition, is better in sample
...but out of sample???
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Financial Risk Management
Prof. Jeff (YuQing) Shen, UC Berkeley
Nonparametric approaches
Examples: neural nets, density estimations,…
Pros: very flexible structure
Cons: data intensive ==> possibly large estimation error with limited data
Example: estimate the changes in interest rates, CONDITIONAL on the level, the spread and vol
...to the extent that level and spread have information on the future path of rates, we “learn” from the past on the conditional distribution of interest rate changes
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Financial Risk Management
Prof. Jeff (YuQing) Shen, UC Berkeley
Weights on past t-i2
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Financial Risk Management
Prof. Jeff (YuQing) Shen, UC Berkeley
Historical Simulation
IN REALITY:
Returns could be fat-tailed and skewed
Correlations at the extremes may be misestimated
It is extremely difficult to model and estimate these effects
====>====> LET THE DATA “TELL” USLET THE DATA “TELL” US
METHODOLOGY:
Recalculate the value of your CURRENT portfolio during the last 100 (or 250) periods
The 5% VaR is – The 5th lowest observations of the recent 100, or– The 12th-13th lowest observation of the recent 250, or ...
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Financial Risk Management
Prof. Jeff (YuQing) Shen, UC Berkeley
Historical Simulation
Pros:
– (almost) assumption-free: we make no distributional assumptions
– (almost) no parameters: no more vol, no more corr
HS works in the presence of skewness, fat tails,...
Cons:
– Very little data is used (e.g, the bi-weekly 1%VaR)– Stale information lingers (long flat VaRs are typical) – Extrapolation from 1-day-VaR to J-day-VaR impossible
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Financial Risk Management
Prof. Jeff (YuQing) Shen, UC Berkeley
OUTLINE
Introduction to VaR
– Statistical framework. Risk and diversification: some examples. Possible applications. Visual interpretation.
The Stochastic Behavior of Asset Returns
– Time variations in volatility. VaR: approaches and comparison. The Hybrid Approach to VaR.
Beyond Volatility Forecasting
– The VaR of derivatives and interest rate VaR. Structured Monte Carlo. Extreme events and correlation breakdown. Stress testing and scenario analysis. Worst case scenario
![Page 78: Section 3: Risk Management for Investment Banks Financial Risk Management UC Berkeley Prof. Jeff (YuQing) Shen.](https://reader031.fdocuments.us/reader031/viewer/2022012922/56649efd5503460f94c1110e/html5/thumbnails/78.jpg)
Financial Risk Management
Prof. Jeff (YuQing) Shen, UC Berkeley
The Hybrid Approach(the best of both worlds)
Estimates VaR by applying exponentially declining weights to the past return series
... and build a nonparametric time-weighted distribution
Intuition:
– If the lowest 5 returns occurred recently (e.g., between t-1 and t-10), VaR should be higher than if they occurred long ago (e.g., t-70 and t-100)
– If the latter is true, give these lowest returns less weight -- keep on aggregating up
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Financial Risk Management
Prof. Jeff (YuQing) Shen, UC Berkeley
The hybrid approach
As in Historical Simulation:
– (+) almost assumption-freealmost assumption-free– (+) OK with fat tails, skewness...
