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Unformatted text preview: UNIVERSITY OF NORTH CAROLINA KENAN-FLAGLER BUSINESS SCHOOL BUSI 580: INVESTMENTS PART I: PORTFOLIO THEORY AND ASSET PRICING Prof. Günter Strobl Spring 2010 Solution to Problem Set 1 A. Empirical Properties of Stock Returns 1. All three indices had their lowest returns in October 1987 (see the Excel spreadsheet PS1_solution.xls), the month of the infamous market crash known as “Black Monday” (October 19, 1987). The lowest return of Coca Cola occurred during the 1973/74 market crash in September 1974. Two of the five stocks (IBM and Eastman Kodak had their lowest returns during the recent financial crisis of 2008/09. Of course, a market crash is by definition a cataclysmic event in which all (or at least most) stocks experience sharp declines, so this result is not surprising. Most of the largest monthly returns occurred during the recovery phase after the 1973/74 crash or the 2000-2002 crash, as stocks experienced a sharp rebound. Of course, all stocks are affected to some extent by market crashes and other significant macroeconomic events. For well-diversified portfolios like the market indices and the S&P 500 index, we would expect macroeconomic events to be the primary (and possibly only) driver of returns. On the other hand, we would expect individual stocks to be much more sensitive to idiosyncratic events. For example, we find that IBM experienced a return of 8.6% in July 1996 after it reported record fourth-quarter revenues of $23.1 billion. By contrast, all other stocks and indices had negative returns in that month. 2. While the average returns of individual stocks and stock indices are comparable, individual stocks are clearly more volatile than the stock indices (see the calculations in the Excel spreadsheet). Only the average returns of the S&P and the T-bills are significantly different from 1% at the 5% level (all other p-values are greater than 0.05). Looking at the skewness, we find that the returns on all indices are left-skewed, which is consistent with sudden stock market crashes and slow booms. For the five individual stocks, the skewness is sometimes positive and sometimes negative, but 1 always fairly close to zero. We therefore cannot reject the hypothesis that the monthly returns are “drawn from a symmetric distribution.” For the kurtosis, we get a different picture. While the values are less dramatic than those for daily returns (see the examples in the lecture notes), they are consistently above 3 (note that Excel normalizes the kurtosis of the normal distribution to 0, so you have to add back 3). Notice that the kurtosis of the equally-weighted returns is much higher than either the value-weighted returns or the S&P 500 returns. This shows that stocks of smaller firms (which have disproportionately more weight in the equally-weighted index) have a higher kurtosis than those of larger firms. While we cannot reject the hypothesis that the returns are drawn from a normal distribution without a formal test, the high estimates for the kurtosis indicate that the return distribution has fat tails. 3. Of the two market indices, the value-weighted index has by far the larger correlation with the S&P 500. This should not come as a surprise, since the S&P 500 is also a value-weighted index, and hence puts more emphasis on the returns of stocks with large market capitalizations. By contrast, an equally-weighted index gives the same weight to all stocks, which means that small stocks get disproportionately more weight while large stocks get disproportionately less weight (relative to a valueweighted index). Of the five individual stocks, IBM has the highest correlation with the S&P 500 index. This suggests that IBM is most representative of the stocks which make up the S&P 500 index (at least within our sample), although even the smallest correlation (for Merck) is still quite large. This reflects the fact that all five stocks have large market capitalizations, which gives them more weight in the S&P 500 index. B. Mean-Variance Portfolio Optimization For the answers to questions 4 – 7, please see the Excel spreadsheet “PS1_solution.xls” posted on the course website. 2 ...
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