Value at Risk

By A V Vedpuriswar

September 15, 2009

What is VAR?

VAR summarizes the worst loss over a target horizon that will not be exceeded at a given level of confidence.

For example, “under normal market conditions, the most the portfolio can lose over a month is about $3.6 billion at the 99% confidence level.”

The main idea behind VAR is to consider the total portfolio risk at the highest level of the institution.

Initially applied to market risk, it is now used to measure credit risk, operational risk and enterprise wide risk.

Many banks can now use their own VAR models as the basis for their required capital for market risk.

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Average revenue = $5.1 million per day

Total no. of observations = 254.

Std dev = $9.2 million

Confidence level = 95%

No. of observations < - $10 million = 11

No. of observations < - $ 9 million = 15

Find VAR.

Illustration

Find the point such that the no. of observations to the left

= (254) (.05) = 12.7

(12.7 – 11) /( 15 – 11 ) = 1.7 / 4 ≈ .4

So required point = - (10 - .4) = - $9.6 million

VAR = E (W) – (-9.6) = 5.1 – (-9.6) = $14.7 million

If we assume a normal distribution,

Z at 95% confidence interval, 1 tailed = 1.645

VAR = (1.645) (9.2) = $ 15.2 million

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Problem% Returns Frequency Cumulative

Frequency- 16 1 1- 14 1 2- 10 1 3- 7 2 5- 5 1 6- 4 3 9- 3 1 10- 1 2 120 3 151 1 162 2 184 1 196 1 207 1 218 1 229 1 23

11 1 2412 1 2614 2 2718 1 2821 1 2923 1 30

What is VAR (90%) ?

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10% of the observations, i.e, (.10) (30)

= 3 lie below -7

So VAR = -7

Solution

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Problem

The VAR on a portfolio using a one day horizon is USD 100 million. What is the VAR using a 10 day horizon ?

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Solution

Variance gets multiplied by 10, std deviation by √10

VAR = 100 √10 = (100) (3.16) = 316

(σN2 = σ1

2 + σ22 ….. = Nσ2)

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Problem

If the daily VAR is $12,500, calculate the weekly, monthly, semi annual and annual VAR. Assume 250 days and 50 weeks per year.

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Solution

Weekly VAR ＝ (12,500) (√5) ＝ 27,951

Monthly VAR ＝ ( 12,500) (√20) ＝ 55,902

Semi annual VAR ＝ (12,500) (√125) ＝ 139,754

Annual VAR ＝ (12,500) (√250) ＝ 197,642

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Problem

Consider a portfolio with a one day VAR of $1 million. Assume that the market is trending with an auto correlation of 0.1. Under this scenario, what would you expect the two day VAR to be?

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Solution

V2 = 2σ2 (1 + ῤ)

= 2 (1)2 (1 + .1) = 2.2

V = √2.2 = 1.4832

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Problem

Based on a 90% confidence level, how many exceptions in back testing a VAR should be expected over a 250 day trading year?

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Solution

10% of the time loss may exceed VAR

So no. of observations = (.10) (250)

= 25

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Problem

Suppose we have a portfolio of $10 million in shares of Microsoft. We want to calculate VAR at 99% confidence interval over a 10 day horizon. The volatility of Microsoft is 2% per day. Calculate VAR.

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Solution

σ = 2% = (.02) (10,000,000) = $200,000

Z (P = .01) = Z (P =.99) = 2.33

Daily VAR = (2.33) (200,000) = $ 466,000

10 day VAR = 466,000 √10 = $ 1,473,621

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Problem

Consider a portfolio of $5 million in AT&T shares with a daily volatility of 1%. Calculate the 99% VAR for 10 day horizon.

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Solution

σ= 1% = (.01) (5,000,000) = $ 50,000

Daily VAR = (2.33) (50,000) = $ 116,500

10 day VAR = $ 111,6500 √10 = $ 368,405

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Problem

Now consider a combined portfolio of AT&T and Microsoft shares. Assume the returns on the two shares have a bivariate normal distribution with the correlation of 0.3. What is the portfolio VAR.?

