Market Risk

1
Market Risk

• From Saunders and Cornett
• Ch. 10 Market Risk
• (Go to www.gloriamundi.org, it has a lot of
  articles on Value at Risk)

2
I. Market Risk Management

• Market risk is defined as the uncertainty of an
  FI's earnings resulting from changes in market
  conditions such as the price of an asset,
  interest rates, market volatility, and market
  liquidity.

3
I. Market Risk Management

• Dennis Weatherstone, former chairman of J. P.
  Morgan (JPM)
• "At close of business each day tell me what the
  market risks are across all businesses
  locations." In a nutshell, the chairman of J. P.
  Morgan wants a single dollar number at 415 PM
  New York time that tells him J. P. Morgan's
  market risk exposure on that day.
• For a FI, it is concerned with how much it could
  potentially lose should market conditions move
  adversely
• Market risk Estimated potential loss under
  adverse circumstances

4
I. Market Risk Management
5
I. Market Risk Management

• Five reasons why market risk measurement is
  important
• 1. Management Information.
• Provides senior management with information on
  the risk exposure taken by traders. This risk
  exposure can then be compared to the capital
  resources of the Fl.
• 2. Setting Limits.
• Measures the market risk of traders' portfolios,
  which will allow the establishment of
  economically logical position limits per trader
  in each area of trading.

6
I. Market Risk Management

• 3. Resource Allocation.
• Compares returns to market risks in different
  areas of trading, which may allow the
  identification of areas with the greatest
  potential return per unit of risk into which more
  capital and resources can be directed.
• 4. Performance Evaluation.
• Calculates the return-risk ratio of traders,
  which may allow a more rational evaluation of
  traders and a fair bonus system to be put in
  place.
• 5. Regulation.
• With the BIS and Federal Reserve proposing to
  regulate market risk through capital
  requirements, private sector benchmarks are
  important if it is felt that regulators are
  overpricing some risks.

7
II. What is VaR?

• VaR can be defined as the loss or change in value
  that is not expected to be exceeded with a given
  degree of confidence over a specified time
  period.
• Example A position has a daily VaR of 10m at
  the 99 confidence level and a holding period of
  5 days using a one-tailed confidence level means
  with a confidence level of 99 that over this
  period the loss in the value of the
  position/portfolio under consideration will not
  exceed 10m.

8
II. What is VaR?

• All components of a portfolio can be verified and
  validated
• VaR methodology historical, parametric
  (variance/covariance), or Monte Carlo
  methodologies are all acceptable.
• Holding period 1 day to 10 days for trading
  activity varying time frame for assets.
• Confidence interval 99 or 95 one-tailed or
  two-tailed approach.

9
II. What is VaR?

• The three main methodologies for calculating VaR
  are
• Parametric, closed form, or variance/covariance
• Monte Carlo
• Historical

10
II. What is VaR?

• Parametric, Closed Form, or Variance/Covariance
  Analysis (e.g., JP Morgans RiskMetric)
• A specific probability distribution on returns is
  assumed.
• Estimated VaR using an equation that specifies
  parameters such as volatility, correlation,
  delta, and gamma.
• Extensive historical data are not required, only
  volatility and a correlation matrix are needed.
• Is accurate for linear instruments but less
  accurate for nonlinear portfolios or for skewed
  distributions.

11
II. What is VaR?

• Monte Carlo Simulation
• Estimated VaR by simulating random scenarios and
  revaluing positions in the portfolio.
• Extensive historical data are not required.
• Is accurate for all instruments and provides a
  full distribution of potential portfolio values,
  not just a specific percentile.
• Monte Carlo simulation permits use of various
  distributional assumptions (normal,
  T-distribution, normal mixture, etc.)
• A disadvantage is that is it computationally
  intensive and time consuming, entailing
  revaluation of the portfolio under each scenario.

12
II. What is VaR?

• Historical Analysis
• Estimated VaR by taking actual historical rates
  and revaluing positions for each change in the
  market.
• Is accurate for all instruments and provides a
  full distribution of potential portfolio values,
  not just a specific percentile.
• The users need not make distribution assumptions,
  although parameter fitting may be performed on
  the resulting distribution.
• Historical analysis is faster than Monte Carlo
  simulation because fewer scenarios are used,
  although it is still somewhat computationally
  intensive and time consuming.
• A disadvantage is that a significantly daily rate
  history is required, and sampling far back can
  create problems if the data are irrelevant to
  current conditions.
• An additional disadvantage is that the results
  are harder to verify at high confidence level
  (99 and beyond).

13
II. What is VaR?

• Holding Period
• Holding period is the time over which the
  variability in the value of a portfolio or
  estimated earnings from an economic activity is
  assessed.
• During the holding period, changes in the market
  prices of the assets or other variables
  underlying the portfolio drive corresponding
  changes in the value or earnings estimates that
  were used at the beginning of the holding period.
• When a one-day holding period is used, the metric
  is commonly referred to as the daily earnings at
  risk (DEaR). DEaR provides a measure of the
  market risk of the portfolio in a short period of
  time.

