Portfolio Modelling:
Applications & Limitations of MPT
M.Sc. Real Estate Management & Construction
Project Management, University of Wuppertal
April 1, 2008
Dr. Franz Fuerst
Investor Risk Tolerance and SML
• To attain a higher expected return than is
available at the market portfolio (in
exchange for accepting higher risk), an
investor can borrow at the risk free-rate.
• Other minimum variance portfolios (on the
efficient frontier) are not considered.
Assumptions / Limitations of
Markowitz Portfolio Theory
• Investors take a single-period perspective
in determining their asset allocation.
– Drawback: Investors seldom have a singleperiod perspective. In a multiple-period
horizon, even Treasury bills exhibit variability
in returns
– Possible Solutions:
• Include the “risk-free asset” as a risky asset class.
• If investors have a liquidity need, construct an efficient
frontier and asset allocation on the funds remaining after
the liquidity need is satisfied.
Assumptions / Limitations of
Markowitz Portfolio Theory
• Investors base decisions solely on expected
return and risk. These expectations are derived
from historical returns.
– Drawback: Optimal asset allocations are highly sensitive
to small changes in the inputs, especially expected
returns. Portfolios may not be well diversified.
– Potential solutions:
• Conduct sensitivity tests to understand the effect on asset
allocation to changes in expected returns.
• Use a more robust approach to developing asset allocations
(Reverse Optimization).
Assumptions / Limitations of
Markowitz Portfolio Theory
• Investors can borrow and lend at the riskfree rate.
– Drawback: Borrowing rates are always higher
than lending rates. Certain investors are
restricted from purchasing securities on margin.
– Potential solutions:
• Differential borrowing and lending rates can be easily
incorporated into MPT. However, leverage may be
practically irrelevant for many investors (liquidity, regulatory
restrictions).
Practical Application of MPT
• MPT can be used to determine optimal
portfolio weights with a certain subset of
all investable assets.
• An efficient frontier can be constructed
with inputs (expected return, standard
deviation and correlations) for the
selected assets.
Practical Application of MPT
• MPT can be either unconstrained, in which
case we do not place any constraints on
the asset weights, or it can be
constrained.
Practical Application of MPT
• Unconstrained Optimization
– The simplest optimization places no
constraints on asset-class weights except that
they add up to 1.
– With unconstrained optimization, the asset
weights of any minimum variance portfolio is a
linear combination of any other two minimum
variance portfolios.
Practical Application of MPT
• Constrained Optimization
– The more useful optimization for strategic
asset allocation is constrained optimization.
– The main constraint is usually a restriction on
short sales.
Practical Application of MPT
• Excel Solver is a powerful tool that can be
used to determine optimal portfolio
weights for a set of assets.
• To use the tool, we need expected returns
and standard deviations for our assets as
well as a set of constraints that are
appropriate for the portfolio.
DIVERSIFICATION ANALYSIS
•
•
•
•
•
GEOGRAPHY
PROPERTY-TYPE
LEASE STRUCTURES
FINANCIAL STRUCTURES
OWNERSHIP STRUCTURES
MSA ECONOMIC
DIVERSIFICATION
ANALYSIS
ECONOMIC PERFORMANCE
Constant
High
Growth
Recent
High
Growth
Diversified
Cyclical
Patterns
Consistent
Low
Growth
Los Angeles
Energy
Denver
Government
Baltimore
Manufacturing
Services
Recent
Low
Growth
Chicago
San Francisco
Philadelphia
ISSUES IN PORTFOLIO
ANALYSIS
• 1. ALLOCATION PARADIGM
– REAL ESTATE
– OTHER ASSETS
•
•
•
2. EX POST VS. EX ANTE ANALYSIS
3. RE-BALANCING
4. REAL ESTATE, INFLATION, AND
OTHER ASSETS
CORRELATION COEFFICIENTS
BASED ON GEOGRAPHIC
DIVERSIFICATION
Naive Diversification (1974:IV - 1983:III)
East
Midwest
West
South
East
1.000
0.576
0.478
0.298
Midwest
West
South
1.000
0.628
0.299
1.000
0.286
1.000
Eight-Region Diversification (1974:IV - 1987:II)
N. E.
New England
1.000
Mid-Atlantic
-0.154
Old South
0.226
Industrial
-0.030
Farm Belt
0.010
Mineral
-0.212
Southern California 0.131
Northern California 0.039
M. A. O. S.
1.000
0.241
0.396
0.304
0.092
0.373
0.268
1.000
0.228
0.209
0.182
0.359
0.089
IND.
F. B.
MIN. S. CA N. CA
1.000
0.389
0.351
0.564
0.372
1.000
0.308 1.000
0.307 0.195 1.000
0.069 0.198 0.312 1.000
Source: "Refining the Analysis of Regional Diversification of Income Producing Real Estate."
