Chebshev's Inequalities

Chebyshev's Inequalities

Chebyshev's Inequality (often referred to as the Chebyshev’s monotonic inequalities) relates to synchronization of the average of the products of two sequences to the product of their averages, based on their relative ordering.

The Formal Definition

Let \( a_1 \ge a_2 \ge \dots \ge a_n \ and \ b_1 \ge b_2 \ge \dots \ge b_n \) be two monotonic sequences of real numbers. If the sequences are similarly ordered (both non-increasing or both non-decreasing), then the following inequality holds: $$\frac{1}{n} \sum_{i=1}^{n} a_i b_i \ge \left( \frac{1}{n} \sum_{i=1}^{n} a_i \right) \left( \frac{1}{n} \sum_{i=1}^{n} b_i \right)$$ If the sequences are oppositely ordered (one is non-increasing while the other is non-decreasing), the inequality is reversed: $$\frac{1}{n} \sum_{i=1}^{n} a_i b_i \le \left( \frac{1}{n} \sum_{i=1}^{n} a_i \right) \left( \frac{1}{n} \sum_{i=1}^{n} b_i \right)$$

Continuous Version

The inequality can also be expressed in terms of integrable functions \( f(x)\ and\ g(x)\) on an interval \( [a,\ b] \). If both functions are monotonic in the same direction, then: $$\frac{1}{b-a} \int_{a}^{b} f(x)g(x) \, dx \ge \left( \frac{1}{b-a} \int_{a}^{b} f(x) \, dx \right) \left( \frac{1}{b-a} \int_{a}^{b} g(x) \, dx \right)$$

Interpretation

The concrete version essentially states that the sum of products is maximized when the largest elements of the first set are paired with the largest elements of the second set.

The Simplified Definition

List A: 1, 10 List B: 2, 20

1. Average of the Products: We multiply the pairs as they are (matched big-with-big), then find the average.

Products: (1 × 2) = 2 and (10 × 20) = 200

Sum of products: 2 + 200 = 202

Average of the products: 202 / 2 = 101

2. Products of the Averages: We find the average of each list first, then multiply those two results together.

Average of List A: (1 + 10) / 2 = 5.5

Average of List B: (2 + 20) / 2 = 11

Products of averages: 5.5 × 11 = 60.5

In this case, 101 ≥ 60.5.

There are many real world applications of Chebyshev’s inequalities and here is an example using stocks.

To identify high performers among the following stock list: EPM, WAL, TPL, VEEV, GOOGL, VRTX, QQQ, and ANET. We can treat two separate financial metrics as our sequences A and B. In healthy, high-performing stock, these metrics should be synchronized.

The Setup

Let us define our two sequences for these stocks:

Sequence A: Momentum (e.g., 6-month price appreciation)

Sequence B: Quality (e.g. Return on Equity or Free Cash Flow growth)

If these stocks are “high performers,” we expect that the stocks with the highest quality (TPL, ANET, VEEV) are also the ones receiving the highest market momentum.

The Chebyshev’s Monotonic Inequalities in Practice

In a random or efficient market, the average of (Momentum x Quality) might be close to the product of their individual averages. However, when we see a strong “Chebysev Gap,” it indicates that the market is rewarding quality with momentum in a synchronized fashion.

If we rank these tickers:

1. ANET & TPL: These often exhibit high synchronization where superior margins (B) are paired with aggressive price action (A).

2. GOOGL & VRTX: These represent stable synchronization; they might not have the highest spikes, but their growth and valuation metrics general move in the same direction.

3. EPM & WAL: These often show “Desynchronization.” For example. If WAL has high earnings but the market suppresses the price due to sector risk, the product of their averages will actually be higher than the average of their products, signaling an “underperformer” or a “value trap.”

Identifying the High Performers

To find the high performers using this logic, you calculate the following:

Find the mean Momentum across all tickers.

Find the mean Quality across all 8 tickers.

Calculate the product of those two means.

Compare this to the actual average of (Momentum x Quality) for each individual ticker.

The “High Performers” are the tickers that contribute the most significantly to the left side of the inequality. In this group, ANET and TPL typically drive the “Average of Products” higher because their high-quality scores are paired with high-performance price weights, whereas EPM might pull the average down by high a high score in one metric but a very low score in the other.

In the context of the stock list, this mathematically proves that the "average performance" of a portfolio is highest when the highest quality weights are aligned with the highest momentum weights.