BUSINESS SCHOOL

Portfolio Selection with Skew Normal Asset Returns

Quan GanDiscipline of Finance, University of Sydney

ApologiesDue to the expended scope of my paper. I change the title of the paper.

The previous title was:

Performance of Equity REITs: Does a Skew Factor Matter?

BUSINESS SCHOOL

Main Contributions

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The main contributions of the paper:1. I show that Azzalini & Dalla Valle (1996)’s

multivariate skew normal distribution is a special case of Simaan (1993)’s three-parameter model.

2. All Simaan (1993)’s results are applicable to the skew normal asset returns.

3. Combining the skew normal asset returns with the CARA utility, I obtain the closed-form certainty equivalent and skewness premium.

Main Contributions (cont.)

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4. I show that the skewness premium is positive (negative) when asset returns have negative (positive) skewness. The magnitude of the skewness premium is increasing in market risk aversion.

5. I find that when investors face broad investment opportunities, the economic value of considering higher moments is negligible under realistic margin requirements.

6. Implication of real estate market (high leverage is needed for exploring higher moments benefit).

Simaan’s three-parameter model

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Simaan (1993)’s three parameter model is a nice extension of Markowitz’s mean-variance framework.

1) it is fully compatible with the expected utility maximization;

2) it nests the mean-variance portfolio selection results; 3) it has nice geometric properties; 4) it considers information of all moments.

Simaan’s three-parameter model (Cont.)

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The data-generating process is written as a regression:

1. where st is a scalar perturbation with any non-elliptical distribution.

2. Conditioning on st, yt is elliptical.

Simaan (1993) meets Engle et al. (1983)

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Simaan (1993)’s construction is related to Engle et al. (1983)’s weak exogeneity concept.Weak exogeneity:

Information on yt|st is enough to conduct statistical inference on parameters of g(yt|st), thus information on marginal distribution g(st) can be safely ignored.

Simaan (1993) meets Engle et al. (1983)

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In Simaan (1993)’s construction, st is weakly exogenous to the parameter set of elliptical distribution.

Simaan (1993) meets Azzalini & Dalla Valle (1996)

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Azzalini & Dalla Valle (1996) proposes the multivariate skew normal distribution.

(yt|st >0) has the density:

where

Simaan (1993) meets Azzalini & Dalla Valle (1996)

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The multivariate skew normal is a special case of Simaan (1993)’s three-parameter model.

st follows a half normal distribution (non-elliptical).Conditioning on st, yt|st follows multivariate normal

(elliptical) distribution.

where st > 0.

Simaan (1993) meets Athayde and Flˆores (2004)

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The three-parameter efficient frontier has linearity property as shown in panel (b) in the next slide.

Simaan (1993) meets Athayde and Flˆores (2004)

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Simaan (1993) meets Markowitz (1952)

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The three-parameter portfolio selection problems can be written as:

Simaan (1993) meets Markowitz (1952)

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The solution is a nice three-fund separation which nests the classic two-fund separation result.

Higher Moments Portfolio Selection Paradigms

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Certainty Equivalent of the CARA utility

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Maximizing the expected utility is equivalent to maximizing the certainty equivalent. For the CARA utility, the certainty equivalent is:

Certainty Equivalent of the CARA utility (normal distribution vs. skew normal distribution)

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Combining the CARA utility with normal distribution, the certainty equivalent is:

Combining the CARA utility with normal distribution, the certainty equivalent is:

Asset Pricing with Skew Normal Returns

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The new asset pricing formula is:

Skewness Premium

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The skewness premium depends on C and risk aversion (a formal proof of the relationships is provided in my paper):

Comparing the certainty equivalents

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Following P´astor & Stambaugh (2000) and Tu & Zhou (2004), I use the certainty equivalents to compare economic values of different portfolio choices.

I find that under reasonable margin requirements, the economic value of higher moments are negligible.

One implication of my empirical results is that the value of higher moments might be better explored by investors with relaxed financial constraints or by those facing a small and special investment universe.

Both are relevant to real estate investors.

More?

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The paper is now available on the conference website and SSRN.

In the paper, I also have:

1) Bayesian algorithm of estimating skew normal distributions

2) A stochastic dominance result of skew normal distribution

3) Skew normal distribution is an unique distribution which can be nested by Simaan (1993)’s three-parameter model.

……