The trade-off between risk and return for a portfolio depends not only on the expected asset returns and variances but also on the correlation of asset returns. The correlation between two assets represents the degree to which assets are related.
The correlation is the engine that drives the whole theory of portfolio diversification. The following figure illustrates minimum-variance frontier of a two-asset portfolio for four different correlations.
The endpoints (X and Y) for all the frontiers are the same, since at each endpoint the expected return and standard deviation are simply the expected return and standard deviation of either asset.
The conclusion: As the correlation between two assets decreases, the diversification benefits increase.
Effect of Number of Assets on Portfolio Diversification
For an equally-weighted portfolio, its variance is
σ2-bar is the average variance of return across all stocks, and Cov-bar is the average covariance of all pairs of two stocks.
Note that if n gets large enough:
Therefore, the variance of an equally-weighted portfolio approximately equals the average covariance as the number of assets becomes large.
Assume portfolio A has 2 assets and portfolio B has 30 assets. They are both equally weighted. The average asset variance is 0.5 and the average covariance is 0.3.
The variance of A is 1/2 0.5 + 1/2 0.3 = 0.4.
Portfolio B which has more assets has a lower variance.
In general, as the number of stocks increases, the variance of the portfolio will decrease.
D. - infinity.
Correlation is a number between -1 and +1. The lower the correlation, the bigger the diversification benefits.
A. Individual asset risks.
B. Average individual asset variance.
C. Average covariance.
D. Market risk.
The variance of a portfolio approaches the average covariance as the number of assets gets large.
A. has returns that are negatively correlated with the other stocks in the portfolio.
B. is from an industry that is not already represented in the portfolio.
C. has returns that are positively correlated with the largest holdings in that portfolio.
A. the identification of unsystematic risk.
B. the elimination of systematic risk.
C. the effect of diversification on portfolio risk.
A. Both Portfolio X and Portfolio Y are poorly diversified.
B. Portfolio X is well diversified and Portfolio Y is poorly diversified.
C. Portfolio X is poorly diversified and Portfolio Y is well diversified.
A. decline more by buying Black.
B. decline more by buying Green.
C. increase by buying either Black or Green.
I. The higher the average correlation, the fewer stocks we need to achieve a target risk reduction.
II. The variance of an equally-weighted portfolio approaches the average variance as n gets large.
I: The lower the correlation, the greater the diversification benefits.
II: It approaches the average covariance.
A. equally invested in stocks A and B.
Everything else being equal, the portfolio that has the lowest correlation, or most negative, will have the lowest risk.
A. Negative; decrease
Diversification benefits: risk reduction through a diversification of investments. Investments that are negatively correlated or that have low positive correlation provide the best diversification benefits. Such benefits may be particularly evident in an internationally diversified portfolio.
A. Diversification reduces the portfolio's expected return because diversification reduces a portfolio's systematic risk.
A. who want to reduce risk would be better off adding B to their portfolios.
Since B and C have the same expected return and standard deviation, the lower correlation of A and B will provide portfolios with superior opportunities to reduce risk.