as in EXP:
– (+) recent observations weigh morerecent observations weigh more– (+) OK for cyclical volatility
– (--) little data is used for low % VaRs– (--) difficult to obtain j-period VaRs
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Financial Risk Management
Prof. Jeff (YuQing) Shen, UC Berkeley
The hybrid approach: implementation Step 1:
– denote by R(t) the realized return from t-1 to t – To the most recent K returns: R(t), R(t-1),...,R(t-K+1),
assign a probability weight C*1, C*, ..., C*
– ( C=[(1-)/(1-)] ensures that the weights sum to 1 )
Step 2:
– Order the returns in ascending order
Step 3:
– To obtain the x% VaR of the portfolio, start from the lowest return and accumulate weights until x% is reached
– Linear interpolation is used between adjacent points to achieve exactly x% of the distribution
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Financial Risk Management
Prof. Jeff (YuQing) Shen, UC Berkeley
Example
Order Return Periods ago
Weight Cumul. weight
Weight Cumul. weight
Initial Date:
1 -3.30% 3 0.0221 0.0221 0.01 0.01 2 -2.90% 2 0.0226 0.0447 0.01 0.02 3 -2.70% 65 0.0063 0.0511 0.01 0.03 4 -2.50% 45 0.0095 0.0605 0.01 0.04 5 -2.40% 5 0.0213 0.0818 0.01 0.05 6 -2.30% 30 0.0128 0.0947 0.01 0.06
25 Days Later:
1 -3.30% 28 0.0134 0.0134 0.01 0.01 2 -2.90% 27 0.0136 0.0270 0.01 0.02 3 -2.70% 90 0.0038 0.0308 0.01 0.03 4 -2.50% 70 0.0057 0.0365 0.01 0.04 5 -2.40% 30 0.0128 0.0494 0.01 0.05 6 -2.30% 55 0.0077 0.0571 0.01 0.06
)
Hybrid: initial 5% VaR ==> 2.73%
25d later 5% VaR ==> 2.34%
HS (k=100d): 5% VaR ==> 2.35%
HybridHybrid H. S.H. S.
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Financial Risk Management
Prof. Jeff (YuQing) Shen, UC Berkeley
Results: BOVESPA
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Financial Risk Management
Prof. Jeff (YuQing) Shen, UC Berkeley
VaR and aggregation
11
TT
ww11 wwnn Portfolio returnsPortfolio returns
Aggregation ==>
“simulated returns”
VAR-COV
Estimation
Weights+
Parameters+
Normality
VaRVaR
VAR only
Estimation+
Normality
Ordered
“simulated”
returns
VaR =
x% observation
D a
t a
D a
t a
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Financial Risk Management
Prof. Jeff (YuQing) Shen, UC Berkeley
Implied vol as a vol predictor
Pros: – Uses all relevant information– Completely structural
Cons:– Not available for all assets, and model is asset-specific– (almost) no correlations– Model (Black-Scholes, HJM, HW) may not apply
Is the model biased?– Often implied > realized
– There is no one “implied”– Option prices compensate for crash premium, stochastic vol risk,...– What do we make of implied as a predictor of
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Financial Risk Management
Prof. Jeff (YuQing) Shen, UC Berkeley
Implied vol: the GBP crash of 1992
DMDM/L/L
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Financial Risk Management
Prof. Jeff (YuQing) Shen, UC Berkeley
VIX, a better risk forecast?
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Financial Risk Management
Prof. Jeff (YuQing) Shen, UC Berkeley
Normalization
Take each IR change and divide it by its pre-estimated vol
it,t+1 / t should be distributed N(0,1)N(0,1)
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Financial Risk Management
Prof. Jeff (YuQing) Shen, UC Berkeley
Long horizon vol
What is the J-period CONDITIONAL variance of St,t+J ?
Recall:St,t+2=St,t+1 +St+1,t+2
(using cont’ comp’ returns)
Under what assumptions do we obtain the SQRT-J rule?
JJ-day VaR = SQRT(-day VaR = SQRT(JJ) *(1day VaR)) *(1day VaR)
Recall:
VAR(St,t+1+St+1,t+2) = VAR(St,t+1) + VAR(St+1,t+2)
+2 COV(St,t+1,St+1,t+2 )
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Financial Risk Management
Prof. Jeff (YuQing) Shen, UC Berkeley
Long horizon vol: assumptions
VAR(St,t+1+St+1,t+2)=VAR(St,t+1)+VAR(St+1,t+2)+2COV(St,t+1,St+1,t+2 )
To obtain the “SQRT-SQRT-JJ rule rule” we need to assume
– A1: COV(St,t+1 ,St+1,t+2 )=0
– A2: VAR(St,t+1)=VAR(St+1,t+2)
With these assumptions:
VAR(St,t+1+St+1,t+2)=2VAR(St,t+1)=2 t2
==> STD(St,t+2)=SQRT(2) t
...and so on for J-day returns
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Financial Risk Management
Prof. Jeff (YuQing) Shen, UC Berkeley
The empirical record of the “SQRT(J) rule”
The reliability of the “SQRT(J) rule” depends on the reliability of the assumptions
A1: COV(St,t+1 ,St+1,t+2 )=0
“no predictability”, or “no mean-reversion”
A2: VAR(St,t+1)=VAR(St+1,t+2)
“constant volatility” or “no mean-reversion in volatility”
We need to determine:
When would you expect A1 or A2 not to work?
