BEAM046 Lecture06 Value at Risk

8/8/2019 BEAM046 Lecture06 Value at Risk

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BEAM046

FINANCIAL MODELLING

Dr J. Shen

Lecture 6

Risk Measurement



2

Intended learning outcomes

By the end of this lecture students should

• Be familiar with the different types of risk in financial markets

•

Understand the concept of value at risk • Be able to calculate value at risk using the variance-covariance approach

• Be able to calculate value at risk using the historical simulation approach

• Be able to undertake back testing of value at risk models

Reading

• S. Benninga, Financial Modeling , MIT Press, 2008, Chapter 15

• Kevin Dowd, Beyond Value at Risk , Wiley, 1998. Chapters 2, 3, 4 and 5.

• Philippe Jorion, 2006, Value at Risk: The New Benchmark for Managing Financial Risk , McGraw Hill,Chapters 5 and 6.

• http://www.gloriamundi.org



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Introduction

• Risk is inherent in any business activity that involves uncertain cash flows

• There are six main types of risk that firms may face

Business risk : uncertainty over the business environment that is specific to the firmMarket risk : uncertainty over the price of financial assetsCredit risk : uncertainty over the repayment of debtOperational risk : uncertainty over the internal systems of a company and the people who operate themLiquidity risk : uncertainty over the ability to find an immediate market for financial assetsLegal risk : uncertainty over the enforceability of legal contracts

• Ignoring risk can have disastrous consequences (e.g. Barings Bank, Orange County, Metallgesellschaft,Sumitomo Corp, Daiwa Bank, Long Term Capital Management)

• Most businesses engage in risk management

• However, before risks can be managed , they must be measured



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Value at Risk

• The single most important risk measurement tool is Value at Risk , or VaR, which embodies the marketrisk of a portfolio, or indeed of a whole firm, in a single number

•

VaR is defined as the maximum loss on a portfolio that can be expected over a certain time interval witha certain degree of confidence

Example

If a portfolio whose current market value is £10 million has a 10-day 95% VaR of £200,000, then over the next ten days, with 95% certainty, the largest loss that should be expected is £200,000.

• VaR is therefore (the negative of) the appropriate quantile (or percentile) of the distribution of profit andloss (or return) of the portfolio

• VaR is a measure of the maximum loss expected from ‘normal’ market movements

• Central to the definition of value at risk is (a) the holding period and (b) the confidence level

• The units of VaR is the units of the portfolio (e.g. GBP, USD or EUR), but that we could write this as a percentage of the initial portfolio value (i.e. as a return)



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The Background to Value at Risk

• VaR was first used in the 1980’s by large financial firms to measure the risk of their trading portfolios

• In 1993, the ‘group of thirty’ (a group of 30 prominent US financiers, bankers and academics)

commissioned a report that recommended the use of VaR for all banks to measure the risk of their derivatives positions

• VaR gained popularity following the launch by JP Morgan in 1994, of Riskmetrics, an integrated packageto enable companies to measure VaR

• In 1996, the Basle Committee on Banking Supervision required banks to set capital requirementsaccording to ‘internal models’ of market risk, leaving the choice of model to each bank, subject to certainstringent criteria. Under BCBS recommendations, the estimated VaR for a bank translates directly intothe capital requirements that the bank must meet. The BCBS recommends that banks hold capital equal tothree times the 99% 10-day VaR

• Depending on the effectiveness of the VaR model used, the bank may be required to increase their capitalrequirement

• VaR is now widely used not just among banks, but by other financial and non-financial companies



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The Different Approaches to Calculating Value at Risk

The Variance-Covariance Approach

• Assumes a particular distribution for the portfolio return distribution, or for the underlying factors that

drive returns, and uses knowledge of that distribution to compute the appropriate quantile

Historical Simulation

• Uses the historical distribution of portfolio returns in order to compute the appropriate quantile

Monte Carlo Simulation

• Assumes a particular distribution for the underlying factors that drive portfolio returns, and then drawsrandomly from these in order to simulate the distribution of portfolio returns

Extreme Value Theory

• Recognises that the tails of all fat-tailed distributions have a common shape, which is parameterised bythe tail index, and once this is estimated for a given portfolio, extreme quantiles of the portfolio returndistribution can be calculated



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The variance-covariance approach

• Consider a portfolio comprising a 1000 USD investment in the S&P500 index

• Assume that continuously compounded (i.e. log) daily returns on the S&P500 are normally distributed

