Market Risk Management
Zvi Wiener
02-588-3049
http://pluto.mscc.huji.ac.il/~mswiener/zvi.html
MRM
FRM-GARP
Oct-2001
Introduction to Market Risk
Measurement
Following Jorion 2001, Chapter 11
Financial Risk Manager Handbook
MRM
FRM-GARP
Oct-2001
Old ways to measure risk
• notional amounts
• sensitivity measures (duration, Greeks)
• scenarios
• ALM, DFA
assume linearity
do not describe probability
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Zvi Wiener - MRM
slide 3
1938
1952
1963
1966
1973
1983
1986
1988
1993
1994
1997
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Bonds duration
Markowitz mean-variance
Sharpe’s CAPM
Multiple risk-factors
Black-Scholes option pricing
RAROC, risk adjusted return
Limits on exposure by duration
Risk-weighted assets for banks;
exposure limits by Greeks
VaR endorsed by G-30
Risk Metrics
CreditMetrics, CreditRisk+
Zvi Wiener - MRM
slide 4
How much can we lose?
Everything
correct, but useless answer.
How much can we lose realistically?
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Zvi Wiener - MRM
slide 5
What is the current Risk?
• Bonds
• Stocks
• Options
• Credit
• Forex
• Total
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duration, convexity
volatility
delta, gamma, vega
rating
target zone
?
Zvi Wiener - MRM
slide 6
Standard Approach
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slide 7
Modern Approach
Financial Institution
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slide 8
Definition
VaR is defined as the predicted worst-case
loss at a specific confidence level (e.g. 99%)
over a certain period of time.
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Zvi Wiener - MRM
slide 9
Definition (Jorion)
VaR is the maximum loss over a target
horizon such that there is a low, prespecified
probability that the actual loss will be larger.
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Zvi Wiener - MRM
slide 10
VaR
1
0.8
0.6
0.4
VaR1%
1%
0.2
Profit/Loss
-3
http://www.tfii.org
-2
-1
1
Zvi Wiener - MRM
2
3
slide 11
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.
VaR
1%
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slide 12
Returns
year
1% of worst cases
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slide 13
Main Ideas
• A few well known risk factors
• Historical data + economic views
• Diversification effects
• Testability
• Easy to communicate
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Zvi Wiener - MRM
slide 14
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!
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Zvi Wiener - MRM
slide 15
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
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Zvi Wiener - MRM
slide 16
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
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Zvi Wiener - MRM
slide 17
VaR caveats
• VaR does not describe the worst loss
• VaR does not describe losses in the left tail
• VaR is measured with some error
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Zvi Wiener - MRM
slide 18
Other Measures of Risk
• The entire distribution
• The expected left tail loss
• The standard deviation
• The semi-standard deviation
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slide 19
Risk Measures
1
0.8
0.6
0.4
0.2
Profit/Loss
-3
http://www.tfii.org
-2
-1
1
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2
3
slide 20
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!
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Zvi Wiener - MRM
slide 21
No subadditivity of VaR
Bond has a face value of $100,000, during the
target period there is a probability of 0.75%
that there will be a default (loss of $100,000).
Note that VaR99% = 0 in this case.
What is VaR99% of a position consisting of 2
independent bonds?
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Zvi Wiener - MRM
slide 22
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
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Zvi Wiener - MRM
slide 23
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
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Zvi Wiener - MRM
slide 24
Confidence level
low confidence leads to an imprecise result.
For example 99.99% and 10 business days
will require history of
100*100*10 = 100,000 days in order to have
only 1 point.
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Zvi Wiener - MRM
slide 25
Time horizon
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.
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Zvi Wiener - MRM
slide 26
Time horizon
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 distributed returns.
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Zvi Wiener - MRM
slide 27
FRM-97, Question 7
To convert VaR from a one day holding period
to a ten day holding period the VaR number is
generally multiplied by:
A. 2.33
B. 3.16
C. 7.25
D. 10
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Zvi Wiener - MRM
slide 28
FRM-97, Question 7
To convert VaR from a one day holding period
to a ten day holding period the VaR number is
generally multiplied by:
A. 2.33
B. 3.16
C. 7.25
D. 10
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Zvi Wiener - MRM
slide 29
Basel Rules
• horizon of 10 business days
• 99% confidence interval
• an observation period of at least a year of
historical data, updated once a quarter
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Zvi Wiener - MRM
slide 30
Basel Rules MRC
Market Risk Charge = MRC
SRC - specific risk charge, k 3.
 k 60

