Market Risk by K61gtQv

VIEWS: 7 PAGES: 51

									                          Market Risk
1.    Definition of Market Risk

2.    Interest Rate Risk: Cash Flow Risk

3.    Motivations to quantify Market Risk

4.    Calculating Market VAR

5.    VAR and Confidence Level

6.    VAR and Diversification Effect

7.    Multiperiod VAR

8.    Historical VAR

9.    Backtesting the VAR

10.   Portfolio Toolkit

11.   Portfolio VAR

12.   Pros and Cons of VAR Measures

13.   Market Risk Regulation

14.   Stress Testing
                Defining Market Risk


   Interest rate shifts
   Foreign exchange shifts         Reduction in the
                                    value of A/L,
   Commodity price shifts          securities, etc.
    portfolios, etc.
   Equity price shifts


   Market risk affects :
       Trading books               mark-to-market
       Banking book
       Derivative trading          mark-to-market
       Credit derivative trading   mark-to-market
                      Interest Rate Risk


   Shifts in the term structure of interest rate have two impacts

       A cash flow effect



       Market value effect



   To manage cash flow risk, banks can calculte the GAP:



                   NII  GAP x i
 Interest Rate Risk:
   Cash flow risk
 ASSETS          LIABILITIES
   RSA               RSL
400 @ 8%          600 @ 6%

   FRA               FRL
600 @ 10%         400 @ 8%

  Interest income = 32 + 60 = 92
Interest expenses = 36 + 32 = 68
                        NII = 24
 Interest Rate Risk:
   Cash flow risk
  ASSETS          LIABILITIES
   RSA                RSL
400 @ 10%          600 @ 8%

   FRA                FRL
600 @ 10%          400 @ 8%

  Interest income = 40 + 60 =100
Interest expenses = 48 + 32 = 80
                         NII = 20
                       Interest Rate Risk:
                         Cash flow risk
   Interest rates increase by 2% such that NII decreases by $4
    million.

   Using the GAP:

                     NII  $200 x 0.02
                     NII  $4

   For all RSA: duration and convexity are used to capture the
    impact of interest rate.
                        Motivations to
                     Quantify Market Risk

   Required by many internal instances such as the Chief Risk Officer,
    various risk management committees, etc, to track at any time the
    exposure of the bank (or other FI’s) with respects to market risk.

   Limit setting.

   Risk components of performance evaluation.

   Economic capital allocation across the different business units.

   Required for regulatory purposes (Since Basel 98).
         Calculating Market VAR
   Recognized as the best measure to capture market risk.

   VAR measure?
       How much could the FI loose on a given position, with
        an x% probability of occurrence over a given time
        frame?

   Example :
       A one-day VAR of $20 million at a probability of 1%,
        (or equivalently, at a confidence level of 99%) means
        that one day out of 100, the bank could loose $20
        million or more

   Statistical Methodology
           Calculating Market Var

Graphically :       Two possible Value-at-Risk measures


                         VAR-measure
     Probability             no 1
     distribution




                    1%



                                 0   Expected         Return profits
                    VAR-measure
                                     return profits
                          no 2
            Calculating Market VAR
           (Normal density function)

   Example :

    A European bank currently has a $120 million in foreign exchange
    position. What is the one-day VAR at a confidence level of 95% (5%
    probability of occurrence) knowing that the current $US/€ is 1.20 ?

    Based on an historical sample, assume that the mean and standard
    deviation of the daily Δ% $US/€ exchange rate are (continuously
    compounding):

                r  0.10%      and      ( r )  0.55%
    Assume further that the exchange rate return (Δ% $US/€)
    follows a normal probability distribution function
          Calculating Market VAR
         (Normal density function)
•   VAR measure no 1: In this case, the deviation is
    calculated from the mean. Therefore:

We look for Probr  VAR    %  1  c%. Thus,
Probz  z c    %
Probz  -1.65  5%
     r - r         
       (r)   -1.65  5%
Prob       
                   
Probr  -1.65  (r)  r   5%
Probr  -1.65 0.0055  0.0010  5%
           Calculating Market VAR
          (Normal density function)
   Example (cont’d):


           Prob r  0.008075   5%
           where VAR  0.8075 %
The VAR can also be expressed in Euro:

