one report

Australian School of Business Australian School of Business

Business Risk Management Week 5

Measuring Financial Risk - 2

Australian School of BusinessObjectives of this Week’s Session

• A refresher on Statistics • An Introduction to Value-at-Risk (VaR)

–The three common calculation methods –The required assumptions –Approaches to rationalising these assumptions

• Calculating Value-at-Risk (VaR) –The Tail of the Distribution –Confidence Intervals

Australian School of BusinessRecap - The Factors in Measuring Risk

• Possibility or Probability of a risk event • The likely Quantum of the Loss arising from the

risk event • The more data we have the better our estimate

of the amount of risk • The quantum of risk can be thought of as the

value of an option – The key element in the valuation of an option involves assessing

volatility (Dembo)

Australian School of BusinessThe Use of Statistics in Risk Measurement

• Statistics aims to provide some tools to help us make sense of some sample data

• The aim of this is broadly twofold: – To describe the data and its structure – To infer meaning and possible broader implications from the data.

• In the context of risk measurement the sample data is intended to provide insight into the amount of risk that may exist. – In particular the amount of variability (or volatility) that exists in the

data – Statistics can help with this.

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Statistics Refresher

• A distribution is a set of observations that are arranged in order, generally by value. • E.g. The following sample of 50 pulse rate values ordered by the number of beats per

minute: 62 64 65 66 68 70 71 71 72 72

73 74 74 75 75 76 77 77 77 78

78 78 79 79 79 80 80 80 80 81

81 81 81 82 82 82 83 83 85 85

86 87 87 88 89 90 90 92 94 96

Australian School of Business Describing the Distribution

• The first thing we note about the distribution is the maximum and minimum observations this is the Range – Thus the range of the distribution is 62 to 96.

• We can also assess the middle value of the distribution – this is the Median – This involves finding the value where there are the same number of

observations above that value as there are below – In the previous distribution, containing 50 values, this would be the

value between the 25th and 26th observations, i.e. between 79 & 80. – We split the difference and the median would be 79.5 – This is one form of the “representative” average for a distribution

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Describing the Distribution (Cont.)

• A far more common measure of the average of a distribution is the Mean – Calculated by adding all of the values of the observations in a

distribution together and dividing by the number of observations • In our previous example the sum of the values is 3,955 and there

are 50 observations • The mean is therefore 79.1

– This is commonly referred to as the “Expected Value” – The Mean is a key measure that will be used in the process of risk

measurement

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The Shape of the Distribution

• Statistics also allow us to get an idea of the shape of the distribution – This will help to identify:-

• Where particular observations lie within the distribution in relation to others or

• What values are common and uncommon – We can do this by viewing the example distribution as a

histogram (with no grouping of observations) as follows:

Australian School of Business

Sample of 50 Students

0

1

2

3

4

5

62 64 65 66 68 70 71 72 73 74 75 76 77 78 79 80 81 82 83 85 86 87 88 89 90 92 94 96

Heart Beats per Minute

N um

be r

of O

bs er

va tio

ns Sample Distribution

Australian School of BusinessSome Observations about the Distribution

• The histogram shows that: – Two observations have highest frequency (80 and 81)

• The distribution therefore has two Modes, which is a third measure of the arithmetic average

– There is a tendency for observations to centre around a particular value • The three “averages” are all measures of this central tendency. • The principle measure of this tendency is however the Mean

– This is the measure of central tendency that we will focus on

Australian School of BusinessStatistical Distributions and Risk

• Last week we saw that Dembo identified that the valuation of risk can be thought of as an option – As we noted last week, the key determinant of the value of an

option is its volatility – A key factor in measuring risk is therefore the degree of variability

(volatility) of potential outcomes • Statistics examines variability in data via measures of dispersion • Measured as (potential) deviation from the measure of central

tendency noted earlier (the key measure of which is the Mean)

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Measures of Dispersion • Variance (σ2)*

– In our example 57.6

• Standard Deviation (σ)

– In our example 7.6 – Thus in our sample data one standard deviation either side of the mean (79.1)

would provide a range of between 71.5 and 86.7.