14
II. What is VaR?

• Confidence Level
• A confidence level of a is defined as the
  probability that, given the underlying
  distribution of the random variable, the set of
  possible outcomes will lie in a range greater
  than or equal to a predetermined value.
• Equivalently, a confidence level of (1- a) is
  defined as the probability that the set of
  outcomes will lie in a range less than or equal
  to a predetermined value.
• For example, a confidence level of 5 is used to
  assess the set of possible outcomes and assessing
  a probability of 1 in 20 that the actual outcome
  will lie below a predetermined value, the latter
  being a function of the underlying distribution
  and the level of confidence being used.

15
II. What is VaR?

• Technical Clarification 1 Normal Return
  Distribution
• F ( R )

16
II. What is VaR?

• If c denotes the confidence level, say 99, then
  R is defined analytically by
• Prob(RltR)
• Prob (Z lt (R- ?)/ ?)
• 1-c

17
II. What is VaR?

• Z (R- ?)/ ? denotes a standard normal variable,
  N(0,1) with mean 1 and unit standard deviation.
• The cut-off return R can be expresses as
• R ? ? ?
• Where the threshold limits, ?, as a function of
  confidence level
• C ? (R- ?)/ ?
• _____________________________________
• 99.97 -3.43
• 99.87 -3.00
• 99 -2.33
• 95 -1.65

18
II. What is VaR?

• Technical Clarification 2 Derive the 10-day VaR
  from the daily VaR
• If assume that markets are efficient and daily
  returns, Rt, are independent and identically
  distributed, then the 10-day return R(10) ?
  Rt, is also normally distributed with mean ?10
  10 ?, and variance ?210 10 ?2, since it is the
  sum of 10 i.i.d. normal variables. It follows
  that
• VaR (10c) ?10 VaR (1 c)

19
II. What is VaR?

• In option pricing we measure volatility per
  year
• In VaR calculations we measure volatility per
  day
• Strictly speaking we should define sday as the
  standard deviation of the continuously compounded
  return in one day
• In practice we assume that it is the standard
  deviation of the percentage change in one day

20
II. What is VaR?

• Back Testing
• Back testing involves testing how well the VaR
  estimates would have performed in the past.
• Suppose that we are calculating a 1-day 99 VaR.
  Back testing would involve looking at how often
  the loss in a day exceeded the 1-day 99 VaR that
  would have been calculated for that day. If this
  happened on about 1 of the days, we can feel
  reasonably comfortable with the methodology. If
  it happened on, say 7 of days, the methodology
  is suspect.

21
II. What is VaR?

• Stress Testing
• This involves testing how well a portfolio
  performs under some of the most extreme market
  moves seen in the last 10 to 20 years
• E.g., to test the impact of an extreme movement
  in the US equity prices, a company might set the
  percentage changes in all market variables equal
  to those on October 19, 1987 (when the SP 500
  moved by 22.3 standard deviations). If this is
  too extreme, the company might choose January 8,
  1988 (when the SP 500 moved by 6.8 standard
  deviations).
• To test the effect of extreme movements in UK
  interest rates, the company might set the
  percentage changes in al market variables equal
  to those on April 10, 1992 (when 10-year bond
  yields moved by 7.7 standard deviations).

22
II. What is VaR?

• Four Approaches for Stress Scenarios
• The first approach uses historical scenarios
• The second shocks market rates to examine
  portfolio sensitivities and concentrations.
• The third considers hypothetical future
  scenarios, based on current market conditions.
• The fourth searches for stress scenarios by
  analyzing portfolio vulnerabilities.

23
II. What is VaR?

• Stress testing can be considered as a way of
  taking into account extreme events that do occur
  from time to time but that are virtually
  impossible according to the probability
  distributions assumed for market variables.
• A 5-standard deviation daily move in a market
  variable is one such extreme event. Under a
  normal distribution, it happens about once every
  7,000 years, but , in practice, it is not
  uncommon to see a 5-standard-deviation daily move
  once or twice every 10 years.

24
III. The Variance-Covariance Approach

• Three measurable components for the FI's daily
  earnings at risk
• Daily earnings at risk (DEAR)
• (Dollar value of the position) (Price
  volatility)
• (Dollar value of the position) ( Price
  sensitivity ) (Potential adverse move in yield)

25
III. The Variance-Covariance Approach

• Example
• Consider a portfolio consists of
• seven-year, zero-coupon, fixed-income (1 million
  market value),
• spot DM (1 million market value), and
• the U.S. stock market index (l million market
  value)
• What is the DEAR for each security included?
• What is the DEAR for the portfolio?