• We can estimate the mean, standard deviation and other statistics of daily returns using historical data

• We could use simple returns, but log returns are more likely to be normally distributed



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• The VaR of the portfolio is found by first finding the appropriate percentile of the log return series, i.e.the log return VaR, under the assumption that log returns are normally distributed

•

This tells us that there is only a five percentchance that the log return on our portfoliowill be less than –1.34%



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• We could also compute this by noting that the percentile of a normal variable is the same as the percentileof a standard normal variable, transformed by the mean and standard deviation of the variable

)*)((%ln µ σ α +−= cVaR

• where )(α c is the α -percent quantile of the

standard normal distribution, σ is the standard

deviation of returns, µ is the mean return



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• We can then use this log return to calculate the simple return VaR of our portfolio using the followingformula

)1(%

%ln−−=

− VaR

eVaR

• This tells us that there is only a five percentchance that dollar loss on out portfolio will

be more than $13.31, or 1.33% of the valueof the portfolio



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The Variance Covariance Approach

• It is common to assume that the mean return is equalto zero, and so that the log return VaR of the portfoliois given by

σ α *)(%ln cVaR −=

• To compute VaR over longer horizons, we can use theresult that if returns are serially uncorrelated and

homoscedastic, σ σ T T =)( and so when 0=µ ,

)1(*)( VaRT T VaR =



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The Variance Covariance Approach

Advantages of the variance covariance approach

• Easy to implement for simple portfolios

• Does not require much historical data

Disadvantages of the variance covariance approach

• Poor approximation for ‘non-linear’ portfolios

• Relies critically on the distributional assumption of normality



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• The variance covariance approach relies heavily on the distributional assumptions about portfolio returns.An alternative approach is to use historical data to compute the quantile of the distribution of the portfoliovalue

• In particular, historical simulation takes the distribution of actual daily returns over the previous period,applies those returns to the current

portfolio to obtain a simulateddistribution of changes in portfolio value

• We can then use the appropriate quantileof this distribution to estimate value atrisk

• Note that we could use simple returns(because the non-normality of simplereturns is irrelevant with historical

simulation), but to be consistent with the previous example, we use log returns

(-) 95% one-day ln% VaR for S&P500 daily returns

Histogram of Daily S&P500 Returns

0

20

40

60

80

100

120

- 3. 8 0

%

- 3. 2 0

%

- 2. 6 0

%

- 2. 0 0

%

- 1. 4 0

%

- 0. 8 0

%

- 0. 2 0

%

0. 4 0

%

1. 0 0

%

1. 6 0

%

2. 2 0

%

2. 8 0

%

3. 4 0

%



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• In order to compute the 95% one-day VaR of the portfolio, we require the 5

thpercentile of

this empirical log return distribution, whichwe can obtain using Excel’s PERCENTILE function

• This can then be used to compute the dollar VaR of the portfolio, as before

• This compares with $VaR of $13.75 using the

variance-covariance approach





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Back Testing

• Firms have a wide range of models to choose from in their estimation of VaR, and many ways of implementing each of these models

• Inevitably, some models will work much better than others

• If a model systematically understates the true VaR of a firm’s portfolio, then it will hold insufficientcapital to cover unexpected losses

• A critical component of VaR implementation is the back testing of VaR models

• Back testing involves comparing the forecasts of a VaR model with actual returns, ex post

• The problem is that we do not observe VaR, and so a direct comparison is not possible

• Instead, there are a range of indirect measures of performance that can be used to evaluate the performance of a VaR model



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Back Testing

• The simplest measure is the unconditional coverage of the VaR model

• Suppose that we have estimated VaR each day using the previous1 year (251 days) of data

• Every day, we have a 95% VaR estimate and an actual return; If the VaR model is accurate, the actual return should exceed theestimated VaR on only 5% (i.e. one minus 95%) of days

• We can create a new series, which is a ‘1’if there is a VaR exception, and a ‘0’ if there is not



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Back Testing

• We can use this series to calculate both the number of exceptions and the percentage of exceptions

• To test whether the unconditional coverage of a VaR model is equal to the nominal confidence level, we can usean LR test

• If p is the nominal significance level (one minus the nominal confidence level), N is the number of exceptions and T is the total number of observations