                                                     1
     VAR in Euro  -0.008075 $US 120 million 
                                                1.20 $US/E
     VAR in Euro   Euro 807 500
          Calculating Market VAR
         (Normal density function)
•   VAR measure no 2: In this case, the VAR is
    calculated only using the standard deviation

We look for Prob r  VAR    %  1  c%. Thus,
Prob z  z c    %
Prob z  -1.65   5%
Prob r  -1.65   (r)   5%
Prob r  -1.65  0.0055   5%
           Calculating Market VAR
          (Normal density function)
   Example (cont’d):


           Prob r  0.009075   5%
           where VAR  0.9075 %
The VAR can also be expressed in Euro:

                                                     1
     VAR in Euro  -0.009075 $US 120 million 
                                                1.20 $US/E
     VAR in Euro   Euro 907 500
           Calculating Market VAR
          (Normal density function)

   If the probability of occurrence is set at 1%, under the
    VAR measure no 2, we obtain :

                (2.33  0.0055 )  Euro 100 million
                 Euro 1 281 500

   The Var measure no 2 closely depends on the
    standard deviation measure:

                Var    N ( x)   -1
              VAR and Confidence Level

                        Probability of occurence
       High                                                Low



  Less conservative:                               More conservative:
  Low capital charges                              High capital charges
                           Increase in VAR value




Often, the level of confidence at which the Var is calculated is
closely related to the credit rating. For instance, a AA bank
with an S&P one-year probability of default of 0.03% might
choose a corresponding level of confidence of 99.97%.
             VAR and Diversification Effect
   Example :

       A European bank has a position in a USD bond with 6 year
       to maturity, annual coupon of 8% and a yield to maturity
       of 12%. The price of the bond is $835.54 while its
       duration is 4.88 years. The bank holds a $120 million
       (market value) position in that bond.
   Given a spot exchange rate of $1.20/€ 1:

                        $120 million
                                      Euro 100 million
                           1.20

   There are two sources of risk: the interest rate and the
    currency risk.

   However, we can calculate individual VAR’s, the first captures
    the interest risk while the second captures the currency risk
    VAR and Diversification Effect
•   Interest rate VAR:

                ΔP              Δy
                      D 
                 P            (1  y)
                                   Δy
                rBond, y  D 
                                (1  y)
                                     σ(Δy)
                σ(rBond, y )  D 
                                   (1  0.12)

•   Assuming  (y)  0.001

                              4.88  0.001
               σ(rBond, y )                0.43%
                                 1.12
         VAR and Diversification Effect

•   Interest rate VAR at 1% (normal density function,
    measure no. 2):

             (2.33  0.0043  $ 120 million )
       VAR                                      Euro 1001900
                       1.20 $ /Euro

•   Currency VAR at 1% (normal density function,
    measure no. 2, see previous example):

                  - Euro 1 281 500
    VAR and Diversification Effect

Total VAR         = Interest rate VAR + currency VAR
                  = € 1001900 + € 1 281 500
                  = € 2 283 400
   But that approach does not account for the diversification effect:
     The return on the total position is :


                        rbond  rbond,y  rbond,ex

             %∆ in the bond price            %∆ in the bond price
             resulting from shifts in        resulting from shifts in $/€
             US interest rate                exchange rate
                  VAR and Diversification Effect
Thus:

 σ 2 (rbond )  σ 2 (rbond,y  rbond,ex )
 σ 2 (rbond )  σ 2 (rbond,y )  σ 2 (rbond,ex )  2  rbond,y ,rbond,ex σ(rbond,y )σ(rbond,ex )

 σ(rbond )  σ 2 (rbond )  σ 2 (rbond,y )  σ 2 (rbond,ex )  2  rbond,y ,rbond,ex σ(rbond,y )σ(rbond,ex )
  VAR and Diversification Effect

Assuming    r  bond,y rbond,ex
                                   0.5


             rp               0.43%2  0.55%2  2  0.5  0.43%  0.55%
                      0.00007239

            0.8508%
Then, the total VAR is (normal density function):
   VAR in Euro  2.33  0.008508  Euro 100 million
   VAR in Euro   Euro 1 982 418





          VAR and Diversification Effect

       The gain from diversification are:

           € 2 283 400      -    € 1 982 418 = € 300 982

           No diversification   Diversification effect
                 effect


                                Correlation effect

       When there are more than one risk factor involves, the correlation
        parameter,  , is responsable for a risk diversification effect. To see
        that, rework the previous example with   0,   1 and   1 and look
        at what happens to the Var calculation.
              Multiperiod VAR


   Multiple-day VAR based on the one-day VAR :

           n-day VAR = n½ x one-day VAR
                       Historical VAR
   The previous approach is based on the assumption of a
    normal probability distribution function

        May be vaild for diversified portfolios with independant risk
         factors
        Unlikely to be valid in many situations (data frequency, type of
         instruments, etc.)

