Guide · Portfolio analysis
Portfolio diversification: how it works and how to measure it
Portfolio diversification means holding investments that do not all move together, so that a loss in one is partly offset by the others. It lowers the risk of the portfolio without lowering its expected return, but only down to a floor set by how correlated the holdings are. What matters is not how many holdings you own, but how differently they behave.
In short
- Risk falls fast, then stops. With an average correlation of 0.3, ten equally weighted holdings keep 61% of the volatility of a single holding, fifty keep 56%, and no number goes below 55%.
- Count is not diversification. Two funds with a correlation of 0.95 or more behave as one position, whatever their names say.
- Weight is not risk. In a 70/30 mix of a stock fund and a bond fund, the stock fund can carry about 97% of the volatility.
- It weakens when it is needed most. Correlations between equity markets increase in bear markets, not in bull markets (Longin and Solnik, 2001).
Why diversification works: the formula
The volatility of a portfolio is not the average of its holdings’ volatilities. It also depends on how the holdings move in relation to each other, which is what Harry Markowitz formalised in 1952.[1] For a portfolio with weights w, volatilities σ and correlations ρ:
σ²(portfolio) = Σ wᵢ² σᵢ² + Σ Σ wᵢ wⱼ σᵢ σⱼ ρᵢⱼ (i ≠ j)
The second term is where diversification happens: the lower the correlations, the smaller it gets. In the simplest case, N equally weighted holdings with the same volatility σ and the same correlation ρ between every pair, the formula reduces to[3]:
σ(portfolio) = σ × √( 1/N + (1 − 1/N) × ρ ) → σ × √ρ as N grows
Two things follow. Adding holdings removes risk quickly at first and then hardly at all. And there is a floor, σ × √ρ, that no number of holdings can break: the part of the risk that the holdings share. Move the sliders to see both.
- Risk left
- 61%
- Floor
- 55%
- With 20
- 58%
10 equally weighted holdings with an average correlation of 0.30 keep 61% of the volatility of a single holding. Doubling to 20 brings it to 58%; no number of holdings takes it below 55%.
Simplified case: equal weights, the same volatility for every holding and the same correlation for every pair. Risk left = √(1/N + (1 − 1/N) × ρ).
Risk left, by number of holdings and correlation
| Holdings | Correlation 0.1 | Correlation 0.3 | Correlation 0.6 |
|---|---|---|---|
| 1 | 100% | 100% | 100% |
| 2 | 74% | 81% | 89% |
| 5 | 53% | 66% | 82% |
| 10 | 44% | 61% | 80% |
| 20 | 38% | 58% | 79% |
| 30 | 36% | 57% | 78% |
| 50 | 34% | 56% | 78% |
| Floor (any number) | 32% | 55% | 77% |
Volatility of the portfolio as a share of the volatility of one holding. Equal weights, equal volatilities, one correlation for every pair. Real portfolios differ on all three, which is why the measures below exist.
How to measure diversification: five numbers
A list of holdings says little about diversification. These five numbers describe it from the returns the holdings actually produced. None is sufficient alone; together they show whether the positions offset each other or only look different. The calculators below compute them.
| Measure | Formula | How to read it |
|---|---|---|
| Average correlation | Mean of ρᵢⱼ over all pairs | From −1 to 1. The closer to 1, the less the holdings diversify each other. It ignores weights, so read it with the next two. |
| Diversification ratio | Σ wᵢ σᵢ ÷ σ(portfolio) | 1 means no benefit; higher means the mix removes more volatility. Defined by Choueifaty and Coignard (2008). |
| Effective number of holdings, by weight | 1 ÷ Σ wᵢ² | How many equal positions the weights are worth. Weights of 50%, 30% and 20% count as 2.63 positions, not 3. |
| Effective number of bets, by risk | 1 ÷ Σ RCᵢ², with RCᵢ the share of volatility from holding i | The same count on risk instead of capital. When it is well below the count by weight, a few holdings drive the result. |
| Correlation on the worst days | ρᵢⱼ on the portfolio’s worst 5% of days | What the holdings do to each other when the portfolio falls. Higher than the full-period figure means the protection fades under stress. |
The two “effective number” measures are the inverse of the Herfindahl-Hirschman Index, the sum of squared shares used to measure market concentration.[10] The diversification ratio is from Choueifaty and Coignard.[8]
Worked example: a 70/30 portfolio is not two positions
Take 70% in a stock fund with 18% volatility and 30% in a bond fund with 6% volatility, with a correlation of 0.1 between them. These are illustrative figures, not a forecast.
