Optimal Portfolio Selection: the role of
illiquidity and investment horizon
Ping Cheng, Florida Atlantic University
Zhenguo Lin, California State University, Fullerton
Yingchun Liu, Laval University
Motivation of this study:
To solve the real estate allocation puzzle
Real estate allocation puzzle in the mixed-asset portfolio:
◦ The academic claim: real estate should constitute
15% - 40% or more of a diversified portfolio.
 Hartzell, Hekman and Miles (1986), Firstenberg, Ross and Zisler (1988), and
Hudson-Wilson, Fabozzi and Gordon (2005), etc.
◦ The market reality: most institutional investors typically have
only about 3 - 5% of their total assets in real estate.
 Two surveys by Pension & Investments (2002) and Goetzmann and Dhar (2005)
2
Cause for Academic findings: the “superior” real estate
performance
Quarterly Returns and Risks of Asset Classes
Mean
Stdev
Sharpe ratio
S&P 500
NASDAQ
Dow Jones Industrial
3.26
2.66
2.68
9.99
7.60
7.04
0.28
0.28
0.31
Private Equity (NCREIF)
Industrial
Office
Retail
Apartment
OFHEO HPI
OFHEO Purchase-only
2.48
2.57
2.33
2.52
2.88
1.35
1.31
1.70
1.65
2.58
1.65
1.62
0.94
0.89
1.17
1.25
0.71
1.23
1.47
0.90
0.91
Note: NCREIF and sub-indices are based on quarterly data from 1978Q1 through 2007Q2. The OFHEO HPI
and OFHEO Purchase only are 1991Q1-2007Q2. The table shows that all categories of private real estate
equity exhibit significantly higher risk-adjusted returns than stocks. The risk-free rate is obtained from Ken
French’s website. For the period analyzed, the average quarterly risk-free rate is 0.496%.
3
Historically low correlation between real estate and
financial assets
•
•
•
•
Ibbotson and Siegel (1984) found real estate's correlation with S&P stocks to
be -0.06 by using annual U.S. data from 1947 to 1982.
Worzala and Vandell (1993) estimate the correlation between NCREIF
quarterly index from 1980 to 1991 and stocks of the same period to be about
-0.0971.
Eichholtz and Hartzell (1996) document correlations between real estate and
stock indexes to be -0.08 for U.S., -0.10 for Canada, and -0.09 for U.K.
Quan and Titman (1999) examined the correlation for 17 countries and, unlike
earlier studies, they find a generally positive correlation pattern in most
countries, but for the U.S. such positive correlation is insignificant.
Table 1.
Correlations between NCREIF and S&P500 (1978-2007)
Holding Period (years)
NCREIF Overall vs. S&P500
NCREIF Apartment vs. S&P500
NCREIF Industrial vs. S&P500
NCREIF Office vs. S&P500
NCREIF Retail vs. S&P500
2
3
4
5
0.116
0.094
0.168
0.149
-0.008
0.114
0.078
0.118
0.165
-0.079
0.074
0.069
0.107
0.142
-0.117
0.028
0.074
0.104
0.115
-0.107
4
What’s behind MPT? I.I.D. condition is critical!
•
Modern Portfolio Theory (MPT)
or
•
N
  N


~
~
max N
 E   w i r    Var (  w i ri ) 
( w , , w 2 ,..., w N ,  w i  1 ) 
i 1
 i 1
i

