ch 20 hull fundamentals 7 the d

7/28/2019 Ch 20 Hull Fundamentals 7 the d

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Fundamentals of Futures and Options Markets, 7th Ed, Ch 20, Copyright John C. Hull 2010

Value at Risk

Chapter 20

1


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Objective of VaR

VaR provides a single number thatsummarizes the total risk in a portfolio offinancial assets

It is easy to understand

It asks the simple question: How bad can

things get?

More specifically: What loss level is such that we areX%

confident it will not be exceeded inNbusinessdays?

2Fundamentals of Futures and Options Markets, 7th Ed, Ch 20, Copyright John C. Hull 2010


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VaR and Regulatory Capital

Example of VaR in practice:

Regulators base the capital they require

banks to keep on VaR The market-risk capital is ktimes the 10-day

99% VaR where kis at least 3.0

3


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VaR

VaR is the loss corresponding to the

(100 x) percentile of the distribution of

the change in the value of a portfolio overN days

If x = 99, VaR is the first percentile of the

distribution If x = 97, VaR is the third percentile of the

distribution



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N-Day VaR

Although the theoretical VaR is based on N

number of days, practitioners usually set N=1and then find longer periods as follows:

N-day VaR = 1-day VaR x N



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Methods for Calculating VaR

Historical Simulation Approach

Model Building Approach



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Historical Simulation

Use past data as a guide as to what mighthappen in the future.

1. Identify the market variables (assets) that affectthe value of the portfolio

2. Create a database of the daily movements in allmarket variables overn days

3. Define each days change in the variables asscenarios:

1. Scenario 1 = Day 1 percent change in each variable

2. Scenario 2 = Day 2 percent change in each variable

3. Etc.



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4. The first scenario trial assumes that thepercentage changes in all market variables areas on the first day. The second scenario trial

assumes that the percentage changes in allmarket variables are as on the second day,and so on. The portfolios value tomorrow iscalculated for each scenario trial

5. For each scenario trial, calculate the dollarchange in portfolio value

6. Rank the outcomes from highest loss to lowestloss.



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7. The X percentile of the distribution is the worst(n*X) observations

8. The X estimate of VaR at the (1-X)% confidencelevel is the loss that occurs at the (n*X)thobservation

9. Update the VaR estimate using a rolling windowofn observations. i.e. Drop Day 1 and add Day

n+1. Drop Day 2 and add Day n+2. Etc.If the last n days is a good representation of whatmight happen in the future, then the manager is (1X)% certain that a loss greater than VaR will not

occur. 9Fundamentals of Futures and Options Markets, 7th Ed, Ch 20, Copyright John C. Hull 2010


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Historical Simulation continued

Suppose we use 501 days of historical data

Let vibe the value of a market variable on day i

There are 500 simulation trials

The ith trial assumes that the value of the

market variable tomorrow is

1

500

i

i

v

vv

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Fundamentals of Futures and Options Markets, 7th Ed, Ch 20, Copyright John C. Hull 2010 11

Sample Data

Day Market Market MarketVariable 1 Variable 2 .. . Variable n

0 20.33 0.1132 .. . 65.37

1 20.78 0.1159 .. . 64.91

2 21.44 0.1162.. .

65.023 20.97 0.1184 .. . 64.90

.. .. .. .. . ..

.. .. .. .. . ..498 25.72 0.1312 .. . 62.22

499 25.75 0.1323 .. . 61.99500 25.85 0.1343 .. . 62.1


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Results for Sample Data

Scenario Market Market Market PortfolioNumber Variable 1 Variable 2 .. .. Variable n Value ($mill)

1 26.42 0.1375 .. .. 61.66 23.71

2 26.67 0.1346.. ..

62.21 23.123 25.28 0.1368 .. .. 61.99 22.94.. .. .. .. .. .. .... .. .. .. .. .. ..

