11

Value at Risk Value at Risk

Chapter 20Chapter 20

22

The Question Being Asked The Question Being Asked in VaRin VaR

““What loss level is such that we are What loss level is such that we are XX% confident it will not be exceeded in % confident it will not be exceeded in NN business days?” business days?”

33

VaR and Regulatory CapitalVaR and Regulatory Capital

Regulators base the capital they Regulators base the capital they require banks to keep on VaRrequire banks to keep on VaR

The market-risk capital is The market-risk capital is kk times the times the 10-day 99% VaR where 10-day 99% VaR where kk is at least is at least 3.03.0

44

VaR vs. C-VaR VaR vs. C-VaR

VaR is the loss level that will not be VaR is the loss level that will not be exceeded with a specified probabilityexceeded with a specified probability

C-VaR is the expected loss given that C-VaR is the expected loss given that the loss is greater than the VaR levelthe loss is greater than the VaR level

Although C-VaR is theoretically more Although C-VaR is theoretically more appealing, it is not widely used appealing, it is not widely used

55

Advantages of VaRAdvantages of VaR

It captures an important aspect of It captures an important aspect of riskrisk

in a single numberin a single number It is easy to understandIt is easy to understand It asks the simple question: “How It asks the simple question: “How

bad can things get?” bad can things get?”

66

Time HorizonTime Horizon

Instead of calculating the 10-day, 99% VaR Instead of calculating the 10-day, 99% VaR directly analysts usually calculate a 1-day directly analysts usually calculate a 1-day 99% VaR and assume99% VaR and assume

This is exactly true when portfolio changes This is exactly true when portfolio changes on successive days come from on successive days come from independent identically distributed normal independent identically distributed normal distributionsdistributions

day VaR1-day VaR-10 10

77

The Model-Building The Model-Building ApproachApproach

The main alternative to historical The main alternative to historical simulation is to make assumptions about simulation is to make assumptions about the probability distributions of return on the probability distributions of return on the market variables and calculate the the market variables and calculate the probability distribution of the change in probability distribution of the change in the value of the portfolio analyticallythe value of the portfolio analytically

This is known as the model building This is known as the model building approach or the variance-covariance approach or the variance-covariance approachapproach

88

Daily VolatilitiesDaily Volatilities

In option pricing we express volatility In option pricing we express volatility as volatility per yearas volatility per year

In VaR calculations we express In VaR calculations we express volatility as volatility per dayvolatility as volatility per day

252year

day

99

Daily Volatility continuedDaily Volatility continued

Strictly speaking we should define Strictly speaking we should define dayday as the standard deviation of the as the standard deviation of the continuously compounded return in continuously compounded return in one dayone day

In practice we assume that it is the In practice we assume that it is the standard deviation of the percentage standard deviation of the percentage change in one daychange in one day

1010

Microsoft ExampleMicrosoft Example

We have a position worth $10 million We have a position worth $10 million in Microsoft sharesin Microsoft shares

The volatility of Microsoft is 2% per The volatility of Microsoft is 2% per day (about 32% per year)day (about 32% per year)

We use We use NN=10 and =10 and XX=99=99

1111

Microsoft Example Microsoft Example continuedcontinued

The standard deviation of the change The standard deviation of the change in the portfolio in 1 day is $200,000in the portfolio in 1 day is $200,000

The standard deviation of the change The standard deviation of the change in 10 days is in 10 days is

200 000 10 456, $632,

1212

Microsoft Example Microsoft Example continuedcontinued

We assume that the expected We assume that the expected change in the value of the portfolio is change in the value of the portfolio is zero (This is OK for short time zero (This is OK for short time periods)periods)

We assume that the change in the We assume that the change in the value of the portfolio is normally value of the portfolio is normally distributeddistributed

Since Since NN((––2.33)=2.33)=0.010.01, the VaR is , the VaR is 2 33 632 456 473 621. , $1, ,

1313

AT&T ExampleAT&T Example

Consider a position of $5 million in Consider a position of $5 million in AT&TAT&T

The daily volatility of AT&T is 1% The daily volatility of AT&T is 1% (approx 16% per year)(approx 16% per year)

The S.D per 10 days isThe S.D per 10 days is

The VaR isThe VaR is

50 000 10 144, $158,

158 114 2 33 405, . $368,

1414

PortfolioPortfolio

Now consider a portfolio consisting of Now consider a portfolio consisting of both Microsoft and AT&Tboth Microsoft and AT&T

Suppose that the correlation Suppose that the correlation between the returns is 0.3between the returns is 0.3

1515

S.D. of PortfolioS.D. of Portfolio

A standard result in statistics states thatA standard result in statistics states that

In this case In this case XX = 200,000 and = 200,000 andYY = = 50,000 and 50,000 and = 0.3. The standard = 0.3. The standard deviation of the change in the portfolio deviation of the change in the portfolio value in one day is therefore 220,227value in one day is therefore 220,227

YXYXYX 222

1616

VaR for PortfolioVaR for Portfolio

The 10-day 99% VaR for the portfolio The 10-day 99% VaR for the portfolio isis

The benefits of diversification areThe benefits of diversification are

(1,473,621+368,405)(1,473,621+368,405)––1,622,657=$219,3691,622,657=$219,369

657,622,1$33.210220,227

1717

The Linear ModelThe Linear Model

We assumeWe assume The daily change in the value of a The daily change in the value of a

portfolio is linearly related to the portfolio is linearly related to the daily returns from market variablesdaily returns from market variables

The returns from the market The returns from the market variables are normally distributedvariables are normally distributed

1818

When Linear Model Can be When Linear Model Can be UsedUsed

Portfolio of stocksPortfolio of stocks Portfolio of bondsPortfolio of bonds Forward contract on foreign currencyForward contract on foreign currency Interest-rate swapInterest-rate swap

1919

The Linear Model and The Linear Model and OptionsOptions

Consider a portfolio of options Consider a portfolio of options dependent on a single stock price, dependent on a single stock price, SS. . DefineDefine

andand

S

P

S

Sx

2020

Linear Model and Options Linear Model and Options continuedcontinued

As an approximationAs an approximation

Similar when there are many underlying Similar when there are many underlying market variablesmarket variables

where where ii is the delta of the portfolio with is the delta of the portfolio with respect to the respect to the iith assetth asset

xSSP

i

iii xSP

2121

ExampleExample Consider an investment in options on Consider an investment in options on

Microsoft and AT&T. Suppose the stock prices Microsoft and AT&T. Suppose the stock prices are 120 and 30 respectively and the deltas of are 120 and 30 respectively and the deltas of the portfolio with respect to the two stock the portfolio with respect to the two stock prices are 1,000 and 20,000 respectivelyprices are 1,000 and 20,000 respectively

As an approximationAs an approximation

where where xx11 and and xx22 are the percentage changes are the percentage changes in the two stock pricesin the two stock prices

21 000,2030000,1120 xxP

2222

ExampleExample Assume that the daily volatilities of Microsoft Assume that the daily volatilities of Microsoft

and ATT are 2% and 1% as before, and the and ATT are 2% and 1% as before, and the correlation is 0.3. The STD of correlation is 0.3. The STD of P (in thousands P (in thousands of $) isof $) is

Because N(-1.65)=0.05, the 5-day 95% VAR isBecause N(-1.65)=0.05, the 5-day 95% VAR is

099.73.0*01.0*600*02.0*120*2)01.0*600()02.0*120( 22

193,26$099.7565.1