   Historial VAR :
     1) Identification of an historical sample for the risk factors, at a
        given frequency.

     2) Application of those changes to the current risk factors.
        Computation of the corresponding position.

     3) Sorting of the payoff vector and identification of the appropriate
        percentile on the empirical distribution.
                   Historical VAR
   An example :
        A European bank has a position in a USD bond with
        6 years to maturity, annual coupon of 8% while the
        yield to maturity is 12%. The price of the bond is $
        835.54 while its duration is 4.88 years. The bank
        holds a $120 million position in that bond.
       The exposure is :
                     $120 million
                                   Euro 100 million
                        1.20
       Current data :
              interest rate   exchange rate duration
                12%                1.20       4.88
                    Historical VAR
The historical changes in the risk factors :

      n = 250        ∆% $/Euro   ∆ 10-year rate
January 2, 2006       - 0.521%        0.02%
January 3, 2006     - 0.4405%         0.02%
January 4, 2006     - 0.0256%        -0.04%
         ▪                 ▪           ▪
         ▪                 ▪           ▪
         ▪                 ▪           ▪
December 23, 2006     - 0.563%        0.03%
December 24, 2006     - 1.004%       -0.05%
                                  Historical VAR
   The return on this position :
                              rbond  rbond, y  rbond,ex

                                               y
                                      D             %$ /E
                                             1  y 
                                           4.88
                                                    y  %$ /E
                                        1  0.12 
   On January 2, 2006 :

                               rbond   4.357  0.0002  0.00521
                               rbond  0.6081%
   Euro variation in the bond position:

                                 Euro 100 million   0.00682
                                  Euro 608 074 .108
                           Historical VAR
   The procedure is repeated for every day of the historical sample :
               Day           Euro gains/losses on the bond
       January 2, 2006              - 608 074.108
       January 3, 2006              - 507 812.391
       January 4, 2006                149 032.398
                ▪                           ▪
                ▪                           ▪
                ▪                           ▪
       December 23, 2006            - 694 284.502
       December 24, 2006           - 786 181. 722



   The vector of gains/losses is then sorted from the highest to the
    lowest : Rank Scenario       Euro gains/losses on the bond
                 1         96                1 691 388.41
                 2         231               1 625 185.38
                 3         127               1 535 152.77
                 ▪          ▪                      ▪
                 ▪          ▪                      ▪
                238        191              - 1 360 529.24
                 ▪          ▪                      ▪
                240        120              - 1 737 629.56
                  Historical VAR


   The 1% VAR is :
        Rank          Scenario   Euro gains/losses on the bond
         238            191              - 1 360 529.24


   The 5% VAR is :

        Rank          Scenario   Euro gains/losses on the bond
         228             12              - 989 650.223
                       Historical VAR:
                     Confidence Interval
   The historical Var measure corresponds to a quantile of
    the empirical distribution. Kandal and Stuart (1972, c.f.
    Hull 2007 p. 220) derive an estimate of the
    corresponding standard error:
                                        1    q(1  q)
                      std(HVar) 
                                      f ( x)    n
            where q : quantile of the distribution
                    n : number of observations
                    f ( x) : postulated density function of the HVar

   Assumed f (x) is the normal probability density function.
    Then, the confidence interval at a 95% confidence level
    is:
            
                                            
                                                                  
       prob HVar - std(Hvar) 1.96  HVar  HVar  std(Hvar) 1.96  95%
           
                                                                  
                                                                   
                    Historical VAR:
                  Confidence Interval
   Based on the previous example:
                               1          0.05  0.95
          std(HVar)                                   81278 .69
                        1.773087 e  07      240


