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
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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- Fall '09
- Normal Distribution, returns, Stock market index, individual stocks