MRC t  Max VaRt i ,VaRt 1   SRC t
 60 i 1

VaRt  VaRt (1d , 99%)  10
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Zvi Wiener - MRM
slide 31
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
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Zvi Wiener - MRM
slide 32
VaR system
Risk factors
Portfolio
Historical data
positions
Model
Mapping
Distribution of
risk factors
VaR
method
Exposures
VaR
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Zvi Wiener - MRM
slide 33
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
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Zvi Wiener - MRM
slide 34
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
http://www.tfii.org
Zvi Wiener - MRM
slide 35
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 structure?
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
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Zvi Wiener - MRM
slide 36
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 structure?
A. Vanilla IR swaps, bonds and IR futures
B. IR futures, vanilla and callable IR swaps
C. Repos and bonds
No limits!
D. FX swaps, back-to-back exotic FX options
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Zvi Wiener - MRM
slide 37
Stress-testing
• scenario analysis
• stressing models, volatilities and correlations
• developing policy responses
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Scenario Analysis
• Moving key variables one at a time
• Using historical scenarios
• Creating prospective scenarios
The goal is to identify areas of potential
vulnerability.
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Zvi Wiener - MRM
slide 39
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
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Zvi Wiener - MRM
slide 40
FRM-98, Question 20
VaR measure should be supplemented by
portfolio stress-testing because:
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 probable events.
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Zvi Wiener - MRM
slide 41
FRM-00, Question 105
VaR analysis should be complemented by
stress-testing because 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%
confidence level.
D. Identifies losses that go beyond the normal
losses measured by VaR.
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Zvi Wiener - MRM
slide 42
Identification of Risk Factors
Following Jorion 2001, Chapter 12
Financial Risk Manager Handbook
MRM
FRM-GARP
Oct-2001
Absolute and Relative Risk
• Absolute risk - measured in dollar terms
• Relative risk - measured relative to
benchmark index and is often called tracking
error.
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Zvi Wiener - MRM
slide 44
Directional Risk
Directional risk involves exposures to the
direction of movements in major market
variables.
beta for exposure to stock market
duration for IR exposure
delta for exposure of options to undelying
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Zvi Wiener - MRM
slide 45
Non-directional Risk
Non-linear exposures, volatility exposures, etc.
residual risk in equity portfolios
convexity in interest rates
gamma - second order effects in options
vega or volatility risk in options
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Zvi Wiener - MRM
slide 46
Market versus Credit Risk
Market risk is related to changes in prices,
rates, etc.
Credit risk is related to defaults.
Many assets have both types - bonds, swaps,
options.
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Zvi Wiener - MRM
slide 47
Risk Interaction
You buy 1M GBP at 1.5 $/GBP, settlement in
two days. We will deliver $1.5M in exchange
for 1M GBP.
Market risk
Credit risk
Settlement risk (Herstatt risk)
Operational risk
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Zvi Wiener - MRM
slide 48
Exposure and Uncertainty
P  ( PD*)  y
Dollar duration
Losses can occur due to a combination of
A. exposure (choice variable)
B. movement of risk factor (external variable)
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slide 49
Exposure and Uncertainty
Ri   i   i  RM   i
Market loss =
Exposure * Adverse movement in risk factor
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Zvi Wiener - MRM
slide 50
Specific Risk
Pi  Pi  RM   i Pi 
Market exposure
Specific risk
Specific risk - risk due to issuer
specific price movements
 Pi   Pi   
2
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2
2
RM     i Pi 
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2
slide 51
FRM-97, Question 16
The risk of a stock or bond which is NOT
correlated with the market (and thus can be
diversified) is known as:
A. interest rate risk.
B. FX risk.
C. model risk.
D. specific risk.
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Zvi Wiener - MRM
slide 52
• Continuous process - diffusion
• Discontinuities
• Jumps in prices, interest rates
• Price impact and liquidity
• market closure
• discontinuity in payoff:
• binary options
• loans
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Zvi Wiener - MRM
slide 53
Emerging Markets
Emerging stock market - definition by IFC
(1981) International Finance Corporation.
Stock markets located in developing countries
(countries with GDP per capita less than
$8,625 in 1993).
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Zvi Wiener - MRM
slide 54
Liquidity Risk
Difficult to measure.
Very unstable.
Market depth can be used as an approximation.
Liquidity risk consists of both asset liquidity
and funding liquidity!
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Zvi Wiener - MRM
slide 55
Funding Liquidity
Risk of not meeting payment obligations.
Cash flow risk!
Liquidity needs can be met by
• using cash
• selling assets
• borrowing
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Zvi Wiener - MRM
slide 56
Highly liquid assets
• tightness - difference between quoted mid
market price and transaction price.
• depth - volume of trade that does not affect
prices.
• resiliency - speed at which price fluctuations
disappear.
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slide 57
Flight to quality
Shift in demand from low grade towards high
grade securities.
Low grade market becomes illiquid with
depressed prices.
Spread between government and corporate
issues increases.
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slide 58
On-the-run
• recently issued
• most active
• very liquid
• after a new issue appears they become offthe-run
• liquidity premium can be compensated by
repos/reverse repos
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Zvi Wiener - MRM
slide 59
FRM-98, Question 7
Which of the following products has the least
liquidity?
A. US on-the-run Treasuries
B. US off-the-run Treasuries
C. Floating rate notes
D. High grade corporate bonds
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Zvi Wiener - MRM
slide 60
FRM-98, Question 7
Which of the following products has the least
liquidity?
A. US on-the-run Treasuries
B. US off-the-run Treasuries
C. Floating rate notes
D. High grade corporate bonds
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Zvi Wiener - MRM
slide 61
FRM-97, Question 54
“Illiquid” describes an instrument which
A. does not trade in an active market
B. does not trade on any exchange
C. can not be easily hedged
D. is an over-the-counter (OTC) product
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Zvi Wiener - MRM
slide 62
FRM-97, Question 54
“Illiquid” describes an instrument which
A. does not trade in an active market
B. does not trade on any exchange
C. can not be easily hedged
D. is an over-the-counter (OTC) product
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Zvi Wiener - MRM
slide 63
FRM-98, Question 6
A finance company is interested in managing its
balance sheet liquidity risk. The most
productive means of accomplishing this is by:
A. purchasing market securities
B. hedging the exposure with Eurodollar futures
C. diversifying its sources of funding
D. setting up a reserve
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Zvi Wiener - MRM
slide 64
FRM-98, Question 6
A finance company is interested in managing its
balance sheet liquidity risk. The most
productive means of accomplishing this is by:
A. purchasing market securities
B. hedging the exposure with Eurodollar futures
C. diversifying its sources of funding
D. setting up a reserve
http://www.tfii.org
Zvi Wiener - MRM
slide 65
FRM-00, Question 74
In a market crash the following is usually true?
I. Fixed income portfolios hedged with short
Treasuries and futures lose less than those hedged
with IR swaps given equivalent duration.
II. Bid offer spreads widen due to less liquidity.
III. The spreads between off the run bonds and
benchmark issues widen.
A. I, II & III
C. I & III
B. II & III
D. None of the above
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Zvi Wiener - MRM
slide 66
FRM-00, Question 74
In a market crash the following is usually true?
I. Fixed income portfolios hedged with short
Treasuries and futures lose less than those hedged
with IR swaps given equivalent duration.
II. Bid offer spreads widen due to less liquidity.
III. The spreads between off the run bonds and
benchmark issues widen.
A. I, II & III
C. I & III
B. II & III
D. None of the above
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Zvi Wiener - MRM
slide 67
Sources of Risk
Following Jorion 2001, Chapter 13
Financial Risk Manager Handbook
MRM
FRM-GARP
Oct-2001
Currency Risk
• free movements of currency
• devaluation of a fixed or pegged currency
• regime change (Israel, Europe)
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Zvi Wiener - MRM
slide 69
Currency Volatility
Argentina
Australia
Canada
Switzerland
Denmark
Britain
Hong Kong
Indonesia
Japan
Korea
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End 99
0.35
7.6
5.1
10
9.8
6.5
0.3
24
11
6.9
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End 96
0.4
8.5
3.6
10
7.8
9.1
0.3
1.6
6.6
4.5
slide 70
Currency Volatility
Mexico
Malaysia
Norway
New Zealand
Philippines
Sweden
Singapore
Thailand
Taiwan
Euro
S. Africa
http://www.tfii.org
End 99
7.5
0.1
7.6
13.4
5.5
8.5
3.8
9.7
1.8
9.8
4.2
Zvi Wiener - MRM
End 96
7
1.6
7.6
7.9
0.6
6.4
1.8
1.2
0.9
8.3
8.4
slide 71
FRM-97, Question 10
Which currency pair would you expect to have
the lowest volatility?
A. USD/DEM
B. USD/CAD
C. USD/JPY
D. USD/ITL
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Zvi Wiener - MRM
slide 72
FRM-97, Question 10
Which currency pair would you expect to have
the lowest volatility?
A. USD/DEM
B. USD/CAD
C. USD/JPY
D. USD/ITL
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Zvi Wiener - MRM
slide 73
FRM-97, Question 14
What is the implied correlation between
JPY/DEM and DEM/USD when given the
following volatilities for foreign exchange rates?
JPY/USD 8%, JPY/DEM 10%, DEM/USD 6%
A. 60%
B. 30%
C. -30%
D. -60%
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Zvi Wiener - MRM
slide 74
Cross Rate volatility
JPY/USD = x
x
y
z
JPY/DEM = y
x  yz
DEM/USD = z
ln x  ln y  ln z
 (ln x)   (ln y)   (ln z)  2 ln y ln z (ln y) (ln z)
2
2
2
0.08  0.1  0.06  2   0.1 0.06
2
2
2
0.01  0.0036  0.0064
0.0072
 

