Diversifying in public real estate: The ex-post performance

Abstract

We calculate the ex-post, realised portfolio performance for an investor who diversifies among US stocks, bonds, real estate indirect investment vehicles (E-REITS), and cash. Simulations are performed for two alternative asset allocation frameworks — classical and Bayesian — and for scenarios involving two different samples and six different investment horizons. Interestingly, the ex-post welfare cost of restricting portfolio choice to traditional financial assets (ie, stocks, bonds, and cash) is only found to be positive in all scenarios for a Bayesian investor. On the contrary, substitution of E-REITS for stocks in optimal portfolios turns out to reduce ex-post portfolio performance over the nineties and for a Classical investor who ignores parameter estimation uncertainty.

Appendix

Classical buy-and-hold investor

Let θ be the vector collecting all the parameters, that is, θ≡[μ′ vech(Σ)′]′. From the assumption in (5), the conditional distribution of cumulative future returns z_t, T≡∑_k=1^Tz_t+k is multivariate normal with mean and covariance matrix given by:

Since the parametric form of the predictive distribution of z_t, T is known, it is simple to approach the problem in (1), or equivalently

where φ(E _t [z_t, T], Var _t [z_t, T]) is a multivariate normal with mean E _t [z_t, T] and covariance matrix Var _t [z_t, T], by simulation methods. This means evaluating the integral by drawing a large number of times (N) from φ(E _t [z_t, T], Var _t [z_t, T]) and then maximising the functional (6). At this stage, the portfolio weight non-negativity constraints are imposed by using a simple two-stage grid search algorithm that sets ω _t ^j to 0, 0.01, 0.02, …, 0.99, 0.9999 for j=s, b, r.

Bayesian buy-and-hold investor

Given the problem

the task is somewhat simplified by the fact that predictive draws can be obtained by drawing from the posterior distribution of the parameters and then, for each set of parameters drawn, by sampling one point from the distribution of returns conditional on past data and the parameters drawn in the first stage. If we consider the following standard uninformative diffuse prior:

then the posterior distribution for the coefficients in θ, p(μ, Σ⁻¹∣Z̈ _t ) can be characterised as:

where Ŝ is the sample covariance of the residuals and μ̂ is the sample mean. Also for the Bayesian case, we adopt a simulation method by which: first, we draw N independent variates from p(μ, Σ⁻¹∣Z̈ _t ). This is done by first sampling from a marginal Wishart for Σ⁻¹ and then (after calculating Σ) from the conditional N(vec(μ̂), Σ). Second, for each set (μ, Σ) obtained, the algorithm samples cumulated returns from a multivariate normal with mean vector and covariance matrix given by first-round draws.