What Is the Harmonic Mean?

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What Is the Harmonic Mean?

You might not realize it, but the harmonic mean is a specific way to average a series of financial ratios. I find it particularly useful when you're dealing with multiples like the price-to-earnings ratio. To calculate it, you divide the number of values in your series by the sum of the reciprocals of each number. Essentially, it's the reciprocal of the arithmetic mean of those reciprocals.

Key Takeaways

Let me break this down for you: the harmonic mean is indeed the reciprocal of the arithmetic mean of the reciprocals. In finance, you use it to average data like price multiples. It weights each value equally, unlike the weighted version which assigns importance-based weights. Market technicians also apply it to spot patterns like Fibonacci sequences.

Formula and Calculation

When you need to calculate the harmonic mean, start by taking the reciprocal of each number in your series. Add those up, then divide the number of values by that total. For example, with numbers 1, 4, and 4, the reciprocals are 1, 0.25, and 0.25. Sum them to 1.5, divide 3 by 1.5, and you get 2. Remember, the reciprocal of n is just 1/n—it's that straightforward.

Using the Harmonic Mean

You'll see the harmonic mean in finance and technical analysis of markets. It helps with multiplicative or divisor relationships in fractions, skipping the hassle of common denominators. It's also great for averaging rates, like travel speeds over trips. Specifically for multiples like P/E ratios, it ensures equal weight to each point, making the result less biased. If some values matter more, switch to the weighted harmonic mean: that's the sum of weights divided by the sum of weights over each x_i.

Arithmetic Mean and Geometric Mean

You should know that harmonic, arithmetic, and geometric means are the Pythagorean means, each suited to different scenarios. The arithmetic mean is the sum of numbers divided by their count—think class average from test scores. A weighted version emphasizes higher points. The geometric mean is the nth root of the product of n numbers, perfect for percentages in investments. Use arithmetic for raw values, geometric for products.

Fast Fact

Here's a quick note: the harmonic mean works best for fractions like rates or multiples.

Example of the Harmonic Mean

Consider two companies: one with $100 billion market cap and $4 billion earnings (P/E 25), another with $1 billion cap and $4 million earnings (P/E 250). If you're indexing with 10% in the first and 90% in the second, the weighted arithmetic mean gives a P/E of 227.5, but the weighted harmonic mean comes to about 131.6. You can see how the arithmetic version overestimates it.

Advantages and Disadvantages

The harmonic mean includes every entry and can't be computed if any is zero. With weighting, it emphasizes smaller values and handles negatives. It produces a straighter curve than arithmetic or geometric means. On the downside, reciprocals make it complex and time-consuming. Extreme values heavily influence it, and zeros make it impossible.

What Is the Difference Between Harmonic Mean and Arithmetic Mean?

The harmonic mean divides the number of values by the sum of reciprocals, while arithmetic is sum divided by count. It's the reciprocal of the arithmetic mean of reciprocals.

When Is the Harmonic Mean Used?

Apply it to fractions like rates or multiples, such as P/E in finance, or for Fibonacci patterns in technical analysis.

What Affects the Calculation of the Harmonic Mean?

It includes all entries, weights smaller values more, handles negatives, but fails with zeros.

The Bottom Line

To wrap this up, you calculate the harmonic mean by dividing the number of entries by the sum of reciprocals. It differs from arithmetic and geometric means by using reciprocals and weighting smaller values, which is key when that's needed.

What Is the Harmonic Mean?

Highlights