Whether you rely on fundamental or technical analysis to identify promising investments, having a working knowledge of statistical terms is helpful.
Not every investor will be competent to do null-hypothesis significance testing. But every investor should be aware of the meaning of terms such as significance levels and p-values.
What is a p-value?
A p-value, or probability value, is a statistical measure used in hypothesis testing to determine the significance of a result. In finance, you might use hypothesis testing to determine if a particular variable (like interest rates) significantly affects stock prices.
A null hypothesis is a statement that there is no effect or difference. For example: "Changes in interest rates do not affect stock prices".
A p-value is a number that quantifies the likelihood that a null hypothesis is true. A small p-value means the null hypothesis probably should be rejected. A more significant number most likely means your alternative hypothesis was on the money.
You may have heard the phrase, "Correlation does not equal causation". How do we know this? In many instances, a p-value reflects the probability that one variable in your data didn't influence another variable — the likelihood that your data occurred randomly.
Which is better: High p-value or low?
Since a p-value is a probability, it will always be between 0 and 1. Generally, a low p-value means rejecting the null hypothesis and accepting your alternative hypothesis.
A low p-value (typically ≤ 0.05) suggests that the observed data is unlikely under the null hypothesis. This is often interpreted as evidence against the null hypothesis.
A high p-value indicates that the observed data is more likely under the null hypothesis. This suggests there is insufficient evidence to reject it.
It helps if you determine a significance level before calculating the p-value. A significance level is an arbitrary number representing the probability of arriving at a result by chance. Although there's no single best significance level, the most typical values are 0.1, 0.05, and 0.01, with 0.05 being the most commonly used figure.
You can use the significance level in conjunction with the p-value to determine the validity of your conclusions. As a general rule, if the significance level exceeds the p-value, you can reject the null hypothesis.
Calculating the p-value
Fifty years ago, you likely relied on a deep knowledge of integral calculus and an expensive engineering calculator to calculate a p-value for your hypothesis. Thankfully, those days are gone, and it's not necessary to hope you used the correct order of operations for the following formula:
You can enter the formula in a spreadsheet. It's worth noting that Excel has a built-in formula that makes things considerably easier. For Excel, you'll need to know the test statistic (t), the degrees of freedom, and whether you have a one- or two-tailed test.
The test statistic is calculated from your sample data. It will be used for comparison when you test your null hypothesis. The most straightforward test statistic involves subtracting a hypothesised mean from the sample mean and dividing it by the standard error of the mean.
A more complex test statistic would yield the chi-squared value by adding the squared differences between observed and expected values and then dividing by expected values.
Calculating the degrees of freedom is more straightforward. It involves simply subtracting one from the sample size (for example, a sample size of 10 would yield 9 degrees of freedom).
A one-tailed test specifies a direction — greater than or less than a number. A two-tailed test provides information in both directions and is often preferred for thoroughness.
The Excel formula is simple:
=tdist(test statistic, degrees of freedom, 1 or 2 for a one- or two-tailed test).
Not all good investors are quantitative analysts, and not all quantitative analysts are good investors. But, relatively simple and accurate tools exist to help investors test hypotheses and draw more informed conclusions about the merits of specific investments.
How to use p-value in investing
How does this play into investing? You might want to see if there's a correlation between the p-value of an individual stock over a specific period and the historical prices of a commodity, such as wheat or alcohol.
Or you may want to test whether a broker's assertion that a particular set of exchange-traded funds (ETFs) has outperformed the S&P/ASX 200 Index (ASX: XJO) over the past five years is valid.
Although an eye test is useful, any investor knows there's seldom such a thing as too much data. So, it's helpful to collect as much relevant data as possible and then use basic statistical tools to verify (or discount) your hypothesis.
What's the significance?
A p-value is used to determine the statistical significance of a result. For instance, a low p-value in a study testing the effectiveness of a new trading algorithm might suggest that the algorithm performs significantly better than the market average.
So, while not all investors will be quantitative analysts, quantitative analysis is a key element of investing in stocks, even at a rudimentary level. Whether it's part of a buy-and-hold strategy or to short-sell a security, investors buy stocks and bonds for one reason: They believe it will build wealth.
The quantitative element comes into play when trying to predict the future price of a security — the heart of any investment. Will its price go up? Will it go down? The right hypothesis can make or break a particular investment.
Frequently Asked Questions
-
A p-value reveals the strength of evidence against a null hypothesis. Essentially, it quantifies the probability that the results from your data could have occurred under random conditions. In finance, a low p-value (usually ≤ 0.05) indicates a significant finding, suggesting that the effect observed in your data (like a trading strategy's performance) is unlikely to be due to chance. Conversely, a high p-value implies that the observed data align more closely with what you would expect under the null hypothesis, suggesting insufficient evidence to claim a significant effect. It's a vital metric for investors, helping to differentiate between strategies or trends that are genuinely effective and those that might just be flukes.
-
Imagine you're testing a new stock-picking strategy, hypothesising that it outperforms the standard market index. After conducting your study, you calculate a p-value of 0.03. This low p-value suggests there's only a 3% probability that the superior performance of your strategy, compared to the market index, occurred by random chance. In simpler terms, it's strong evidence that your strategy genuinely outperforms the market rather than the outperformance being a result of luck. In this way, p-values can provide statistical backing to investment strategies, adding a layer of confidence to financial decisions.
-
A high p-value is a bit of a double-edged sword. It suggests that the data observed is more likely to have occurred under the null hypothesis, meaning there's insufficient evidence to support a significant effect or difference. For an investor, this can mean that a proposed trading strategy might not be as effective as anticipated or that a presumed market trend isn't statistically significant. While this might seem disappointing, it's crucial information. It helps investors avoid making decisions based on unreliable data or erroneous assumptions. In essence, a high p-value acts as a cautionary signal, guiding investors to reevaluate their strategies or look for more compelling evidence before making investment decisions. In contrast, a low p-value suggests that the observed data is unlikely under the null hypothesis, and your alternative hypothesis is more acceptible.
