CHAPTER 10 Edited by Bo Sjö Overview This chapter discusses the nature of market risk and measures of market risk: 1) Value at Risk (VaR) or RiskMetrics 2) Historic or Back Simulation 3) Monte Carlo simulation - briefly > Finally, consider the link between market risk (VaR) and capital requirements according to Basel. 10-2 McGraw-Hill/Irwin Copyright 2011 by The McGraw-Hill Companies, Inc. All Rights Reserved. Recall the basic accounting identity E = A -D If market foces changes the prices of assets and liabilities it will affect equity (and thus reserves): E = A D If E < 0: You are out of buiness! You are also out when E is less than required minum reserves. We need to know (i) how big are normal day-to-day fluctuations, and (ii) how big are possible abnormal fluctuations and (iii) not totally unlikely extremely abnormal fluctuations? The answer is the basis for risk management and the level of reserves needed to survive. Need not necessarily survive (iii) above 10-3 10-4 Implications Emphasizes importance of: Measurement of exposure to risk factors Control mechanisms for direct market risk Hedging mechanisms And, of interest to regulators (i) Need to monitor the exposure (ii) Set limits to excessive exposure Market risk is the uncertainty resulting from unpredicted changes in market prices of assets Consider changes in interest rates, exchange rates, equity and other assets Can be measured over periods as short as one day Usually measured (i) in terms of dollar, exposure amount or (ii) as a relative amount against some benchmark. 10-5 10-6 1
Ask the Following Questions What are normalday-to-day fluctuations on the market? What are abnormal fluctuations? To sort out the normal from the abnormal -look at the historical distribution of changes in asset prices (i) Calculate the sample Mean and variance of r (ii) Calculate a confidence intervall around the mean fluctuations of r! (iii) Outside the confidence intervall consider it Abnormal (adverse movements) Measurement Important in terms of: Management information (control & management) Setting limits to risk exposure Resource allocation (the risk-return tradeoff) Performance evaluation 10-7 10-8 Measurement Regulation: -BIS (Bank of International Settlements) and Fed (Federal Reserve Bank of USA) regulate market risk via capital requirements leading to potential for overpricing of risks -Allowances for the use of internal models to calculate capital requirements (=Set necessary reserves) Calculating Exposure What is the estimated loss under adverse (=extreme) circumstances? Three major approaches: 1) VaRor JPM RiskMetrics (or variance/ covariance approach) 2) Historic or Back Simulation 3) Monte Carlo Simulation 10-9 10-10 JP Morgan VaR JP Morgan developed the Value at Risk VaR concept into a commerical idea. They called it RiskMetrics (Trade Mark) Now RiskMetrics is run by separate company owned by JPM & Reuters The RiskMetrics Model The Idea is to determine DEAR(Daily Earnings At Risk) = dollar value of position (price sensitivity potential adverse move in yield) DEAR = dollar market value of position price volatility price volatility = price sensitivity of position (Duration) potential adverse move in yield 10-11 10-12 2
VaR Daily Earnings at Risk DEAR can be stated as: DEAR = (MD) (potential adverse daily yield move) where, MD = Modified duration (D(1+r) D = Macaulay duration, instead of r = interest rate we can use y = yield. To Calculate DEAR: 10-13 10-14 Here is the answer to our?:s Confidence Intervals ±(Mean of r) 1.96 Standard deviation of r Here we have assumed that r has a normal frequency distribution, with 5% risk at both sides => 2.5% is abnormal (bad). Other risk levels: ±1.65 σ 90% confidence interval, 5% adverse moves ± 1.96 σ 95 % confidence interval, 2.5 % adverse moves ± 2.33 σ 98 % confidence interval, 1% adverse moves ± 2.66 σ 99% confidence interval, 0.5% adverse moves 10-15 10-16 Confidence Intervals If we assume that changes in r (and thus the yield) are normally distributed => we can construct confidence intervals for DEAR Assuming normality, 90% of the time the disturbance will be within ±1.65 standard deviations of the mean 5% of the extreme values remain in each tail of the distribution, see next slide which shows the distribution (possible values) of r for a sevenyear zero coupon bondwith mean = 0. Adverse 7-Year Rate Move 10-17 10-18 3
Basis Points Changes in interest rates are measured in Basis Points Decimals of per cent. One basis point is 0.01% or 0.0001 100 basis points = 1%. Confidence Intervals: Example Suppose that we are long in 7-year zerocoupon bonds and we define bad yield changes such that there is only a 5% chance of the yield change being exceeded in either direction. Assuming normality, 90% of the time yield changes will be within 1.65 standard deviations of the mean. If the standard deviation is 10 basis points (0.0010 or 0.1%), this corresponds to 1.65 0.0010=0.00165. Probability of yield increases greater than 16.5 basis points is 5%. 10-19 10-20 N Confidence Intervals: Example D = 7 years. Yield on the bond = 7.243%, so MD = 6.527 years Price volatility = (MD) (Potential adverse change in yield) = (6.527) (0.00165) = 1.077% DEAR = Market value of position (Price volatility) = ($1,000,000) (.01077) = $10,770 Confidence Intervals: Example To calculate the potential loss for more than one day (say a trading week): Market value at risk (VaR N ) = DEAR N Example: For a five-day period, VaR 5 = $10,770 5 = $24,082 10-21 10-22 Foreign Exchange In the case of foreign exchange, DEAR is computed in the same fashion we employed for interest rate risk, but there is a correlation across exchange rates. DEAR = dollar value of position FX rate volatility, where the FX rate volatility is taken as 1.65 σ FX Statistical Formulas Calculate the (arithmetic) mean Variance (and standard deviation) Correlation coefficient 10-23 10-24 4
