Backtest Results
The backtest on the three-asset portfolio is complete for all five scenarios. The high-dimensional test at N=30, which is the decisive case for the convergence correction, is reported below once the run completes.
Three-asset portfolio (AAPL, JPM, CVX)
Each window contains 389 observations, with approximately 19.45 exceedances expected at the 95% level. Likelihood-ratio statistics are compared against their χ² critical values (LRuc and LRind at 3.84, LRcc at 5.99).
| Scenario | Phase | Hits | LRuc | Kupiec | LRind | LRcc | Christoffersen |
|---|---|---|---|---|---|---|---|
| calm | baseline | 13 | 2.54 | pass | 0.59 | 3.12 | pass |
| calm | fixed | 12 | 3.46 | pass | 0.80 | 4.25 | pass |
| volatile1 | baseline | 17 | 0.34 | pass | 0.09 | 0.43 | pass |
| volatile1 | fixed | 17 | 0.34 | pass | 0.09 | 0.43 | pass |
| volatile2 | baseline | 19 | 0.01 | pass | 10.71 | 10.72 | reject |
| volatile2 | fixed | 19 | 0.01 | pass | 10.71 | 10.72 | reject |
| volatile3 | baseline | 14 | 1.77 | pass | 0.97 | 2.75 | pass |
| volatile3 | fixed | 13 | 2.54 | pass | 0.83 | 3.37 | pass |
| crash | baseline | 6 | 13.27 | reject | 0.16 | 13.43 | reject |
| crash | fixed | 7 | 11.01 | reject | 0.22 | 11.23 | reject |
The three-asset crash is a secondary result. A well-conditioned 3×3 correlation matrix does not degenerate, so convergence was never the failure mode at this dimension. The portfolio is instead too conservative, producing six to seven exceedances against an expected 19.45, and rejects both tests. This motivates testing the convergence claim at the dimension where the degeneracy arises.
Crash convergence at N=30
The hypothesis is registered before the result is examined. At N=30 the crash-window correlation matrix is expected to degenerate, with |Rt| approaching zero and a large conditioning number, so that the uncorrected optimizer fails to converge. The halt masking and robust optimizer should restore convergence and allow the window to be scored. A run on the calm window at N=30 serves as a control. A null result, in which the degeneracy does not generalize beyond the original configuration, is a legitimate finding and is reported as such.
| Scenario (N=30) | Converged | r_cond (med / max) | LRuc | LRcc |
|---|---|---|---|---|
| crash · baseline | — | — | — | — |
| crash · fixed | — | — | — | — |
| calm · control | — | — | — | — |