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Risk Management Kolb

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Risk Management Kolb Powered By Docstoc
					        Financial Risk Management
                     Zvi Wiener
                     02-588-3049
      http://pluto.mscc.huji.ac.il/~mswiener/zvi.html
FRM
                        Risk
             • Business Risk
             • Operational Risk
             • Financial Risk
                credit risk
                market risk
                liquidity risk

             • Legal Risk

Zvi Wiener               FRM-1    slide 2
• Crouhy, Galai, Mark, Risk Management, McGraw Hill,
2000.
• Golub, Tilman, Risk Management Approaches for Fixed
Income Markets, Wiley, 2000.
• Jorion, Value at Risk, McGraw Hill, 1997.


• http://www.gloriamundi.org
• http://www.riskmetrics.com
• http://www.bis.org
• http://www.garp.com
 Zvi Wiener                FRM-1                  slide 3
             Derivatives 1993-1995
                              ($ million)
   •Shova Shell, Japan        1,580
   •Kashima Oil, Japan        1,450
   •Metallgesellschaft        1,340
   •Barings, U.K.             1,330
   •Codelco, Chile            200
   •Procter & Gamble, US      157

Zvi Wiener            FRM-1                 slide 4
                        Barings
             • February 26, 1995
             • 233 year old bank
             • 28 year old Nick Leeson
             • $1,300,000,000 loss
             • bought by ING for $1.5


Zvi Wiener                 FRM-1         slide 5
                Public Funds
                               ($ million)
 • Orange County               1,640
 • San Diego                   357
 • West Virginia               279
 • Florida State Treasury      200
 • Cuyahoga County             137
 • Texas State                 55

Zvi Wiener             FRM-1                 slide 6
               Orange County

  • Bob Citron, the county treasures
  • $7.5B portfolio (schools, cities)
  • borrowed $12.5B, invested in 5yr. notes
  • interest rates increased
  • reported at cost - big mistake!
  • realized loss of $1.64B

Zvi Wiener               FRM-1                slide 7
             Financial Losses
  • Barings                    $1.3B
  • Bank Negara, Malaysia 92   $3B
  • Banesto, Spain             $4.7B
  • Credit Lyonnais            $10B
  • S&L, U.S.A.                $150B
  • Japan                      $500B



Zvi Wiener            FRM-1            slide 8
              Metallgesellshaft
   • 14th largest industrial group
   • 58,000 employees
   • offered long term oil contracts
   • hedge by long-term forward contracts
   • short term contracts were used (rolling hedge)
   • 1993 price fell from $20 to $15
   • $1B margin call in cash

Zvi Wiener              FRM-1                   slide 9
Zvi Wiener   FRM-1   slide 10
             Risk Management and
              Risk Measurement




Zvi Wiener          FRM-1          slide 11
               Basic Statistics
   • Certainty and uncertainty
   • Probabilities, distribution, PDF, CDF
   • Mean, variance
   • Multivariable distributions
   • Covariance, correlation, beta
   • Quantile


Zvi Wiener              FRM-1                slide 12
       A                100 km.             B


                       100 km/hr



                        50 km/hr




      1 – 100           2 – 50       3 – 50
             (100+50+50)/3 = 66.67 km/hr.

Zvi Wiener                 FRM-1                slide 13
      1.     -2%             1.   +40%
      2.     +1%             2.   +10%
      3.     -1%             3.   -50%
      4.     +1%             4.   +20%
 0.98*1.01*0.99*1.01 =       1.4*1.1*0.5*1.2 =
                0.9897                  0.924

Zvi Wiener           FRM-1                   slide 14
                  Probabilities

  Certainty

  Uncertainty

  Probabilities




Zvi Wiener             FRM-1      slide 15
             Probabilities


  Mean

  Variance




Zvi Wiener        FRM-1      slide 16
                         Probabilities
             0.3
                         30% 30%
             0.2
                                           10% 10%
                   20%
             0.1

                   1      2        3        4   5

                           p i
                                       i   1
Zvi Wiener                        FRM-1              slide 17
                       Probabilities
             0.3

             0.2

             0.1

                   1    2        3      4   5
                            

                             dp  1
                            0
Zvi Wiener                      FRM-1           slide 18
                 Probabilities
                               N
             mean  X   X i pi
                               i 1
                                   N
 Variance   ( X )   ( X  X i ) pi
                   2                      2

                                   i 1




Zvi Wiener             FRM-1                  slide 19
                Probabilities
                               
             mean  X   Xdp
                               
                                
   Variance   ( X )   ( X  X ) dp
                   2                 2

                                

              Variance   ( X )
Zvi Wiener             FRM-1             slide 20
                   Sample Estimates