N Xi 

2 2 )( 

2 

* The unbiased estimate uses n-1 as the denominator

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Calculations Value Freq-ency Value x Freq (Value-µ)2

(Value-µ)2 x Freq

62 1 62 292.41 292.41 64 1 64 228.01 228.01 65 1 65 198.81 198.81 66 1 66 171.61 171.61 68 1 68 123.21 123.21 70 1 70 82.81 82.81 71 2 142 65.61 131.22 72 2 144 50.41 100.82 73 1 73 37.21 37.21 74 2 148 26.01 52.02 75 2 150 16.81 33.62 76 1 76 9.61 9.61 77 3 231 4.41 13.23 78 3 234 1.21 3.63 79 3 237 0.01 0.03 80 4 320 0.81 3.24 81 4 324 3.61 14.44 82 3 246 8.41 25.23 83 2 166 15.21 30.42 85 2 170 34.81 69.62 86 1 86 47.61 47.61 87 2 174 62.41 124.82 88 1 88 79.21 79.21 89 1 89 98.01 98.01 90 2 180 118.81 237.62 92 1 92 166.41 166.41 94 1 94 222.01 222.01 96 1 96 285.61 285.61

Total 50 3955 2882.5 Mean (µ) 79.1 Variance (σ2) 57.7 Standard Deviation (σ) 7.6

Mean The product of the observation and the frequency divided by the total number of observations Variance The sum difference between each value and the mean squared multiplied by the frequency of the observations divided by the number of observations Standard Deviation The square root of the variance

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Standard Normal Distributions

• If we take many observations there is a tendency for resultant distribution to acquire the shape of a Normal Distribution as follows:

• Standard Normal Distributions have very useful properties in relation to their Standard Deviation (as shown above).

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Using Normal Distributions

• One reason that the normal distribution is so important is that they are easy for mathematical statisticians to work with. This means that many kinds of statistical tests and inferences can be derived for normal distributions.

• If we assume normality the distribution can be defined by reference to its mean and standard deviation making inferences from the data simpler.

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Using Normal Distributions (cont.)

• Statisticians who wish to draw inferences about a broad “population” from a sample typically assume that the distribution of observations in the population is normally distributed. – In the context of risk measurement we are trying to

assess the “population” of potential future outcomes based on a “sample” of actual historical observed outcomes.

– Hence the assumption that distributions are normally distributed is commonly made

– Note an assumption that future outcomes “mirror exactly” past observations, i.e. have an identical distribution can also be made.

Australian School of Business Examining the One-sided (single) Tail of

a Standard Normal Distribution

50% + 34.1% 13.6%

2.3% Standard Deviations Confidence level (one tail) 85% 90% 95% 97.5% 99% 99.9%

Approx. Std. Dev’s 1 1.25 1.65 1.95 2.33 3

Australian School of BusinessNormal Distributions and Risk Measurement

• When looking at risk we want to understand with some degree of certainty or confidence what the worst possible situation might be.

• The statistical principles above are utilised in calculating Value-at-Risk (VaR) which is defined as follows:

Value at Risk (VAR) • Definition

“VAR measures the worst expected loss (measured in terms of value changes*) that

a firm is likely to suffer over a given time interval under normal market conditions

at a given confidence level” » Cormac Butler, Mastering Value at Risk, Financial Times Publishing 1999. * Added

Australian School of BusinessPhillipe Jorion* on the benefits of Value at Risk (VaR)

“The greatest benefit of VAR lies in the imposition of a structured methodology for critically thinking about risk. Institutions that go through the process of computing their VAR are forced to confront their exposure to financial risks and to set up a proper risk management function. Thus the process of getting to VAR may be as important as the number itself.” It is therefore important to not put excessive reliance on

the result itself* P Jorion*, Value at Risk, McGraw Hill

* e.g. LTCM’s over-reliance on the result

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The Foundations of VaR

• Initially developed as a measure of market risk for banks – But relates to any frequently observable event or series of events

• Reflects the Risk Manager’s: – Beliefs about the future distribution of changes to the

value of an investment or group of investments, • as we have seen these can be individual investments or in fact the firm

as a whole; and

– Relative aversion to the risk of loss (Dembo’s lambda) • If we are risk averse we will want an estimate of risk that encompasses

more potential outcomes (ie a higher confidence level)

• VaR recognises that actual future loss is unpredictable but the “worst expected loss” can be estimated

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VaR Fundamentals

• The Expectation of loss relates to the probability of the event occurring in the future

• We therefore need to assess the probability of the event occurring – How to calculate probability? VaR uses one of three methods or a

combination of them to derive the sample data. (see later slide)

• What is the appropriate time frame to consider the risk event?

• This is a vital consideration.

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VaR - The Basics

Change In Value -30 -20 -10 0 10 20 30

Probability (%) 5 10 20 30 20 10 5

Cum. Probability(%) 100 95 85 65 35 15 5

What is the VaR (worst expected loss) of this Distribution if the risk manager wants to be 95% confident about the estimate?

