Understanding Price Earnings Ratio and Normal Distribution

The question asks us to determine the percentage of firms in a certain industry that have a price-earnings (P/E) ratio equal to or higher than a firm with a standardized value (Z-score) of 1.0. We are told that the P/E ratios follow a normal distribution.

In a normal distribution, a Z-score measures how many standard deviations an observation is away from the mean. A Z-score of 1.0 means the firm's P/E ratio is one standard deviation above the industry average P/E ratio.

The normal distribution is symmetric around its mean (which corresponds to a Z-score of 0). The total area under the normal distribution curve is 1, or 100%. This area represents the total probability or the total percentage of firms.

Finding the Percentage Using Z-score

We need to find the percentage of firms that have a Z-score greater than or equal to 1.0. This corresponds to the area under the normal distribution curve to the right of Z = 1.0.

Key properties of the standard normal distribution (Z-distribution) are helpful here:

• The area to the right of the mean (Z = 0) is 50%.
• The area between Z = 0 and Z = 1.0 is approximately 34.13%.

To find the area to the right of Z = 1.0, we can subtract the area between Z = 0 and Z = 1.0 from the total area to the right of Z = 0:

Area (Z > 1.0) = Area (Z > 0) - Area (0 < Z < 1.0)

Area (Z > 1.0) $\approx$ 50% - 34.13%

Area (Z > 1.0) $\approx$ 15.87%

Alternatively, using a standard Z-table, the cumulative area to the left of Z = 1.0 is approximately 0.8413 or 84.13%. The area to the right is then:

Area (Z > 1.0) = 1 - Cumulative Area (Z $\le$ 1.0)

Area (Z > 1.0) $\approx$ 1 - 0.8413

Area (Z > 1.0) $\approx$ 0.1587 or 15.87%

This 15.87% represents the proportion of firms with a P/E ratio that is one standard deviation or more above the mean P/E ratio. Since the question asks for the percentage of firms included in the highest percentage, this 15.87% is the correct value.

Conclusion

A firm with a Z-score of 1.0 for its price-earnings ratio is at a point where 15.87% of firms in the industry have a P/E ratio equal to or higher than this firm. Therefore, it is included in the highest 15.87% of firms.

Revision Table: Normal Distribution and Z-scores

Additional Information: Applying Normal Distribution in Finance

The normal distribution is widely used in finance to model the distribution of various variables like stock returns, asset prices, and financial ratios such as the price-earnings ratio. Understanding the normal distribution and Z-scores helps analysts:

• Assess the probability of extreme events (very high or very low values).
• Compare different financial metrics or assets on a standardized scale.
• Build statistical models for forecasting and risk management.
• Understand where a particular firm or asset stands relative to its peers or historical data.

For instance, knowing a firm's P/E ratio has a Z-score of 1.0 helps contextualize its valuation within the industry – it indicates the firm is valued higher than a significant portion of its peers, assuming the P/E ratios are normally distributed.