Download overhead - 23 Portfolio, Risk Stationarity

Portfolio Analyzer and Risk Stationarity Lecture 23
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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