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Market Risk Management
Value-at-Risk
Market Risk Management
What is Risk ?
The possibility of suffering harm or, loss.
In financial parlance, risk is the chance that expected investment returns will not be materialised. Sources of risk are manifold
Market Risk is the risk of not realising the expected profit due to unfavourable market movements.
Market Risk Management
Market Risk: Definition
Market risk is the uncertainty resulting from changes in market prices
Market Risks arises due to movements in variables such as : Interest rates Currencies Equity Commodities
Arises due to directional risks from taking a net long/short position in a given asset class
Market Risk Management
Assessment of Market Risk
Important in terms of: Management information Setting limits Resource allocation (risk/return tradeoff) Performance evaluation Regulation
Market Risk Management
Measurement of Market Risk
Quantify the risk of losses due to movements in financial market variables
Various measures to assess market risk: Investment Limits Risk Factors ( PV01, Delta , Gamma etc) Value at Risk (VaR)
Market Risk Management
Value at Risk (VaR)
Statistical estimate of
The maximum amount of money
that may be lost on a portfolio
over a given period of time
with a given level of confidence
under normal market conditions
Market Risk Management
Definition Value-at-Risk is defined as a loss level that will
not be exceeded at some specified confidence level for a specified time horizon under normal market conditions “What loss level is such that we are X% confident
it will not be exceeded in N business days?”
If the VaR for one day horizon and at a confidence level of 95% is Rs.10 mn, that means: The likelihood that our losses will exceed Rs.10
mn over the next 24 hours is 5%
Concept of “Tail Risk”
Market Risk Management
VAR VAR asks “How bad things can get?”
VAR : Summarizes worst loss over a target horizon with a given level of confidence
VAR describes the down side quantile of the projected distribution of gains/losses over target horizon.
A single number ( currency amount) which estimates expected maximum loss (worst loss) over a given time horizon and at a given confidence level
Describes the probability boundary of potential losses
VAR an estimate of likely losses: Actual loss may differ
BIS accepted VAR as Market Risk measure & for provision of capital adequacy
Market Risk Management
VAR
Denotes impact of normal market risk events
Provides predictive, aggregate view of portfolio risk in terms of probable loss
Complements stop-loss limits and cumulative loss limits
Integrates risk management architecture
Intended to be used in future as primary input for capital allocation, risk adjusted performance measurement and creating a risk limit framework
VaR model validated by Back Testing
Complemented by Stress Testing.
Market Risk Management
BIS Guidelines
“…Each bank must meet, on a daily basis, a capital requirement expressed as the higher of (i) its previous day’s VaR number measured according to the parameters specified and (ii) an average of the daily VaR measures on each of the preceding sixty business days, multiplied by a multiplication factor.” This multiplying factor is minimum 3.
Market Risk Management
Why VaR is Popular(1) Representation through a single no.
Common measure across various products – helps in comparative analysis
It provides a control measure Limits can be set up based on VaR. VaR trends can be monitored and any unusual move
can act as a pointer for further examination
Leading Banks in US in 1998 and European Banks in 1997 were allowed to use internal models to calculate Capital Charge for Market Risk
Central banks have made it mandatory to their supervised banks for quantifying market risk through VaR and to maintain minimum required capital for this quantified risk
Market Risk Management
Why VaR is Popular(2)
Choosing appropriate VaR model Required capital charge for market risk is linked
to VaR estimates Higher VaR means higher Capital Charge
Banks may have a tendency/ preference towards a model that produces lower VaR Exposed to risk beyond their capacity and may be
vulnerable to the shocks arising out of market swings
Market Risk Management
Why VaR is Popular(3) Regulators provide certain norms (such as back
testing, data period and other factors for VaR estimations, etc.) to be satisfied by the VaR estimates.
