| Pair | Portfolio 1: return / variance | Portfolio 2: return / variance |
|---|---|---|
| A / B | 14% / 400 | 13% / 441 |
| J / K | 20% / 529 | 20% / 400 |
| R / S | 8% / 225 | 9% / 225 |
| X / Y | 12% / 380 | 15% / 460 |
Variances are in squared percentage-point units.
A dominates B; K dominates J; S dominates R. X and Y involve a risk–return trade-off, so the choice depends on preferences and the rest of the available opportunity set.
Asset A has expected return 6 and variance 1.5; asset B has expected return 3.5 and variance 4. Covariance is −0.35. Returns use a common unit, and variances use its square. Portfolio weights sum to one.
70.161290% in A.
29.838710% in B.
Variance = 0.947983871; standard deviation = 0.973644633. Expected return = 5.254032258.
Asset A takes returns −1, 1 and 3 with probability 1/3 each. Asset B takes returns −1, 1 and 2 with probabilities 1/4, 1/4 and 1/2. Weights sum to one. First compare their moments; then consider the stipulated correlation −1 and check whether it is consistent with these marginal distributions.
B dominates A under the mean–variance criterion: identical expected return and lower variance.
A: 42.857143%; B: 57.142857%.
This zero-risk result is conditional. Perfect negative correlation requires B to be an affine decreasing function of A almost surely.
A values −1, 1 and 3 would require B values 2.5, 1 and −0.5, each with probability 1/3. Those are not the supplied B distribution. Therefore no joint distribution with the stated marginals can have correlation −1; the formal portfolio is not feasible for those exact distributions.
A has expected return 0.09 and variance 0.14; B has expected return 0.13 and variance 0.18. Covariance is 0.10. Weights sum to one and portfolio variance must not exceed 0.15. Short selling is permitted; also report whether the optimum is long-only.
Feasible A weights lie between 0.225708115 and 1.107625219. Expected return decreases with x, so choose the lower endpoint.
22.570811%.
77.429189%.
12.097168%.
Variance equals 0.15 and standard deviation equals 0.387298335. Both weights are positive, so the same optimum is feasible without short selling.
A has expected return 1 and variance 4; B has expected return 2 and variance 9. Correlation is −1/2 and weights sum to one. Use the stated common return unit and squared-unit variances. Short selling is permitted; the reported optima also have non-negative weights.
A: 63.157895%; B: 36.842105%. Variance 1.421052632; standard deviation 1.192079121. Compared with the lower standalone standard deviation of 2, diversification reduces risk by 0.807920879 return units.
Choose the upper-return branch, with variance at least 27/19 and expected return at least 26/19. The lower-return branch is inefficient. With long-only weights, also restrict expected return to at most 2.
1.542980252 return units.