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Risk Management Session 4 Analytics of Risk Management II: Statistical Measures of Risk Lecturer: Mr. Frank Lee Overview Quantitative measures of risk - 3 main types: 1. Sensitivity – derivative based measures 2. Volatility & Statistical measures of risk: ◦ ◦ ◦ ◦ Attitudes to risk Relation with Finance Theory Portfolio Theory, CAPM & APT Post-modern Portfolio Theory Downside Risk measures 3. ◦ ◦ Statistical underpinning Value at Risk Value = Expected Net Present Value n E(NPV) = Σ E(Rt) - E(Ct) t=1 (1 + R)t A Set of Definitions Risk - 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 are risky and which are uncertain Risky situations can form the basis for tractable financial analysis Uncertain situations are considerably less analytically tractable Examples of Risk and Uncertainty Risk - Prices in Financial Markets Share Prices Interest Rates Exchange Rates Uncertainty Terrorist Attacks Wars Financial Crises Statistics 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 Distributions We generally focus on normal distributions Normal Distributions are entirely specified by Mean and Standard Deviation Normal Distribution Frequency 34.13% 34.13% -s +s m Return Risk and Return Return - Average or Expected Return Risk ◦ 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 Risk Utility Function - Defined over a probability distribution of returns. Mean and Variance approach Higher Moments Time Attitudes to Risk Risk Averse - Like Return Dislike Risk Risk Neutral - Like Return and totally unconcerned about Risk Risk Loving - Like Return and Like Risk Utility and Risk Utility U(B) U(*) U(A) W(A) W(*) W(B) Wealth Risk Aversion and Utility W(A) + W(B) = W(*) 2 U(*) > U(A) + U(B) 2 Prefer W(*) to bet with 0.5 probability of W(A) and 0.5 probability of W(B) which has the same expected value of wealth of W(*) Preferences for Risk and Return Return B A X C D Risk For Risk Averse Investor A Preferred to X X Preferred to D C and X no ordering B and X no ordering Indifference Curves for the Risk Averse Investor Return U1 U2 U3 x Risk Indifference Curve Slope is a Measure of Risk Aversion Return a b c d ab > cd xy xy x y Risk Measures of Risk Standard Deviation Combination of Moments Value at Risk Expected Tail Loss Moments Relative to Benchmarks - Risk Free Rate, Zero Return, Capital Asset Pricing Model, Arbitrage Pricing Theory Measuring Risk Variance - Average value of squared deviations from mean. A measure of volatility. n 2 = pi (ri E (r ))2 i 1 Standard Deviation - Square root of variance (square root of average value of squared deviations from mean). A measure of volatility. = n 2 pi (ri E (r )) i 1 2 Standard Deviation Square root of variance Equates risk with uncertainty Implies symmetric, normal return distribution Upside volatility penalized same as downside volatility Measures risk relative to the mean Same risk for all goals Moments 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 Assumptions Investors Risk Averse Investors 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 Assets At Least One Riskless Asset Portfolio standard deviation Measuring Risk Unique risk Market risk 0 5 10 Number of Securities 15 Portfolio Risk Covariance = n Covxy pi (rxi rx )( ryi ry ) i 1 Correlation = xy Covxy x y (-1 < xy <1) Variables Trend Together Y Figure 1 X Variables Trend in Opposite Directions Y Figure 2 X Correlation Values One - Perfect linear relation Between zero and one variables trend together Zero - No relation between variables Between zero and minus one variables trend in opposite directions Minus one - variables have negative perfect linear relation Portfolio Return & Risk Expected Portfolio Return (x1 r1 ) (x 2 r2 ) Portfolio Variance x12σ 12 x 22σ 22 2(x1x 2ρ 12σ 1σ 2 ) Portfolio Risk The shaded boxes contain variance terms; the remainder contain covariance terms. 1 2 3 STOCK To calculate portfolio variance add up the boxes 4 5 6 N 1 2 3 4 5 6 STOCK N Hedging with a Portfolio Return A B Time Simple Portfolio Impacts Return/SD Asset y Return/SD Asset 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 Revisited Chart 1: Portfolio Opportunity Sets: different correlations 30 all investment in N 25 expected return (%) 20 correlation=0 correlation=-1 15 correlation = 1 all investment in M correlation=0.25 10 5 0 0.0 5.0 10.0 15.0 risk (standard deviation (%)) 20.0 25.0 30.0 The Set of Risky Assets E(R) A SD(R) Optimal Set of Risky Assets E(R) y x SD(R) Adding the Riskless Asset E(R) Rf SD(R) The Capital Market Line E(R) G E The Capital Market Line H D SD(R) Investors Choice E(R) y x SD(R) Portfolio Theory Conclusions All Investors hold the same set of risky assets if they hold risky assets All investors must hold the market portfolio Their risk preferences determine whether they gear up or down by borrowing or lending Security Market Line Return Market Return = rm . Efficient Portfolio Risk Free Return = rf 1.0 BETA Security Market Line Return SML rf 1.0 BETA SML Equation = rf + B ( rm - rf ) Beta and Unique Risk im Bi 2 m Covariance with the market Variance of the market Downside Risk Measures Expert 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 meanvariance approach leads to unsatisfactory predictions of investor behavior. Post-Modern Portfolio Theory Two Fundamental Advances on MPT: Downside risk replaces standard deviation PMPT permits non-normal return distributions ‘PMPT’ 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 measures Shortfall probability Average shortfall Semi-variance LPM 0 LPM 1 LPM 2 p r r 0 r p r r 1 r p r r 2 r Downside Risk Defined by below-target semideviation Standard deviation of below-target returns Differentiates between risk and uncertainty Naturally incorporates skewness Recognizes that upside volatility is better than downside volatility Combines frequency and magnitude of bad outcomes No single riskless asset Value at Risk (VAR) Value at Risk VaR is a potential loss The ‘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 losses relative to 0 or the Mean which will only be exceeded in a particular proportion of instances over a particular time period. Value at Risk Frequency RVAR = |E(R) - x| AVAR = |0 - x| a x 0 E(R) Risk Definition of VAR Absolute VAR = (0 - x) Relative VAR = (y - x) Probability Distribution and Value at Risk Key Choice Parameters Time Period Confidence Level Time Period Liquidity of Portfolio Regulatory Framework (10 Days) Measurement Technique - Does one Assume Normality ? How does one deal with changing composition of Portfolio Required Data for Testing Confidence Level Risk Management/Capital Requirement Regulatory Requirement (1% VAR) Testing - Higher so more extreme observations Accounting and Comparison Measuring Value at Risk Variance/Covariance Historical Simulation Monte Carlo Simulation Stress Testing 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 cont’d The 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 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 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 = £11.65million VaR 10 days = £11.65 x 100.5=£36.84m Can adjust for expected return if any E.g. if E(z) = 0.1 percent per day, daily VaR in the example becomes £11.55m Because small, daily E(z) is ignored in practice Summary of Conclusions on Risk and Return I Investors Choose Between Distributions of Returns Return is Mean or Expected Return With Normal Distributions of Returns Mean and Standard Deviation of Returns totally summarise the information in the Distribution of Returns Summary of Conclusions on Risk 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 and 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.