Value At Risk (VAR) Models - MIT OpenCourseWare

Lecture 7: Value At Risk (VAR) ModelsKen AbbottDeveloped for educational use at MIT and for publication through MIT OpenCourseware.No investment decisions should be made in reliance on this material.

Risk Management's Mission1. To ensure that management is fully informedabout the risk profile of the bank.2. To protect the bank against unacceptablylarge losses resulting from concentration ofrisks3. In other words:NO SURPRISES Two analogies: Spotlight coloring bookDeveloped for educational use at MIT and for publication through MIT OpenCourseware.No investment decisions should be made in reliance on this material.2

MethodologyDeveloped for educational use at MIT and for publication through MIT OpenCourseware.No investment decisions should be made in reliance on this material.

Different Methodologies Example of one-asset VaR Price-based instrumentsYield-based instrumentsVariance/CovarianceMonte Carlo SimulationHistorical SimulationDeveloped for educational use at MIT and for publication through MIT OpenCourseware.No investment decisions should be made in reliance on this material.4

Variables in the methods1.2.3.4.5.6.7.Interest rate sensitivity – duration, PV01,Equity exposureCommodity exposureCredit – spread durationDistribution/Linearity of price behaviorRegularity of cash flow/prepaymentCorrelation across sectors and classesDeveloped for educational use at MIT and for publication through MIT OpenCourseware.No investment decisions should be made in reliance on this material.5

Methodology: What Are We Tryingto We want to estimate the worst 1% of the possible outcomes.Developed for educational use at MIT and for publication through MIT OpenCourseware.No investment decisions should be made in reliance on this -1.00-1.50-2.00-2.50-3.00-3.50-4.000.006

Methodology: VaR 1.03Consider one time series, the MexicanIPC stock index,IPCfor /01/9607/01/9612/30/96We want to know how much this market could possibly move against us, so weknow how much capital we need to support the position .Developed for educational use at MIT and for publication through MIT OpenCourseware.No investment decisions should be made in reliance on this material.7

Methodology: Why We 201003400Frequency Distribution of IPC Levels: 1995-19961800 1600 Most financial time series follow random walks, which means, among otherthings, that the best estimate of tomorrow’s value is today’s value.Since random walks are not bounded, predicting the future path isdifficult if we focus only on the levels.A frequency distribution of IPC levels from 1995-1996 illustrates thedifficulty:1400 Developed for educational use at MIT and for publication through MIT OpenCourseware.No investment decisions should be made in reliance on this material.8

Methodology: Why We UseReturns Consider the returns of the IPC over the same period, wherereturns are defined as the percentage change in the index: , the frequency distribution falls into a familiar pattern.Developed for educational use at MIT and for publication through MIT OpenCourseware.No investment decisions should be made in reliance on this material.9

Methodology: Estimating Volatility Once we have a time series of returns, we can gauge theirrelative dispersion with a measure called variance.Variance is calculated by subtracting the average returnfrom each individual return, squaring that figure, summingthe squares across all observations, and dividing the sum bythe number of observations.The square root of the variance, called the standarddeviation or the volatility, can be used to estimate risk.In a normal distribution, 2.33 * the standard deviationrepresents the largest possible movement 99% of the time(1.64 * the standard deviation for 95%).Developed for educational use at MIT and for publication through MIT OpenCourseware.No investment decisions should be made in reliance on this material.10

Methodology: Estimating Volatility Mathematically, variance is:n i 1 xi - x 2(n - 1) The standard deviation is the square root of this term. The Excel functions for these two are var() and stdev()Developed for educational use at MIT and for publication through MIT OpenCourseware.No investment decisions should be made in reliance on this material.11

Methodology: Using Volatilityto Estimate Value at Risk The variance of the daily IPC returns between1/95 and 12/96 was 0.000324The standard deviation was 0.018012 or1.8012%2.33 * 1.8012% 0.041968 or 4.1968%We can conclude that we could expect to lose nomore than 4.1968% of the value of our position,99% of the time.Developed for educational use at MIT and for publication through MIT OpenCourseware.No investment decisions should be made in reliance on this material.12

Methodology: Using Volatilityto Estimate Value at Risk This means that an investment in the IPC of MXP 100investment would suffer daily losses over MXP 4.2 onlyabout 1% of the time.In fact, the IPC lost more than 4.2% 8 times since 1/1/95, orabout 1.5% of the time.While this figure is approximately accurate, it illustrates aproblem VaR has in certain markets, that it occasionallyunderestimates the number of large market moves.This problem, while frequent at the security or desk level,usually disappears at the portfolio level.Developed for educational use at MIT and for publication through MIT OpenCourseware.No investment decisions should be made in reliance on this material.13

Methodology: Review of OneAsset VaR1.2.3.4.Collect price dataCreate return seriesEstimate variance of return seriesTake square root of variance to get volatility(standard deviation )5. Multiply volatility by 2.33 by position size toget estimate of 99% worst case loss.Developed for educational use at MIT and for publication through MIT OpenCourseware.No investment decisions should be made in reliance on this material.14

Methodology: Caveats Longs vs. Shorts: sign is important– simple for equities– requires thought for FX One-sided vs. two-sided confidenceintervalsBad dataPercentage changes vs. log changesDeveloped for educational use at MIT and for publication through MIT OpenCourseware.No investment decisions should be made in reliance on this material.15