 0.6
2  0.06  0.1
0.012
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Zvi Wiener - MRM
slide 75
Fixed Income Risk
Arises from potential movements in the level
and volatility of bond yields.
Factors affecting yields
• inflationary expectations
• term spread
• higher volatility of the low end of TS
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Zvi Wiener - MRM
slide 76
Volatilities of IR/bond prices
Price volatility in %
Euro 30d
Euro 180d
Euro 360d
Swap 2Y
Swap 5Y
Swap 10Y
Zero 2Y
Zero 5Y
Zero 10Y
Zero 30Y
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End 99
0.22
0.30
0.52
1.57
4.23
8.47
1.55
4.07
7.76
20.75
Zvi Wiener - MRM
End 96
0.05
0.19
0.58
1.57
4.70
9.82
1.64
4.67
9.31
23.53
slide 77
Duration approximation
 P 

  D *  (y )
 P 
What duration makes bond as volatile as FX?
What duration makes bond as volatile as stocks?
A 10 year bond has yearly price volatility of 8%
which is similar to major FX.
30-year bonds have volatility similar to equities
(20%).
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Zvi Wiener - MRM
slide 78
Models of IR
Normal model (y) is normally distributed.
Lognormal model (y/y) is normally distributed.
Note that:
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 y 
 (y )  y    
 y 
Zvi Wiener - MRM
slide 79
Time adjustment
Square root of time adjustment is more
questionable for bond prices than for other
assets
• there is a strong evidence of mean reversion
• bond prices converge approaching maturity
(bridge effect) - strong for short bonds, weak for
long.
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Zvi Wiener - MRM
slide 80
Volatilities of yields
Yield volatility in %, 99 (y/y)
Euro 30d
45
Euro 180d
10
Euro 360d
9
Swap 2Y
12.5
Swap 5Y
13
Swap 10Y
12.5
Zero 2Y
13.4
Zero 5Y
13.9
Zero 10Y
13.1
Zero 30Y
11.3
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Zvi Wiener - MRM
(y)
2.5
0.62
0.57
0.86
0.92
0.91
0.84
0.89
0.85
0.74
slide 81
FRM-99, Question 86
For computing the market risk of a US T-bond
portfolio it is easiest to measure:
A. yield volatility, because yields have positive
skewness.
B. price volatility, because bond prices are
positively correlated.
C. yield volatility for bonds sold at a discount
and price volatility for bonds sold at a premium.
D. yield volatility because it remains more
constant over time than price volatility, which
must approach zero at maturity.
http://www.tfii.org
Zvi Wiener - MRM
slide 82
FRM-99, Question 86
For computing the market risk of a US T-bond
portfolio it is easiest to measure:
A. yield volatility, because yields have positive
skewness.
B. price volatility, because bond prices are
positively correlated.
C. yield volatility for bonds sold at a discount
and price volatility for bonds sold at a premium.
D. yield volatility because it remains more
constant over time than price volatility, which
must approach zero at maturity.
http://www.tfii.org
Zvi Wiener - MRM
slide 83
FRM-99, Question 80
You have position of $20M in the 6.375% Aug-27
US T-bond. Calculate daily VaR at 95% assume
that there are 250 business days in a year.
Price 98 8/32
Accrued 1.43%
Yield 6.509%
Duration 13.133
Modified Dur. 12.719 Yield volatility 12%
A. $291,400
B. $203,080
C. $206,036
D. $206,698
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Zvi Wiener - MRM
slide 84
FRM-99, Question 80
Value of the position
8

