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

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

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


• 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?


• Portfolio analysis with three stocks or investments

• Find the best combination of the three• Note Corr Coef.


• Nine portfolios analyzed, expressed as percentage combinations of Investments 1-3


• 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


• Preferred is 100% invested in Invest 1• Next best thing is P6, then P5

Stochastic Efficiency with Respect to A Function (SERF) Under a Neg. Exponential Utility Function

P1

P2

P3

P4

P5

P6

P7P8

P9

0.0970

0.0980

0.0990

0.1000

0.1010

0.1020

0.1030

0.1040

0.1050

0.1060

0.1070

0.1080

0.000000 0.000002 0.000004 0.000006 0.000008 0.000010 0.000012

ARAC

P1 P2 P3 P4 P5 P6 P7 P8 P9


• Next how does the preferred portfolio change as the investor considers investments with low correlation


• The results for simulating 9 portfolios where the individual investments have low correlation and near equal means

• Portfolios still have lower relative risk


• A portfolio (P6) is ranked second followed by P5

Stochastic Efficiency with Respect to A Function (SERF) Under a Neg. Exponential Utility Function

P1

P2

P3

P4

P5

P6

P7P8

P9

0.08

0.09

0.09

0.09

0.09

0.09

0.09

0.09

0.09

0.09

0.09

0 1E-06 2E-06 3E-06 4E-06 5E-06 6E-06 7E-06 8E-06

ARAC

P1 P2 P3 P4 P5 P6 P7 P8 P9


• 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

Portfolio Objective

HighCurrentIncome

ConservativeIncome

Income with

Growth

Growth with

Income

Growth Aggressive

Growth

Asset Class

Cash Equivalent 5% 5% 5% 5% -- --

Short/Intermediate Investment-Grade 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%


• 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)


• 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 constantSimulation Results

Mean 2.73 3.00 3.25 4.00 4.50 5.00Std. Dev. 1.84 1.84 1.84 1.84 1.84 1.85CV 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

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 = 2.0Test Values for Stoch 2 Applying the correction for the EMP simulationMean 0.253571 Theoretical mean for the the simulation is J * historical meanStd Dev 0.02327 =F3*D5*2/100 Theoretical std dev for the the simulation is

Historical Mean * J * Simulated CVTest of Hypothesis for Parameters for Stoch 2Confidence Level 95.0000%

Given ValueTest ValueCritical ValueP-Valuet-Test 0.253571 2.03 2.25 0.04 Fail to Reject the Ho that the Mean is Equal to 0.253571428571429Chi-Square Test 0.02327 507.41 LB: 439.00 0.78 Fail to Reject the Ho that the Standard Deviation is Equal to 0.0232702471108615

CV Stationarity and Empirical Distribution

CR 1 CR 2 CR 3 CR 4 CR 5 CR 6 CR 7 CR 8 CR 9 CR 100

5

10

15

20

25

30

35Normal for 10 Years with No Risk Adjustment

Average 5th Percentile 25th Percentile

75th Percentile 95th Percentile

CR 1 CR 2 CR 3 CR 4 CR 5 CR 6 CR 7 CR 8 CR 9 CR 100

5

10

15

20

25

30

35Empirical Fan Graph with No Risk Adjustment

Average 5th Percentile 25th Percentile

75th Percentile 95th Percentile

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 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

Documents

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