26
III. The Variance-Covariance Approach

• A. The Market Risk of Fixed -Income Securities
• Suppose an FI has a 1 million market value
  position in zero-coupon bonds of seven years to
  maturity with a face value of 1,631,483. Today's
  yield on these bonds is 7.243 percent per annum.
  These bonds are held as part of the trading
  portfolio. Thus
• Dollar value of position 1 million

27
III. The Variance-Covariance Approach

• The FI manager wants to know the potential
  exposure faced by the FI should a scenario occur
  resulting in an adverse or reasonably bad market
  move against the FI.
• How much will be lost depends on the price
  volatility of the bond. From the duration model
  we know that

28
III. The Variance-Covariance Approach

• Daily price volatility (Price sensitivity to a
  small change in yield) (Adverse daily yield
  move)
• (-MD) (Adverse daily yield move)
•   
• The modified duration (MD) of this bond is
• D 7
• MD --------- -- ----------- 6.527
• 1R (1.07243)
•  
• given the yield on the bond is R 7.243 percent.

29
III. The Variance-Covariance Approach

• Suppose we want to obtain maximum yield changes
  such that there is only a 5 percent chance the
  yield changes will be greater than this maximum
  in either direction.
• Assuming that yield changes are normally
  distributed, then 90 percent of the area under
  normal distribution is to be found within ?1.65
  standard deviations from the mean-that is, 1.65?.
  
• Suppose over the last year the mean change in
  daily yields on seven-year zeros was 0 percent
  while the standard deviation was 10 basis points
  (or 0.1), so 1.65? is 16.5 basis points (bp).

30
III. The Variance-Covariance Approach

• Then
• Price volatility (-MD) (Potential adverse
  move in yield)
• (-6.527) (.00165)
• .01077 or 1.077
• and 
• Daily earnings at risks (Dollar value of
  position) (Price volatility)
• (l,000,000) (.01077)
• 10,770

31
III. The Variance-Covariance Approach

• Extend this analysis to calculate the potential
  loss over 2, 3, ....., N days. Assuming that
  yield shocks are independent, then the N-day
  market risk (VAR) is related to daily earnings at
  risk (DEAR) by
•   VAR DEAR x ?N
• If N is 5 days, then
•   VAR 10,770 x ?5
• 24,082
•  If N is 10 days, then
•   VAR 10,770x ?10
• 34,057

32
III. The Variance-Covariance Approach

• B. Foreign Exchange
• Suppose the bank had a DM 1.6 million trading
  position in spot German Deutsch marks. What is
  the daily earnings at risk?
• The first step calculate the dollar amount of
  the position
• Dollar amount of position
• (FX position) (DM/ spot exchange rate)
• (DM 1.6 million) (0.625/DM)
• 1 million 

33
III. The Variance-Covariance Approach

• Suppose that the ? of the daily changes on the
  spot exchange rate was 56.5 bp over the past
  year.
• We are interested in adverse moves--that is, bad
  moves that will not be exceeded more than 5
  percent of the time or 1.65 ?.
•   FX volatility 1.65 x 56.5 bp 93.2 bp
• Thus
• DEAR (Dollar amount of position) (FX
  volatility)
• (1 million)x (.00932) 9,320

34
III. The Variance-Covariance Approach

• C. Equities
• From the Capital Pricing Model (CAPM)  
• Total risk Systematic risk Unsystematic risk
• ?2it ?2 it ?2 mt ?2 eit
• Systematic risk reflects the movement of that
  stock with the market (reflected by the stock's
  beta ( ?it ) and the volatility of the market
  portfolio (? mt), while unsystematic risk is
  specific to the firm itself (? eit)

35
III. The Variance-Covariance Approach

• In a very well-diversified portfolio,
  unsystematic risk can be largely diversified
  away, leaving behind systematic (undiversifiable)
  market risk.
• Suppose the FI holds a 1 million trading
  position in stocks that reflect a U.S. market
  index (e.g., the Wilshire 5000). Then DEAR would
  be
•   DEAR (Dollar value of position) (Stock
  market return volatility)
• (l,000,000) (1.65 ? m).

36
III. The Variance-Covariance Approach

• If, over the last year, the ? m of the daily
  changes in returns on the stock index was 2
  percent, then 1.65 ?m 3.3 percent.  
• DEAR (1,000.000) (0.033)
• 33,000
• In less well-diversified portfolios, the effect
  of unsystematic risk ? eit, on the value of the
  trading position would need to be added.
• Moreover, if the CAPM does not offer a good
  explanation of asset pricing say, multi-index
  arbitrage pricing theory (APT), a degree of error
  will be built into DEAR calculation.