- Portfolio volatility
- 12.9%
- Diversification ratio
- 1.12
- Risk from the stock fund
- 97%
- Positions by weight
- 1.72
- Bets by risk
- 1.07
By weight the portfolio holds 1.72 equal positions. By risk it holds 1.07: the stock fund produces 97% of the volatility and the bond fund 3%. The bonds lower the total, from 14.4% to 12.9%, but what happens to the portfolio is decided almost entirely by the stocks. This is the gap that risk parity allocations are built to close.
Calculators: try your own numbers
Both run in your browser on the numbers you type; nothing is sent or saved. The first starts from the 70/30 example above.
Two holdings: volatility, diversification ratio and risk shares
Set the weight, the volatility of each holding and the correlation between them. Move the correlation towards 1 and the benefit disappears; move it below zero and the second holding starts to cancel the first.
- Portfolio volatility
- 12.9%
- Without diversification
- 14.4%
- Diversification ratio
- 1.12
- Positions by weight
- 1.72
- Bets by risk
- 1.07
- Risk removed
- 10%
Holding A
Holding B
Holding A has 70% of the money and 97% of the risk. The mix has a volatility of 12.9% instead of the 14.4% weighted average: correlation below 1 removes 10% of it.
Effective number of holdings: how concentrated are your weights?
Paste the weights of your portfolio. The result is the number of equal positions they are worth: 1 divided by the sum of the squared weights.
- Holdings
- 5
- Effective number
- 3.77
- Largest holding
- 40%
5 holdings with these weights are spread like 3.77 equal positions. The largest positions weigh more than the count suggests. This counts money only: to see how the risk is spread, use the calculators around it.
Diversification calculator: a mix of ETFs on real data
Pick up to six US-listed ETFs and their weights. The calculator measures the mix on the funds’ real monthly total returns: its volatility, the diversification ratio, the correlation of every pair and how the risk is split compared with the money.
- Portfolio volatility
- 11.1%
- Without diversification
- 11.9%
- Diversification ratio
- 1.07
- Average correlation
- 0.61
- Positions by weight
- 1.92
- Bets by risk
- 1.42
SPY
vol 15%
AGG
vol 7%
Correlations
- SPY–AGG 0.61
SPY has 60% of the money and 82% of the risk. The mix removed 7% of the weighted average volatility.
Monthly total returns in US dollars, Dec 2021 to Nov 2026 (60 months). Volatility is annualised. Historical figures, not a forecast. Source: MEDGE Capital.
15 funds across stocks, bonds, gold, real estate and commodities. For European ETFs and up to ten funds, use the ETF correlation matrix.
How many holdings do you need?
The research on randomly chosen US stocks does not give one number, and the answers have moved over time.
| Study | Finding |
|---|---|
| Evans and Archer, 1968[2] | The dispersion of returns falls quickly as stocks are added; little further benefit beyond about ten. |
| Statman, 1987[4] | A well-diversified portfolio of randomly chosen stocks needs at least 30 for an investor who borrows and 40 for one who lends. |
| Campbell, Lettau, Malkiel and Xu, 2001[5] | From 1962 to 1997 the volatility of individual firms rose relative to the market, correlations among stocks declined, and the number of stocks needed for a given level of diversification increased. |
| Bessembinder, 2018[9] | Since 1926 most US common stocks returned less than one-month Treasury bills over their lifetime; the best-performing 4% of listed companies account for the net gain of the whole market. |
The last finding explains why the count matters for individual stocks: a portfolio of a few names can easily miss the small group that produced the market’s gain. With funds the question changes. One broad index fund already holds hundreds or thousands of stocks, so adding more equity funds mostly adds the same companies again. Past that point, diversification comes from holdings that respond to different forces, such as bonds, gold or cash, and the number to watch is the correlation between them, not the count.