                      N
                1
             ˆ
             X
                N
                     X
                     i 1
                            i



                                           
                                    N
                       1
                                ˆ          2
              (X ) 
              ˆ2
                                 X  Xi
                      N  1 i 1
      Sometimes one can use weights

Zvi Wiener                  FRM-1               slide 21
             Normal Distribution N(, )




Zvi Wiener               FRM-1             slide 22
             Normal Distribution N(, )

                                    




                                     


Zvi Wiener               FRM-1             slide 23
                  Normal Distribution




             1%



                  quantile    

Zvi Wiener                   FRM-1      slide 24
             Lognormal Distribution
0.6

0.5

0.4

0.3

0.2

0.1


                1        2     3         4

Zvi Wiener            FRM-1           slide 25
                Covariance
Shows how two random variables are connected
For example:
          independent
          move together
          move in opposite directions

covariance(X,Y) =   E  X  X Y  Y 

Zvi Wiener            FRM-1               slide 26
                      Correlation

                      E  X  X Y  Y 
              XY   
                          ( X )  (Y )
                       -1    1
=0           independent
=1           perfectly positively correlated
 = -1        perfectly negatively correlated
Zvi Wiener                FRM-1                 slide 27
                 Properties

E (A  B)  E ( A)  E ( B)

 (A  B) 
  2


     ( A)    ( B)  2Cov( A, B)
       2     2   2       2



 (A  B) 
  2


    ( A)    ( B)  2( A) ( B) 
       2     2   2   2

Zvi Wiener               FRM-1      slide 28
             Time Aggregation

              T   annualT
              T   annual T
             Assuming normality



Zvi Wiener          FRM-1         slide 29
                 Time Aggregation
   • Assume that yearly parameters of CPI are:
     mean = 5%, standard deviation (SD) = 2%.
   • Then daily mean and SD of CPI changes are:
                       1
             d   y      0.02 %
                      250
                        1
             d  y         0.1265 %
                        250
Zvi Wiener               FRM-1                   slide 30
                  Portfolio
     2(A+B) = 2(A) + 2(B) + 2(A)(B)
 
                    A


rf
             B

                                 
Zvi Wiener           FRM-1             slide 31
                                    

                                 ¥$£
                   $¥                  £¥


                                 £$¥
                                        £
                 $£¥
                         £$
             $
Zvi Wiener               FRM-1                slide 32
    X  X  X  2 X 1 X 2Cos
          2
         12       1
                   2       2
                           2


        2
         12        2 1 2 12
                   2
                   1
                               2
                               2

                 12 ~ Cos
                                   2         12

John Zerolis                       
"Triangulating Risk",
Risk v.9 n.12, Dec. 1996
                                        1
Zvi Wiener                 FRM-1             slide 33
                 Useful Books

• Duffie D., Dynamic Asset Pricing Theory.
• Duffie D., Security Markets, Stochastic Models.
• Shimko D. Finance in Continuous Time, A
        Primer. Kolb Publishing Company, 1992.




Zvi Wiener              FRM-1                 slide 34
                   Binomial Tree
                                0.125

                       0.25

             0.5                0.375

  1.0                  0.5
             0.5                0.375
                       0.25
                                0.125
Zvi Wiener              FRM-1           slide 35
Zvi Wiener   FRM-1   slide 36
                   Example
We will receive n dollars where
n is determined by a die.

What would be a fair price for
participation in this game?




Zvi Wiener             FRM-1      slide 37
                  Example 1
Score        Probability
1            1/6
2            1/6
                   1 2 3 4 5 6
3            1/6            3.5
4            1/6   6 6 6 6 6 6
5            1/6   Fair price is 3.5 NIS.
6            1/6   Assume that we can play
                 the game for 3 NIS only.
Zvi Wiener            FRM-1                 slide 38
                      Example
If there is a pair of dice the
mean is doubled.

What is the probability to
gain $5?




Zvi Wiener                FRM-1   slide 39
                       Example
All combinations:

1,1      2,1   3,1   4,1   5,1      6,1
1,2      2,2   3,2   4,2   5,2      6,2
1,3      2,3   3,3   4,3   5,3      6,3
1,4      2,4   3,4   4,4   5,4      6,4
1,5      2,5   3,5   4,5   5,5      6,5
1,6      2,6   3,6   4,6   5,6      6,6

36 combinations with equal probabilities
Zvi Wiener                  FRM-1          slide 40
                       Example
All combinations:

1,1      2,1   3,1   4,1   5,1      6,1
1,2      2,2   3,2   4,2   5,2      6,2
1,3      2,3   3,3   4,3   5,3      6,3
1,4      2,4   3,4   4,4   5,4      6,4
1,5      2,5   3,5   4,5   5,5      6,5
1,6      2,6   3,6   4,6   5,6      6,6