499 25.88 0.1354 .. .. 61.87 23.63500 25.95 0.1363 .. .. 62.21 22.87


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The Model-Building Approach

The main alternative to historical simulation is tomake assumptions about the probabilitydistributions of return on the market variables

and calculate the probability distribution of thechange in the value of the portfolio analytically

This is known as the model building approach orthe variance-covariance approach

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Daily Volatilities

Since we are working with variances, weneed to define the appropriate volatilityestimate.

In option pricing we express volatility asvolatility per year

In VaR calculations we express volatility

as volatility per day

day

year

252



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Daily Volatility continued

Strictly speaking we should define dayasthe standard deviation of the continuouslycompounded return in one day

In practice we assume that it is thestandard deviation of the percentagechange in one day

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Microsoft Example

One Asset Case

We have a position worth $10 million inMicrosoft shares

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

The standard deviation of the change in

the portfolio in 1 day is $10 Mill x 2% =$200,000

We useN= 10 andX= 99

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Microsoft Example

Over a one-day period, the expectedchange in the value of Microsoft is zero.(This is OK for short time periods)

We assume that the change in the value ofthe portfolio is normally distributed

Review normal distributions using theTable on p. 590-591.



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Microsoft Example

We want to compute a 10-day 99% VaR.

UseN= 10 andX= 99

We need 0.01 in the left tail and 0.99 in the rest

of the distribution. How many standarddeviations is this?

Table Entries

.00 .01 .02 .03 .04

-2.3 .0107 .0104 .0102 .0099 .0096



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Microsoft Example continued

The 1-day 99% VaR for the $10 millionposition is

The 10-day 99% VaR is

Were 99% sure that we wont lose over$1,473,621 over the next 10 days



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AT&T Example

Find 10-day 99% VaR

Consider a position of $5 million in AT&T

The daily volatility of AT&T is 1% (approx16% per year)

The Std. Dev. for 1 day is

The S.D per 10 days is

The VaR is

20


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Portfolio

Now consider a portfolio consisting of bothMicrosoft and AT&T

Suppose that the correlation between thereturns is 0.3

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S.D. of Portfolio

A standard result in statistics states that

Where X= 200,000, Y= 50,000, and r =0.3.

The standard deviation of the change inthe portfolio value in one day is:

YXYXYX

r

222

22


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VaR for Portfolio

The 10-day 99% VaR for the portfolio is

The individual VaRs were:

Microsoft $1,473,621

AT&T $ 368,405

So, the benefits of diversification are

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

657,622,1$33.2 10xx$220,227



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Diversification Benefits

The diversification benefit is due to acorrelation coefficient less than +1. This isthe same result that you were introducedto in your investments classes. Assecurities are added to a portfolio, the riskdeclines.



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Generalizing The Linear Model

We assume

The daily change in the value of a portfolio

is linearly related to the daily returns frommarket variables

The returns from the market variables are

normally distributed

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Markowitz Result for Variance of

Return on Portfolio


sinstrumentthandthofreturnsbetweenncorrelatiois

portfolioininstrumentthonreturnofvarianceis

portfolioininstrumentthofweightis

ReturnPortfolioofVariance

ij

2i

j

i

i

iw

ww

i

n

i

n

j

jijiij

r

r 1 1

26


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VaR Result for Variance of

Portfolio Value (ai= wiP)

daypervalueportfoliotheinchangetheofSDtheis

return)dailyofSD(i.e.,instrumentthofvolatilitydailytheis

P

i

n

i

jiji

ji

ijiiP

n

i

n

jjijiijP

n

i

ii

i

xP

aara

aar

a

1

222

1 1

2

1

2

27


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Microsoft and AT&T Example

p2 = 1

212 + 2

222 + 2[1,21212]

The 10-day 99% VaR is:



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Diversification Benefits

The diversification benefit is due to acorrelation coefficient less than +1. This isthe same result that you were introducedto in your investments classes. Assecurities are added to a portfolio, the riskdeclines.



