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Chapter 7: Problem 29

In economic theory, a "hurdle rate" is the minimum return that a person requires before he or she will make an investment. A research report says that annual returns from a specific class of common equities are distributed according to a normal distribution with a mean of \(12 \%\) and a standard deviation of \(18 \%\). A stock screener would like to identify a hurdle rate such that only 1 in 20 equities is above that value Where should the hurdle rate be set?

Short Answer

Expert verified
Set the hurdle rate at approximately 29.61%.

Step by step solution

01

Understanding the Problem

We are asked to determine a hurdle rate for annual returns from a specific class of equities. These returns follow a normal distribution with a mean of 12% and a standard deviation of 18%. We aim to set a hurdle rate such that only 5% (or 1 in 20) of equities exceed this return rate.
02

Identifying the Probability Target

Since we want only 5% of the values to be greater than our hurdle rate, this implies we need to find a value on the normal distribution curve where the cumulative distribution function (CDF) has a value of 95%.
03

Finding Z-Score for 95%

To find the z-score corresponding to the 95th percentile of a standard normal distribution, we look it up in a standard normal distribution table or use a calculator. The z-score for 95% is approximately 1.645.
04

Applying the Z-Score Formula

Use the z-score formula for a normal distribution to find the corresponding value of the hurdle rate. The formula is: \[ z = \frac{X - \mu}{\sigma} \]where \(X\) is the hurdle rate, \(\mu\) is the mean (12%), and \(\sigma\) is the standard deviation (18%).
05

Calculating the Hurdle Rate

Rearrange the z-score formula to solve for \(X\): \[ X = z \cdot \sigma + \mu \]Substitute \(z = 1.645\), \(\mu = 12\%\), and \(\sigma = 18\%\) to get:\[ X = 1.645 \times 18\% + 12\% = 29.61\% \]
06

Conclusion

The hurdle rate should be set at approximately 29.61% to ensure that only about 5% of equities exceed this return rate.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Normal Distribution
Normal distribution is at the heart of many statistical concepts and is essential for analyzing data trends. It's a continuous probability distribution, often represented as a bell-shaped curve.
  • Symmetrical: The curve is symmetrical around the mean, which is the peak of the curve.
  • Mean, Median, and Mode: In a normal distribution, the mean, median, and mode are all the same.
  • Standard Deviation: Determines the spread or width of the curve. Smaller standard deviations mean a steeper curve, while larger ones create a flatter curve.
Normal distribution is vital in investment analysis because it helps investors understand how returns could vary. They can predict, assess risks, and make informed decisions based on statistical expectations.
Exploring the Z-Score
The z-score is a statistical measure that tells you how many standard deviations a data point is from the mean. It鈥檚 useful for determining probabilities in a normal distribution.Here's the formula for calculating it:\[ z = \frac{X - \mu}{\sigma} \]
  • X is the data point or hurdle rate you are assessing.
  • \mu is the mean of the distribution.
  • \sigma is the standard deviation.
Investors use the z-score to identify where a specific performance stands compared to the average. For example, a high z-score means the value is much higher than the mean, indicating rare performance in this context. This insight is crucial for setting precise expectations for returns.
Cumulative Distribution Function (CDF) Insights
The cumulative distribution function (CDF) is a function that indicates the probability that a random variable will take a value less than or equal to a specific value. Here鈥檚 why CDF is significant:
  • Probability Calculation: CDF provides the probability that the value is below a specific point in the distribution. This is used to understand overall distribution behavior.
  • Visual Representation: At any given point along the CDF, you can see the probability coverage up to that point, making it easier to visualize probabilities.
In investment analysis, CDF helps investors set and analyze hurdle rates, as it enables them to precisely calculate the percentage of returns falling below or exceeding particular values. For instance, when only 5% of the returns exceed a hurdle rate, we focus on the 95th percentile in the CDF.
Investment Analysis and Hurdle Rate
In investment analysis, the hurdle rate is crucial as it represents the minimum acceptable return on an investment necessary before a decision to proceed is made. Key points about hurdle rate:
  • Risk Assessment: It helps in determining the minimum return required to justify the risk of a particular investment.
  • Investment Decisions: Companies and investors use the hurdle rate to make decisions about projects or assets. If projected returns are below the hurdle rate, the investment may be considered unattractive.
Using normal distribution and related statistical tools, such as z-score and CDF, allows investors to base this rate on statistical analysis. They can gauge realistic expectations and apply these insights to potential opportunities, ensuring smarter investment choices.

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Most popular questions from this chapter

Most four-year automobile leases allow up to 60,000 miles. If the lessee goes beyond this amount, a penalty of 20 cents per mile is added to the lease cost. Suppose the distribution of miles driven on four-year leases follows the normal distribution. The mean is 52,000 miles and the standard deviation is 5,000 miles. a. What percent of the leases will yield a penalty because of excess mileage? b. If the automobile company wanted to change the terms of the lease so that \(25 \%\) of the leases went over the limit, where should the new upper limit be set? c. One definition of a low-mileage car is one that is 4 years old and has been driven less than 45,000 miles. What percent of the cars returned are considered low-mileage?

The annual commissions earned by sales representatives of Machine Products Inc., a manufacturer of light machinery, follow the normal probability distribution. The mean yearly amount earned is \(\$ 40,000\) and the standard deviation is \(\$ 5,000 .\) a. What percent of the sales representatives earn more than \(\$ 42,000\) per year? b. What percent of the sales representatives earn between \(\$ 32,000\) and \(\$ 42,000 ?\) c. What percent of the sales representatives earn between \(\$ 32,000\) and \(\$ 35,000 ?\) d. The sales manager wants to award the sales representatives who earn the largest commissions a bonus of \(\$ 1,000\). He can award a bonus to \(20 \%\) of the representatives. What is the cutoff point between those who earn a bonus and those who do not?

Suppose the Internal Revenue Service reported that the mean tax refund for the year 2017 was \(\$ 2,800 .\) Assume the standard deviation is \(\$ 450\) and that the amounts refunded follow a normal probability distribution. a. What percent of the refunds are more than \(\$ 3,100 ?\) b. What percent of the refunds are more than \(\$ 3,100\) but less than \(\$ 3,500 ?\) c. What percent of the refunds are more than \(\$ 2,250\) but less than \(\$ 3,500 ?\)

Assume that the hourly cost to operate a commercial airplane follows the normal distribution with a mean of \(\$ 2,100\) per hour and a standard deviation of \(\$ 250 .\) What is the operating cost for the lowest \(3 \%\) of the airplanes?

The manufacturer of a laser printer reports the mean number of pages a cartridge will print before it needs replacing is \(12,200 .\) The distribution of pages printed per cartridge closely follows the normal probability distribution and the standard deviation is 820 pages. The manufacturer wants to provide guidelines to potential customers as to how long they can expect a cartridge to last. How many pages should the manufacturer advertise for each cartridge if it wants to be correct \(99 \%\) of the time?
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