i 1
In general, asset allocations should be related to expected holding period
or
Known issues with the application of MPT to
the mixed asset portfolio
• Real estate does not fit into the financial paradigm
1.
Overwhelming empirical evidence suggesting that real
estate returns are not i.i.d.
•
•
•
•
2.
Case and Shiller (AER1989)
Englund, Gordon and Quigley (JREFE 1999)
Gao, Lin and Na (JHE 2010)
…
Expensive trading with high transaction cost:
• Collett, Lizieri and Ward (2003) report “the round-trip lump-sum
costs” were approximately 7-8% of the asset value based on U.K. data.
• Infrequent trading – hold for multiple periods
Known issues with the application of MPT to the
mixed asset portfolio (cont’d)
• Real estate does not fit into the financial paradigm.
3.
Difficult trading – illiquidity:
• TOM is both substantial and uncertain.
• But TOM is assumed nonexistent in finance theories such as MPT.
Figure 2
The Real Estate Transaction Process
Two risks at time of decision:
 random selling price
 random time-on-market.
0
P0
Random Time-on-market
( TOM )
t
t+ TOM
Time
Pt  TOM
Immediate sale is not optimal
Random selling price
~
rt  TOM TOM
Return upon successful sale
These issues are often dismissed or downplayed when MPT is applied to
the mixed-asset portfolio including real estate.
Two presidential addresses called for new theories for real estate
 1987 Presidential Address to American Real Estate and Urban
Economics Association by Ken Lusht
“The Real Estate Pricing Puzzle” argues that “available pricing
models are not up to the task”.
 2003 Presidential Address to American Real Estate and Urban
Economics Association by Jim Shilling
“Is There A Risk Premium Puzzle in Real Estate?” argues that “the
risk premium on real estate based on financial models is
misleading”.
8
New approach to the old question
• Previous efforts focus on ad hoc solutions.
o De-smoothing (if appraisal-based data are used)
o Additional constraints on real estate weight
• Rather than fine-tune the way MPT is applied, we modify the
theory itself.
• Objective: extending the classical MPT by incorporating unique
real estate features:
o Non-i.i.d. nature of real estate returns
o Liquidity risk - substantial and uncertain TOM
o High transaction cost
The importance of investment horizon
• MPT is a single-period model. It assumes all assets are to be
held for “one-period”. The reality: most investments are
multi-period.
• What bridges the single-period theory and multi-period
reality?
o asset returns over time are i.i.d. (Merton (1969), Samuelson (1969), and
Fama (1970))
• But real estate is not i.i.d. The non-i.i.d. nature of real estate
returns implies
• the real estate performance is horizon-dependent
• single-period financial models such as MPT cannot be applied to
real estate.
Non-i.i.d. implies real estate performance is horizon-dependent
The Holding Period Effect on Periodic Risk and Returns
NCREIF Property Index (1978Q1 - 2007Q2)
(B)
(C)
(D)
(E)
Average
Average
Average
NCREIF
Holding period (T)
periodic
Periodic
periodic
Sharpe
(number of quarters) return (%) Variance (%) St. dev. (%)
Ratios
uT/T2T/(A)
TT2 / T T2 / TT2 / T
Computation
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
Equation (3)
2.48
2.25
2.28
2.56
2.35
2.39
2.42
2.65
2.49
2.52
2.56
2.71
2.63
2.67
2.72
2.76
0.03
0.05
0.07
0.10
0.11
0.13
0.16
0.21
0.20
0.22
0.25
0.30
0.30
0.32
0.35
0.38
1.71
2.15
2.60
3.17
3.34
3.67
3.97
4.54
4.49
4.74
4.98
5.47
5.44
5.68
5.92
6.15
1.17
0.77
0.65
0.62
0.53
0.49
0.46
0.45
0.42
0.41
0.40
0.39
0.38
0.37
0.36
0.35
S&P 500 Index (1978Q1 - 2007Q2)
(F)
(G)
(H)
(I)
Average
Average
Average
S&P 500
periodic
Periodic
periodic
Sharpe
return (%) Variance (%) St. dev. (%)
Ratios
uT / T
2.72
2.81
2.81
2.79
2.83
2.86
2.90
2.93
2.99
3.04
3.09
3.13
3.18
3.20
3.23
3.28
Equation (3)
59.64
64.74
69.81
68.62
77.05
79.28
81.63
76.82
84.86
90.68
99.27
103.87
114.96
120.31
125.95
131.88
7.72
8.05
8.36
8.28
8.78
8.90
9.03
8.76
9.21
9.52
9.96
10.19
10.72
10.97
11.22
11.48
0.27
0.27
0.26
0.27
0.26
0.26
0.26
0.27
0.26
0.26
0.25
0.25
0.24
0.24
0.24
0.24
11
As expected, holding-period matters a lot to real estate but not much to stocks.
How far are real estate returns from the i.i.d. condition?
Risk Curves of Real Estate (NCREIF) vs. S&P500

40.00

i .i .d . :    
2
35.00
2





Standardized Risk
30.00
Alternative risk structure:     
25.00
20.00
NCREIF
15.00
10.00
S&P 500
5.00
(1,1)
Brownian motion:     
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36
Holding-periods (number of quarters)
12
If real estate is not i.i.d., what is it?
An empirical approach

40.00
35.00

~

E
[
R

 ]  u
i .i .d .
2
2

   
30.00
This is reasonable.

 1   T (  1)  

Standardized Risk
This is not
25.00
NCREIF
20.00
15.00
10.00
S&P 500
5.00
(1,1)
Brownian motion:   

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36
Holding-periods (number of quarters)
13
If real estate is not i.i.d., what is it?
An empirical approach (cont’d)
The risk curves of real estate
Model assumptions for real estate asset

The non-i.i.d. replaces the i.i.d. assumption for real estate with
R T  TOM  (T  TOM ) u RE
 T  TOM  [  (T  TOM ) RE ]
2
2
Note that upon successful sale, the actual holding-period is  = T + TOM.

