The results and the discussion of them are presented in two parts.
Initially, the rôle of real estate within an unconstrained portfolio is
considered, together with the effect of unsmoothing real estate returns
on its inclusion within a multi-asset portfolio. There follows a
discussion of how a positive constraint on investment in real estate
might affect the composition and performance of the optimal portfolio.
Finally, the effect of including Far East equities on the proportion of
real estate held is examined.

## The Naïve Model

The following assumptions were made, that:

- returns are normally distributed;
- there are no
transaction costs; and
- investors reassess and consequently
rebalance their portfolios at the end of each period.

This
model might be used to describe semi-passive investment in a stock
market. As

with such a policy turnover is low, transaction costs are
ignored. However, due to the nature of real estate investment -
illiquidity^{21} and high transaction costs^{22} - the model is used as a
benchmark.

The results suggest that real estate returns are
almost orthogonal with respect to financial assets. In this
unconstrained model, real estate behaves like any other risky
investment. The more risk-averse the investor, the less real estate is
willingly held. A negative relationship between risk-aversion and the
holding of real estate was obtained.This relationship held for all but
one of the periods considered. Tables 3.3 and 3.4 show the portfolio composition and realised returns
for
investors with risk aversion factors of, respectively, *γ*-50 and *γ*0.
Real estate was unsmoothed at 0.625.

As expected, figures
3.9 and 3.10 below show that over the period considered,
unsmoothing of the real estate returns series influenced portfolio
composition.

Unsmoothing the series as above resulted in a
large reduction in the percentage of real estate held by more
risk-averse investors. Less risk-averse investors often increased their
exposure. However, the overall effect on allocation was limited, a
result principally due to the effect of the unsmoothing method utilised
on real estate's covariance with the other asset classes. The covariance
increased slightly, while the correlation coefficient

remained practically unchanged;^{23} see, respectively, tables 3.5 and 3.6.

There are several commonly accepted methods of testing for abnormal investment performance:

- Jensen's [1968] test of selectivity, or microforecasting;
- Treynor & Mazuy's [1966] and Henriksson & Merton's [1981] tests of market timing, or macroforecasting; and
- a paired
*t*-statistic of the difference in investment returns.

In view of the relatively long holding period (one year), the small number of observations, and the lack of a well-defined market portfolio, only a paired t-statistic was employed. The null hypothesis, that the returns and variances of the portfolios were equal, was tested using a *t*-statistic and an *f*-statistic respectively. That is,

against a single-sided alternative in each case.

Table 3.7 shows the results of the paired *t*-statistics and *f*-statistics when returns with RE-0.75 and RE-0.5 are compared for each risk attitude. The results are supported by that of Webb & Rubens [1988] and Grauer & Hakansson [1995] and suggest that the issue of unsmoothing real estate returns, although important, may not unduly affect the portfolio diversification benefits of real estate. This point is particularly true of the more risk-averse investor. This strengthens the argument that

within a multi-asset portfolio, direct real estate provides a good instrument for diversification.

Table 3.8 presents the results when returns with and without real estate are compared for each risk attitude. The inclusion of real estate for more risk-averse strategies improves returns, but it also increases the variance of portfolios.The presence of real estate has a positive impact for less risk-averse strategies, although this is never significant. These results are consistent with the findings of previous studies i. e. the

more risk-averse an investor, the greater the benefit of diversification. However, they are qualified in their degree, as a highly risk-averse investor, *γ*-50, benefits less than a *γ*-10 investor. The results are also shown in figure 3.11 below, which compares the returns on portfolios with and without RE-0.625, with those on equally-weighted portfolio.

____________________________

^{21} See section 2.4.4

^{22}See section 2.4.8.

^{23}The correlation of real estate with the main asset classes has been well researched. References include MacGregor [1990], Baum & Schofield [1991], Brown [1991b] and MacGregor & Nanthakumaran [1992]. Also see section 2.4.6.