We discuss portfolio allocation when one asset exhibits phases of locally explosive behavior. We model the conditional distribution of such an asset through mixed causal-non-causal models which mimic well the speculative bubble behaviour. Relying on a Taylor-series-expansion of a CRRA utility function approach, the optimal portfolio(s) is(are) located on the mean-variance-skewness-kurtosis efficient surface. We analytically derive these four conditional moments and show in a Monte-Carlo simulations exercise that incorporating them into a two-assets portfolio optimization problem leads to substantial improvement in the asset allocation strategy. All performance evaluation metrics support the higher out-of-sample performance of our
investment strategies over standard benchmarks such as the mean-variance and equally-weighted portfolio. An empirical illustration using the Brent oil price as the speculative asset confirms these findings.