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Solution

σ2 = w12 σ1

2 + w22 σ2

2 + 2 ῤPw1 W2 σ1 σ2

= (200,000)2 + (50,000)2 + (2) (.3) (200,000) (50,000)

σ = 220,277

Daily VAR = (2.33) (220,277) = 513,129

10 day VAR = (513,129) √10 =$1,622,657

Effect of diversification = (1,473,621 + 368,406) – (1,622,657)

= 219,369

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VAR can be used as a company wide yardstick to compare risks across different markets.

VAR can also be used to understand whether risk has increased over time.

VAR can be used to drill down into risk reports to understand whether the higher risk is due to increased volatility or bigger bets.

VAR as a benchmark measure

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VAR can also give a broad idea of the worst loss an institution can incur.

The choice of time horizon must correspond to the time required for corrective action as losses start to develop.

Corrective action may include reducing the risk profile of the institution or raising new capital.

Banks may use daily VAR because of the liquidity and rapid turnover in their portfolios.

In contrast, pension funds generally invest in less liquid portfolios and adjust their risk exposures only slowly.

So a one month horizon makes more sense.

VAR as a potential loss measure

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VAR can be used to determine the amount of capital needed.

The VAR measure should adequately capture all the risks facing the institution - market risk, credit risk, operational risk and other risks.

The higher the degree of risk aversion of the company, the higher the confidence level chosen.

If the bank is targeting a particular credit rating, the expected default rate can be converted directly into a confidence level.

Higher credit ratings should lead to a higher confidence level.

VAR as equity capital

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Banks can disclose their aggregated risk without revealing their individual positions.

Ideally, institutions should provide summary VAR figures on a daily, weekly or monthly basis.

Disclosure of information is an effective means of market discipline.

VAR as an information reporting tool

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Position limits alone do not give a complete picture.

The same limit on a 30 year treasury, (compared to 5 year treasury) may be more risky.

VAR limits can supplement position limits.

In volatile environments, VAR can be used as the basis for scaling down positions.

VAR acts as a common denominator for comparing various risky activities.

VAR as a risk control tool.

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VAR can be viewed as a measure of risk capital or economic capital required to support a financial activity.

The economic capital is the aggregate capital required as a cushion against unexpected losses.

VAR helps in measuring risk adjusted return.

Without controlling for risk, traders may become reckless.

A trader making a large profit, receives a large bonus.

If the trader makes a loss, the worst that can happen is a small fine.

VAR as a measure of risk adjusted performance

Parametric and non parametric methods

Non parametric method : This is the most general method which does not make any assumption about the shape of the distribution of returns.

Parametric method: VAR computation becomes much easier if a distribution, such as normal, is assumed.

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The historical simulation method consists of going back in time and applying current weights to a time series of historical asset returns.

This method makes no specific assumption about return distribution, other than relying on historical data.

This is an improvement over the normal distribution because historical data typically contain fat tails.

The main drawback of this method is its reliance on a short historical moving window to infer movements in market prices.

Historical simulation method

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The sampling variation of historical simulation VAR is greater than for a parametric method.

Longer sample paths are required to obtain meaningful quantities.

The dilemma is that this may involve observations that are no longer relevant.

Banks use periods between 250 and 750 days.

This is taken as a reasonable trade off between precision and non stationarity.

Many institutions are now using historical simulation over a window of 1-4 years, duly supplemented by stress tests.

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The Monte Carlo Simulation Method is similar to the historical simulation, except that movements in risk factors are generated by drawings from some pre-specified distribution.

The risk manager samples pseudo random numbers from this distribution and then generates pseudo-dollar returns as before.

Finally, the returns are sorted to produce the desired VAR.

This method uses computer simulations to generate random price paths.

Monte Carlo Simulation Method

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Monte Carlo methods are by far the most powerful approach to VAR.

They can account for a wide range of risks including price risk, volatility risk, fat tails and extreme scenarios and complex interactions.

Non linear exposures and complex pricing patterns can also be handled.

Monte Carlo analysis can deal with time decay of options, daily settlements & associated cash flows and the effect of pre specified trading or hedging strategies.

