How many stocks make a diversified portfolio in a continuous-time world?

Abstract

Abstract. This thesis aims to answer how many stocks make a diversified portfolio in a continuous-time world. The study investigates what are the factors determining diversification effects in a real, continuous-time, world as opposed to thoroughly studied theoretical single period world. Continuous-time world investors care about geometric, instead of arithmetic, rate of return. We show how methodology based on information theory can be utilized in investing context. Geometric risk premium is explained by the Shannon limit and its derivative, fractional Kelly criterion. Investing world counterpart for the Shannon limit, compounding process capacity, is derived. Geometric risk premium is decomposed to single stock risk premium and diversification premium. Method for estimating diversification premium is provided. Concept of realizable risk premium is derived and used in risk averse investor diversification metrics. Diversification effect is measured as a (realizable) risk premium ratio and as a (realizable) gross compound excess wealth ratio. Both ratios are between a randomly selected portfolio of selected size and fully diversified benchmark. We show, both analytically and empirically, that diversification in a continuous-time world is a negative price lunch as opposed to free lunch in a single period world. Investor is paid a diversification premium, implying higher geometric risk premium, for consuming a lunch. The magnitude of diversification premium difference to benchmark, the opportunity cost of foregone diversification, is shown to be equal to one half of portfolio’s idiosyncratic variance scaled by squared investment fraction. To maintain a constant wealth ratio, required level of diversification for a long-term risk neutral investor is approximately directly proportional to investment time horizon length. The factors determining required level of diversification in a continuous-time world are number of stocks in the benchmark, Sharpe ratio and variance of the benchmark, idiosyncratic variance of an average stock, investment fraction and time. At investment fraction 1.0, risk averse investor requires more than 100, 200 or 1000 stocks to achieve 90%, 95% or 99% of the maximum diversification benefit, respectively. For short-term risk neutral investor, the corresponding numbers are about 20, 40 or 200 stocks and yet significantly more for long-term risk neutral investor. The numbers increase and decrease as investment fraction increase and decrease, respectively. We find that small firms require substantially more diversification compared to large firms and that there are substantial and consistent differences in diversification premiums between investing styles