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MARKET RISK MANAGEMENT: VAR Market Risk Measurement BAHATTIN BUYUKSAHIN, CELSO BRUNETTI 1 Slide 2 ANALYTIC RISK MANAGEMENT TOOLS Risk-weighted assets (banks)1988 Stress Testing1992 Value-at-Risk, VaR1993 RiskMetrics1994 CreditMetrics1997 Integration of credit and market1998- Enterprisewide RM2000- BAHATTIN BUYUKSAHIN, CELSO BRUNETTI 2 Slide 3 HOW MUCH CAN WE LOSE? Everything correct, but useless answer How much can we lose realistically? BAHATTIN BUYUKSAHIN, CELSO BRUNETTI 3 Slide 4 DEFINITION VaR is defined as the predicted worst-case loss at a specific confidence level (e.g. 99%) over a certain period of time BAHATTIN BUYUKSAHIN, CELSO BRUNETTI 4 Slide 5 DEFINITION (JORION) VaR is the worst loss over a target horizon with a given level of confidence BAHATTIN BUYUKSAHIN, CELSO BRUNETTI 5 Slide 6 VAR BAHATTIN BUYUKSAHIN, CELSO BRUNETTI 6 Profit/Loss 1% VaR 1% Slide 7 MEANING OF VAR A portfolio manager has a daily VaR equal $1M at 99% confidence level This means that there is only one chance in 100 that a daily loss bigger than $1M occurs BAHATTIN BUYUKSAHIN, CELSO BRUNETTI 7 1% VaR under normal market conditions Slide 8 RETURNS BAHATTIN BUYUKSAHIN, CELSO BRUNETTI 8 year 1% of worst cases Slide 9 MAIN IDEAS A few well known risk factors Historical data + economic views Diversification effects Testability Easy to communicate BAHATTIN BUYUKSAHIN, CELSO BRUNETTI 9 Slide 10 CONVENTIONAL ANALYSIS BAHATTIN BUYUKSAHIN, CELSO BRUNETTI 10 Risk factor $ value sensitivity scenarios Slide 11 VAR APPROACH BAHATTIN BUYUKSAHIN, CELSO BRUNETTI 11 Risk factor $ yield price Slide 12 IMPORTANT VaR is a necessary, but not sufficient procedure for controlling risk It must be supplemented by limits and controls, in addition to an independent risk-management function Sound risk-management practices BAHATTIN BUYUKSAHIN, CELSO BRUNETTI 12 Slide 13 MARKET RISK MANAGEMENT Introduction to Market Risk BAHATTIN BUYUKSAHIN, CELSO BRUNETTI 13 Slide 14 VALUE AT RISK (VAR) In theory, risk managers should consider the entire distribution of profits and losses over the specified horizon In practice, this distribution is summarized by one number: the worst loss at a specified confidence level, such as 99% VAR is only one of the measures that risk managers focus on It should be complemented by stress testing, which identifies potential losses under extreme market conditions BAHATTIN BUYUKSAHIN, CELSO BRUNETTI 14 Slide 15 MARKET RISK & OTHER RISKS Three main types of risks: Market Risk, Credit Risk, Operational Risk These three types of risks are interconnected sometimes it is difficult to distinguish among them! Example: Corporate Bond Bond price movements may be due to: Risk-free interest rate movements (FOMC meeting) Market Risk Likelihood of default of the company Credit Risk BAHATTIN BUYUKSAHIN, CELSO BRUNETTI 15 Slide 16 RISK MANAGEMENT TOOLS (1) Example: five-year inverse-floater, coupon = 16% - 2 LIBOR (if positive), on a notional principal of $100M, current price of the bond = $100M very risky bond If y goes up, the coupon rate will drop and discount rate will increase the value of the bond will decrease sharply How can we measure the risk of this bond? Notional Amount: $100M this is the potential loss and does not give any indication about a realistic loss Worst Case Scenario: interest rate rises at 8% or above coupon = 0 The bond becomes a zero-coupon bond whose value is $68M (at 8% rate) Loss = $100M - $68M = $32M better than Notional Amount BAHATTIN BUYUKSAHIN, CELSO BRUNETTI 16 Slide 17 RISK MANAGEMENT TOOLS (2) Sensitivity Measure: Duration Modified Duration: 13.5 years very high duration (as expected) Duration does not tell us whether the movement in the interest rate is likely and does not take into account of the non-linearity of the relation between interest rates and bond price Scenario Analysis: it allows to investigate nonlinear, extreme effects but does not tell us the probabilities of these extreme events BAHATTIN BUYUKSAHIN, CELSO BRUNETTI 17 Slide 18 RISK MANAGEMENT TOOLS (3) All the above tools do not account for aggregation and