37
III. The Variance-Covariance Approach

• More Examples
• Example 1 We have a position worth 10 million
  in E-Bay shares
• The standard deviation of E-Bay is 2 per day
  (about 32 per year)
• We use N10 and X99
• The standard deviation of the change in the
  portfolio in 1 day is 200,000
• The standard deviation of the change in 10 days
  is

38
III. The Variance-Covariance Approach

• We assume that the expected change in the value
  of the portfolio is zero (This is OK for short
  time periods)
• We assume that the change in the value of the
  portfolio is normally distributed
• Since N(2.33)0.01, the VaR is

39
III. The Variance-Covariance Approach
Example 2 Consider a position of 5 million in
Citigroup The daily volatility of Citigroup is 1
(approx 16 per year) The S.D per 10 days
is The VaR is
40
III. The Variance-Covariance Approach

• D. Portfolio Aggregation
• Consider a portfolio consists of
• seven-year, zero-coupon, fixed-income (1 million
  market value),
• spot DM (1 million market value), and
• the U.S. stock market index (l million market
  value).
•  The individual DEARS were
• 1. Seven-year zero 10,770
• 2. DM spot 9,320
• 3. U.S. equities 33,000

41
III. The Variance-Covariance Approach

• Correlations ( ?ij ) among Assets
•  
• Seven-year DM/ U.S. stock Zero index
• ___________________________________________
• Seven-year - -.2 .4
• DM/ - .1
• U.S stock index - -
• ___________________________________________
•  

42
III. The Variance-Covariance Approach

• Using this correlation matrix along with the
  individual asset DEARs, we can calculate the risk
  of the whole trading portfolio
• DEAR portfolio (DEARZ) 2 (DEARDM) 2
• (DEARU.S) 2 (2 ?Z,DM DEARZ
• DEARDM) (2 x ?Z,U.S DEARZ DEARU.S)
• (2 ?U.S,DM DEARUS DEARDM )1/2
• (10.77)2 (9.32)2 (33)2
• 2(.2)(10.77)(9.32) 2(.4)(10.77)(33)
• 2(.1)(9.32)(33) 1/2
• 39,969

43
III. The Variance-Covariance Approach

• In actuality, the number of markets covered by
  JPMs traders and the correlations among those
  markets require the daily production and updating
  of over volatility estimates ((T) and 53,628
  correlations (P).

44
III. The Variance-Covariance Approach

• RiskMetrics Volatilities and Correlations
• Number of Number of Total Markets Point
  s
• __________________________________________________
  __
• Term structures
• Government bonds 14 7-10 120
• Money markets and 15 12 180
• and swaps
• Foreign exchange 14 1 14
• Equity indexes 14 1 14
• Volatilities 328
• Correlations 53,628
• ____________________________________ 

45
IV. Historical or Back Simulation Approach

• A major criticism of RiskMetrics is the need to
  assume a symmetric (normal) distribution for all
  asset returns.
• The advantages of the historical approach
• (1) it is simple,
• (2) it does not require that asset returns be
  normally distributed, and
• (3) it does not require that the correlations or
  standard deviations of asset returns be
  calculated.

46
IV. Historical or Back Simulation Approach

• The essential idea is to take the current market
  portfolio of assets and revalue them on the basis
  of the actual prices that existed on those assets
  yesterday, the day before that, and so on.
• The FI will calculate the market or value risk of
  its current portfolio on the basis of prices
  that existed for those assets on each of the last
  500 days. It would then calculate the 5 percent
  worst case, that is, the portfolio value that has
  the 25th lowest value out of 500.

47
IV. Historical or Back Simulation Approach

• Example At the close of trade on December 1,
  2000, a bank has a long position in Japanese yen
  of 500,000,000 and a long position in Swiss
  francs of 20,000,000. If tomorrow is that one bad
  day in 20 (the 5 percent worst case), how much
  does it stand to lose on its total foreign
  currency position?
• Step 1 Measure exposures.
• Convert today's foreign currency positions into
  dollar equivalents using today's exchange rates.

48
IV. Historical or Back Simulation Approach

• Step 2 Measure sensitivity.
• Measuring sensitivity of each FX position by
  calculating its delta, where delta measures the
  change in the dollar value of each FX position if
  the yen or the Swiss franc depreciates by 1
  percent.
• Step 3 Measure risk.
• Look at the actual percentage changes in exchange
  rates, yen/ and Swf/, on each of the past 500
  days.
• Combining the delta and the actual percentage
  change in each FX rate means a total loss of
  47,328.9 if the FI had held the current Y
  500,000,000 and Swf 20,000,000 positions on that
  day (November 30, 2000).

49
IV. Historical or Back Simulation Approach

• Step 4 Repeat Step 3.
• Step 4 repeats the same exercise for the
  positions but using actual exchange rate changes
  on November 29, 2000 November 28, 2000 and so
  on. For each of these days the actual change in
  exchange rates is calculated and multiplied by
  the deltas of each position.
• Step 5 Rank days by risk from worst to best.
• The worst-case loss would have occurred on May 6,
  1999, with a total loss of 105,669.
• We are interested in the 5 percent worst case.
  The 25th worst loss out of 500 occurred on
  November 30, 2000. This loss amounted to
  47,328.9.