Selection of an appropriate VaR model in reality is important Produces as minimum VaR as possible and also
satisfy the regulatory requirements/norms prescribed by the regulators/Basle Committee
To minimise certain loss functions while making a choice of a VaR model from various alternatives
Market Risk Management
Drawbacks of VAR VAR system subject to
Model Risk:
Risk of errors arising from inappropriate assumptions on which models are based
Implementation Risk: risk of error arising from the manner n which the model has been implemented; common to all risk model
Assumptions : portfolio return normally distributed: existence of unusual or extreme events in market not captured by Normal Distribution
Some VAR models use historical return data: presumption that past is reliable guide to the future: Not always the case
A single VAR figure may give can give misleading information: two positions of same VAR (same confidence level and holding period
Market Risk Management
Drawbacks of VAR
Difficulties associated with capturing of reliable data
Some methods costly and difficult to set up Different method can give different VAR
estimates on daily basis for the same portfolio VAR itself is not risk management It is a tool for measuring market risk part of complete range of activities /duties of
involved in managing and minimizing financial institution’s risk exposure
Market Risk Management
VaR computation approaches
Historical Simulation
Analytic
Monte Carlo Simulation
Market Risk Management
Historical Simulation
Create a database of the daily movements in all market variables.
The first simulation trial assumes that the percentage changes in all market variables are as on the first day
The second simulation trial assumes that the percentage changes in all market variables are as on the second day
and so on
Market Risk Management
Historical Simulation continued
Suppose we use m days of historical data Let vi be the value of a market variable on
day i There are m-1 simulation trials The ith trial assumes that the value of
the market variable tomorrow (i.e., on day m+1) is
1i
im v
vv
Market Risk Management
Quantiles
Defined as values ‘q’ such that areas to their right (or left) represents a given probability “c”:
It is method of sorting the data and finding at any point how much data is their to the either side.
c=Prob (X≥q)=∫f(x).dx For normal distribution quantiles can be
found from statistical tables For a R.V Normal (0,1) : to find q for c=.95
we can use the table
Market Risk Management
Example Historical Simulation
Data for VaR historical simulation calculation
DayMarket
Variable 1Market
Variable 2…
Market Variable n
0 20.33 0.1132 … 65.37
1 20.78 0.1159 … 64.91
2 21.44 0.1162 … 65.02
. . . . .
. . . . .
499 25.75 0.1323 … 61.99
500 25.85 0.1343 … 62.1
Value of portfolio today is equal to $ 23.50 Mn
Market Risk Management
Example Historical Simulation Contd.Scenarios generated for tomorrow (Day
501)
Scenario No.
Variable 1
Variable 2
…Variab
le n
Portfolio
Value ($
mn)
Change in
value ($ mn)
1 26.42*0.137
5 … 61.66 23.71 0.21
2 26.670.134
6 … 62.21 23.12 -0.38
. . . … . . .
. . . … . . .
. . . … . . .
499 25.880.135
4 … 61.87 23.63 0.13
500 25.950.136
3 … 62.21 22.87 -0.63
*25.85 × 20.78/20.33= 26.42
Market Risk Management
Information Decay Give more weight to new information Referred as EWMA method Exponentially declining weights on historical data Smoothing is achieved by setting Lambda between
0 and 1 (Goes with our previous point) Weights totaled to be 1 Lambda at 0.97 means the weight given to the
latest price movement as 3% and declines at the rate of 97%.
JP Morgan document advocated the lambda to be at 0.94 for daily VaR and 0.97 for monthly var.
Exponential Smoothing
0*)1(
Market Risk Management
Calculation of VaR in EWMA
Apply the weight as per the lambda factor with a declining rate from recent to distant returns
Sort the returns from low to high Find the cumulative weight at any
simulated price The quintile has to be computed from the
cumulative of the weight.
Market Risk Management
Calculation under EWMADATE PRICE RETURN SIMULATED
PRICES (FOR 501)
WEIGHT (0.97) WEIGHT (0.94) Equal Weight
11/06/2008 741.20 1.40% 751.59 0.03000000 0.060000000 0.002
10/06/2008 730.95 -2.58% 722.08 0.02910000 0.056400000 0.002
09/06/2008 750.30 -2.48% 722.80 0.02822700 0.053016000 0.002
06/06/2008 769.40 -1.13% 732.82 0.02738019 0.049835040 0.002
05/06/2008 778.20 2.73% 761.40 0.02655878 0.046844938 0.002
04/06/2008 757.55 -0.39% 738.28 0.02576202 0.044034241 0.002
………. ………. ………. ………. ………. ………. ……….
