or compactness is not, however, an innocuous assumption. Portfolio revi- sions entail transaction costs, meaning that rebalancing must of necessity be somewhat lim- ited and that skewness and other higher moments cannot entirely be ignored. Compactness also rules out such phenomena as the major stock price jumps that occur in response to takeover attempts. It also rules out such dramatic events as the 25% one-day decline of the stock market on October 19, 1987. Except for these relatively unusual events, however, mean-variance analysis is adequate. In most cases, if the portfolio may be revised fre- quently, we need to worry about the mean and variance only. Portfolio theory, for the most part, is built on the assumption that the conditions for mean-variance (or mean-standard deviation) analysis are satisfied. Accordingly, we typi- cally ignore higher moments.   CONCEPT C H E C K ☞ QUESTION A.1 How does the simultaneous popularity of both lotteries and insurance policies confirm the notion that individuals prefer positive to negative skewness of portfolio returns?       3 Paul A. Samuelson, "The Fundamental Approximation Theorem of Portfolio Analysis in Terms of Means, Variances, and Higher Moments," Review of Economic Studies 37 (1970). II. Portfolio Theory 6. Risk and Risk Aversion The McGraw−Hill Companies, 2001           CHAPTER 6 Risk and Risk Aversion 175     Table 6A.1 Frequency Distribution of Rates of Return from a One-Year Investment in Randomly Selected Portfolios from NYSE-Listed Stocks     N 1 N 8 N 32 N 128   Statistic Observed Normal Observed Normal Observed Normal Observed Normal   Minimum 71.1 NA 12.4 NA