Is there a predictable bias?
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Financial Risk Management
Prof. Jeff (YuQing) Shen, UC Berkeley
No predictability assumption
A1: COV( St,t+1 , St+1,t+2 )=0
Holds true for most financial series (e.g., stock prices, FX)
However, interest rates DO exhibit mean reversion
==> COV( St,t+1 , St+1,t+2 ) <?> 0
==> 44-quarter VaR <?> SQRT(-quarter VaR <?> SQRT(JJ) *(1qtr VaR)) *(1qtr VaR)
Long Run MeanLong Run Mean
ii
tt
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Financial Risk Management
Prof. Jeff (YuQing) Shen, UC Berkeley
Constant vol assumption
A2: VAR(St,t+1)=VAR(St+1,t+2)
Most financial assets exhibit mean-reverting volatility
==> VAR(St,t+1)<?>VAR(St+1,t+2)
==> 44-quarter VaR <?> SQRT(-quarter VaR <?> SQRT(JJ) *(1qtr VaR)) *(1qtr VaR)
Long Run MeanLong Run Mean
volatilityvolatility
tt
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Financial Risk Management
Prof. Jeff (YuQing) Shen, UC Berkeley
Mean reversion: example
Xt+1=a+bXt+et+1
STDt(Xt,t+1)= STDt(a+bXt+et+1 - Xt)= t
SPSE b=0.9, t=10%
==> STDt(Xt,t+1)=10%
Xt,t+2= ... (write Xt+2 in terms of Xt+1, then in terms of Xt)
VAR t(Xt,t+2)=(1+b2) t2 =(1+0.81)*(10%)2
==> STDt(Xt,t+2)=1.34*(10%)
Lower than the SQRT-J rule volatility:
1.41* (10%)
Especially relevant with short term arbitrage strategies
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Financial Risk Management
Prof. Jeff (YuQing) Shen, UC Berkeley
Regulatory backtesting
Intended to satisfy regulatory requirements and minimize the chance of an equity penalty
Based on “theoretical P&L”
==> intraday results have no effect on backtesting
Regulatory equity =
= MAX [ VaRt ; (avg of 60 past days VaR)*(Mult_factor) ]
Mult_factor is a function of the number of violations
==> no use of SIZE of large P&L
==> no use of P&L when VaR limit not violated
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Financial Risk Management
Prof. Jeff (YuQing) Shen, UC Berkeley
Internal backtesting: goals
Ensure sucessful regulatiry backtest – ...dynamic and path dependent control of the prob of a Mult_factor increase – VaR manipulation/management industry should arise
Ensure true risk control– Test distributional assumptions (vol, corr, normality, iid-ness)– Update and maintain VaR model – Help in monitoring trading groups’ pricing models (quotes, curves, factors,
approximations,...)– Prevent possible stress (identify “holes”, assess liquidity and ability to reverse
positions, detect “digital exposures”)
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Financial Risk Management
Prof. Jeff (YuQing) Shen, UC Berkeley
Benchmarking & backtesting VaR: Methodology
By definition, at any given period the following must hold:
Prob[R(t+1)<-VaR(t)]=x%
Benchmarking and backtesting is done by observing the properties of the frequency and size of VaR violations
Define: I(t)=1I(t)=1 if the if the VaR(t) VaR(t) is exceeded, 0 otherwise is exceeded, 0 otherwise
Attributes:
Unbiasedness:
– Unconditional: avg[I(t)]=x%– Conditional: low Mean Absolute Error
Proper Updating: I(t) should be i.i.d..
==> Autocorr[ I(t) ] = 0
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Financial Risk Management
Prof. Jeff (YuQing) Shen, UC Berkeley
OUTLINE
Introduction to VaR
– Statistical framework. Risk and diversification: some examples. Possible applications. Visual interpretation.
