1. Financial Risk Management
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Following
P. Jorion, Financial Risk Manager Handbook
FRM

2. Following P. Jorion 2001 Financial Risk Manager Handbook
Chapter 17 VaR Methods
Following P. Jorion 2001
Financial Risk Manager Handbook
FRM

3. 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
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4. How to measure VaR Historical Simulations Variance-Covariance
Monte Carlo
Analytical Methods
Parametric versus non-parametric approaches
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5. Historical Simulations
Fix current portfolio.
Pretend that market changes are similar to those observed in the past.
Calculate P&L (profit-loss).
Find the lowest quantile.
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6. Example
Assume we have $1 and our main currency is SHEKEL. Today $1=4.30.
Historical data:
4.00
4.20
4.10
4.15
4.30*4.20/4.00 = 4.515
4.30*4.20/4.20 = 4.30
4.30*4.10/4.20 = 4.198
4.30*4.15/4.10 = 4.352
P&L
0.215
-0.112
0.052
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7. USD NIS
today
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8. today Changes USD: +1% +1% +1% +1% in IR NIS: +1% 0% -1% -1%
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9. Returns
year
1% of worst cases
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10. VaR
Profit/Loss 1%
VaR1%
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11. Variance Covariance Means and covariances of market factors
Mean and standard deviation of the portfolio
Delta or Delta-Gamma approximation
VaR1%= P – 2.33 P
Based on the normality assumption!
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12. Variance-Covariance  1%
2.33
-2.33
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13. Monte Carlo
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14. Monte Carlo Distribution of market factors
Simulation of a large number of events
P&L for each scenario
Order the results
VaR = lowest quantile
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15. Monte Carlo Simulation
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16. 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.
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17. Stock Portfolio Single risk factor or multiple factors
Degree of diversification
Tracking error
Rare events
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18. Bond Portfolio Duration Convexity Partial duration Key rate duration
OAS, OAD
Principal component analysis
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19. Options and other derivatives
Greeks
Full valuation
Credit and legal aspects
Collateral as a cushion
Hedging strategies
Liquidity aspects
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20. Credit Portfolio rating, scoring credit derivatives reinsurance
probability of default
recovery ratio
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21. Reporting
Division of VaR by business units, areas of activity, counterparty, currency.
Performance measurement - RAROC (Risk Adjusted Return On Capital).
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22. Backtesting Verification of Risk Management models.
Comparison if the model’s forecast VaR with the actual outcome - P&L.
Exception occurs when actual loss exceeds VaR.
After exception - explanation and action.
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23. Backtesting OK Green zone - up to 4 exceptions increasing k
Yellow zone exceptions
Red zone - 10 exceptions or more OK
increasing k
intervention
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24. Stress Designed to estimate potential losses in abnormal markets.
Extreme events
Fat tails
Central questions:
How much we can lose in a certain scenario?
What event could cause a big loss?
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25. Local Valuation Full Valuation
Simple approach based on linear approximation.
Full Valuation
Requires repricing of assets.
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26. Delta-Gamma Method
The valuation is still local (the bond is priced only at current rates).
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27. FRM-97, Question 13
An institution has a fixed income desk and an exotic options desk. Four risk reports were produced, each with a different methodology. With all four methodologies readily available, which of the following would you use to allocate capital?
A. Simulation applied to both desks.
B. Delta-Normal applied to both desks.
C. Delta-Gamma for the exotic options desk and the delta-normal for the fixed income desk.
D. Delta-Gamma applied to both desks.
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28. FRM-97, Question 13 Bad question!
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!
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29. Mapping
Replacing the instruments in the portfolio by positions in a limited number of risk factors.
Then these positions are aggregated in a portfolio.
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30. Delta-Normal method Assumes linear exposures
risk factors are jointly normally distributed
The portfolio variance is
Forecast of the covariance matrix for the horizon
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31. 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
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32. FRM-97, Question 12
Delta-Normal, Historical-Simulations, and MC are various methods available to compute VaR. If underlying returns are normally distributed, then the:
A. DN VaR will be identical to HS VaR.
B. DN VaR will be identical to MC VaR.
C. MC VaR will approach DN VaR as the number of simulations increases.
D. MC VaR will be identical to HS VaR.
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33. FRM-97, Question 12
Delta-Normal, Historical-Simulations, and MC are various methods available to compute VaR. If underlying returns are normally distributed, then the:
A. DN VaR will be identical to HS VaR.
B. DN VaR will be identical to MC VaR.
C. MC VaR will approach DN VaR as the number of simulations increases.
D. MC VaR will be identical to HS VaR.
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34. 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.
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35. 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.
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36. FRM-99, Questions 15, 90
The VaR of one asset is 300 and the VaR of another one is If the correlation between changes in asset prices is 1/15, what is the combined VaR?
A. 525
B. 775
C. 600
D. 700
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37. FRM-99, Questions 15, 90
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38. 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
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40. The forward contract is equivalent to
a long position of SP* on the spot rate
a long position of SP* in the foreign bill
a short position of KP in the domestic bill
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41. On the valuation date we have S = 1.6595, r = 4.9375%, r* = 5.9688%
Vt = $93,581 - the current value of the contract
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