Optimal Portfolio Selection: the role of illiquidity and investment

horizon

Ping Cheng, Florida Atlantic UniversityZhenguo Lin, California State University, FullertonYingchun 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)

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Cause for Academic findings: the “superior” real estate performance

Mean Stdev Sharpe ratio

S&P 500 3.26 9.99 0.28NASDAQ 2.66 7.60 0.28Dow Jones Industrial 2.68 7.04 0.31

Private Equity (NCREIF) 2.48 1.70 1.17Industrial 2.57 1.65 1.25

Office 2.33 2.58 0.71Retail 2.52 1.65 1.23

Apartment 2.88 1.62 1.47OFHEO HPI 1.35 0.94 0.90OFHEO Purchase-only 1.31 0.89 0.91

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Quarterly Returns and Risks of Asset Classes

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%.

Historically low correlation between real estate and financial assets

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• 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) 2 3 4 5

NCREIF Overall vs. S&P500 0.116 0.114 0.074 0.028 NCREIF Apartment vs. S&P500 0.094 0.078 0.069 0.074 NCREIF Industrial vs. S&P500 0.168 0.118 0.107 0.104 NCREIF Office vs. S&P500 0.149 0.165 0.142 0.115

NCREIF Retail vs. S&P500 -0.008 -0.079 -0.117 -0.107

What’s behind MPT? I.I.D. condition is critical!

• Modern Portfolio Theory (MPT)

or

• In general, asset allocations should be related to expected holding period

or

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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.

• Case and Shiller (AER1989)• Englund, Gordon and Quigley (JREFE 1999)• Gao, Lin and Na (JHE 2010)• …

2. 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.

These issues are often dismissed or downplayed when MPT is applied to the mixed-asset portfolio including real estate.

Figure 2 The Real Estate Transaction Process

0 t t+TOM Time

Random Time-on-market (TOM )

Two risks at time of decision: random selling price random time-on-market.

P0

Immediate sale is not optimal TOMtP

Random selling price

TOMr TOMt~

Return upon successful sale

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”.

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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 returnso Liquidity risk - substantial and uncertain TOMo 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-dependentThe Holding Period Effect on Periodic Risk and Returns

As expected, holding-period matters a lot to real estate but not much to stocks.11

(A) (B) (C) (D) (E) (F) (G) (H) (I)

Holding period (T) (number of quarters)

Average periodic

return (%)

Average Periodic

Variance (%)

Average periodic

St. dev. (%)

NCREIF Sharpe Ratios

Average periodic

return (%)

Average Periodic

Variance (%)

Average periodic

St. dev. (%)

S&P 500 Sharpe Ratios

Computation Equation (3) Equation (3)

1 2.48 0.03 1.71 1.17 2.72 59.64 7.72 0.272 2.25 0.05 2.15 0.77 2.81 64.74 8.05 0.273 2.28 0.07 2.60 0.65 2.81 69.81 8.36 0.264 2.56 0.10 3.17 0.62 2.79 68.62 8.28 0.275 2.35 0.11 3.34 0.53 2.83 77.05 8.78 0.266 2.39 0.13 3.67 0.49 2.86 79.28 8.90 0.267 2.42 0.16 3.97 0.46 2.90 81.63 9.03 0.268 2.65 0.21 4.54 0.45 2.93 76.82 8.76 0.279 2.49 0.20 4.49 0.42 2.99 84.86 9.21 0.2610 2.52 0.22 4.74 0.41 3.04 90.68 9.52 0.2611 2.56 0.25 4.98 0.40 3.09 99.27 9.96 0.2512 2.71 0.30 5.47 0.39 3.13 103.87 10.19 0.2513 2.63 0.30 5.44 0.38 3.18 114.96 10.72 0.2414 2.67 0.32 5.68 0.37 3.20 120.31 10.97 0.2415 2.72 0.35 5.92 0.36 3.23 125.95 11.22 0.2416 2.76 0.38 6.15 0.35 3.28 131.88 11.48 0.24

NCREIF Property Index (1978Q1 - 2007Q2) S&P 500 Index (1978Q1 - 2007Q2)

TT /2

TT /2

TT /2

TT /2

TuT /

TuT /

-

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10.00

15.00

20.00

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30.00

35.00

40.00

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

Stan

dard

ized

Ris

k

Holding-periods (number of quarters)

NCREIF

S&P 500

Brownian motion:(1,1)

Alternative risk structure:

How far are real estate returns from the i.i.d. condition?

Risk Curves of Real Estate (NCREIF) vs. S&P500

22 :... dii

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If real estate is not i.i.d., what is it? An empirical approach

-

5.00

10.00

15.00

20.00

25.00

30.00

35.00

40.00

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

Sta

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isk

Holding-periods (number of quarters)

NCREIF

S&P 500

Brownian motion:(1,1)

)1(1 T

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This is reasonable.

This is not

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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

Note that upon successful sale, the actual holding-period is t = T + TOM.

With regard to illiquidity risk

The task: extend the MPT by incorporating non-i.i.d. nature of real estate returns, its illiquidity risk, and its high transaction cost.

),(~ 2TOMTOMtTOM (No particular distribution specified.)

22 ])([

)(

RETOMT

RETOMT

TOMT

uTOMTR

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

]]~

[[]~

[ TOMREERE PortfolioTOMT

PortfolioTOMT

anteex

]~[]

~[)

~( TOMRVarETOMREVarRVar Portfolio

TOMTPortfolio

TOMTPortfolio

TOMTanteex

By Law of Iterated Expectations

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

17

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)1(max

2222),( ,LLLRELRERERE

fLRELLRERE

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The alternative model

• Optimal allocation to the mixed-asset portfolio can be obtained by solving following mean-variance optimization:

• Optimal allocations are affected by not only single-period return, risk, and correlations between them, but also affected byo The non-i.i.d. real estate returns – the slope of risk curve (

)o The expected length of TOM ( ) o The uncertainty of TOM ( )o Transaction cost ( C ).o Holding period ( )

]2])()

[([)1()(max

2

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022

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REREREfLRELLRETOMRE

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Application of the alternative model

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we need to know the following information:

Application of the alternative model (cont’d)

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Application of the alternative model (cont’d)

21

Application of the alternative model (cont’d)

22

Application of the alternative model (cont’d)

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The optimal allocation of real estate

= 1.0 = 0.9 = 0.8Transaction Cost

( C ) 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%

10% 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%

8% 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%

6% 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 = 0.9 = 0.8Transaction Cost

( C ) 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%

10% 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%

8% 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%

6% 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 = 0.9 = 0.8Transaction Cost

( C ) 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%

10% 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%

8% 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%

6% 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 = 0.9 = 0.8

Transaction Cost ( C ) 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%

10% 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%

8% 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%

6% 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 = 0.9 = 0.8

Transaction Cost ( C ) 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%

10% 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%

8% 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%

6% 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 wordsReal 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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