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.
Edited by Bo Sjö
Copyright © 2011 by The McGraw-Hill Companies, Inc. All Rights Reserved.
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
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.
Emphasizes importance of:
Market risk is the uncertainty resulting from unpredicted changes in market prices of assets
– Measurement of exposure to risk factors – Control mechanisms for direct market risk – Hedging mechanisms
– 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.
And, of interest to regulators – (i) Need to monitor the exposure – (ii) Set limits to ‘excessive’ exposure
Ask the Following Questions
Market Risk Measurement Important in terms of:
What are normal day-to- day fluctuations on the market? What are abnormal fluctuations?
– Management information (control & management) – Setting limits to risk exposure – Resource allocation (the risk-return tradeoff) – Performance evaluation
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)
Market Risk Measurement
Calculating Market Risk Exposure
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)
What is the estimated loss under adverse (=extreme) circumstances? Three major approaches: – 1) VaR or JPM RiskMetrics (or variance/ covariance approach) – 2) Historic or Back Simulation – 3) Monte Carlo Simulation
JP Morgan VaR
The RiskMetrics Model
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 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 10-12
To Calculate DEAR:
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.
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 ± 1.96×σ ± 2.33×σ ± 2.66×σ
90% confidence interval, 5% adverse moves 95 % confidence interval, 2.5 % adverse moves 98 % confidence interval, 1% adverse moves 99% confidence interval, 0.5% adverse moves
Adverse 7-Year Rate Move
– 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 bond with mean = 0. 10-17
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%.
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 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 (VaRN) = DEAR × N Example: For a five-day period, VaR5 = $10,770 × 5 = $24,082
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 σFX 10-23
Calculate the (arithmetic) mean Variance (and standard deviation) Correlation coefficient
Example: One-day 90 per cent VaR of portfolio, showing next day positions, (with skewed distributionan 10 % risk at the lower end.
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 σm If not well diversified, a degree of error will be built into the DEAR calculation
Estimation FX VaR
Aggregating DEAR Estimates
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
Three assets case
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 (ρac) between the DEARs Two-asset case: DEAR portfolio = [DEARa2 + DEARb2 + 2ρab × DEARa × DEARb + 2ρac]1/2 10-28
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 = [DEARa2 + DEARb2 + DEARc2 + 2ρab × DEARa × DEARb + 2ρac × DEARa × DEARc + 2ρbc × DEARb × DEARc]1/2 10-29
Problems with VaR
The Advantage of 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.
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.
Historic or Back Simulation
Historic or Back Simulation
Basic idea: Revalue the current portfolio with historical prices. Say 500 days back. Then calculate 5% worst-case outcomes (25th lowest value of 500 days). Only 5% of the outcomes are lower in terms of less value.
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
Monte Carlo Simulation
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. 10-35
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 distribution of observed outcomes with synthetic data 10-36
BIS (including FED) approach:
– Specific risk charge:
– Market risk may be calculated using standard BIS model
Risk weights × absolute ‘dollar values’ of long and short positions
Specific risk charge depending who is the borrower and time to maturity Different reserve % for different type (above)
– General market risk charge:
– Subject to regulatory permission, large banks may be allowed to use their internal models as the basis for determining capital requirements (VaR)
reflect modified durations × expected interest rate shocks for each maturity
– Vertical offsets: Adjust for basis risk
– Horizontal offsets within/between time zones
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:
For information on the BIS framework, visit: Bank for International Settlement www.bis.org Federal Reserve Bank www.federalreserve.gov
Previous day’s VAR (or DEAR × 10 ) Average Daily VAR over previous 60 days times a multiplication factor ≥ 3 10-39
Pertinent Websites American Banker Banker of America Bank for International Settlements Federal Reserve J.P. Morgan Chase RiskMetrics
www.americanbanker.com www.bankofamerica.com www.bis.org
www.federalreserve.gov www.jpmorganchase.com www.riskmetrics.com