therefore correlation VaR: One number aggregates the risks across the whole portfolio, taking into account leverage and diversification, and providing a risk measure with an associated probability If the worst increase in yield at the 95% level is 1.65%, we can compute VaR as VaR = Market value Modified duration Worst yield increase (10.1) VAR = $100 13.5 0.0165 = $22 million The investor can now make a statement such as the worst loss at the 95% confidence level is approximately $22M BAHATTIN BUYUKSAHIN, CELSO BRUNETTI 18 Slide 19 VAR: EXAMPLE Downside Risk: Maximum loss over a target horizon such that there is a low, prespecified probability that the actual loss will be greater Position: $4 B short the Yen, long the Dollar How much this position could lose over a day? Use 10-year historical data to simulate a daily return where Q 0 is the current dollar value of the position For instance, for two hypothetical days S 1 = 112.0 and S 2 = 111.8. The simulated return is R 2 ($) = $4,000 million [111.8 112.0]/112.0 = $7.2 M Repeating this operation over the whole sample, or 2,527 trading days, creates a time series of fictitious returns BAHATTIN BUYUKSAHIN, CELSO BRUNETTI 19 Slide 20 VAR: EXAMPLE We now wish to summarize the distribution by one number Quantile VaR: typically reported as a positive number, even if it is a loss BAHATTIN BUYUKSAHIN, CELSO BRUNETTI 20 Slide 21 VAR: EXAMPLE BAHATTIN BUYUKSAHIN, CELSO BRUNETTI 21 Slide 22 VAR: EXAMPLE VaR is sometimes reported as deviation from the mean: VaR(c) = E(X) Q(X, c) (2.9) If time horizon is short, daily expected returns for financial assets are zero the two definitions are identical With a total of T=2,537 observations and c=95%, p=1 c = 5% there are pT = 0.05 2,527 = 126 observations in the left tail. We pick from the ordered distribution the cutoff value, which is R * = $47.1 million We can now make a statement such as: The maximum loss over one day is about $47 million at the 95% confidence level i.e. there is 5% probability that the loss will be greater than $47M BAHATTIN BUYUKSAHIN, CELSO BRUNETTI 22 Slide 23 VAR: CAVEATS VaR does not describe the worse loss actual loss might be larger than VaR VaR does not describe the losses in the left tail. VaR does not say anything about the distribution of losses in its left tail VAR fails to qualify as a subadditive risk measure, which is a very desirable properties of risk measures Subadditivity implies that the risk of a portfolio must be less than the sum of risks of the portfolio components VaR merging portfolios may increase risk peculiar result! BAHATTIN BUYUKSAHIN, CELSO BRUNETTI 23 Slide 24 VAR: CAVEATS BAHATTIN BUYUKSAHIN, CELSO BRUNETTI 24 Slide 25 VAR: CAVEATS VAR is measured with some error The VAR number itself is subject to normal sampling variation. In our example, we used 10 years of daily data. Another sample period, or a period of different length, will lead to a different VAR number What matters is the first-order magnitude BAHATTIN BUYUKSAHIN, CELSO BRUNETTI 25 Slide 26 VAR: ALTERNATIVE MEASURES Entire Distribution Conditional VaR: it is an average of tail loss i.e. computes the average of the tail it tells us how much we can lose if we are hit beyond VaR For the Yen example, the average loss beyond VaR (= $47M) is CVaR = $74M BAHATTIN BUYUKSAHIN, CELSO BRUNETTI 26 Slide 27 VAR: ALTERNATIVE MEASURES Standard Deviation Advantage: takes into account all observation Disadvantage: Symmetrical, it cannot distinguish between large losses or gains, computing VAR requires distributional assumption. Where is z value, is the volatility of the rate of return, is the expected value and W is the amount invested. BAHATTIN BUYUKSAHIN, CELSO BRUNETTI 27 Slide 28 SEMI STANDARD DEVIATION Semi-Standard Deviation Considers only losses. Advantage: accounts for asymmetries in the distribution BAHATTIN BUYUKSAHIN, CELSO BRUNETTI 28 Slide 29 DRAWDOWN Drawdown: decline from peak over a fixed time interval. Of course maximum drawdown is the decline from peak to though. If returns are not independent from period to period or if the portfolio is actively managed. Disadvantage: It is backward looking. Time interval inconsistencies. BAHATTIN BUYUKSAHIN, CELSO BRUNETTI 29 Slide 30 CASH FLOW AT RISK Cash Flow at Risk (CFAT) measures the worst shortfall in cash flows due to unfavorable movements in market risk factors CF = Q (P C) where Q is quantity, P is unit revenue (price), C is unit cost Example: value of a farmers harvest, say corn At the beginning of the year, costs are fixed and do not contribute to risk The price of corn and the size of the harvest in the fall are unknown. Price movements are driven by supply shocks weather If there is a drought during the summer, quantities will fall and prices will increase, and conversely if there is an exceptionally abundant harvest Negative correlation between Q and P total revenues will fluctuate less Such relationships need to be factored into the risk measurement system because they will affect the hedging program VaR 1) model the relationship between Q, P and risk factors 2) Simulation VaR BAHATTIN BUYUKSAHIN, CELSO BRUNETTI 30 Slide 31 VAR PARAMETERS: C The higher the confidence level c, the greater the VAR measure c: 95%, 99%, 99.9%, 99.99% Each value will produce an increasingly larger, but less likely, loss As c increases, the number of observations below VaR shrinks it is problematic to compute quantiles accurately With 1,000 observations and c = 99.9%, the quantile has only 1 observation VaR is computed based only on 1 observation Solution to the problem: Consistency of c across time and firm, choose c according to the objective If VaR is used for Capital Adequacy: how much capital to be set aside to avoid bankruptcy c should be very high Backtesting: checking that the frequency of losses exceeding VaR is in line with p = 1 - c BAHATTIN BUYUKSAHIN, CELSO BRUNETTI 31 Slide 32 VAR PARAMETERS: HORIZON The longer the horizon T, the greater the VAR measure To compute VaR for longer horizons from a daily VaR, we need to assume: R t ~ i.i.d. (, 2 ) and () is stable i.e. does not change for longer horizons (Normal) VaR(T days) = VaR(1 day) T (10.10) Key concept: VaR can be extended from a one-day horizon to T days by multiplication by the square root of time. This adjustment is valid with i.i.d. returns that have a normal distribution BAHATTIN BUYUKSAHIN, CELSO BRUNETTI 32 Slide 33 VAR PARAMETERS: HORIZON The choice of the horizon also depends on the characteristics of the portfolio: if the positions change quickly VaR with short horizon The choice of the horizon depends on the use of VAR: if the purpose is to provide an accurate benchmark measure of downside risk, the horizon should be short; if VaR is used for Capital Adequacy, then a long horizon is advisable In practice, the horizon cannot be less than the frequency of reporting of profits and losses: Daily for banks, trading desks; corporations use a longer interval (ranging from daily to monthly). This interval is the minimum horizon for VaR Backtesting: power of the statistical tests increases with the number of observations, it is advisable to have a horizon as short as possible BAHATTIN BUYUKSAHIN, CELSO BRUNETTI 33 Slide 34 BASEL RULES The Basel market risk charge requires VAR to be computed with the following parameters: 10 trading days horizon (or two calendar weeks) 99% confidence interval A year of observations (historical data) and updated at least once a quarter Under the internal models approach (IMA), the market risk charge (MRC) is k: supervisor determined multiplier; SRC: specific risk charge BAHATTIN BUYUKSAHIN, CELSO BRUNETTI 34 Slide 35 BASEL RULES: SPECIFIC RISK RULE The specific risk charge is designed to provide a buffer against losses due to idiosyncractic factors Example: Corporate Bond by Ford Motor: credit rating BBB The market risk component should capture the effect of movements in yields for an index of BBB-rated corporate bonds SRC should capture the effect of credit downgrades for Ford SRC can be computed using VaR of portfolios that contain specific risk BAHATTIN BUYUKSAHIN, CELSO BRUNETTI 35 Slide 36 BASEL RULES: VAR The Basel Committee allows the 10-day VAR to be obtained from an extrapolation of 1-day VAR