1. Portfolio Analyzer and Risk Stationarity Lecture 23 Read Chapters 13 and 14 Lecture 23 Portfolio Analyzer Example.xlsx Lecture 23 Portfolio Analyzer 2015.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

2. 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

3. 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

4. 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 raice corn and soybeans –Irrigated west grew cotton and alfalfa Undiversified portfolio is 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?

5. Portfolio Analysis Models Portfolio analysis with three stocks or investments Find the best combination of the three Note Corr Coef.

6. Portfolio Analysis Models Nine portfolios analyzed, expressed as percentage combinations of Investments 1-3

7. 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-P9, it is minimized with portfolio P7

8. Portfolio Analysis Models Preferred is 100% invested in Invest 1 Next best thing is P6, then P5

9. Portfolio Analysis Models Next how does the preferred portfolio change as the investor considers investments with low correlation

10. Portfolio Analysis Models The results for simulating 9 portfolios where the individual investments have low correlation and near equal means Portfolios still have lower relative risk

11. Portfolio Analysis Models A portfolio (P6) is ranked second followed by P5

12. 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

13. Strategic Asset Allocation Guidelines Portfolio Objective High Current Income Conservative Income Income with Growth Growth with Income Growth Aggressive Growth Asset Class Cash Equivalent5% -- Short/Intermediate Investment- Grade Bonds 20%30%20%10%-- Long Investment-Grade Bonds50%40%25%20%-- Speculative Bonds15%-- Real Estate10 %5% -- U.S. Large-Cap Stocks--20%30% 55%40% U.S. Mid-Cap Stocks-- 10%15%20% U.S. Small-Cap Stocks-- 10%15%20% Foreign Developed Stocks-- 5% 10%15% Foreign Emerging Market Stocks-- 5%

14. 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 Ỹ i ) –Multiply returns by portfolio j fractions or R ij = F j * Ỹ i –Sum returns across investments for portfolio j or P j = ∑ R ij sum across i investments for portfolio j –Simulate on the total returns (P j ) for all j portfolios –SERF ranking of distributions for total returns (P j )

15. Portfolio Analysis Models Typical portfolio analysis might look like: Assume 10 investments so stochastic returns are Ỹ i for i=1,10 Assume two portfolios j=1,2 Calculate weighted returns R ij = Ỹ i * F ij where F ij is fraction of funds invested in investment i for portfolio j Calculate total return for each j portfolio as P j = ∑ R ij

16. 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 –R t = (Price t – Price t-1 ) / Price t-1 –Temptation is to use the prices directly rather than percentage returns Brokerage houses provide prices on web in downloadable format to Excel

17. 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, CV Sim, to differ from historical CV Hist : CV Hist = σ H / Ῡ H Not Equal CV Sim = σ H / Ῡ S

18. 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 CV Sim = σ H / Ῡ S –If Mean for X decreases, CV increases, which implies more relative risk about the mean as we get farther out with the forecast CV Sim = σ H / Ῡ S See Chapter 9

19. 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 Mean2.733.003.254.004.505.00 Std. Dev.1.84 1.85 CV67.2461.4856.6546.0240.8837.04 Min-3.00-3.36-2.83-1.49-1.45-1.03 Max8.108.318.5910.509.8111.85

20. 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 J t+i value for each period (t+i) to simulate as: J t+i = Ῡ t+i / Ῡ history The J t+i value is then used to simulate the random variable in period t+i as: Ỹ t+i = Ῡ t+i + (Std Dev history * J t+i * SND) Ỹ t+i = NORM(Ῡ t+i, Std Dev * J t+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

21. 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 Ỹ t+i = Ῡ t+i * [1 + Empirical(S j, F(S j ), USD)] Simulation Results Mean2.743.003.254.004.505.00 Std Dev1.731.902.052.532.843.16 CV63.19 63.1863.19 Min0.00 Max5.155.656.127.538.479.42

22. 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

23. CV Stationarity and Empirical Distribution

24. Add Heteroskedasticy to Simulation Sometimes we want the CV to change over time –Change in policy could increase the relative risk –Change in management strategy could change relative risk –Change in technology can change relative risk –Change in market volatility can change relative risk Create an Expansion factor or E t+i value for each year to simulate –E t+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

25. 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 E t+i values for years 1-5 are, respectively, 0.0, 1.0, 1.15, 1.25, 1.25 Apply the E t+i expansion factors as follows: –Normal distribution Ỹ t+i = Ῡ t+i + ( Std Dev history * J t+i * E t+i * SND ) Ỹ t+i =NORM (Ῡ t+i, Std Dev history * J t+i * E t+i ) –Empirical Distribution if S i are deviations from mean Ỹ t+i = Ῡ t+i * { 1 + [Empirical(S j, F(S j ), USD) * E t+I ]}

26. Example of Expansion Factors