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Portfolio Analyzer and Risk Stationarity Lecture 23 • • • • • • • Read Chapters 13 and 14 Lecture 23 Portfolio Analyzer Example.xlsx Lecture 23 Portfolio Analyzer 2016.XLSX Lecture 23 Portfolio Low Correlation.XLSX Lecture 23 Portfolio High Correlation.XLSX Lecture 23 Changing Risk Over Time.XLSX Lecture 23 CV Stationarity.XLSX Portfolio and Bid Analysis Models • Many business decisions can be couched in a portfolio analysis framework • A portfolio analysis refers to comparing investment alternatives • A portfolio can represent any set of risky alternatives the decision maker considers • For example an insurance purchase decision can be framed as a portfolio analysis if many alternative insurance coverage levels are being considered Portfolio Analysis Models • Basis for portfolio analysis – overall risk can be reduced by investing in two risky instruments rather than one IF: – This always holds true if the correlation between the risky investments is negative – Markowitz discovered this result 50+ years ago while he was a graduate student! – Old saw: “Don’t put all of your eggs in one basket” is the foundation for portfolio analysis Portfolio Analysis Models • Application to business – given two enterprises with negative correlation on net returns, then we want a combination of the two rather than specializing in either one – Mid West used to raise corn and feed cattle, now rice corn and soybeans – Irrigated west grew cotton and alfalfa • Undiversified portfolio is to grow only corn • Thousands of investments, which ones to include in the portfolio is the question? – Own stocks in IBM and Microsoft – Or GMC, Intel, and Cingular • Each is a portfolio, which is best? Portfolio Analysis Models • Portfolio analysis with three stocks or investments • Find the best combination of the stocks Correlation Matrix Invest 1 Invest 2 Invest 3 Invest 4 Invest 5 Invest 6 Invest 7 Invest 8 Invest 9 Invest 1 1 0.94 0.82 -0.07 0.77 0.48 0.97 0.34 0.91 Invest 2 1 0.75 0.00 0.79 0.45 0.97 0.41 0.99 Invest 3 1 0.12 0.67 0.39 0.86 0.44 0.68 Invest 4 1 0.55 0.68 0.00 0.84 0.01 Invest 5 1 0.85 0.80 0.81 0.80 Invest 6 1 0.44 0.86 0.49 Invest 7 1 0.42 0.94 Invest 8 1 0.43 Invest 9 1 Portfolio Analysis Models • Nine portfolios analyzed, expressed as percentage combinations of Investments Step 9 Define the portfolios to be tested, as fractions of each instrument. Invest 1 Invest 2 Invest 3 Invest 4 Invest 5 Invest 6 Invest 7 Invest 8 Invest 9 Sum Check Portfolio 1 1 1 Portfolio 2 1 1 Portfolio 3 1 1 Portfolio 4 1 1 Portfolio 5 1 1 Portfolio 6 1 1 Portfolio 7 1 1 Portfolio 8 1 1 Portfolio 9 1 1 Portfolio 10 0.1111 0.1111 0.1111 0.1111 0.1111 0.1111 0.1111 0.1111 0.1111 1 Portfolio 11 0.2 0.2 0.2 0.2 0.2 1 Portfolio 12 0.2 0.2 0.2 0.2 0.2 1 Portfolio 13 0.2 0.2 0 0.2 0.2 0.2 1 Portfolio 14 0.4 0.4 0.2 1 Portfolio 15 0.333 0.334 0.333 1 Portfolio 16 0.5 0.5 1 Portfolio Analysis Models • The statistics for 9 simulated portfolios show variance reduction relative to investing exclusively in one instrument • Look at the CVs across Portfolios P1-P6, it is minimized with portfolio P11 Mean StDev CV Min Max Iteration 0.101 0.080 0.045 0.182 0.151 0.282 0.067 0.193 0.099 0.133 0.112 0.158 0.159 0.224 0.117 0.073 0.073 0.064 0.037 0.240 0.108 0.223 0.055 0.220 0.086 0.096 0.077 0.120 0.112 0.206 0.083 0.052 71.956 79.763 82.002 131.544 71.726 78.977 82.802 114.113 87.241 71.745 68.467 75.675 70.628 91.944 70.625 71.205 -0.013 -0.010 -0.027 -0.234 -0.058 -0.061 -0.017 -0.189 -0.025 -0.070 -0.068 -0.070 -0.075 -0.156 -0.032 -0.020 0.223 0.195 0.114 0.748 0.317 0.678 0.163 0.537 0.269 0.360 0.318 0.393 0.432 0.678 0.270 0.168 P1 P2 P3 P4 P5 P6 P7 P8 P9 P10 P11 P12 P13 P14 P15 P16 Portfolio Analysis Models • Preferred is 100% invested in Invest 6 • Next best thing is P14, then