The Stochastic Behavior of Asset Returns
– Time variations in volatility. VaR: approaches and comparison. The Hybrid Approach to VaR.
Beyond Volatility Forecasting–The VaR of derivatives and interest rate VaR.
Structured Monte Carlo. Extreme events and correlation breakdown. Stress testing and scenario analysis. Worst case scenario
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Financial Risk Management
Prof. Jeff (YuQing) Shen, UC Berkeley
The VaR of derivatives: introduction
A derivative is priced off of an underlying asset
==> Changes in the value of a derivatives are derived from changes in the underlying (and the “factor(s)”moving the underlying)
Linear derivatives: derivative is linear in factor(s) – P= + Delta f ==> P = Delta *f – forwards, futures, swaps
NON-linear derivatives: P is Nonlinear in factor(s)
– P = Delta(X t) f ) , where is a state dependent variable (e.g., the level of interest rates, the “moneyness” of the option)
– options, MBSs, Bradys, caps/floors
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Financial Risk Management
Prof. Jeff (YuQing) Shen, UC Berkeley
How to calculate the VaR of derivatives?
If linear: straightforward – P= + Delta f
==> P = Delta *f ==> VaRP = Delta * VaRF
Every asset is LOCALLY linear
…but for large moves (long horizon VaRs, stress scenarios,...) nonlinearity matters
Two methods/approaches to address nonlinearity
– Full valuation (usually in conjunction with structured Monte Carlo)
– Approximation to the nonlinearity effect (“the Greeks” using Taylor expansion)
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Financial Risk Management
Prof. Jeff (YuQing) Shen, UC Berkeley
Linear derivatives: the VaR of FX forwards
FX forward contract: exchange $F for DM1 in at t+T
Forwards are priced via covered parity: Ft,T = St (It,T/I*t,T)
It is derived by arbitrage, using the fact that the following are equivalent: – purchase DM forward – short $bond at It,T, convert into DM, long DMbond at I*
t,
In log terms F = S + I - I*
– in words: the change in the value of a forward contract is equivalent to (by arbitrage) the change in the spot rate, plus the change in the $ bond, less the change in the DM bond
==> The VaR of the forward depends linearly on the vol and corr across the three variables: [St , $bond , DMbond ]
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Financial Risk Management
Prof. Jeff (YuQing) Shen, UC Berkeley
The VaR of an FX forward: example
Recall: F = St + I - I*
Hence: 2F= 2
S + 2I
+ 2I*
+2cov(S,I)-2cov(S,I*)-2cov(I, I*)
Example:– S=$0.555/DM, I=5%, I*=3%, T=1yr ==> F=$.566/DM– Notional amount DM1.8MM=$1MM– Suppose Sbp/day, 2
I2I*
bp/day, CORR=0
– VaRS= $7000*1.65=11,550,
– VaRI=VaRI*= 700*1.65=1,155
– VaRF = sqrt(11,550 2 + 1,155 2 + 1,155 2) =$11,664
– Why is VaRF so close to VaRS ?
(NOTE: these calculations are approximate! we can be more precise with respect to the underlying bond’s notional etc. The are correct to a first approximation, though)
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Financial Risk Management
Prof. Jeff (YuQing) Shen, UC Berkeley
Linear derivatives: The VaR of Swaps
An FX swap is also a linear derivative. Why?
A long swap position can be decomposed into– a sequence (hence, the linear sum) of FX forwards – a portfolio of:
long a domestic coupon bond, short an FX coupon bond, and a spot position
Hence the linearity in the spot rate and bond prices
The VaR of a 5yr semi annually paying swap may involve 10 domestic rates, 10 foreign rates, and spot the rate
...hence 21 volatilities and 210 correlations (20*21/2)
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Financial Risk Management
Prof. Jeff (YuQing) Shen, UC Berkeley
Nonlinear derivatives
Recall linear derivatives: P = DELTA *F
In the case of nonlinear derivatives, the DELTA is state dependent:
P = DELTA(state variables) *F
Examples (by increasing complexity):– Bond are nonlinear in interest rates– Options are nonlinear in the underlying– Convertibles are nonlinear in the underlying– callable & convertible bonds are nonlinear in the underlying and in interest rates– Defaultable (e.g., Brady) bonds are nonlinear in the default probabilities – MBSs are nonlinear in interest rates
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Financial Risk Management
Prof. Jeff (YuQing) Shen, UC Berkeley
The problem with duration
PP
YYYY00 Y’Y’
PP00
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Financial Risk Management
Prof. Jeff (YuQing) Shen, UC Berkeley
Duration + convexity
PP
YYYY00 Y’Y’
PP00
Duration + Duration +
convexityconvexity Full valuation / true priceFull valuation / true price
Duration
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Financial Risk Management
Prof. Jeff (YuQing) Shen, UC Berkeley
The VaR of options: small moves
xxsst+Tt+T
for small changes in the underlying, for small changes in the underlying,
the option is nearly linear, and delta approx. the option is nearly linear, and delta approx.