 1
$20 98   1.43 
 $19.936
32

 100
Daily yield volatility
 y  1
 (y)  y   annual 
 0.000494
 y  250
VaR  D * P 1.645   (y )
VaR  12.719  $19.936M 1.645  0.000494  $206,055
http://www.tfii.org
Zvi Wiener - MRM
slide 85
Correlations
Eurodeposit block
zero-coupon Treasury block
very high correlations within each block and
much lower across blocks.
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Zvi Wiener - MRM
slide 86
Principal component analysis
• level risk factor 94% of changes
• slope risk factor (twist) 4% of changes
• curvature (bend or butterfly)
See book by Golub and Tilman.
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Zvi Wiener - MRM
slide 87
FRM-00, Question 96
Which statement about historic US Treasuries
yield curves is TRUE?
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Zvi Wiener - MRM
slide 88
FRM-00, Question 96
A. Changes in the long-term yield tend to be
larger than in short-term yield.
B. Changes in the long-term yield tend to be
approximately the same as in short-term yield.
C. The same size yield change in both long-term
and short-term rates tends to produce a larger
price change in short-term instruments when all
securities are traded near par.
D. The largest part of total return variability of
spot rates is due to parallel changes with a smaller
portion due to slope changes and the residual due
to curvature changes.
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Zvi Wiener - MRM
slide 89
FRM-00, Question 96
A. Changes in the long-term yield tend to be
larger than in short-term yield.
B. Changes in the long-term yield tend to be
approximately the same as in short-term yield.
C. The same size yield change in both long-term
and short-term rates tends to produce a larger
price change in short-term instruments when all
securities are traded near par.
D. The largest part of total return variability of
spot rates is due to parallel changes with a smaller
portion due to slope changes and the residual due
to curvature changes.
http://www.tfii.org
Zvi Wiener - MRM
slide 90
FRM-97, Question 42
What is the relationship between yield on the
current inflation-proof bond issued by the US
Treasury and a standard Treasury bond with
similar terms?
A. The yields should be about the same.
B. The yield on the inflation protected bond
should be approximately the yield on treasury
minus the real interest.
C. The yield on the inflation protected bond
should be approximately the yield on treasury plus
the real interest.
D. None of the above.
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Zvi Wiener - MRM
slide 91
• Credit Spread Risk
• Prepayment Risk (MBS and CMO)
• seasoning
• current level of interest rates
• burnout (previous path)
• economic activity
• seasonal patterns
• OAS = option adjusted spread = spread over
equivalent Treasury minus the cost of the
option component.
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Zvi Wiener - MRM
slide 92
FRM-99, Question 71
You held mortgage interest only (IO) strips backed by
Fannie Mae 7 percent coupon. You want to hedge this
by shorting Treasury interest strips off the 10-year
on-the-run. The curve steepens as 1 month rate drops,
while the 6 months to 10 year rates remain stable.
What will be the effect on the value of your
portfolio?
A. Both IO and the hedge appreciate in value.
B. Almost no change in both (may be a small
appreciation).
C. Not enough information to find changes in both.
D. The IO will depreciate, the hedge will appreciate.
http://www.tfii.org
Zvi Wiener - MRM
slide 93
FRM-99, Question 71
You held mortgage interest only (IO) strips backed by
Fannie Mae 7 percent coupon. You want to hedge this
by shorting Treasury interest strips off the 10-year
on-the-run. The curve steepens as 1 month rate drops,
while the 6 months to 10 year rates remain stable.
What will be the effect on the value of your
portfolio?
A. Both IO and the hedge appreciate in value.
B. Almost no change in both (may be a small
appreciation).
C. Not enough information to find changes in both.
D. The IO will depreciate, the hedge will appreciate.
http://www.tfii.org
Zvi Wiener - MRM
slide 94
FRM-99, Question 73
A fund manager attempting to beat his LIBOR
based funding costs, holds pools of adjustable
rate mortgages and is considering various
strategies to lower the risk. Which of the
following strategies will NOT lower the risk?
A. Enter a total rate of return swap swapping
the ARMs for LIBOR plus a spread.
B. Short US government bonds
C. Sell caps based on the projected rate of
mortgage paydown.
D. All of the above.
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Zvi Wiener - MRM
slide 95
FRM-99, Question 73
A fund manager attempting to beat his LIBOR
based funding costs, holds pools of adjustable
rate mortgages and is considering various
strategies to lower the risk. Which of the
following strategies will NOT lower the risk?
A. Enter a total rate of return swap swapping
the ARMs for LIBOR plus a spread.
B. Short US government bonds.
C. Sell caps based on the projected rate of
mortgage paydown. He should buy caps, not sell!
D. All of the above.
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Zvi Wiener - MRM
slide 96
Fixed income portfolio risk
• Yield curve component (government)
• Credit spread (of the class of similar rating)
• Specific spread
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Zvi Wiener - MRM
slide 97
Equity risk
• Market risk (beta based relative to an index)
• Specific risk
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Zvi Wiener - MRM
slide 98
FRM-97, Question 43
Which of the following statements about SP500 is
true?
I. The index is calculated using market prices as
weights.
II. The implied volatilities of options of the same
maturity on the index are different.
III. The stocks used in calculating the index remain
the same for each year.
IV. The SP500 represents only the 500 largest US
corporations.
A. II only.
B. I and II.
C. II and III.
D. III and IV only.
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Zvi Wiener - MRM
slide 99
FRM-97, Question 43
Which of the following statements about SP500 is
values
true?
I. The index is calculated using market prices as
weights.
II. The implied volatilities of options of the same
maturity on the index are different.
III. The stocks used in calculating the index remain
the same for each year.
IV. The SP500 represents only the 500 largest US
corporations.
A. II only.
B. I and II.
C. II and III.
D. III and IV only.
http://www.tfii.org
Zvi Wiener - MRM
slide 100
Forwards and Futures
Ft e
 rt
 St e
 yt
The forward or futures price on a stock.
e-rt the present value in the base currency.
e-yt the cost of carry (dividend rate).
For a discrete dividend (individual stock) we
can write the right hand side as St- D, where D
is the PV of the dividend.
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Zvi Wiener - MRM
slide 101
FRM-97, Question 44
A trader runs a cash and future arbitrage book
on the SP500 index. Which of the following
are the major risk factors?
I. Interest rate
II. Foreign exchange
III. Equity price
IV. Dividend assumption risk
A. I and II only.
B. I and III only.
C. I, III, and IV only.
D. I, II, III, and IV.
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Zvi Wiener - MRM
slide 102
FRM-97, Question 44
A trader runs a cash and future arbitrage book
on the SP500 index. Which of the following
are the major risk factors?
I. Interest rate
II. Foreign exchange
III. Equity price
IV. Dividend assumption risk
A. I and II only.
B. I and III only.
C. I, III, and IV only.
D. I, II, III, and IV.
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Zvi Wiener - MRM
slide 103
CAPM
Ri   i   i  RM   i
i 
Cov( Ri , RM )

2
M
 i ,M
 ( Ri )
 ( RM )
In an equilibrium the following holds (Sharpe)
E Ri   R f   i E RM   R f 
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Zvi Wiener - MRM
slide 104
APT
Arbitrage Pricing Theory
Ri   i   i1  y1    iK  yK   i
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Zvi Wiener - MRM
slide 105
FRM-98, Question 62
In comparing CAPM and APT, which of the
following advantages does APT have over
CAPM?
I. APT makes less restrictive assumptions about
investor preferences toward risk and return.
II. APT makes no assumption about the
distribution of security returns.
III. APT does not rely on the identification of
the true market portfolio, and so the theory is
potentially testable.
A. I only.
B. II and III only.
C. I, and III only.
D. I, II, and III.
http://www.tfii.org
Zvi Wiener - MRM
slide 106
FRM-98, Question 62
In comparing CAPM and APT, which of the
following advantages does APT have over
CAPM?
I. APT makes less restrictive assumptions about
investor preferences toward risk and return.
II. APT makes no assumption about the
distribution of security returns.
III. APT does not rely on the identification of
the true market portfolio, and so the theory is
potentially testable.
A. I only.
B. II and III only.
C. I, and III only.
D. I, II, and III.
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Zvi Wiener - MRM
slide 107
Commodity Risk
Base metal - aluminum, copper, nickel, zinc.
Precious metals - gold, silver, platinum.
Energy products - natural gas, heating oil,
unleaded gasoline, crude oil.
Metals have 12-25% yearly volatility.
Energy products have 30-100% yearly
volatility (much less storable).
Long forward prices are less volatile then
short forward prices.
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Zvi Wiener - MRM
slide 108
FRM-97, Question 12
Which of the following products should have the
highest expected volatility?
A. Crude oil
B. Gold
C. Japanese Treasury Bills
D. DEM/CHF
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Zvi Wiener - MRM
slide 109
FRM-97, Question 12
Which of the following products should have the
highest expected volatility?
A. Crude oil
B. Gold
C. Japanese Treasury Bills
D. DEM/CHF
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Zvi Wiener - MRM
slide 110
FRM-97, Question 23
Identify the major risks of being short $50M of
gold two weeks forward and being long $50M of
gold one year forward.
I. Spot liquidity squeeze.
II. Spot risk.
III. Gold lease rate risk.
IV. USD interest rate risk.
A. II only.
B. I, II, and III only.
C. I, III, and IV only.
D. I, II, III, and IV.
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Zvi Wiener - MRM
slide 111
FRM-97, Question 23
Identify the major risks of being short $50M of
gold two weeks forward and being long $50M of
gold one year forward.
I. Spot liquidity squeeze.
Spot risk is eliminated
II. Spot risk.
by offsetting positions
III. Gold lease rate risk.
IV. USD interest rate risk.
A. II only.
B. I, II, and III only.
C. I, III, and IV only.
D. I, II, III, and IV.
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Zvi Wiener - MRM
slide 112
Hedging Linear Risk
Following Jorion 2001, Chapter 14
Financial Risk Manager Handbook
MRM
FRM-GARP
Oct-2001
Hedging
Taking positions that lower the risk profile of
the portfolio.
• Static hedging
• Dynamic hedging
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Zvi Wiener - MRM
slide 114
Unit Hedging with Currencies
A US exporter will receive Y125M in 7
months.
The perfect hedge is to enter a 7-months
forward contract.
Such a contract is OTC and illiquid.
Instead one can use traded futures.
CME lists yen contract with face value
Y12.5M and 9 months to maturity.
Sell 10 contracts and revert in 7 months.
http://www.tfii.org
Zvi Wiener - MRM
slide 115
Market data
time to maturity
US interest rate
Yen interest rate
Spot Y/$
Futures Y/$
0
9
6%
5%
125.00
124.07
7m
2
6%
2%
150.00
149.00
P&L
1 
 1
Y 125M  