Financial Research. Vol. 2. No. 2 (1987), pg. 90-91.
Journal of
IMPACT OF ADDING
APARTMENTS
ON PORTFOLIO:
RISK/RETURN RELATIONSHIP
Expected Return
10.8
10.9
11.0
11.1
11.2
Standard Deviation Standard Deviation
of 3-Property
of Portfolio with Basis Point Decline
Portfolio
Apartments
in Risk (%)
3.41
3.59
3.88
4.24
4.67
3.29
3.33
3.45
3.63
3.87
Source: Paul Firstenberg and Charles Wurtzebach. "Managing Portfolio and Reward."
Review . Vol. 19. No. 2 (Summer 1989). Exhibit 2. pg. 64.
0.12
0.19
0.43
0.61
0.80
Real Estate
Correlation of asset types
1970-1997(Ambrose, 2003)
v s. RSTOC
Stocks & RBOND
Bonds:
+44%
Stocks & Real
RREST vEstate:
s. RSTOC +7%
0.5
Real Estate&
RREST vBonds:
s. RBOND -34%
0.3
0.3
0.2
0.2
0.1
0.1
0.4
0.2
0.1
0.0
-0.1
0.0
-0.1
-0.4
RREST
RREST
RBOND
0.3
-0.2
0.0
RSTOC
0.2
0.4
-0.2
-0.4
0.0
-0.1
-0.2
0.0
RSTOC
0.2
0.4
-0.2
-0.1
0.0
0.1
0.2
RBOND
0.3
0.4
0.5
Investor Utility and the Efficient
Frontier
•
E(R)
Efficient Frontier
σp
Efficient Markets & Equilibrium
Portfolios
E(Rm)
M
.
Z
Rf
σm
If Z is not held in
equilibrium then
price will fall and
E(Rz) will rise until
it is bought by some
investor.
MPT and the Aggregate Capital
market
•
•
•
What happens if everyone does it (MPT)?
What happens in equilibrium?
Can we find pricing model?
Refinements of MPT: Sharpe,
Lintner, Mossin, Treynor
•
•
•
•
Assume homogenous expectations
Risk free rate available
Identical Efficient Frontier for all investors
Geared up or down by different amounts
Sharpe’s Integrated Asset Allocation Model
C1
Capital Market
Conditions
I1
Investor Assets, Liabilities
and Net Worth
C2
Prediction
Procedure
I2
Investor's Risk Tolerance
Function
C3
Expected Returns, Risk
and Correlations
I3
Investor's Risk Tolerance
M1
Optimizer
M2
Investor's Asset Mix
M3
Returns
Sharpe’s Integrated Asset Allocation Model
(cont.)
• Notice that the feedback loops after the performance
assessment box (M3) make the portfolio management
process dynamic in nature.
• The strategic asset allocation process can be viewed
as going through the model once and then removing
boxes (C2) and (I2), thus removing any temporary
adjustments to the baseline allocation.
• Tactical asset allocation effectively removes box (I2),
but allows for allocation adjustments due to perceived
changes in capital market conditions (C2).
2 - 22
Applying MPT to real estate
• Characteristics of real estate make MPT
applications less accurate (illiquidity,
infrequent trading, market fragmentation,
effects of valuation etc.)
• Future returns difficult to predict, depend
on a number of factors:
FACTORS AFFECTING REAL
ESTATE VALUE RISK:
Micro-Analysis
VALUE OF THE INVESTMENT
BUSINESS-ASSET
RISK
CAP RATES
NOI
FINANCIAL
STRUCTURE RISK
CASHFLOW
COLLATERAL
WHAT AFFECTS THE CAP RATE?
•
•
•
INTEREST RATES
INFLATION EXPECTATIONS
MARKET CONDITIONS
WHAT AFFECTS PROSPECTIVE NOI?
1. GROSS INCOME
– LEASE ARRANGEMENTS
– MARKET EXPOSURE
– EFFICIENCY OF MANAGEMENT / MARKETING
2. OPERATING EXPENSES
–
–
–
–
LEASE PASS-THROUGH PROVISIONS
EXPENSE-SERVICE CONTRACTS
VACANCY EXPOSURE
EFFICIENCY OF MANAGEMENT
Behavioral Finance and MPT
Risk/Return Revisisted
Risk and reward:
The CAPM equation
E(Ri) = Rf + βi(E(Rm) - Rf ))
where:
E(Ri) = the required return on security i,
Rf = the risk-free rate of interest,
βi = the beta of security i, and
E(Rm) = the return on the market index
Downside Risk
• A fundamental tenet of Financial
Economics:
– There is a trade-off between reward and risk
• But how do we measure risk?
