Value-at-Risk and Expected Shortfall
Presentation at Finansforening’s Network for Performance Measurement
Copenhagen, June 8, 2015
Søren Plesner, CFA
[email protected]
1
The Moderator
•
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•
•
•
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Soren Plesner
M.Sc. (Economics)
CFA, FRM & PRM
External Lecturer at the Copenhagen Business School
Founder of Upside (SPFK Financial Knowhow)
Previously
–
–
–
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BASISPOINT
SimCorp
Danske Bank
IBM
• www.spfk.dk
• [email protected]
2
Some people Don’t Like CFA’s!
•
So when you see a quantitative “expert”, shout for help, call for his disgrace, make
him accountable. Do not let him hide behind the diffusion of responsibility. Ask for
the drastic overhaul of business schools (and stop giving funding). Ask for the
Nobel prize in economics to be withdrawn from the authors of these theories, as
the Nobel’s credibility can be extremely harmful. Boycott professional associations
that give certificates in financial analysis that promoted these methods
–
Nassim Nicholas Taleb and Pablo Triana i Financial Times den 7. december 2008
3
Not Much Confidence in VaR Either!
•
Remove Value-at-Risk books from the shelves – quickly. Do not be afraid for your
reputation. Please act now. Do not just walk by. Remember the scriptures: “Thou
shalt not follow a multitude to do evil”
–
Nassim Nicholas Taleb and Pablo Triana i Financial Times den 7. december 2008
4
Outline
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•
•
•
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What Is Value at Risk (VaR)?
Problems with VaR
Value at Risk in Financial Regulation
The Move to Expected Shortfall
Performance Measurement
– In banks
– In asset portfolios
5
Value-at-Risk
• VaR is defined as the maximum potential change in value of a portfolio of
financial instruments with a given probability over a certain horizon –
under “normal” market conditions!
Distribution of
portfolio’s value
5% probability
of loss > VaR
VaR (95%)
Current value
6
VaR – Important Applications
• Risk management
– Risk assessment
– Limit setting
• Regulatory requirements
– Trading book capital charges
• Evaluating the performance of risk takers
–
–
–
–
RAPM
RAROC
Reward to VaR ratio
………
7
Steps in Constructing VaR
•
•
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Mark-to-market of current position/portfolio
Measure the variability of the risk factor(s)
Set the time horizon
Set the confidence level
Calculate VaR for single position(s)
Calculate total portfolio VaR using correlation matrix
Report maximum potential loss (VaR)
Perform stress testing of assumptions
8
Financial Markets – Empirical Facts
• Financial return distributions are leptokurtotic
– have heavier tails and a higher peak than a normal distribution.
• Returns are typically negatively skewed
• Squared returns have significant autocorrelation
– i.e. volatilities of market factors tend to cluster.
9
A “Typical” Financial Time Series of Returns
10
Problems with VaR
•
•
VaR measures maximum loss at a specified confidence level under normal
market conditions
VaR does not measure the possible extent of losses beyond VaR – and
very often, this is where the action is!
?
VaR
11
Surprised?
• ”We are seeing things that were 25-standard deviation moves,
several days in a row”
– David Viniar, CFO of Goldman Sachs, August 2007
12
Even the Best May Fail!
• May, 2012
13
VaR Methodologies – Broad Categories
• Parametric (RiskMetrics and GARCH)
• Extreme Value Theory
• Nonparametric
– Historical Simulation
– Monte Carlo simulation
14
VaR in Financial Banking Regulation (Basel)
• VaR was introduced I banking regulation in the “market risk amendment”
in 1996
• Banks’ balance sheets were divided into “banking books” and a “trading
books”
• 99% 10-day VaR  capital charge for “market risk”
– Subject to backtesting an stress testing
• But VaR (and stress testing failed completely to capture risks in the run-op
to the financial crisis
• Regulatory reactions
– Basel 2.5
– Trading Book Review
15
16
Regulatory Tsunami?
# of Pages
1200
1000
800
600
400
200
0
Basel I
Basel II
Basel III
Basel IV?
Basel V?
17
Basel III Framework
18
The Basel III Framework – New Developments
TBR 
Basel
3.5?
Basel 3.5?
19
Market Risk Capital under Basel III
20
Stressed VaR
• VaR (Internal Model) where under Basel II:
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(a) VaR computed on a daily basis.