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The Monte Carlo approach requires users to make assumptions about the stochastic process and to understand the sensitivity of the results to these assumptions.

Different random numbers will lead to different results.

A large number of iterations may be needed to converge to a stable VAR measure.

When all the risk factors have a normal distribution and exposures are linear, the method should converge to the VAR produced by the delta-normal VAR.

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The Monte Carlo approach is computationally quite demanding.

It requires marking to market the whole portfolio over a large number of realisations of underlying random variables.

To speed up the process, methods, have been devised to break the link between the number of Monte Carlo draws and the number of times the portfolio is repriced.

In the grid Monte Carlo approach, the portfolio is exactly valued over a limited number of grid points.

For each simulation, the portfolio is valued using a linear interpolation from the exact values at adjoining grid points.

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Backtesting is done to check the accuracy of the model.

It should be done in such a way that the likelihood of catching biases in VAR forecasts is maximized.

Too high a confidence level reduces the expected number of observations in the tail and thus the power of the tests.

There is no simple way to estimate a 99.99% VAR from the sample because it has too few observations.

Shorter time intervals create more data points and facilitate more effective back testing.

Backtesting

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Simulation methods are quite flexible.

They can either postulate a stochastic process (Monte Carlo) or resample from historical data ( Historical)

Monte Carlo methods are prone to model risk and sampling variation.

Greater precision can be achieved by increasing the number of replications but this may slow the process down.

Choosing the method

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If the stochastic process chosen for the price is unrealistic, so will be the estimate of VAR.

For example, the geometric Brownian motion model adequately describes the behaviour of stock prices and exchange rates but not that of fixed income securities.

In Brownian motion models, price shocks are never reversed and prices move as a random walk.

This cannot be the price process for default free bond prices which must converge to their face value at expiration.

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EVT extends the central limit theorem which deals with the distribution of the average of identically and independently distributed variables from an unknown distribution to the

distribution of their tails.

The EVT approach is useful for estimating tail probabilities of extreme events.

For very high confidence levels (>99%), the normal distribution generally underestimates potential losses.

Extreme Value Theory (EVT)

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Empirical distributions suffer from a lack of data in the tails.

This makes it difficult to estimate VAR reliably.

EVT helps us to draw smooth curves through the extreme tails of the distribution based on powerful statistical theory.

In many cases the t distribution with 4-6 degrees of freedom is adequate to describe the tails of financial data.

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Fitting EVT functions to recent historical data is fraught with the same pitfalls as VAR.

Once in a lifetime events cannot be taken into account even by powerful statistical tools.

So they need to be complemented by stress testing.

The goal of stress testing is to identify unusual scenarios that would not occur under standard VAR models.

Stress testing

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The problem with stress testing is the stress needs to be pertinent to the type of risk the institution

has.

Otherwise, the complex portfolio models banks generally employ may give the illusion of accurate simulation at the expense of substance.

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During the credit crisis risk models of many banks were unable to predict the likelihood , speed or severity of the crisis.

There were several exceptions.

Goldman Sachs’ chief financial officer David Viniar once described the credit crunch as “a 25-sigma event”

Why?

There was a major paradigm shift.

How effective are VAR models? VAR and sub prime

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Exceptions

A few VAR exceptions are expected.

A properly working model would still produce two to three exceptions a year.

But – the existence of clusters of exceptions indicated that something was seriously wrong.

Credit Suisse reported 11 exceptions at the 99% confidence level in the third quarter, Lehman brothers three at 95%, Goldman Sachs five at 95%, Morgan Stanley six at 95%, Bear Stearns 10 at 99% and UBS 16 at 99%.

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What window?

VAR models failed especially as the environment was emerging from a period of relatively benign volatility.

The models were clearly reacting not fast enough.

What kind of models would have worked best?

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What models work best? With the benefit of hindsight, the type of VAR

model that would actually have worked best in the second half of 2007 would most likely have been a model driven by a frequently updated short data history.

Or any frequently updated short data history that weights more recent observations more heavily than

more distant observations.

In the wake of the recent credit crisis, there is a strong case for increasing the frequency of updating.