P8 Stochastic Efficiency with Respect to A Function (SERF) Under a Power Utility Function 0.30 P6 0.25 P14 P8 P4 P13 P12 P5 P10 P15 P11 P1 P9 P2 P16 P7 P3 0.20 0.15 0.10 0.05 0.00 0 1 2 3 4 5 RRAC P1 P2 P3 P4 P5 P6 P7 P8 P9 P10 P11 P12 P13 P14 P15 P16 Portfolio Analysis Models • How are portfolios observed in the investment world? • The following is a portfolio mix recommendation prepared by a major brokerage firm • The words are changed but see if you can find the portfolio for extremely risk averse and slightly risk averse investors Strategic Asset Allocation Guidelines High Current Income Conservative Income Income with Growth Growth with Income Growth Aggressive Growth Cash Equivalent 5% 5% 5% 5% -- -- Short/Intermediate InvestmentGrade Bonds 20% 30% 20% 10% -- -- Long Investment-Grade Bonds 50% 40% 25% 20% -- -- Speculative Bonds 15% -- -- -- -- -- Real Estate 10 % 5% 5% 5% -- -- U.S. Large-Cap Stocks -- 20% 30% 30% 55% 40% U.S. Mid-Cap Stocks -- -- 10% 15% 20% 20% U.S. Small-Cap Stocks -- -- -- 10% 15% 20% Foreign Developed Stocks -- -- 5% 5% 10% 15% Foreign Emerging Market Stocks -- -- -- -- -- 5% Portfolio Objective Asset Class Portfolio Analysis Models • Simulation does not tell you the best portfolio, but tells you the rankings of alternative portfolios • Steps to follow for portfolio analysis – Select investments to analyze – Gather returns data for period of interest – annual, monthly, etc. based on frequency of changes – Simulate stochastic returns for investment i (or Ỹi) – Multiply returns by portfolio j fractions or Rij= Fj * Ỹi – Sum returns across investments for portfolio j or Pj = ∑ Rij sum across i investments for portfolio j – Simulate on the total returns (Pj) for all j portfolios – SERF ranking of distributions for total returns (Pj) Portfolio Analysis Models • Typical portfolio analysis might look like: • Assume 10 investments so stochastic returns are Ỹi for i=1,10 • Assume two portfolios j=1,2 • Calculate weighted returns Rij = Ỹi * Fij where Fij is fraction of funds invested in investment i for portfolio j • Calculate total return for each j portfolio as Pj = ∑ Rij Data for a Portfolio Analysis Models • Gather the prices of the stocks for the time period relevant to frequency of your investment decision – Monthly data if adjust portfolio monthly, etc. – Annual returns if adjust once a year • Convert the prices to percentage changes – Rt = (Pricet – Pricet-1) / Pricet-1 – Temptation is to use the prices directly rather than percentage returns • Brokerage houses provide prices on web in downloadable format to Excel Covariance Stationary & Heteroskedasticy • Part of validation is to test if the standard deviation for random variables match the historical std dev. – Referred to as “covariance stationary” • Simulating outside the historical range causes a problem in that the mean will likely be different from history causing the coefficient of variation, CVSim, to differ from historical CVHist: CVHist = σH / ῩH Not Equal CVSim = σH / ῩS Covariance Stationary • CV stationarity is likely a problem when simulating outside the sample period: – If Mean for X increases, CV declines, which implies less relative risk about the mean as time progresses CVSim = σH / ῩS – If Mean for X decreases, CV increases, which implies more relative risk about the mean as we get farther out with the forecast CVSim = σH / ῩS • See Chapter 9 CV Stationarity • The Normal distribution is covariance stationary BUT it is not CV stationary if the mean differs from historical mean • For example: – Historical Mean of 2.74 and Historical Std Dev of 1.84 • Assume the deterministic forecast for mean increases over time as: 2.73, 3.00, 3.25, 4.00, 4.50, and 5.00 • CV decreases while the std dev is constant Simulation Results Mean Std. Dev. CV 2.73 1.84 67.24 3.00 1.84 61.48 3.25 1.84 56.65 4.00 1.84 