Archive / Investment Adviser (Level 1) / Introduction to Modern Portfolio Theory

nism-series-xa-mpt-two-asset-portfolio-variance-covariance[ INVESTMENT ADVISER (LEVEL 1) ]

An adviser structures a portfolio with 70% in Asset X (Standard Deviation = 18%) and 30% in Asset Y (Standard Deviation = 8%). The correlation coefficient (ρ_XY) between Asset X and Y is 0.20. What is the Portfolio Standard Deviation?

[ A ]15.00%

[ B ]13.29%[ CORRECT ]

[ C ]17.52%

[ D ]11.18%

Takeaway

Portfolio Standard Deviation is approximately 13.29%, lower than the simple weighted average of 15.0% due to diversification benefits.

Derivation

Asset X's variance contribution: (0.70)^2 × (0.18)^2 = 0.49 × 0.0324 = 0.015876. Asset Y's variance contribution: (0.30)^2 × (0.08)^2 = 0.09 × 0.0064 = 0.000576. The covariance term captures how the two assets move together: 2 × 0.70 × 0.30 × 0.20 × 0.18 × 0.08 = 0.0012096. Summing all three gives total portfolio variance: 0.015876 + 0.000576 + 0.0012096 = 0.0176616, and taking the square root gives Portfolio Standard Deviation ≈ 13.29%. Because the correlation of 0.20 is well below 1.0, the portfolio's actual volatility comes in meaningfully below the naive weighted average of 15.00%, which is the entire point of the diversification benefit this formula quantifies.

Why the others fail

A — 15.00%

This is the simple weighted average of the two standard deviations, which ignores the partial offsetting effect of imperfect correlation between the assets.

C — 17.52%

Assumes the two assets move in perfect lockstep (correlation of +1.0), which overstates portfolio risk versus the actual 0.20 correlation given in the question.

D — 11.18%

Assumes the two assets move in perfect opposition (correlation of -1.0), which understates portfolio risk versus the actual 0.20 correlation given in the question.