Monthly, quarterly or even weekly updating of the data series would improve the responsiveness of the model to a sudden change of conditions.

Problem on VAR cash flow mapping

Consider a long position in a $1 million Treasury bond.

Maturity : 0.8 years

Coupon : 10% payable semiannually Annualized yield & volatility3 Month 6 Month 1 Year

Annualised yield 5.50 6.00 7.00Volatility 0.06 0.10 0.20

Correlations between daily returns3 Month 6 Month 1 Year

3 month 1.0 0.9 0.66 month 0.9 1.0 0.71 year 0.6 0.7 1.0

Explain how mapping can be done while calculating VaR,

Solution

The current position involves the following:

Cash flow of $50,000 in .3 years

Cash flow of $1,050,000 in .8 years

So the position can be considered a combination of two zero coupon bonds, maturity 0.3, 0.8 years .

Let us write the position as equivalent to a combination of standard 3 month, 6 month and 1 year bonds.

3 month interest rate = 5.50%

6 month interest rate = 6.00%

.3 years = (.3) (12) = 3.6 months.

Solution Cont…

Effective interest rate for 3.6 months zero coupon bond = 5.50 + .6/3(.5) = 5.60%

Present value = = 49,189

Volatility = = .068%.

Let us allocate to a 3 month bond and 1 - of the present value to a 6 month bond.

Then we can write: 2 = 12 + 2

2 + 2 12

Here = .068 1 = .06 2 = .10 = .90

or .0682 = 2 (.06)2+ (1-)2(.10)2 + 2 (.9)() (1-)(.06)(.10)

3.)0560.1(

000,50

)04(.3

6.06.

Solution Cont…

or .0682 = 2 (.06)2 + (1-)2 (.10)2 + 2(.9) ()(1-)(.06)(.10)

Putting = .7603

LHS = .00462

RHS = .00208 + .00057 + .001968

= .00462

So we can write the position as equivalent to

$ (.7603) (49,189) = $37,399 in 3 month bond

$ (.2397) (49,189) = $11,791 in 6 month bond

Solution Cont…

Now consider $1,050,000 received after 0.8 years.

It can be considered a combination of 6 month and 12 month positions.

Interpolating the interest rate we get: =.066

Volatility = [.1 + (3.6/6)(0.1) ] = 0.16

Present value of cash flows = =$997,662

)01(.6

6.306.

8.)066.1(

000,050,1

Let be the position in the 6 month bond and (1-) in the 12 month bond. Then we can write:

2 = 2 12 + (1-)2 2

2 + 2 (1-) 12

Or (.16)2 = 2 (.1)2 + (1-)2 (.2)2 + 2 (.7) () (1-) (.1)(.2)

LHS = .0256 Put = .320337

We get RHS =.001026 + .01848 + .006096

≈ .0256

Solution Cont…

So the position is equivalent to

(.320337) (997,662) = $319,589 in 6 month bond

(.679663) (997,662) = $678,074 in 12 month bond

We can now write the portfolio in terms of 3 month, 6 month, 12 month zero coupon bonds.

$50,000 $1,050,000 Total

t = .3 t = .8

3 month bond 37,399 -- 37,399

6 month bond 11,791 319,589 331,380

12 month bond -- 678,074 678,074

Solution Cont…

Let 1, 2, 3 be the volatilities of the 3 month, 6 months, 12 months bonds and 12, 13, 23 be the respective correlations.

Then 2 = 12 + 2

2 + 32 + 21212 + 22323 +

21313

= [(37,399)2 (.06)2 + (331,380)2 (.10)2 + (678,074)2 (.20)2

+ (2) (37,399) (331,380) (.06) (.10) (.90)

+ (2) (331,380) (678,074) (.10) (.20) (.70)

+ (2) (37,399) (678,074) (.06) (.20) (.60)] x 10-

4

= [5,035,267 + 1,098,127,044 + 18,391,373,980 + 133,847,431+6,291,604,539+365,173,769]x10-4

Solution Cont…

≈ 2,628,516

=

= $1621.3

10 day 99% VAR

= 1621.3 x 10 x 2.33

= $11,946

516,628,2