Questions: Diversification Benefits and Correlation Effects

Question 1

Two assets each have 20% volatility. You hold them in equal (50/50) weights, and their correlation is ρ = 1. What is the portfolio's volatility?

• A. 14.1% — combining any two assets reduces variance due to diversification
• B. 20% — perfect positive correlation means combining them gives no diversification benefit
• C. 10% — equal weighting always halves the portfolio volatility
• D. 28.3% — correlated assets amplify each other's risk

Answer: (B)

When ρ = 1, the portfolio variance formula simplifies to σ_p = wσ_A + (1−w)σ_B — a straight-line blend with no risk reduction. At 50/50 weights with both at 20%, portfolio volatility is exactly 20%. There is zero diversification benefit. The assets move in perfect lockstep, so combining them is like holding more of the same asset. Option A represents the most common misconception: that any combination of assets reduces risk. Diversification only helps to the extent that ρ < 1.

Question 2

You want to add a second asset to your portfolio to maximize risk reduction. Holding individual volatility and expected return constant across candidates, which asset provides the most benefit?

• A. The asset most highly correlated with your existing holdings — it tracks your portfolio's performance closely
• B. The asset with the highest expected return, regardless of its correlation
• C. The asset least correlated with your existing holdings — it provides the most genuine risk reduction per unit of expected return
• D. The asset with the lowest individual volatility, regardless of correlation

Answer: (C)

Correlation with your existing holdings is the key input to how much diversification benefit an asset provides. Two assets with identical volatilities and expected returns are not interchangeable: the one with lower correlation with your portfolio contributes more to risk reduction. The cross-term in the portfolio variance formula — 2w(1−w)σ_Aσ_Bρ — is what captures this. A highly correlated asset (high ρ) leaves this term large and positive, providing little reduction. A negatively correlated asset makes this term negative, actively reducing variance below the weighted average of the individual volatilities.

Question 3

With perfect negative correlation (ρ = −1) and appropriately chosen weights, two individually risky assets can be combined into a portfolio with zero variance.

• T. True
• F. False

Answer: True

When ρ = −1, the portfolio standard deviation formula reduces to σ_p = |wσ_A − (1−w)σ_B|. Setting this to zero: wσ_A = (1−w)σ_B, which gives w = σ_B/(σ_A + σ_B). At these weights, the assets' fluctuations cancel perfectly — when A goes up, B goes down by exactly the offsetting amount. This is the mathematical basis for hedging: constructing a risk-free position from risky components. In practice, perfect negative correlation is extremely rare, but the logic underlies real hedging strategies.

Question 4

Diversification usually reduces a portfolio's volatility below the volatility of the least-risky individual asset in the portfolio.

• T. True
• F. False

Answer: False

Diversification reduces portfolio volatility below the *weighted average* of individual volatilities (when ρ < 1), but not necessarily below the *minimum* individual volatility. When correlations are moderately positive — which is typical for most real-world assets — the diversification benefit reduces variance but not to the level of the best single asset. Only with negative or zero correlation can combining assets potentially produce a portfolio less volatile than the lowest-volatility component. The claim in the statement conflates 'reduces the weighted average' with 'reduces below the minimum.'

Question 5

Explain why two assets with identical expected returns and identical individual volatilities are not necessarily equally valuable additions to a portfolio.

Model answer: Their value depends on their correlation with the existing portfolio. The portfolio variance formula's cross-term — 2w(1−w)σ_Aσ_Bρ — scales directly with correlation. An asset that is highly correlated with what you already own adds little risk reduction (large positive cross-term). An asset with low or negative correlation reduces portfolio variance substantially (small or negative cross-term). Since the two candidates have equal expected returns and equal individual risk, the one with lower correlation with your portfolio delivers the same return at lower combined risk — it is strictly more valuable as an addition.

This is the core insight of portfolio theory: individual asset properties (return, volatility) are not sufficient to evaluate an asset's contribution to a portfolio. Correlation — how the asset co-moves with what you already hold — determines how much genuine risk diversification it provides. Two identical assets viewed in isolation become very different viewed as portfolio additions if their correlations differ.