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# Alpha Risk

## What Is Alpha Risk?

Alpha risk is the risk that in a statistical test a null hypothesis will be rejected when it is actually true. This is also known as a type I error, or a false positive. The term "risk" refers to the chance or likelihood of making an incorrect decision. The primary determinant of the amount of alpha risk is the sample size used for the test. Specifically, the larger the sample tested, the lower the alpha risk becomes.

Alpha risk can be contrasted with beta risk, or the risk of committing a type II error (i.e., a false negative).

Alpha risk, in this context, is unrelated to the investment risk associated with an actively managed portfolio that seeks alpha, or excess returns above the market.

### Key Takeaways

• Known as a type I error, alpha risk occurs during hypothesis testing when a null hypothesis is rejected, even though it is accurate and should not be rejected.
• The null hypothesis assumes no cause and effect relationship between the tested item and the stimuli applied during the test.
• A type I error is essentially a "false positive," leading to an incorrect rejection of the null hypothesis.
• Alpha, or the active return from investing, is not related to alpha risk in statistical decision-making.

## Understanding Alpha Risk

The null hypothesis in a statistical test usually states that there is no difference between the value being tested and a particular number, such as zero or one. When the null hypothesis is rejected, the person conducting the test is saying there is a difference between the tested value and the particular number.

Alpha risk is the risk that a difference will be detected when no difference actually exists. It may be explained as the risk found in incorrectly rejecting the null hypothesis when an alternative hypothesis is, in fact, false. This is a false positive, put simply, it is taking the position that there is a difference when, in fact, there is none. A statistical test should be employed to detect differences between a hypothesis and the null, and the alpha risk is the probability that such a test will report one when there is really nothing there. If alpha risk is 0.05, there is a 5% likelihood of inaccuracy.

The best way to decrease alpha risk is to increase the size of the sample being tested with the hope that the larger sample will be more representative of the population.

## Hypothesis Testing

Hypothesis testing is a process of testing a conjecture by using sample data. The test is designed to provide evidence that the conjecture or hypothesis is supported by the data being tested. A null hypothesis is the belief that there is no statistical significance or effect between the two data sets, variables, or populations being considered in the hypothesis. Typically, a researcher would try to disprove the null hypothesis.

For example, let's say the null hypothesis states that an investment strategy doesn't perform any better than a market index, such as the S&P 500. The researcher would take samples of data and test the historical performance of the investment strategy to determine if the strategy performed at a higher level than the S&P. If the test results showed that the strategy performed at a higher rate than the index, the null hypothesis would be rejected.

This condition is often denoted as "n=0." If—when the test is conducted—the result seems to indicate that the stimuli applied to the test subject cause a reaction, the null hypothesis stating that the stimuli do not affect the test subject would, in turn, need to be rejected.

Ideally, a null hypothesis should never be rejected if it's found to be true, and it should always be rejected if it's found to be false. However, there are situations when errors can occur.

## Examples of Alpha Risk

An example of alpha risk in finance would be if one wanted to test the hypothesis that the average yearly return on a group of equities was greater than 10%. So the null hypothesis would be if the returns were equal to or less than 10%. In order to test this, one would compile a sample of equity returns over time and set the level of significance.

If, after statistically looking at the sample, you determine that the average yearly return is higher than 10%, you would reject the null hypothesis. But in reality, the average return was 6% so you have made a type I error. The probability that you have made this error in your test is the alpha risk. This alpha risk could lead you to invest in a group of equities when the returns do not actually justify the potential risks.

In medical testing, a type I error would cause the appearance that a treatment for a disease has the effect of reducing the severity of the disease when, in fact, it does not. When a new medicine is being tested, the null hypothesis will be that the medicine does not affect the progression of the disease. Let's say a lab is researching a new cancer drug. Their null hypothesis might be that the drug does not affect the growth rate of cancer cells.

After applying the drug to the cancer cells, the cancer cells stop growing. This would cause the researchers to reject their null hypothesis that the drug would have no effect. If the drug caused the growth stoppage, the conclusion to reject the null, in this case, would be correct. However, if something else during the test caused the growth stoppage instead of the administered drug, this would be an example of an incorrect rejection of the null hypothesis, i.e., a type I error.

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