What diversification did in practice
These are MEDGE’s own backtests of well-known allocations, from a single stock fund to mixes of four and five funds: total return in US dollars, rebalanced once a year. All cover the same period, Nov 2011 to Nov 2026.
| Portfolio | Funds | Return a year | Volatility | Worst fall | 2022 |
|---|---|---|---|---|---|
| 100% Stocks | 1 | +14.1% | 16.9% | −35.0% | −19.5% |
| 60/40 Portfolio | 2 | +9.3% | 10.3% | −21.8% | −16.9% |
| Three-Fund Portfolio | 3 | +8.1% | 10.0% | −22.3% | −16.3% |
| All Weather Portfolio | 5 | +5.7% | 8.0% | −23.8% | −19.2% |
| Permanent Portfolio | 4 | +6.0% | 7.1% | −17.5% | −12.5% |
| Golden Butterfly Portfolio | 5 | +7.0% | 8.3% | −19.5% | −13.3% |
Source: MEDGE Capital backtests on end-of-day fund prices, updated Nov 2026. “Worst fall” is the largest peak-to-trough decline in the period. Past performance says nothing certain about the future. All portfolios and method.
The all-stock portfolio fell 35.0% at its worst and the 60/40 portfolio 21.8%, with volatility of 16.9% and 10.3%. Over the same years stocks returned +14.1% a year and the 60/40 mix +9.3%: in this period the smoother ride had a price. Diversification changes the path, not the direction of the assets inside it.
When diversification fails
Diversification relies on correlations staying below 1, and correlations are not constant. Longin and Solnik studied international equity markets and found that correlation increases in bear markets, but not in bull markets.[6] Ang and Chen found the same asymmetry inside the US market: stocks move much more closely with the market in falls, especially large ones, than in rises.[7] A correlation measured over a calm period overstates the protection available in a bad one.
2022 is the recent example in the table above: the all-stock portfolio returned −19.5% and the 60/40 portfolio −16.9%. The bond part lost money in the same year, so it could not offset the fall in stocks.
Two checks make this visible before it matters. The first is the correlation on the portfolio’s worst days, measure five above. The second is a stress test: apply a specific shock, such as a fall in stocks together with a rise in rates, and see what each holding does.
Five common mistakes
- 1Counting holdings. Twenty positions with an average correlation of 0.6 keep 79% of the volatility of one. Five with a correlation of 0.1 keep 53%.
- 2Owning the same thing twice. Funds that track overlapping indices move almost identically. A correlation of 0.95 or more between two holdings means they add a line to the statement, not diversification.
- 3Measuring by weight instead of by risk. A holding’s share of the capital and its share of the risk can be far apart, as in the 70/30 example where 70% of the money carries 97% of the volatility.
- 4Treating correlations as fixed. They change with the market regime and tend to rise in falls. A figure from one calm period is a best case.
- 5Diversifying within one risk only. Many stocks from one sector or one country still share that sector’s or country’s risk: the floor in the formula stays high because ρ is high.
How to check your own portfolio
1List holdings and weights
Tickers and the share of the portfolio in each. Everything else is computed from their returns.
2Look at the correlation matrix
Find the pairs above 0.9 and the holdings that are low against everything else.
Free correlation matrix3Compare weight with share of risk
For each holding, its share of the capital next to its share of the volatility, plus the diversification ratio and the effective number of bets.
Portfolio backtesting in MEDGE4Check the worst days
Correlations and each holding’s contribution on the worst 5% of days and in past falls.
Period and regime analysis5Run a scenario
Apply the shock you fear and read the result holding by holding.
Stress test a portfolio
MEDGE Capital computes these on your own holdings: each holding’s share of risk next to its weight, the diversification ratio, the effective number of independent bets, the correlations on the worst days, and a warning when two holdings move almost identically.
Frequently asked questions
What is portfolio diversification?
Portfolio diversification is the practice of holding investments that do not all move together, so that losses in some are partly offset by others. It brings the volatility of the portfolio below the weighted average of its holdings’ volatilities, as long as the correlation between them is below 1.
How many stocks do you need for a diversified portfolio?
Studies of randomly chosen US stocks give different answers. Evans and Archer (1968) found little further benefit beyond about ten stocks, Statman (1987) concluded that at least 30 to 40 are needed, and Campbell, Lettau, Malkiel and Xu (2001) found the number had risen as individual stocks became more volatile. A broad index fund holds hundreds or thousands of stocks in a single position.
Is one ETF enough to be diversified?