With regard to illiquidity risk
TOM ~ ( t TOM ,  TOM )
2

(No particular distribution specified.)
The task: extend the MPT by incorporating non-i.i.d. nature of real
estate returns, its illiquidity risk, and its high transaction cost.
15
Ex ante performance is a forward-looking that is unconditional
upon a successful sale
Mathematically, we can express the ex ante return and risk as follows
E
Var
ex  ante
ex  ante
~ Portfolio
~ Portfolio
[ RT  TOM ]  E [ E [ RT  TOM TOM ]]


By Law of Iterated Expectations


~ Portfolio
~ Portfolio
~ Portfolio
( RT  TOM )  Var E [ RT  TOM TOM ]  E [Var RT  TOM TOM ]
By conditional variance formula
16
A Case of Three Assets Portfolio:
Real Estate (RE), Financial Asset (L) and Risk-free Asset (f)
Modern Portfolio Theory
 w RE u RE  w L u L  (1  w RE  w L ) r f  
max 

2
2
2
2
( w RE , , w L )
  ( w RE  RE  2  w RE w L  RE  L  w L  L ) 
17
The alternative model
• Optimal allocation to the mixed-asset portfolio can be obtained
by solving following mean-variance optimization:
2
2
 w RE (T  t TOM ) u RE  w L Tu L  (1  w RE  w L )Tr f  w RE C   [ w RE
[( u RE  


max 
3

w RE , w L
2
2
2
2
2
2
2
2
2
   )

   RE (T  t TOM ) ]  w L T  L  2  w RE w L  T  RE  L ]
RE
T 0M


• Optimal allocations are affected by not only single-period return,
risk, and correlations between them, but also affected by
o
o
o
o
o
The non-i.i.d. real estate returns – the slope of risk curve (  )
The expected length of TOM (t TOM )
2
The uncertainty of TOM ( TOM
)
Transaction cost ( C ).
Holding period (T )
Application of the alternative model
we need to know the following information:
19
Application of the alternative model (cont’d)
20
Application of the alternative model (cont’d)
21
Application of the alternative model (cont’d)
22
Application of the alternative model (cont’d)
23
The optimal allocation of real estate
 = 1.0

Transaction Cost
(C )
10%
8%
6%

TO
 = 0.9
 = 0.8
M
2
5
8
2
5
8
2
5
8
10%
4.28%
4.25%
4.20%
5.34%
5.30%
5.22%
6.84%
6.77%
6.65%
5%
4.49%
4.46%
4.41%
5.58%
5.53%
5.45%
7.09%
7.03%
6.90%
0%
4.73%
4.69%
4.63%
5.83%
5.79%
5.70%
7.38%
7.31%
7.18%
-5%
4.98%
4.94%
4.88%
6.11%
6.06%
5.97%
7.70%
7.62%
7.49%
-10%
5.25%
5.22%
5.15%
6.42%
6.37%
6.28%
8.05%
7.97%
7.83%
10%
4.39%
4.36%
4.30%
5.47%
5.43%
5.35%
7.00%
6.93%
6.81%
5%
4.60%
4.57%
4.51%
5.71%
5.66%
5.58%
7.26%
7.19%
7.06%
0%
4.83%
4.80%
4.74%
5.96%
5.92%
5.83%
7.55%
7.47%
7.34%
-5%
5.08%
5.05%
4.98%
6.25%
6.19%
6.10%
7.86%
7.79%
7.65%
-10%
5.36%
5.32%
5.25%
6.56%
6.50%
6.40%
8.22%
8.14%
7.99%
10%
4.49%
4.46%
4.40%
5.61%
5.56%
5.48%
7.17%
7.10%
6.97%
5%
4.71%
4.67%
4.61%
5.84%
5.79%
5.70%
7.43%
7.35%
7.22%
0%
4.94%
4.90%
4.84%
6.09%
6.05%
5.95%
7.71%
7.64%
7.50%
-5%
5.19%
5.15%
5.09%
6.38%
6.32%
6.23%
8.03%
7.95%
7.81%
-10%
5.47%
5.43%
5.36%
6.69%
6.63%
6.53%
8.39%
8.30%
8.15%
The optimal allocation of real estate
 = 1.0