50
IV. Historical or Back Simulation Approach

• Step 6. VAR. If assumed that the recent past
  distribution of exchange rates is an accurate
  reflection of the likely distribution of FX rate
  changes in the future--that exchange rate changes
  have a "stationary" distribution--then the
  47,328.9 can be viewed as the FX value at risk
  (VAR) exposure of the FI on December 1, 2000.
  This VAR measure can then be updated every day as
  the FX position changes and the delta changes.

51
IV. Historical or Back Simulation Approach

• Table Hypothetical Example of the Historical or
  Back Simulation Approach
• Yen Swiss Franc
• __________________________________________________
  __
• Step 1 Measure Exposure
• 1. Closing position on Dec. 1, 2000 500,000,000
  20,000,000
• 2. Exchange Rate on Dec. 1, 2000 Y130/1
  Swf1.4/1
• 3. U.S. equivalent position
• on Dec. 1, 2000 3,846,154 14,285,714
•  
• Step 2 Measuring Sensitivity
• 4. 1.01current exchange rate Y131.3/1
  Swf1.414/1
• 5. revalued position in 3,808,073
  14,144,272
• 6. Delta of position -38,081 -141,442

52
IV. Historical or Back Simulation Approach

• Step 3 Measuring risk of Dec. 1, 2000, closing
  position using exchange rates that existed on
  each of the last 500 days
• November 30, 2000 Yen Swiss Franc
• __________________________________________________
  __
• 7. Change in exchanger rate
• () on Nov. 30, 2000 0.5 0.2
• 8. Risk (deltachange in
• exchange rate) -19,040.5 -28,288.4
• 9. Sum of risks -47,328.9
• __________________________________________________
  __
• Step 4 Repeat Step 3 for each of the remaining
  499 days

53
IV. Historical or Back Simulation Approach

• Step 5 Rank days by risk from worst to best
• Date Risk ()
• __________________________________________________
  
• 1. May 6, 1999 -105,669
• 2. Jan 27, 2000 -103,276
• 3. Dec 1, 1998 -90,939
• .
• 25. Nov 30, 2000 -47,329
• .
• 500 July 28, 1999 -108,376
• __________________________________________________
  __
• Step 6 VAR (25th worst day out of last 500)
•   VAR -47,328.9 (Nov. 30, 2000)

54
IV. Historical or Back Simulation Approach

• Advantages of the Historic (Back Simulation)
  Model versus RiskMetrics
• No need to calculate standard deviations and
  correlations to calculate the portfolio risk
  figures.
• It directly provides a worse-case scenario
  number. RiskMetrics, since it assumes asset
  returns are normally distributed--that returns
  can go to plus and minus infinity--provides no
  such worst-case scenario number.

55
IV. Historical or Back Simulation Approach

• The disadvantage
• The degree of confidence we have in the 5 percent
  VAR number based on 500 observations.
• Statistically speaking, 500 observations are not
  very many, and so there will be a very wide
  confidence band (or standard error) around the
  estimated number (47,328.9 in our example).
• One possible solution is to go back in time more
  than 500 days and estimate the 5 percent VAR
  based on 1,000 past observations (the 50th worst
  case) or even 10,000 past observations (the 500th
  worst case). The problem is that as one goes back
  farther in time, past observations may become
  decreasingly relevant in predicting VAR in the
  future.

56
V. The Monte Carlo Simulation Approach

• We can implement the model-building approach
  using Monte Carlo simulation to generate the
  probability distribution for DP, the dollar
  change on the portfolio.
• To calculate VaR using M.C. simulation we
• 1. Value portfolio today using the current values
  of market variables
• 2. Sample once from the multivariate normal
  distributions of the Dxi
• 3. Use the Dxi to determine market variables at
  end of one day
• 4. Revalue the portfolio at the end of day
• 5. Subtract the value calculated in Step 1 from
  the value in Step 4 to determine a sample DP
• 6. Repeat many times to build up a probability
  distribution for DP

57
V. The Monte Carlo Simulation Approach

• VaR is the appropriate fractile of the
  distribution times square root of N
• For example, with 5,000 trial, the 1-day 99 VaR
  is the value of DP for the 50th worst outcome
  the 1-day VaR 95 is the value of DP for the
  250th worst outcome, and so on.

58
V. The Monte Carlo Simulation Approach

• The drawback of Monte Carlo simulation is tha tit
  tends to be slow because a companys complete
  portfolio (which might consist of more than 1,000
  instruments) has to be revalued many times.
• If a relationship between DP and Dxi can be
  specified, we can then jump straight from step 2
  to step 5 in the Monte Carlo simulation and avoid
  the need for a complete revaluation of the
  portfolio. This is sometimes referred to as the
  partial simulation approach.

