1. 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 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 they raise 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?

5. Portfolio Analysis Models
Portfolio analysis with nine stocks or investments
Find the best combination of the stocks
In reality most stocks move together (positive correlation coefficients) so Markowitz’s rule does not work

6. Portfolio Analysis Models
15 portfolios analyzed and expressed as percentage combinations of Investments

7. Portfolio Analysis Models
The statistics for 16 simulated portfolios show variance reduction relative to investing exclusively in one instrument
Look at the CVs across Portfolios P1-P16, it is minimized with portfolio P11

8. Portfolio Analysis Models
Preferred is 100% invested in Invest 6
Next best thing is P14, then P8

9. 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

10. Strategic Asset Allocation Guidelines
Portfolio Objective
High
Current
Income
Conservative
Income with
Growth
Growth with
Aggressive
Asset Class
Cash Equivalent 5% --
Short/Intermediate Investment-Grade Bonds
20%
30%
10%
Long Investment-Grade Bonds
50%
40%
25%
Speculative Bonds
15%
Real Estate
10 %
U.S. Large-Cap Stocks
55%
U.S. Mid-Cap Stocks
U.S. Small-Cap Stocks
Foreign Developed Stocks
Foreign Emerging Market Stocks

11. 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 you wish to make in your portfolio mix
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)

12. Portfolio Analysis Models
Typical portfolio analysis might look like:
Consider 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

13. Data for a Portfolio Analysis Models
Gather the prices of the stocks for the time period relevant to frequency of updating 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

14. 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

15. 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

16. 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
2.73
3.00
3.25
4.00
4.50
5.00
Std. Dev.
1.84
1.85 CV
67.24
61.48
56.65
46.02
40.88
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

17. 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

18. 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
63.18
Min
0.00
Max
5.15
5.65
6.12
7.53
8.47
9.42

19. 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

20. CV Stationarity and Empirical Distribution
Empirical distribution does not narrow over time whereas the Normal distribution gets narrower over time

21. Add Heteroskedasticy to Simulation
Sometimes we want the CV to change over time, different that what we saw in history
Change in policy could increase the relative risk (Brexit, New president in the US, Greece defaulting on loans, etc.)
Change in management strategy could change relative risk
Change in technology can change relative risk
Change in commodity market volatility can change relative risk
Create an Expansion factor or Et+i for each year
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 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

22. Add Heteroskedasticy to Simulation
Example of simulating 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

23. Add Heteroskedasticy to Simulation
Apply the Et+i expansion factors as follows:
Normal distribution
Ỹt+i = Ῡt+i + (Std Devhistory * Jt+i * Et+i * SND) Or
Ỹt+i =NORM (Ῡt+i , Std Devhistory * Jt+i * Et+i )
Empirical Distribution if Si are fractional deviations from the mean or trend
Ỹt+i = Ῡt+i * { 1 + [Empirical(Sj , F(Sj), USD) * Et+I ]}

24. Example of Expansion Factors for Simulation