Investments
Portfolio risk and diversification
Combining assets changes risk in a way that depends on correlation, not on the number of holdings.
Definition
Portfolio theory studies how the risk and return of a combination of assets relate to the risk and return of the individual holdings. Because asset returns are not perfectly correlated, a portfolio's volatility is generally lower than the weighted average of its components.
The two results that matter in practice are diversification, which removes idiosyncratic risk, and the separation of the remaining systematic risk into beta, which the market prices through CAPM.
Intuition
Risk does not add up like weights. It adds up like a conversation: when assets do not move together, part of the disagreement cancels out.
When to use it
- Deciding how many holdings a portfolio actually needs.
- Comparing a concentrated book with a diversified index at the same return.
- Understanding why a portfolio beta drives its required return.
When not to use it
- Treating historical correlations as stable: they rise sharply in a crisis.
- Assuming diversification removes market risk, which it cannot.
Formula
Two-asset portfolio variance
σp² = w1²σ1² + w2²σ2² + 2·w1·w2·ρ·σ1·σ2
Correlation decides how much of the two risks survives in the portfolio.
- w
- Weight of each asset
- ρ
- Correlation between the two returns
Formula
Portfolio beta
βp = Σ wi × βi
Beta is linear, so a portfolio's beta is the weighted average of its holdings.
Worked example
Two assets, one portfolio
Assets A and B each have a volatility of 20% and are held in equal weights.
- 1
Perfect correlation
ρ = 1: σp = 20%, exactly the weighted average.
- 2
Some diversification
ρ = 0.3: σp² = 0.02 + 0.006 = 0.026, so σp = 16.1%.
- 3
No correlation
ρ = 0: σp² = 0.02, so σp = 14.1%.
The same two assets carry 20%, 16.1%, or 14.1% volatility depending only on how they move together. Correlation is the free lunch.
Common mistakes
- Averaging volatilities: the average ignores correlation and overstates portfolio risk whenever correlation is below one.
- Counting holdings instead of checking correlation; twenty shares in one sector is one bet.
- Confusing a low-beta portfolio with a low-risk one: idiosyncratic risk may still be large.
How it connects
is an advanced form of
The source extends the target once the basics are secure.
- Risk and return: Portfolio risk applies single-asset risk and return to a combination of holdings.
relates to
Useful to study together, in either order.
- CAPM and beta: Beta is the bridge from a portfolio to its required return.
- Bond pricing and yield: A bond sleeve is one component of a diversified portfolio.