figures. Thus VAR is really VAR t (10, 99%) = 10 VAR t (1, 99%) The 10-day period corresponds to the time required for corrective action by bank regulators should an institution start to run into trouble The 99% confidence level corresponds to a low probability of bank failure due to market risk Even so, one occurrence every 100 periods implies a high frequency of failure there are 52/2 = 26 two-week periods in one year one failure should be expected to happen every 100/26 = 3.8 years, which is still much too frequent This explains why the Basel Committee has applied a multiplier factor, k 3, to guarantee further safety BAHATTIN BUYUKSAHIN, CELSO BRUNETTI 36 Slide 37 VAR: 3 STEPS 1. From market data, construct the distribution of risk factors (e.g., normal, empirical, or other) 2. Collect the portfolio positions and map them to the risk factors 3. Based on a VaR method (normal, historical, Monte Carlo), construct the portfolio VaR BAHATTIN BUYUKSAHIN, CELSO BRUNETTI 37 Slide 38 VAR: 3 STEPS STEP 1 The risk factors represent a subset of all market variables that adequately represent the risks of the current portfolio Simplification: There are literally tens of thousands of securities available, but a much more restricted set of useful risk factors choose market factors that are adequate for the portfolio Example 1: fixed-income portfolio one bond market risk factor may be enough to represent the risk of the whole portfolio Example 2: option portfolio volatilities should be added as risk factors In general, the more complex the strategies, the greater the number of risk factors that should be used BAHATTIN BUYUKSAHIN, CELSO BRUNETTI 38 Slide 39 VAR: 3 STEPS STEP 2 Portfolio positions assumption: all positions are constant over the horizon (this is a simplification) The true risk can be greater or lower than the VaR measure It can be greater if VaR is based on close-to-close positions if traders take more risks during the day, the true risk will be greater than indicated by VaR Conversely, the true risk can be lower if management enforces loss limits limits the risk that traders can take if losses develop BAHATTIN BUYUKSAHIN, CELSO BRUNETTI 39 Slide 40 VAR: 3 STEPS STEP 3 The choice of the VaR method depends on the nature of the portfolio Example 1: for a fixed-income portfolio a linear method may be enough Example 2: portfolio containing options we need to include nonlinear effects Risk management is as much an art as a science Risk managers need to make reasonable approximations to come up with a cost-efficient measure of risk Traders could be induced to find holes in the risk management system A VaR system alone will not provide effective protection against market risk It needs to be used in combination with limits on notionals and on exposures and, in addition, should be supplemented by stress tests BAHATTIN BUYUKSAHIN, CELSO BRUNETTI 40 Slide 41 FRM-99, QUESTION 89 What is the correct interpretation of a $3 overnight VaR figure with 99% confidence level? A. expect to lose at most $3 in 1 out of next 100 days B. expect to lose at least $3 in 95 out of next 100 days C. expect to lose at least $3 in 1 out of next 100 days D. expect to lose at most $6 in 2 out of next 100 days BAHATTIN BUYUKSAHIN, CELSO BRUNETTI 41 Slide 42 STRESS TESTING VaR does not account for extreme losses stress testing: identifies situations that could create extraordinary losses Identify areas of potential vulnerability The goal of stress testing is to identify areas of potential vulnerability Stress testing is a key risk management process that includes (1) scenario analysis; (2) stressing models, volatilities, and correlations; and (3) policy response development Scenario Analysis: Moving key variables one at a time; using historical scenarios; Creating prospective scenarios Event Risk: Changes in government; changes in economic policies; coups, civil wars, invasions; currency devaluations BAHATTIN BUYUKSAHIN, CELSO BRUNETTI 42 Slide 43 EXAMPLE: TURMOIL IN ARGENTINA Up to 2001, the Argentine peso was fixed to the U.S. dollar at a one-to-one The Argentinean Government had promised to defend the