59
VI. Regulatory Models The BIS Standardized
Framework

• The 1993 BIS proposals regulate the market risk
  exposures of banks by imposing capital
  requirements on their trading portfolios.
• Since January 1998 the largest banks in the world
  are allowed to use their own internal models to
  calculate exposure for capital adequacy purposes,
  leaving the standardized framework as the
  relevant model for smaller banks.

60
VI. Regulatory Models The BIS Standardized
Framework

• 1. Fixed Income
• 1. The specific risk charge is meant to measure
  the risk of a decline in the liquidity or credit
  risk quality of the trading portfolio over the
  FI's holding period.
• Treasury's have a zero risk weight, while junk
  bonds have a risk weight of 8 percent.
• Multiplying the absolute dollar values of all the
  long and short positions in these instruments by
  the specific risk weights produces a total
  specific risk charge of 229.

61
VI. Regulatory Models The BIS Standardized
Framework

• 1. Fixed Income
• 1. The specific risk weights

62
VI. Regulatory Models The BIS Standardized
Framework

• 2. The general market risk charges or weights
  reflect the same modified durations and interest
  rate shocks for each maturity in the BIS model
  for total gap exposure.
• This results in a general market risk charge of
  66.

63
VI. Regulatory Models The BIS Standardized
Framework

• Panel A FI Holdings and Risk Charges
• Specific Risk General Market Risk
• (1) (2) (3) (4) (5) (6) (7)
• Time Band Issuer Position Weight Charge Weight Cha
  rge
• __________________________________________________
  _______________________
• 0-1 month Treasury 5,000 0.00 0.00 0.00 0.00
• 1-3 month Treasury 5,000 0.00 0.00 0.20 10.00
• 3-6 month Qual Corp 4,000 0.25 10.00 0.40 16.0
  0
• 6-12 month Qual Corp (7,500) 1.00 75.00 0.70 (52
  .50)
• 1-2 years Treasury (2,500) 0.00 0.00 1.25 (31.2
  5)
• 2-3 years Treasury 2,500 0.00 0.00 1.75 43.75
• 3-4 years Treasury 2,500 0.00 0.00 2.25 56.25
• 3-4 years Qual Corp (2,000) 1.60 32.00 2.25 (45.0
  0)
• 4-5 years Treasury 1,500 0.00 0.00 2.75 41.
  25
• 5-7 years Qual Corp (1,000) 1.60 16.00 3.25 (32.
  50)

64
VI. Regulatory Models The BIS Standardized
Framework

• Panel A FI Holdings and Risk Charges
• Specific Risk General Market Risk
• (1) (2) (3) (4) (5) (6) (7)
• Time Band Issuer Position Weight Charge Weight Cha
  rge
• __________________________________________________
  _______________
• 7-10 years Treasury (1,500) 0.00 0.00 3.75 (
  56.25)
• 10-15 years Treasury (1,500) 0.00 0.00 4.50 (67.5
  0)
• 10-15 years Non Qual 1,000 8.00 80.00 4.50 45.0
  0
• 15-20 years Treasury 1,500 0.00 0.00 5.25 78.
  75
• gt 20 years Qual corp 1,000 1.60 16.00 6.00 60.0
  0
• __________________________________________________
  _______________
• Specific Risk 229.00
• Residual General Market Risk 66.00

65
VI. Regulatory Models The BIS Standardized
Framework

• 3. Offsets or Disallowed Factors The BIS model
  assumes that long and short positions, in the
  same maturity bucket but in different
  instruments, cannot perfectly offset each other.
  Thus, this 66 general market risk tends to
  underestimate interest rate or price risk
  exposure.
• For example, the FI is short 10-15 year U.S.
  Treasuries with a market risk charge of 67.50
  and is long 10-15 year junk bonds with a risk
  charge of 45. However, because of basis
  risk--that is, the fact that the rates on
  Treasuries and junk bonds do not fluctuate
  exactly together---we cannot assume that a 45
  short position in junk bonds is hedging an
  equivalent (45) value of U.S. Treasuries of the
  same maturity.

66
VI. Regulatory Models The BIS Standardized
Framework

• Vertical Offsets
• Thus, the BIS requires additional capital charges
  for basis risk, called vertical offsets or
  disallowance factors. In our case, we disallow 10
  percent of the 45 position in junk bonds in
  hedging 45 of the long Treasury bond position.
  This results in an additional capital charge of
  4.5.

67
VI. Regulatory Models The BIS Standardized
Framework

• Horizontal Offsets within Time Zones
• The debt portfolio is divided into three maturity
  zones
• zone 1 (1 month to 12 months),
• zone 2 (over 1 year to 4 years), and
• zone 3 (over 4 years to 20 years plus).
• Because of basis risk, long and short positions
  of different maturities in these zones will not
  perfectly hedge each other.
• This results in additional (horizontal)
  disallowance factors of
• 40 percent (zone 1),
• 30 percent (zone 2), and
• 30 percent (zone 3).