A single broad equity index fund is diversified across companies, because it holds hundreds or thousands of them. It is not diversified across asset classes: all its holdings are stocks and tend to fall together in a stock market decline. Whether that is enough depends on how large a fall the investor can accept.
How do you measure diversification?
Five numbers cover it: the average correlation between holdings; the diversification ratio, which is the weighted average volatility divided by the portfolio volatility; the effective number of holdings by weight, which is 1 divided by the sum of squared weights; the effective number of bets by risk, the same calculation on each holding’s share of volatility; and the correlation on the portfolio’s worst days.
What is a good diversification ratio?
A diversification ratio of 1 means no benefit: the holdings move as one. The higher the ratio, the more the mix reduces volatility; two equally weighted, equally volatile holdings with zero correlation have a ratio of 1.41. There is no universal threshold, because the ratio depends on the assets available, so it is most useful for comparing versions of the same portfolio.
Does diversification reduce returns?
Not by itself. The expected return of a portfolio is the weighted average of its holdings’ expected returns, while its volatility is lower than the weighted average whenever correlations are below 1. What diversification gives up is the chance of matching the single best holding.
Why does diversification fail in a crisis?
Because correlations are not constant. Longin and Solnik (2001) found that the correlation between international equity markets increases in bear markets but not in bull markets, and Ang and Chen (2002) found that US stocks move much more closely with the market in falls than in rises. Holdings that looked independent in calm periods can fall together.
What is the difference between diversification and asset allocation?
Asset allocation is the decision of how much to hold in each asset class, such as stocks, bonds, gold or cash. Diversification is the result: how far those holdings offset each other. An allocation can contain many funds and still be poorly diversified if they are highly correlated.
Is there a free portfolio diversification calculator?
Yes, this guide has four, with no sign-up. One shows how much risk is left as holdings are added at a given correlation. One computes the volatility, diversification ratio and risk shares of a two-holding mix. One turns a list of weights into the effective number of holdings. One measures a mix of US-listed ETFs on their real monthly returns. MEDGE Capital’s portfolio analyzer does the same on any portfolio of stocks and funds.
Sources
- Markowitz, H. (1952). Portfolio Selection. The Journal of Finance, 7(1), 77–91. doi.org/10.1111/j.1540-6261.1952.tb01525.x
- Evans, J. L., & Archer, S. H. (1968). Diversification and the Reduction of Dispersion: An Empirical Analysis. The Journal of Finance, 23(5), 761–767. doi.org/10.1111/j.1540-6261.1968.tb00315.x
- Elton, E. J., & Gruber, M. J. (1977). Risk Reduction and Portfolio Size: An Analytical Solution. The Journal of Business, 50(4), 415–437. doi.org/10.1086/295964
- Statman, M. (1987). How Many Stocks Make a Diversified Portfolio? The Journal of Financial and Quantitative Analysis, 22(3), 353–363. doi.org/10.2307/2330969
- Campbell, J. Y., Lettau, M., Malkiel, B. G., & Xu, Y. (2001). Have Individual Stocks Become More Volatile? An Empirical Exploration of Idiosyncratic Risk. The Journal of Finance, 56(1), 1–43. doi.org/10.1111/0022-1082.00318
- Longin, F., & Solnik, B. (2001). Extreme Correlation of International Equity Markets. The Journal of Finance, 56(2), 649–676. doi.org/10.1111/0022-1082.00340
- Ang, A., & Chen, J. (2002). Asymmetric Correlations of Equity Portfolios. Journal of Financial Economics, 63(3), 443–494. doi.org/10.1016/S0304-405X(02)00068-5
- Choueifaty, Y., & Coignard, Y. (2008). Toward Maximum Diversification. The Journal of Portfolio Management, 35(1), 40–51. doi.org/10.3905/jpm.2008.35.1.40
- Bessembinder, H. (2018). Do Stocks Outperform Treasury Bills? Journal of Financial Economics, 129(3), 440–457. doi.org/10.1016/j.jfineco.2018.06.004
- U.S. Department of Justice, Antitrust Division. Herfindahl-Hirschman Index. www.justice.gov/atr/herfindahl-hirschman-index
Keep reading
Source: MEDGE Capital calculations and backtests on end-of-day fund prices; academic sources listed above. Figures are historical and descriptive; they are not forecasts and not investment advice. MEDGE Capital does not recommend buying or selling any instrument. Risk disclaimer.