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

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

VaR Roadmap Introduction Methods –Reading: Linsmeier and Pearson Easy example Harder example: –Linsmeir and Pearson Monte-carlo methods and even harder examples –Jorion

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

Why VaR? Risk summary number –Relatively simple –Relatively standardized Give high level management risk in 1 number

What is VaR? Would like to know maximum amount you stand to lose in portfolio However, the max might too large 5% VaR is the amount that you would lose such that 5% of outcomes will lose more

5% VaR = 84: 100 Start Value Normal Distribution (std = 10)

Value at Risk: Methods Methods (Reading: Linsmeier and Pearson) –Historical –Delta Normal –Monte-carlo –Resampling

Historical 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 Kaplan: Appendix C 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: histdvar.m

Value at Risk: Methods Methods (Reading: Linsmeier and Pearson) –Historical –Delta Normal –Monte-carlo –Resampling

Delta Normal Make key assumptions to get analytics –Normality –Linearization Dow example: –Assume returns normal mean = m, std = s 5% return = -1.64*s + m 1% return = -2.32*s + m –Use these returns to find VaR –matlab: dnormdvar.m

Compare With Historical Fatter tails Plot Comparison: twodowh.m

Longer Horizon: 10 Days Matlab: hist10d.m

Value at Risk: Methods Methods (Reading: Linsmeier and Pearson) –Historical –Delta Normal –Monte-carlo –Resampling

Monte-Carlo VaR Make assumptions about distributions Simulate random variables matlab: mcdow.m Results similar to delta normal Why? –More complicated portfolios and risk measures –Confidence intervals: mcdow2.m

Value at Risk: Methods Methods (Reading: Linsmeier and Pearson) –Historical –Delta Normal –Monte-carlo –Resampling

Resampling (bootstrapping) Historical/Monte-carlo hybrid –Also known as bootstrapping We’ve done this already –data = [ ]; –sample(n,data); Example –rsdow.m

VaR Roadmap Introduction Methods –Reading: Linsmeier and Pearson Easy example Harder example: –Linsmeir and Pearson Monte-carlo methods and even harder examples –Jorion

Harder Example Foreign currency forward contract 91 day forward 91 days in the future –Firm receives 10 million BP (British Pounds) –Delivers 15 million US $

Mark to Market Value (values in millions)

Risk Factors Exchange rate ($/BP) r(BP): British interest rate r($): US interest rate Assume: –($/BP) = –r(BP) = 6% per year –r($) = 5.5% per year –Effective interest rate = (days to maturity/360)r

Find the 5%, 1 Day VaR Very easy solution –Assume the interest rates are constant Analyze VaR from changes in the exchange rate price on the portfolio

Mark to Market Value (current value in millions $)

Mark to Market Value (1 day future value) X = % daily change in exchange rate

X = ? Historical Normal Montecarlo Resampled

Historical Data: bpday.dat Columns –1: Matlab date –2: $/BP –3: British interest rate (%/year) –4: U.S. Interest rate (%/year)

BP Forward: Historical Same as for Dow, but trickier valuation Matlab: histbpvar1.m

BP Forward: Monte-Carlo Matlab: mcbpvar1.m

BP Forward: Resampling Matlab: rsbpvar1.m

Harder Problem 3 Risk factors –Exchange rate –British interest rate –U.S. interest rate

3 Risk Factors 1 day ahead value

Daily VaR Assessment Historical Historical VaR Get percentage changes for –$/BP: x –r(BP): y –r($): z Generate histograms matlab: histbpvar2.m

Daily VaR Assessment Resample Historical VaR Get percentage changes for –$/BP: x –r(BP): y –r($): z Resample from these matlab: rsbpvar2.m

Resampling Question: Assume independence? –Resampling technique differs –matlab: rsbpvar2.m

Risk Factors and Multivariate Problems Value = f(x, y, z) Assume random process for x, y, and z Value(t+1) = f(x(t+1), y(t+1), z(t+1))

New Challenges How do x, y, and z impact f()? How do x, y, and z move together? –Covariance?

Delta Normal Issues Life is more difficult for the pure table based delta normal method It is now involves –Assume normal changes in x, y, z –Find linear approximations to f() This involves partial derivatives which are often labeled with the Greek letter “delta” This is where “delta normal” comes from We will not cover this

Monte-carlo Method Don’t need approximations for f() Still need to know properties of x, y, z –Assume joint normal –Need covariance matrix ie var(x), var(y), var(z) and cov(x,y), cov(x,z), cov(y,z) Next section, and Jorion