   The value of f ( x)  1.756E  07 comes from Excel
    assuming that the mean payoff is 0 and its std is
    $595860.22 and using the following command:

            NORMINV(0.05,0,595860.22)=-980102.84
        NORMDIST(-980102.84,0, 595860.22,False)=1.736E-07


Then, the confidence for the HVar is:
             $989650.22- 81278.49 1.96  HVar 
         prob                                     95%
             $989650.22 81278.49 1.96         
                    Historical VAR:
                  Confidence Interval
   Remark: It is also possible to produce a confidence
    interval via bootstrapping:
   Since n  250 , it is possible to sample 250 000 scenarios
    of 1000  250 samples
   For each sample, calculate the HVar. With all the HVar
    estimates, rank them from the Highest to the lowest
   This provides an empirical distribution of Var based on
    1000 HVar estimates. Using quantiles, we know that:


           
                                       
                                                
      prob HVar (at 97.5%)  HVar  HVar (at 2.5%)  95%
          
                                                  
                                                   
                   Historical VAR
   Advantages :
       Method completely nonparametric, no distributional
        assumption
       Easy to apply
       Captures complex volatility and correlation effects


   Disavantages :
       Method is very sensitive to the choice of historical
        data
       Historical sample may not account for structural
        events
       In some cases, the data availability is a problem
                   Backtesting the Var
   Example:
      N=600 observations

      99%-day VAR. Exception occurs at 1% of the days

      0.01 x 600 = 6 is the expected number of exceedances

      Actual number of exceedances is 8. Is this statistically significant?

      Define x, a bernoulli variable, the number of exceedances. Given the
       large number of observations, we can use the central limit theorem such
       that

                                   x  pN
                          z                  N(0,1)
                                   p(1  p)N
   Hypothesis testing:

                      H o : x  pN  6
                      H1 : x  8
                                86          2
                      z                         0.82
                           0.01(1 0.01)600 2.43
At a 5% level of confidence, the critical value is 1.95. Therefore, we cannot not
reject H0.
                    Portfolio Toolkit
   Consider a portfolio composed of one stock (for a value of $6000) and
    one bond for (for a value of $4000). The value of the portfolio is:
       $10000  $6000  $4000
       p  sb
       p  s   b
       p    s  b
                 
        p     p      p
            s s     b b
       rp         
            p s      p b
       rp  ws rs  wb rb

   This last equation says that the return on the portfolio, r p , is a
    weighted average of the return on the stock, rs , and the bond,
                 2                                                         rb.
    Notice that  wi  1, with:
                  i 1
                $6000                       $4000
        ws            60%   and   wb            40%
               $10000                      $10000
                   Portfolio Toolkit
The   return on a portfolio is given by:
                        m                    m
                  rp   x i ri   with      x       i   1
                       i 1                  i 1


where ri , is the return on asset i, xi , the proportion of asset i
in the portfolio and m , the number of assets in the portfolios.

   The corresponding variance of the portfolio is (say m  1000)

                      1000               1000 1000
             V(rp )   x i (ri )    x i x j  ij  (ri ) (r j )
                                   2

                       i 1              i 1 j 1

                                          i j
                        Portfolio VAR

   Under the normality assumption:
        Var %   p N -1 (rp )


but the term  (r )  V(r ) requires the
                        p         p


calculation of (1000 x 999)/2 terms  ij.
The problem is impractical!
                        Portfolio VAR

   Consider the quadratic approximation:
              s          2s
         s     F  0.5 2 F 2
              F         F

where s and F are respectively the value of
an asset and a market factor.

   Consider a portfolio of 500 stocks and 500
    bonds:
              500          500
         rp   w si rsi   w bi rbi
              i 1         i 1
                   Portfolio VAR
   From portfolio theory, we know that:
                       rsi   0  1 rm   si

Such that the second derivative is equal to 0. In
this case, F=market index.

   For bonds, based on duration and convexity, we
    have:
                        b i   2
                                       bi
              b i          y  0.5      y 2
                        y            y 2
               b i   b i y           2bi 1
                             y  0.5          y 2
                bi    y b i           y 2 b i
              rbi  D m oiy  0.5C xr y 2
in this case, F=y.
                 Portfolio VAR
   The 500 beta measures can be aggregated to
    generate the portfolio beta.