4 out of 36 give $5, probability = 1/9
Zvi Wiener                  FRM-1         slide 41
                                    Additional information:
                                    the first die gives 4.
All combinations:
1,1      2,1   3,1   4,1   5,1      6,1
1,2      2,2   3,2   4,2   5,2      6,2
1,3      2,3   3,3   4,3   5,3      6,3
1,4      2,4   3,4   4,4   5,4      6,4
1,5      2,5   3,5   4,5   5,5      6,5
1,6      2,6   3,6   4,6   5,6      6,6

1 out of 9 give $5, probability = 1/9
Zvi Wiener                  FRM-1                     slide 42
                                    Additional information:
                                    the first die gives 4.
All combinations:
1,1      2,1   3,1   4,1   5,1      6,1
1,2      2,2   3,2   4,2   5,2      6,2
1,3      2,3   3,3   4,3   5,3      6,3
1,4      2,4   3,4   4,4   5,4      6,4
1,5      2,5   3,5   4,5   5,5      6,5
1,6      2,6   3,6   4,6   5,6      6,6

4 out of 24 give $5, probability = 1/6
Zvi Wiener                  FRM-1                     slide 43
                   Example 1
                               1
                                  16.67%
                               6




             -2   -1   0       1   2    3


Zvi Wiener             FRM-1                slide 44
                     Example 1

             1   2   3    4      5    6    we pay
   1         2   3   4    5      6    7    6 NIS.
   2         3   4   5    6      7    8
   3         4   5   6    7      8    9
   4         5   6   7    8      9    10
   5         6   7   8    9      10   11
   6         7   8   9    10     11   12

Zvi Wiener               FRM-1                 slide 45
                       P&L

             1    2    3     4      5   6
   1         -4   -3   -2    -1     0   1
   2         -3   -2   -1    0      1   2
   3         -2   -1   0     1      2   3
   4         -1   0    1     2      3   4
   5         0    1    2     3      4   5
   6         1    2    3     4      5   6

Zvi Wiener                  FRM-1           slide 46
                Example 1 (2 cubes)
        0.15

       0.125

         0.1

       0.075

        0.05

       0.025


               -4   -3   -2   -1   0       1   2   3   4   5   6




Zvi Wiener                         FRM-1                           slide 47
                Example 1 (5 cubes)
 0.1


0.08


0.06


0.04


0.02



       -10-9-8-7-6-5-4-3-2-1 0 1 2 3 4 5 6 7 8 9 101112131415




Zvi Wiener                     FRM-1                       slide 48
                   Breakfast




             $2                $4
             50%               50%

Zvi Wiener            FRM-1          slide 49
                   Lunch




             $5             $11
             50%            50%

Zvi Wiener          FRM-1         slide 50
Zvi Wiener   FRM-1   slide 51
                       Breakfast
                       $2            $4

             $5        $7            $9       50%
Lunch
             $11       $13           $15      50%

                     50%            50%

                    = $11            = ??
Zvi Wiener                  FRM-1                   slide 52
                   Correlation =+1
                        Breakfast
                       $2            $4

             $5        $7            $9        50%
Lunch        $11       $13           $15
                                               50%


                       50%           50%

                     = $11            = $4
Zvi Wiener                   FRM-1                   slide 53
                   Correlation =-1
                        Breakfast
                       $2           $4

             $5        $7           $9       50%
Lunch        $11       $13          $15      50%


                      50%           50%

                    = $11           = $2
Zvi Wiener                  FRM-1                  slide 54
                   Correlation =0
                        Breakfast
                       $2           $4

             $5        $7           $9
                                           50%
Lunch        $11       $13          $15
                                           50%


                      50%           50%

                    = $11           = $3.16
Zvi Wiener                  FRM-1                slide 55
             How much can we lose?
                    Everything

              correct, but useless answer.

   How much can we lose realistically?



Zvi Wiener                FRM-1              slide 56
              What is the current Risk?
             • Bonds     duration, convexity
             • Stocks    volatility
             • Options   delta, gamma, vega
             • Credit    rating
             • Forex     target zone
             • Total        ?

Zvi Wiener                 FRM-1               slide 57
             Standard Approach




Zvi Wiener          FRM-1        slide 58
             Modern Approach




             Financial Institution




Zvi Wiener           FRM-1           slide 59
                 Value
                  8.25
                      8                                          4.3
                   7.75
                     7.5                                       4.25
                    7.25
                      10                                  4.2
                           11
                                 12                     4.15
                                        13               dollar
                       Interest Rate         14
                                                  4.1




             interest rates and dollar are
                  NOT independent
Zvi Wiener                      FRM-1                                  slide 60

				
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