(b) At a 99% confidence level.
(c) Over a 10 day holding period.
(d) With an overall multiplier of 3 times VaR imposed.
• Basel 2.5
– Banks must also calculate a “Stressed Value at Risk measure” based on a 10
day holding period, 99th percentile VaR with model inputs based on a 12
month Period (250 days) of a period of continuous stress (e.g., 2007, Russian
crisis etc…)
21
The Capital Requirement with Stressed VaR
22
Stressed VaR - Example
• Portfolio of 2 stocks
– IBM , MV = 20M
– Ford, MV = 10m
• VaR calculated used daily return data 23 May 2011 – 23 May 2012
• Stressed VaR calculated used daily return data 27 May 2008 – 27
May 2009
– ”the worst period in living memory”
23
Daily Return Data
Daily Returns IBM, 25 May 2008 - 23 May 2012
Daily Returns Ford, 25 May 2008 - 23 May 2012
12,000%
15,000%
10,000%
10,000%
8,000%
5,000%
6,000%
0,000%
4,000%
-5,000%
2,000%
-10,000%
0,000%
-15,000%
-2,000%
-4,000%
-20,000%
-6,000%
-25,000%
-8,000%
27-05-2008
27-05-2009
27-05-2010
27-05-2011
-30,000%
27-05-2008
27-05-2009
27-05-2010
27-05-2011
24
VaR for ”Past Year”
Past Year
IBM
Mean
StDev
Ford
0,07%
1,38%
Correlation
0,38
Confidence
H
99%
10
VaR (%)
Market Value
VaR
Undiversified VaR
Diversified VaR
Obeservations
Quantile
-0,02%
2,31%
10,155%
17,028%
20.000.000
2.031.060
3.733.863
3.110.984
253
-10,09%
10.000.000
1.702.803
10,4%
25
Stressed VaR
Mean
StDev
Stressed VaR (2008-2009)
IBM
Ford
-0,07%
-0,23%
2,52%
4,12%
Correlation
0,56
Confidence
H
99%
10
VaR (%)
18,574%
30,273%
Market Value
VaR
Undiversified VaR
Diversified VaR
20.000.000
3.714.798
6.742.072
5.970.424
10.000.000
3.027.274
Quantile
Capital charge (2.5)
-18,64%
9.081.408
19,9%
26
”Value-at-Risk” and Alternative Measures
”Expected Shortfall” and ”Extreme VaR
are increasingly used to measure tail risk
Source: Stress testing at major financial institutions: survey results and practice
Report by a working group established by the Committee on the Global Financial System. BIS January 2005
27
Expected Shortfall vs. VaR
The ”tail” of the distribution
Expected loss,
conditional on
losses exceeding
the 95% threshold!
VaR (95%)
ES (95%)
5% probability of
loss > VaR
VaR (95%)
Current value
28
Coherent Risk Measures
A coherent risk measure  for two random loss variables X and Y
has the followingproperties :
Subadditivity
 (X  Y)   (X)   (Y)
Monotonicity. If X  Y then
 (X)   (Y)
Positive homogeniety. For all   0
 (X)   (X)
Translation invariance
 (X   )   (X) - 
where  is a constant
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Which approaches are coherent?
• VAR measures are not coherent
– Not sub-additive
• Expected shortfall (mean excess, tail VAR) is coherent
30
Example
• Two different bonds A and B with non–overlapping default
probabilities
• For example: two bonds issued by Nokia and Motorola: If one
defaults the other will not and vice versa.
• A portfolio that contains both bonds may have a global VaR which is
bigger than the sum of the two VaR’s.
• Numerical example.
– The two bonds have two different default states each with recovery values at
70 and 90 and probabilities 3% and 2% respectively.
– Otherwise they will redeem at 100.