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Handling Interest Rates

Bond Portfolios

We do not want to define every bond as adifferent market variable

We therefore choose as assets zero-coupon

bonds with standard maturities: 1-month, 3months, 1 year, 2 years, 5 years, 7 years, 10years, and 30 years

Cash flows from instruments in the portfolio aremapped to bonds with the standard maturities

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Cash Flow Mapping

Assume a portfolio containing two bonds:

Bond #1 Bond #2

1.2 Yr Maturity 0.8 Yr Maturity

Semi-annual pmts Semi-annual pmts

Cash Flows at: Cash Flows at:

0.2 Yrs (2.4 months) 0.3 Yrs (3.6 months)

0.7 Yrs (8.4 months) 0.8 Yrs (9.6 months)

1.2 Yrs (14.4 months)



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Cash Flow Mapping

Bond 1 CFs 1 Month Bond 2 CFs

2.4 months 3 Months 3.6 months

8.4 months 6 Months 9.6 months

14.4 months 1 Year2 Years

5 Years

7 Years10 Years

30 Years



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Cash Flow Mapping

The values assigned to each standardcategory are based upon interpolations ofthe actual interest and principal payments.

For a change in market interest rates, theresulting change in bond values iscomputed for the 9 categories shown.

Only 9 standard deviations are requiredwith cash flow mapping.



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Cash Flow Mapping

Correlation Coefficients

1 Mo 3 Mo 6 Mo 1 Yr 2 Yr 5 Yr 7 Yr 10 Yr 30 Yr

1 Mo 1m,1m 1m,3m 1m,6m 1m,1y 1m,2y 1m,5y 1m,7y 1m,10y 1m,30y

3 Mo 3m,3m 3m,6m 3m,1y 3m,2y 3m,5y 3m,7y 3m,10y 3m,30y

6 Mo 6m,6m 6m,1y 6m,2y 6m,5y 6m,7y 6m,10y 6m,30y

1 Yr 1y,1y 1y,2y 1y,5y 1y,7y 1y,10y 1y,30y

2 Yr 2y,2y 2y,5y 2y,7y 2y,10y 2y,30y

5 Yr 5y,5y 5y,7y 5y,10y 5y,30y

7 Yr 7y,7y 7y,10y 7y,30y

10 Yr 10y,10y 10y,30y

30 Yr 30y,30y



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When Linear Model Can be Used

Portfolio of stocks

Portfolio of bonds

Cash flow mapping Forward contract on foreign currency

Value as bonds and use cash flow mapping

Interest-rate swap Value as bonds and use cash flow mapping

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The Linear Model is Only an

Approximation for Options

TdT

TrKSd

T

TrKSdwhere

dNSdNeKp

dNeKdNSc

rT

rT

10

2

01

102

210

)2/2()/ln(

)2/2()/ln(

)()(

)()(



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The Greek Letters

Delta is the rate of change of the option valuewith respect to the price

Gamma is the rate of change of delta with

respect to the price of the underlying asset Theta is the rate of change of the option value

with respect to the passage of time

Vega is the rate of change of the option valuewith respect to volatility

Rho is the rate of change of the option valuewith respect to the interest rate



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Delta ()

Option value function is not linear.

Option

price

AB

Slope =

Stock price


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The Linear Model and Options

To calculate the VaR of an option portfoliousing the linear model, we need the standarddeviation of the price changes. We obtain

the by utilizing the delta that is defined inChapter 17

Consider a portfolio of options dependent on

a single stock price, S. Define and

S

P

S

Sx



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Linear Model and Options

continued (equations 20.3 and 20.4)

As an approximation

For a portfolio of many underlying marketvariables

where i is the delta of the portfolio withrespect to the ith asset

xSSP

i

iii xSP

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Example

Consider an investment in options on Microsoft andAT&T.

MSFT AT&T

S = $120 $30

= 1,000 20,000

= 2% 1%

= 0.3

As an approximation

where x1 and x2 are the percentage changes inthe two stock prices

21 000,2030000,1120 xxP



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Example

Using the equation for variance and substituting 1 forx1 and 2 forx2, the variance of the portfolio (inthousands) is:

2

port = 1202

(.02)2

+ 6002

(.01)2

+ 2(.3)(120)(600)(.02)(01)2port = 50.4 and port = 7.09929574

Find the 5-day 95% VaR of the portfolio.