68
VI. Regulatory Models The BIS Standardized
Framework

• Horizontal Offsets between Time Zones
• Finally, any residual long or short position in
  each zone can only partly hedge an offsetting
  position in another zone. This leads to a final
  set of offsets or disallowance factors between
  time zones.
•  
• Summing the specific risk charges (229), the
  general market risk charge (66), and the basis
  risk or disallowance charges (75.78) produces a
  total capital charge of 370.78.

69
VI. Regulatory Models The BIS Standardized
Framework

• Panel B Calculation of Capital Charge
• 1. Specific Risk 229.00
• 2. Vertical Offers within Same Time Bands
• (1) (2) (3) (4) (5) (6) (7)
• Time Band Longs Shorts Residual Offset
  Disallowance Charge
• __________________________________________________
  _______________
• 3-4 years 56.25 (45.00) 11.25 45.00 10.00 4.50
• 10-15 years 45.00 (67.50) (22.50) 45.00 10.00 4.50
  
• __________________________________________________
  _______________

70
VI. Regulatory Models The BIS Standardized
Framework

• Panel B Calculation of Capital Charge
• 3. Horizontal Offers within Same Time Bands
• (1) (2) (3) (4) (5) (6) (7)
• Time Band Longs Shorts Residual Offset
  Disallowance Charge
• __________________________________________________
  ________________________
• Zone 1
• 0-1 month 0.00
• 1-3 month 10.00
• 3-6 months 16.00
• 6-12 months (52.50)
• Total Zone 1 26.00 (52.50) (26.50) 26.00 40.00 10
  .40
• Zone 2
• 1-2 years (31.25)
• 2-3 years 43.75
• 3-4 years 11.25
• Total Zone 2 55.00 (31.25) 23.75 31.25 30.00 9.38
  

71
VI. Regulatory Models The BIS Standardized
Framework

• Panel B Calculation of Capital Charge
• 3. Horizontal Offers within Same Time Bands
• (1) (2) (3) (4) (5) (6) (7)
• Time Band Longs Shorts Residual Offset
  Disallowance Charge
• __________________________________________________
  _______________________
• Zone 3
• 4-5 years 41.25
• 5-7 years (31.50)
• 7-10 years (56.25)
• 10-15 years (22.50)
• 15-20 years 78.75
• gt 20 years 60.00
• Total Zone 3 180.00 (111.25) 68.75 111.25 30.00 3
  3.38
• __________________________________________________
  _______________________

72
VI. Regulatory Models The BIS Standardized
Framework

• Panel B Calculation of Capital Charge
• 4. Horizontal Offers between Time Zones
• (1) (2) (3) (4) (5) (6) (7)
• Time Band Longs Shorts Residual Offset
  Disallowance Charge
• __________________________________________________
  _______________
• Zones 1 and 2 23.75 (26.50) (2.75) 23.75 40.00 9.
  50
• Zones 1 and 3 68.75 (2.75) 66.00 2.75 150 4.12
• 5. Total Capital Charge
• Specific Risk 229.00
• Vertical disallowances 9.00
• Horizontal disallowances
• Offsets within same time zones 53.1
• Offsets between time zones 13.62
• Residual general marker risk after all offsets
  66.00
• Total 370.78

73
VI. Regulatory Models The BIS Standardized
Framework

• 2. Foreign Exchange
• The BIS originally proposed two alternative
  methods to calculate FX trading exposure--a
  shorthand and a longhand method
• The shorthand method requires the FI to calculate
  its net exposure in each foreign currency and
  then convert this into dollars at the current
  spot exchange rate.
• As shown in Table below, the FI is net long
  (million dollar equivalent) 50 yen, 100 DM, and
  150 pounds while being short 20 French francs
  and 180 Swiss francs.

74
VI. Regulatory Models The BIS Standardized
Framework

• Table Example of the Shorthand Measure of
  Foreign Exchange Risk
• Once a bank has calculated its net position in
  each foreign currency, it converts each position
  reporting currency and calculates the shorthand
  measure as in the following example, in which
  position in the reporting currency has been
  excluded
•  
• Yen DM GBE Fr fr SW fr
  Gold Platinum
• __________________________________________________
  __
•   50 100 150 -20 -180
  -30 5
•   (300) (-200)
  (35)
• __________________________________________________
  __
• The capital charge would be 8 percent of the
  higher of the longs and shorts (i.e., 300) plus
  positions in precious metals (35) 335 x 8
  26.8.

75
VI. Regulatory Models The BIS Standardized
Framework

• The BIS proposes a capital requirement equal to 8
  percent times the maximum absolute value of
  either aggregate long or short positions.
• In this example, 8 percent times 300 million
  24 million. This assumes some partial but not
  complete offsetting of currency risk by holding
  opposing long or short positions in different
  currencies.