Transaction Cost
(C )
10%
8%
6%

TO
 = 0.9
 = 0.8
M
2
5
8
2
5
8
2
5
8
10%
4.28%
4.25%
4.20%
5.34%
5.30%
5.22%
6.84%
6.77%
6.65%
5%
4.49%
4.46%
4.41%
5.58%
5.53%
5.45%
7.09%
7.03%
6.90%
0%
4.73%
4.69%
4.63%
5.83%
5.79%
5.70%
7.38%
7.31%
7.18%
-5%
4.98%
4.94%
4.88%
6.11%
6.06%
5.97%
7.70%
7.62%
7.49%
-10%
5.25%
5.22%
5.15%
6.42%
6.37%
6.28%
8.05%
7.97%
7.83%
10%
4.39%
4.36%
4.30%
5.47%
5.43%
5.35%
7.00%
6.93%
6.81%
5%
4.60%
4.57%
4.51%
5.71%
5.66%
5.58%
7.26%
7.19%
7.06%
0%
4.83%
4.80%
4.74%
5.96%
5.92%
5.83%
7.55%
7.47%
7.34%
-5%
5.08%
5.05%
4.98%
6.25%
6.19%
6.10%
7.86%
7.79%
7.65%
-10%
5.36%
5.32%
5.25%
6.56%
6.50%
6.40%
8.22%
8.14%
7.99%
10%
4.49%
4.46%
4.40%
5.61%
5.56%
5.48%
7.17%
7.10%
6.97%
5%
4.71%
4.67%
4.61%
5.84%
5.79%
5.70%
7.43%
7.35%
7.22%
0%
4.94%
4.90%
4.84%
6.09%
6.05%
5.95%
7.71%
7.64%
7.50%
-5%
5.19%
5.15%
5.09%
6.38%
6.32%
6.23%
8.03%
7.95%
7.81%
-10%
5.47%
5.43%
5.36%
6.69%
6.63%
6.53%
8.39%
8.30%
8.15%
The optimal allocation of real estate
 = 1.0

Transaction Cost
(C )
10%
8%
6%

TO
 = 0.9
 = 0.8
M
2
5
8
2
5
8
2
5
8
10%
4.28%
4.25%
4.20%
5.34%
5.30%
5.22%
6.84%
6.77%
6.65%
5%
4.49%
4.46%
4.41%
5.58%
5.53%
5.45%
7.09%
7.03%
6.90%
0%
4.73%
4.69%
4.63%
5.83%
5.79%
5.70%
7.38%
7.31%
7.18%
-5%
4.98%
4.94%
4.88%
6.11%
6.06%
5.97%
7.70%
7.62%
7.49%
-10%
5.25%
5.22%
5.15%
6.42%
6.37%
6.28%
8.05%
7.97%
7.83%
10%
4.39%
4.36%
4.30%
5.47%
5.43%
5.35%
7.00%
6.93%
6.81%
5%
4.60%
4.57%
4.51%
5.71%
5.66%
5.58%
7.26%
7.19%
7.06%
0%
4.83%
4.80%
4.74%
5.96%
5.92%
5.83%
7.55%
7.47%
7.34%
-5%
5.08%
5.05%
4.98%
6.25%
6.19%
6.10%
7.86%
7.79%
7.65%
-10%
5.36%
5.32%
5.25%
6.56%
6.50%
6.40%
8.22%
8.14%
7.99%
10%
4.49%
4.46%
4.40%
5.61%
5.56%
5.48%
7.17%
7.10%
6.97%
5%
4.71%
4.67%
4.61%
5.84%
5.79%
5.70%
7.43%
7.35%
7.22%
0%
4.94%
4.90%
4.84%
6.09%
6.05%
5.95%
7.71%
7.64%
7.50%
-5%
5.19%
5.15%
5.09%
6.38%
6.32%
6.23%
8.03%
7.95%
7.81%
-10%
5.47%
5.43%
5.36%
6.69%
6.63%
6.53%
8.39%
8.30%
8.15%
The optimal allocation of real estate
 = 1.0

