Portfolio & Asset Pricingintermediate

Diversification

Combining assets that do not move in lockstep lowers portfolio risk without sacrificing expected return, the core idea of diversification. The link between two assets is measured by their covariance or, in scale-free form, their correlation. Total risk splits into systematic risk, the market-wide component that affects all assets, and idiosyncratic risk, the firm-specific component unique to each asset. Diversification cancels idiosyncratic risk as the number of holdings grows, but systematic risk cannot be diversified away and is what investors are ultimately paid to bear.

Try it yourself

The efficient frontier

Sweep the weight w on asset A to trace the frontier of risk and return. A lower correlation ρ bows it toward the return axis, the diversification benefit. The minimum-variance portfolio is its leftmost point.

Current portfolio (w_A = 0.50)11.0% · σ 13.5%

MVP risk σ 11.3%MVP weight w* 0.83

Weight on A, w0.50

Correlation ρ0.20

E(R) of A8.0%

σ of A12.0%

E(R) of B14.0%

σ of B22.0%

Risk is below the weighted-average volatility whenever ρ < 1. The frontier bends most as ρ falls toward −1, where the right mix drives risk close to zero. The MVP holds w* = 0.83 in A.

Try this. Set ρ = +1 — the frontier collapses to a straight line, so there is no diversification benefit. Now set ρ = −1 — it bends sharply and the right mix can push risk almost to zero.

Why it matters

Risk falls when you mix assets because their bad surprises do not all arrive at once. The lower the correlation, the more one holding cushions another, and only when correlation is a perfect positive one does diversification stop working. Firm-specific shocks, a factory fire or a failed product, wash out across many names because they are unrelated. Economy-wide shocks, a recession or a rate hike, hit every name together and therefore survive diversification, which is why the market rewards systematic risk and not the idiosyncratic kind you could have removed for free.

Formulas

Covariance and correlation

\rho_{12} = \dfrac{\sigma_{12}}{\sigma_1\,\sigma_2}, \qquad -1 \le \rho_{12} \le 1

Correlation rescales covariance

\sigma_{12}

to lie between

-1

and

+1

. Diversification benefits grow as

\rho_{12}

falls toward

-1

Total risk decomposition

\sigma_i^2 = \beta_i^2\,\sigma_m^2 + \sigma_{\varepsilon,i}^2

Systematic variance

\beta_i^2\sigma_m^2

plus idiosyncratic variance

\sigma_{\varepsilon,i}^2

. Only the second term shrinks under diversification.

Worked examples

Scenario

Two assets each have a standard deviation of 20%. Compare the risk of a fifty-fifty portfolio when their correlation is $+1$ versus zero.

Solution

With $\rho=+1$ the portfolio standard deviation is the simple average, 20%, so nothing is gained. With $\rho=0$ it is $\sqrt{0.5^2(0.2^2)+0.5^2(0.2^2)}=\sqrt{0.02}\approx 14.1\%$ . The same two assets, merely uncorrelated, cut risk by almost a third with no change in expected return.

Scenario

A portfolio of one stock has 30% volatility. Why does adding 30 more roughly halve total volatility but not eliminate it?

Solution

Each stock carries idiosyncratic risk that is unrelated across names, so averaging many of them drives that component toward zero. What remains is the shared systematic component, the floor that diversification cannot pierce. Risk falls steeply at first then flattens to that systematic floor as holdings rise.

Common mistakes

✗Diversification eliminates all risk. It removes only idiosyncratic risk, while systematic, market-wide risk remains no matter how many assets you hold.
✗Holding more assets is always better. Once idiosyncratic risk is largely gone, extra names add little benefit and may add cost, so beyond a few dozen holdings the gain is marginal.
✗Two assets with high returns make a good diversifier pair. What matters for risk reduction is low correlation, not high individual returns, since two highly correlated winners still fall together.
✗A negative correlation is required for diversification to help. Any correlation below a perfect positive one reduces risk, so even modestly positive correlations still deliver a diversification benefit.

Revision bullets

•Diversification lowers risk by combining imperfectly correlated assets
•Correlation rescales covariance to the range from minus one to plus one
•Total risk splits into systematic and idiosyncratic components
•Idiosyncratic risk diversifies away, systematic risk does not
•Investors are compensated for systematic risk, not for diversifiable risk

Quick check

Which type of risk can be removed by holding a well-diversified portfolio?

Combining two risky assets reduces portfolio standard deviation most when their correlation is

Connected topics

Return & Risk Portfolio Variance Mean-Variance CAPM & SML Asset Allocation

Sources

Brailsford, Heaney & Bilson (2015), Ch. 7
Brailsford, T., Heaney, R., & Bilson, C. Investments: Concepts and Applications. 5th ed. Cengage Learning Australia, 2015.
Covers covariance, correlation, and the split of total risk into systematic and diversifiable parts.
Markowitz (1952)
Markowitz, H. Portfolio Selection. Journal of Finance, 7(1), 77-91, 1952.
The foundational argument that correlation, not individual variance alone, governs portfolio risk.

How to cite this page

Dr. Phil's Quant Lab. (2026). Diversification. Derivatives Atlas. https://phucnguyenvan.com/concept/im-diversification

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