   The 500 duration and convexity measures can be
    aggregated to generate the portfolio duration
    and convexity. Therefore after a few
    manipulations:


         rp   p rm  Dmopy  0.5C xrp y   2
                       Portfolio VAR
   Under an historical simulation, the previous
    equation is applied directly.

   Under the assumption that the returns on the
    portfolio are drawn from a normal distribution,
    we can use the variance equation of the
    portfolio:
                                                                      0.5
                 p  (rm )  D m op  (y )  0.25C xrp  (y ) 
                 2       2        2       2            2      2 2

                                                                 
                         2  p D m opcov(rp,rm)                 
      (rp )                                                    
                         2  p C xrp cov(rp,y 2 )              
                                                                 
                         2D m opC xrp cov(y, y 2 )            
                  Portfolio Var

   Exemple: In class
         Pros and Cons of VAR measures
   Pros :
      Provide a risk measure that is easly aggregated

      Captures netting effects and risk diversification

      Consistent measure across risk factors, securities, etc.

      Direct translation into risk limits

      Facilitate the communication amongst managers



   Cons :
      Require a choice of methodology. Different approaches
       may lead to different VAR estimates
      VAR measures are not made to capture extreme, unlikely
       events. Other approaches are required
      Estimates only valid over short horizion
Market Risk-VAR
 Deutsche Bank
             Market Risk Regulation

   Basel 1998 :
       Banks must have capital charges to cover exposures
        with respect to market risk within the trading book,
        and with respect to the credit risk involved with the
        counterparties in the trading book.

       The regulation affects both on- and off- balance
        sheets.

       Large banks are allowed to use their own-internal
        model based on the VAR methodology. Other banks
        must use the standard model proposed by BIS.
              Market Risk Regulation

   Capital charges :

                                  Credit risk of individual issuers


        K  (10  day VAR, 1%)  4  (10  day specific risk VAR)

    May be changed by the regulator. K = 3 (minimum value)




   Total capital = 0.08 x (Credit risk RWA + Market risk RWA)

       where market risk RWA = 12.5 x capital charges
        above
Market Risk Regulation: Basel 98

                       Capital charges for specific risks (%)
Example :                                Internal   Standardized
Portfolio of 100                          model       approach
1 € bonds
diversified           AAA                  0.26         1.6
across                AA                   0.77         1.6
industries            A                    1.00         1.6
                      BBB                  2.40         1.6
                      BB                   5.24          8
                      B                    8.45          8
                      CCC                 10.26          8




 Source: Crouhy, Galai and Mark (2001)
              Market Risk Regulation
   The Group of 30 policy’s recommendations
    (1993) :

       Series of recommendations on price risk management for
        dealers, end-users of derivatives, legislators and regulators.

            Recommendation   2 : Mark-to-market process
            Recommendation   5 : Measuring market risk using a VAR
             metho.
            Recommendation   6 : Importance of stress testing
            Recommendation   8 : Banks should have an independant risk
                                    management function
                         Stress Testing
        VAR                      Market                 Stress
                                  Risk                  testing



Standard environments                               Extreme events


   Stress testing: assessments of potential losses under specific
    scenarios

   Scenarios from the Derivative Policy Group (1995) :

       Shifts in the yield curve ± 100 bps
       Twist risk of ± 25 bps
       Etc.
   Basel requires stress-testing in addition to Var calculation
                                                    Stress Testing
               Example: Stress testing at BMO:


                   Stress-testing with respect to interest rate at BMO
                                                Year 2003                                  Year 2002
                             Economic Value           Earnings sensitivity    Economic Value      Earnings sensitivity
                                sensitivity               (12 months)            sensitivity          (12 months)
100 bps increase            - $ 202.3 millions          $ 10.8 millions      - $ 152.7 millions      $ 1.1 millions
100 bps decease              $ 142.7 millions          - $ 17.6 millions      $ 123.8 millions      - $ 0.1 millions
200 bps increase            - $ 431.8 millions         $ 431.8 millions      - $ 354.1 millions     - $ 3.9 millions
200 bps decease             - $ 181.2 millions         - $ 61.2 millions      - $ 180 millions     - $ 70.7 millions


    Source: Bank of Montreal, annual report, 2003

								
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