Financial Risk Management
Introduction to Market Risk Management
Following P. Jorion,
Financial Risk Management Chapter 11
Old ways to measure risk
sensitivity measures (duration, Greeks)
do not describe probability
1938 Bonds duration
1952 Markowitz mean-variance
1963 Sharpe’s CAPM
1966 Multiple risk-factors
1973 Black-Scholes option pricing
1983 RAROC, risk adjusted return
1986 Limits on exposure by duration
1988 Risk-weighted assets for banks;
exposure limits by Greeks
1993 VaR endorsed by G-30
1994 Risk Metrics
1997 CreditMetrics, CreditRisk+
How much can we lose?
correct, but useless answer.
How much can we lose realistically?
What is the current Risk?
Bonds duration, convexity
Options delta, gamma, vega
Forex target zone
VaR is defined as the predicted worst-case loss at a
specific confidence level (e.g. 99%) over a certain period
VaR is the maximum loss over a target horizon such that
there is a low, prespecified probability that the actual
loss will be larger.
-3 -2 -1 1 2 3
Meaning of VaR
A portfolio manager has a daily VaR equal $1M at
99% confidence level.
This means that there is only one chance in 100
that a daily loss bigger than $1M occurs,
under normal market conditions.
1% of worst cases
A few well known risk factors
Historical data + economic views
Easy to communicate
History of VaR
80’s - major US banks - proprietary
93 G-30 recommendations
94 - RiskMetrics by J.P.Morgan
98 - Basel
SEC, FSA, ISDA, pension funds, dealers
Widely used and misused!
FRM-99, Question 89
What is the correct interpretation of a $3 overnight VaR
figure with 99% confidence level?
A. expect to lose at most $3 in 1 out of next 100 days
B. expect to lose at least $3 in 95 out of next 100 days
C. expect to lose at least $3 in 1 out of next 100 days
D. expect to lose at most $6 in 2 out of next 100 days
VaR does not describe the worst loss
VaR does not describe losses in the left tail
VaR is measured with some error
Other Measures of Risk
The expected left tail loss
The standard deviation
The semi-standard deviation
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Properties of Risk Measure
Monotonicity (X<Y, R(X)>R(Y))
Translation invariance R(X+k) = R(X)-k
Homogeneity R(aX) = a R(X) (liquidity??)
Subadditivity R(X+Y) ≤ R(X) + R(Y)
the last property is violated by VaR!
No subadditivity of VaR
FRM-98, Question 22
Consider arbitrary portfolios A and B and their combined
portfolio C. Which of the following relationships always
holds for VaRs of A, B, and C?
A. VaRA+ VaRB = VaRC
B. VaRA+ VaRB ≥ VaRC
C. VaRA+ VaRB ≤ VaRC
D. None of the above
low confidence leads to an imprecise result.
For example 99.99% and 10 business days will require
100*100*10 = 100,000 days in order to have only 1
long time horizon can lead to an imprecise result.
1% - 10 days means that we will see such a loss
approximately once in 100*10 = 3 years.
5% and 1 day horizon means once in a month.
Various subportfolios may require various horizons.
When the distribution is stable one can translate VaR
over different time periods.
VaR (T days ) = VaR (1 day ) T
This formula is valid (in particular) for iid normally
FRM-97, Question 7
To convert VaR from a one day holding period to a ten
day holding period the VaR number is generally
horizon of 10 business days
99% confidence interval
an observation period of at least a year of historical
data, updated once a quarter
Basel Rules MRC
Market Risk Charge = MRC
SRC - specific risk charge, k ≥3.
MRCt = Max ∑ VaRt −i , VaRt −1 + SRCt
60 i =1
VaRt = VaRt (1d , 99%) × 10
FRM-97, Question 16
Which of the following quantitative standards is NOT
required by the Amendment to the Capital Accord to
Incorporate Market Risk?
A. Minimum holding period of 10 days
B. 99% one-tailed confidence interval
C. Minimum historical observations of two years
D. Update the data sets at least quarterly
Risk factors Portfolio
Historical data positions
Distribution of VaR
risk factors method
FRM-97, Question 23
The standard VaR calculation for extension to multiple
periods also assumes that positions are fixed. If risk
management enforces loss limits, the true VaR will be:
A. the same
B. greater than calculated
C. less then calculated
D. unable to determine
FRM-97, Question 9
A trading desk has limits only in outright foreign
exchange and outright interest rate risk. Which of the
following products can not be traded within the current
A. Vanilla IR swaps, bonds and IR futures
B. IR futures, vanilla and callable IR swaps
C. Repos and bonds
D. FX swaps, back-to-back exotic FX options
stressing models, volatilities and correlations
developing policy responses
Moving key variables one at a time
Using historical scenarios
Creating prospective scenarios
The goal is to identify areas of potential vulnerability.
FRM-97, Question 4
The use of scenario analysis allows one to:
A. assess the behavior of portfolios under large moves
B. research market shocks which occurred in the past
C. analyze the distribution of historical P&L
D. perform effective back-testing
FRM-98, Question 20
VaR measure should be supplemented by portfolio
A. VaR measures indicate that the minimum is VaR,
they do not indicate the actual loss
B. stress testing provides a precise maximum loss level
C. VaR measures are correct only 95% of time
D. stress testing scenarios incorporate reasonably
FRM-00, Question 105
VaR analysis should be complemented by stress-testing
A. Provides a maximum loss in dollars.
B. Summarizes the expected loss over a target horizon
within a minimum confidence interval.
C. Assesses the behavior of portfolio at a 99%
D. Identifies losses that go beyond the normal losses
measured by VaR.