Analyzing Risk in Timberland Portfolio
Allocations
A Coherent Risk Measure Approach
Stanislav Petrasek
Hancock Timber Resource Group
Boston, MA
Southern Forest Economics Workshop
Charlotte, North Carolina
March 20-21, 2012
Outline
1
Introduction
Motivation
Overview of Risk Measures
2
Asset Class Properties
Asset Classes
Normality Tests
3
Portfolio Optimization Results
Capital Allocation Scenarios
Asset Class Portfolio Weights
Re-sampling Timberland Allocations
Motivation
The mean-variance portfolio optimization model uses only
returns mean and variance to allocate capital.
The model implies that either
1
2
Investors’ preferences can be characterized by quadratic
utility functions, or
Asset returns are normally distributed.
In either case, the asset returns covariance matrix is the
appropriate risk measure.
However, empirical studies show that returns on financial
assets are frequently not normally distributed.
Therefore, the covariance matrix may be an inadequate
measure of risk.
Alternative risk measures are provided by Value at Risk
(VaR) and Conditional Value at Risk (CVaR).
VaR and CVaR Definitions & Illustration
Distribution of Returns
0.04
Definitions
0.02
1
CVaRα = −
α
α
0.01
CVaRα
VaRα
0.00
Density
0.03
VaRα = − inf{x ∈ R : FX (x) > α}
−30
−20
−10
0
Return (%)
10
20
30
Z
α
VaRs (X ) ds
0
Risk Measures
A risk measure is a function that calculates total portfolio risk
from individual asset returns.
Any coherent risk measure must be:
1
Monotonous – Larger losses translate to higher risks.
2
Positive Homogeneous – Increasing weights by a linear
factor increases risk by the same factor.
3
Translation Invariant – Adding a constant to losses
leaves risk unchanged.
4
Sub-additive – Total portfolio risk is no greater than the
sum of individual asset risks.
Risk Measure Coherency Test — Part 1
Simple investment scenario1 to test Covariance, VaR, and
CVaR:
Scenario
Asset 1
%
Asset 2
%
Portfolio
%
Probability
%
1
−20
2
−9
3
2
−3
2
−0.5
2
3
2
−20
−9
3
4
2
−3
−0.5
2
5
2
2
2
90
Large potential losses in the tails.
Individual losses diversified in the portfolio.
1
Adopted from Scherer and Martin, 2005
Risk Measure Coherency Test — Part 2
Risk Measure Performance Results
Risk Measure
Covariance
VaR
CVaR
Asset 1
%
Asset 2
%
Portfolio
%
3.80
3.80
2.63
−3
−13.20
−3
−13.20
−9
−5.60
Both Covariance and CVaR are coherent risk measures.
VaR attains a higher value for the portfolio of assets 1 and
2—this violates 4th Property of coherent risk measures.
Because VaR is not a coherent risk measure, we will not
calculate its values.
Annual Asset Class Returns
Mean Asset Returns
1969-2011 Returns Statistics
Asset
Mean
SD
Timberland, PNW
Timberland, South
Treasury Bills
S&P 500
Small Cap Stocks
Real Estate
Intl. Stocks
U.S. Bonds
U.S. Farmland
18.20
10.77
5.49
10.87
13.70
9.32
11.14
9.37
11.02
27.10
12.09
3.15
17.79
23.81
7.26
22.50
12.10
8.44
pnw
south
usTbills
sp500
smallCapEquities
realEstate
nonUsEquities
ltGovtBonds
farmlandUS
0
5
10
15
ltGovtBonds
usTbills
sp500
nonUsEquities
smallCapEquities
realEstate
pnw
south
farmlandUS
farmlandUS
south
pnw
realEstate
smallCapEquities
nonUsEquities
sp500
usTbills
ltGovtBonds
Asset Correlation Matrix Plot
Asset Class Returns Normality Tests
Jarque−Bera Test p−Values
Normality Test p-Values
Univariate Tests
p-Value
Timberland, PNW
Timberland, South
Treasury Bills
S&P 500
Small Cap Stocks
Real Estate
Intr. Stocks
U.S. Bonds
U.S. Farmland
7.95e−07
9.49e−01
3.77e−01
2.39e−01
6.81e−01
2.86e−07
9.74e−01
5.10e−01
4.53e−01
Multivariate Tests
p-Value
pnw
south
Shapiro Test
Energy Test
1.54e−06
2.70e−03
usTbills
sp500
smallCapEquities
realEstate
nonUsEquities
ltGovtBonds
farmlandUS
0.0
0.2
0.4
0.6
0.8
Timberland Returns in U.S. South
U.S. South Timberland Returns
1969−2011
0.03
0.02
0.01
0.00
Density
0.04
0.05
Summary Statistics
−20
−10
0
10
Returns (%)
20
30
40
Statistic
Value
Mean
Std. Deviation
Skewness
Excess Kurtosis
10.77
12.09
−0.11
−0.09
Timberland Returns in U.S. Pacific Northwest
Summary Statistics
0.025
U.S. Pacific Northwest Timberland Returns
1969−2011
Statistic
0.005
0.010
0.015
0.020
Mean
18.20
Std. Deviation
27.10
Skewness
1.52
Excess Kurtosis
2.16
0.000
Density
Value
−20
0
20
40
60
Returns (%)
80
100
120
