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, 5.2.5-5.2.6

• 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 ValueNormal 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?

DataDow 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 FilesKaplan: 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 = [5 3 -6 9 0 4 6 ];– 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)

)1($15

)1(10

)$

($rr

BPBP

valBP

Risk Factors

• Exchange rate ($/BP)• r(BP): British interest rate• r($): US interest rate• Assume:

– ($/BP) = 1.5355– r(BP) = 6% per year– r($) = 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 $)

300,331$

)055.0)360/91(1(

$15

)06.0)360/91(1(

10)5355.1(

BPval

Mark to Market Value(1 day future value)

)055.0)360/90(1(

$15

)06.0)360/90(1(

10)5355.1)(1(

BPxval

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 Factors1 day ahead value

)055.0)1)(360/90(1(

$15

)06.0)1)(360/90(1(

10)5355.1)(1(

z

y

BPxval

Daily VaR AssessmentHistorical

• Historical VaR

• Get percentage changes for – $/BP: x– r(BP): y– r($): z

• Generate histograms

• matlab: histbpvar2.m

Daily VaR AssessmentResample

• 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