46.02 4.50 1.84 40.88 5.00 1.85 37.04 Min -3.00 -3.36 -2.83 -1.49 -1.45 -1.03 Max 8.10 8.31 8.59 10.50 9.81 11.85 CV Stationarity for Normal Distribution • An adjustment to the Std Dev can make the simulation results CV stationary if you are simulating a Normal dist. • Calculate a Jt+i value for each period (t+i) to simulate as: Jt+i = Ῡt+i / Ῡhistory • The Jt+i value is then used to simulate the random variable in period t+i as: Ỹt+i = Ῡt+i + (Std Devhistory * Jt+i * SND) Ỹt+i = NORM(Ῡt+i , Std Dev * Jt+i) • The resulting random values for all years t+i have the same CV but different Std Dev than the historical data – This is the result desired when doing multiple year simulations CV Stationarity and Empirical Distribution • Empirical distribution automatically adjusts so the simulated values are CV stationary if the distribution is expressed as deviations from the mean or trend Ỹt+i = Ῡt+i * [1 + Empirical(Sj , F(Sj), USD)] Simulation Results Mean 2.74 3.00 3.25 4.00 4.50 5.00 Std Dev 1.73 1.90 2.05 2.53 2.84 3.16 CV 63.19 63.19 63.18 63.19 63.19 63.19 Min 0.00 0.00 0.00 0.00 0.00 0.00 Max 5.15 5.65 6.12 7.53 8.47 9.42 Empirical Distribution Validation • Empirical distribution as a fraction of trend or mean automatically adjusts so the simulated values are CV stationary – This poses a problem for validation • The correct method for validating Empirical distribution is: – Calculate the Mean and Std Dev to test against as follows – Mean = Historical mean * J – Std Dev = Historical mean * J * CV for simulated values / 100 • Here is an example for J = 2.0 Test Values for Stoch 2 Applying the correction for the EMP simulation Mean 0.253571 Theoretical mean for the the simulation is J * historical mean Std Dev 0.02327 =F3*D5*2/100 Theoretical std dev for the the simulation is Historical Mean * J * Simulated CV Test of Hypothesis for Parameters for Stoch 2 Confidence Level 95.0000% Given Value Test ValueCritical Value P-Value t-Test 0.253571 2.03 2.25 0.04 Fail to Reject the Ho that the Mean is Equal to 0.25357142857 Chi-Square Test 0.02327 507.41 LB: 439.00 0.78 Fail to Reject the Ho that the Standard Deviation is Equal to 0 CV Stationarity and Empirical Distribution Normal for 10 Years with No Risk Adjustment 35 30 25 20 15 10 5 0 CR 1 CR 2 CR 3 CR 4 CR 5 CR 6 Average 5th Percentile 75th Percentile 95th Percentile CR 7 CR 8 CR 9 CR 10 25th Percentile Empirical Fan Graph with No Risk Adjustment 35 30 25 20 15 10 5 0 CR 1 CR 2 CR 3 CR 4 CR 5 CR 6 Average 5th Percentile 75th Percentile 95th Percentile CR 7 CR 8 CR 9 25th Percentile CR 10 Add Heteroskedasticy to Simulation • Sometimes we want the CV to change over time – – – – Change in Change in Change in Change in policy could increase the relative risk management strategy could change relative risk technology can change relative risk market volatility can change relative risk • Create an Expansion factor or Et+i value for each year to simulate – – – – – – Et+i is a fractional adjustment to the relative risk Here are the rules for setting and Expansion Factor 0.0 results in No risk at all for the random variable 1.0 results in same relative risk (CV) as the historical period 1.5 results in 50% larger CV than historical period 2.0 results in 100% larger CV than historical period • Chapter 9 Add Heteroskedasticy to Simulation • Simulate 5 years with no risk for the first year, historical risk in year 2, 15% greater risk in year 3, and 25% greater CV in years 4-5 – The Et+i values for years 1-5 are, respectively, 0.0, 1.0, 1.15, 1.25, 1.25 • Apply the Et+i expansion factors as follows: – Normal distribution Ỹt+i = Ῡt+i + (Std Devhistory * Jt+i * Et+i * SND) Ỹt+i =NORM (Ῡt+i , Std Devhistory * Jt+i * Et+i ) – Empirical Distribution if Si are deviations from mean Ỹt+i = Ῡt+i * { 1 + [Empirical(Sj , F(Sj), USD) * Et+I ]} Example of Expansion Factors