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Example (continued)
32
Ways of Calculating ES
• Analytical Metods
– Normal distribution
– Extreme VaR (GPD)
• Numerical Methods
– Estimate full distribution using
• Historical simulation
• Monte Carlo simulation
33
“Extreme Value Theory”
• Extreme Value Theory (EVT) focuses on estimation of tails of
probability distributions
• Two main approaches in EVT
– Peaks-over-threshold (POT)
– Block maxima approach
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”Peak over Threshold” - Illustration
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Generalized Pareto Distribution (GPD)
Expression:

 x  u 
 
FGPD x   1  1   
  

=0
>0
<0
1 / 
Exponential
Pareto
Pareto II
36
GPD distributions
Pareto
(concave)
Exponential
(straight)
Pareto Il
(convex)
threshold
37
Expected Shortfall (ES)
• ES for normal distribution:
( z)
ES q     
1 - ( z)
Varq  
z

• ES for GDP:
VaR q ˆ (u) - ˆ  u
ESq 

1- ˆ
1- ˆ
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ES vs. VaR: Normal and GDP Distributions
3.50
3.00
2.50
VaR (normal)
VaR (GPD)
2.00
ES (normal)
ES (GPD)
1.50
1.00
95.0%
96.0%
97.0%
98.0%
99.0%
100.0%
Confidence Level
39
ES vs. VaR: Normal and GDP Distributions
(continued)
Confidence Level VaR (normal) VaR (GPD) ES (normal) ES (GPD)
95,0%
1,64
1,63
2,06
2,15
96,0%
1,75
1,75
2,15
2,27
97,0%
1,88
1,89
2,27
2,42
98,0%
2,05
2,10
2,42
2,64
99,0%
2,33
2,47
2,66
3,01
99,5%
2,58
2,84
2,88
3,39
99,9%
3,09
3,77
3,34
4,32
99,95%
3,29
4,19
4,08
7,55
99,97%
3,43
4,51
4,14
8,08
99,99%
3,89
5,70
4,31
8,69
Difference - ES Difference - VaR
4,347%
-0,63%
5,272%
-0,27%
6,610%
0,57%
8,688%
2,23%
12,541%
5,83%
16,436%
9,91%
25,728%
19,94%
61,550%
24,27%
66,863%
27,43%
70,116%
38,23%
40
Risk-Adjusted Performance
Measurement in Banks
41
Self-Insurance with Economic Capital
Frequency
99.95% Confidence Level
Expected
Loss
Unexpected Loss
Stress Loss
Loss
Operating
Expense
Economic Capital
Transfer/accept
Value-at-Risk
42
Optimal Capital Allocation
• Key problem: How to aggregate the risks attributed to the bank’s various
business lines
• Some banks choose simply to add risks up and thus recognize none of the
benefits of risk diversification
• Others use flat percentage reduction in risk capital to reflect
diversification
43
Risk Capital Allocation - Example
• Stand-alone Capital
Business Econ. Capital Marginal EC
A+B
A
B
Div. Effect
100
60
70
30
30
40
– A: 60
– B: 70
• Fully Diversified
– A: 46 (60 – 30*60/130)
– B: 54 (70 – 30*70/130)
• Marginal
– A: 30
– B: 40
44
Which Method to Use?
• Choice of a capital measure depends on the desired objective:
– Assessing solvency of the firm and minimum risk pricing:
• Fully diversified capital
– Active portfolio management and business mix decisions
• marginal risk capital
– Performance measurement
• stand-alone capital for incentive compensation
• fully diversified risk capital to assess extra performance from diversification
45
Risk Adjusted Performance Measurement
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External Performance Measurement
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–
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Internal Performance Measurement
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Goal: to reward units that produce best performance within allowed parameters
Relevant risk measure: Absolute (undiversified) VaR
Traditional Performance Measures
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–
•
Measuring the degree to which shareholder value is created
Relevant risk measure: marginal (diversified) VaR
ROA
ROE
Risk-Adjusted Performance Measures (RAPM)
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–
–
–
ROC
RORAC, RAROC and RARORAC
EVA
MVA
46
RAROC
• Most common definition of RAROC is simply ROC with an adjustment for
expected loss:
RAROC 
Revenues - Expenses  Expected Losses Income from Capital
Capital
• Expected loss is the mean of the loss distribution
– Expected loss from defaulting loans
47
Balance Sheet Management: Loan Pricing
• Credit models are useful for loan pricing in banks
• Loans should be priced as to promise the same ”Risk-adjusted
Return” (RAROC) across all loans
RAROC 
Net Profits  EXPECTED Credit Losses Income on Capital
Economic Capital
Unexpected loss
Interest income
minus funding
Loan rates should be set so that ”profits” (NII), adjusted for
expected losses yield a return on economic capital of X%
48
RAROC - Example
Funding rate
Loan
Loan 1
Loan 2
2.0%
NomAmnt Maturity (yrs)
100
1
200
1
Riskfree rate
PDF
2%
5%
LGD
50%
60%
1.0%
Exp. Loss Unexp. loss
1.0
10.00%
6.0
15.00%
EC
Loan Rate
10.0
5.00%
30.0
9.00%
Raroc
21%
28%
49
Risk-Adjusted Performance
Measurement of Investment
Portfolios
50
Categorization of Performance Measures
•
Measures based on
– “Simple” volatility
– Value-at-Risk
– Lower Partial Moments
– Drawdown
•
Assumptions about distribution
– Normal distribution
• Sharpe Ratio, Excess Return on VaR, Conditional Sharpe Ratio
– Accounting for higher moments (skewness, kurtosis)
• Adjusted Sharpe Ratio
• Modified Sharpe Ratio
– No assumptions about distribution at all!
