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Measurement of Market Risk 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. Value-at-Risk X 95% 5% (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 S t z S dS dt dz 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(t1/ 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 Sz 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 (multiplication factor equals 3). a multiplication factor 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 V10 = 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 straightforward • Depends on a large number of practitioner choices • Choice of risk factors to stress • Methods of combining factors stressed • Range of values considered nor 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