to the VaR is enoughto the VaR is enough
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Financial Risk Management
Prof. Jeff (YuQing) Shen, UC Berkeley
The VaR of options: large moves
xxsst+Tt+T
For large changes in the underlying, For large changes in the underlying,
the option is not linear, and delta approx. the option is not linear, and delta approx.
to the VaR is not enoughto the VaR is not enough
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Financial Risk Management
Prof. Jeff (YuQing) Shen, UC Berkeley
The VaR of options: convexity correction
xxsst+Tt+T
for large changes in the underlying, for large changes in the underlying,
the option is nonlinear in the underlying, the option is nonlinear in the underlying,
==> use delta+gamma approximation, ==> use delta+gamma approximation,
or full revaluationor full revaluation
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Financial Risk Management
Prof. Jeff (YuQing) Shen, UC Berkeley
The VaR of a portfolio of derivatives
As the previous example shows, the problem is not only nonlinearities, but also
nonmonotonicities (17 letters)
The problem with full the revaluation approach is its computational cost/time– we need to cover the entire range of the distribution ==> simulation– with N state variables (interest rates, exchange rates, default spreads, etc), we need to
revalue the portfolio thousands of times.
e.g., revalue MBSs, caps, swaptions etc for 10,000 scenarios– solution: reduce the number of states
e.g., the level and spread (= 2 factors) may suffice to describe the entire term structure
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Financial Risk Management
Prof. Jeff (YuQing) Shen, UC Berkeley
The Risk of Derivatives -Summary
Approaches
The “Delta-Normal” approach– The “Greeks”
Full revaluation– Historical simulation– Stress testing and scenario analysis– Structured Monte Carlo
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Financial Risk Management
Prof. Jeff (YuQing) Shen, UC Berkeley
The “Delta-Normal” approach
Assumes all returns are normal
Pros– relatively easy and computationally efficient
Cons– Delta-approx’ driven -- needs some fixing– Inaccurate when no closed form– Highly model dependent
no event risk no fat tails non-factor changes not accounted for
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Financial Risk Management
Prof. Jeff (YuQing) Shen, UC Berkeley
Full Revaluation
May be computationally intensive for large portfolios and many factors ( => need to simulate the “full distribution”)
Historical simulation apply current portfolio weights to past values of factor changes fully revalue get the empirical tail …this is the basis for the Basel standard
– Pros realistic tails, accounting for “true dist’n, correlation breakdowns, nonlinearities computationally not as bad as SMC
– Cons Weighting (hybrid is a solution long horizons very problematic
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Financial Risk Management
Prof. Jeff (YuQing) Shen, UC Berkeley
OUTLINE
Introduction to VaR
– Statistical framework. Risk and diversification: some examples. Possible applications. Visual interpretation.
The Stochastic Behavior of Asset Returns
– Time variations in volatility. VaR: approaches and comparison. The Hybrid Approach to VaR.
