BPS Variance Calculator: Formula & Examples

This calculator works out the variance of a set of numbers expressed in basis points (bps), a common unit in finance for describing percentage changes. You enter a list of values (each already in bps, e.g., 15 for 0.15%), and the calculator returns the sample variance and population variance in bps², along with the standard deviation in bps. The first output is the sample variance, the most commonly used measure in financial analysis.

Formula

Variance measures how spread out a set of numbers is. For a sample of n values, the sample variance (s²) is:

s² = Σ(xᵢ – x̄)² / (n – 1)

where xᵢ is each individual value (in bps), x̄ is the mean of all values (in bps), and n is the number of values. The population variance (σ²) uses n instead of n – 1 in the denominator. The standard deviation is simply the square root of the variance, bringing the units back to bps.

Worked Example

Using the default values: 10, 12, 9, 11, 13, 8 (all in bps).

1. Calculate the mean: (10+12+9+11+13+8) / 6 = 63/6 = 10.5 bps
2. Compute each deviation from the mean and square it: (10-10.5)²=0.25, (12-10.5)²=2.25, (9-10.5)²=2.25, (11-10.5)²=0.25, (13-10.5)²=6.25, (8-10.5)²=6.25
3. Sum of squared deviations: 0.25+2.25+2.25+0.25+6.25+6.25 = 17.5
4. Sample variance: 17.5 / (6-1) = 3.5 bps²
5. Population variance: 17.5 / 6 ≈ 2.9167 bps²
6. Sample standard deviation: √3.5 ≈ 1.87 bps

The calculator displays sample variance as 3.50 bps², population variance as 2.92 bps², and sample standard deviation as 1.87 bps.

Why Use Basis Points?

Basis points (bps) are a unit of measure equal to 1/100th of a percentage point. One bps = 0.01% = 0.0001 in decimal. Financial professionals use bps to avoid confusion when talking about small percentage changes. For example, a yield change from 2.50% to 2.75% is often described as a 25 bps increase, not a 0.25% increase (which could be misread as a relative change). When you calculate variance on values already expressed in bps, the result is in bps², which is not intuitive. That’s why the standard deviation (in bps) is often reported alongside variance.

How to Read the Result

A higher variance means the values are more spread out from the mean. For instance, if you are analyzing daily changes in a bond yield (in bps), a sample variance of 3.5 bps² corresponds to a standard deviation of about 1.87 bps. This tells you that on a typical day, the yield change deviates from the average by roughly 1.87 bps. The sample variance is used when your data is a sample of a larger population (e.g., 30 days of data out of a year). The population variance is used when you have the entire dataset (e.g., all trading days in a year).

Common Mistakes and Limits

• Unit mix-up: Ensure your input values are already in bps. If you have percentages (e.g., 0.15%), multiply by 100 to get bps (15). Do not mix units.
• Sample vs. population: Using the wrong denominator changes the result. For small samples, the difference is large. Always decide whether your data is a sample or the full population.
• Interpreting bps²: Variance in bps² is not directly comparable to variance in percentage². To convert variance from bps² to percentage², divide by 10,000 (since 1 bps = 0.0001, so 1 bps² = 0.00000001, and 1 percentage point² = 0.0001, so 1 bps² = 0.0001 percentage²).
• Outliers: Variance is sensitive to extreme values. A single large deviation can inflate the variance significantly.
• Zero variance: If all values are identical, variance is 0. This means no spread.

Alternatives to Variance

Variance is a measure of dispersion, but it’s not the only one. The standard deviation (square root of variance) is more interpretable because it is in the same units as the original data. The interquartile range (IQR) is less sensitive to outliers and measures the spread of the middle 50% of the data. The mean absolute deviation (MAD) is another option, but it is less common in finance. For financial risk, the variance of returns (often in bps) is a key input to the Sharpe ratio and Value at Risk (VaR) calculations.

When to Use This Calculator

Use this calculator when you have a set of values already expressed in bps and you need to quantify their variability. Common scenarios include:

• Analyzing daily yield changes on a bond or Treasury bill.
• Measuring the volatility of credit spreads.
• Comparing the variability of expense ratios or fee changes.
• Evaluating the consistency of a trading strategy’s returns (in bps).

For raw percentages, you can still use this calculator by first converting each percentage to bps (multiply by 100). The resulting variance will be in bps², which you can convert back if needed.

Frequently Asked Questions

What is the difference between sample and population variance?

Sample variance uses n-1 in the denominator to correct for the fact that you are estimating the variance of a larger population from a sample. Population variance uses n, assuming you have every data point. For large samples (n>30), the difference is negligible.

Can I enter negative values?

Yes, negative values are allowed. In finance, negative bps can represent a decrease or a negative spread. The variance calculation squares deviations, so negative values are handled correctly.

How do I convert variance from bps² to percentage²?

Divide the variance in bps² by 10,000. For example, 3.5 bps² = 0.00035 percentage². To get the standard deviation in percentage, take the square root of that, or simply divide the standard deviation in bps by 100.

Why is variance not in the same unit as my data?

Because variance is the average of squared deviations, the unit is squared. That is why the standard deviation, which is the square root of variance, is often preferred for interpretation. This calculator gives you both.

What if my list has only one value?

If you enter a single value, the sample variance is undefined because the denominator (n-1) becomes zero. The calculator will show an error for sample variance. Population variance would be 0, but with one data point, variance is meaningless. Add at least two values.

Can I use this for percentages directly?

Yes, but be careful. If you enter percentages (e.g., 0.15, 0.12), the variance will be in percentage², not bps². To get bps², multiply each percentage by 100 first. The calculator does not auto-convert.

Related

• Correlation Coefficient: Formula and Calculation
• Specificity From a Confusion Matrix: Formula and Example
• Calculate Relative Error: Formula and Examples

in list | Values (bps) | 10, 12, 9, 11, 13, 8 | 0
out var_sample | Sample variance (bps²) | (sum((x - mean)^2) / (count - 1)) | 2
out var_pop | Population variance (bps²) | (sum((x - mean)^2) / count) | 2
out sd_sample | Sample std dev (bps) | sqrt(var_sample) | 2