  $166,667
 150 125 
1 
 1
 10  Y 12.5M  

  $168,621
 149 124.07 
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Zvi Wiener - MRM
slide 116
Stacked hedge - to use a longer horizon and
to revert the position at maturity.
Strip hedge - rolling over short hedge.
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Zvi Wiener - MRM
slide 117
Basis Risk
Basis risk arises when the characteristics of
the futures contract differ from those of the
underlying.
For example quality of agricultural product,
types of oil, Cheapest to Deliver bond, etc.
Basis = Spot - Future
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Zvi Wiener - MRM
slide 118
Cross hedging
Hedging with a correlated (but different) asset.
In order to hedge an exposure to Norwegian
Krone one can use Euro futures.
Hedging a portfolio of stocks with index future.
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Zvi Wiener - MRM
slide 119
FRM-00, Question 78
What feature of cash and futures prices tend to make
hedging possible?
A. They always move together in the same direction
and by the same amount.
B. They move in opposite direction by the same
amount.
C. They tend to move together generally in the same
direction and by the same amount.
D. They move in the same direction by different
amount.
http://www.tfii.org
Zvi Wiener - MRM
slide 120
FRM-00, Question 78
What feature of cash and futures prices tend to make
hedging possible?
A. They always move together in the same direction
and by the same amount.
B. They move in opposite direction by the same
amount.
C. They tend to move together generally in the same
direction and by the same amount.
D. They move in the same direction by different
amount.
http://www.tfii.org
Zvi Wiener - MRM
slide 121
FRM-00, Question 17
Which statement is MOST correct?
A. A portfolio of stocks can be fully hedged by
purchasing a stock index futures contract.
B. Speculators play an important role in the futures
market by providing the liquidity that makes
hedging possible and assuming the risk that hedgers
are trying to eliminate.
C. Someone generally using futures contract for
hedging does not bear the basis risk.
D. Cross hedging involves an additional source of
basis risk because the asset being hedged is exactly
the same as the asset underlying the futures.
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Zvi Wiener - MRM
slide 122
FRM-00, Question 17
Which statement is MOST correct?
A. A portfolio of stocks can be fully hedged by
purchasing a stock index futures contract.
B. Speculators play an important role in the futures
market by providing the liquidity that makes
hedging possible and assuming the risk that hedgers
are trying to eliminate.
C. Someone generally using futures contract for
hedging does not bear the basis risk.
D. Cross hedging involves an additional source of
basis risk because the asset being hedged is exactly
the same as the asset underlying the futures.
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Zvi Wiener - MRM
slide 123
FRM-00, Question 79
Under which scenario is basis risk likely to exist?
A. A hedge (which was initially matched to the
maturity of the underlying) is lifted before expiration.
B. The correlation of the underlying and the hedge
vehicle is less than one and their volatilities are
unequal.
C. The underlying instrument and the hedge vehicle
are dissimilar.
D. All of the above.
http://www.tfii.org
Zvi Wiener - MRM
slide 124
FRM-00, Question 79
Under which scenario is basis risk likely to exist?
A. A hedge (which was initially matched to the
maturity of the underlying) is lifted before expiration.
B. The correlation of the underlying and the hedge
vehicle is less than one and their volatilities are
unequal.
C. The underlying instrument and the hedge vehicle
are dissimilar.
D. All of the above.
http://www.tfii.org
Zvi Wiener - MRM
slide 125
The Optimal Hedge Ratio
S - change in $ value of the inventory
F - change in $ value of the one futures
N - number of futures you buy/sell
V  S  N  F

2
V

2
S
N 
2
2
F
 2 N S ,F

2
 2 N F  2 S ,F
N
2
V
http://www.tfii.org
Zvi Wiener - MRM
slide 126
The Optimal Hedge Ratio

2
 2 N F  2 S ,F
N
2
V
N opt
 S ,F
 S
  2    S ,F
 F
 F
Minimum variance hedge ratio
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Zvi Wiener - MRM
slide 127
Hedge Ratio as Regression Coefficient
The optimal amount can also be derived as the
slope coefficient of a regression s/s on f/f:
s
f
    sf

s
f
 sf
s
 sf  2   sf
f
f
http://www.tfii.org
Zvi Wiener - MRM
slide 128
Optimal Hedge
One can measure the quality of the optimal
hedge ratio in terms of the amount by which
we have decreased the variance of the original
portfolio.
2
2
R 
2
( s   V * )

2
s
V*   s 1 R

2
sf
2
If R is low the hedge is not effective!
http://www.tfii.org
Zvi Wiener - MRM
slide 129
Optimal Hedge
At the optimum the variance is

http://www.tfii.org
2
V*

 

