In computing portfolio variance using a covariance matrix, why can we use only the upper (or lower) triangular covariances plus diagonal variances instead of all n^2 entries?
Explanation
Use symmetry of covariance matrix to reduce distinct terms needed; diagonal are variances, off-diagonals pairwise covariances. (chapter 5: Portfolio Expected Return and Variance of Return)
Other questions
Given three assets with weights 0.4, 0.3, and 0.3, individual variances 100, 81, and 144 (in % squared) and covariances Cov12=20, Cov13=10, Cov23=30 (all in % squared), what is the portfolio variance (in % squared)?
If two assets have standard deviations of 10% and 20% and their covariance is 12 (% squared), what is the correlation between them?
Portfolio expected return for a portfolio with weights 0.5 in Asset A (E[R]=8%) and 0.5 in Asset B (E[R]=12%) is:
If two assets are independent, which statement is always true?
Given joint probabilities of two assets across three states: P(state1)=0.2 with (R_A=25%, R_B=20%); state2 P=0.5 with (12%,16%); state3 P=0.3 with (10%,10%). The expected returns are 14% for A and 15% for B. What is Cov(RA,RB)?
A portfolio has expected return 12%, standard deviation 15%. Investor's minimum acceptable return RL is 3%. What is the safety-first ratio?
If RL equals the risk-free rate, what classic performance ratio is equivalent to Roy's safety-first ratio?
For a covariance matrix of three assets given as variances on diagonal [400, 81, 441] and covariances [45, 189, 38] corresponding to Cov(S&P, Bonds)=45, Cov(S&P, EAFE)=189, Cov(Bonds, EAFE)=38 (units percent squared), and weights 0.5, 0.25, 0.25, which portfolio standard deviation (approx) results?
If two assets have covariance 0 and nonzero variances, what is their correlation?
A two-asset portfolio has weights 0.3 and 0.7, standard deviations 12% and 25%, and correlation 0.2. What is the portfolio standard deviation (approx)?
How many distinct covariance terms must be estimated for a portfolio of 20 assets (excluding variances)?
If a portfolio's covariances with other holdings are largely negative, what effect does this have on portfolio variance all else equal?
Using a joint probability table, why is covariance calculated by summing P(Ri, Rj)*(Ri - E(Ri))*(Rj - E(Rj)) across all cells?
A fund has expected return 6% and standard deviation 9%. The minimum acceptable return RL is 2%. What is approximate probability the fund returns below 2% if returns are normal? (Use Normal table: N(-0.444)~0.328.)
In the 3-asset covariance matrix example, if all off-diagonal covariances were set to zero, the portfolio standard deviation would be smaller. This illustrates:
Which of the following statements about correlation is correct?
If you increase the number of independent holdings in a portfolio (with identical individual variances and near-zero covariances), what happens to portfolio variance?
Given a covariance matrix entry Cov(Ri,Rj)=0.20 (in decimal return units), sigma_i=0.10 and sigma_j=0.25, what is corr(Ri,Rj)?
Which term increases in importance for portfolio variance as the number of assets increases (assuming nonzero correlations)?
If two assets have a correlation of -1, what is the implication for a portfolio constructed as a weighted combination of the two (ignoring leverage limitations)?
You estimate expected returns for three assets as 13%, 6%, and 15% and set weights 0.50, 0.25, 0.25. What is expected portfolio return?
Which of the following is true about covariance signs?
If portfolio variance equals 195.875 (% squared) what is standard deviation (percent)?
What is covariance if correlation is 0.25 and standard deviations are 8.2% and 3.4% (use percent units)?
Which of the following best describes Roy's safety-first criterion?
If expected return on portfolio A is 25% with sd 27% and portfolio B is 11% with sd 8% and RL=3.75%, which has lower shortfall probability?
Given covariance matrix entry Cov(S&P, EAFE)=189 (%^2), S&P variance=400, EAFE variance=441, what is correlation between S&P and EAFE?
Covariance between two assets is negative. Which statement is true?
You have a portfolio with expected return 8% and standard deviation 8%. RL for investor is 2%. Which probability (approx) of shortfall applies? (z=(2-8)/8=-0.75; Normal(-0.75)=0.2266).
When constructing a two-asset plot of portfolio expected return vs standard deviation as weights change from all in asset 1 to all in asset 2, the shape is typically:
If the covariance between portfolio returns and bond index returns is +18.9 (percent squared) and both series have positive standard deviations, the correlation is:
Which of the following is true about the safety-first ratio and the probability of shortfall under normal returns?
In Example 3, when comparing portfolio variances before and after a regulation change with sample variances 4.644 and 3.919 over equal sample sizes (418), the F-statistic was 1.185. If the two-tailed critical interval is (0.8251,1.2119), what is the inference?
A portfolio manager says 'if we diversify across many securities, portfolio risk will approach zero.' The correct response is:
Which numeric operation yields a portfolio's expected return from component expected returns and weights?
Which of these choices best explains why covariance estimates can be unreliable with limited historical data?
If portfolio expected return equals weighted sum of expected returns, which input change will NOT change expected portfolio return?
A covariance estimate between two assets is negative. To improve overall portfolio variance, an investor should:
You compute portfolio variance using percent units. Why is it important to be consistent with percent vs decimal usage?
If two assets have expected returns 5% and 9% and weights 0.6 and 0.4, what is the portfolio expected return?
Which of the following components of portfolio variance grows most rapidly as number of assets increases?
In a joint probability function with mutually exclusive states that sum to 1, how do you compute E(R) for a single asset?
When using a covariance matrix expressed in percent squared, portfolio variance computed with percent-based weights yields result in:
What is a practical reason portfolio managers prefer stratified sampling for bond index replication rather than pure simple random sampling?
A portfolio has expected return 6% and sd 10%. If investor wants 99% confidence that return >= RL, what RL approximately equals? (z for 99% one-sided is 2.33 so RL = E - z*sigma = 6% - 2.33*10% = -16.3%)
If portfolio A has SFratio 0.5 and portfolio B has SFratio 0.8 under normality, which has lower probability of falling below RL?
Which is the correct formula for portfolio variance in matrix notation using weight vector w and covariance matrix Sigma?
Which action would increase the SFratio for a portfolio (all else equal)?
Practical recommendation: when estimating a large covariance matrix for 50 assets using historical returns, an analyst should: