Tài chính doanh nghiệp - Topic 2: Measuring portfolio risk and return

1A Course in Risk ManagementInstructor: Lou GattisTopic #2: Measuring Portfolio Risk and ReturnLearning ObjectivesMeasure the risks of a stock investmentMeasure the risks of a portfolio of stocks, bills, and bonds2Risk Measurement - MeasuresWhat is the risk of a $100,000 investment in domestic stocks?Stress TestsHypothetical Stress TestE.g. +/- 10%, E.g. Market Crash: -30%Historical Stress TestE.g. Great Depression, Housing Crash, Black Monday (10/19/87)Value-at-Risk (VaR)Historical Simulated VaRE.g. 1-year, 90%, 95%, 99%,Variance-Covariance VaR (Assumes Normal Distribution)E.g. 1-year Horizon, 90% Confidence IntervalE.g. 1-day Horizon, 95% Confidence IntervalE.g. 5-day Horizon, 95% Confidence Interval3Hypothetical Stress Tests$100,000 investment in domestic stocksA hypothetical stress test specifies a specific loss and/or market environment+/- 10%: $10,00030% Market Crash: $30,000ProsTransparent, SimpleConsNo horizon specifiedDo not know likelihood of lossDoes not use historical record4Historical Stock Returns5Historical Stress TestsA historical stress test computes the loss from a historical time period and/or eventExamples for $100,000 stock investmentGreat Depression 1929-1931: $61,582Housing Crash 2008: $36,550Black Monday (October 19, 1987): $22,680ProsTransparent, SimpleUses the historical recordHorizon specifiedConsDo not know likelihood of future lossCannot specify horizonHistorical sample is limited61928-2011Historical Simulated VaRVaR is the maximum loss not exceeded with given a horizon and confidence levelExample: What is the maximum loss over 1-year with a 90% confidence level.Historical Simulated VaR: Simulates portfolio losses using historical asset returns.VaR(1-Year, 90%) = 10th percentile largest lossUsing the 84 years supplied, record the 8th (.1*84) largest loss7Returns Sorted Low to HighPercent = # / 84 obsVariance-Covariance VaRVaR is the maximum loss not exceeded with given a horizon and confidence levelThe Historical Simulated VaR uses a historical distributionVaR is flexible enough to use other distributions, such as the Normal DistributionA VaR that assumes asset returns are normally distribution is often call a variance-covariance VaR (VC VaR)It is called the VC VaR because the portfolio variance (or standard deviation) is a parameterCovariance terms are needed to compute portfolio variance when there is more than one variableAlso known as Parametric VaR or Analytical VaR89History of Normal DistributionsDe Moivre (18th Century statistician and gambling consultant) finds the distribution of coin flips has orderDist. of heads in 12 coin flipsGalton BoardThis order is considered nature’s “normal” orderAlso called a bell curve, or Gaussian distributionCentral Limit Theory: A large sample of independent, random observations is often assumed to be normally distributed (under certain conditions)Normal curves are used inLiving organisms’ height, weight, mass, mortalityE.g.; Male height μ=5’10”, σ =3”Intensity of lightWeather – rain, snow, temperaturesIQ Test ScoresAsset returns10Return Mean, Variance and Standard DeviationArithmetic Mean measures the central tendency of the return.Variance measures the average squared deviation from the mean or expected return. Standard Deviation is the square root of variance and is used to compare with returns11Standard deviation fully describes risk if returns are normally distributed Asset returns are often assumed to be normally distributed, independent events as a computational short-cut. However, the historical record shows some deviation from normality. For example, large cap U.S. stock returns from 1928-2011 exhibit:Negative Skew: Fatter left-tailed distributionNegative Kurtosis: Less peaked “flatter” distributionPositive Serial Correlation: returns follow trends (although weakly)Standard deviation is often simply called volatility or “vol” in financeNote: If returns were normally distributed, skew and kurtosis would be zeroHistorical Record1213Variance-Covariance VaRIf portfolio returns are normally distributed with expected return, µ, and standard deviation, σ, then the X% Confidence Interval VaR for a portfolio value of V$ is:Where