Value At Risk IEF 217a: Lecture Section 5 Fall 2002 Jorion Chapter 5

Value-at-Risk (VaR) Probabilistic worst case Almost “perfect storm” 1/100 year flood level

VaR Advantages Risk -> Single number Firm wide summary –Handles futures, options, and other complications Relatively model free Easy to explain Deviations from normal distributions

Value at Risk (VaR) History Financial firms in the late 80’s used it for their trading portfolios J. P. Morgan RiskMetrics, 1994 Currently becoming: –Wide spread risk summary –Regulatory

Value at Risk: Methods Methods –Delta Normal –Historical –Monte-carlo –Bootstrap

Outline Computing VaR Interpreting VaR Time Scaling Regulation and VaR –Jorion 3, Estimation errors

Computing VaR 1.Mark to market (value portfolio) Identify and measure risk factor variability Normal: mean = 0, std. = 0.1 over 1 month 2.Set time horizon 1 month 3.Set confidence level 5%

Portfolio value today = 100, Normal returns (mean = 0, std = 10 per month), time horizon = 1 month, 5% VaR = 16.4

Normal Distributions Many VaR calculations can be done using tables Find percentile value for confidence level for normal, mean 0, std = 1 using standard tables For 0.05 level, this is –1.64 Critical return (R*) = std(percentile value) = 0.1*(-1.64) = W* = W(1+R*) = 100( ) = 83.6 VaR = Loss = W – W* = = 16.4

Normal Distributions Nonzero Mean (Absolute VaR) Assume std = 0.10, mean = 0.05 Critical return (R*) = mean + std(percentile value) = *(-1.64) = W* = W(1+R*) = 100( ) = 88.6 VaR = Loss = W – W* = = 11.4 This is known as Absolute VaR Absolute dollar loss

Normal Distributions Nonzero Mean (Relative VaR) Assume std = 0.10, mean = 0.05 Critical return (R*) = mean + std(percentile value) = *(-1.64) = W* = W(1+R*) = 100( ) = 88.6 Relative VaR is measured relative to expected wealth in the future VaR = Loss = E(W) – W* = 100(1.05)-88.6 = 16.4 This is known as Relative VaR

Absolute versus Relative VaR Absolute –Measure total loss possible against today’s wealth Relative –Measure loss against expected increases in today’s wealth. –If portfolio is expected to grow by 10 percent, measure loss relative to this growth If means are positive, then relative VaR will be larger (more conservative) If means are near zero (short horizons) then they are the same

Normal Distributions in Practice Assume returns are normal Estimate mean and std using data Then get VaR using tables or monte-carlo

Historical VaR Use past data to build histograms Method: –Gather historical prices/returns –Use this data to predict possible moves in the portfolio over desired horizon of interest

Easy Example Portfolio: –$100 in the Dow Industrials –Perfect index tracking Problem –What is the 5% and 1% VaR for 1 day in the future?

Data Dow Industrials dow.dat (data section on the web site) File: –Column 1: Matlab date (days past 0/0/0) –Column 2: Dow Level –Column 3: NYSE Trading Volume (1000’s of shares)

Matlab and Data Files All data in matrix format “Mostly” numerical Two formats –Matlab format filename.mat –ASCII formats Space separated Excel (csv, common separated)

Loading and Saving Load data –“load dow.dat” –Data is in matrix dow Save data – ASCII save -ascii filename dow –Matlab save filename dow

Example: Load and plot dow data Matlab: pltdow.m Dates: –Matlab datestr function

Back to our problem Find 1 day returns, and apply to our 100 portfolio Matlab: dnormdvar.m Methods used –Delta normal (tables) –Historical –Note difference

Outline Computing VaR Interpreting VaR Time Scaling Regulation and VaR –Jorion 3, Estimation errors

Interpreting VAR Benchmark measure –Compare risks across markets in company –Flag risks appearing over time Potential loss measure –Worst loss Equity capital

Outline Computing VaR Interpreting VaR Time Scaling Regulation and VaR –Jorion 3, Estimation errors

Time Scaling VaR calculations can be made beyond 1 period in the future Time scaling –Analytic –Monte-carlo

Scale Factors and Analytics (Jorion) Reminder Let r(t) be a random return (independent over time)

Scale Factors and Analytics

Scaling in Words Mean: scales with T Std. : scales with sqrt(T) Reminder: needs independence

