Market Risk
• Directional risk
• Relative value risk
• Price risk
• Liquidity risk
Type of measurements
– scenario analysis
– statistical analysis

Scenario Analysis
• A scenario analysis measures the change in market value that would result if market factors were changed from their current levels, in a specified way. No assumption about probability of changes is made.
• A stress test is a measurement of the change in the market value of a portfolio that would occur for a specified unusually large change in a set of market factors.

Value at Risk
• A single number that summarizes the likely loss in value of a portfolio over a given time horizon with specified probability.
• C-VaR states expected loss conditional on change in value in the left tail of the distribution.
• Three approaches
– Historical simulation
– Model-building approach
– Monte Carlo simulation

Historical Simulation
• Identify market variables that determine the portfolio value
• Collect data on movements in these variables for a reasonable number of historical days
• Build scenarios that mimic changes over the historical period
• For each scenario calculate the change in value of the portfolio over the specified time horizon
• From this empirical distribution of value changes calculate VaR

Model Building Approach
• Portfolio of n-assets
• Calculate mean and standard deviation of change in the value of portfolio for one day
• Assume normality
• Calculate VaR

Monte Carlo Simulation
• Value of the portfolio today
• Draw samples from the probability distribution of changes of the market variables
• Using the sampled changes calculate the new portfolio value and its change
• From the simulated probability distribution of changes in portfolio value calculate VaR

Pitfalls of Normal Distribution Based VaR
• Normality assumption may not be valid for tail part of the distribution
• VaR of a portfolio is not less than weighted sum of VaR of individual assets (not sub-additive)
• Expected shortfall conditional on the fact that loss is more than VaR is a sub-additive measure of risk

Pitfalls of Value-at-Risk
• VaR is a statistical measurement of price risk
• VaR assumes a static portfolio. It does not take into account
– Structural change in the portfolio that would contractually occur during the period
– Dynamic hedging of the portfolio
• VaR calculation has two basic components
– Simulation of changes in market rates
– Calculation of resultant changes in the portfolio value

Value-at-Risk
VaR (Value-at-Risk) is a measure of the risk in a portfolio over time.
Quoted in terms of a time horizon and a confidence level.
Example: 10 day 95% VaR is the size of loss X that will not happen 95% of the time over the next 10 days.
5%
Value-at-Risk
X
95%
(Profit/Loss Distribution)

Value-at-Risk Levels
Two standard VaR levels are 95% and 99%.
95% is 1.645 standard deviations from the mean
99% is 2.33 standard deviations from the mean mean

Value-at-Risk Assumptions
1) Percentage change (return) of assets is Gaussian: dS
 
Sdt
 
Sdz or dS
  dt
  dz
S

S
   t
   z
S
Normal Distribution

Value-at-Risk Assumptions
2) Mean return m is zero:

S
   t
   z
S
Mean of
 t is .
  t ~ O (
 t )
Standard deviation of ∆t is.
  z ~ O (
 t
1 / 2
)
Time is measured in years, hence t or change in time is insignificant. Hence the mean μ is not taken into consideration and the mean return is stated as:

S

S
  z

VaR and Regulatory Capital
Regulators require banks to keep capital for market risk equal to the average of VaR estimates for past 60 trading days using confidence level of 99% and number of days (N) =10, times a multiplication factor
(multiplication factor equals 3).

Advantages of VaR
• Captures an important aspect of risk in a single number
• Easy to understand
• Indicates the worst loss that could happen

Daily Volatilities
• Option pricing (volatility is express as volatility per year)
• aR calculations (volatility is express as volatility per day)
 day

 year
252

0 .
063
  year

6 %
  year

Daily Volatility
• day is defined as the standard deviation of the continuously compounded return in one day
• In practice it is also assumed that it is the standard deviation of the proportional change in one day

Example
• Based on 60 days prior trading data the following computations have been made
• Volatility of a bank is 2% per day (about 32% per year)
• Assume N =10 and confidence level is 99 %
• Standard deviation of the change in the market price (₹
60,000) in 1 day is ₹ 1,200 (2% x 60,000)
• Standard deviation of the change in 10 days is
1,200 x = 3,794.733 (1,200 x )
10

Example (continued)
• Assume that the expected change in the value of the bank’s share is zero
• Assume that the change in the value of the bank’s share is normally distributed
• Since N(0.01)= -2.33, ({Z<-2.33}=0.01) the VaR is
2.33 x 3,794.733 = ₹ 8,846.728.

Example (continued)
• VaR for one year (252 days) = ₹ 44,385.12
• Bank’s Gross Income = ₹ 1,869,906
• 15% of Gross Income = ₹ 280,485.
• Capital charge for operational risk = ₹ 280,097.
• Bank’s current share capital will be related to risk weights assessed by the capital charge.

