Oct2001

1
Market Risk Management

• Zvi Wiener
• 02-588-3049
• http//pluto.mscc.huji.ac.il/mswiener/zvi.html

2
Introduction to Market Risk Measurement

• Following Jorion 2001, Chapter 11
• Financial Risk Manager Handbook

3
Old ways to measure risk

• notional amounts
• sensitivity measures (duration, Greeks)
• scenarios
• ALM, DFA
• assume linearity
• do not describe probability

4

• 1938 Bonds duration
• 1952 Markowitz mean-variance
• 1963 Sharpes 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

5
How much can we lose?

• Everything
• correct, but useless answer.
• How much can we lose realistically?

6
What is the current Risk?

• duration, convexity
• volatility
• delta, gamma, vega
• rating
• target zone

• Bonds
• Stocks
• Options
• Credit
• Forex

• Total ?

7
Standard Approach
8
Modern Approach
Financial Institution
9
Definition

• VaR is defined as the predicted worst-case loss
  at a specific confidence level (e.g. 99) over a
  certain period of time.

10
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.

11
VaR
12
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.
13
Returns
year
14
Main Ideas

• A few well known risk factors
• Historical data economic views
• Diversification effects
• Testability
• Easy to communicate

15
History of VaR

• 80s - 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!

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

17
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

18
VaR caveats

• VaR does not describe the worst loss
• VaR does not describe losses in the left tail
• VaR is measured with some error

19
Other Measures of Risk

• The entire distribution
• The expected left tail loss
• The standard deviation
• The semi-standard deviation

20
Risk Measures
21
Properties of Risk Measure

• Monotonicity (XltY, R(X)gtR(Y))
• Translation invariance R(Xk) R(X)-k
• Homogeneity R(aX) a R(X) (liquidity??)
• Subadditivity R(XY) ? R(X) R(Y)
• the last property is violated by VaR!

22
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?

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

24
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

25
Confidence level

• low confidence leads to an imprecise result.
• For example 99.99 and 10 business days will
  require history of
• 10010010 100,000 days in order to have only 1
  point.

26
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 10010 3 years.
• 5 and 1 day horizon means once in a month.
• Various subportfolios may require various
  horizons.

27
Time horizon

• When the distribution is stable one can translate
  VaR over different time periods.

This formula is valid (in particular) for iid
normally distributed returns.
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

29
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

30
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

31
Basel Rules MRC

• Market Risk Charge MRC
• SRC - specific risk charge, k ?3.

32
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

33
VaR system
Risk factors
Portfolio
Historical data
positions
Model
Mapping
Distribution of risk factors
VaR method
Exposures
VaR
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

35
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

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
• D. FX swaps, back-to-back exotic FX options

37
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

38
Stress-testing

• scenario analysis
• stressing models, volatilities and correlations
• developing policy responses

39
Scenario Analysis

• Moving key variables one at a time
• Using historical scenarios
• Creating prospective scenarios
• The goal is to identify areas of potential
  vulnerability.

40
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 PL
• D. perform effective back-testing

41
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.

42
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.

43
Identification of Risk Factors

• Following Jorion 2001, Chapter 12
• Financial Risk Manager Handbook

44
Absolute and Relative Risk

• Absolute risk - measured in dollar terms
• Relative risk - measured relative to benchmark
  index and is often called tracking error.

45
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

46
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

47
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.

48
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

49
Exposure and Uncertainty

• Losses can occur due to a combination of
• A. exposure (choice variable)
• B. movement of risk factor (external variable)

50
Exposure and Uncertainty

• Market loss
• Exposure Adverse movement in risk factor

51
Specific Risk
Specific risk - risk due to issuer specific price
movements
52
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.

53

• Continuous process - diffusion
• Discontinuities
• Jumps in prices, interest rates
• Price impact and liquidity
• market closure
• discontinuity in payoff
• binary options
• loans

54
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).

55
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!

56
Funding Liquidity

• Risk of not meeting payment obligations.
• Cash flow risk!
• Liquidity needs can be met by
• using cash
• selling assets
• borrowing

57
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.

58
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.

