Box 1. The classical portfolio theory with entry costs Consider the simplest, static mean-variance portfolio model where investors decide how to allocate their wealth on the basis of the expected return and variance of their portfolios. Suppose that there is one risky security (stocks) and a safe asset, whose gross return is Rf. Letting Rr and denote the expected return and variance of the risky assets, a the consumer's degree of relative risk aversion, and assuming quadratic preferences, the share of wealth invested in the risky asset is
Provided Rr–Rf > 0, all investors participate in the stock market. If instead of only one risky asset there were n, we know since Tobin (1958) that investors would combine the safe asset with the portfolio of risky assets with the largest Sharpe ratio (the ratio of the average excess return to the portfolio standard deviation) and end up having the same portfolio of risky assets. In reality, access to the stock market is costly due to information and trading costs. In the presence of entry costs it is difficult for a single investor to achieve the best allocation. Suppose, in the two assets example, that investors incur a fixed cost K to buy stocks (or to obtain the best portfolio in the n risky assets case). Then, for a consumer it will pay to invest in the risky asset only if EU(RfW+λW(R̃r −Rf) –K) > U(RfW), where expectations are taken over the risky assets return R̃. Furthermore, let RfW+λW(R̂r− Rf) denote the certainty equivalent level of final wealth and R̂r the certainty equivalent return on stocks, where clearly Rf < R̂r < Rr. Then a consumer with wealth W will invest if λW(R̂r− Rf) > K. The left hand side is the (certainty equivalent) extra flow of interest that the investor would obtain if he invested in stocks a share λ of his wealth in case he participates; we call λ the conditional share. It is then clear that the higher the investor's wealth, the more likely is that he invests in stocks. Furthermore, the larger the conditional share, the larger the potential gains from the equity premium and the more likely is participation. More generally, any factor that increases the share invested conditional on participation would also make participation more likely. A higher equity premium affects participation in two ways: because it raises the conditional share and because it increases the certainty equivalent premium. In particular, a lowering of stocks riskiness would increase the conditional share and raise participa-tion; for instance, in the multi securities case this could be brought about by the development of the mutual funds industry and their ability to offer a diversified portfolio. Finally, holding other factors constant, a decline in fixed entry costs, while leaving conditional shares unaffected, would raise participation by lowering the wealth threshold that triggers entry into the stock market. Thus, following a decline in K, the new entrants will be on average less wealthy than the incumbents.