Analytics of Risk Management II:

Statistical Measures of Risk

Risk Management

Lecturer:

Mr. Frank Lee

Session 4

OverviewOverview

Quantitative measures of risk - 3 main types:

1. Sensitivity – derivative based measures2. Volatility & Statistical measures of risk:

◦ Attitudes to risk◦ Relation with Finance Theory◦ Portfolio Theory, CAPM & APT◦ Post-modern Portfolio Theory

3. Downside Risk measures◦ Statistical underpinning ◦ Value at Risk

Value = Expected Net Present Value = Expected Net Present ValueValue

E(NPV) = Σ E(Rt) - E(Ct) (1 + R)tt = 1

n

A Set of DefinitionsA Set of DefinitionsRisk - The outcome in a particular

situation is unknown but the probability distribution from which the outcome is to drawn is known

Uncertainty - The outcome in a particular situation is unknown as is the probability distribution from which the outcome is drawn.

Problems of which situations Problems of which situations are risky and which are are risky and which are uncertainuncertainRisky situations can form the basis

for tractable financial analysisUncertain situations are

considerably less analytically tractable

Examples of Risk and Examples of Risk and UncertaintyUncertaintyRisk - Prices in Financial Markets

Share PricesInterest Rates

Exchange RatesUncertainty -

Terrorist AttacksWarsFinancial Crises

Statistics and EstimatesStatistics and Estimates

Mean - Measure of Central Tendency, like Median and Mode.

Variance - Measure of dispersion round the mean the squared term ensures that Variance is positive and gives extra weight to observations furthest from the mean.

Standard Deviation - Square Root of Variance. It is in same metric as Mean

Probability DistributionsProbability Distributions

We generally focus on normal distributions

Normal Distributions are entirely specified by Mean and Standard Deviation

Normal DistributionNormal DistributionFrequency

Return-s +s m

34.13%34.13%

Risk and ReturnRisk and Return

Return - Average or Expected ReturnRisk

◦ For Normal Distributions Standard Deviation is totally satisfactory

◦ For Non-normal Distributions there may be a diversity of statistical and psychological measures or risks

Attitudes to RiskAttitudes to Risk

Utility Function - Defined over a probability distribution of returns.

Mean and Variance approachHigher MomentsTime

Attitudes to RiskAttitudes to Risk

Risk Averse - Like Return Dislike RiskRisk Neutral - Like Return and totally

unconcerned about RiskRisk Loving - Like Return and Like

Risk

Utility and RiskUtility and Risk

Utility

U(B)

U(A)

W(A)

W(*) W(B) Wealth

U(*)

Risk Aversion and UtilityRisk Aversion and Utility

W(A) + W(B) = W(*) 2

U(*) > U(A) + U(B) 2

Prefer W(*) to bet with 0.5 probabilityof W(A) and 0.5 probability of W(B) which has the same expected value of wealth of W(*)

Preferences for Risk and Preferences for Risk and ReturnReturn

X

A B

C D

Return

Risk

For Risk Averse InvestorFor Risk Averse InvestorA Preferred to XX Preferred to DC and X no orderingB and X no ordering

Indifference Curves for the Indifference Curves for the Risk Averse InvestorRisk Averse Investor

x

U1U2

U3

Return

Risk

Indifference Curve Slope is a Indifference Curve Slope is a Measure of Risk AversionMeasure of Risk Aversion

a

bcd

x y

Return

Risk

ab > cdxy xy

Measures of RiskMeasures of Risk

Standard DeviationCombination of MomentsValue at RiskExpected Tail LossMoments Relative to Benchmarks - Risk

Free Rate, Zero Return, Capital Asset Pricing Model, Arbitrage Pricing Theory

Measuring RiskMeasuring Risk

Variance - Average value of squared deviations from mean. A measure of volatility.

2 =

Standard Deviation - Square root of variance (square root of average value of squared deviations from mean). A measure of volatility.

