Results
In order to understand how the uncertainty set impacts the solution we will solve the portfolio allocation problem for various protection levels k and compare the proposed budget allocation against the conservative approach.
The trader knows that she can expect a portfolio value in the range of 0 to 100. She therefore wishes to calculate for various ranges the probability that the value of the portfolio will belong to this range. She also wants to reach a portfolio value of 60 with high probability.
Input Data
Table Asset value distribution shows the asset mean value which is also the market price (Mean), the standard variation of the value (S. Dev), and the considered worst case value with N=1.5 (Worst value).
| Shares | Mean | S.Dev. | Worst value |
|---|---|---|---|
| 1 | 100 | 65.99 | 1.015 |
| 2 | 96 | 62.69 | 1.965 |
| 3 | 92 | 59.43 | 2.855 |
| 4 | 88 | 56.22 | 3.670 |
| 5 | 84 | 53.04 | 4.440 |
| 6 | 80 | 49.91 | 5.135 |
| 7 | 76 | 46.83 | 5.755 |
| 8 | 72 | 43.79 | 6.315 |
| 9 | 68 | 40.80 | 6.800 |
| 10 | 64 | 37.86 | 7.210 |
| 11 | 60 | 34.97 | 7.545 |
| 12 | 56 | 32.13 | 7.805 |
| 13 | 52 | 29.34 | 7.990 |
| 14 | 48 | 26.61 | 8.085 |
| 15 | 44 | 23.94 | 8.090 |
| 16 | 40 | 21.33 | 8.005 |
| 17 | 36 | 18.79 | 7.815 |
| 18 | 32 | 16.31 | 7.535 |
| 19 | 28 | 13.91 | 7.135 |
| 20 | 24 | 11.58 | 6.630 |
| 21 | 20 | 9.33 | 6.005 |
| 22 | 16 | 7.18 | 5.230 |
| 23 | 12 | 5.13 | 4.305 |
| 24 | 8 | 3.20 | 3.200 |
| 25 | 4 | 1.33 | 2.005 |
A quick glance at the input data reveals that asset #15 would be the best choice for the conservative trader because it has the highest worst case value, whereas asset #1 would be the preferred investment for an optimistic trader since it has the largest best case value.
Analysis
In order to understand how the robust optimization behaves compared to the nominal problem, we solve the nominal problem (A) and 5 robust optimization problems parameterized by the protection level k (ranging from 0 to 1=100%). With a protection level of k=0 the deviation budget is also zero, and hence the solution is a very optimistic one.
In a second step, we use a Monte-Carlo method to simulate the actual value of the assets selected by these solutions and calculate the probability with which the portfolio value lies in the various revenue ranges.
Table Results of the parameterized robust optimization presents the portfolio selection suggestion for each of these problems. For the sake of simplicity we list only those assets that are selected in at least one solution. The conservative solution (A) is to allocate all budget to the asset with highest worst case value. The optimistic solution (k=0.0) is to allocate the whole budget to the asset with the highest expected value. With an increasing protection level the suggested solutions improve the balance between assets with a high largest expected value and those with a high worst case value.
| Shares | Wst price | Price | A | k=0.0% | k=1% | k=10% | k=25% | k=100% |
|---|---|---|---|---|---|---|---|---|
| 1 | 1 | 100 | 100% | 100% | 60% | 33% | 11% | |
| 2 | 2 | 96 | 35% | 28% | 11% | |||
| 3 | 3 | 92 | 5% | 22% | 11% | |||
| 4 | 4 | 88 | 14% | 11% | ||||
| 5 | 4 | 84 | 3% | 11% | ||||
| 6 | 5 | 80 | 10% | |||||
| 7 | 6 | 76 | 10% | |||||
| 8 | 6 | 72 | 9% | |||||
| 9 | 7 | 68 | 7% | |||||
| 10 | 7 | 64 | 5% | |||||
| 11 | 8 | 60 | 2% | |||||
| 15 | 8 | 44 | 100% |
Table Results of the Monte-Carlo simulation presents the results of the Monte-Carlo simulation of the portfolio value for each portfolio selection suggestion. The conservative solution (A) is the one with highest worst case value, and as expected the probability of this portfolio's value to go above 60 is very low. From the trader's point of view this is not a good solution, even in the average case for which she will get a value of 44.
The optimistic solution (k=0%) and the solution with the lowest protection level (k=1%) have the largest expected value. However, the trader cannot use these two solutions because she won't be able to commit to achieving a minimum value of 60 since the probability of obtaining a portfolio value greater than 60 is only 73% for these two solutions.
The behavior of the other solutions (k≥10%) shows that with an increasing deviation budget, the probability of achieving a portfolio value higher than 60 also increases. The worst case value of the portfolio equally increases while the expected value decreases. The volatility of the distribution of the values against the three ranges tends to reduce.
| A | k=0.0% | k=1% | k=10% | k=25% | 100% | |
|---|---|---|---|---|---|---|
| Expected value | 44 | 100 | 100 | 98 | 95 | 83 |
| Standard deviation | 24 | 66 | 66 | 45 | 32 | 17 |
| P (value≥0) | 97% | 94% | 93% | 98% | 100% | 100% |
| P (value≥20) | 84% | 89% | 89% | 96% | 99% | 100% |
| P (value≥40) | 57% | 82% | 82% | 90% | 96% | 99% |
| P (value≥60) | 25% | 73% | 73% | 80% | 86% | 91% |
| P (value≥80) | 07% | 62% | 62% | 66% | 68% | 58% |
| P (value≥100) | 01% | 50% | 50% | 48% | 44% | 17% |
The solution with k=100% protects the trader against important losses and allows her to commit to a minimum portfolio value of 60 with a likelyhood of more than 90% and a total expected value of 83. A more risk-prone trader may want to reduce the protection level to say, 10% resulting in a considerably higher average outcome (98) and still having a 90% chance of achieving a portfolio value greater than 40.