76
VI. Regulatory Models The BIS Standardized
Framework

• The alternative longhand method First, the FI
  calculates its net position in each foreign
  currency. The BIS assumes that the FI will hold
  its position for a maximum of 14 days (10 trading
  days). Exposure is measured by the possibility of
  an outcome occurring over the holding (trading)
  period. As in the JPM model, the worst outcome
  is a simulated loss that will occur in only 1 of
  every 20 days or exceeded only 5 percent of the
  time.

77
VI. Regulatory Models The BIS Standardized
Framework

• To estimate its potential exposure, the FI looks
  back at the history of spot exchange rates over
  the last five years and--assuming overlapping
  10-day holding periods-simulates the gains and
  losses on the 10 million short position. Over the
  five years, this will involve approximately 1,300
  simulated trading period gains and losses. The
  worst-case scenario (95 percent) is the 65th
  worst outcome of the 1,300 simulations. If the
  worst-case scenario is a loss of 2 million, the
  FI would be required to hold a 2 percent capital
  requirement against that loss or
• 2 million x .02 40,000

78
VI. Regulatory Models The BIS Standardized
Framework

• Table Simulation of Gains/Losses on a Position
• Current Position Net Short 10 Million
• __________________________________________________
  __________
• Date(-t) Rate Position Value Value at
  Profit/
• () at t -(t-10) Loss
• __________________________________________________
  __________
• -1 1.2440
• -2 1.2400
• -3 1.2350
• .
• .
• -11 1.2350 -10 12.35 12.44 -.09
• -12 1.2400 -10 24.00
  24.00 -
• -13 1.2500 -10 12.5 12.35 .15
• .
• __________________________________________________
  ___________

79
VI. Regulatory Models The BIS Standardized
Framework

• 3. Equities
• X factor The BIS proposes to charge for
  unsystematic risk by adding the long and short
  positions in any given stock and applying a 4
  charge against the gross position in the stock.
• Suppose stock number 2 in the following table,
  the FI has a long 100 million and short 25
  million position in that stock. Its gross
  position that is exposed to unsystematic (firm
  specific) risk is 125 million, which is
  multiplied by 4 percent, to give a capital charge
  of 5 million.

80
VI. Regulatory Models The BIS Standardized
Framework

• Y factor Market or systematic risk is reflected
  in the net long or short position.
• In the case of stock number 2, this is 75
  million (100 long minus 25 short). The capital
  charge would be 8 percent against the 75
  million, or 6 million.
• The total capital charge (x factor y factor) is
  11 million for this stock.

81
VI. Regulatory Models The BIS Standardized
Framework

• Table BIS Capital Requirement for Equities
•   x Factor_ y Factor
• Stock Sum of Sum of Gross 4 Percent Net
  8 percent Capital
• Long Short position of Gross position of Net
  Requirement
• Position Position
• __________________________________________________
  _______________
• 1 100 0 100 4 100 8 12
• 2 100 25 125 5 75 6 11
• 3 100 50 150 6 50 4 10
• 4 100 75 175 7 25 2 9
• 5 100 100 200 8 0 0 8
• 6 75 100 275 7 25 2 9
• 7 50 100 150 6 50 4 10
• 8 25 100 125 5 75 6 11
• 9 0 100 100 4 100 8 12
• __________________________________________________
  _______________

82
VII. Large Bank Internal Models

• Starting from April 1998, large banks are allowed
  to use their own internal models to calculate
  risk. The required capital calculation had to be
  relatively conservative
• 1. An adverse change in rates is defined as being
  in the 99th percentile rather than in the 95th
  percentile (multiply a by 2.33 rather than by
  1.65)
• 2. The minimum holding period is 10 days (this
  means that RiskMetrics' daily DEAR would have to
  be multiplied by ? 10).
• 3. Empirical correlations are to be recognized in
  broad categories--for example, fixed income--but
  not between categories---for example, fixed
  income and FX--so that diversification is not
  fully recognized.

83
VII. Large Bank Internal Models

• The proposed capital charge will be the higher
  of
• 1. The previous day's VAR (value at risk or DEAR
  ? 10)
• 2. The average daily VAR over the previous 60
  days times a multiplication factor with a minimum
  value of 3 (i.e., Capital change (DEAR) (?
  10) (3)).
• In general, the multiplication factor will make
  required capital significantly higher than VAR
  produced from private models. 

84
VII. Large Bank Internal Models

• An additional type of capital can be raised
• Tier 1 retained earnings and common stock
• Tier 2 long-term subordinated debt (gt 5 years)
• Tier 3 short-term subordinated debt (lt 2 years)
  .
• Limitations
• Tier 3 capital is limited to 250 of Tier 1
  capital
• Tier 2 capital can be substituted for Tier 3
  capital up to the same 250 limit.

85
VII. Large Bank Internal Models