59
On-the-run

• recently issued
• most active
• very liquid
• after a new issue appears they become
  off-the-run
• liquidity premium can be compensated by
  repos/reverse repos

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

61
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

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

63
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

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

65
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

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

67
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

68
Sources of Risk

• Following Jorion 2001, Chapter 13
• Financial Risk Manager Handbook

69
Currency Risk

• free movements of currency
• devaluation of a fixed or pegged currency
• regime change (Israel, Europe)

70
Currency Volatility

• End 99 End 96
• Argentina 0.35 0.4
• Australia 7.6 8.5
• Canada 5.1 3.6
• Switzerland 10 10
• Denmark 9.8 7.8
• Britain 6.5 9.1
• Hong Kong 0.3 0.3
• Indonesia 24 1.6
• Japan 11 6.6
• Korea 6.9 4.5

71
Currency Volatility

• End 99 End 96
• Mexico 7.5 7
• Malaysia 0.1 1.6
• Norway 7.6 7.6
• New Zealand 13.4 7.9
• Philippines 5.5 0.6
• Sweden 8.5 6.4
• Singapore 3.8 1.8
• Thailand 9.7 1.2
• Taiwan 1.8 0.9
• Euro 9.8 8.3
• S. Africa 4.2 8.4

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

73
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

74
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

75
Cross Rate volatility

• JPY/USD x JPY/DEM y DEM/USD z

76
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

77
Volatilities of IR/bond prices

• Price volatility in End 99 End 96
• Euro 30d 0.22 0.05
• Euro 180d 0.30 0.19
• Euro 360d 0.52 0.58
• Swap 2Y 1.57 1.57
• Swap 5Y 4.23 4.70
• Swap 10Y 8.47 9.82
• Zero 2Y 1.55 1.64
• Zero 5Y 4.07 4.67
• Zero 10Y 7.76 9.31
• Zero 30Y 20.75 23.53

78
Duration approximation

• 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).

79
Models of IR

• Normal model ?(?y) is normally distributed.
• Lognormal model ?(?y/y) is normally distributed.
• Note that

80
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.

81
Volatilities of yields

• Yield volatility in , 99 ?(?y/y) ?(?y)
• Euro 30d 45 2.5
• Euro 180d 10 0.62
• Euro 360d 9 0.57
• Swap 2Y 12.5 0.86
• Swap 5Y 13 0.92
• Swap 10Y 12.5 0.91
• Zero 2Y 13.4 0.84
• Zero 5Y 13.9 0.89
• Zero 10Y 13.1 0.85
• Zero 30Y 11.3 0.74

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.

83
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.

84
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

85
FRM-99, Question 80

• Value of the position

Daily yield volatility
86
Correlations

• Eurodeposit block
• zero-coupon Treasury block
• very high correlations within each block and much
  lower across blocks.

87
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.

88
FRM-00, Question 96

• Which statement about historic US Treasuries
  yield curves is TRUE?

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.

90
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.

91
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.

92

• 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.

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.

94
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.

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.
• D. All of the above.

96
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.

He should buy caps, not sell!
97
Fixed income portfolio risk

• Yield curve component (government)
• Credit spread (of the class of similar rating)
• Specific spread

98
Equity risk

• Market risk (beta based relative to an index)
• Specific risk

99
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.

100
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.

101
Forwards and Futures

• 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.

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.

103
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.

104
CAPM

• In an equilibrium the following holds (Sharpe)

105
APTArbitrage Pricing Theory
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.

107
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.

108
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.

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

110
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

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.
• 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.

112
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.

Spot risk is eliminated by offsetting positions
113
Hedging Linear Risk

• Following Jorion 2001, Chapter 14
• Financial Risk Manager Handbook

114
Hedging

• Taking positions that lower the risk profile of
  the portfolio.
• Static hedging
• Dynamic hedging

115
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.

116

• Market data 0 7m PL
• time to maturity 9 2
• US interest rate 6 6
• Yen interest rate 5 2
• Spot Y/ 125.00 150.00
• Futures Y/ 124.07 149.00

117

• Stacked hedge - to use a longer horizon and to
  revert the position at maturity.
• Strip hedge - rolling over short hedge.

118
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

119
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.

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.

121
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.

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.

123
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.

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.

125
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.

126
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

127
The Optimal Hedge Ratio
Minimum variance hedge ratio
128
Hedge Ratio as Regression Coefficient

• The optimal amount can also be derived as the
  slope coefficient of a regression ?s/s on ?f/f

129
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.

If R is low the hedge is not effective!
130
Optimal Hedge

• At the optimum the variance is

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

132
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

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

134
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

135
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.

136
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.

137
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.

138
Solution

• The notional is Qs2,770,000, the SD in is
• ?(?s/s)sQs0.2117?277 ?10,000 586,409
• the SD of one futures contract is
• ?(?f/f)fQf0.1859?0.6903?42,000 5,390
• with a futures notional
• fQf 0.6903?42,000 28,993.

139
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.

140
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 ( 0.82432)

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 long-term 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

142
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 long-term 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

143
Duration Hedging
144
Duration Hedging
If we have a target duration DV we can get it by
using
145
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.