2

1

))(( rErpn

iii

= 22

1))(( rErp

n

iii

Standard DeviationStandard Deviation

Square root of variance Equates risk with uncertaintyImplies symmetric, normal return

distributionUpside volatility penalized same as

downside volatilityMeasures risk relative to the meanSame risk for all goals

MomentsMoments

ith Moment around a = E(R - a)i Measure of Skewness = E(R - E(R))3

(Minus value skewed to left, Positive Value skewed to right)

Measure of Kurtosis = E(R - E(R))4

(Larger value flatter the distribution)

Modern Portfolio Theory

Portfolio Theory Portfolio Theory AssumptionsAssumptionsInvestors Risk AverseInvestors only interested in the Mean

and Standard Deviation of the Distribution of Returns on an Asset

Investors have knowledge of Mean and Standard Deviation of Returns

A Range of Risky AssetsAt Least One Riskless Asset

Measuring RiskMeasuring Risk

05 10 15

Number of Securities

Po

rtfo

lio

sta

nd

ard

dev

iati

on

Market risk

Uniquerisk

Portfolio RiskPortfolio Risk

Covariance =

Correlation = (-1 < xy <1)

n

iyyxxixy rrrrpCov

ii1

))((

yx

xyxy

Cov

Variables Trend TogetherVariables Trend Together

Y

X

Figure 1

Variables Trend in Opposite Variables Trend in Opposite DirectionsDirections

Y

X

Figure 2

Correlation ValuesCorrelation Values

One - Perfect linear relationBetween zero and one variables trend

togetherZero - No relation between variablesBetween zero and minus one variables

trend in opposite directionsMinus one - variables have negative

perfect linear relation

Portfolio Return & RiskPortfolio Return & Risk

)rx()r(x Return Portfolio Expected 2211

)σσρxx(2σxσxVariance Portfolio 21122122

22

21

21

Portfolio RiskPortfolio RiskThe shaded boxes contain variance terms; the remainder contain covariance terms.

1

2

3

4

5

6

N

1 2 3 4 5 6 N

STOCK

STOCKTo calculate portfolio variance add up the boxes

Hedging with a PortfolioHedging with a Portfolio

Return

Time

A

B

Simple Portfolio ImpactsSimple Portfolio Impacts

Return/SDAsset y

Return/SDAsset x

Correlation

Case 1 20/10 10/5 -1

Case 2 20/10 10/5 0

Case 3 20/10 10/5 1

Correlation Coefficients Correlation Coefficients RevisitedRevisited

Chart 1: Portfolio Opportunity Sets: different correlations

0

5

10

15

20

25

30

0.0 5.0 10.0 15.0 20.0 25.0 30.0

risk (standard deviation (%))

exp

ecte

d r

etu

rn (

%)

correlation=0

correlation=-1

correlation = 1

correlation=0.25

all investment in N

all investment in M

The Set of Risky AssetsThe Set of Risky Assets

E(R)

SD(R)

A

Optimal Set of Risky Optimal Set of Risky AssetsAssets

E(R)

SD(R)

x

y

Adding the Riskless AssetAdding the Riskless Asset

E(R)

SD(R)

Rf

The Capital Market LineThe Capital Market Line

E(R)

SD(R)

The CapitalMarket Line

D

E

G

H

Investors ChoiceInvestors Choice

x

y

E(R)

SD(R)

Portfolio Theory Portfolio Theory ConclusionsConclusionsAll Investors hold the same set of risky

assets if they hold risky assetsAll investors must hold the market

portfolioTheir risk preferences determine

whether they gear up or down by borrowing or lending

Security Market LineSecurity Market Line

Return

BETA

.

rf

Risk Free

Return =

Market Return = rm

Efficient Portfolio

1.0

Security Market LineSecurity Market LineReturn

BETA

rf

1.0

SML

SML Equation = rf + B ( rm - rf )

Beta and Unique RiskBeta and Unique Risk

2m

imiB

Covariance with the market

Variance of the market

Downside Risk Measures

Expert OpinionsExpert Opinions

Markowitz (1992): Since an investor worries about underperformance rather than over-performance, semi-deviation is a more appropriate measure of investor's risk than variance.

Sharpe (1963): Under certain conditions the mean-variance approach leads to unsatisfactory predictions of investor behavior.

Post-Modern Portfolio Post-Modern Portfolio TheoryTheory

Two Fundamental Advances on MPT:

Downside risk replaces standard deviation

PMPT permits non-normal return

distributions

‘‘PMPT’ v MPTPMPT’ v MPT

• Risk measure:

Downside Risk vs. Standard Deviation

• Probability distribution:Lognormal vs. Normal.

