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 - illiquidity21 and high transaction costs22 - 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 γ0. 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  test of selectivity, or microforecasting;
- Treynor & Mazuy's  and Henriksson & Merton's  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  and Grauer & Hakansson  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
22See section 2.4.8.
23The correlation of real estate with the main asset classes has been well researched. References include MacGregor , Baum & Schofield , Brown [1991b] and MacGregor & Nanthakumaran . Also see section 2.4.6.