Zx is from the normal distribution table Z90% = 1.28; Z95% = 1.65; Z99% = 2.3310%VaR(1Yr, 90%)=V(μ-1.28σ)VaR(1Yr, 95%)=V(μ-1.65σ)VaR(1Yr, 99%)=V(μ-2.33σ)5%1%14=normsdist(σ)1.6451.282.33Variance-Covariance VaRIf portfolio returns are normally distributed with expected return, µ, and standard deviation, σ, then the X% Confidence Interval VaR for a portfolio value of V$ is:VC VaR Assuming stock returns are normally distributed with μ=11.2% and σ=20%; Stock Returns ~ N(.112,.20)Assume 252 trading days in a yearVaR(90%,1-year)=-100,000(.112-1.28*.2)=$14,400VaR(95%,1-year)=-100,000(.112-1.65*.2)=$21,800VaR(99%,1-year)=-100,000(.112-2.33*.2)=$35,400VaR(95%,1-day)=-100,000(.112/252-1.65*.2/sqrt(252))=$2,034VaR(99%,5-day)=-100,000(.112/252*5-2.33*.2/sqrt(252)*sqrt(5))=$6,324Risk Report – U.S. Equity16Portfolio Risk What is the risk of a $100,000 portfolio invested 50% in domestic Stocks, 30% in Treasury Bonds, and 20% invested in Treasury Bills.Risk MeasuresHypothetical Stress TestHistorical Stress TestHistorical VaRVC VaR17Hypothetical Stress TestsA hypothetical stress test specifies a specific loss and/or market environmentDeflationary Recession (interest rates and equities fall) Assume: Stock -30%, Bonds +10%, Bills +0%=100000*(.5*-.3+.3*.1)=-$12,000Inflationary Recession (Interest rates rise, equities fall)(Stock -30%, Bonds -10%, Bills +0)=100000*(.5*-.3+.3*-.1)=-$18,00018Historical Stress TestsA historical stress test computes the loss from a historical time period and eventExamples for $100,000 stock investmentGreat Depression 1929-1931: -$31,8001929: Stock/Bond/Bill -8%/+4%/+3%1930: Stock/Bond/Bill -25%/+5%/+5% 1931: Stock/Bond/Bill -44%/-3%/+2%Housing Crash 2008:-$11,9302008: Stock/Bond/Bill -37%/+20%/+2%19Historical Simulated 1-Yr VaRs20Variance-Covariance VaRThe VC VaR requires portfolio mean and standard deviation21Expected rate of return of a portfolio:The variance of the portfolio rate of return:Expected Return for a portfolio with N-assetsExpected rate of return of the portfolio:i=1 to Nwiri1w1r12w2r23w3r3Sum=w1r1+w2r2+w3r3e.g., N=32223Variance of a 3-asset Portfolioi=1 to nj =1 to nwiwjσij11w1w1σ11= w12σ1212w1w2σ1213w1w3σ1321w2w1σ2122w2w2σ22= w22σ2223w2w3σ2331w3w1σ3132w3w2σ3233w3w3σ33= w32σ32Sum=w12σ12 +.+ w32σ3224Correlation and covariance are measures of diversificationCovariance is the average of the product of two securities’ deviations. (Actual – Expected)Where i is the period, and there are n period returnsIf both securities returns tend to be above or below their expected return at the same time, the covariance will be a positive value (-,- or +,+)Low covariances are an indicator of diversification, but its hard to compare because covariances also depend on the relative size of each assets standard deviationCorrelation normalizes covariances and are bounded by 1 and -1Stock and Bond Covariance and Correlation2526The Covariance Matrix shows all the correlations among assetsThe covariance matrix shows the covariances ( ij) between the returns on any pair of securities. The diagonal simply shows all the variances.Note: σ11= σ12 and σ12= σ2127The Covariance Matrix makes it easier to compute portfolio varianceTo find the variance of this portfolio, we need the portfolio weights and the variances and covariances:Portfolio Risk and Return28Variance-Covariance VaRIf portfolio returns are normally distributed with expected return, µ, and standard deviation, σ, then the X% Confidence Interval VaR for a portfolio value of V$ is:VC VaR Assuming stock returns are normally distributed with μ=7.96% and σ=10.3%Assume 252 trading days in a yearVaR(90%,1-year)=-100,000(.0796-1.28*.103)=$5,224VaR(95%,1-year)=-100,000(.0796-1.65*.103)=$9,035VaR(99%,1-year)=-100,000(.0796-2.33*.103)=$16,039VaR(95%,1-day)=-100,000(.0796/252-1.65*.103/sqrt(252))=$1,039VaR(99%,5-day)=-100,000(.0796/252*5-2.33*.103/sqrt(252)*sqrt(5))=$3,223Risk Report – 50/30/20 Portfolio3031Implementing VaRV: Positions to Includeμ: Expected Returnσ: Return Standard Deviationt: Time PeriodX: Probability, which determines Z(X)32VaR: Positions (V)Must choose whether to apply the VaR to entire portfolio or a subset of portfoliosMust choose which portfolio losses affect the viability of the firmTrading Portfolio: Losses reduce earningsAvailable-for-Sale: Losses reduce equityHold-to-Maturity: Losses no recordedCombinations or other33VaR: Expected ReturnExpected returns could be calculated using historical averages or model, such as CAPM or APT (Arbitrage pricing theory)Daily VaRs often assume zero daily expected return34VaR: Standard DeviationStandard deviations are often calculated using historical standard deviationsMust choose how far to go backAnswer: which ever is a better indicator of future riskIf not sure use more dataHistorical data is often adjusted to add weight to the most recent dataOne often used historical method is the exponentially weighted moving average (EWMA).GARCHAlternative is to use Black-Scholes implied volatilities computed using option prices35VaR: Time Period (t)Time period often reflectsthe time it takes to hedge or unwind the portfolioavailability of new parameters (V, Vol, Mean)Reporting horizonChoose 1-day if position could be easily hedged or unwound on a daily basis and new portfolio value and sigma are available dailyUnwinding: depends on market liquidity and sizeHedging: depends on availability of derivativesLonger term VaRs may have little significance if the positions change and are hedged frequently36VaR DistributionStock and bond returns are often assumed to be normally distributedOther securities, such as options have non-normal distributionVaR is general enough to use any distributionNormal, Lognormal, HistoricalSoftware packages like @Risk and Crystal Ball have many other distributions37VaR: probability and limitThe choice of confidence interval and maximum loss limit depends on the firms risk appetite and the consequences of a lossThe standard implementation of VaR is to use 95% or 99% Confidence IntervalLimit is level of VaR that requires management actionE.g.: Set VaR Limit to $1M, if VaR>$1M, requires actionSet limit equal to loss that could trigger significant negative consequences for the firmE.g., miss earnings target, trigger margin call, ratings downgrade 38Advantages of VaRCan summarize the total risk of a corporation’s vast operations in one numberAccounts for expected variability of risk factorsAccounts for the expected correlations between risk factorsCan be used to make estimates of the probability of loss (e.g., normal distribution cumulative distribution function, CDF)Can be scaled across time by using the sqrt(t)39Criticisms of VaRReliance on historical correlations and volatilitiesNon-normal DistributionsFat TailsAsymmetrical payoffs (Option Payoffs, Bond Defaults)Black Swan CritiqueVaR does not provide information about how big losses can get if VaR is exceededConditional VaR or Expected Tail Loss is a measure of average loss if VaR is exceededIn practice, managers often set capital cushion equal to 2-3 times VaR to protect against expected tail events (larger that Zσ moves) and non-normality.40Risk Management with VaRVaR is a measures of exposure in portfolio valueManagers must determine the optimal amount of riskManagers can adjust risk levels byUnwinding positionsTaking on risky projects with a negative covariance to existing portfolioUsing DerivativesRaise Capital so losses do not cause distressSetting capital cushion equal to 99% VaR would in theory set the probability of bankruptcy to 1%. VaR, Leverage, and Capital41Use Goal Seek to compute the probability of default --- 2.01% probability of defaultAnalytical Solution100,000=VaR100,000=-1,000,000*(.0796/252*100-Z*.1030/sqrt(252)*sqrt(100)Z=2.03From Normal Table = Z(2.03)=.979FormulasProblemsFor Assets A, B, and C above-- compute asset return means, standard deviations, correlation matrix, and covariance matrix. Compute by hand and verify using a computer. Show Work.What is the 90- business day, 99% VaR of a $1M portfolio that invests 50% in A, 40% in B, and 10% in C. Compute by hand and verify using a computer. Show Work. (assume 252 business days/year)? Suppose the above portfolio was funded with $950,000 in debt and $50,000 in equity. What is the probability of default in 90 business days. Compute by hand and verify using a computer. Show Work. 43Problems4. Using Excel, Compute the Mean, Standard Deviation, and 95%, 1-yr VaR of the above portfolio. 44