Australian School of BusinessThe Resultant Loss Distribution (Speculative Risks)

0

10

20

30

40

50

60

70

80

90

100

Pr ob

ab ilit

y of

L os

s (%

)

Amount of Loss

The Probability of Loss

-30 -20 -10 0 10 20 30

Cumulative Distribution

Note the Amount of Loss represents the potential change in Value

Australian School of BusinessCalculating Mean and Standard Deviation

∆r Frequency ∆r x

Frequency (Deviation from

µ) 2 (Deviation from

µ) 2 x Freq. 30 5% 1.5 900 45 20 10% 2 400 40 10 20% 2 100 20 0 30% 0 0 0

-10 20% -2 100 20 -20 10% -2 400 40 -30 5% -1.5 900 45

Mean 0 Variance 210 Std. Dev 14.49 bps

Australian School of BusinessCalculating VaR at 95% Confidence Interval (CI)

• Percentile Method – What are the worst 5% of the observations? – Assumes that the observations in the distribution are the only “possible” set of

future outcomes. – This must be the deviation from the Mean (using the previous data the Mean is

0, the worst expected value change is -20, hence the VaR is -20 - 0 = -20) • Statistical Method

– Uses the mean and standard deviation? – Assumes that the future outcomes are reflective of the sample and are normally

distributed – What is a 95% CI assuming a standard normal distribution

• One tailed CI for 95% is 1.65σ (1.65 x 14.49 = 23.9) – What is the 99% CI assuming a standard normal distribution

• One tailed CI for 99% is 2.33σ (2.33 x 14.49 = 33.8) – Note these results are the deviation from the Mean

• Why do the results differ?

Australian School of BusinessThree Ways to Derive the Potential Loss Distribution

• Historical Simulation

• Estimated Variance - Co-Variance

• Monte Carlo Simulation

Australian School of Business Historical Simulation

• Estimate the current value of the asset and assess its what is value would have been due to past business risk events on a number of dates:

• Based on this estimate the distribution of value changes can be drawn from this sample

• Derive VaR from cumulative distribution – Strengths and weaknesses of this method

• Assumes that the past reflects the future • No estimate of parameters required • Reasonably heavy computational costs

tnti VVV  

Where i = the observations t = current value n = the time horizon

or timeframe

Australian School of BusinessEstimated Variance/Co- Variance

• Establish key risk variables (benchmarks) – Standard deviations and correlations of other factors compared to the

benchmarks are obtained from historical data

• Derive standardised risk bases – e.g. the return on a project might be a function of sales therefore

model sales and other variables are correlated with that the sales variable

• The resultant parameters provide an estimate of VaR • Strengths and weaknesses of this method

– Selected component (standardised) securities (cashflows) reflects all market changes

– Future is like the past, therefore parameters can be extrapolated into the future

– Less computationally intensive than historical method – Largely discredited following Asian Currency Crisis

Australian School of Business Monte Carlo Simulation

• Generate a series of possible price changes based on – The type of process normal, lognormal etc. – Estimate parameters of price or value change – Estimate correlations between key variables

• Strengths and weaknesses of this method – Maps a broader range of possible value changes – Model must reflect reality – Requires a heavy computational capabilities

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Determining the Timeframe

• Depends on – The nature of the risks – The likely impact (severity) of risk events – How many such events could be borne in any one period – Likely time to immunise against the effects of the risk

• Commonly VaR for market risk is: – Daily or 1 week to one month

• Commonly VaR for credit risk is: – One year

• Business Risk – Operational risks require regular review, the time horizon is likely to

be short term – Longer term/strategic risks would be assessed over a one-year

timeframe

Australian School of BusinessStress Testing – The Crisis or Catastrophic Scenario

• Estimate likely worst case scenario – Extreme events such as:

• The Russian crisis – during which LTCM failed • The Global Financial Crisis – during which Lehman Brothers failed • The 2011 Japanese earthquake and the explosion at the Fukushima nuclear reactor

• Will normally involve applying a higher confidence level than used in the VaR calculation

• Problem with “tail events” – Few observations no matter how the distribution is derived

• These can be seen as the catastrophic events that Crockford identifies

Australian School of BusinessSome Important Points to note:

• Models rely on a high frequency of observations • Modeling or similar techniques are only one

element in a risk management process: – Rigorous risk assessment – Sound policies and procedures – Strong control processes

• This is the application of the risk cycle that we have discussed

• No algorithm will solve all risk control problems – e.g. in LTCM the event was seen as an “8 std. deviation”

event, but was it really?

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The Use of VaR

• Traditionally VaR has been used to measure Market Risks and Credit Risks in banks and other financial institutions

• The next stage, which we will consider, is to apply this technique to the measurement of operational and project risks

• Has potential for all risks for which are reasonably observable, i.e. high frequency (Crockford’s “better”) data

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Class Workshop

Project Risk Spreadsheet of Prior Losses to be

Examined in Week 6