We investigate whether a benchmark and non-constant risk aversion affect the probability density distribution of optimal wealth at retirement. We maximize the expected utility of the ratio of pension wealth at retirement to an inflation-indexed benchmark. Together with a threshold and a lower bound, we are able to generate closed-form solutions. We find that this non-constant risk aversion type of utility could shift the probability density distribution of optimal wealth more towards the benchmark, and that the probability of achieving a certain percentage of the desired benchmark could be increased. The probability density distribution generated under constant relative risk aversion (CRRA) risk preference is more widely spread along the benchmark.

The institutional environment of Portuguese banking during the golden age years of economic growth (1950–73) has been criticised in many instances, at the time and in recent literature. Direct observers of the period as well as historians have stressed two main aspects of that environment: excessive protection of existing banks, allowing them to obtain high rents, which represented a disincentive for them to compete and innovate; excessive concentration of their activity on short-term commercial paper, thus preventing them from contributing to finance growth. There seems to be a contradiction here, however, with the high growth rates of the years 1950 to 1973. The apparent contradiction is not limited to Portugal, in fact, as rapid growth in many economies in that period occurred within a framework of heavily regulated financial systems. This is the ‘financial paradox’ of the golden age. Portugal is an interesting case in the international perspective. As in the rest of the western world, legislation repressed banking quite tightly, but banks circumvented the law and competed with each other. The signs of competition were visible mostly in two dimensions: the growth of time deposits and geographical expansion.