Beyond Volatility Forecasting
– The VaR of derivatives and interest rate VaR. Structured Monte Carlo. Extreme events and correlation breakdown. Stress testing and scenario analysis. Worst case scenario
![Page 114: Section 3: Risk Management for Investment Banks Financial Risk Management UC Berkeley Prof. Jeff (YuQing) Shen.](https://reader031.fdocuments.us/reader031/viewer/2022012922/56649efd5503460f94c1110e/html5/thumbnails/114.jpg)
Financial Risk Management
Prof. Jeff (YuQing) Shen, UC Berkeley
Structured Monte Carlo: basic intuition
Generating scenarios for one variable which is N( , )– generate 10,000 simulations of N(0,1). Denote: Z1 , ... Z10,000
– calculate scenarios St+1,i = St *exp { + Zi}
– Revalue the derivative for each St+1,i
Generating scenarios for K variable which are N(M,)
(where M is a vector of length K, and is a K by K matrix)– generate 10,000 K-vectors N(0,IK)’s: Z1 , ... Z10,000
(where IK is K by K unit matrix)– calculate scenarios St+1,i = St *exp {M + A’ Zi }
(where A is the “square root matrix” of , namely A’A= )– Revalue the derivatives for each St+1,i set of values
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Financial Risk Management
Prof. Jeff (YuQing) Shen, UC Berkeley
Structured Monte Carlo: discussion
The main advantage: correlated scenarios
Compare to independent scenario analysis– a 200bp shift in interest rates, a 25% decline in equities, a 500bp increase in the Brady
strip spread,...– what do we make of such isolated scenarios? do they make economic sense?
The main disadvantage: correlation breakdown– what happens to global yield correlation during an oil crisis?– what happens to strip spreads during an EM currency crisis?– what happens to corporate--equity correlation during an equity crisis?
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Financial Risk Management
Prof. Jeff (YuQing) Shen, UC Berkeley
Structured Monte Carlo
Simulated scenarios according to current var-cov matrix
Full revaluation each time
Pros role to probabilities
Cons var-cov matrix unstable computationally time-consuming
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Financial Risk Management
Prof. Jeff (YuQing) Shen, UC Berkeley
OUTLINE
Introduction to VaR
– Statistical framework. Risk and diversification: some examples. Possible applications. Visual interpretation.
The Stochastic Behavior of Asset Returns
– Time variations in volatility. VaR: approaches and comparison. The Hybrid Approach to VaR.
Beyond Volatility Forecasting
– The VaR of derivatives and interest rate VaR. Structured Monte Carlo.
Extreme events and correlation breakdown. Stress testing and scenario analysis. Worst case scenario
![Page 118: Section 3: Risk Management for Investment Banks Financial Risk Management UC Berkeley Prof. Jeff (YuQing) Shen.](https://reader031.fdocuments.us/reader031/viewer/2022012922/56649efd5503460f94c1110e/html5/thumbnails/118.jpg)
Financial Risk Management
Prof. Jeff (YuQing) Shen, UC Berkeley
Correlation breakdown
Event Date Variables Prior to During
ERM Sep92 GBP/$ , GBP LIBOR -0.10 0.75
Mexico Dec94 Peso/$ , 1mo Cetes 0.30 0.80
87 crash Oct87 Junk yield , 10yr Treasury 0.80 -0.70
Iraq Aug90 10yr JGBs , 10yr Treasury 0.20 0.80
AsianCrisis
1997/8 Brady debt of Bulgaria andthe Philippines
0.04 0.84
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Financial Risk Management
Prof. Jeff (YuQing) Shen, UC Berkeley
Generating scenarios: “reasonable stress”
SDs Normaldist’n
S&P500
Yen/$ 10yr Rate
2 4500 3700 5600 5700
3 270 790 1300 1300
4 6.4 440 310 240
5 0 280 78 79
6 0 200 0 0
Odds in 100,000
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Financial Risk Management
Prof. Jeff (YuQing) Shen, UC Berkeley
OUTLINE
Introduction to VaR
– Statistical framework. Risk and diversification: some examples. Possible applications. Visual interpretation.
The Stochastic Behavior of Asset Returns
– Time variations in volatility. VaR: approaches and comparison. The Hybrid Approach to VaR.
Beyond Volatility Forecasting
– The VaR of derivatives and interest rate VaR. Structured Monte Carlo.