Transaction Cost
(C )
10%
8%
6%

 = 0.9
 = 0.8
TO
M
2
5
8
2
5
8
2
5
8
10%
4.28%
4.25%
4.20%
5.34%
5.30%
5.22%
6.84%
6.77%
6.65%
5%
4.49%
4.46%
4.41%
5.58%
5.53%
5.45%
7.09%
7.03%
6.90%
0%
4.73%
4.69%
4.63%
5.83%
5.79%
5.70%
7.38%
7.31%
7.18%
-5%
4.98%
4.94%
4.88%
6.11%
6.06%
5.97%
7.70%
7.62%
7.49%
-10%
5.25%
5.22%
5.15%
6.42%
6.37%
6.28%
8.05%
7.97%
7.83%
10%
4.39%
4.36%
4.30%
5.47%
5.43%
5.35%
7.00%
6.93%
6.81%
5%
4.60%
4.57%
4.51%
5.71%
5.66%
5.58%
7.26%
7.19%
7.06%
0%
4.83%
4.80%
4.74%
5.96%
5.92%
5.83%
7.55%
7.47%
7.34%
-5%
5.08%
5.05%
4.98%
6.25%
6.19%
6.10%
7.86%
7.79%
7.65%
-10%
5.36%
5.32%
5.25%
6.56%
6.50%
6.40%
8.22%
8.14%
7.99%
10%
4.49%
4.46%
4.40%
5.61%
5.56%
5.48%
7.17%
7.10%
6.97%
5%
4.71%
4.67%
4.61%
5.84%
5.79%
5.70%
7.43%
7.35%
7.22%
0%
4.94%
4.90%
4.84%
6.09%
6.05%
5.95%
7.71%
7.64%
7.50%
-5%
5.19%
5.15%
5.09%
6.38%
6.32%
6.23%
8.03%
7.95%
7.81%
-10%
5.47%
5.43%
5.36%
6.69%
6.63%
6.53%
8.39%
8.30%
8.15%
The optimal allocation of real estate
 = 1.0

Transaction Cost
(C )
10%
8%
6%

TO
 = 0.9
 = 0.8
M
2
5
8
2
5
8
2
5
8
10%
4.28%
4.25%
4.20%
5.34%
5.30%
5.22%
6.84%
6.77%
6.65%
5%
4.49%
4.46%
4.41%
5.58%
5.53%
5.45%
7.09%
7.03%
6.90%
0%
4.73%
4.69%
4.63%
5.83%
5.79%
5.70%
7.38%
7.31%
7.18%
-5%
4.98%
4.94%
4.88%
6.11%
6.06%
5.97%
7.70%
7.62%
7.49%
-10%
5.25%
5.22%
5.15%
6.42%
6.37%
6.28%
8.05%
7.97%
7.83%
10%
4.39%
4.36%
4.30%
5.47%
5.43%
5.35%
7.00%
6.93%
6.81%
5%
4.60%
4.57%
4.51%
5.71%
5.66%
5.58%
7.26%
7.19%
7.06%
0%
4.83%
4.80%
4.74%
5.96%
5.92%
5.83%
7.55%
7.47%
7.34%
-5%
5.08%
5.05%
4.98%
6.25%
6.19%
6.10%
7.86%
7.79%
7.65%
-10%
5.36%
5.32%
5.25%
6.56%
6.50%
6.40%
8.22%
8.14%
7.99%
10%
4.49%
4.46%
4.40%
5.61%
5.56%
5.48%
7.17%
7.10%
6.97%
5%
4.71%
4.67%
4.61%
5.84%
5.79%
5.70%
7.43%
7.35%
7.22%
0%
4.94%
4.90%
4.84%
6.09%
6.05%
5.95%
7.71%
7.64%
7.50%
-5%
5.19%
5.15%
5.09%
6.38%
6.32%
6.23%
8.03%
7.95%
7.81%
-10%
5.47%
5.43%
5.36%
6.69%
6.63%
6.53%
8.39%
8.30%
8.15%
Main findings: MPT -- around 27% ; The Alterative Model –- 4%-8%
Final words

Real estate is different from financial assets.

The mixed-asset portfolio that includes real estate
needs its own portfolio theory.

The “real estate allocation puzzle” is an illusion
caused by inappropriate application of modern
portfolio theory.
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