146
Example 1

• The notional is
• (932/32)/100?100,000 93,062.5
• The optimal number to sell is

Note that DVBP of the futures is
9.2?93,062?0.0185
147
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.

148
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.

149
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

Note that DVBP of the futures is
0.25?1,000,000?0.0125
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

151
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

152
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

153
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

154
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

155
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?0.0186,000.
• 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

156
Beta Hedging

• ? represents the systematic risk, ? - the
  intercept (not a source of risk) and ? - residual.

A stock index futures contract
157
Beta Hedging
The optimal N is
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.
158
Example

• A portfolio manager holds a stock portfolio worth
  10M, with a beta of 1.5 relative to SP500. The
  current SP 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.

159
Example

• The notional of the futures contract is
• 250?1,400 350,000
• The optimal number of contracts for hedge is

The quality of the hedge will depend on the size
of the residual risk in the portfolio.
160

• A typical US stock has correlation of 50 with
  SP.
• 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 SP
  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.

161
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

162
FRM-00, Question 93

• The optimal hedge ratio is
• N -1.8?50,000,000/(0.623?500,000)289

163
VaR methods

• Following Jorion 2001, Chapter 17
• Financial Risk Manager Handbook

164
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

165
How to measure VaR

• Historical Simulations
• Variance-Covariance
• Monte Carlo
• Analytical Methods
• Parametric versus non-parametric approaches

166
Historical Simulations

• Fix current portfolio.
• Pretend that market changes are similar to those
  observed in the past.
• Calculate PL (profit-loss).
• Find the lowest quantile.

167
Example
Assume we have 1 and our main currency is
SHEKEL. Today 14.30. Historical data
PL 0.215 0 -0.112 0.052

• 4.00
• 4.20
• 4.20
• 4.10
• 4.15

4.304.20/4.00 4.515 4.304.20/4.20
4.30 4.304.10/4.20 4.198 4.304.15/4.10 4.352
168
USD NIS 2000 100 -120 2001 200
100 2002 -300 -20 2003 20 30
today
169
today
Changes in IR
USD 1 1 1 1 NIS 1 0
-1 -1
170
Returns
year
171
VaR
172
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!

173
Variance-Covariance
?-2.33?
174
Monte Carlo
175
Monte Carlo

• Distribution of market factors
• Simulation of a large number of events
• PL for each scenario
• Order the results
• VaR lowest quantile

176
Monte Carlo Simulation
177
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.

178
Stock Portfolio

• Single risk factor or multiple factors
• Degree of diversification
• Tracking error
• Rare events

179
Bond Portfolio

• Duration
• Convexity
• Partial duration
• Key rate duration
• OAS, OAD
• Principal component analysis

180
Options and other derivatives

• Greeks
• Full valuation
• Credit and legal aspects
• Collateral as a cushion
• Hedging strategies
• Liquidity aspects

181
Credit Portfolio

• rating, scoring
• credit derivatives
• reinsurance
• probability of default
• recovery ratio

182
Reporting

• Division of VaR by business units, areas of
  activity, counterparty, currency.
• Performance measurement - RAROC (Risk Adjusted
  Return On Capital).

183
Backtesting

• Verification of Risk Management models.
• Comparison if the models forecast VaR with the
  actual outcome - PL.
• Exception occurs when actual loss exceeds VaR.
• After exception - explanation and action.

184
Backtesting
OK increasing k intervention

• Green zone - up to 4 exceptions
• Yellow zone - 5-9 exceptions
• Red zone - 10 exceptions or more

185
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?

186
Local Valuation

• Simple approach based on linear approximation.

Full Valuation
Requires repricing of assets.
187
Delta-Gamma Method

• The valuation is still local (the bond is priced
  only at current rates).

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.

189
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.

Bad question!
190
Mapping

• Replacing the instruments in the portfolio by
  positions in a limited number of risk factors.
• Then these positions are aggregated in a
  portfolio.

191
Delta-Normal method

• Assumes
• linear exposures
• risk factors are jointly normally distributed
• The portfolio variance is

192

• Delta-normal Histor. MC
• Valuation linear full full
• Distribution normal actual general
• Extreme events low prob. recent possible
• Ease of comput. Yes intermed. No
• Communicability Easy Easy Difficult
• VaR precision Bad depends good
• Major pitalls nonlinearity unstable model
• fat tails risk

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.

194
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.

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.

196
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.

197
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

198
FRM-99, Questions 15, 90
199
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

200
(No Transcript)
201

• 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

202

• On the valuation date we have
• S 1.6595, r 4.9375, r 5.9688
• Vt 93,581 - the current value of the contract