………. ………. ………. ………. ………. ………. ……….
15/06/2006 482.55 6.95% 792.70 0.00000001 0.000000000 0.002
14/06/2006 451.20 -1.84% 727.57 0.00000001 0.000000000 0.002
13/06/2006 459.65 -3.07% 718.46 0.00000001 0.000000000 0.002
12/06/2006 474.20 1.05% 748.98 0.00000001 0.000000000 0.002
Market Risk Management
Calculation of VaR
Market Risk Management
Weights on Past Observations
100 75 50 25 0
Days in the Past
0.03
0.06
Exponential Model Lambda =0.94
Exponential Model Lambda =0.97
• Higher the Lambda the process of Information decay is lesser.
Market Risk Management
Lambda and VaR
Time
Va
R
Exponential Model Lambda =0.97Exponential Model
Lambda =0.94
Higher the lambda the impact of any sudden fall in the market will have long term effect on VaR.
Market Risk Management
Impact of Assumptions
Higher the confidence level higher will be the VaR
Advantage : lesser no of failure in backtesting. Disadvantage : Overestimated VaR will have
negative impact on capital adequacy. Higher the Lambda factor the VaR
movement will be low.Once there is a sudden crash in a market the
VaR will jump to higher level. This shift will stay for longer period if Lambda is high.
Lower Lambda can also cause for higher back testing failures.
Basel recommendation is to keep CI at 99%. No specification about lambda factor.
Market Risk Management
Effective Lambda
Keeping a higher lambda is defensive in nature Lower lambda may lead to frequent back
testing failures. Lambda should be in consistent with the
volatility and time taken by the market to become stabilise.
Higher lambda with short history and lower lambda with large history is a very good cushion.
JP Morgan technical doccument advocated 0.94 as the lambda for one day VaR and 0.97 for one month VaR
Market Risk Management
Effective Risk Model
If the history is 450 days then lambda should be kept at 0.94, So the chances more than 4 failures in a year is minimal.
10 day VaR should be computed from the time scaling method instead of moving window method.
The Back testing should be done with confidence level of 95% and lambda at 0.94. If the number of breaches is higher than one in a quarter the lambda should be increased to 0.95.
Monte-Carlo simulation should be done for the entire portfolio in order to expand the number of simulations and a better result.
Market Risk Management
Advantages and Limitations - Historical SimulationAdvantages
No need to assume normality No need to forecast volatility and
correlation Effective in measuring non-linear risks Based entirely on historical data, objective
Limitations Computationally more intensive vs.
variance-covariance Implied volatility and correlation can be
reversed in tail events
Market Risk Management
Stress testing and Back-testing
Market Risk Management
Back Testing Methodologies
Comparison of VaR model generated P/L against the actual P/L
Comparison of VaR model generated P/L against the theoretically calculated P/L based on the actual positions
Market Risk Management
Back-Testing A VAR Model Calculate 1-Day 95% VAR for a (changing) portfolio each day for
some substantial period of time (e.g., 100 Days) Compare the P/L on the succeeding trading day with the previous
close of business day’s VAR
Count the number of times the loss exceeds the VAR
(25,000,000)
(20,000,000)
(15,000,000)
(10,000,000)
(5,000,000)
-
5,000,000
10,000,000
15,000,000
20,000,000
25,000,000
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49
P/L
VAR
95% 1 day VAR
Market Risk Management
Back Testing CL – 95% Lambda = 0.94 Exceptions
Market Risk Management
Back Testing CL – 95% Lambda = 0.94
Market Risk Management
Backtesting and AssumptionsDate P&L VaR
95/94EXCEPTION
VaR 95/97
EXCEPTION
VaR 99/94
EXCEPTION
VaR 99/97
EXCEPTION
28/07/2008 -540.59 330.50 -210.10 331.64 -208.95 409.39 -131.20 412.94 -127.65
02/07/2008 -357.27 341.41 -15.86 335.95 -21.32 434.04 76.77 432.11 74.84
30/06/2008 -352.13 261.84 -90.29 261.74 -90.39 435.40 83.27 433.14 81.01
19/06/2008 -453.58 236.43 -217.15 239.42 -214.16 242.12 -211.46 340.65 -112.93