Value-at-Risk
• An estimate of potential loss in a
– Position
– Asset
– Liability
– Portfolio of assets
– Portfolio of liabilities
• During a given holding period at a given level of certainty

Value-at-Risk
• Probability of the unexpected happening
• Probability of suffering a loss
• Estimate of loss likely to be suffered
• VaR is not the actual loss
• VaR measures potential loss and not potential gain
• VaR measures the probability of loss for a given time period over which the position is held

Bank for International Settlement (BIS)
• VaR is a measurement of market risk
• Provision of capital adequacy for market risk, subject to approval by banks' supervisory authorities
• Computation of VaR changes based on the estimated time period
– One day
– One week
– One month
– One year

Bank for International Settlement (BIS)
• Holding period for an instrument will depend on liquidity of the instrument
• Varying degrees of certainty changes potential loss
• VaR estimates that the loss will not exceed a certain amount
• VaR will change with different levels of certainty

VaR Methodology
• Computed as the expected loss on a position from an adverse movement in identified market risk parameter(s)
• Specified probability over a nominated period of time
• Volatility in financial markets is calculated as the standard deviation of the percentage changes in the relevant asset price over a specified asset period
• Volatility for calculation of VaR is specified as the standard deviation of the percentage change in the risk factor over the relevant risk horizon

VaR Computation Method
• Correlation Method
– Variance – covariance method
– Deterministic approach
– Change in value of the position computed by combining the sensitivity of each component to price changes in the underlying assets

VaR Computation Method
• Historical Simulation
– Change in the value of a position using the actual historical movements of the underlying assets
– Historical period has to be adequately long to capture all possible events and relationships between the various assets and within each asset class
– Dynamics of the risk factors captured since simulation follows every historical move

VaR Computation Method
• Monte Carlo Simulation
– Calculates the change in the value of a portfolio using a sample of randomly generated price scenarios
– Assumptions on market structures, correlations between risk factors and the volatility of these factors

VaR Application
• Basic parameters
– Holding period
– Confidence interval
– Historical time period (observed asset prices)
• Closer the models fit economic reality, more accurate the estimated
• There is no guarantee that the numbers returned by each VaR method will be near each other

VaR Application
• VaR is used as a Management Information System (MIS) tool in the trading portfolio
• Risk by levels
• Products
• Geography
• Level of organisation
• VaR is used to set risk limits
• VaR is used to decide the next business

VaR Limitation
• VaR does not substitute
– Management judgement
– Internal control
• VaR measures market risk
– Trading portfolio
– Investment portfolio
• VaR is helpful subject to the extent of
– Measurement parameters

Back Testing
• Backtests compare realized trading results with model generated risk measures
• Evaluate a new model
• Reassess the accuracy of existing models
• Banks using internal VaR models for market risk capital requirements must backtest their models on a regular basis

Back Testing
• Banks back test risk models on a monthly or quarterly basis to verify accuracy
• Observe whether trading results fall within pre-specified confidence bands as predicted by the VaR models
• If the models perform poorly establish cause for poor performance
– Check integrity of position
– Check market data
– Check model parameters
– Check methodology

Stress Testing
• Banks gauge their potential vulnerability to exceptional, but plausible, events
• Stress testing addresses the large moves in key market variables that lie beyond day to day risk monitoring but that could potentially occur

Stress Testing
• Process of stress testing involves
– Identifying potential movements
– Market variables to stress
– How much to stress them
– What time frame to run the stress analysis
– Shocks are applied to the portfolio
• Revaluing the portfolios
– Effect of a particular market movement on the value of the portfolio
– Profit and Loss
– Effects of different shocks of different magnitudes

Stress Testing Technique
• Scenario analysis
• Evaluating the portfolios
– under various expectations
– evaluating the impact
• changing evaluation models
• volatilities and correlations
• Scenarios requiring no simulations
– analyzing large past losses

Stress Testing Technique
• Scenarios requiring simulations
– Running simulations of the current portfolio subject to large historical shocks
• Bank specific scenario
– Driven by the current position of the bank rather than historical simulation
• Subjective than VaR
• Identify undetected weakness in the bank's portfolio

Efficiency of a Stress Test
• Relevant to the current market position
• Consider changes in all relevant market rates
• Examine potential regime shifts (whether the current risk parameters will hold or break down)
• Consider market illiquidity
• Consider the interrelationship between market and credit risk

Application of Stress Tests
• Stress tests produce information summarising the bank’s exposure to extreme but possible circumstances
• Role of risk managers in the bank is gathering and summarising information to enable senior management to understand the strategic relationship between the bank’s risk taking
– Extent and character of financial leverage employed
– Risk appetite
– Stress scenarios created on a regular basis
– Stress scenarios monitored over time

Application of Stress Tests
• Influence decision-making
• Manage funding risk
• Provide a check on modelling assumptions
• Set limits for traders
• Determine capital charges on trading desks’ positions

Limitations of Stress Test
• Stress tests are often neither transparent nor straightforward
• Depends on a large number of practitioner choices
• Choice of risk factors to stress
• Methods of combining factors stressed
• Range of values considered

Limitations of Stress Test
• Time frame to analyse
• Risk manager is faced with the considerable tasks of analyzing the results and identifying implications
• Stress test results interpretation for the bank is based on qualitative criteria
• Manage bank’s risk-taking activities is subject to interpretations