The price earnings ratio for firms in a certain industry follows the normal distribution. A firm’s price-earnings ratio has a standardized value (Z) = 1.0 and it is included in the highest _____ of firms in the industry.
15.87%
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.
| Z-score | Area between Mean (0) and Z | Cumulative Area to the Left of Z | Area to the Right of Z |
|---|---|---|---|
| 0.0 | 0.00% | 50.00% | 50.00% |
| 1.0 | 34.13% | 84.13% | 15.87% |
| 2.0 | 47.72% | 97.72% | 2.28% |
| 3.0 | 49.87% | 99.87% | 0.13% |
Revision Table: Normal Distribution and Z-scores
| Concept | Explanation | Relevance to Problem |
|---|---|---|
| Normal Distribution | A symmetric, bell-shaped probability distribution. Many natural and financial variables follow this distribution. | The firm's P/E ratio follows this distribution, allowing us to use Z-scores. |
| Z-score (Standardized Value) | Measures how many standard deviations an observation is from the mean. Calculated as $\frac{(X - \mu)}{\sigma}$. | Given as 1.0 for the firm's P/E ratio. |
| Area under the Curve | Represents probability or proportion/percentage. Total area is 1 (100%). | We need to find the area to the right of Z=1.0 to find the percentage of firms with higher P/E ratios. |
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.
Important Questions from Miscellaneous
A stone is thrown horizontally from the top of a 20 m high building with a speed of 12 m/s. It hits the ground at a distance R from the building. Taking g = 10 m/s2 and neglecting air resistance will give :
A sphere of volume V is made of a material with lower density than water. While on Earth, it floats on water with its volume f1V (f1 < 1) submerged. On the other hand, on a spaceship accelerating with acceleration a < g (g is the acceleration due to gravity on Earth) in outer space, its submerged volume in water is f2V. Then:
A railway wagon (open at the top) of mass M1 is moving with speed v1 along a straight track. As a result of rain, after some time it gets partially filled with water so that the mass of the wagon becomes M2 and speed becomes v2. Taking the rain to be falling vertically and the water stationery inside the wagon, the relation between the two speeds v1 and v2 is :
Consider the following statements:
1. Distance between the longitudes becomes zero on North Pole and South Pole.
2. Distance between the longitudes is maximum on the Equator.
3. Number of longitudes is more than number of latitudes.
Which of the statements given above is/are correct?
One block of 2⋅0 kg mass is placed on top of another block of 3⋅0 kg mass. The coefficient of static friction between the two blocks is 0⋅2. The bottom block is pulled with a horizontal force F such that both the blocks move together without slipping. Taking acceleration due to gravity as 10 m/s2, the maximum value of the frictional force is :