Determine the number of standard deviations using

the normal distribution table. Value of 0.05 fallsbetween 1.64 and 1.65. Using 1.65 results in:

VaR = 1.65 x 7.09929574 x = $26.193 (000)5


B t th di t ib ti f th d il


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But the distribution of the daily

return on an option is not normal

(See Figure 20.4, page 444)


Positive Gamma Negative Gamma

43

Sk


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Skewness

For options, the linear model is an approximation.It fails to capture the skewness in the probabilitydistribution of the portfolio value.

As chapter 17 points out, the delta is computed ata single point. Gamma provides a measure ofcurvature and thus, the degree of skewness.

The VaR is dependent on the left tail of the

probability distribution. When skewness occurs,there is an error in the VaR estimate.



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Skewness

When Gamma is positive, the left tail ofthe distribution is less heavy than normal,and the VaR estimate is too high.

When Gamma is negative, the left tail ofthe distribution is more heavy than normal,and the VaR estimate is too low.

How is the error corrected? By using aquadratic equation (rather than linear)



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Quadratic Model

to correct skewness error

For a portfolio dependent on a singlestock price

where g is the gamma of the portfolio.This becomes

2)(

2

1SSP g

22 )(21 xSxSP g

46


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Use of Quadratic Model

Analytic results are not as readily available

Monte Carlo simulation can be used in

conjunction with the quadratic model (Thisavoids the need to revalue the portfolio foreach simulation trial)

The quadratic model is also sometimesused in conjunction with historicalsimulation



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Estimating Volatility for Model

Building Approach (equation 20.6)

Define n as the volatility per day on day n, asestimated at end of dayn-1

Define Si as the value of market variable at end ofday i

Define ui= ln(Si/Si-1)

The usual estimate of volatility from m observationsis:

m

i

in

m

i

inn

um

u

uum

1

1

22

1

)(1

1

48


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Simplifications(equations 20.7 and 20.8)

Defineui as (SiSi-1)/Si-1Assume that the mean value ofui is zero

Replace m1 by m

This gives

n n iim

mu2 2

1

1

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Weighting Scheme

Instead of assigning equal weights to theobservations we can set

a

a

n i n ii

m

ii

m

u2 21

1

1

where

50


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EWMA Model (equation 20.10)

In an exponentially weighted movingaverage model, the weights assigned tothe u2 decline exponentially as we moveback through time

This leads to

21

21

2 )1( nnn u

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Attractions of EWMA

Relatively little data needs to be stored

We need only remember the currentestimate of the variance rate and the most

recent observation on the market variable Tracks volatility changes through time

The same method can be used to find

correlations between market variableswhen computing VaR



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Correlations

Define ui=(Ui-Ui-1)/Ui-1 and vi=(Vi-Vi-1)/Vi-1 Also

u,n: daily vol ofUcalculated on day n-1v,n: daily vol ofVcalculated on day n-1

covn: covariance calculated on day n-1

covn= rnu,nv,nwhere rnis the correlation between Uand V

53

Correlations continued


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Correlations continued

(equation 20.12)

Using the EWMA

covn= covn-1+(1-)un-1vn-1

RiskMetrics uses = 0.94 for dailyvolatility forecasting

54


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Comparison of Approaches

Advantages Disadvantages

Model

Building

Results producedquickly

Accommodatesvolatility estimatingapproach like EWMA

Assumes multivariatenormal distributions

Gives poor results forlow delta portfolios

Historical

Simulation

Determines the jointprobabilitydistributions of thevariables

Avoids the need forcash flow mapping

Computationally slow

Doesnt easily allow

volatility updatingschemes such as EWMA


Back Testing


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Back-Testing

Tests how well VaR estimates wouldhave performed in the past

We could ask the question: How oftenwas the loss greater than the VaRlevel

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Stress Testing


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F d t l f F t d O ti M k t 7th Ed Ch 20 C i ht J h C H ll 2010

Stress Testing

This involves testing how well aportfolio would perform under someof the most extreme market moves

seen in the last 10 to 20 years

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ch 20 hull fundamentals 7 the d

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