What determines the risk of a portfolio?

Portfolio variance: how individual asset risk, weights and correlations combine to determine overall portfolio risk.

Play with it

Diversification and portfolio risk. Mix two assets, trace the efficient frontier, then switch on a crisis and watch correlations jump.

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Weight in asset A60% A, 40% B

Correlation between A and B-0.20

A volatility18%

B volatility7%

A expected return8.0%

B expected return3.5%

Portfolio volatility10.6%

Expected return6.2%

If risk just added up13.6%no diversification

Where the portfolio's variance comes from

Risk from A

117

Risk from B

Interaction

-12

Total variance

112

Variance in squared percentage points. Negative interaction cancels risk.

Holding 60% in A and 40% in B gives volatility of 10.6%, not the 13.6% you'd get if risks simply added up. The interaction term is negative, so the two risks partly cancel out. The lowest-risk mix holds 18% in A.

Try this

Set correlation to +1. The curve becomes a straight line: no free reduction in risk.
Set it to −1 and find the weight where portfolio volatility drops to zero.
Start at 100% in B (bonds) and add a little A (stocks). Does risk go up or down at first?
Any point below the lowest-risk dot is inefficient: more risk for less return.

The math

\sigma_p^2 = w^2\sigma_A^2 + (1-w)^2\sigma_B^2 + 2w(1-w)\,\rho\,\sigma_A\sigma_B

$w$: weight in asset A
$\sigma_A, \sigma_B$: each asset's volatility
$\rho$: their correlation
$2w(1-w)\rho\sigma_A\sigma_B$: the interaction term

Quick check

Two questions to see if it clicked. Answers explain themselves.

Portfolio A returns 10% every year. Portfolio B averages about 10% but swings between +30% and −10%. What differs?

Which term in the portfolio variance formula decides whether risks cancel or amplify?