18/06/2008 -239.42 197.41 -42.01 242.66 3.24 242.70 3.28 342.55 103.13
06/06/2008 -203.99 192.75 -11.23 244.00 40.01 276.08 72.09 354.24 150.25
21/05/2008 -221.63 136.33 -85.30 237.38 15.75 315.18 93.55 381.23 159.60
14/03/2008 -329.15 313.87 -15.29 286.97 -42.18 362.86 33.71 390.97 61.82
12/03/2008 -352.99 258.92 -94.08 258.73 -94.26 333.66 -19.33 391.41 38.42
29/02/2008 -266.16 241.34 -24.82 244.57 -21.59 400.82 134.66 447.15 180.99
21/01/2008 -402.25 352.27 -49.97 277.16 -125.09 581.81 179.56 524.44 122.19
18/01/2008 -627.90 268.17 -359.73 235.24 -392.66 283.00 -344.90 279.29 -348.61
17/01/2008 -284.75 191.55 -93.20 187.69 -97.06 250.59 -34.16 253.60 -31.15
09/01/2008 -253.15 149.58 -103.57 149.59 -103.56 182.25 -70.90 213.15 -40.00
08/01/2008 -211.11 117.26 -93.85 148.41 -62.70 184.26 -26.85 214.82 3.71
07/01/2008 -118.18 114.65 -3.53 151.38 33.20 186.10 67.92 216.21 98.03
Market Risk Management
Stress TestingObjective
To capture exposures of a portfolio to adverse discontinuous market events which are extreme but possible Historical Scenarios
To isolate exposures to the extreme historical events which exceeds the loss threshold determined based on statistical measures
Hypothetical ScenariosTo identify exposure to the extreme but possible future market events which have not yet occurred in the past
Performed at different levels (individual asset classes to portfolio level to Bank’s portfolio as a whole)
Market Risk Management
Setting VaR Limit
Market Risk Management
Evolution of VaR Applications
Passive
Defensive
Active
Reporting Risk
• Managing Reports• Disclosure to Share Holders• Regulatory Requirements
Controlling Risk
• Setting Risk Limits
Allocating Risk
• Performance Evaluation• Capital Allocation• Strategic Business decisions
Market Risk Management
Need For VaR limits
To complement other cut loss triggers. Comparing VaR Limit with stress testing. Accommodate the hedging benefits. Monitor performance of the Dealers. Capital Allocation. VaR limit is a part of the Advanced
measurement method under Basel – II Management information.
Market Risk Management
Approaches
Top Down Approach
Set the risk level from capital Set the VaR Limit for the Enterprise Set the Sub limits for the Business Groups Set the Limit for Individual Asset classes
Bottom up Approach
Set the VaR limit for individual asset class depending on volatility, current profit and unrealised profit
Analyse the correlation among scrips and asset classes
Apply the diversification benefits and set the limit for the entire enterprise.
Market Risk Management
Existing Policy
Overall VaR limit is set on the basis of the loss limit prescribed by the ALM.
The VaR limit is for the asset class is given after considering the diversification benefit
The allocation of VaR Limit is based on Volatility and maximum exposure. This limit is given in absolute amount.
Operating VaR limit is given as percentage of MTM expecting the actual exposure is less than the maximum exposure.
Market Risk Management
Performance of VaR limit
The Overall VaR limit has breached from 22nd January due to sharp decline in the Equity market. This had continued till 30th March 08.
The operative VaR limit breached from 21st January for equity and mutual funds.
The bond Market which was stable upto May started declining due to rising inflation and RBI policy to change the key rates.
Since operative VaR limit is in percentage only complete exit from the market can restore the VaR within the limit.
Bond Market is volatile and inclusion of long duration bond the effect was further worse.
Market Risk Management
Setting New Limit for VaR
Basis for the VaR Limit Capital of the bank
The capital of the bank has undergone a change so linking the VaR to overall Capital or the regulatory capital is a good measure
Volatility in the market. The allocation can be base on the volatility
of the indices and the volatility of the key securities in our portfolio.
Compare the VaR limit with the stress testing figures
Current realised and unrealised profit/loss of the portfolio.
Market Risk Management
Thank You