Skewed distribution Example: One-day 90 per cent VaR of portfolio, showing next day positions, (with skewed distributionan 10 % risk at the lower end. Equities For equities, total risk = systematic risk + unsystematic risk If the portfolio is well diversified, then DEAR = dollar value of position stock market return volatility, where market volatility taken as 1.65 σ m If not well diversified, a degree of error will be built into the DEAR calculation 10-25 10-26 Estimation FX VaR Convert today s FX positions into dollar equivalents at today s FX rates Measure sensitivity of each position Calculate its delta Measure risk Actual percentage changes in FX rates for each of past 500 days Rank days by risk from worst to best 10-27 Aggregating DEAR Estimates Cannot simply sum up individual DEARs, since the individual DEAR are correlated (ρ) with each other. To aggregate two DEARs we require the correlation coefficient (ρ ac ) between the DEARs Two-asset case: DEAR portfolio= [DEAR a2 + DEAR b2 + 2ρ ab DEAR a DEAR b + 2ρ ac ] 1/2 10-28 Three assets case DEAR: Large US Banks 2005 & 2008 In order to aggregate the DEARs from individual exposures we require the correlation matrix, the correlations among all pairs of DEARs. The three-asset case: DEAR portfolio= [DEAR a2 + DEAR b2 + DEAR c2 + 2ρ ab DEAR a DEAR b + 2ρ ac DEAR a DEAR c + 2ρ bc DEAR b DEAR c ] 1/2 10-30 10-29 5
Problems with VaR 1) Historical data (means and variances), used from 100 days rolling windows might not reflect the future. The worst day is yet to come perhaps. => Remember to predic future volatility. 2) Volatility is persistent, high volatility today will be followed by high volatity tomorrow. Changes in prices, exchange rates,interesr rates etc are not indedendent. => GARCH models to predict future volatility. 3) The Normal distribution will under-estimate the frequency of abnormal changes. Real world data show much fatter tails than the Normal distribution. => Look for other distributios than the normal. The Advantage of VaR It is, or should be, a forward looking measure, using predicted future volatility. Not be based on historical data (backward looking) only. With VaR it is possible to measure risk when risk is taken! The risk manager can interact with traders immediately to measure and control risk. If your predictions are good, VaR works fine. Works well in the short-run, in periods with small changes in prices and interest rates. VaR is a simple question with a simple answer. The weakness, which is difficult calculate when the economy is changing. 10-31 10-32 Historic or Back Simulation Basic idea: Revalue the current portfolio with historical prices. Say 500 days back. Then calculate 5% worst-case outcomes (25 th lowest value of 500 days). Only 5% of the outcomes are lower in terms of less value. Historic or Back Simulation Advantages: Simplicity Does not need correlations or standard deviations of individual asset returns Does not require normal distribution of returns (which is a critical assumption for RiskMetrics) Directly provides a worst case value 10-33 10-34 Weaknesses Disadvantage: 500 observations is not very many from a statistical standpoint Increasing number of observations by going back further in time is not desirable Could weight recent observations more heavily and go further back Backward looking only, focus on historical data. Monte Carlo Simulation To overcome problem of limited number of observations, simulate different possible prices. Start by estimating the historical variance-covariance matrix and use a random number generator to synthesize observations Objective is to replicate the distributionof observed outcomes with synthetic data 10-35 10-36 6
Regulatory Models BIS (including FED) approach: Market risk may be calculated using standard BIS model Specific risk charge depending who is the borrower and time to maturity Different reserve % for different type (above) Subject to regulatory permission, large banks may be allowed to use their internal models as the basis for determining capital requirements (VaR) 10-37 BIS Model Specific risk charge: Risk weights absolute dollar values of long and short positions General market risk charge: reflect modified durations expected interest rate shocks for each maturity Vertical offsets: Adjust for basis risk Horizontal offsets within/between time zones 10-38 Web Resources For information on the BIS framework, visit: Bank for International Settlement www.bis.org Federal Reserve Bank www.federalreserve.gov Large Banks: Using Internal Models In calculating DEAR, adverse change in rates defined as 99th percentile (rather than 95th under RiskMetrics) Minimum holding period is 10 days (means that RiskMetrics DEAR multiplied by 10 ). Capital charge will be higher of: Previous day s VAR (or DEAR 10) Average Daily VAR over previous 60 days times a multiplication factor 3 10-39 10-40 American Banker Banker of America Bank for International Settlements Federal Reserve J.P. Morgan Chase RiskMetrics Pertinent Websites www.americanbanker.com www.bankofamerica.com www.bis.org www.federalreserve.gov www.jpmorganchase.com www.riskmetrics.com 10-41 7