Extreme events and correlation breakdown. Stress testing and scenario analysis. Worst case scenario
![Page 121: Section 3: Risk Management for Investment Banks Financial Risk Management UC Berkeley Prof. Jeff (YuQing) Shen.](https://reader031.fdocuments.us/reader031/viewer/2022012922/56649efd5503460f94c1110e/html5/thumbnails/121.jpg)
Financial Risk Management
Prof. Jeff (YuQing) Shen, UC Berkeley
Generating stress scenarios in practice
Common practice: examine historical events
Links to the historical simulation methodology:– use HS to generate the empirical distribution for the current portfolio– use HS to examine the “five worst weeks” given the current portfolio, when did they occur,
and what were the circumstances
Remember: there is no way to apply common statistical techniques, with so few (and economically different) data points
However: extreme value theory is now commonly applied to the problem
Its usefulness is very questionable
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Financial Risk Management
Prof. Jeff (YuQing) Shen, UC Berkeley
Stress Testing at XYZ
Regular + ad hoc stress tests
Regular tests– Everything falls in value
yields go up, IR spreads widen, currencies fall vs USD, Vols riseEverything rises in value
yields go down, spreads narrow, currencies rise, Vols rise“SUB-Scenarios”
- EM Crisis: all equities down, EM yields up, Spread widen, industrialized yields down + (1) currency pegs stable, (2) currency pegs collapse:
- General Recovery/G7 Bond Crash: all equities up, EM yields down, Spread narrow, industrialized yields up + (1) currency pegs stable, (2) currency pegs collapse:
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Financial Risk Management
Prof. Jeff (YuQing) Shen, UC Berkeley
(*) Structural economic models
Consider the following structural model for bond yields
DOM yield: Yt= Dt+Wt , FOR yield: Yt*= Dt
*+Wt
With orthogonal factors,
corr(Y, Y*)= 2W / sqrt{(2
D+2W)*(2
D*+2W)}
what happens to corr when:– DOM factor volatility explodes?
– WORLD factor volatility explodes?
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Financial Risk Management
Prof. Jeff (YuQing) Shen, UC Berkeley
(*) Structural models for derivatives
S&P500: S10/87=330, S11/87=250
Vola: 10/87=.125, 11/87=.25
Other data: Rf=.075, T=1yr, X=330
– C(S=330, X=330, =.125) = 30– C(S=250, X=330, =.125) = 0.76– C(S=250, X=330, =.25)= 8.12
==> In extreme market conditions it is crucial
to account for changes in Vola
This could be achieved by correlating vola and prices
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Financial Risk Management
Prof. Jeff (YuQing) Shen, UC Berkeley
Asset & credit concentration
Another critical aspect is diversification– Consider, for example, a portfolio of MBSs, Bradys & junk– We can quantify the systematic ($ Interest rate) and total risk– These depend on modeling assumptions
Some asset classes have key variables which are not well understood– CMOs -- prepayment -- 1994– Bradys -- comovements -- 1994, 1997– Junk -- correlations -- 1987
==> Asset concentration is an issue due to model risk or liquidity risk, outside the common set of VaR models
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Financial Risk Management
Prof. Jeff (YuQing) Shen, UC Berkeley
Asset & credit concentration: solutions
The treatment of asset-class concentration is by – monitoring large exposures – conducting stress tests– (similar to credit risk monitoring)
The assumption is that model/liquidity risk is diversifiable
Do we leave systematic risk out (e.g., MBSs) ?
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Financial Risk Management
Prof. Jeff (YuQing) Shen, UC Berkeley
Stress testing and scenario analysis
Subjective scenarios taken out of “thin air”+full revaluation yield curve shifts and twists, crashes, …
Recommended by all standards/proposals
Pros covered rare/unseen events
Cons no economic guidance no sense of probabilities questionable for multiple factors
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Financial Risk Management
Prof. Jeff (YuQing) Shen, UC Berkeley
OUTLINE
Introduction to VaR
– Statistical framework. Risk and diversification: some examples. Possible applications. Visual interpretation.
The Stochastic Behavior of Asset Returns
– Time variations in volatility. VaR: approaches and comparison. The Hybrid Approach to VaR.
Beyond Volatility Forecasting
– The VaR of derivatives and interest rate VaR. Structured Monte Carlo. Extreme events and correlation breakdown. Stress testing and scenario
analysis. Worst case scenario
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Financial Risk Management
Prof. Jeff (YuQing) Shen, UC Berkeley
The Distribution of the Worst Case
A major critique of VaR: it simply asks the wrong question.
