Intro To VaR, Distributions, KRIs And Logic Test

Treating Customers FairlyTreating Customers FairlyTechnical Guide to Value at Risk, Key Risk Indicators and Logic TestsTechnical Guide to Value at Risk, Key Risk Indicators and Logic Tests

by Jon Beckett, BA, ASI, by Jon Beckett, BA, ASI, For the CCCFor the CCC

IntroductionIntroduction

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Bell CurveBell CurveIndexIndex TCFTCF VaRVaR ProcessProcess KRIsKRIs

1. Value at Risk, Volatility Spikes1. Value at Risk, Volatility Spikes11 and Treating Customers Fairly and Treating Customers Fairly

• Value at Risk (‘VaR’) is a statistical risk measure that originated with derivatives traders but now common place amongst trading desks, group risk and compliance

• Until TCF, VaR had been primarily used for trading, capital adequacy and liquidity controls

• Prolonged market volatility up to and following the credit crunch had cast increasing scrutiny on Firm-wide risk cultures and whether products perform, as promoted, during “volatility spikes”1 :

• Increasing supervision and development of internal risk systems up to 2007, saw regulator initiatives to carry this risk culture from back to front office in 2008-2009, consistent with MiFID and RDR

• Treating Customers Fairly (‘TCF’) has led to the inference of using VaR as Management Information (‘TCF MI’) to demonstrate “Outcome 5” to FTIML Board, investors and FSA from Quarter 4 2008*:

FSA, Outcome

5

The almost unprecedented nature, depth, and duration of the current market turmoil have raised major challenges for nearly all significant participants in financial markets. In this environment, participants face increasing pressure to understand the risks they face, to measure and assess such risks appropriately, and to take the necessary steps to reduce, hedge, or otherwise manage such risk exposures.

1 ‘Observations on Risk Management Practices during the Recent Market Turbulence’, Senior Supervisors Group; (inc. FSA) March 6, 2008

Consumers are provided with products that perform as firms have led them to expect and the associated service is both of an acceptable standard and as they have been led to expect

‘Treating Customers Fairly: Measuring Outcomes’, Financial Services Authority, November, 2007, *Progress Update June 2008

Volatility (related)Volatility (related)

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Introduction to VaR: Unexpected Risks for the Investor?Introduction to VaR: Unexpected Risks for the Investor?

Most investors have probably never heard of Value at Risk (‘VaR’)

• Value at Risk helps investment managers gauge the amount of their assets at risk from the market

• Until now most of the UK Retail market had focussed on 3 year returns and 3 year volatility

• Investors expect a Fund to follow a normal range of returns around its historical Mean

• Occasionally a strategy may experience an abnormal ‘event’ creating ‘volatility spikes’ (see below)

• Traditional measures such as Standard Deviation (Volatility) do not quantify these unexpected risks

• We review a Fund’s risk against 3 tests: investor expectations, investment guidelines and basis sold

Unexpected Unexpected Losses/VolatilityLosses/Volatility

Expected Normal Range of Return

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What is Volatility: What does Standard Deviation tell us about a Fund?What is Volatility: What does Standard Deviation tell us about a Fund?

• Standard Deviation measures the volatility of a Fund’s returns Standard Deviation measures the volatility of a Fund’s returns around the Mean Return around the Mean Return

• The Mean Return is the average return over a specific period – The Mean Return is the average return over a specific period – think of it as the thick centre line on the chart rightthink of it as the thick centre line on the chart right

• Standard Deviation is the moving line above and below the Standard Deviation is the moving line above and below the centre Mean Return line, we call this ‘dispersion’centre Mean Return line, we call this ‘dispersion’

• Standard Deviation is normally considered more reliable over Standard Deviation is normally considered more reliable over longer time periods; (such as 3 or 5 years) using monthly longer time periods; (such as 3 or 5 years) using monthly returnsreturns

• The volatility of returns can also be shown in a histogram of The volatility of returns can also be shown in a histogram of returns (right) – the greater the frequency of returns away from returns (right) – the greater the frequency of returns away from the Mean the higher the Volatilitythe Mean the higher the Volatility

Fund XYZ Returns over 36 Months

Standard Deviation (‘Volatility’)

Negative Deviation Positive Deviation

Mean Return

Up

Down

Volatility Spikes (related)Volatility Spikes (related)

VolatilityVolatility

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Volatility Spikes: Shortcomings of Standard DeviationVolatility Spikes: Shortcomings of Standard Deviation

• Standard Deviation is a good single measure for a Fund’s Standard Deviation is a good single measure for a Fund’s Volatility but can be misleading in terms of all risksVolatility but can be misleading in terms of all risks

• Volatility does not actually tell you how much a Fund has Volatility does not actually tell you how much a Fund has lost or gained; simply how volatile its returns were between lost or gained; simply how volatile its returns were between two pointstwo points

• Standard Deviation, in isolation, does not indicate the Standard Deviation, in isolation, does not indicate the proportion of positive or negative volatilityproportion of positive or negative volatility

• Volatility alone does not reliably convey short-term risk or Volatility alone does not reliably convey short-term risk or changes in risk (‘volatility spikes’)changes in risk (‘volatility spikes’)

• Volatility does not indicate the likely frequency, Volatility does not indicate the likely frequency, magnitude or expected likelihood of lossesmagnitude or expected likelihood of losses

Standard Deviation = (Expected Risk)

?

Normal Volatility

Volatility Volatility Spike: Spike: Unexpected Unexpected Risk?Risk?

Unexpected Unexpected Risk?Risk?

??

Hypothetical example (related)Hypothetical example (related)

Vol. SpikesVol. Spikes

Average/Mean Return

Bell Curve

Bell CurveBell CurveIndexIndex TCFTCF VaRVaR ProcessProcess KRIsKRIs

2. Expected Returns, Distribution and the Bell Curve?2. Expected Returns, Distribution and the Bell Curve?

Histogram of Returns of XYZ Fund

Left-tail

Returns less than Mean (‘downside’)

Bell-Curve:Bell-Curve: The normal bell-shaped distribution of statistics plots all of its values in a symmetrical fashion, with the majority of returns centred around the mean/or median.

The expected outcome is that returns following the Bell Curve will equally plot either above or below the mean; with diminishing occurrences of returns away from the mean. We call those less-likely returns as ‘tails’.

This model expects returns to revert to their Mean; in reality investments rarely follow the perfect distribution, as Risk is changing and often unpredictable..

Returns greater than Mean (‘upside’)

Right-tail

Distribution Analysis:Distribution Analysis: You may not know that the industry has analysed distributions for many years?

Often these key indicators are termed regression or returns-based analysis:

• Standard Deviation

• % Positive v Negative Returns

• Maximum Loss/Drawdown

• Capital Asset Pricing Model

• Capture Rates

• Attribution analysis

Distribution Debate (related)Distribution Debate (related)

Bell CurveBell CurveIndexIndex TCFTCF VaRVaR ProcessProcess KRIsKRIs

Maximum Loss or Drawdown

Median/Mean Return

Bell Curve

Value at RiskValue at Risk (‘VaR’) is a downside estimate of how much a Fund could lose within a given investment horizon and at a given confidence level.

95% Confidence means that the Value at Risk will not exceed a maximum loss for 95% of the time but for 5% it could.

VaR does not indicate the likelihood nor probability of a maximum loss occurring..

Flag VaR and other Key Risk Indicators (‘KRIs’):

• 1 and 3 month VaR values expressed as a %, for the open-ended investor

• Firm-wide methodology by PAIR

• 95% model creates early signals

• Rolling 36 month horizons (up to 10yrs)

• De minimis of 12 month history

• 1Kurtosis indicates depth and length of a Fund’s ‘tails’

• 2Skewness indicates likely trend left or right back to the Mean

• Magnitude and occurrence of VaR is directly related to its distribution pattern

11KurtosisKurtosis

Maximum Gain or Bull Rally

22Positive SkewnessPositive Skewness

22 Negative Skewness Negative Skewness

3. How does Value at Risk (‘VaR’) fit into the Distribution?3. How does Value at Risk (‘VaR’) fit into the Distribution?

Hypothetical example (related)Hypothetical example (related)

? =VaR%? =VaR%

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Distribution Debate: So who cares what pattern a Fund’s returns make?Distribution Debate: So who cares what pattern a Fund’s returns make?

DistributionDistribution

XYZ Fund holds a mixed portfolio of risk assets with an expected moderate level of Standard DeviationStandard Deviation and Return.

Distribution Risk is the risk of returns away from the Mean (or expected) return.

The patterns of returns could look like any of those below; the majority of patterns are grouped into a few broad categories..

Downside (related)Downside (related)

Believe it or not – all 3 patterns could display the same Standard Deviation and the same Mean Return..

The same Fund could exhibit all of these patterns, if the manager changes the Fund RiskFund Risk by changing strategy, or the Fund’s sensitivity to Market RiskMarket Risk by changing the asset allocation.

However the Risk presented to the investor, by each of these distributions, is very different.

There are 2 key questions: the frequencyfrequency (likelihood of loss) and the severity severity of maximum loss.

A. This is the Normal distribution of returns; (otherwise known as the Gaussian or Bell Curve).

The frequency of returns are well spaced with the greatest around the Mean.

When a Fund returns slightly outside the normal distribution it is usually referred to as a Bi-nominal or non-normal distribution.

This Fund’s expected VaR and maximum loss will be moderate/variable and frequency regular.

Balanced Managed and Asset Allocation Funds often strive for this..

B. This Fund follows a Uniform Distribution (Sometimes known as a Leptokurtic pattern).

The range of return is narrower around the Mean and the frequency of returns higher.

That means fewer different returns and more of the same sort of return, both negative and positive.

This Fund’s expected VaR and maximum loss will occur frequently but will be less severe.

Bond Funds often display this type of pattern..

C. The last is the Negative, Platykurtic pattern; (commonly referred to as the ‘Left-tailed’ Distribution).

Often there is a high concentration of returns around the Mean; much like the Uniform distribution, but usually all positive and in the higher return ranges.

To the left there is then a long series of acute negative returns - this is the left-tail.

The Fund’s expected VaR and maximum loss occurs infrequently but will be more severe on those occasions; (boom-bust cycles).

Higher risk funds (E.g. Emerging Markets) often have left-tailed distributions and Funds with low correlated returns to core markets such as REITs and High Yield Bonds..

Fre

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Severity Severity Severity

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Downside Patterns: So what frequency and severity would we expect?Downside Patterns: So what frequency and severity would we expect?

DownsideDownside

Bell Curve (related)Bell Curve (related)

A.A. Normal Distribution B.B. Uniform Distribution C.C. Left-tailed Distribution

Mean Return over time

Mean Return over time

Mean Return over time

NormalNormal – E.g. Core Equity

B.UniformUniform – E.g. Bond

VolatilityVolatilityA.

C.Left-tail Left-tail – E.g. Emerging Market or Specialist

tt

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Hypothetical Example: How Risk changed through the ‘Credit Crunch’Hypothetical Example: How Risk changed through the ‘Credit Crunch’

Credit Crunch:Credit Crunch: Volatility Spikes lead to increased

Maximum Losses, returns fall outside of the Expected Range

Expected Upper Range of Return

Standard Deviation

Expected Lower Range of Return

VaR HypoVaR Hypo

XYZ Fund holds a conventional portfolio of equity and bonds with expected moderate Standard Deviation and Return

The Manager’s Mean Return and Standard Deviation are based on the previous 36 month period and won’t immediately reflect the rising VaR in the portfolio. VaR values and rolling short-term KRIs, through the investment period, flag the abnormal risk sooner!

Mean Return

Value at Risk:Value at Risk: indicates the likely maximum size of volatility spikes if they occurred – VaR can quickly

flag changes in short-term volatility

Portfolio Managers began to invest aggressively into derivatives, high yield and securitised debt securities – this changed the future distribution/risk of the portfolio but will not impact the Standard Deviation of the Fund in the short-term..

Fund Returns

Volatility (STDEV)

Value at Risk (VaR)

Bell CurveBell CurveIndexIndex TCFTCF VaRVaR ProcessProcess KRIsKRIs

4. Process: From Distribution to Outcome 4. Process: From Distribution to Outcome

Step1. PAIR Stress-testingStep1. PAIR Stress-testing –Indices/Sectors are tested for historical indicators once per

annum

Step1. PAIR Stress-testingStep1. PAIR Stress-testing –Indices/Sectors are tested for historical indicators once per

annum

Step2. Fund PatternsStep2. Fund Patterns – PAIR test ongoing data and supply fund level reports on a regular

basis

Step2. Fund PatternsStep2. Fund Patterns – PAIR test ongoing data and supply fund level reports on a regular

basis

Left-tail Distribution FundsLeft-tail Distribution Funds

Positive Distribution FundsPositive Distribution Funds

Uniform Distribution FundsUniform Distribution Funds

Normal (Bell Curve) Distribution FundsNormal (Bell Curve) Distribution Funds

Marginal RiskMarginal Risk Variable Risk!Variable Risk!

Moderate RiskModerate Risk Higher Risk!Higher Risk!

Step3. Analysis of Outcomes –Step3. Analysis of Outcomes – Designated Manager, or analyst, analyses Key Risk

Indicators (‘KRIs’) against expected outcomes from Step1. Set of logical tests are applied for

each distribution type

Step3. Analysis of Outcomes –Step3. Analysis of Outcomes – Designated Manager, or analyst, analyses Key Risk

Indicators (‘KRIs’) against expected outcomes from Step1. Set of logical tests are applied for

each distribution type

Step4. Flag Outcomes and Actions – Step4. Flag Outcomes and Actions – Score funds as either ‘Green’, ‘Amber’ or ‘Red’ based on Step3 and prepare the health monitor report for peer review and escalates alerts or action

through Global Product Strategy.

Step4. Flag Outcomes and Actions – Step4. Flag Outcomes and Actions – Score funds as either ‘Green’, ‘Amber’ or ‘Red’ based on Step3 and prepare the health monitor report for peer review and escalates alerts or action

through Global Product Strategy.

Framework (related)Framework (related)

Bell CurveBell CurveIndexIndex TCFTCF VaRVaR ProcessProcess KRIsKRIs

5. Example Key Risk Indicators (KRIs) applicable to Outcome 55. Example Key Risk Indicators (KRIs) applicable to Outcome 5

The list of indicators is atypical and not exhaustive: scoring will take into account specific circumstancesThe list of indicators is atypical and not exhaustive: scoring will take into account specific circumstances

Left-tail Distribution FundsLeft-tail Distribution FundsUniform Distribution FundsUniform Distribution Funds

Marginal RiskMarginal Risk Variable Risk!Variable Risk!

Moderate RiskModerate Risk Higher Risk!Higher Risk!

Positive Distribution FundsPositive Distribution Funds Normal (Bell Curve) Distribution FundsNormal (Bell Curve) Distribution Funds

(R)(R) Negative losses

(R)(R) Mean Return below cash rate (e.g. LIBOR)

(A)(A) Rising Volatility

(A)(A) Excessive positive returns

(A)(A) High Negative Skewness

(A)(A) Rising or Moderate Value at Risk

(A)(A) Redemptions or Falling Assets

(A)(A) Above average TERs

(R)R) Mean Return below cash rate AND sector

(R)(R) Maximum Loss in excess of expected outcomes

(R)(R) Rising Correlation to Equities and/or High Yield

(A)(A) Value at Risk above expected outcomes

(A)(A) Violations above permitted

(A)(A) Falling Yields/Narrowing Spreads

(A)(A) High or Rising Negative Skewness

(A)(A) Redemptions or Falling Assets

(R)(R) Maximum Loss in excess of expected outcomes

(R)(R) Value at Risk above expected outcomes

(A)(A) High or Rising Negative Skewness/Deviation

(A)(A) Violations above permitted

(A)(A) Very low/high Correlation to Benchmark

(A)(A) Negative Alpha to Benchmark/Sector

(A)(A) Displays increasing left-tailed returns

(A)(A) Redemptions or Falling Assets

(R)(R) Unsystematic (market) rises in negative volatility

(R)(R) High XS Value at Risk, frequent violations

(R)(R) Prolonged drawdowns – poor long-term return

(A)(A) High or Rising Negative Skewness

(A)(A) Underperforms index over expected horizon

(A)(A) Excessive Liquidity into/out of Sector

(A)(A) Large Redemptions or Falling Assets

(A)(A) Poor Performance from higher TER Funds

Flags:

(R) Red (Action to remedy)(R) Red (Action to remedy)

(A) Amber (Alert for issues)(A) Amber (Alert for issues)

(G) Green (Business as usual)(G) Green (Business as usual)

3 Tests in relation to Outcome 5:3 Tests in relation to Outcome 5: Products not performing in line with:

1. Investor expectations2. Investment guidelines or 3. On the basis sold

TimePeriodLevelConfidence **

LevelConfidence

TimePeriod

Value at Risk Factor =

Normal Distribution Confidence Level; E.g. 95% (1.96)

Standard deviation of historical fund returns.

= The number of months (36) or days (20) measured

Value at Risk Amount = VaR Factor * NAV NAV = Net Asset Value of the Fund

Value at RiskValue at Risk (‘VaR’) is a downside estimate of how much a Fund could lose within a given investment horizon and at a given confidence level. 95% Confidence means that the Value at Risk will not exceed a maximum loss for 95% of the time but for 5% it could. VaR does not indicate the likelihood nor probability of a maximum loss occurring..

Confidence % (Intervals) =About 68% of values in a normal distribution are within one standard deviation (sigma, σ) away from the Mean (μ); about 95% of the values are within two standard deviations and about 99.7% lie within three standard deviations. Each standard deviation multiple (sigma , σ) then provides confidence intervals of expected ranges of return.

To be more precise, the area under the bell curve between Mean and +/− each sigma multiple (σ) in terms of the cumulative normal distribution function is given by the error function (erf). To 12 decimal places, the 1 to 6 sigma multiples are:

You can then reverse the relation of sigma multiples for a few associated indicators used to describe the area under the bell curve. These values are useful to determine (asymptotic) confidence intervals for other indicators based on a normally distributed curve.

The sigma multiples and confidence intervals are shown right.

The left table then indicates the proportion (Confidence%) of a given interval and n is a multiple of the standard deviation that specifies the width of each interval. 0.9999999980276

0.9999994266975

0.9999366575164

0.9973002039373

0.9544997361040.95449973610422

 0.682689492137 1

σ Interval

4.41720.99999

3.89060.9999

3.290520.999

3.090230.998

2.807030.995

2.575830.99

2.326350.98

1.959961.959960.950.95

1.644850.90

 1.28155 0.80

Interval Width (n)

68%

99.999%

99.99%

99,9%

99.8%

99.5%

99%

98%

95%95%

90%

80%

Confidence%