Three Methods Approximate scaling Exact (log normal) scaling Bootstrap/monte-carlo

Approximate Assume that long horizon returns are the sum of the short horizon returns

Computing VaR 1.Mark to market (value portfolio) Identify and measure risk factor variability Normal: mean = 0, std. = 0.1 over 1 month 2.Set time horizon 6 months (before 1 month) Std = sqrt(6)0.1= Set confidence level 5%

6 Month VaR Many VaR calculations can be done using tables Find percentile value for confidence level for normal, mean 0, std = 1 using standard tables For 0.05 level, this is –1.64 Critical return (R*) = std(percentile value) = sqrt(6)*0.1*(-1.64) = W* = W(1+R*) = 100(1-0.40) = 60 VaR = Loss = W – W* = = 40

Exact Methods Assume that prices are a “geometric random walk” with normal increments

Value of Portfolio at T

Critical Return Let R* be the alpha critical value for the T period log return Now define the future wealth level at the alpha level by

Computing VaR 1.Mark to market (value portfolio) Identify and measure risk factor variability. Assume log returns are distributed: Normal: mean = 0, std. = 0.1 over 1 month 2.Set time horizon 6 months (before 1 month) Std = sqrt(6)0.1= Set confidence level 5%

6 Month VaR Exact (approximate numbers) Find percentile value for confidence level for normal, mean 0, std = 1 using standard tables For 0.05 level, this is –1.64 Critical return (R*) = std(percentile value) = sqrt(6)*0.1*(-1.64) = W* = W(1+R*) = 100*exp(-0.40) = 67 (60) VaR = Loss = W – W* = = 33 (40)

Bootstrap Methods If the 1 period return distribution is unknown, and you don’t want to hope the central limit theorem is working at T periods, then a bootstrap might be a good way to go Resample 1 period returns, T at a time, and build a histogram for the T period returns Use this to find the alpha critical value for wealth

Examples From Data Matlab: – hist10d.m – hist10dln.m

Outline Computing VaR Interpreting VaR Time Scaling Regulation and VaR –Jorion 3, Estimation errors

Regulation and Basel Capital Accord 1988 Minimum capital requirements Agreed minimum for signing central banks Why? –Avoid global systemic risk

The Early Basel Formulas Capital back must be at least 8% of “risk weighted” assets Risk weighting increases arbitrarily across asset classes

Criticism Ignores risk mitigation (hedging) methods Ignores diversification effects Ignores term structure effects Too few risk classes Ignores market risk

Standardized Model (1993) More classes New formulaic risk measures Problems –Still arbitrary formulas and classes –Misses diversification effects –Ignores internal risk management methods

Internal Models Approach 1995 Radical Change Core component (VaR) –10 trading day VaR –99 percent confidence –Max ( last 60 days VaR, today’s VaR) –Use at least 1 year of historical data –Scale factor (3 or more) –Plus factor if bank’s numbers look unreliable

Scale Adjustment Find 99% quantile return for 10 day period R* Adjust this by a factor of 3 3*R* Why 3? –Trying to eliminate failures

An Example Using the Delta- Normal Approximation Estimate distribution of 1 day returns –Normal, mean = 0, std = 0.01 Find the 10 day std. – sqrt(10)*0.01 = –Mean = 0*10 = 0 Get the 99% return level from tables –2.33*0.032 = 0.075

An Example Using the Delta- Normal Approximation Get the 99% return level from tables –2.33*0.032 = Critical R* = (k)*0.075 = (3)* = % loss Basel requires cushion for $100 portfolio -> Capital required = $22.5 All is standard VaR except for k

Outline Computing VaR Interpreting VaR Time Scaling Regulation and VaR –Jorion 3, Estimation errors

Estimation Errors Value at Risk is only an estimate What are its “confidence bands”? Methods –Analytics (Jorion 5) –Monte-carlo

Precision of Mean and Std Estimators (Jorion page 123)

Quantile Std. Errors

Normal Quantile Estimates 99%95% Exact quantile Sample (T=250)[1.85, 2.80][1.38, 1.91] Use std and table [2.24, 2.42][1.50, 1.78]

Precision Note: mean more precise than std Can use as input into VaR estimates to get confidence bounds We wont do this. Monte-carlo methods mcdow2.m

Outline Computing VaR Interpreting VaR Time Scaling Regulation and VaR –Jorion 3, Estimation errors