currency at all costs Argentina Experienced the worst economic crisis in decades excessive borrowing In December 2001, Argentina announced it would stop paying interest on its $135 billion foreign debt This was the largest sovereign default recorded so far January 6 devaluation: the exchange rate promptly moved from 1 peso to more than 3 pesos per dollar Such moves could have been factored into risk management systems by scenario analysis What was totally unexpected, however, was the governments announcement that it would treat bank loans and deposits differentially Dollar-denominated bank deposits were converted to devalued pesos, but dollar-denominated bank loans were converted to pesos at a one-to-one rate Banking system technically insolvent loans (bank assets) overnight became less valuable than deposits (bank liabilities) Nobody predicted this move In 2005, the Argentinean government proposed to pay back about 30% of the face value of its debt. This recovery rate was very low by historical standards BAHATTIN BUYUKSAHIN, CELSO BRUNETTI 43 Slide 44 LIQUIDITY RISK Component of market risk Liquidity risk consists of both asset liquidity risk and funding liquidity risk Asset liquidity risk, also called market/product liquidity risk, arises when transactions cannot be conducted at quoted market prices due to the size of the required trade relative to normal trading lots Funding liquidity risk, also called cash-flow risk, arises when the institution cannot meet payment obligations BAHATTIN BUYUKSAHIN, CELSO BRUNETTI 44 Slide 45 ASSET LIQUIDITY RISK Example 1: A bank has a large position in IBM shares The bank would like to sell the position - IBM close price $79.36 - The market is illiquid At what price would the bank be able to sell the large IBM position? Example 2: A bank has a large portfolio The bank would like to assess the risk of the portfolio The market is illiquid How should the bank compute volatilities and correlations? BAHATTIN BUYUKSAHIN, CELSO BRUNETTI 45 Slide 46 ASSET LIQUIDITY RISK Aggregate versus stock specific liquidity Aggregate liquidity is market-wide, fluctuates over time, and has different impacts on different stocks (October 1987, August 1998 LTCM) Stock specific liquidity is a property of a stock Definition: Liquidity is the ability to trade quickly any amount at the market price with no additional cost BAHATTIN BUYUKSAHIN, CELSO BRUNETTI 46 Slide 47 ASSET LIQUIDITY RISK: STYLIZED FACTS Expected returns increase with illiquidity (liquidity premium): - Amihud (2002), Amihud & Mendelson (1986) Volatility increases with illiquidity: Pastor & Stambaugh (2003), Papanicolaou & Sircar (1998), Schoenbucher & Wilmott (2000) Correlation increases with illiquidity: Bhansali & Wise (2001) Sharpe Ratios increase with illiquidity: Lo, Petrov & Wierbicki (2003) BAHATTIN BUYUKSAHIN, CELSO BRUNETTI 47 Slide 48 LIQUID ASSETS Tightness, which is a measure of the divergence between actual transaction prices and quoted mid-market prices Depth, which is a measure of the volume of trades that can be made without affecting prices too much (e.g., the bid/offer prices) and is in contrast to thinness Resiliency, which is a measure of the speed at which price fluctuations from trades are dissipated In contrast, illiquid markets are those where transactions can quickly affect prices BAHATTIN BUYUKSAHIN, CELSO BRUNETTI 48 Slide 49 LIQUIDITY RISK When the market is illiquid flight to quality: there is a shift in demand away from low-grade securities toward high-grade securities Flight to quality may even affect government bonds: The yield spread can widen between off-the-run securities and corresponding on-the-run securities On the-run securities are those issued most recently and hence are more active and liquid. Other securities are called off-the-run The latest-issued 30-year U.S. Treasury bond This bond is called on- the-run until another 30-year bond is issued, at which time it becomes off-the-run Because these securities are very similar in terms of market and credit risk, the yield spread is a measure of the liquidity premium BAHATTIN BUYUKSAHIN, CELSO BRUNETTI 49