2
S
Zvi Wiener - MRM
2
SF
2
F
slide 130
FRM-99, Question 66
The hedge ratio is the ratio of the size of the position taken in the
futures contract to the size of the exposure. Denote the standard
deviation of change of spot price by 1, the standard deviation of
change of future price by 2, the correlation between the changes in
spot and futures prices by . What is the optimal hedge ratio?
A. 1/1/2
B. 1/2/1
C. 1/2
D. 2/1
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Zvi Wiener - MRM
slide 131
FRM-99, Question 66
The hedge ratio is the ratio of the size of the position taken in the
futures contract to the size of the exposure. Denote the standard
deviation of change of spot price by 1, the standard deviation of
change of future price by 2, the correlation between the changes in
spot and futures prices by . What is the optimal hedge ratio?
A. 1/1/2
B. 1/2/1
C. 1/2
D. 2/1
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Zvi Wiener - MRM
slide 132
FRM-99, Question 66
The hedge ratio is the ratio of derivatives to a spot position (vice
versa) that achieves an objective such as minimizing or eliminating
risk. Suppose that the standard deviation of quarterly changes in the
price of a commodity is 0.57, the standard deviation of quarterly
changes in the price of a futures contract on the commodity is 0.85,
and the correlation between the two changes is 0.3876. What is the
optimal hedge ratio for a three-month contract?
A. 0.1893
B. 0.2135
C. 0.2381
D. 0.2599
http://www.tfii.org
Zvi Wiener - MRM
slide 133
FRM-99, Question 66
The hedge ratio is the ratio of derivatives to a spot position (vice
versa) that achieves an objective such as minimizing or eliminating
risk. Suppose that the standard deviation of quarterly changes in the
price of a commodity is 0.57, the standard deviation of quarterly
changes in the price of a futures contract on the commodity is 0.85,
and the correlation between the two changes is 0.3876. What is the
optimal hedge ratio for a three-month contract?
A. 0.1893
B. 0.2135
C. 0.2381
D. 0.2599
http://www.tfii.org
Zvi Wiener - MRM
slide 134
Example
Airline company needs to purchase 10,000
tons of jet fuel in 3 months. One can use
heating oil futures traded on NYMEX.
Notional for each contract is 42,000 gallons.
We need to check whether this hedge can be
efficient.
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Zvi Wiener - MRM
slide 135
Example
Spot price of jet fuel $277/ton.
Futures price of heating oil $0.6903/gallon.
The standard deviation of jet fuel price rate of
changes over 3 months is 21.17%, that of
futures 18.59%, and the correlation is 0.8243.
http://www.tfii.org
Zvi Wiener - MRM
slide 136
Compute
• The notional and standard deviation f the
unhedged fuel cost in $.
• The optimal number of futures contracts to
buy/sell, rounded to the closest integer.
• The standard deviation of the hedged fuel
cost in dollars.
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Zvi Wiener - MRM
slide 137
Solution
The notional is Qs=$2,770,000, the SD in $ is
(s/s)sQs=0.2117$277 10,000 = $586,409
the SD of one futures contract is
(f/f)fQf=0.1859$0.690342,000 = $5,390
with a futures notional
fQf = $0.690342,000 = $28,993.
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Zvi Wiener - MRM
slide 138
Solution
The cash position corresponds to a liability
(payment), hence we have to buy futures as a
protection.
sf= 0.8243  0.2117/0.1859 = 0.9387
sf = 0.8243  0.2117  0.1859 = 0.03244
The optimal hedge ratio is
N* = sf Qss/Qff = 89.7, or 90 contracts.
http://www.tfii.org
Zvi Wiener - MRM
slide 139
Solution
2unhedged = ($586,409)2 = 343,875,515,281
- 2SF/ 2F = -(2,605,268,452/5,390)2
hedged = $331,997
The hedge has reduced the SD from $586,409
to $331,997.
R2 = 67.95%
http://www.tfii.org
(= 0.82432)
Zvi Wiener - MRM
slide 140
FRM-99, Question 67
In the early 90s, Metallgesellshaft, a German oil company, suffered a
loss of $1.33B in their hedging program. They rolled over short
dated futures to hedge long term exposure created through their longterm fixed price contracts to sell heating oil and gasoline to their
customers. After a time, they abandoned the hedge because of large
negative cashflow. The cashflow pressure was due to the fact that
MG had to hedge its exposure by:
A. Short futures and there was a decline in oil price
B. Long futures and there was a decline in oil price
C. Short futures and there was an increase in oil price
D. Long futures and there was an increase in oil price
http://www.tfii.org
Zvi Wiener - MRM
slide 141
FRM-99, Question 67
In the early 90s, Metallgesellshaft, a German oil company, suffered a
loss of $1.33B in their hedging program. They rolled over short
dated futures to hedge long term exposure created through their longterm fixed price contracts to sell heating oil and gasoline to their
customers. After a time, they abandoned the hedge because of large
negative cashflow. The cashflow pressure was due to the fact that
MG had to hedge its exposure by:
A. Short futures and there was a decline in oil price
B. Long futures and there was a decline in oil price
C. Short futures and there was an increase in oil price
D. Long futures and there was an increase in oil price
http://www.tfii.org
Zvi Wiener - MRM
slide 142
Duration Hedging
dP   D * P  dy
Dollar duration
S   DS*  S  y
F   DF*  F  y
  D  S   
2
S
2
*
S
2
y
  D  F   
2
F
*
F
2
2
y
 SF  D  F D  S  
*
F
http://www.tfii.org
*
S
Zvi Wiener - MRM
2
y
slide 143
Duration Hedging
 SF
D S
N*   2  
F
D F
*
S
*
F
If we have a target duration DV* we can get it by using
D V  D  S
N
*
DF  F
*
V
http://www.tfii.org
Zvi Wiener - MRM
*
S
slide 144
Example 1
A portfolio manager has a bond portfolio worth
$10M with a modified duration of 6.8 years, to
be hedged for 3 months. The current futures
prices is 93-02, with a notional of $100,000.
We assume that the duration can be measured
by CTD, which is 9.2 years.
Compute:
a. The notional of the futures contract
b.The number of contracts to by/sell for optimal
protection.
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Zvi Wiener - MRM
slide 145
Example 1
The notional is:
(93+2/32)/100$100,000 =$93,062.5
The optimal number to sell is:
D S
6.8  $10,000,000
N*  

 79.4
D F
9.2  $93,062.5
*
S
*
F
Note that DVBP of the futures is 9.2$93,0620.01%=$85
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Zvi Wiener - MRM
slide 146
Example 2
On February 2, a corporate treasurer wants to
hedge a July 17 issue of $5M of CP with a maturity
of 180 days, leading to anticipated proceeds of
$4.52M. The September Eurodollar futures trades
at 92, and has a notional amount of $1M.
Compute
a. The current dollar value of the futures contract.
b. The number of futures to buy/sell for optimal
hedge.
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Zvi Wiener - MRM
slide 147
Example 2
The current dollar value is given by
$10,000(100-0.25(100-92)) = $980,000
Note that duration of futures is 3 months,
since this contract refers to 3-month LIBOR.
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Zvi Wiener - MRM
slide 148
Example 2
If Rates increase, the cost of borrowing will
be higher. We need to offset this by a gain, or
a short position in the futures. The optimal
number of contracts is:
D S
180  $4,520,000
N*  

