- A.Is portfolio variance simply the sum of the entries of a covariance matrix?
Covariance matrices and many-asset diversification
A.
Portfolio variance is the sum of the weight-adjusted covariance terms wᵢwⱼCov(Rᵢ,Rⱼ), including the diagonal variance terms. Summing an ordinary unweighted covariance matrix is generally wrong. The statement is correct only if the displayed matrix already includes portfolio weights. Portfolio standard deviation is the square root of this weighted sum.
Covariance matrices and many-asset diversification
| Asset | Weight | Expected return | Volatility |
|---|---|---|---|
| A | 30% | 5% | 20% |
| B | 30% | 8% | 25% |
| C | 40% | 10% | 35% |
- A.What can be said about portfolio volatility without knowing correlations?
A.
Volatility is at most 27.5%, hence certainly below 35%.
The exact volatility cannot be calculated without the correlations.
Covariance matrices and many-asset diversification
- A.Must the correlation between two assets be negative to reduce portfolio volatility below the weighted average of their individual volatilities? Can removing an asset increase risk?
A.
No. For positive weights and nonzero volatilities, any correlation below +1 produces a strict diversification benefit relative to the weighted average of individual volatilities. Negative correlation is helpful but not necessary. Removing a diversifying asset may increase volatility after the remaining weights are reallocated.
Covariance matrices and many-asset diversification
- A.Which inputs are needed to calculate a security’s contribution to portfolio volatility?
A.
You need its weight, the full portfolio weights, its variance and its covariances with the other holdings. Variance and covariances determine Cov(Rᵢ,Rᴘ); individual standard deviations alone are insufficient. The risk contribution also requires total portfolio volatility.
Covariance matrices and many-asset diversification
There are N assets, each with expected return r and variance v. Every distinct pair has covariance c. Each weight is 1/N.
- A.Derive portfolio mean and variance.
- B.What remains as N grows?
A.
B.
The individual-variance contribution shrinks; common covariance remains. This limit presumes a feasible sequence of covariance matrices. In a well-diversified portfolio, covariance with the portfolio is what drives an asset’s marginal risk contribution.
Covariance matrices and many-asset diversification
Discuss portfolio risk for a risk-averse investor.
- A.Can a long-only portfolio return exceed every component return or fall below every component return?
- B.Can volatility be below every component’s volatility? What happens to beta?
- C.How does diversification work?
- D.Can a risky asset have beta zero?
- E.Should all oil stocks be rejected solely because their individual volatility is high?
A.
No. Its expected return is a convex weighted average, bounded by the smallest and largest component expected returns.
B.
Yes, sufficiently imperfect or negative correlations can reduce total volatility below every individual volatility. Portfolio beta is always the weighted average of component betas.
C.
Combining imperfectly correlated positions offsets firm-specific surprises. It does not eliminate common systematic risk; correlation and position weights matter.
D.
Yes. It can have nonzero variance but zero covariance with the market. CAPM assigns it the risk-free expected return, not necessarily a risk-free realised payoff.
E.
No. Assess their contribution to the investor’s diversified portfolio, systematic risk, expected return and correlations. Standalone volatility alone is insufficient.
Covariance matrices and many-asset diversification
Discuss long-only portfolios and risk sources.
- A.Can a long-only portfolio return exceed every component return or fall below every component return?
- B.Can volatility be below every component’s volatility? What happens to beta?
- C.How does diversification work?
- D.Can a risky asset have beta zero?
- E.Compare an unexpected ECB rate rise with a lender increasing only Delta’s borrowing rate.
A.
No. Its expected return is a convex weighted average, bounded by the smallest and largest component expected returns.
B.
Yes, sufficiently imperfect or negative correlations can reduce total volatility below every individual volatility. Portfolio beta is always the weighted average of component betas.
C.
Combining imperfectly correlated positions offsets firm-specific surprises. It does not eliminate common systematic risk; correlation and position weights matter.
D.
Yes. It can have nonzero variance but zero covariance with the market. CAPM assigns it the risk-free expected return, not necessarily a risk-free realised payoff.
E.
A broad rate shock is a systematic factor. A Delta-specific lender repricing can be diversifiable, although it may also reflect a broader systematic shock; identify the underlying cause.