Capital Allocation Scenarios
Scenario 1
Scenario 2
0 ≤ wi ≤ 1
Scenario 3
0 ≤ wi
0 ≤ wi ≤ 1
wpnw + wsouth ≤ 0.3
wreal
estate
wfarmland ≤ 0.05
≤ 0.2
wreal estate ≤ 0.1
wfarmland ≤ 0.1
wsouth ≤ 0.05
wpnw ≤ 0.05
wUS bonds ≥ 0.1
P9
i=1 wi
= 1 for all scenarios
wnonUS stocks ≥ 0.05
Scenario 1: No short sales
Scenario 2: Constrained alternative asset
Scenario 3: "Typical"2 wealth manager allocation
2
Constructed from surveys in Pensions & Investments
wS&P 500 ≥ 0.2
Weights
Target Risk
3.15
6.43
14
27.1
0.8
farmlandUS
ltGovtBonds
nonUsEquities
realEstate
smallCapEquities
sp500
usTbills
south
pnw
0.0
0.4
Weight
3.15
5.49
8.67
11.8
15
MV | solveRquadprog | minRisk = 2.47083
Long-Only Portfolio Allocations
18.2
Target Return
Weights
6.89
15
27.1
0.4
0.8
farmlandUS
ltGovtBonds
nonUsEquities
realEstate
smallCapEquities
sp500
usTbills
south
pnw
5.49
8.67
11.8
15
18.2
Target Return
CVaR | solveRglpk | minRisk = 2.70892
Target Risk
3.5
0.0
Weight
3.15
Weights
Target Risk
3.32
6.73
17.5
0.8
farmlandUS
ltGovtBonds
nonUsEquities
realEstate
smallCapEquities
sp500
usTbills
south
pnw
0.0
0.4
Weight
3.15
5.49
8.67
11.8
MV | solveRquadprog | minRisk = 2.51925
Portfolio Allocations with Real Asset Ceiling
15
Target Return
Weights
7.31
17.5
0.4
0.8
farmlandUS
ltGovtBonds
nonUsEquities
realEstate
smallCapEquities
sp500
usTbills
south
pnw
5.49
8.67
11.8
15
Target Return
CVaR | solveRglpk | minRisk = 2.70083
Target Risk
3.97
0.0
Weight
3.15
"Typical" Wealth Manager Portfolio Allocations
Weights
0.8
farmlandUS
ltGovtBonds
nonUsEquities
realEstate
smallCapEquities
sp500
usTbills
south
pnw
0.0
7.61
MV | solveRquadprog | minRisk = 4.8285
Target Risk
7.71
0.4
Weight
4.92
10.8
Target Return
Weights
0.4
0.8
farmlandUS
ltGovtBonds
nonUsEquities
realEstate
smallCapEquities
sp500
usTbills
south
pnw
7.61
10.8
Target Return
CVaR | solveRglpk | minRisk = 5.45773
Target Risk
7.71
0.0
Weight
5.46
Mean-Variance Portfolio Weights: Scenario 1
Considerable timberland
allocation variability
0.08
0.10
0.12
0.14
0.16
0.18
0.20
High fraction of zero
allocations to Southern
timberland
0.00
0.05
0.10
0.15
0.20
CVaR Portfolio Weights: Scenario 1
Lower allocation variability
to Pacific Northwest
timberland
0.1
0.2
0.3
0.4
Reduced fraction of zero
allocations to Southern
timberland
0.0
0.1
0.2
0.3
0.4
0.5
Mean-Variance Portfolio Weights: Scenario 2
Non-zero allocations to
timberland in both regions
Active ceiling constraints
0.08
0.10
0.12
0.14
Allocation variability
still high
0.10
0.15
0.20
CVaR Portfolio Weights: Scenario 2
High timberland allocation
variability
Active ceiling constraints
0.00
0.05
0.10
0.15
0.20
0.25
0.30
Zero allocations to
timberland in both regions
0.00
0.05
0.10
0.15
0.20
0.25
0.30
Covariance & CVaR Portfolio Weights: Scenario 3
Under Scenario 3, timberland allocations in U.S. South and
Pacific Northwest constrained 5% or less
Re-sampling results showed:
Both constraints active for all runs
No solution variability under either risk measure
That is, for both Covariance and CVaR:
i
i
wsouth
= wpnw
= 0.05
∀ i ∈ {1, . . . , 25 000}
This result implies that:
Historically, timberland asset class has performed well.
Choice of risk measure has limited impact on portfolio risk
due to small allocations and limited number of investment
opportunities.
Summary
Covariance is a coherent risk measure but should be used
with caution because asset returns are typically not
normally distributed
VaR is not a coherent risk measure and CVaR should be
used instead
In principle, allocations to timberland differ in response to
the choice of a risk measure
In practice, timberland allocations within institutional
portfolios have been too small for these differences to
matter
References4
Bernd Scherer and R. Douglas Martin
Introduction to Modern Portfolio Optimization With NUOPT
and S-PLUS
Springer, 2005
Stanislav Petrasek, Brent J. Keefer, Mary Ellen Aronow,
and Courtland L. Washburn
Statistical Distributions of Timberland Returns
HTRG Research Report, 2011, Available on HTRG Site3
Pensions & Investments
Surveys of Investment Managers’ Portfolio Allocations
3
4
Please contact me to request a copy if you are not a current investor
This version created on 2012-03-13 13:23:06 -0400 (Tue, 13 Mar 2012)