• The same application: Asset allocation/portfolio optimalisation and performance measurement

Downside risk measuresDownside risk measures

rr

rr

rr

rpLPM

rpLPM

rpLPM

22

11

00Shortfall

probability

Average shortfall

Semi-variance

Downside RiskDownside Risk

Defined by below-target semideviationStandard deviation of below-target returnsDifferentiates between risk and uncertaintyNaturally incorporates skewnessRecognizes that upside volatility is better than

downside volatilityCombines frequency and magnitude of bad

outcomesNo single riskless asset

Value at Risk (VAR)Value at Risk (VAR)

Value at RiskValue at Risk

VaR is a potential lossThe ‘maximum’ loss at a present

confidence level.The confidence level is the probability

that the loss exceeds this upper bound.VaR applies to all risks – market, credit,

default…VaR applies as long as we can build up a

distribution of future values of transactions or losses

Value at Risk (VAR) - The level of Value at Risk (VAR) - The level of losses relative to 0 or the Mean losses relative to 0 or the Mean which will only be exceeded in a which will only be exceeded in a particular proportion of instances particular proportion of instances over a particular time period.over a particular time period.

Value at RiskValue at Risk

a

x 0 E(R)

RVAR = |E(R) - x|AVAR = |0 - x|

Frequency

Risk

Definition of VARDefinition of VAR

Absolute VAR = (0 - x)

Relative VAR = (y - x)

Probability Distribution and Value at Risk

Key Choice ParametersKey Choice Parameters

Time PeriodConfidence Level

Time PeriodTime PeriodLiquidity of PortfolioRegulatory Framework (10 Days)Measurement Technique - Does one

Assume Normality ?How does one deal with changing

composition of PortfolioRequired Data for Testing

Confidence LevelConfidence Level

Risk Management/Capital Requirement

Regulatory Requirement (1% VAR)Testing - Higher so more extreme

observationsAccounting and Comparison

Measuring Value at RiskMeasuring Value at RiskVariance/CovarianceHistorical SimulationMonte Carlo SimulationStress Testing

Issues in Modelling VaR Issues in Modelling VaR

Need to move from a ‘standalone’ VaR (distribution of losses on individual assets) to the portfolio loss distribution (combines losses from all individual assets in the portfolio).

Difficult to model the loss distribution of a portfolio.

Issues in Modelling VaR Issues in Modelling VaR cont’dcont’dThe focus on high losses implies

modelling of the ‘fat tail’ of the distribution rather than looking at the central tendency.

Expected Tail Loss - ETL is quintile average; expected loss if we get a loss in excess of VAR.

The normal distribution does a poor job of modelling distribution tails (e.g. for credit risk the loss distributions are highly skewed.

Issues with VARIssues with VAR

How does one deal with Non-normality ?

How does one deal with financial crises ?

How does one deal with shifting parameter values ?

What types of risks is it best applied to ?

If normal just a multiple of Standard Deviation !

VAR ExampleVAR Example

Portfolio value is £100million, volatility (st. deviation) is 5%. Assuming normality, what is the 1% VaR of the portfolio over the next 10 days?

VaR= 2.33 x 5% x £100m = £11.65millionVaR 10 days = £11.65 x 100.5=£36.84mCan adjust for expected return if anyE.g. if E(z) = 0.1 percent per day, daily

VaR in the example becomes £11.55mBecause small, daily E(z) is ignored in

practice

Summary of Conclusions on Summary of Conclusions on Risk and Return IRisk and Return IInvestors Choose Between

Distributions of ReturnsReturn is Mean or Expected ReturnWith Normal Distributions of Returns

Mean and Standard Deviation of Returns totally summarise the information in the Distribution of Returns

Summary of Conclusions on Summary of Conclusions on Risk and Return IIRisk and Return II

The appropriate measures depend on investor preferences

Investor Preferences seem to concentrate on returns relative to the market, making losses and worst case options

The falling cost of computation and the increasing risks associated with financial markets have driven the increasing focus on these issues.

Summary of Conclusions on Risk Summary of Conclusions on Risk and Return IIIand Return III

We mainly consider the standard approaches to Risk and Return but these have a restricted view of risk and return.

Many developments in finance related to the risk management issues raised have been or will be discussed further in other courses on the programme.