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Portfolio theory reduces investments to two dimensions: risk and return. Both variables are forward looking and hence difficult to assess without perfect foresight. Therefore, analysis relies on historical relationships that are extrapolated to the future. It is assumed, or rather hoped, that the historical relationships will also hold in the future. This may or may not be the case.

Risk is a variable that is particularly difficult to define. The most common substitute for risk is price volatility. An asset's historical price fluctuations are observed, and it is assumed that these historical fluctuations incorporate all the risks that stockholders faced in the past. This historical volatility is then used in forward-looking analysis, and it is assumed that any risks that this stock faces have already occurred in the past and hence are incorporated in the historical volatility. The length of time over which historical volatility is calculated is the most important determinant of whether there is any validity to this approach. It clearly makes no sense to produce 10-year forecasts based on historical volatilities observed over only one or two years.

More fundamentally, it is a strong assumption that all risks inherent in a stock have already manifested themselves in the past. The economy evolves constantly and markets are in flux; assuming that future fluctuations will somehow resemble those of the past is not obvious. However, it is the only practical approach that can be taken when forecasts are made.

An improvement over a static forecast can be achieved through the use of GARCH models. In these models, volatility is autocorrelated. These models are better at replicating some of basic observations about volatility, notably that volatility occurs in clusters and is mean reverting. Volatility clusters are periods in which markets are highly volatile for longer periods of time or exhibit low volatility for long periods of time.8 In the words of B. Mandelbrot, “Large changes tend to be followed by large changes, of either sign, and small changes tend to be followed by small changes.”9

In the long run, however, volatility tends to revert to a mean value.

Other than volatility, another statistical term that plays an important role in the construction of portfolios is correlation. It is just as important as return and risk. Even though the risk/return trade-off has become a household term, correlation somehow has been left out. The popular business press does not refer to risk/return/correlation trade-offs.

Correlation describes the comovement of two different assets and can range from to . A perfect correlation of means that prices of the assets move exactly in parallel, whereas means that they move exactly in the opposite direction. When building financial portfolios, it is best to have assets that have no correlation at all.

Volatility of Stocks Going through a Merger

Once a merger is announced, the volatility of a stock declines markedly. Figure 3.1 shows the stock prices of Autonomy Corporation before and after the announcement of its cash merger. It can be seen that price fluctuations following the announcement of the merger are much smaller than before. Figure 3.1 shows daily price changes of Autonomy Corporation, a U.K. – based infrastructure software firm, that was acquired by Hewlett-Packard Co in 2011. This merger was discussed in more detail in Chapter 2. It is clear from the picture that daily price variations are much smaller following the announcement of the merger on August 11, 2011, than prior to that date.

Figure 3.1 Daily Price Changes (in pence) of Autonomy Corporation before and after the Merger Announcement

Daily total returns differ from daily stock price returns in that they incorporate dividends. For dividend paying stocks, price return will be negative on the ex-date of a dividend, even though the investor receives a separate cash flow from the dividend payment. Therefore, total returns are the appropriate measure that will be used for the remainder of this book.

To demonstrate that this is not just an effect in isolated cases, pre- and post-announcement volatilities of 258 mergers were analyzed. The data set consists of 258 cash mergers for the period of March 31, 2010, until March 31, 2014, retrieved from the Bloomberg database. Total returns for these stocks were retrieved also from Bloomberg. The calculation for premerger volatilities and returns starts 60 trading days before the announcement, and post-merger returns and volatilities are calculated from the day following the announcement for 60 days or until completion, whichever came first. The histogram in Figure 3.2 shows the resulting cross-sectional distribution of returns. The height of the bars in these histograms shows the number of stocks whose returns fell within a given range. The distribution of dark bars is that of pre-announcement returns, that of light bars of the returns post-announcement.

8

As an aside, periods of high volatility tend to be accompanied by high trading volumes, at least in developed markets.

9

B. B. Mandelbrot, “The Variation of Certain Speculative Prices,” Journal of Business 36 (1963): 392–417.