 9.2
D F
90  $980,000
*
S
*
F
Note that DVBP of the futures is 0.25$1,000,0000.01%=$25
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Zvi Wiener - MRM
slide 149
FRM-00, Question 73
What assumptions does a duration-based hedging
scheme make about the way in which interest rates
move?
A. All interest rates change by the same amount
B. A small parallel shift in the yield curve
C. Any parallel shift in the term structure
D. Interest rates movements are highly correlated
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Zvi Wiener - MRM
slide 150
FRM-00, Question 73
What assumptions does a duration-based hedging
scheme make about the way in which interest rates
move?
A. All interest rates change by the same amount
B. A small parallel shift in the yield curve
C. Any parallel shift in the term structure
D. Interest rates movements are highly correlated
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Zvi Wiener - MRM
slide 151
FRM-99, Question 61
If all spot interest rates are increased by one basis
point, a value of a portfolio of swaps will increase
by $1,100. How many Eurodollar futures contracts
are needed to hedge the portfolio?
A. 44
B. 22
C. 11
D. 1100
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Zvi Wiener - MRM
slide 152
FRM-99, Question 61
The DVBP of the portfolio is $1,100.
The DVBP of the futures is $25.
Hence the ratio is 1100/25 = 44
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Zvi Wiener - MRM
slide 153
FRM-99, Question 109
Roughly how many 3-month LIBOR
Eurodollar futures contracts are needed to
hedge a position in a $200M, 5 year, receive
fixed swap?
A. Short 250
B. Short 3,200
C. Short 40,000
D. Long 250
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Zvi Wiener - MRM
slide 154
FRM-99, Question 109
The dollar duration of a 5-year 6% par bond is
about 4.3 years. Hence the DVBP of the fixed
leg is about
$200M4.30.01%=$86,000.
The floating leg has short duration - small
impact decreasing the DVBP of the fixed leg.
DVBP of futures is $25.
Hence the ratio is 86,000/25 = 3,440. Answer A
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Zvi Wiener - MRM
slide 155
Beta Hedging
Rit   i   i Rmt   it
 represents the systematic risk,  - the
intercept (not a source of risk) and  - residual.
S
M

S
M
A stock index futures contract
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Zvi Wiener - MRM
F
M
1
F
M
slide 156
Beta Hedging
M
M
V  S  NF  S
 NF
M
M
S
The optimal N is N *  
F
The optimal hedge with a stock index futures
is given by beta of the cash position times its
value divided by the notional of the futures
contract.
http://www.tfii.org
Zvi Wiener - MRM
slide 157
Example
A portfolio manager holds a stock portfolio
worth $10M, with a beta of 1.5 relative to
S&P500. The current S&P index futures price
is 1400, with a multiplier of $250.
Compute:
a. The notional of the futures contract
b. The optimal number of contracts for hedge.
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Zvi Wiener - MRM
slide 158
Example
The notional of the futures contract is
$2501,400 = $350,000
The optimal number of contracts for hedge is
N*  
 S
F
1.5  $10,000,000

 42.9
1 $350,000
The quality of the hedge will depend on the
size of the residual risk in the portfolio.
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Zvi Wiener - MRM
slide 159
A typical US stock has correlation of 50% with S&P.
Using the regression effectiveness we find that the
volatility of the hedged portfolio is still about
(1-0.52)0.5 = 87% of the unhedged volatility for a
typical stock.
If we wish to hedge an industry index with S&P
futures, the correlation is about 75% and the
unhedged volatility is 66% of its original level.
The lower number shows that stock market hedging
is more effective for diversified portfolios.
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Zvi Wiener - MRM
slide 160
FRM-00, Question 93
A fund manages an equity portfolio worth $50M
with a beta of 1.8. Assume that there exists an
index call option contract with a delta of 0.623 and
a value of $0.5M. How many options contracts are
needed to hedge the portfolio?
A. 169
B. 289
C. 306
D. 321
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Zvi Wiener - MRM
slide 161
FRM-00, Question 93
The optimal hedge ratio is
N = -1.8$50,000,000/(0.623$500,000)=289
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Zvi Wiener - MRM
slide 162
VaR methods
Following Jorion 2001, Chapter 17
Financial Risk Manager Handbook
MRM
FRM-GARP
Oct-2001
Risk Factors
There are many bonds, stocks and currencies.
The idea is to choose a small set of relevant economic
factors and to map everything on these factors.
• Exchange rates
• Interest rates (for each maturity and indexation)
• Spreads
• Stock indices
http://www.tfii.org
Zvi Wiener - MRM
slide 164
How to measure VaR
• Historical Simulations
• Variance-Covariance
• Monte Carlo
• Analytical Methods
• Parametric versus non-parametric approaches
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Zvi Wiener - MRM
slide 165
Historical Simulations
• Fix current portfolio.
• Pretend that market changes are
similar to those observed in the past.
• Calculate P&L (profit-loss).
• Find the lowest quantile.
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Zvi Wiener - MRM
slide 166
Example
Assume we have $1 and our main currency is
SHEKEL. Today $1=4.30.
Historical data:
P&L
4.00
4.20
4.30*4.20/4.00 = 4.515
0.215
4.20
4.30*4.20/4.20 = 4.30
0
4.10
4.30*4.10/4.20 = 4.198
-0.112
4.15
4.30*4.15/4.10 = 4.352
0.052
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Zvi Wiener - MRM
slide 167
USD
NIS
2000
100
-120
2001
200
100
2002
-300
-20
2003
20
30
today
100
200
 300
20



2
3
1  0.06 (1  0.061) (1  0.062) (1  0.063) 4
 120
100
 20
30



2
3
1  0.1 (1  0.11)
(1  0.12)
(1  0.13) 4
http://www.tfii.org
Zvi Wiener - MRM
slide 168
today
100
200
 300
20



2
3
1  0.06 (1  0.061) (1  0.062) (1  0.063) 4
 120
100
 20
30



2
3
1  0.1 (1  0.11)
(1  0.12)
(1  0.13) 4
USD:
NIS:
Changes
in IR
+1% +1%
+1% 0%
+1%
-1%
+1%
-1%
100
200
 300
20



2
3
1  0.07 (1  0.071) (1  0.072) (1  0.073) 4
 120
100
 20
30



2
3
1  0.11 (1  0.11)
(1  0.11)
(1  0.12) 4
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Zvi Wiener - MRM
slide 169
Returns
year
1% of worst cases
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Zvi Wiener - MRM
slide 170
VaR
1
0.8
0.6
0.4
VaR1%
1%
0.2
Profit/Loss
-3
http://www.tfii.org
-2
-1
1
Zvi Wiener - MRM
2
3
slide 171
Variance Covariance
• Means and covariances of market factors
• Mean and standard deviation of the portfolio
• Delta or Delta-Gamma approximation
• VaR1%= P – 2.33 P
• Based on the normality assumption!
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Zvi Wiener - MRM
slide 172
Variance-Covariance VaR1%  V  2.33 V
1%
2.33
-2.33
http://www.tfii.org