( ) ( ) N N T U pT N pT N LR /)/()1/()/1(ln2

−

−−=



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Back Testing

• Under the null hypothesis of correct unconditional coverage, the LR statistic has a chi-squareddistribution with one degree of freedom; the critical values for the LR statistic can be obtained from excelusing the CHIINV function

• In this case, the LR statistic is lessthan the critical value, and so we cannot reject at the 5% significancelevel the hypothesis that the VCVmodel has correct unconditionalcoverage

• In addition to the measures of unconditional coverage andindependence, there are many other measures that can be used to evaluatea VaR model

• Some of these measures evaluate the conditional coverage of the VaR model, in other words whether VaR exceptions are forecastable. Others evaluate the forecast of the entire density of portfolio returns,rather than just a single quantile



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Back Testing

• Back testing results are used by the BCBS in conjunction with the bank’s estimate of VaR in order to setthe bank’s capital requirements

• The benchmark capital requirement is three times the estimated 99% 10-day VaR, but a ‘penalty’ capitalrequirement that depends on its unconditional coverage

• Using a back testing sample of 250 days, the capital requirement for a bank is increased from three to upto four times the estimated VaR depending on the number of exceptions recorded

Exceptions Capital requirement factor

0 to 4 3.005 3.406 3.507 3.658 3.75

9 3.8510 or more 4.00



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Volatility Clustering

• A particular feature of virtually all short horizon financial asset returns is that they display volatility

clustering: large returns (of either sign) tend to be followed by more large returns; small returns (of either sign) tend to be followed by more small returns

• Allowing for volatility clustering should improve VaR estimates, since if we know we are currently in ahigh volatility period, our VaR estimates should be correspondingly higher

• In order to incorporate volatility clustering into the calculation of VaR, we need a model of time-varying,or conditional, volatility

• The simplest model of time-varying volatility is a rolling window estimator; more sophisticated modelsinclude EWMA and GARCH



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Incorporating Volatility Clustering into VaR estimates

• To incorporate volatility clustering into the variance-covariance approach involves replacing theunconditional estimate of volatility with the conditional volatility from the model

t t cVaR σ α *)(%ln −=

• To incorporate volatility clustering into historical simulation, we compute the empirical quantile of thehistorical standardised distribution (i.e. the distribution of returns scaled by each day’s estimated standarddeviation from the model) and then scale this up by the model’s forecast of tomorrow’s standarddeviation

• To incorporate volatility clustering into Monte Carlo simulation, we simply draw randomly from a pre-specified distribution with an estimate of the conditional volatility from the model



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Practical exercise 6

You are a risk management analyst for an investment bank. Your manager wants you to compute the 60%,90% and 99% VaR for a position in 50,000 December 2009 call options on Apple, with an exercise price of 185.00. The VaR horizon is the maturity of the option contract. She would also like you to compute the VaR for a portfolio of the underlying shares.

You decide to estimate VaR using Monte Carlo simulation. Monte Carlo simulation proceeds by generatingsimulated price paths for the underlying asset, valuing the option portfolio at the VaR horizon for each of these simulated price paths, and using the distribution of portfolio values across all the simulated price pathsin order to estimate the portfolio VaR.

1. Obtain the option price, associated underlying share price and time to expiry from www.cboe.com.

2. Obtain daily data for the underlying share from finance.yahoo.com. Use this to calculate daily log returnsand estimate daily volatility.

3. To simulate the stock price on day t , t S , you decide to use the following process

t t t S S σε +=−1lnln



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where σ is the daily volatility of log returns and ε is a standard normal variable (i.e. a normallydistributed variable with a zero mean and unit variance). Note that we are assuming that the mean logreturn is equal to zero as we are considering only a very short horizon.

We can therefore simulate the stock price by generating a series of standard normal variables, andapplying the above formula. Use the random number generator in Excel (Data/Data Analysis/Random

Number Generation) to generate 1000 realizations of a standard normal random number for each day.

4. Using the simulated values of ε , the initial stock price, 0S , and your estimate of daily volatility, σ ,

compute the series for t S ln for each realisation. Compute also the series t S . Plot all the simulated series t S .

5. Using the simulated value of T S (the price of the share at the expiry date of the option), compute the value

of the option, T C at expiry.

6. Using the simulated values of T S and T C , compute the simple return on the share portfolio and the option

portfolio for each realisation. Use these returns to compute the VaR for the share portfolio and the option portfolio. Plot the empirical distributions of the share and option returns.

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BEAM046 Lecture06 Value at Risk