Question

Returns on common stocks The return on a stock is the change in its market price plus Page Number: 51 any dividend payments made. Return is usually expressed as a percent of the beginning price. The figure shows a histogram of the distribution of monthly returns for the U.S. stock market over a 273-month period.

Part (a). Describe the overall shape of the distribution of monthly returns.

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

Part (c). Explain why you cannot find the exact value for the minimum return. Between what two values does it lie?

Part (d). A return less than 0 means that stocks lost value in that month. About what percent of all months had returns less than 0?

Short answer

Part (a)

Skewed to the left

2 gaps

3 Potential outlier

Part (b)To be between 0 and 2.5 percent

Part (c)

We can't figure out the exact value of the minimum return because we haven't been given it, and we only know a range of possible values for the minimum return from the corresponding bar in the histogram.

The minimum return ranges between -25 and -22.5 percent.

Part (d) 37.36 %

Step-by-step solution

Part (a) Step 1 Given information. 

A histogram of the monthly return distribution for the US stock market over a 273-month period.

Part (a) Step 2. The overall shape of the distribution of monthly returns.  

Because the highest bars in the histogram are to the right, with a tail of smaller bars to the left, the distribution is skewed to the left.

The distribution appears to have three outliers because there are two bars (one with frequency 2 and one with frequency 1) that are separated from the other bars of the histogram by a gap.

The gaps are roughly at -20 and -13, with two potential outliers between -25 and -22.5 and one other between -17.5 and -15.

As a result:

Skewed to the left

2 gaps

3 Potential outlier

Part (b) Step 1. The distribution's approximate center

We anticipate that the distribution's center will be near the top of the histogram.

Because the highest bar has monthly percent returns ranging from 0% to 2.50%, we estimate the center of this distribution to be between 0% and 2.50%.

As a result:

To be between 0 and 2.5 percent

Part (c) Step 1. The exact value for the lowest possible return What two values does it fall between

The leftmost bar in the histogram represents the lowest return.

We cannot, however, determine the exact value of the minimum return because we have not been given its exact value and only know a range of possible values for the minimum return from the corresponding bar in the histogram.

The leftmost bar in the histogram represents monthly percent returns on common stocks ranging from -25 to -22.5 percent, implying that the minimum return is between -25 and -22.5 percent.

A s a result:

We can't figure out the exact value of the minimum return because we haven't been given it, and we only know a range of possible values for the minimum return from the corresponding bar in the histogram.

The minimum return ranges between -25 and -22.5 percent.

Part (d) Step 1. What percentage of all months had returns less than zero

Let us first estimate the frequency of each bar's interval, which is given by the height of the bar.

Interval	Frequency
-25<-25	2
-22.5<-20	0
-20<-17.5	0
17.5<-15	1
-15<-12.5	0
-12.5<-10	4
-10<-7.5	6
-7.5<-5	11
-5<-2.5	28
-2.5<0	50
0<2.5	78
2.5<5	58
5<7.5	24
7.5<10	8
10<12.5	3

Add the frequencies of all intervals with negative returns next.

$2+0+0+1+0+4+6+11+28+50=102$

As a result, we can see that 102 of the 273 months in the sample have a return less than zero.

$$\begin{gathered} \frac{102}{273}=\frac{34}{91} \\ \frac{34}{91}\approx 0.3736 \\ 0.3736=37.36\% \end{gathered}$$

This means that 37.36 percent of the three months had returns less than zero.