Zvi Wiener - MRM
slide 173
Monte Carlo
1
0.5
-1
0.5
-0.5
1
-0.5
-1
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Zvi Wiener - MRM
slide 174
Monte Carlo
• Distribution of market factors
• Simulation of a large number of events
• P&L for each scenario
• Order the results
• VaR = lowest quantile
http://www.tfii.org
Zvi Wiener - MRM
slide 175
Monte Carlo Simulation
15
10
5
10
20
30
40
-5
-10
-15
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Zvi Wiener - MRM
slide 176
Weights
Since old observations can be less relevant,
there is a technique that assigns decreasing
weights to older observations. Typically the
decrease is exponential.
See RiskMetrics Technical Document for
details.
http://www.tfii.org
Zvi Wiener - MRM
slide 177
Stock Portfolio
• Single risk factor or multiple factors
• Degree of diversification
• Tracking error
• Rare events
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Zvi Wiener - MRM
slide 178
Bond Portfolio
• Duration
• Convexity
• Partial duration
• Key rate duration
• OAS, OAD
• Principal component analysis
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Zvi Wiener - MRM
slide 179
Options and other derivatives
• Greeks
• Full valuation
• Credit and legal aspects
• Collateral as a cushion
• Hedging strategies
• Liquidity aspects
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Zvi Wiener - MRM
slide 180
Credit Portfolio
• rating, scoring
• credit derivatives
• reinsurance
• probability of default
• recovery ratio
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Zvi Wiener - MRM
slide 181
Reporting
Division of VaR by business units, areas of
activity, counterparty, currency.
Performance measurement - RAROC (Risk
Adjusted Return On Capital).
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Zvi Wiener - MRM
slide 182
Backtesting
Verification of Risk Management models.
Comparison if the model’s forecast VaR with
the actual outcome - P&L.
Exception occurs when actual loss exceeds
VaR.
After exception - explanation and action.
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Zvi Wiener - MRM
slide 183
Backtesting
Green zone - up to 4 exceptions
OK
Yellow zone - 5-9 exceptions
increasing k
Red zone - 10 exceptions or more
intervention
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Zvi Wiener - MRM
slide 184
Stress
Designed to estimate potential losses in abnormal
markets.
Extreme events
Fat tails
Central questions:
How much we can lose in a certain scenario?
What event could cause a big loss?
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Zvi Wiener - MRM
slide 185
Local Valuation
Worst dP  ( D * P)  (Worst dy )
Simple approach based on linear approximation.
Full Valuation
Worst dP  P( y0  Worst dy)  P( y0 )
Requires repricing of assets.
http://www.tfii.org
Zvi Wiener - MRM
slide 186
Delta-Gamma Method
2
dP
1d P
2
dP 
dy 
(dy)
2
dy
2 dy
dP   D * Pdy  0.5CP(dy)
2
The valuation is still local (the bond is priced
only at current rates).
http://www.tfii.org
Zvi Wiener - MRM
slide 187
FRM-97, Question 13
An institution has a fixed income desk and an exotic
options desk. Four risk reports were produced, each
with a different methodology. With all four
methodologies readily available, which of the
following would you use to allocate capital?
A. Simulation applied to both desks.
B. Delta-Normal applied to both desks.
C. Delta-Gamma for the exotic options desk and the
delta-normal for the fixed income desk.
D. Delta-Gamma applied to both desks.
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Zvi Wiener - MRM
slide 188
FRM-97, Question 13
An institution has a fixed income desk and an exotic
options desk. Four risk reports were produced, each
with a different methodology. With all four
methodologies readily available, which of the
following would you use to allocate capital?
A. Simulation applied to both desks.
B. Delta-Normal applied to both desks.
C. Delta-Gamma for the exotic options desk and the
delta-normal for the fixed income desk.
D. Delta-Gamma applied to both desks.
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Zvi Wiener - MRM
slide 189
Mapping
Replacing the instruments in the portfolio by
positions in a limited number of risk factors.
Then these positions are aggregated in a
portfolio.
http://www.tfii.org
Zvi Wiener - MRM
slide 190
Delta-Normal method
Assumes
• linear exposures
• risk factors are jointly normally distributed
The portfolio variance is
 (returns)  x x
2
T
Forecast of the covariance matrix for the horizon
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Zvi Wiener - MRM
slide 191
Delta-normal
Valuation
linear
Distribution
normal
Extreme events
low prob.
Ease of comput. Yes
Communicability Easy
VaR precision
Bad
Major pitalls
nonlinearity
fat tails
http://www.tfii.org
Zvi Wiener - MRM
Histor.
full
actual
recent
intermed.
Easy
depends
unstable
MC
full
general
possible
No
Difficult
good
model
risk
slide 192
FRM-97, Question 12
Delta-Normal, Historical-Simulations, and MC are
various methods available to compute VaR. If
underlying returns are normally distributed, then
the:
A. DN VaR will be identical to HS VaR.
B. DN VaR will be identical to MC VaR.
C. MC VaR will approach DN VaR as the number
of simulations increases.
D. MC VaR will be identical to HS VaR.
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Zvi Wiener - MRM
slide 193
FRM-97, Question 12
Delta-Normal, Historical-Simulations, and MC are
various methods available to compute VaR. If
underlying returns are normally distributed, then
the:
A. DN VaR will be identical to HS VaR.
B. DN VaR will be identical to MC VaR.
C. MC VaR will approach DN VaR as the number
of simulations increases.
D. MC VaR will be identical to HS VaR.
http://www.tfii.org
Zvi Wiener - MRM
slide 194
FRM-98, Question 6
Which VaR methodology is least effective for
measuring options risks?
A. Variance-covariance approach.
B. Delta-Gamma.
C. Historical Simulations.
D. Monte Carlo.
http://www.tfii.org
Zvi Wiener - MRM
slide 195
FRM-98, Question 6
Which VaR methodology is least effective for
measuring options risks?
A. Variance-covariance approach.
B. Delta-Gamma.
C. Historical Simulations.
D. Monte Carlo.
http://www.tfii.org
Zvi Wiener - MRM
slide 196
FRM-99, Questions 15, 90
The VaR of one asset is 300 and the VaR of
another one is 500. If the correlation between
changes in asset prices is 1/15, what is the
combined VaR?
A. 525
B. 775
C. 600
D. 700
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Zvi Wiener - MRM
slide 197
FRM-99, Questions 15, 90


2
A B
2
A B
     2  A B
2
A
300  500
 300  500 2
15
2

http://www.tfii.org
2
B
2
2
AB
 600
Zvi Wiener - MRM
2
slide 198
Example
On Dec 31, 1998 we have a forward contract
to buy 10M GBP in exchange for delivering
$16.5M in 3 months.
St - current spot price of GBP in USD
Ft - current forward price
K - purchase price set in contract
ft - current value of the contract
rt - USD risk-free rate, rt* - GBP risk-free rate
 - time to maturity
http://www.tfii.org
Zvi Wiener - MRM
slide 199
1
1
*
Pt  PV ($1) 
, Pt  PV (1GBP ) 
*
1  rt
1  rt 
St
K
*
ft 

 St Pt  KPt
*
1  rt  1  rt
df t
df t
df t
*
df t  dS  dP  * dP
dS
dP
dP
*
*
 P dS  SdP  KdP
http://www.tfii.org
Zvi Wiener - MRM
slide 200
*
dS
dP
* dP
df  SP
 SP *  KP
S
P
P
*
The forward contract is equivalent to
a long position of SP* on the spot rate
a long position of SP* in the foreign bill
a short position of KP in the domestic bill
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Zvi Wiener - MRM
slide 201
GBP10M  St $16.5M
Vt  Qf t 

*
1  rt 
1  rt
On the valuation date we have
S = 1.6595, r = 4.9375%, r* = 5.9688%
Vt = $93,581 - the current value of the
contract
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Zvi Wiener - MRM
slide 202