Methodology: Fixed Income Fixed income instruments require an adjustmentto this method.This is because time series generally availablefor fixed income securities are yield series, whilewe are concerned with price behavior.The adjustment requires expressing the volatilityin of basis points and the position in terms ofsensitivity to a 1 basis point movement in yields.Developed for educational use at MIT and for publication through MIT OpenCourseware.No investment decisions should be made in reliance on this material.16

Methodology: Fixed IncomeEquities, Foreign Exchange, Commodities:position * price * 2.33 (or 1.64 for 95%)Fixed Income:position * PV01 * close * yield * 2.33 *100}position sensitivity to a oneBP movement in yieldspotential movement inyields measured in BPDeveloped for educational use at MIT and for publication through MIT OpenCourseware.No investment decisions should be made in reliance on this material.17

Methodology: Fixed IncomeExample Consider a USD 100 position in 10 year UST: Vol of the percentage changes in UST yields since1/1/03 1.312%DV01 change in price at 12/31/96 yield if yieldswere to increase by 1 basis point(Duration may also be used here)Developed for educational use at MIT and for publication through MIT OpenCourseware.No investment decisions should be made in reliance on this material.18

Methodology: Fixed IncomeExample Price at yield of 4.644% 100Price at yield of 4.654% 99.92077DV01 99.920765-100 0.07923 per 100This sensitivity changes with the level ofyields, but provides a good approximationDeveloped for educational use at MIT and for publication through MIT OpenCourseware.No investment decisions should be made in reliance on this material.19

Methodology: Duration vs. DV01 Duration measures the weighted average time to a security’scash flows, where the weighting is the cash flow.Duration also shows the percentage change in price perchange in yield.DV01 provides a similar measure, but often per 1 million offace value.Bond traders think in DV01’s; portfolio managers think interms of duration.Either measure is effective but BE CAREFUL OF THEUNITS. This is one of the easiest errors to make!Developed for educational use at MIT and for publication through MIT OpenCourseware.No investment decisions should be made in reliance on this material.20

Methodology: Duration & DV01Examples UST Example:Use Excel PRICE( ) functionRemember to divide coupon/yield by100 (4.644% .0644)Assume redemption at parNote similarity of PV01 to DurationBasisDay count basis0 or omittedUS (NASD) n 30/360Developed for educational use at MIT and for publication through MIT OpenCourseware.No investment decisions should be made in reliance on this material.21

Methodology: Putting It AllTogether Position details: VaR -.07923*0.04644*.01312*2.33*100 1.12479–––––Size: USD 100Maturity: 10 YearsVol: 0.01312DV01: -.07923 (per 100)12/27/06 Close 4.644%Developed for educational use at MIT and for publication through MIT OpenCourseware.No investment decisions should be made in reliance on this material.22

Spread PV01 For credit-risky securities, we should distinguish betweeninterest rate risk and credit risk The credit spread takes default (and recovery) intoconsideration We usually consider these separately Often, we assume PV01 CSPV01– If recovery 0, then this is true– Otherwise, it is not There are different sources for spreads– Calculated– CDS– Asset swap spreadsDeveloped for educational use at MIT and for publication through MIT OpenCourseware.No investment decisions should be made in reliance on this material.23

Spread PV01CreditSpreadRisk-FreeRateDeveloped for educational use at MIT and for publication through MIT OpenCourseware.No investment decisions should be made in reliance on this material.24

Methodology:Adding Additional Assets The notion of covariance allows us toconsider the way assets’ prices behave withrespect to each other.Technically:n xi - x ( yi - y )i 1( n - 1) Developed for educational use at MIT and for publication through MIT OpenCourseware.No investment decisions should be made in reliance on this material.25

Methodology: Covariance What does this mean?–It gives an indication of how far one variable isfrom its mean when we observe another variable acertain distance from its mean.– In other words, it says how much (and in whichdirection) y moves when x moves.– It provides a measure for every variable withrespect to every other variable.Developed for educational use at MIT and for publication through MIT OpenCourseware.No investment decisions should be made in reliance on this material.26

Methodology: Portfolios Some Basic Statistical Principles:variance(a b) variance(a) variance(b) 2*covariance (a,b) a2 .27-0.231.26 0.181.34 a2 b2a -0.412.60 2 Var(b)0.469281Cov(ab)0.349660Var(a) Var(b) 2*cov(ab)1.501660Var(a b)estimated directly1.501660Developed for educational use at MIT and for publication through MIT OpenCourseware.No investment decisions should be made in reliance on this material.27

Methodology: Covariance Why is this important?–––If we know the variances and covariances ofall of the securities in a portfolio, we canassess the risk of the entire portfolio.We can also assess the risk of anysubportfolio.This is the basis of much of modern portfoliotheory.Developed for educational use at MIT and for publication through MIT OpenCourseware.No investment decisions should be made in reliance on this material.28

Methodology: Correlation How is correlation different from covariance?–we can scale covariance to get correlation:––covariance is not unit freecorrelation is an index of linearityCov( a , b) a bDeveloped for educational use at MIT and for publication through MIT OpenCourseware.No investment decisions should be made in reliance on this material.29