Returns on common stocks The return on a stock is the change in its market price plus any dividend payments made. The total return is usually expressed as a percent of the beginning price. The figure below shows a histogram of the distribution of the monthly returns for all common stocks listed on U.S. markets from January 1985 to September 2007 (273 months).28 The extreme low outlier represents the market crash of October 1987, when stocks lost 23% of their value in one month.

(a) Ignoring the outliers, describe the overall shape of the distribution of monthly returns.

(b) What is the approximate center of this distribution?

(c) Approximately what were the smallest and largest monthly returns, leaving out the outliers?

(d) A return less than zero means that stocks lost value in that month. About what percent of all months had returns less than zero?

Consider a return on common stocks. A histogram is given representing the distribution of the monthly returns for all common stocks listed on US markets from January 1985 to September 2007.

It can be observed from the given histogram that at -25% and -15%, monthly return is very rare. There is almost no percentage of monthly return is close to them. So, the monthly return -25% and -15% are the outlier.

It can be observed that the tail is at the left and the maximum value lies around the center which refers to the skewed distribution towards the left.

The histogram is distributed over the approximated range of -12.5% to 12.5%. The approximated center of the distribution will be in the interval of 0% to 2.5%.

The smallest monthly return can be observed from the histogram graph excluding the outlier is the leftmost bar, that is, between -12.5% to 10%. Similarly, the largest monthly return can be observed from the given histogram graph excluding the outlier in the rightmost bar, that is, between 10% to 12.5%.

A monthly return less than 0 implies that the return is in the negative direction and there is a loss in investment.

The total number of months the investment is done can be obtained by observing the total number of months in each bar and summing them up.

The total sum of the number of months is given as follows:

$\begin{array}{rcl}\mathrm{Sum}& =& 1+2+4+5+10+30+50+80+60+20+8+3\\ & =& 273\end{array}$

Compute the sum of the number of months of all intervals lies between -25 to 0.

$1+2+4+5+10+30+50=102$

Compute the ratio of the number of months in the interval -25 to 0 and the total number of months invested in the stock. $\frac{102}{273}=0.3736$

Compute the percentage of months that have been in loss.$\left(0.3736\times 100\right)\%=37.36\%$