– According to the CAPM, risk is measured by
beta, or covariance with the market
– But, the CAPM assumes a symmetric
distribution and equal treatment of risk across
down markets and up markets
Downside versus Upside Risk
• Downside and upside risk are usually not
perceived as equal by investors:
¾ Markowitz (1959, 1992) advocates the use of
semi-variance as a risk measure
¾ In loss aversion utility, investors place greater
weight on losses relative to gains
¾ Investors more sensitive to downside risk
require a premium for holding assets with
large exposure to downside risk
Loss Aversion (example)
Flip a coin.
- Heads? You lose €10,000.
- Tails? You win €X!
How much would you have to win (X=?)
before you take the bet?
Loss Aversion (example)
• Most people want to gain between
2 and 2.5 times as much as they
put at risk.
• Most people will want a chance
to win at least €20,000 before they
will play.
• Simply put, pain from losing money is
stronger than pleasure in winning money.
Loss aversion
Tversky, A. & Kahneman, D. (1991)
published seminal work on loss aversion
One who loses €100 will lose more
satisfaction than another person will gain
satisfaction from a €100 windfall.
In marketing, the use of trial periods and
rebates try to take advantage of the
buyer's tendency to value the good more
after he incorporates it in the status quo.
Loss aversion
• "Disposition effect":
– Propensity of investors to lock in sure gains than
to lock in a sure loss
• Investors are more likely to sell stocks that
have risen in price rather than those that
have fallen in price (Odean 2001)
– Stock that is up in value is 70% more likely to be
sold than a stock that is down
– Average holding period:
Losing stock: 124 days
Winning stock: 102 days
– Stocks that investors sold outperformed the
stocks they kept by 3.4% over the next 12 months
Downside Beta
•CAPM Measure of Risk
– Beta, β = cov( ri , rm ) / var(rm ) , is not a sufficient
statistic
•Downside Beta: β =
−
cov(ri , rm rm < µ m )
var(rm rm < µ m )
•Relative Downside Beta: ( β − β )
−
Potential Bias?
−
• Is there a bias induced by computing β and
average returns over the same period?
– “High betas are related to high average returns by
construction"
• By conditioning on down moves of the market, we pick
−
up periods of low returns => A priori expect high β to be
associated with lower, not higher, returns
−
• β can be computed using demeaned market returns to
correct for bias (averaging out to zero).
Forecasting Downside Risk
• Predicting Future Downside Risk and
Future Returns
¾ There usually exists a strong relationship
between downside risk and returns.
¾ If we can predict future downside risk
exposure, we could form trading strategies to
predict future expected returns.
¾ Which assets have highest downside risk?
¾ Is past downside beta persistent?
Downside Risk: Contagion risk
• When markets decline together more than can be
expected from past correlation patterns, this is
called CONTAGION.
• Due to financial, economic (fundamental) or
political links
• Example: Common movements of the Russian
and Brazilian stock markets (Sull 2006):
Stable Period
Crisis Period
Both recorded
positive returns
23.3%
28.0%
Both negative
40.0%
60.0%
Total common
movements
63.3%
88.0%
Summary
• Diversification among risky assets allows:
– Greater expected return to be obtained for any
given risk exposure, &/or;
– Less risk to be incurred for any given expected
return target.
– (This is called getting on the "efficient
frontier".)
Summary
• Portfolio theory allows us to:
– Quantify this effect of diversification
– Identify the "optimal" (best) mixture of risky
assets
Summary : Problems with MPT
MPT OPTIMIZATION
¾ Counter-Intuitive Allocations
¾ Corner Solutions
¾ Impact of Outliers
¾ Data Sensitivity & Stability
MISSING VARIABLES
¾ Marketable Assets
¾ Global Asset Allocation?
¾ Flaws in Top-Down Allocation
MODEL ASSUMPTIONS
¾ Market Efficiency & Information
¾ Market Structure
¾ Pricing => Investment Risk & Return
¾ Investors’ Time Horizons
Summary: Problems with Real Estate
REAL ESTATE CHARACTERISTICS
• Heterogeneity
• Indivisibility
• Large Lot Size
MARKET CHARACTERISTICS
• Entry Barriers
• Liquidity & Timing
• Transaction Costs
• Sale by Private Treaty
• Segmentation & Thin Markets
INFORMATION & UNCERTAINTY
• Valuations & Prices
• Returns
• Monitoring Costs
• Benchmarking
Summary
• Risk-reward models are useful but highly
context-specific
• MPT ignores systematic/specific risk
• Some observed effects (loss aversion,
disposition effect, contagion etc.) may not be
reflected in the standard MPT models
• If investors are more risk averse to down
markets, then assets that have high exposure
to downside risk must command a higher risk
premium