Why the Number of Holdings Matters
An equally weighted portfolio assigns the same fraction of its value to every holding. Increasing the number of holdings can reduce portfolio risk because each asset then has less weight. The reduction depends not only on how many assets are included, but also on how their returns move together. If the assets remain positively correlated, adding holdings cannot eliminate all portfolio risk under the equal-correlation model.
The result in this tutorial is a model-based calculation, not a guarantee that a larger portfolio will always be safer. It assumes equal individual variances and the same correlation for every pair of assets. Real portfolios may not satisfy either condition. The model is useful because it isolates the effect of the number of holdings and makes the role of common movement explicit.
Deriving the Portfolio Variance
The portfolio return is the average of the \(n\) asset returns. The variance of a sum includes each asset's own variance and the covariance between every distinct pair. There are \(n\) individual variance terms and \(\frac{n(n-1)}{2}\) distinct pairs. Each pair contributes twice in the variance-of-a-sum formula, which is why the total covariance contribution has a factor of \(n(n-1)\).
The second line follows from the general identity \(\operatorname{Var}(\sum_i R_i)=\sum_i\operatorname{Var}(R_i)+2\sum_{i<j}\operatorname{Cov}(R_i,R_j)\). Under the model, each variance is \(\sigma^2\), while each covariance is \(\rho\sigma^2\). Substituting those values gives \(n\sigma^2\) from the individual terms and \(2\cdot\frac{n(n-1)}{2}\rho\sigma^2=n(n-1)\rho\sigma^2\) from the pairs. Dividing by \(n^2\), because each asset has weight \(1/n\), gives the stated result.
The variance formula has two useful terms. The quantity \(\frac{\sigma^2}{n}\) is the contribution from the assets' individual variances after equal weighting. The remaining contribution, \(\frac{n-1}{n}\rho\sigma^2\), comes from their pairwise covariances. As \(n\) increases, the first term shrinks toward zero. When \(\rho\) is positive, the covariance contribution approaches \(\rho\sigma^2\), rather than disappearing.
How the Diversification Benefit Changes
For a fixed feasible \(\rho<1\), the variance difference between portfolios with \(n\) and \(n+1\) holdings is
This difference is positive when \(\rho<1\), so adding a holding reduces variance under the model. It also gets smaller as \(n\) grows: the denominator \(n(n+1)\) increases. This is the mathematical reason that the benefit of each additional holding diminishes. If \(\rho=1\), the difference is zero, because perfectly correlated assets with equal variances move together and equal weighting does not reduce their variance.
When \(\rho>0\) is held fixed, the variance approaches \(\rho\sigma^2\) as \(n\) grows, and the standard deviation approaches \(\sigma\sqrt{\rho}\). This is the model's common-risk floor. It is a limit, not generally the exact risk of any finite portfolio. When \(\rho=0\), variance instead approaches zero. A fixed negative \(\rho\) cannot be maintained for arbitrarily many assets: for each finite \(n\), feasibility requires \(\rho\geq-\frac{1}{n-1}\). In particular, a negative common correlation that is feasible for one portfolio size may not be feasible for a larger one.
Worked Example: Four Holdings With Positive Correlation
A hypothetical portfolio holds four assets in equal proportions. Each asset has return variance \(0.04\), corresponding to a standard deviation of \(20\%\). Every pair has correlation \(0.25\). Treat these as model inputs, with finite return variances and no distributional assumption. Calculate the portfolio variance and standard deviation, separating the individual and covariance contributions.
Method. Use the variance-of-a-sum derivation, or substitute \(n=4\), \(\sigma^2=0.04\), and \(\rho=0.25\) into the equal-weight formula. The implied covariance of each pair is \(0.25(0.04)=0.01\).
There are four individual variance terms. After weighting by \(1/4\) per asset, their combined contribution is
There are \(\frac{4(3)}{2}=6\) distinct pairs, or 12 covariance terms when each pair is counted twice. Their combined contribution is
Adding the two contributions gives portfolio variance \(0.0100+0.0075=0.0175\). Equivalently, the formula gives \(0.04\left(0.25+\frac{0.75}{4}\right)=0.04(0.4375)=0.0175\). The standard deviation is \(\sqrt{0.0175}=0.132287\ldots\), or 13.23% rounded to two decimal places.
Check. The result is below the individual standard deviation of \(20\%\), but above the model's limiting standard deviation of \(20\%\sqrt{0.25}=10\%\). The positive covariance contribution explains why diversification does not reduce risk to zero. A common mistake is to use only \(0.04/4\) and report a standard deviation of \(10\%\); that calculation omits all six pairwise covariances.
Comparing Different Portfolio Sizes
The formula can compare portfolios directly when the individual risk and pairwise correlation are held constant. The variance reduction from adding a particular number of holdings can look small in absolute terms, especially after the portfolio is already diversified. It is important to distinguish a variance difference from a standard-deviation difference: standard deviation is the square root of variance, so the two differences are not interchangeable.
Worked Example: Comparing Five and Twenty Holdings
Consider two hypothetical equally weighted portfolios. Every asset in both has variance \(0.0225\), or standard deviation \(15\%\), and every pair has correlation \(0.40\). Assume finite variances and the same model inputs at both portfolio sizes; no distributional assumption is made. Calculate each portfolio's variance and standard deviation, then find the variance reduction from five to six holdings and from twenty to twenty-one holdings.