VaR(5%) happens on average 5 in 100 periods
… but perhaps more, perhaps less,
…and when it does, how bad does it look ?
http://www.faculty.idc.ac.il/kobi/wcsrisk.pdf
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Financial Risk Management
Prof. Jeff (YuQing) Shen, UC Berkeley
VaR and the worst case scenario
Over the next 100 bi-weekly periods, what is the worst that will happen to the value of the firm's trading portfolio (or the collateral value of a SPV)?
Assume: trading portfolios are adjusted to maintain the same fraction of capital invested (bet more when you make money)
VaR tells us that losses greater than -2.33, the 1%tile of the portfolio's value, will occur, on average, once over the next 100 trading periods
Unanswered questions:
– what is the size of these losses?– how often may they occur
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Financial Risk Management
Prof. Jeff (YuQing) Shen, UC Berkeley
The worst will happen
WCS analyzes the distribution of losses during the worst trading period
Key conceptual point: aa worst period will occur worst period will occur with probability
one! The only question is how bad will it be?
!!!!!!
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Financial Risk Management
Prof. Jeff (YuQing) Shen, UC Berkeley
Analysis
Generate 10,000 vectors of size H of random N(0,1)'s (interpreted as the normalized trading returns)
Analyze the distribution the worst observation Zi of each vector
......
zz1 1 zz2 2 zz3,3,...,..., zz10,00010,000
11
HH
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Financial Risk Management
Prof. Jeff (YuQing) Shen, UC Berkeley
Results
H=100 (other results in the paper)
The expected number of Zs < -2.33 is 1.00 (1% VaR)
The distribution of the worst case:
AVG(Z)=-2.5150% 10% 5% 1%
-2.47 -3.08 -3.28 -3.72
In words: Over the next 100 trading periods a return worse than -2.33 is expected to occur once, when it does, it is expected to be of size -2.51, but with probability 1% it might be -3.72 or worse (i.e., we focus on the 1%tile of the Z's).
Note: WCS=VaR*1.6WCS=VaR*1.6
Results on bonds and bond options: see enclosed paper
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Financial Risk Management
Prof. Jeff (YuQing) Shen, UC Berkeley
Applications
Relevant for “prudence multipliers”, applied by the regulator: (2week VaR) * 3 = Capital (2week VaR) * 3 = Capital requirementrequirement
– The * 3 * 3 multiplier is pulled out of thin air...
Need to account for time varying vol, fat tails, corr breakdown,...
Can provide – a better understanding of the riskiness of financial institutions– a control over desired levels of “prudence” and systemic risk– a safer and more capital efficient financial institutions
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Financial Risk Management
Prof. Jeff (YuQing) Shen, UC Berkeley
JP Morgan Risk Management Case
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Financial Risk Management
Prof. Jeff (YuQing) Shen, UC Berkeley
JP Morgan Chase
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Financial Risk Management
Prof. Jeff (YuQing) Shen, UC Berkeley
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Financial Risk Management
Prof. Jeff (YuQing) Shen, UC Berkeley
VaR
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Financial Risk Management
Prof. Jeff (YuQing) Shen, UC Berkeley
Trading revenue revenue
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Financial Risk Management
Prof. Jeff (YuQing) Shen, UC Berkeley
Stress tests
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Financial Risk Management
Prof. Jeff (YuQing) Shen, UC Berkeley
Limits and models
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Financial Risk Management
Prof. Jeff (YuQing) Shen, UC Berkeley
142
The BIG Picture
• Theory
•Efficient Frontier
•Asset/Liability Framework
•Alpha/Beta risk budgeting
• Practice
•Pension Crisis
•GM Pension Plan
•Yale Endowment
• Theory
•VaR
•Stochastic behavior of asset
returns
•Worst case scenario
• Practice
•Investment bank VaR
•Bear Stearns
• The Theory
•Active Portfolio Management
•Quantitative equity models
•Hedge fund investment
• The Practice
•Hedge fund industry
•Aug 2007 crisis for quant
Investment BankInstitutional Investor Hedge Fund