• LPM and Drawdown
51
Measuring Risk Adjusted Performance (1)
•
Sharpe Ratio
SP 
•
•
•
RP  RF
P
The Sharpe ratio measures excess return (over the risk-free rate)
per unit of risk measured as standard deviation
Principal drawback: assumption of normality in the excess return
distribution.
This is a problem when the portfolio contains options and other
instruments with non-symmetric payoffs
52
Reward-to-VaR Ratio
• A portfolio’s Reward-to-VaR Ratio (RV) is the additional average rate of
return investors would have earned if they had borne an additional
percentage point of VaR by moving a fraction of wealth from the risk-free
security to portfolio
RVP 
RP  RF
VaRP  RF
53
Reward-to-VaR Ratio – am Illustration
Average Rate of Return
RB
B
M
RM
A
RA
F
-Rf
0
VaRA
VaRM
VaRB
VaR at a given
confidence level
54
Example – Normally Distributed Returns
0.57
2.33  0.57
Portfolio Average Return Standard Dev. Sharpe ratio RV (99%)
A
18.23%
23.29%
0.57
0.32
B
10.44%
13.93%
0.39
0.20
M
12.78%
18.15%
0.43
0.23
Riskfree
5.00%
55
Example – “Unknown” (Empirical) Distribution
10%  5%
9.60%  5%
Portfolio Average Return Standard Dev. Sharpe ratio VaR(99%) RV(99%)
C
10%
11.55%
0.43
9.60%
0.34
D
12%
14.80%
0.47 18.00%
0.30
56
Different Confidence Levels Can
Lead to Different Portfolio Performance Rankings!
12%  5%
10%  5%
Portfolio Average Return Standard Dev. Sharpe ratio VaR(95%) RV(95%)
C
10%
11.55%
0.43
8.00%
0.38
D
12%
14.80%
0.47 10.00%
0.47
57
Measuring Risk-Adjusted Performance (2)
• Risk-Adjusted return on Invested Capital (RAROC)
– The ratio of the portfolio’s expected return to some measure of risk, such as VAR
– Management can then compare the managers RAROC to his historical or expected
RAROC or to a benchmark
• Return over Maximum Drawdown
– Drawdown is the difference between a portfolio’s highest and lowest sub- period
values over a given measurement period.
– The maximum drawdown is the largest drawdown over the total period.
– To calculate return on maximum drawdown (RoMAD), the analyst divides the average
portfolio return in the period by the maximum drawdown.
ROMAD 
RP
Maximum Drawdown
58
Measuring Risk-Adjusted Performance (3)
•
RoMAD vs. Sharpe Ratio and Information Ratio
–
–
–
–
–
•
RoMAD is similar in concept to the Sharpe and information ratios
All three measure average return as a percentage of risk.
Maximum drawdown is considered more intuitive than standard deviation as a measure of risk,
however, because it deals with more concrete” numbers.
Maximum drawdown measures the maximum range of the changes in portfolio value, which
investors can easily visualize.
Like standard deviation, however, using RoMAD also makes an implicit assumption that historical
return patterns will continue.
Sortino Ratio
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–
–
Ratio of excess return to risk
Excess return = portfolio return minus minimum acceptable return (MAR)
Denominator is standard deviation of returns calculated using only returns below MAR
Sortino P 
RP  MAR
 P MAR
59