introduction to value at risk

INTRODUCTION TO

VALUE AT RISK (VaR)

ALAN ANDERSON, Ph.D.

ECI Risk Training

www.ecirisktraining.com

(c) ECI Risk Trainingwww.ecirisktraining.com 2

Value at Risk (VaR) is a statisticaltechnique designed to measure themaximum loss that a portfolio of assetscould suffer over a given time horizonwith a specified level of confidence

Value at Risk was originally used tomeasure market risk

It has since been extended to othertypes of risk, such as credit risk andoperational risk

EXAMPLE

Suppose that it is determined that a$100 million portfolio could potentiallylose $20 million (or more) once every20 trading days

The VaR of this portfolio equals $20million with a 95% level of confidenceover the coming trading day; 19 out of20 trading days (95% of the time),losses are less than $20 million

At the 95% confidence level, VaR representsthe border of the 5% “left tail” of the normaldistribution, also known as the fifth percentileor .05 quantile of the normal distribution

This diagram shows that:

95% of the time, the portfolio’svalue remains above $80 million

5% of the time, the portfolio’svalue falls to $80 million or less

The VaR of this portfolio is therefore

$100 million - $80 million = $20 million

VaR is based on the assumption that therates of return of the assets held in aportfolio are jointly normally distributed

VaR has the advantage that the risksof different assets can be combined toproduce a single number that reflectsthe risk of a portfolio

Further, the probability of a givenloss can be calculated using VaR

VaR can also be used to determinethe impact on risk of changes in aportfolio’s composition

VaR has the disadvantage that itis computationally intensive andrequires major adjustments fornon-linear assets, such as options

COMPUTING VaR

Value-at-Risk is based on the work ofHarry Markowitz, who was awardedthe Nobel Prize in Economics in 1990for his pioneering research in the areaof portfolio theory

Portfolio theory shows howrisk can be reduced by holdinga well-diversified set of assets

A collection of assets is considered to be well-diversified if the assets are affected differentlyby changes in economic variables, such asinterest rates, exchange rates, etc.

As a result, a well-diversified portfolio isless likely to experience extreme changesin value; in this way, risk is reduced

In statistical terms, a well-diversified portfoliocontains assets whose rates of return havevery low or negative correlations with eachother

EXAMPLE

A portfolio consisting exclusively of oilstocks would not be well-diversified, sincechanges in the price of oil would have ahuge impact on the portfolio’s value

A portfolio invested in both oil stocksand automotive stocks would be farmore diversified:

Rising oil prices would hurt the automotivestocks while helping the oil stocks

Falling oil prices would hurt the oil stockswhile helping the automotive stocks

As a result, the impact of oil priceswings would be offset by changes inthe value of the automotive stocks

On balance, risk would be reduced

The risk of holding a portfolio containing twoassets, X and Y, is measured by its standarddeviation, as follows:

2 2 2 22

P X X Y Y X Y X Yw w w w= + +

where:

P = the standard deviationof the returns to the portfolio

X = standard deviation ofthe returns to asset X

Y = standard deviation ofthe returns to asset Y

wX = weight of asset X

wY = weight of asset Y

The weights represent the proportion

of the portfolio invested in each asset;the sum of the weights is one

If short-selling is not possible, then:

0 wX 1

0 wY 1

If short-selling is possible, theweights can be negative

= “rho”

this represents the correlation

between the returns to assetsX and Y; -1 1

The lower is the correlationbetween assets, the lower willbe the risk of the portfolio

The Value at Risk of aportfolio is a function of:

the dollar value of the portfolio

the portfolio standard deviation

the confidence level

the time horizon

COMPUTING VaR FOR

A SINGLE ASSET

For a single asset, using dailyreturns data at a confidence levelof c, the VaR is computed as:

where:

V0 = initial value of the asset

= standard deviation of the asset’s daily returns

= the number of standard deviationsbelow the mean corresponding tothe (1-c) quantile of the standardnormal distribution

EXAMPLE

For a 95% confidence level, c = 0.95

(1-c) is the fifth quantile (1-.95 = .05 =5%) of the standard normal distribution

The corresponding value of is 1.645

The value of corresponding to anyconfidence level can be found with anormal table or with the Excel functionNORMSINV

EXAMPLE

For a 99% confidence level, the valueof can be determined as follows:

c = 0.99

(1-c) = 0.01 = 1%

NORMSINV(0.01) = -2.33

= 2.33

EXAMPLE

Suppose that an investor’s portfolio consistsentirely of $10,000 worth of IBM stock.

Since the portfolio only contains IBM stock,it can be thought of as a single asset

Assume that the standard deviation of thestock’s returns are 0.0189 (1.89%) per day

If the investor wants to know hisportfolio’s VaR over the comingtrading day at the 95% confidencelevel, this would be calculated asfollows:

V0 = (10,000)(1.645)(0.0189)

= $310.905

This means that over the coming day,there is a 5% chance that the investor’slosses could reach $310.905 or more(i.e., the portfolio’s value could fall to$9,689.095 or less)

VaR can be extended to differenttime horizons by applying the square

root of time rule

According to this rule, the standarddeviation increases in proportion tothe square root of time:

t periods = t 1 period

If the investor wants to know hisportfolio’s VaR over the comingmonth at the 95% confidence level,based on the assumption that thereare 22 trading days in a month, thiswould be calculated as follows:

0 (10,000)(1.645)(0.0189 22)V =

(10,000)(1.645)(0.0189 22) $1,458.27=

Similarly, if the investor wants to knowwhat his portfolio’s VaR is over the comingyear, assuming that there are 252 tradingdays in a year, the calculations would be:

0 (10,000)(1.645)(0.0189 252)V =

(10,000)(1.645)(0.0189 252) $4,935.46=

COMPUTING PORTFOLIO VaR

In order to compute the Value atRisk of a portfolio of two or moreassets, the correlations among theassets must be explicitly considered

The lower these correlations, thelower will be the resulting VaR

The Value at Risk of a portfolio

is calculated by determining the:

weight (proportion of the totalinvested) of each asset in theportfolio

standard deviation of each asset’srate of return in the portfolio

correlations among the assets’ ratesof return in the portfolio

Once a confidence level and a timehorizon have been chosen, theweights, volatilities and correlationscan be combined using Markowitz’sapproach to derive the portfolio’s VaR

EXAMPLE

Assume that a $100,000 portfoliocontains $60,000 worth of Stock Xand $40,000 worth of Stock Y.

Given the following data, computethe VaR of this portfolio with a 95%confidence level over the coming:

wX = 0.60 wY = 0.40

X = 0.016284 Y = 0.015380

= -0.19055

= 0.01144627 = 1.144627%

2(0.6)(0.4)( 0.19055)(0.016284)(0.015380)

2 2 2 2(0.6) (0.016284) (0.4) (0.015380)P= + +

The portfolio VaR over the coming day is:

= $1,882.91

0 (100,000)(1.645)(0.01144627)P

The portfolio VaR over the coming month is:

= $8,831.638

0 (100,000)(1.645)(0.01144627 22)P

The portfolio VaR over the coming year is:

= $29,890.29

0 (100,000)(1.645)(0.01144627 252)P

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