Investing Guide: How to Calculate Covariance — A Practical Guide for UK Portfolio Builders

Covariance is a statistical measure of how two variables move together over time. According to FCA guidance on investment risk, in investing, those variables are typically the returns of two assets — say, FTSE 100 shares and UK government gilts. A positive covariance means that when one asset's returns are above its average, the other's tend to be as well. A negative covariance means they tend to move in opposite directions.

The formula for sample covariance between two assets X and Y over n periods is:

Cov(X, Y) = Σ [(Xᵢ − X̄) × (Yᵢ − Ȳ)] / (n − 1)

Where Xᵢ and Yᵢ are the individual observations, X̄ and Ȳ are the means, and n is the number of observations. The denominator uses (n − 1) rather than n for sample covariance, applying what statisticians call Bessel's correction to produce an unbiased estimate.

It is important to understand that covariance on its own does not tell you the strength of the relationship — only the direction. A covariance of 0.013 between gilt yields and interest — see the DMO for current gilt data (dmo.gov.uk), part of GOV.UK rates is positive, but is that strong or weak? To answer that question you need correlation, which normalises covariance by dividing it by the product of the two standard deviations. We will return to this distinction later. For more on portfolio construction and diversification, see our dedicated guide.

Let us work through a concrete example using twelve months of real data. We will calculate the covariance between UK long-term gilt yields and the US Federal Funds rate from February 2025 to January 2026 — two interest rate benchmarks that influence UK mortgage pricing and savings rates.

The monthly data points are as follows. UK gilt yields (percent): 4.51, 4.64, 4.58, 4.60, 4.52, 4.59, 4.64, 4.69, 4.57, 4.50, 4.48, 4.45. Federal Funds rate (percent): 4.33, 4.33, 4.33, 4.33, 4.33, 4.33, 4.33, 4.22, 4.09, 3.88, 3.72, 3.64.

Step 1 — Calculate the means. The mean UK gilt yield is 4.564% (54.77 ÷ 12). The mean Federal Funds rate is 4.155% (49.86 ÷ 12).

Step 2 — Calculate deviations from the mean. For each month, subtract the mean from the observed value. For example, February 2025 gilt yield deviation is 4.51 − 4.564 = −0.054, while the Fed Funds deviation is 4.33 − 4.155 = +0.175.

Step 3 — Multiply the paired deviations. For February 2025: (−0.054) × (+0.175) = −0.00945. Repeat for all twelve months.

Step 4 — Sum the products and divide by (n − 1). The sum of all twelve products equals approximately 0.1439. Dividing by 11 gives us a sample covariance of 0.0131.

The positive result confirms that UK gilt yields and the Federal Funds rate tended to move in the same direction over this period — both drifted lower in the final quarter of 2025 as central banks eased monetary policy.

A raw covariance of 0.0131 is difficult to interpret without context. According to Bank of England yield curve data, is that a strong relationship or a weak one? This is where the Pearson correlation coefficient becomes essential. Correlation standardises covariance to a scale between −1 and +1 by dividing by the product of both assets' standard deviations.

Correlation(X, Y) = Cov(X, Y) / (σX × σY)

Using our data, the standard deviation of UK gilt yields over the twelve months is approximately 0.076, and for the Federal Funds rate it is approximately 0.256. This gives a correlation of 0.0131 / (0.076 × 0.256) ≈ 0.67.

A correlation of 0.67 indicates a moderately strong positive relationship — when one rate moves, the other tends to follow, though not in lockstep. For portfolio construction purposes, correlations above 0.7 suggest limited diversification benefit, while correlations below 0.3 or negative values indicate strong diversification potential.

This is directly relevant for UK investors. If your portfolio holds both UK gilts and assets sensitive to US interest rates, you are getting less diversification than you might assume. This is why many financial advisers recommend including asset classes with low or negative correlation to gilts — such as UK commercial property REITs, commodities, or emerging market equities — within a Stocks and Shares ISA or SIPP. For more on understanding covariance in portfolio context, see our dedicated guide.

Understanding covariance and correlation has direct implications for how you structure investments within your ISA or SIPP wrapper. The core principle is straightforward: combining assets with low or negative covariance reduces overall portfolio volatility without necessarily sacrificing returns.

Consider a simple example relevant to many UK investors. The FTSE 100 and FTSE 250 indices are heavily correlated — they tend to rise and fall together because they are both UK equity benchmarks heavily influenced by the same economic forces. Adding more FTSE 250 exposure to an existing FTSE 100 allocation provides limited diversification benefit.

By contrast, UK gilts have historically shown low or negative correlation with UK equities, particularly during market downturns. When equity markets fall sharply, investors often flee to government bonds, pushing gilt prices up. This negative covariance during stress periods is precisely why balanced funds typically hold a mix of equities and bonds.

For UK investors using the £20,000 annual ISA allowance for 2025/26, the practical takeaway is clear. Rather than concentrating in a single asset class or geographic region, aim for a mix of holdings where the portfolio-level covariance is as low as possible. A global index tracker combined with a UK gilt fund and perhaps a small commodity allocation provides structurally better risk-adjusted returns than putting everything into FTSE All-Share.

For a deeper look at this area, read our guide to How to Calculate Fixed Asset Depreciation Using Excel.

Covariance is a powerful tool, but it comes with important limitations that UK investors should understand before relying on it too heavily. (Source: MoneyHelper diversification guide.)

Covariance is backward-looking. The calculation uses historical data, but past relationships between assets are not guaranteed to persist. During the 2022 gilt crisis, for example, UK government bonds and equities fell simultaneously — a positive covariance event that caught many supposedly diversified portfolios off guard. The relationship between gilts and equities can shift dramatically during regime changes in monetary or fiscal policy.

It assumes a linear relationship. Covariance only captures linear co-movement. Two assets might have a complex, nonlinear relationship that covariance misses entirely. For instance, gold might show near-zero covariance with equities during normal markets but strongly negative covariance during crises — a pattern that a single covariance number smooths away.

The time period matters enormously. Calculate covariance over three months and you may get a very different result than over three years. Short-term noise can dominate, while very long-term calculations may average away important structural changes. Most professional portfolio managers use rolling windows of three to five years for strategic asset allocation, with shorter windows for tactical adjustments.

It does not account for magnitude. A positive covariance tells you two assets tend to move in the same direction, but nothing about how far they move. Two assets might both fall during a downturn, but one might drop 5% while the other drops 40%. For a complete risk picture, you also need to consider individual asset volatility alongside the covariance matrix.

Despite these limitations, covariance and its normalised cousin, correlation, remain the foundation of portfolio construction. Most UK investment platforms that offer model portfolios — from Vanguard LifeStrategy to Nutmeg's managed portfolios — use covariance matrices as a core input to their allocation algorithms.

This article is for informational purposes only and does not constitute regulated financial advice. The value of investments can go down as well as up, and you may get back less than you invest. For personalised advice, consult a qualified financial adviser.. For more on asset allocation strategies for UK investors, see our dedicated guide. For more on tax-efficient investing through ISAs, see our dedicated guide.

For a deeper look at this area, read our guide to Cash Flow Statements Explained.