For five holdings, the factor multiplying \(\sigma^2\) is \(0.40+\frac{0.60}{5}=0.52\). Thus the variance is \(0.0225(0.52)=0.011700\), and the standard deviation is \(\sqrt{0.011700}=0.108167\ldots\), or 10.82% to two decimal places.
For twenty holdings, the factor is \(0.40+\frac{0.60}{20}=0.43\). The variance is \(0.0225(0.43)=0.009675\), and the standard deviation is \(\sqrt{0.009675}=0.098361\ldots\), or 9.84% to two decimal places.
From five to six holdings, the variance reduction is
From twenty to twenty-one holdings, it is
The first addition reduces variance by \(0.000450\), while the later addition reduces it by about \(0.00003214\). Both are beneficial in this model, but the later reduction is much smaller. Check: the calculated variances also decrease from \(0.011700\) to \(0.011250\) when moving from five to six holdings, and from \(0.009675\) to \(0.00964286\) when moving from twenty to twenty-one.
Common mistake avoided. Do not call \(0.000450\) the reduction in standard deviation; it is a reduction in variance. To compare standard deviations, take the square root of each portfolio variance first.
Interpreting the Common-Risk Floor
The positive-correlation limit is a useful way to distinguish reducing asset-specific variation from eliminating all risk. Under the equal-correlation model, each asset's return contains variation that is not perfectly shared with every other asset, as well as variation represented by the positive pairwise covariance. Equal weighting progressively averages out the former contribution. The latter remains in the portfolio as holdings increase.
This interpretation is about the covariance structure, not proof of a particular economic cause. A positive correlation alone does not establish that a specific market event or common factor caused the assets to move together. The model says that, if the pairwise covariance remains positive and constant as the portfolio grows, a positive variance floor remains.
Worked Example: Calculating the Limiting Risk
A hypothetical set of equally weighted assets has individual standard deviation \(10\%\), so \(\sigma^2=0.01\), and common pairwise correlation \(\rho=0.30\). Assume these common inputs remain fixed as holdings increase and that returns have finite variances, with no distributional assumption. Calculate the variance and standard deviation for ten holdings, then find the limiting standard deviation as the number of holdings grows.
For ten holdings, the variance factor is \(0.30+\frac{0.70}{10}=0.37\). Therefore, portfolio variance is \(0.01(0.37)=0.0037\), and portfolio standard deviation is \(\sqrt{0.0037}=0.060828\ldots\), or 6.08% to two decimal places.
As \(n\) grows, the term \(\frac{1-\rho}{n}\) approaches zero. The limiting variance is therefore \(0.01(0.30)=0.0030\), and the limiting standard deviation is \(\sqrt{0.0030}=0.054772\ldots\), or 5.48% to two decimal places.
The ten-asset portfolio remains above that limit: its variance is \(0.0037\), compared with the limiting variance \(0.0030\). The gap is the portion that continues to shrink as more holdings are added. The limit does not mean that ten assets have already reached the floor, nor does it mean that diversification has removed every source of risk.
Assumptions, Limits, and an Exam Trap
Before using the formula, confirm that the portfolio and inputs match the model. The assets must have equal weights; their individual variances must be equal; and every distinct pair must have the same correlation. If asset variances or pairwise correlations differ, the equal-correlation formula does not apply directly. The general portfolio-variance calculation must instead include the individual weights, variances, and covariances.
The common correlation must also be mathematically feasible for the number of holdings. For example, an assumed pairwise correlation of \(-0.40\) is not feasible for four equal-variance assets, because the lower bound is \(-1/(4-1)=-1/3\), and \(-0.40<-1/3\). A result with negative variance is a warning that the inputs or calculation are invalid, not a meaningful portfolio risk estimate.
A frequent trap is to calculate only the average individual variance and stop. Equal weighting does divide the individual-variance contribution by \(n\), but it does not remove the pairwise covariance terms. Another trap is to say that more holdings always reduce risk to zero. The model supports a decreasing variance when \(\rho<1\), but for positive \(\rho\) the variance approaches \(\rho\sigma^2\), not zero. Finally, if an answer requests standard deviation, take the square root only after calculating the total variance.
Check Your Understanding
For each problem, assume the stated returns have finite variances and that no normal-distribution assumption is needed. Round standard deviations to two decimal places where requested.
- An equally weighted portfolio has nine assets, each with standard deviation \(12\%\), and common pairwise correlation \(0.20\). Find its variance and standard deviation. (Answer: variance \(0.004160\); standard deviation \(6.45\%\).)
- Six equally weighted assets each have standard deviation \(18\%\), and every pair has correlation \(0.50\). Calculate portfolio variance and standard deviation. (Answer: variance \(0.018900\); standard deviation \(13.75\%\).)
- Four equal-variance assets each have standard deviation \(10\%\), with common pairwise correlation \(0.10\). Find the portfolio variance and the limiting standard deviation if the same nonnegative correlation and individual variance are maintained as the number of holdings increases. (Answer: variance \(0.003250\); limiting standard deviation \(3.16\%\).)
- Assets each have standard deviation \(20\%\), and common pairwise correlation \(0.20\). What is the smallest integer number of equally weighted holdings for which portfolio standard deviation is at most \(12\%\)? (Answer: 5 holdings; at \(n=5\), variance is \(0.0144\), giving a \(12.00\%\) standard deviation.)
- A portfolio has ten equal-weighted assets, each with standard deviation \(16\%\), and common pairwise correlation \(0.30\). Using the same inputs, find the reduction in portfolio variance when moving from ten to eleven holdings. (Answer: \(0.00016291\) in return-variance units.)