Analytics of Risk Management II: Statistical Measures of Risk
description
Transcript of Analytics of Risk Management II: Statistical Measures of Risk
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.